Journal of Automation, Mobile Robotics and Intelligent Systems, vol. 13, no. 3 (2019) by Ł-PIAP

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Journal of Automation, Mobile Robotics and Intelligent Systems pISSN 1897-8649 (PRINT) / eISSN 2080-2145

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Journal of Automation, Mobile Robotics and Intelligent Systems A peer-reviewed quarterly focusing on new achievements in the following fields: • Fundamentals of automation and robotics • Applied automatics • Mobile robots control • Distributed systems • Navigation • Mechatronic systems in robotics • Sensors and actuators • Data transmission • Biomechatronics • Mobile computing Editor-in-Chief

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Journal of Automation, Mobile Robotics and Intelligent Systems Volume 13, N° 3, 2019 DOI: 10.14313/JAMRIS/3-2019

Contents 49

Mechanical Properties of Modern Wheeled Mobile Robots Maciej Trojnacki, Przemysław Dąbek DOI: 10.14313/JAMRIS/3-2019/21 14

Low Cost Soft Robotic Gloves for At-home Rehabilitation and Daily Living Activities Amir Souhail, Passakorn Vessakosol DOI: 10.14313/JAMRIS/3-2019/22 27

Evaluation of User Interface Performing a DVZ-Fuzzy Logic Pilot for Powered Wheelchair Lobna Amouri, Mohamed Jallouli, Cyril Novales, Gerard Poisson, Nabil Derbel DOI: 10.14313/JAMRIS/3-2019/23 34

Heat Transfer Model of a Small Size Satellite on Geostationary Orbit Philippe Preumont, Roman Szewczyk, Pawel Wittels, Filip Czubaczyński DOI: 10.14313/JAMRIS/3-2019/24 39

Preface to Special Issue on Contemporary Problems of Computer Science, Physics and Applied Mathematics Piotr A. Kowalski, Szymon Łukasik, Piotr Kulczycki DOI: 10.14313/JAMRIS/3-2019/25 41

Decomposition Integral without Alternatives, its Equivalence to Lebesgue Integral, and Computational Algorithms Adam Šeliga DOI: 10.14313/JAMRIS/3-2019/26 2

Articles

Statistical Analysis of Models for Punching Resistance Ensuring Jana Kalická, Mária Minárová, Jaroslav Halvoník, Lucia Majtánová DOI: 10.14313/JAMRIS/3-2019/27 56

Global and Local Trend Analysis and Change-Point Analysis of Selected Financial and Market Indices Dominika Ballová DOI: 10.14313/JAMRIS/3-2019/28 64

Probability Measures and Logical Connectives on Quantum Logics Oľga Nánásiová, Ľubica Valášková, Viera Čerňanová DOI: 10.14313/JAMRIS/3-2019/29 74

2D-Raman Correlation Spectroscopy as a Method to Recognize of the Interaction at the Interface of Carbon Layer and Albumin Anna Kołodziej, Aleksandra Wesełucha-Birczyńska, Paulina Moskal, Ewa Stodolak-Zych, Maria Dużyja, Elżbieta Długoń, Julia Sacharz, Marta Błażewicz DOI: 10.14313/JAMRIS/3-2019/30 84

State-of-the-art in Modeling Nonlinear Dependence Among Many Random Variables with Copulas and Application to Financial Indexes Tomáš Bacigál, Magdaléna Komorníková, Jozef Komorník DOI: 10.14313/JAMRIS/3-2019/31

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Mechanical Properties of Modern Wheeled Mobile Robots Submitted: 29th July 2019; accepted: 14th September 2019

Maciej Trojnacki, Przemysław Dąbek

DOI: 10.14313/JAMRIS/3-2019/21 Abstract: The paper discusses mechanical properties of modern wheeled mobile robots including aspects of kinematics and dynamics. Relevant features of these robots and of used types of wheels are considered. Robots are categorized into six groups according to kinematic structures, which can be obtained using various types of wheels. For each group mechanical properties, which characterize the robots are discussed. Various variants of the robots within particular groups are described and some examples of existing solutions are given. Individual variants of the robots are compared and assessed taking into account the possessed features. Keywords: wheeled mobile robot, kinematics, dynamics, kinematic structure, wheel

1. Introduction Wheeled mobile robots (WMRs) are often unmanned vehicles, which can be teleoperated or have various levels of autonomy. They can be dedicated to perform dangerous or tedious tasks thus enhancing workers health and safety or releasing people to more creative tasks. WMRs find more and more practical applications in manufacturing, civil engineering, transportation, agriculture, space exploration, help for disabled and in other sectors of science and technology. One of the fundamental problems is choice of optimal kinematic structure of a robot for a given application. This problem is associated with selection of appropriate types of wheels, their mutual arrangement, number of used drives etc. The choice of optimal solution is usually a compromise between mechanical properties and complexity, which is directly connected with cost. In the available literature there are few items that provide guidance for designers and comparison of various solutions of modern robots. One can mention among others works [9] and [10], but they are not comprehensive or are partially outdated. Therefore, the objective of this paper is comparative analysis of WMRs with various kinematic structures. This analysis takes into account various features of WMRs for which the examples of actual

solutions are given. Primarily commercial robots and other selected designs having high Technology Readiness Level are considered. The analysis is limited to solutions of wheeled robots with fixed kinematic structure. The paper does not include reconfigurable and hybrid robots (i.e. robots that combine features of continuous and discrete locomotion). Some examples of hybrid solutions one can find in work [13] and on webpage [32].

2. Features of Wheeled Mobile Robots Particular solutions of mobile robots are discussed taking into consideration the following features: – number of control degrees of freedom, – mobility, – maneuverability, – stability of motion, – dead reckoning, – complexity of design, – environment of operation. Features such as mobility, maneuverability, stability of motion, dead reckoning and complexity of design are described by 3 linguistic levels, that is: bad (or low), medium, and good (or high). Degrees of Freedom. The important characteristic of mobile robots is their number of Degrees of Freedom (DoFs). Number of DoFs of a mechanical system is the number of independent parameters that define its configuration. There are also known concepts of representational DoFs (related to number of coordinates) and control DoFs (reflecting the number of actuators). The vehicles for which the number of control DoFs is less than representational DoFs are called underactuated, whilst those for which is higher are named overactuated. Considerations concerning the number of DoFs in WMRs will be limited to the mobile platform only. It means that DoFs related to e.g. manipulators and other non-wheeled effectors are not taken into account. For analyzed examples of robots are given only the numbers of control DoFs. Since motion of WMRs is realized in practice most often in plane, therefore the number of representational DoFs is equal to 3. Mobility. The term mobility can be defined as robot ability to move with desired parameters of motion

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in defined conditions of environment, with limitations of the robot itself taken into account [9]. For determination of the mobility, motions of a robot on the ground with various mechanical properties and inclinations are analyzed. Robot ability of negotiation of environment obstacles of various shapes and heights (for example curbs, stairs) is also important. Robot mobility on particular terrain depends on number of factors, including: geometry of locomotion system, center of mass coordinates, properties of wheels (e.g. adhesion, rolling resistance, area of contact with the ground), constraints resulting from characteristics of drives (e.g. power, maximum rotational speed, maximum driving torque), battery parameters (e.g. maximum continuous discharge current) and other [12]. For these reasons, it is difficult to objectively assess the mobility of the robot without realization of simulation and experimental studies. In this paper the assumption is adopted that the robot possesses good mobility if it meets two of the following requirements: – is able to overcome obstacles with a height, which is close to the radius of the driven wheels, both in forward and backward directions, – has the ability of movement in both directions on the ground surface which is inclined at an angle of at least 30 deg and for which are present high values of friction coefficients for wheel-ground pair (e.g. asphalt, concrete, unpaved road). The robot is characterized by medium mobility, if it meets one of the above mentioned requirements and by bad mobility if none of these conditions is met. Some of the solutions are assessed individually, in which case it is supported by justification. Maneuverability. The maneuverability is robot ability to change its direction of motion. Maneuverability is especially important in case of movement in narrow spaces, in which the robot should e.g. be able to rotate in place. The robot is characterized by good maneuverability if it has both the possibility of linear motion in any direction and rotation in place. The robot has medium maneuverability if it meets one of the following requirements: – is able to rotate in place and perform linear motion in direction which depends on current pose or turning with defined radius, – is not able to rotate in place and can realize linear motion in any direction independent of the current pose. Finally, the robot will have bad maneuverability if none of the mentioned requirements is met.

Stability of motion. The stability of motion in case of WMRs is understood as robot resistance to unevenness of the ground during its movement. The robot has bad stability if as a result of unevenness of the ground it loses contact of one of the driven wheels with the ground and then unintentionally and significantly changes its previous direction of motion. If the impact of the loss of contact by one of the driven Articles

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wheels causes small change in direction of movement, it is characterized by medium stability, and if this impact is negligible – by good stability. Dead reckoning. Dead reckoning of a vehicle is the ability of estimation of its location based on estimated speed, direction and time of travel with respect to a previous estimate [2]. In case of WMRs dead reckoning is typically based on odometry of the robot [1]. Therefore, in this study the assessment criterion of dead reckoning will be accuracy of determination of motion parameters of robot mobile platform on the basis of measured angular parameters of spin and turning angle of its wheels.

Complexity of design and cost. The complexity of design of WMRs is first of all related to mechanical and control system design. The important factor is also the number of controlled drives, hence the number of control DoF. Nowadays, implementation of motion even in case of complex robot kinematic structures does not present major difficulties considering current computing power of controllers, therefore it does not significantly affect the hardware complexity of the solution. It should be mentioned, that hardware complexity of the design is related to higher cost of manufacturing of the vehicle. The use of a large number of drives requires also the use of highly efficient power supply systems. Therefore, when assessing the level of complexity of WMRs, first of all the number of drives of locomotion system will be taken into account. Moreover, it will be assumed, that higher complication of design is caused by adding some steering mechanism to a wheel than by adding wheel driving mechanism and that the most complex is the case in which the wheel is both steered and driven. This reasoning is justified in particular in case of small-size robots. Finally, higher complexity of the design causes use of Mecanum wheels or omni-wheels in comparison to standard wheels. Environment of operation. Type of environment in which the vehicle operates is limited to division into indoor and outdoor environment. The use of indoor can be associated with solutions of at least medium maneuverability, while outdoor with solutions of at least medium mobility and stability of motion. Exceptions to this rule of classification are discussed in case of specific solutions. When it comes to indoor use of a mobile robot the stability of motion is less important than in case of outdoor applications, because indoor environment is typically characterized by smaller unevenness of the ground.

3. Classification of Wheeled Mobile Robots The considerations concerning classification of WMRs will be started from used types of wheels. Generally, wheels with the following features are used:

Journal of Automation, Mobile Robotics and Intelligent Systems

– driven and non-driven, – steered and non-steered. By the steered wheel one means such a wheel, which direction can be changed with respect to chassis by control system. In turn, the non-steered wheel is not able to turn (fixed wheels) or it has this capacity, but its turning is not the result of control but is forced by external forces (caster wheels). Next, depending on the combination of aforementioned features one can distinguish the following types of wheels: – castor (non-steered and non-driven), – fixed and non-driven (free wheels), – fixed and driven, – steered (steerable) and non-driven, – steered (steerable) and driven. In steered wheels the axis of turning generally intersects the axis of spin of wheel, whilst in case of caster wheel these axes are placed in some distance relative to each other. This displacement enables selfturning of wheel under the influence of lateral force acting on it. In addition, the wheels can be divided into single and twin (dual). This division has generally no influence on the kinematic structure of the robot, but it affects the distribution of stresses between wheels and the ground, and is not taken into account in further considerations. Special type of fixed and driven wheels represent Mecanum wheels (also named as Swedish) and omniwheels (or omni-directional wheels). The Mecanum wheels are characterized by the fact that on the circumference they have non-driven rollers, whose axes of rotation are rotated by an angle of 45° in relation to the axis of spin of that wheels (see Fig. 1a). In turn, the omni-wheels have also free rollers, but this time their axes of rotation form 90° angle with respect to the wheel spin axis. Moreover, they are typically dual wheels (see Fig. 1b). a)

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They may have a filling in form of a foam. They geometric parameters and properties are different with respect to those of cars. In case of rovers, wheels are made of metals. Flexibility is obtained by appropriate design of the inner part of the wheel. What is more, one can notice that in case of WMRs suspension systems are sometimes introduced. Because this type of vehicles do not carry people, vibrations of mobile platform may be acceptable. In some solutions of robots intended for movement in open and unstructured environments the control arms are used to improve mainly robot mobility. WMRs are typically non-holonomic vehicles, which are subjected to velocities constraints on their motion. In practice, this means that the number of control DoF is less than the number of representational DoF, which is equal to 3. The exceptions are omnidirectional robots, which are considered holonomic vehicles, since they can realize 3 independent movements: in longitudinal and lateral direction as well as rotation about vertical axis. Depending on used wheel arrangement and wheel types one can distinguish the following kinematic structures of WMRs: – differentially driven, – skid-steered, – car-like, – omnidirectional (Mecanum drive, holonomic drive), – rover-type (robots with high number of driven and steered wheels), – all wheels steered and driven by only two motors (also called synchro drive). Particular types of kinematic structures of WMRs are characterized in detail in the next section. The synchro drive structure is not discussed, because nowadays it is rarely used. One of known, older solutions of this type is Nomad 200 robot of Nomadic Technologies, Inc.

4. Mechanical Properties of WMRs The mechanical properties of various solutions of WMRs are discussed taking into account previously defined features.

4.1. Differentially Driven WMRs Fig. 1. Example of: Mecanum wheel (a) and double omni-wheel (b) [28] Wheels have a significant impact on vehicle motion and should be characterized by: – high adhesion to the surface and resistance to lateral drift, – adequate load capacity (the ability to carry the load), – the ability of damping vibrations and shocks, – low rolling resistance, – durability (resistance to wear and impact). As noted in work [3], in the case of lightweight wheeled robots, wheels are usually non-pneumatic.

Differentially driven robots typically have 2 fixed and (differentially) driven wheels. They are characterized by simplicity of mechanical design and therefore low cost. They have also medium maneuverability. In order to change direction of motion they can rotate in place. Therefore they are usually applied in closed areas (e.g. rooms). This kind of robots is characterized by bad stability of motion, because they may have tendency to unintended changes of direction of desired movement due to unevenness of the ground and different adhesion of the driven wheels to this ground. For this reason they usually do not have outdoor applications. Regarding the kinematic and dynamic properties of this type of robots, if they move on an even ground, Articles

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their motion can be accomplished with negligible slips of wheels. This allows the solution of forward and inverse kinematics problems. Moreover, modeling of dynamics of such systems does not cause much difficulty. In this case, due to good dead reckoning, robot movement can be carried out on the basis of odometry only, so by controlling the wheels [1]. Example of such approach one can find among others in work [4]. One can distinguish solutions of differentially driven robots with 2, 3, 4 or even 6 wheels. The selected solutions of this kind of robots are described below. Two-wheeled differentially driven robots. Twowheeled robots can be further divided into those in which the body has to maintain vertical orientation, and those which have horizontal orientation and there is a supporting element e.g. in form of the tail. The robots, in which the body has to keep vertical posture operate based on a principle of the inverted pendulum. They require additional stabilizer module (control system), similarly like in case of Segways. An example of modern solution of this type is the FLASH robot [6]. Such robots should move on even grounds, which limits their use essentially to the rooms. Two-wheeled robots with the supporting element have better stability of motion, which makes them suitable for use both indoors and outdoors. Examples of such solutions are the Recon Scout IR robot [30], in which as the supporting element an elastic tail with ball was used. The disadvantage of such solutions is that the tail may introduce additional forces disturbing the movement of robot, which affects the dead reckoning.

Three-wheeled differentially driven robots. In this type of robots 2 fixed (differentially) driven wheels and 1 castor are applied. Kinematic structure of this kind of solution is illustrated in Fig. 2. With two independently driven wheels can be achieved fairly good maneuverability and negligible slips of wheels in case of movement on even ground. The vehicle can overcome more difficult field obstacles if the movement is carried out in the direction of the driven wheels. Such solutions, however, are the most suitable indoors. An example of such solution is Pioneer 2-DX robot (Fig. 3) and its successor Pioneer 3-DX [29].

Fig. 2. Kinematic structure of three-wheeled differentially driven robot 6

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Fig. 3. Three-wheeled differentially driven Pioneer 2-DX robot Four-wheeled and six-wheeled differentially driven robots. In this case, beside 2 fixed (differentially) driven wheels appear also 2 or 4 castors. There are solutions in which 2 castors are in front or rear of the vehicle and the rest of wheels are driven (e.g. TUG [23]) and those in which one castor is in front and one in rear of the vehicle, whilst driven wheels are arranged on sides (e.g. AGVS [25]). The example of sixwheeled solution with 2 castor in front, 2 castors in rear and 2 fixed driven wheels is MiR100 robot [24]. In such solutions an uneven ground can cause robot motion in an unintended direction. Therefore, such robots are designed primarily to move indoors. One of the exceptions is Tango E5 robot [31], which is used outdoors as a lawn mower.

4.2. Skid-steered WMRs

The second discussed in this paper group of robots are the solutions with all wheels fixed. Such robots are called skid-steered due to the fact that, during turning and rotating in place always occurs lateral slip of wheels, that is skid. This is a major drawback of this type of vehicles due to poor energy efficiency and increased wear of wheels (tires) and degradation of the ground. They include typically 4- or 6-wheeled solutions. They have usually simple and compact design, therefore they are relatively cheap. They are characterized by good mobility and medium maneuverability, i.e. comparable to differentially driven robots. They have also good stability of motion, i.e., robots of this type do not change unintentionally and significantly direction of their motion in the case of uneven ground. For this reason they are often used in open terrain, but also are suitable for closed spaces like rooms, especially in case of fourwheeled solutions. In skid-steered robots all the wheels are independently driven or wheels on each side of the vehicle are driven by one motor. In the first case, because of use of higher number of drives than representational DoFs, this kind of robots are overactuated. In the second case, the drive on each side of the vehicle is distributed to individual wheels via gears or chains or is transmitted to one wheel and then via toothed belts to the remaining wheels. Skid-steered robots are complex objects from the point of view of modeling and control. Modeling and control of motion of this type of robots are, among others, the subject of works [7, 11, 15].

Journal of Automation, Mobile Robotics and Intelligent Systems

Due to slips of wheels occurring during robot turning and rotation in place, in this case one cannot explicitly solve forward and inverse kinematics problems. For the same reason, also more complex is development of dynamics models of such vehicles. Especially complicated is the case when the robot rotates in place on a deformable ground. In such case beside slips of wheels also bulldozing effect appears. Moreover, due to lateral slips, the wheels are under the influence of lateral forces associated with their contact with the ground. The values of the forces can be high therefore this solution requires the use of strengthened in lateral direction means of mechanical connection of wheels to the mobile platform. Motion control of this type of robot, due to bad dead reckoning cannot be performed only on the basis of their odometry, that is by controlling the spin of wheel only. In this case it is necessary to measure actual velocities of the robot mobile platform or its pose (posture). For such robots it is advisable to introduce the hierarchical control system, consisting of kinematic (pose) and dynamic controllers. This approach is used, among others, in the works [5, 15]. In turn, in the paper [11] for motion control of the robot the Nonlinear Model Predictive Control (NMPC) method was used.

Four-wheeled skid-steered robots. In these robots all the wheels are driven independently or – more frequently – independently driven are only two wheels, one on each side, and the drive is transmitted to the other wheel on a particular side by toothed belt. The kinematic structure of such solution is illustrated in Fig. 4. The examples of such robots are Rex [11] and PIAP GRANITE [14] with independent drive on all wheels, as well as PIAP SCOUT and PIAP Fenix [26] (Fig. 5) with 2 driven wheels (one on each side of the robot). Thanks to the use of only 2 drives, one can reduce the size and mass of the robot. This is particularly important if it is necessary to carry a robot by a human.

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Fig. 5. Four-wheeled skid-steered PIAP Fenix robot [25] Six-wheeled skid-steered robots. In case of such robots the most typical is independent drive on all wheels (as for IBIS robot [PIAP] shown in Fig. 6) or one motor is used for each side of the robot and drive is then transmitted to the wheels via gears or chains. In an independent drive typically each wheel on a given side of the robot is controlled with control signal of the same value. Drawback of this solution is that the angular velocities of spin of the individual wheels differentiate depending on the conditions of interaction of the wheels with the ground.

Fig. 6. Six-wheeled skid-steered IBIS robot [26]

Fig. 4. Kinematic structure of four-wheeled skid-steered robot

The application of higher number of wheels decreases the values of forces exerted on the ground by a single wheel and increases robot mobility among others on a deformable terrain. In six-wheeled solutions the control arm systems are often introduced, to guarantee continuous contact of all wheels with the ground. In the control arm systems of WMRs usually shock absorbers are not applied, because, as noticed in work [9], they are justified in case of motion of the vehicles with velocities higher than 8 m/s. Articles

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If control arms are not used, one can place axles of center wheels a little lower, which reduces the resistance of outer wheels (i.e. wheels located at the corners of a mobile platform) during robot turning, thus decreasing also lateral and longitudinal slips of wheels.

4.3. Car-like WMRs

The next group of wheeled mobile robots includes vehicles whose kinematic structure is similar to a car. The main characteristic features of car-like robots are typically limited steering range of steered wheels. The consequence of such solution is the inability of rotation of the robot in place. Due to this fact such solutions are used mostly outdoors. Similarly to cars, in mobile robots the differential mechanism is sometimes used for drive of the wheels. The most common robotic vehicles belonging to this group have exactly the same kinematic structure as a car, that is, they are equipped with four wheels, two of which are steered, and they can be driven by two front wheels, by two rear wheels or by all wheels. In such solutions for steering the wheels usually Ackerman mechanism is used, whilst for driving the wheels often differential mechanism is applied. Instead of Ackerman mechanism one can use two motors to independently set the steering angles of the steered wheels. This kind solution presented in Fig. 7 can be mechanically simpler then Ackerman mechanism and is especially preferred in case of small-size robots.

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wheels the car-like robots are characterized by medium or good mobility, good stability of motion and good dead reckoning. In such applications the differentially driven solutions are inefficient because of too low stability when moving on uneven terrain and the possibility of sinkage of castor wheels in soft ground. The skidsteered robots cause devastation of soil and may bury themselves during cornering. The omnidirectional robots are not suitable due to the loss of characteristics of Mechanum or omni-wheels in case of contamination by soil. Finally, the rover-type robots are too complex and therefore too expensive. The example of such solution is Vine robot [17] shown in Fig. 8 dedicated for application in vineyards. Another example is represented by Trakür robot [19], designed to apply pesticides in greenhouses.

Fig. 8. Car-like Vine robot [17] This group of robots includes also three-wheeled solutions, which are however rarely used. The worst mobility have three-wheeled mobile robots with one driven wheel, which have difficulty in overcoming even small obstacles.

4.4. Omnidirectional WMRs

Fig. 7. Kinematic structure of four-wheeled car-like robot

Car-like robots are solutions of medium complexity, but by virtue of using known and proven automotive solutions, design of this type of robots does not cause major difficulties. Furthermore, when using two drives – one for driving and the other one for steering the wheels, such solution may also be relatively cheap. Such solutions of robots are used primarily in agriculture, examples of which can be found in [16]. This is not accidental, because if one compares mechanical properties of four-wheeled car-like robots with other solutions one can notice that this solution is optimal for such applications. Depending on the number of driven Articles

Another group of wheeled mobile robots are omnidirectional robots also called holonomic robots, because they have zero nonholonomic constraints imposed on vehicle velocities. This means that the robot can move independently in longitudinal and lateral directions and rotate around a vertical axis at the same time. This is achieved by using Mecanum wheels or omni-wheels, which were described earlier. Omnidirectional robot wheels are relatively mechanically complex as compared to designs of other wheels which at the same time makes them relatively expensive. Moreover, in this type of robots all the wheels are independently driven and controlled, which requires at least three motors. In case of omnidirectional robots a little more complicated is also solution of kinematics problems and consequently the control of motion of the mobile platform. The omnidirectional robots are characterized by very good maneuverability – they can simultaneously rotate in place and move in any direction. They do not have good stability of motion because they motion depends critically on velocity of each wheel. If one of them loses contact with the ground then the robot

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moves in nonintentional direction. Therefore, they should move on even terrain. Moreover, they have inferior dead reckoning to differentially driven, carlike, rover-type and synchro drive robots but superior to skid-steered ones. Therefore, in order to ensure high accuracy of movement, the robot motion control should not be based on its odometry only. Such solutions are very good choice in indoor applications but cannot be used outdoors due to the loss of wheel omnidirectional characteristics in case of roller contamination, for example by sand, mud or snow. The kinematic structure of the robot with omni-wheels is named the holonomic drive, whilst with Mecanum wheels the Mecanum drive.

Holonomic drive. In the case of use of so-called holonomic drive, the wheels are arranged in such a way that their axes of rotation intersect at a single point. Robots with holonomic drive usually have three and less often four wheels. Kinematic structure of holonomic robot with three wheels is illustrated in Fig. 9.

Fig. 9. Kinematic structure of three-wheeled robot with holonomic drive The examples of such solutions of NEXUS robot company [28] are kits 10003 (three-wheeled drive) shown in Fig. 10 and 10008 (four-wheeled drive).

Fig. 10. Three-wheeled robot with holonomic drive by Nexus robot [28] Mecanum drive. This type of robots are usually four-wheeled solutions with Mecanum wheels. In this case wheel arrangement can be the same as for skidsteered robots, that is, rotation axes of wheels are parallel to each other. Kinematic structure of this kind is shown in Fig. 11.

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Both four-wheeled holonomic and Mecanum drives result in vehicles being overactuated because they use four drives in locomotion system.

Fig. 11. Kinematic structure of four-wheeled robot with Mecanum drive The examples of omnidirectional robots with Mecanum drive are kit 10011 of Nexus robot company [28] and MPO-500 robot of Neobotix GmbH company [27] shown in Fig. 12. MPO-500 robot is dedicated for autonomous transport systems in industrial environments.

Fig. 12. MPO-500 robot with Mecanum drive by Neobotix GmbH [27]

4.5. Rover-type WMRs The last discussed in this paper group of WMRs covers rover-type robots. Not only solutions of rovers will be discussed but also vehicles that have similar kinematic structures. The rover-type WMRs have high number of motors for driving and steering the wheels, therefore they are classified as highly overactuated vehicles. For this reason they are usually the most complex, hence expensive WMRs. However, they are solutions with very good kinematic and dynamic properties. They are characterized by the best mobility among all types of WMRs. Thanks to the possibility of independent steering of multiple wheels they can rotate in place and also in some solutions, that is with all steered wheels, they can move in any direction. This kind of the vehicles have therefore good maneuverability. What is more, they can move with small slips of wheels, hence they are characterized by good dead reckoning. For safety reasons and in order to reduce slip and sinkage of wheels on soft grounds they usually move at low velocity. Articles

Journal of Automation, Mobile Robotics and Intelligent Systems

The example kinematic structure of four-wheeled rover-type robot is shown in Fig. 13.

Fig. 13. Kinematic structure of four-wheeled rover-type robot The rovers are usually six-wheeled solutions with independently driven and steered all wheels or with independently driven all wheels and steered only outer wheels. For driving the wheels often direct-drive motors are used, i.e., they are placed in the wheel hubs. This is due to the use of control arm systems and the fact that often their wheels are both driven and steered. The examples of such vehicles are Curiosity [21] shown in Fig. 14 and ExoMars rover [22]. It should be noted that the rovers are also developed for various student competitions. In this case, primarily because of a limited budget, simple solutions are predominant, usually containing six independently driven and non-steered wheels. For this reason, the kinematic structures of such solutions are typical for previously discussed skid-steered robots, so they have typical for them mechanical properties. One of many examples of such vehicles are rovers designed by students of the Bialystok University of Technology, Poland [20].

Fig. 14. Six-wheeled Curiosity rover [20]

The rovers are dedicated for outdoor applications, but due to their characteristics such as good maneuverability and dead reckoning, in case of relatively small designs they could be also used indoors. It is also possible to find rover-type robots with four wheels. The example of them is AZIMUT research robot [8] shown in Fig. 15, dedicated for indoor apArticles

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plications. The other examples include vehicles of American Robot Company (AMBOT) [18], such as GRP 4400 (four-wheel independent drive, four-wheel steering) and GRP 2400 (two-wheel independent drive, four-wheel steering), which are dedicated for outdoor applications. In turn, the example of modern rover with four wheels is Interact Centaur [18], which is a customized version of the remote controlled platform manufactured by AMBOT.

Fig. 15. Four-wheeled rover-type AZIMUT robot [8]

5. Conclusion The discussed in the previous sections various types of WMRs are compared taking into account selected features associated with their mechanical properties. The result of this comparison is given in Table 1. Five types of kinematic structures were taken into considerations, for which are specified the variants of drive, steering and the number of wheels. Examples of robots are given for each variant. The selected features used for comparison of individual solutions include: number of control DoFs, environment of operation, mobility, maneuverability, stability of motion, dead reckoning and complexity (directly associated with cost of particular solution). The conducted assessment of particular types of WMRs is qualitative but justified by previous discussion based on the professional experience of authors and supported by available literature. After analysis of information presented in Table 1, one can notice among others that: – differentially driven robots are generally low-cost solutions dedicated for indoor applications, having medium performance in this kind of environment; – skid-steered robots can operate both indoor and outdoor having at least medium performance but their main disadvantage is large slip of wheels, hence they have bad dead reckoning and tend to degrade the ground; – car-like robots understood as unmanned vehicles are dedicated for outdoor applications, especially in agriculture, however due to bad maneuverability they are rarely used in robotics; – omnidirectional robots are used in indoor applications and in this environment they have the best maneuverability; – the rover-type solutions have the best performance of locomotion system but their disadvantages are high complexity of design and cost. To sum up, it can be concluded that taking into account all the analyzed features of WMRs there is no single ideal solution in all aspects. Selection of a kine-

Holonomic drive

Omnidirectional

Rover-type

2 steered & 2 (or 4) driven wheels

Car-like

All wheels steered and driven

All wheels steered and 2 of which driven

All wheels steered and driven

All driven wheels, 4 (outer) of which steered

Holonomic drive

Mecanum drive

All fixed & driven wheels

2 fixed, driven wheels and tail

2 fixed, driven wheels and vertical body

2 fixed, driven wheels and 1 castor

2 fixed, driven wheels in center and 4 castors

2 fixed, driven wheels in center and 2 castors

No. of wheels

2 fixed, driven wheels in rear and 2 castors in front

Variant of drive

Skid-steered

Differentially driven

Type of kinematic structure

Interact Centaur and GRP 4400 [18], Azimut [8]

GRP 2400 [18]

ExoMars [22]

Curiosity [21]

3WD (No. 10003) [28]

4WD (No. 10008) [28]

MPO-500 [27], 4WD (10011) [28]

Vine [17], Trakür [19]

PIAP Fenix [26], Rex [11]

IBIS [26]

Recon Scout IR [30]

FLASH [6]

Pioneer 3-DX [29]

MiR100 [24]

TUG [23], AGVS [25]

Examples Tango E5 Series II [31]

Tab. 1. Comparison of modern WMRs according to selected features

2 (3)

2 (4)

2 (6)

No. of control DoF

Indoor & outdoor

Outdoor (indoor)

Indoor

Outdoor

Indoor & outdoor

Indoor

Outdoor

Environment of operation

Good

Medium

Medium (Good)

Good

Medium

Bad

Medium

Mobility

Good

Medium

Good

Medium

Good

Bad

Medium

Maneuverability

Selected features

Good

Bad

Good

Bad

Medium

Stability of motion

Good

Medium

Good

Bad

Medium

Good

Dead reckoning

High

Medium

High

Medium

Low (Medium)

Low

Medium

Low

Complexity of design

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matic structure should be based on intended application of a robot and its planned market price. The authors hope that this study will be useful for designers of WMRs and help them to make an informed choice of solution for a given application of a robot.

AUTHORS

Maciej Trojnacki* – ŁUKASIEWICZ Research Network – Industrial Research Institute for Automation and Measurements PIAP, Warsaw, 02‑486, POLAND, mtrojnacki@piap.pl.

Przemysław Dąbek – ŁUKASIEWICZ Research Network – Industrial Research Institute for Automation and Measurements PIAP, Warsaw, 02‑486, POLAND, pdabek@piap.pl. *Corresponding author

References

[1] J. Borenstein, H. R. Everett, and L. Feng. “”Where am I?” Sensors and Methods for Mobile Robot Positioning”. Technical report, The University of Michigan, 1996.

[2] P. Corke, Robotics, Vision and Control: Fundamen tal Algorithms In MATLAB®, Springer Tracts in Advanced Robotics, Springer International Publishing, 2017.

[3] P. Dąbek and M. Trojnacki, “Requirements for Tire Models of the Lightweight Wheeled Mobile Robots”. In: J. Awrejcewicz, K. J. Kaliński, R. Szewczyk, and M. Kaliczyńska, eds., Mechatronics: Ide as, Challenges, Solutions and Applications, 2016, 33–51 DOI: 10.1007/978-3-319-26886-6_3. [4] Z. Hendzel, “An adaptive critic neural network for motion control of a wheeled mobile robot”, Nonlinear Dynamics, vol. 50, no. 4, 2007, 849– –855 DOI: 10.1007/s11071-007-9234-1. [5] Z. Hendzel and M. Trojnacki, “Neural Network Control of a Four-Wheeled Mobile Robot Subject to Wheel Slip”. In: J. Awrejcewicz, R. Szewczyk, M. Trojnacki, and M. Kaliczyńska, eds., Mecha tronics – Ideas for Industrial Application, 2015, 187–201 DOI: 10.1007/978-3-319-10990-9_19.

[6] J. Kędzierski and M. Janiak, “Budowa robota społecznego FLASH (Construction of FLASH social robot)”. In: K. Tchon and C. Zielinski, eds., Prace Naukowe Politechniki Warszawskiej. Elektronika, vol. 182, 2012, 681–694 (In Polish). [7] K. Kozlowski and D. Pazderski, “Practical Stabilization of a Skid-steering Mobile Robot – A Ki12

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nematic-based Approach”. In: 2006 IEEE In ternational Conference on Mechatronics, 2006, 519–524 DOI: 10.1109/ICMECH.2006.252581.

[8] M. Lauria, F. Michaud, M. Legault, D. Letourneau, P. Retornaz, I. Nadeau, P. Lepage, Y. Morin, F. Gagnon, P. Giguere, J. Fremy, and L. Clavien, “Elastic locomotion of a four steered mobile robot”. In: 2008 IEEE/RSJ International Conference on Intel ligent Robots and Systems, 2008, 2721–2722 DOI: 10.1109/IROS.2008.4650759.

[9] P. Sandin, Robot Mechanisms and Mechanical De vices Illustrated, McGraw-Hill/TAB Electronics: New York, 2003.

[10] R. Siegwart, I. R. Nourbakhsh, and D. Scaramuzza, Introduction to autonomous mobile robots, MIT Press: Cambridge, 2011.

[11] K. Tchoń, K. Zadarnowska, Ł. Juszkiewicz, and K. Arent, “Modeling and control of a skid-steering mobile platform with coupled side wheels”, Bulletin of the Polish Academy of Sciences. Technical Sciences, vol. 63, no. 3, 2015 DOI: 10.1515/bpasts-2015-0092.

[12] M. Trojnacki, “Analysis of influence of drive system configurations of a four wheeled robot on its mobility”, Journal of Automation, Mobile Ro botics and Intelligent Systems, vol. 6, no. 4, 2012, 65–70. [13] M. Trojnacki, “Modelling the motion of the mobile hybrid robot”, International Journal of Applied Mechanics and Engineering, vol. 15, no. 3, 2010, 885–893. [14] M. Trojnacki and P. Dąbek, “Studies of dynamics of a lightweight wheeled mobile robot during longitudinal motion on soft ground”, Mechanics Research Communications, vol. 82, 2017, 36–42 DOI: 10.1016/j.mechrescom.2016.11.001.

[15] M. Trojnacki, P. Dąbek, J. Kacprzyk, and Z. Hendzel, “Comparative Analysis of Posture Controllers for Tracking Control of a Four-Wheeled Skid-Steered Mobile Robot – Part 1. Theoretical Considerations”. In: R. Jabłoński and T. Brezina, eds., Advanced Mechatronics Solutions, 2016, 583–604 DOI: 10.1007/978-3-319-23923-1_85. [16] “Robots in agriculture”. c2015, Into Robotics, www.intorobotics.com/35-robots-in-agriculture/. Updated on: 2018-01-14, accessed on: 2019-10-02. [17] “A robot to help improve agriculture and wine production”. c2015, ScienceDaily, www.sciencedaily.com/releases/2015/01/150128113713. htm. Accessed on: 2019-10-02.

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[18] “American Robot Company Homepage”, c2015, American Robot Company, www.ambot.com. Accessed on: 2019-10-02. [19] “Argentine greenhouse robot brings automation to the masses”. c2012, CBS Interactive, www.zdnet.com/article/argentine-greenhouserobot-brings-automation-to-the-masses/. Accessed on: 2019-10-02.

[20] “The argo project”. Bialystok University of Technology, http://argo.pb.edu.pl/en/. Accessed on: 2019-10-02. [21] “Home | Curiosity – NASA’s Mars Exploration Program”. National Aeronautics and Space Administration, https://mars.nasa.gov/msl/. Accessed on: 2019-10-02.

[22] “ESA – Robotic Exploration of Mars”. c2000-2019, European Space Agency, https://exploration.esa. int/web/mars/. Accessed on: 2019-10-02. [23] “Mobile robots for healthcare – Pharmacy, Laboratory, Nutrition and EVS”. c2018, Aethon, https://aethon.com/mobile-robots-for-healthcare/. Accessed on: 2019-10-02. [24] “Frontpage | Mobile Industrial Robots”. c20192020, Mobile Industrial Robots A/S, www.mobile-industrial-robots.com. Accessed on: 2019-10-02.

[25] “AGVS | Robotnik”. Robotnik Automation S.L.L., www.robotnik.eu/mobile-robots/agvs/. Accessed on: 2019-10-02. [26] “Counter terrorism robots – special vehicles, equipment and tools by PIAP”. c2015, ŁUKASIEWICZ – Instytut PIAP, www.antiterrorism.eu. Accessed on: 2019-10-02. [27] “Neobotix: Homepage”. Neobotix GmbH, www. neobotix-robots.com. Accessed on: 2019-10-02.

[28] “NEXUS Robot”. c2012, Nexus Automation Limited, www.nexusrobot.com. Accessed on: 201910-02. [29] “Pioneer 3-DX”. Generation Robots, www.generationrobots.com/ media/Pioneer3DX-P3DX-RevA.pdf. Accessed on: 2019-10-02.

[30] “Recon Scout IR”. c2018, ReconRobotics, Inc., https://reconrobotics.com. Accessed on: 201910-02. [31] “Tango E5 Series II | Robotic Mower | John Deere UK & IE”. c2019, Deere & Company, www.deere. co.uk/en/mowers/robotic-mower/tango-e5-series-ii/. Accessed on: 2019-10-02.

[32] “Special Off-road Vehicles”. Jean-Marc Maclou, www.unusuallocomotion.com. Accessed on: 2019-10-02. Articles

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Low Cost Soft Robotic Gloves for At-home Rehabilitation and Daily Living Activities Submitted: 10th April 2019; accepted: 15th August 2019

Amir Souhail, Passakorn Vessakosol

DOI: 10.14313/JAMRIS/3-2019/22 Abstract: Stroke is one of the major reasons which affect the human hand functionality and lead to disability. Different repetitive exercises are used to regain the hand functionality which involves robotic exoskeleton. Soft pneumatic actuators are one of the good alternatives to rigid and fixed exoskeletons for rehabilitation. This paper presents soft robotic gloves fabricated with two different lowcost silicones which can be used in daily living activities and rehabilitation purpose. Soft robotic gloves are light weight and compact. These robotic gloves utilize the pneumatic pressure to flex and extend the human hand. Soft robotic gloves were tested on a healthy object for grasping and rehabilitation ability. Results showed that robotic glove was able to grasping and do the Kapandji test. This work presents an important step toward low cost efficient soft robotic devices for rehabilitation of stroke patients. Keywords: Stroke, rehabilitation devices, pneumatic actuators, low cost silicones, soft robotic glove, Kapandji test

1. Introduction Human hand is the one of the most useful part in daily living activities. Hand disability caused by stroke effects the quality of life and cause depression and anxiety [1]. Fifteen million people around the world experience stroke annually [2]. Number of stroke patients are increasing in developing countries like Pakistan [3] and Thailand [4]. According to World Health Organization, Kazakhstan has the highest ratio of stroke patients per 100,000 while gulf countries have lowest per 100,000 people [5]. There is high chance of impairment in these patients as compared to death [6]. 60% of the stroke patients do not fully recover at 3 to 6 months after stroke attack [7]. There are different rehabilitation therapies and programs are designed for hand and upper limb disabilities that involve manual and device-based techniques. The developed arm and hand rehabilitation programs are playing significant role in recovery of hand disability [8]. Robot assisted therapy, constraint-induced movement theory, virtual reality training, mental practice and mirror therapy are some of the rehabilitation methods currently being used [9–10]. With these techniques, which

Fig. 1. Rigid body rehabilitation robots: (a) The Rutgers Master II [23], (b) Interactive rehabilitation robot [24], (c) HandCARE [25], (d) Electromechanical trainer [26], (e) HEXORR [27], (f) Finger exoskeleton [24], (g) Thumb exoskeleton [30], (h) Haptic Knob [34], (i) EMG-driven exoskeleton hand robotic [31], (j, n) HANDEXOS [38], (k) Electromyographically driven hand orthosis [42], (l) iHandRehab [39], (m) EMG driven exoskeleton for rehabilitation training, (o) orthotic hand-assistive exoskeleton [33], (p) Cable driven robotic system to train finger after stroke [40] 14

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Reference, Developer

Actuation system

Type of usage

System sensors

Supported movements

The Rutgers Master II-ND [23]

Pneumatic

Rehabilitation, Virtual reality trainings

Hall-effect sensors, infrared sensor

index, middle, ring Finger and thumb

HandCARE [25]

Cable Driven , Clutch system

Rehabilitation

HEXORR [27]

DC Brushless motor

Rehabilitation

Optical encoder, torque sensor

Servo motors, gears

Rehabilitation

Force sensor, Data Glove (Immersion Inc.)

four fingers and a thumb

Linear Firgelli L12

Rehabilitation

surface EMG

Full Hand

M. Chen [24]

Reha-Digit [26]

Firgelli linear Actuators

Electromechanical , Vibration engine

Ismail Hakan Ertas [28]

DC motor , Mechanical designed finger

AFX [30]

DC AKM motors , cable driven mechanism

Mulas [32]

Hitec servos HS-805BB, Pulleys, springs

H. Kawasaki [29]

K.Y. Tong [31] Rotella [33]

Interactive rehabilitation

Force sensors, EMG’s sensor

index, middle, ring and pinky Fingers

Rehabilitation

Switches

index, middle, ring and pinky Fingers

optical encoder, sEMG

One finger at one time

Rehabilitation

Force Sensors

Optical encoders, tension sensors EMG

Bowden cables

Grasping and pinching

EMG sensor , Force sensor

J-Glove [35]

Bowden cable, servomotor,

Rehabilitation

EMG sensor

Yamaura [37]

Pulleys, RC Servo motor

Rehabilitation

iHandRehab [39]

RE25, RE36 motors, cables

Rehabilitation

Digit leaf springs, Tension cords

Rehabilitation (Prototype)

Haptic Knob [34] HIFE [36]

HANDEXOS [38]

Dovat [40]

SCRIPT project [41]

Haptic Knob

Shaft, Motor, gear

Pulleys, DC motor

Cable driven, Clutch system, DC motor

Rehabilitation

Full hand

One finger at one time Full Hand Full hand

force sensors,

Full Hand

Rehabilitation

Data acquisition card, computer application

One finger at one time

Rehabilitation

Mechanical switches

Rehabilitation

Full hand

Mechanical switches

One finger at one time

Angle and force sensors

Full Hand

“MilliNewton 2 N” force sensors

Bending sensor, Electric force

One finger at one time Full Hand Full Hand

Tab. 1. Previously developed hardware systems and their specification are hand disability can be recovered significantly as compared to manual therapy sessions [11–12]. Robotics assisted rehabilitation techniques have significant result of recovery as compared to manual therapy performed by therapist [13]. Robot assisted training can be used for patient with different level of motor impairment and recovery stage. These training can help patient to get back his muscles power [14]. As compared to conventional therapy, robotic device can provide higher number of dosage (like number of repetitions or practice movements) and high intensity which can be a critical factor in rehabilitation [15]. Robot aided therapy have positive influence on stroke patients and it can improve motor control aspects for long term effects [16]. Recent studies support this hypothesis that rehabilitation with robotic devices is a promising approach in hand therapy [17]. Different studies [18–19–13] showed that the higher number of repetitions increased the speed of rehabilitation and robotic device can provide more speed with

accuracy. Rehabilitation robotic device can provide higher number of intense practice sessions with minimum supervision of therapist [19]. Robotic assistance therapy also has a great influence on behavioral gains which can faster the speed of motor recovery after stroke [20]. A study showed that within 3 weeks from starting of robot assisted rehabilitation, patient force generation from effected hand increased by 13.7% [18]. Robotic devices for rehabilitation is a fast growing field in recent years. A lot of robotic devices can be found in literature which show the progress in robotic rehabilitation devices [21] as shown in figure 1. Greater number of these robotic devices are rigid body, heavy and not easy to use as portable device [22]. Some of the devices in literature are presented in table 1. Rigidness, complexity and heaviness of these systems is an obstacle in using them out of rehabilitation centers and without help of therapist. Lack of compliance (stiffness and softness) is always a problem in rehabilitation devices [43]. Bulkiness of these devices Articles

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Fig. 2. soft robotic gloves found in literature: (a) Exo-Glove wearable glove base on tendon routing system [45], (b) AirExglove A pneumatic and tendon routing system wearable glove [58], (c) A pneumatic wearable soft robotic glove [59], (d) Shape memory alloys (SMA’s) glove [60] (e) Pneumatic actuator with origami shell [53], (f) Kirigami-inspired Flexible robotic hand [52], (g) A pneumatic glove for rehabilitation training [49], (h) gait rehabilitation soft robot [47], (i) Pneumatic robotic glove for rehabilitation [44], (j) Wearable haptic device [61], (k) Pneumatically actuated robotic glove controlled with EMG [62], (l) fluidic actuated soft robotic glove for rehabilitation [56], (m) Exo-Glove poly actuated by tendon driving system [63].

made it uncomfortable for using them for stroke effected hand. Components of these devices like motor and material put more stress on effected parts of the hand. While considering these limitations, various soft exoskeleton robotic devices have been proposed for rehabilitation [44]. In last decade, soft exoskeleton and artificial muscles are being developed and improving the quality of rehabilitation and make it more safe for human-machine interaction. Actuator material and actuation method make it light weight and easy to use for stroke patients. In the literature, these robots are referred as Exo-Glove [45], Exo-Glove Poly [46], Gait Rehabilitation soft robot [47], MR glove [48], PneuGlove [49], RARD [50], GRIPIT [51], Anthropomorphic Robotic Hand [52], Origami Shell based pneumatic actuator [53], Yu She (Actuator) [54], Hong Kai Yap [55], fluidic pressured glove [56] and Soft Robotic Glove [57] as shown in figure 2. A wearable soft robotic glove can lead to greater improvements of rehabilitation process at home by providing enormous number of degree of freedom and large bending by single input (e.g. fluidic pressure, air pressure). It can provide the safe human-machine interaction because of its soft material used for actuator fabrication and actuation system away from patient body. It can be cheaper as compared to rigid body devices as its material is cheap. Wearable soft glove can be easily portable as it has single actuation energy source (Fluidic reservoir, Air pressure pump) [56]. Moreover, it can be easily used as rehabilitation mode and daily activity mode just by adding a switching for mode change. Different designs and actuation methods like shape memory alloys (SMA) driven actuators [64], tendon Articles

driven actuators [65], Fluid driven actuator [66] and pneumatic actuators [67] has been developed for rehabilitation robotic gloves. Most of the tendon driven cables support only daily living activities and have limited output force and hyperextension. SMA actuators have high operating temperatures ranging from (100°C–500°C). Complex design of SMA actuators made it difficult to use in rehabilitation purposes and daily living activities [64–68]. Pneumatic actuators were selected due to higher stiffness, low weight and simpler design as compared to above mentioned actuators. This paper presents soft robotic gloves for rehabilitation which used the inexpensive silicon for fabrication of these actuators. Air pressure used as actuation method in these actuators. Air pressure will help the robotic fingers for flexing and extending the hand. Glove will be attached on the dorsal side of hand which helps the patient to feel the objects more naturally. Actuation energy source and electromechanical components are mounted separately to make sure to put the as low as possible burden on human finger.

2. Design Soft robotic gloves presented in this work operates with pneumatic (air) pressure which provides the grasping and releasing (extension and flexion) of the human hand for rehabilitation practice and daily living activities. The gloves are assembled with soft pneumatic actuators fabricated with low cost silicones and a cotton glove which gives the support for human hand to make it wearable.

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Properties

Silicon (RTV 225)

Silicon (RTV4503)

Elastosil M4600

Tensile strength (N/mm2)

≥3.43

Viscosity at 23° C (mPa s)

15000-17000

35000

12000-20000

Tear Strength (N/mm2)

≥22

>20

Hardness Shore A

Operating time (Hour) Curing time (Hour)

28-30 0.5

0.5-12

0.25

8-12

Approximately 1.16

1.1

≥420

400

Adjustable

Density at 23° in water (g/cm3)

Elongation at break (%)

0.5

0.5-12

Mixing ratio

Density (g/cm3)

Tab. 2. Properties of silicon used in fabrication process

Adjustable 1.16

2019

10:1 1.1

800

2.1. Single Pneumatic Actuator

3. Fabrication

Soft Pneumatic actuator is based on the design presented by [69] which shows fast actuation. Pneumatic actuator consists of three layers which include extensible top layer which have air chambers, inextensible layer and extensible base layer to enclose both layers as shown in figure (3-a). Molds with required dimensions were designed in Solidworks (Dassault Systèmes, Waltham, MA, USA) and printed by using 3D printer (Prusa I3) with Polylactic acid (PLA). 3D printed molds are shown in fi gure 4.

Soft pneumatic actuator was fabricated with three different locally available low cost materials which include RTV 225(GGC, Taiwan), RTV 4503(GGC, Germany) and Elastosil M4600(WACKER CHEMIE AG, Germany). All three silicones are room temperature curing silicones. Elastosil M4600 have fixed mixing ratio of 10:1 by weight while RTV 225 and RTV 4503 mixing ratios of part A and part B can be adjusting as application requirement, curing time, stiffness and viscosity. Part A is flow-able silicon while Part B is curing agent while both have directly proportional ratio for stiffness, curing time and inversely proportional for viscosity. The material properties comparison is shown in table 2. Different mixing ratios were experimented for getting the required stiffness, curing time and viscosity for RTV 225 and RTV 4503. Paraffin oil was used to control the concentration level of part A. Table 3 and table 4 shows the mixing ratios of RTV 4503 and RTV 225 respectively for part A, part B (curing agent) and paraffin oil. A range of mixing ratios for part B (curing agent) with part A and paraffin oil were found during the experiments which varies from 1.99–2.98 (g) for RTV 4503 while 1.16–1.55 (g) for RTV 225 presented in table 3 and table 4. Part A, part B and paraffin oil was stirred with electric mixer (EMS-52, Thai city electric co. ltd, Thailand). The mixture was poured directly into the mold as shown in figure 5-a. Stirring of silicon parts usually produce the bubbles in mixture which cause the leaking in actuator’s body. Vacuum chambers or sharp edge objects like needle used for degassing depending on the quantity of air bubbles. In this study, large number of air bubbles were observed after stirring the mixture. A custom made vacuum chamber was used for degassing the mixture. The molds were left for curing for 8–12 hours. Fabrication process of actuators is shown in figure 5. Fabrication process of the pneumatic actuators is described here in details [69].

Fig. 3. Soft pneumatic actuator design configuration: (a) different layers of the actuators are labelled, [71] (b) expected bending behavior of the actuator [69] they do so relatively slowly (over seconds.

Fig. 4. 3D printed molds for fabrication of pneumatic actuators: (a) upper and lower molds (b) upper and lower molds fitted together (c) Base mold.

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Silicon (g)

Paraffin Oil (g)

Curing agent (g)

87.31

9.05

3.64

87.52

9.50

2.98

87.94

8.78

88.57

11.03

90.01

8.81

89.51

2.31 1.99

8.98

Tab. 3. Mixing ratios of RTV 4503 Silicon elastomer

1.51 1.18

Silicon (g)

Paraffin Oil (g)

Curing agent (g)

86.34

11.63

2.03

89.12

9.42

1.46

89.56 89.59 88.56 89.96 89.67

Tab. 4. Mixing ratios of RTV 225 Silicone elastomer

2019

3.28

9.10

86.98

N° 3

8.80

1.64

8.86

1.55

10.03

1.41

8.52

1.16

9.32

1.01

4. Results For Blocked force test, free travel trajectory test and glove grasping test, a low-cost air pump was used for constant air pressure supply with a pressure sensor (Honeywell, ASDXAVX100PGAA5) and solenoid valve (SMC pneumatics, VDW31-5G-3-01).

4.1. Blocked Test Force Measurement Fig. 5. Fabrication process of single actuator: (a) stirred mixture was poured directly into mold and left for curing (b) fabricated air chamber (top layer) after demolding, (c) base mold, some mixture was poured directly into base and left for curing. Then a paper layer was inserted and poured some more silicon which glued the upper layer. (d) side view of fabricated actuator after inserting the pipe for air supply (e) bottom view of actuator.

Blocked force test was conducted to measure the force generated by soft actuators at tip. Blocked force test for evaluating the interaction force of pneumatic actuator is shown in figure 6 where bending end of actuator was blocked by sensor and blocked force was measured. All three actuators were tested. The experimental results are shown in figure 7 where soft actuator fabricated with Elastosil M4600, RTV 225 and RTV 4503 is generating the output force of 1.36N, 1.15N and 1.03N respectively. One of the parameter in blocked force is stiffness of actuator which can be increase by input pressure of the actuator, geometry of actuator and stiffness of material [72]. RTV 4503 and RTV 225 stiffness can be increased or decreased by increasing or decreasing the weight percentage of curing agent. The experimental result of blocked force and input pressure relationship for all three silicones is shown in figure 6.

4.2. Free Travel Trajectory Tracking Fig. 6. Experimental setup for blocked force test.

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The setup for free travel trajectory is shown in figure 8 (a). One of the end was fixed with connecter and air pressure was supplied. There was an initial free travel bending of 49° for Elastosil M4600 fabri-

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Fig. 7. Pressure and blocked force test for RTV 225, RTV 4503 and Elastosil M4600 fabricated actuators

Fig. 8. setup for free travel trajectory tracking test: (a) Full bending for RTV 225 fabricated actuator, (b) Full bending for Elastosil M4600 fabricated actuator, (c) Full bending for RTV 4503 fabricated actuator, (d) Some of the Elastosil M4600 actuator shows extra inflation at some chamber.

Fig. 9. setup for estimating bending angle variation with pressure: (a) Experimental setup and bending angle definition, (b) Pressure and bending angle for RTV 225, (c) Pressure and bending angle for RTV 4503, (d) Pressure and bending angle for Elastosil M4600. cated actuator while RTV series fabricated actuators showed free travel bending of 59° and 53° for RTV 225 and RTV 4503 respectively. Higher free travel bending of RTV series actuators were observed and it can be explained by the lower stiffness of materials.

The soft robotic actuators were pressurized until the full bending at constant pressure as shown in figure 8 and deformation was recorded by high resolution camera (iSight camera, iPhone 5) then bending angle was analyzed with tracker (https://physlets. Articles

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Fig. 10. Grasping ability test for soft robotic glove: (a) Pinching pose with thumb and index finger (RTV 225) (b) Index finger bending (RTV 4503) (c) Full hand flexion with robotic glove (RTV 225) (d) Full hand flexion with robotic glove (RTV 4503) (f-1, e-1) Grasping a small bottle with pinching posture (f-2, e-2) Grasping ability of small water bottle with full hand (f-3, e-3) Grasping ability of robotic glove for coffee cup (f-4, e-4) picking up the telephone receiver without small finger actuated.

Fig. 11. Kapandji test for rehabilitation purpose: (a) thumb contact with index finger (b) thumb contact with middle finger (c) thumb contact with ring finger (d) thumb contact with small finger (e) full hand flexion (f-1 – f-4) soft robotic glove assembled with RTV 225 performing all above posture (g-1 – g-4) robotic glove performing the human hand posture for rehabilitation standardized test.

org/tracker/). The experimental results of free travel trajectory over input pressure is shown in figure 9. The definition of free travel trajectory (bending angle) is defined in figure 9a. The relationship of free travel trajectory and input pressure is almost linear. Initial angle was deducted and then response was plotted as shown in figure 9(b), 9(c) and 9(d). All three soft robotic fingers were tested for deflection angle while going under grasping process. RTV Articles

225 fabricated soft finger shows the 87° deflection at 48 kPa while RTV 4503 fabricated finger shows the deflection of 92° bending deflection under 51 kPa and Elastosil M4600 fabricated finger goes under the deflection of 124° bending deflection when 58 kPa pressure is applied. While testing the Elastosil M4600 fabricated finger, it has been observed that some finger shows extra inflation (from desired).

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4.3. Grasping Ability Grasping ability of Pneumatic robotic glove was tested with healthy subject wearing the glove. Soft robotic glove consisted of a woven glove on which pneumatic actuators were glued gives the maximum comfort to user’s hand. The total weights are 157.82 g and 160.17g respectively for gloves assembled with actuators fabricated of RTV 225 and RTV 4503. The ideal soft robotic hand should not exceed from 0.5 kg [67]. Glove can be easily mounted and dismounted as it fits the human hand easily. The healthy subject was instructed to relax his muscles and air pressure was inserted in pneumatic actuators to assist the hand for grasping the objects as shown in figure 10.

4.4. Kapandji Test

Finger opposition using thumb is one of the more difficult exercise for people having grasping difficulties. There are some standardized tests to evaluate the ability of affected hand where Kapandji test [66] is one of these test which implemented on healthy subject. Figure 11 shows the hand postures for standardize Kapandji test and hand flexion.

5. Discussion and Conclusion In this paper, wearable soft robotic gloves design, fabrication and testing has been presented. Pneumatic actuators were designed and fabricated with three different materials. These actuators are one of the best alternatives for rigid and fixed actuators being used in rehabilitation devices. Results shows that glove have capability to replicate the rigid and fixed devices for rehabilitation exercise and can help for daily living activities and rehabilitative exercises. Pneumatic actuators were designed base on PneuNets architecture. The selected geometry was printed with 3D printer and fabricated using three different low cost soft silicon materials with high elongation properties. Actuation pressure was measured with pressure sensor while actuation speed and bending angle was measured using Tracker. Some of the Pneumatic actuators fabricated with Elastosil M4600 shows extra inflation in some chamber upon pressurizing above from 30kPa which can affects the bending of robotic glove. Due to unwanted inflation, Elastosil M4600 fabricated actuators were not chosen for assembling the soft robotic glove. Different mixing ratios were experimented to find the desired curing time and stiffness for RTV 4503 and RTV 225 silicones. Gloves assembled with RTV series silicones shows reliable grasping and continuous flexion and extension of human hand. Rehabilitation device for hand should not be more than 0.5 kg as a standard for getting the better result for rehabilitation where robotic gloves proposed in this work weighs 157g and 160g as compared to 220g [73] and 230g [74-75]. A rehabilitation finger should

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generate block force of 1N magnitude to facilitate the rehabilitation process. Results shows that robotic finger fabricated with these low cost silicones generate more than 1N. Results shows the human hand flexion, extension with one of the standardized rehabilitation test and grasping of daily living activities object. By comparing the test results and observations, robotic glove assembled with RTV 225 silicon actuators shows more reliable grasping and fast rehabilitation exercise movements as compared to robotic glove assembled with RTV 4503 silicone actuator. Currently soft robotic gloves are pressurized with single input air source with open loop strategy. All the actuators are being actuators with single air source resulting uniform air pressure to all actuated. Introducing different sensors with close loop strategy can lead toward the better control and reliable actuation. There were some observations made during the testing of pneumatic actuators. – Increasing the curing agent can lead to higher stiffness, higher pressure, durability, and lower curing time. – It has been observed that direct stirring causes a large number of air bubbles inside the mixed solution for which higher vacuum inside the chamber needed to remove the air bubbles. Trying different mixing method can reduce the air bubble production inside the solution. – Block force magnitude can be increased by increasing the hardness of materials which result in bearing higher pressure to generate higher force. – It has been observed that some actuator fabricated with Elastosil M4600 shows extra inflation at some chamber which make it inadequate where full bending of actuator is required. This paper presents low cost soft robotic glove with open loop control strategy for assisting at-home rehabilitation and daily living activities. Soft robotic gloves presented in this paper are assembled with pneumatic actuators fabricated with low cost silicones (RTV 225 and RTV 4503). These actuators show the fast response on low pressure [76]. Experimental results show the ability of grasping and rehabilitation test with passive healthy human hand. The proposed robotic glove with low cost material exhibits the potential for low cost solution of human hand rehabilitation. In future, closed loop strategy with feedback from different sensors like force sensor and elastic joint angle sensor can lead to better performance of these gloves. Furthermore, increasing the material stiffness with different curing agent ratio can lead the actuators to withhold higher pressure.

ACKNOWLEDGMENTS

This work was supported by Higher Education Research Promotion of the Higher Education Commission and the Education Hub Program for the Southern Region of ASEAN countries. Authors are also thankful to Department of Mechanical Engineering and Faculty of Engineering, Prince of Songkla for providing the resources to carry out this research. Articles

Journal of Automation, Mobile Robotics and Intelligent Systems

AUTHORS Amir Souhail – Department of Mechanical Engineering, Faculty of Engineering, Prince of Songkla University, Hatyai, Thailand, e-mail: amir.souhail@gmail. com.

Passakorn Vessakosol* – Department of Mechanical Engineering, Faculty of Engineering, Prince of Songkla University, Hatyai, Thailand, e-mail: passakorn. vessakosol@gmail.com. * Corresponding author

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[51] B. Kim, H. In, D.-Y. Lee, and K.-J. Cho, “Development and assessment of a hand assist device: GRIPIT”, Journal of Neuroengineering and Reha bilitation, vol. 14, no. 1, 2017 DOI: 10.1186/s12984-017-0223-4. [52] Y. H. Chan, Z. Tse, and H. Ren, “Design evolution and pilot study for a kirigami-inspired flexible and soft anthropomorphic robotic hand”. In: 2017 18th International Conference on Advanced Robotics (ICAR), 2017, 432–437 DOI: 10.1109/ICAR.2017.8023645.

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[60] J. Shintake, V. Cacucciolo, D. Floreano, and H. Shea, “Soft Robotic Grippers”, Advanced Materials, 2018 DOI: 10.1002/adma.201707035. [61] C. Pacchierotti, S. Sinclair, M. Solazzi, A. Frisoli, V. Hayward, and D. Prattichizzo, “Wearable Haptic Systems for the Fingertip and the Hand: Taxonomy, Review, and Perspectives”, IEEE Transac tions on Haptics, vol. 10, no. 4, 2017, 580–600 DOI: 10.1109/TOH.2017.2689006. [62] P. Polygerinos, K. C. Galloway, S. Sanan, M. Herman, and C. J. Walsh, “EMG controlled soft robotic glove for assistance during activities of daily living”. In: 2015 IEEE International Conference on Rehabilitation Robotics (ICORR), 2015, 55–60 DOI: 10.1109/ICORR.2015.7281175.

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[56] P. Polygerinos, Z. Wang, K. C. Galloway, R. J. Wood, and C. J. Walsh, “Soft robotic glove for combined assistance and at-home rehabilitation”, Robotics and Autonomous Systems, vol. 73, 2015, 135–143 DOI: 10.1016/j.robot.2014.08.014.

[58] A. Stilli, A. Cremoni, M. Bianchi, A. Ridolfi, F. Gerii, F. Vannetti, H. A. Wurdemann, B. Allotta, and K. Althoefer, “AirExGlove — A Novel Pneumatic Exoskeleton Glove for Adaptive Hand Rehabilitation in Post-Stroke Patients”. In: 2018 IEEE Inter national Conference on Soft Robotics (RoboSoft), 2018, 579–584 DOI: 10.1109/ROBOSOFT. 2018.8405388. [59] T. Jiralerspong, K. H. L. Heung, R. K. Y. Tong, and Z. Li, “A Novel Soft Robotic Glove for Daily Life Assistance”. In: 2018 7th IEEE International Con

[65] J. P. King, D. Bauer, C. Schlagenhauf, K.-H. Chang, D. Moro, N. Pollard, and S. Coros, “Design. Fabrication, and Evaluation of Tendon-Driven MultiFingered Foam Hands”. In: 2018 IEEE-RAS 18th International Conference on Humanoid Robots (Humanoids), 2018, 1–9 DOI: 10.1109/HUMANOIDS. 2018.8624997.

[67] J. Wang, Z. Liu, and Y. Fei, “Design and Testing of a Soft Rehabilitation Glove Integrating Finger and Wrist Function”, Journal of Mechanisms and Robotics, vol. 11, no. 1, 2019 DOI: 10.1115/1.4041789. [68] Y. Yang, Y. Chen, Y. Li, M. Z. Q. Chen, and Y. Wei, “Bioinspired Robotic Fingers Based on Pneumatic Actuator and 3D Printing of Smart Material”, Soft Robotics, vol. 4, no. 2, 2017, 147–162 DOI: 10.1089/soro.2016.0034. Articles

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[70] B. Mosadegh, P. Polygerinos, C. Keplinger, S. Wenn stedt, R. F. Shepherd, U. Gupta, J. Shim, K. Bertoldi, C. J. Walsh, and G. M. Whitesides, “Pneumatic Networks for Soft Robotics that Actuate Rapidly”, Ad vanced Functional Materials, vol. 24, no. 15, 2014, 2163–2170 DOI: 10.1002/adfm.201303288. [71] P. Polygerinos, B. Mosadegh, and A. Campo, “Design | Soft robotics toolkit”. https://softroboticstoolkit.com/book/pneunets-design. Accessed on: 2019-10-16.

[72] D. Drotman, M. Ishida, S. Jadhav, and M. T. Tolley, “Application-Driven Design of Soft, 3-D Printed, Pneumatic Actuators with Bellows”, IEEE/ASME Transactions on Mechatronics, vol. 24, no. 1, 2019, 78–87 DOI: 10.1109/TMECH.2018.2879299.

[73] C. G. Rose and M. K. O’Malley, “Hybrid Rigid- Soft Hand Exoskeleton to Assist Functional Dexterity”, IEEE Robotics and Automation Letters, vol. 4, no. 1, 2019, 73–80 DOI: 10.1109/LRA.2018.2878931.

[74] P. M. Aubin, H. Sallum, C. Walsh, L. Stirling, and A. Correia, “A pediatric robotic thumb exoskeleton for at-home rehabilitation: The Isolated Orthosis for Thumb Actuation (IOTA)”. In: IEEE 13th International Conference on Rehabilitation Robotics, 2013 DOI: 10.1109/ICORR.2013.6650500. [75] H. Zhang, A. S. Kumar, F. Chen, J. Y. H. Fuh, and M. Y. Wang, “Topology Optimized Multimaterial Soft Fingers for Applications on Grippers, Rehabilitation, and Artificial Hands”, IEEE/ASME Transactions on Mechatronics, vol. 24, no. 1, 2019, 120–131 DOI: 10.1109/TMECH.2018.2874067.

[76] A. Souhail and P. Vassakosol, “Low Cost Soft Robotic Grippers for Reliable Grasping”, Journal of Mechanical Engineering Research and Develop ments, vol. 41, no. 4, 2018, 88–95 DOI: 10.26480/jmerd.04.2018.88.95.

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�� I�� D��-�� S�bm��ed: 19th March 2019; accepted: 2nd September 2019

Lobna Amouri, Mohamed Jallouli, Cyril Novales, Gérard Poisson, Nabil Derbel DOI: 10.14313/JAMRIS/3-2019/23 Abstract: This paper presents the preliminary tests of an adapted user interface that performs an hybrid fuzzy-Deformable Virtual Zone(DVZ) pilot. The proposed concept uses a safely guidance algorithm for the powered wheelchair user and a laser range sensor to avoid collision. An adapted user interface is developed so that the accessibility and the mobility of disable or aged people especially those suﬀering from low cogni�ve abili�es will be enhanced. Trials with the proposed algorithm detected obstacles and avoid them in 80% of trials with diﬀerent objects and generated safe paths for the interface user. Keywords: Assis�ve control, user interface, DVZ �bstacle avoidance, fuzzy guidance, improving mobility, disable people.

�. �ntroduc�on Aiming to better the life quality of the older adults or the disable people, much focus must be upon the assistive technologies. The use of these technologies is became an important challenge because of the growing proportion of the disabilities : cognitive ability, sensory impairments and behavioral skills. Older adults are generally excluded from assitive technologies that deals with the problems resulted from these impairments. This is due to the need of the older adults to some additional safely measures. While people with disabilities have a great challenge to manipulate the variety of existing devices as they consider the differences of disabilities. Several alternative ways were used in order to finalize one task. For example adapted interfaces were used to compensate for motor dexterity. In general, we can’t design a unique assitive technique for the hole disable population as each one has her own solution that depends on the kind of disability : cognitive, sensory impairment or behavioral skills. We propose the following guidelines needed to concept an adapted user interface, basing on the research fields of human computer interaction [6], [4] and [15] : - interface should communicate with multiple devices. Each one could compensate for a kind of disability. - Interface should be simple to use : a short number of steps is needed especially for the person suffering from cognitive impairments.

- Interface should consider different forms of prompting that helps the user to realise a process. The system presented in this paper is an adapted user interface for controlling robotic wheelchair. This interface is used to improve mobility of disable persons by applying intelligent technology. A DVZfuzzy logic pilot is developed to perform the user guidance past obstacles. An study of related works to build these kinds of autonomous platforms for disable people will be presented. A description of the robotic system and the main programs used will be projected. The obtained results additionally to a detailed discussion will be presented. Finally, future research will be provided.

2. Background During the last decades, many researches had been focused on tailoring the control system to the user of a robotic system. This research belonging to the field of artificial intelligence, aims to electricallypower a wheelchair so that it brings independence suffered by the mobility-impaired or older adults. Traditionally, powered wheelchairs have been based on an intuitive solution (joystick). Nevertheless, different interaction methods shall be applied to ensure efficiency and safety. Preliminary researches are carried out basing on the Smart wheelchair [13], then researches [10] were investigating in the fields of face and gaze interface and hand-gesture recognition [7]. More novel fields are later being investigated to compensate the poor reactions of the joystick control. These range from those that used a high level of autonomy as Taha et al. [14] to the Millan et al. [12] brain machine interface offering a very low user input resolution. In these platforms, the actor usually supervises while the wheelchair is reaching the target. This kind of control method is suitable for users afflicted by hard physical disabilities. There are further mixed control systems that swap between distinct mode of operations such as the ”Navchair” robotic system [9], or the controller proposed by Carlson [3] that used a collaborative technique basing on secondary task experiments mobility : they keep the control user-initiated and only adapt signals where necessary. Unfortunately, this type of assistance requires a significant effort from the actor which is not suitable for people suffering from cognitive harm. Other researchers [11] proposed both an anticollision and a prompting system that helps older adults 1

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guiding safely past obstacles. There are also works dealing with the accessibility of people with disabilities by developing four versions of flexible interfaces for assistive devices [15], [16]. These interfaces are based on input device and a camera. In this paper, we suggest a mixed control system that switches from a guidance system to an anticollision pilot. An adapted user interface was developed so that the user accessibility is improved. The robotic wheelchair currently employs the underlying three operating modes : - goal-seeking and path following of a unicycle mobile robot : navigator - obstacle avoidance architecture using Laser sensor : reacting pilot

Wheelchair CB-405 Unit

Laptop Joystick

- the computation between the two objectives using a fuzzy switching - an adapted user interface that shows different forms of prompting that helps the user to adapt the wheelchair control.

Ultrasonic sensors

Laser scanner

3. System Architecture 3.1. Overview of the System The studied vehicle is a robotic wheelchair with two caster wheels and two independent driving wheels that provide the mobile base with two degrees of mobility [2]. We have mounted a Laptop upon our system and connected it to a joystick and motor control unit using a CB-405 bloc system as shown in Fig.1. This allows us to intercept joystick signals and alter them (where necessary), before sending them to the wheelchair�s motor control unit (Fig.2). We have also developed a computer laser-based localization system that works in unknown indoor environments. In order to be aware of its surroundings, the wheelchair must know its relation to a coordinate system. Two encoders ”Easy Roller ENC300CPR” are mounted on the chair’s wheels to determine its location. These encoders provide 300 impulses per tour and output signals used to measure the linear and angular displacement of each wheel. The studied system consists of a robot controlled via a serial communication from the PC. The terminal configuration for the host computer PC must be set to ”9600 Bauds, 8 bit, 1 start bit, 2 stop bit and no parity”. The laptop executes the task of calculating the optimized trajectory using the fuzzy logic controller, determining the robot relative position using odometer measurements and avoiding obstacles. The communication with the laser sensor is ensured via USB port, with a simple protocol to acquire data. [5] 3.2. So�w�re ��terf�ce The wheelchair command system running on the Laptop is operated through a 2D graphical user interface (GUI) presented in Fig.3 and Fig. 4. There are three operation modes in this interface : goalseeking, path following and obstacle avoidance. The user can choose one or more running mode in the 2

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Fig. 1. ”Current con�gura�on of the wheelchair. �he so�ware on the �a�to� uses the measures from the CB405 and the laser to control the wheelchair mo�on” [2].

Laptop

Laser telemeter

CB-405 bloc system

Wheelchair Joystick

Motor Control Unit

Fig. 2. ”�iagram highligh�ng the user interac�on methods � through the �o�s�c� or the �a�to�. �ll the �o�s�c� commands are �rocessed b� the CB�405 module before being sent to the wheelchair motors” [2].

same test. First, the user can interactively place the start and the end points on the displayed environment. Second, he can place different obstacle modes (corridor, with/without corner situation). Third, he can also place ”waypoints which are automatically interpolated using B-splines, to create a smooth path”. These waypoints and obstacles are easily deleted or dragged around the environment at any time to amend the desired driving trajectory. The chair can then follow the given path or attempt the desired target by making use of the fuzzy logic control-

Journal Journal of of Automation, Automation,Mobile MobileRobotics Roboticsand andIntelligent IntelligentSystems Systems

ler we have developed in previous works. It will be able also to avoid unknown obstacles basing on its reactive behaviours presented in Section III-C.

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Fig. 3. The 2D-interface, displaying start and end points with 3 user-waypoints, which have been interpolated with B-splines. The obstacle mode chosen is a corridor with corner situa�on.

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distances to the obstacles present in the robot’s surrounding area with maximum four meters distance to the chair’s center. For a more comprehensive review of the application of this method on the robotic wheelchair, refer to [1] and [2]. 3.4. Fuzzy Switching In this switching method we introduce the concept of safe mini-intrusion. This intrusion information generated with the DVZ controller provides a safe commutation from the current wheelchair position to a sub-goal in the case of corner situation as shown in Fig.5. This problem occurs when the deformed DVZ becomes symmetric with respect to the linear velocity direction. The reaction to this situation is to reduce the velocity, and to rotate until the obstacle is present in one side. In consequence, the speed will be reduced to zero or a negative value. To avoid this local minima problem [8], we have considered a left in front intrusion Il calculated for θ ≤ α < θ + 60◦ , a right in front one Ir calculated for θ − 60◦ ≤ α < θ , a left side intrusion Ils calculated for θ + 60◦ ≤ α < θ + π and a right side intrusion Irs calculated for θ + π ≤ α < θ − 60◦ as presented in Fig.6. Y

α d(α) Y1

dh(α)

Fig. 4. The 2D-interface, displaying the wheelchair naviga�on.

3.3. �e�c��e ��st�c�e ��oi��nce �i�ot In our architecture the wheelchair must attempt a target or a suit of targets while avoiding obstacles assumed to be unknown. Consequently, we decided to use a reactive pilot based on the DVZ as described in [8]. This method is built using a virtual risk zone surrounding the robot which can be deformed due to the proximity information. The risk zone deformation is due to the proximity information. The system reaction drives the robot velocities (linear and angular velocities, V and ω ) function of this deformation calculation. In general, the risk zone deformed by an obstacle can be reformed by reacting on the robot velocities. A laser sensor was then positioned looking directly in front of the chair. This sensor provides

Fig. 5. The ellip�c risk zone is surrounding the wheelchair. This risk zone is a func�on of dh and α .The deformed risk zone is the product of two parts : the ﬁrst is computed using the distance measured by the laser sensor to the obstacles, the second is basing only on distances inside the undeformed risk zone and noted d(α ) The basic idea of this switch is the product of two parts. In the first one, obstacles are present in front of the wheelchair (Il or Ir or both), so that it avoids them. In the second one, obstacles are present in one or two sides (Ils or Irs or both), the robot can attempt its targets without avoiding obstacles. To do so, we have developed a fuzzy switch [2]. 3.5. Hypotheses Experimental and simulation tests was designed to investigate the hypotheses about the adapted user interface. These hypotheses targets to judge the accessibility of laser-sensor as well as encoders inputs and the complexity of the steps needed to perform the user objective. Articles

Journal Journal of of Automation, Automation,Mobile MobileRobotics Roboticsand andIntelligent IntelligentSystems Systems

θ−

θ+

60 ◦

Ils

Irs

θ+

π X

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4.2. Experiment Concept and Requirements Four conditions were tested : corridor guidance and a totally autonomous wheelchair versus a trajectory choosed by the user, and guidance with obstacles in one side versus two sides. Condition A: the user had an interface showing his surrounding environment with corridor situation as well as his initial position (Fig. 7 or Fig. 8). He was to move the mouse in order to select the target location that he wants to achieve and press a button in the interface to let the wheelchair starting the navigation autonomously.

Fig. 6. This ﬁgure highlights the four safe mini-intrusion : Il , Ir , Ils and Irs . �ules are designed for mul�ple scenario that can be helpful for the user.

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Hypothesis 1: The proposed user interface is easy to use but needs some prompting levels. It accommodates several devices : encoders measures, laser sensor and ultrasonic proximity informations (guideline1). Furthermore, it has two step process for target selection (guideline2) and one step to start the guidance. However, the selection of the obstacle disposal is not adapted by the user. Hypothesis 2: It should be more confident for the user to navigate in environment with obstacles present in one side than in two sides. Hypothesis 3: The selection of waypoints and the generation of trajectory should perform the desired target safely and in few moment. One objective of this research is to understand which guidance controller works well for the user : totally autonomous navigation or partial navigation in which the user intercept by choosing the trajectory he wants to follow in order to achieve his target.

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Start point

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Fig. 7. �orridor na�iga�on : the user select onl� the target point and ini�ated the na�iga�on.

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Trials was run in Janvier 2012 to evaluate the effectiveness of the user interface used to control the wheelchair. The participants are student researchers.

10000

Trajectory2

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The participants were invited to participate in the experiments. Four candidates accepted : they were students of the Engineering School of Sfax and aged between twenty and thirty two years old. All of them had a high cognitive ability, and were able to manipulate an interface (selection of points using the mouse). The candidates were two men and two women. The four participants were able to use the power wheelchair and the interface as an access method. Table 2 describes the participants’ cognitive abilities, behavioral abilities, and interface operating. Each participant was given an initial profile of a guidance problem to deal with. A profile contains steps for using the interface based. Different disposal of the obstacles in the envioronment surrounding the user were used : corridor, objects in one side and others in two sides.

End point

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4. Experiment Methodology

4.�. �ar�cipants

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Fig. 8. �orridor na�iga�on : di�erent star�ng orienta�on. Condition B the user had the same interface as that displayed in condition A (Fig. 9). The participant was to use the mouse to move the different points shown on the screen in order to generate his

14000

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Tab. 1. �andidate abili�es Age

Cognition

Behavior

Vision

Inter f ace

able to learn new skills

None

Functional

Standard

standard

Sociable

Functional

Standard

Good memory

Cooperative

Functional

Standard

distractible

None

Functional

Standard

own trajectory. Then he was to press a button to initiate the wheelchair to follow the designed path.

choose either a trajectory to follow either a target to reach.

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Fig. 9. Path following : series of waypoints points selected by the user.

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Fig. 11. �a�iga�on with obstacles on two sides.

Condition C The user had different objects disposed on one side in the environment and displayed on the interface screen (Fig. 10). Then he was to choose either a trajectory to follow either a target to reach.

10000

−2000 −2000

10000

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Fig. 10. �a�iga�on with obstacles on one side. Condition D The user had different objects disposed on two sides (Fig. 11). Then he was also to

In this experiment, the tasks was chosen in order to evaluate the intention of the user when the wheelchair is navigating autonomously using a reactive obstacle avoidance algorithm. The intention means the feelings of fear, encourage, frustration. That’s has a great importance especially when the interface shows in real time the current position of the wheelchair beyond the obstacles. The obstacles used in the experiment are wooden objects. For example, the corridor is composed from four wooden walls in order to prevent injury to the user and equipment. In the first test, the user places the target point in the interface and initiated the navigation of the wheelchair in environment with a corridor. Three trials were performed and are presented in Figures. 7 and 8. while in figure Fig. 10, the participant placed two waypoints in the environment added to the previous start and end point. The trials are carried out by adopting different start points defined by the coordinates (X = 13000mm,Y = 0mm) and three orientations (θ = 0,θ = π2 ,θ = π ). The driver aims to reach the target defined by the coordinates (XT = 1000mm,YT = 4000mm). The adopted route is complicated with obstacles on both sides. In Fig. 7 where the robot initial orientation is 0, we notice that the robot is initially reaching the target. When the detected obstacles become inside of the security zone, the robot starts to avoid them, follows-up the Articles

Journal Journal of of Automation, Automation,Mobile MobileRobotics Roboticsand andIntelligent IntelligentSystems Systems

walls until reaching the target. In the second scenario presented in Fig. 8, the user has changed his initial orientation and attempts to reach the same target. Initially, the experimenter positioned the interface towards the participant at an appropriate position. Then he initiated the communication between the laptop and the wheelchair and display the interface to the user. The experimenter explained the task to execute in the current experiment condition and trained the participant to operate with the interface. The execution began when the experimenter prompted the user to choose a target to achieve and ended when the wheelchair reached it. The duration of each trial averaged forty five minutes. Data was collected basing on post-trial questionnaires and the computer files. The questionnaire asked questions about which kind of navigation (totally autonomous or path following) the user liked most, which navigation they liked least, and suggestions for improving the proposed algorithm. The computer files saved : wheelchair velocities and position, crossed distance and the time needed from target prompt to user selection and the time from the wheelchair movement to the target reached.

5. Results and Discussion Hypothesis 1 (ease of use): The target selection time and the initiate of move was be used as a measure of ease of use of the interface. The average participant’s from prompt to pressing the button of navigation ranged from 9 seconds to 21.12 seconds (table. 2). Participants have operated better in Condition A (moving in corridor with a totally autonomous wheelchair) than Interface B (selection of trajectory). As for the condition C and D, the participants were operated easily for the both cases. For each participant and in the six experiments (three trials for condition A and three trial for the three remaining conditions), they expressed their preference for particular condition B where they choose their own trajectory. Hypothesis 2 (obstacles in one side preference to two sides: Basing on selection time analysis (table. 2) and the level of fear summarized in (table. 3), we notice that participants liked the navigation through obstacles in one side than in two sides. Moreover, the first open-ended questions (Which obstacles disposal did you like more? Why? and Which disposal did you like less ? Why?). Disposal in the condition C was the most liked with four positive comments. Hypothesis 3 (totally autonomous navigation preference to partially guidance): In statistical analysis over fear level (table. 3), we notice that users were found to have more confident in navigation with condition B ( partially autonomous navigation) than navigation with condition A (totally autonomous navigation), and condition C than D. This due to the use of a reactive obstacle avoidance algorithm that supposed the obstacles are unknown. When the user is near to the obstacle, he feels fearing and have not 6

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confident on the wheelchair autonomous navigation, despite it succeeded to avoid the obstacles and reached the target. Besides, we notice that the greater the number of trials the least level of fear. In fact, due to the little sample size and minor abilities problems of users in this experiment, we do not want to make generic claims. In future works, the users will contain a more important disabilities.

6. Conclusion The proposed pilot system have a series of assumptions to be taken into account in future works. Despite it ensures an important level of autonomy to the robot the level of the user safe navigation is yet minor. The time selection and the level of fear Sproved that the participant feel more confident where he interacts with the interface by choosing her own path to achieve the target. Despite these limitations, this paper has proposed generic guidelines for reactive guidance with user interaction in the case of disable people.

AUTHORS

Lobna Amouri∗ – Computer Department, Community College, Imam Abdulrahman Bin Faisal University, P.O Box 1982, Dammam 31441, Saudi Arabia, e-mail: lmamouri@iau.edu.sa. Mohamed Jallouli – Engineering School of Sfax (ENIS), University of Sfax, Tunisia. Cyril Novales – Institut PRISME, 63 av. de Lattre de Tassigny, F-18020 Bourges cedex, France. Gérard Poisson – Institut PRISME, 63 av. de Lattre de Tassigny, F-18020 Bourges cedex, France. Nabil Derbel – Engineering School of Sfax (ENIS), University of Sfax, Tunisia. ∗ Corresponding

author

REFERENCES [1] L. Amouri, C. Novales, G. Poisson, M. Njah, M. Jallouli, and N. Derbel, “DVZ-based obstacle avoidance control of a wheelchair mobile robot”. In: 2011 IEEE International Conference on Mechatronics, 2011, 911–915, 10.1109/ICMECH.2011.5971244. [2] L. Amouri, C. Novales, M. Jallouli, G. Poisson, and N. Derbel, “An effective DVZ-fuzzy logic pilot for a mobile robot using generic architecture”, International Journal of Vehicle Autonomous Systems, vol. 12, no. 3, 2014, 201–220, 10.1504/IJVAS.2014.062977. [3] T. Carlson and Y. Demiris, “Increasing robotic wheelchair safety with collaborative control: Evidence from secondary task experiments”. In: 2010 IEEE International Conference on Robotics and Automation, 2010, 5582–5587, 10.1109/ROBOT.2010.5509257. [4] C. Dune, C. Leroux, and E. Marchand, “Intuitive human interaction with an arm robot for

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Tab. 2. �ser ��e to tar�et selec�on and na�i�a�on star�n�� rials are �ro��ed into t��les� �or exa��le ��x�y� �eans t�e trial x take y seconds ConditionA

ConditionB

ConditionC

ConditionD

T (1, 15), T (2, 12), T (3, 10)

T (1, 18)

T (1, 10)

T (1, 11)

T (1, 21.12), T (2, 15), T (3, 9)

T (1, 21)

T (1, 12)

T (1, 17), T (2, 15), T (3, 12)

T (1, 19)

T (1, 12)

T (1, 16.32), T (2, 12), T (3, 9)

T (1, 18.2)

T (1, 9)

Tab. 3. �ar�ci�ant �earin� le�el ��ero lo� to ten �i�� ConditionA

ConditionB

ConditionC

ConditionD

T (1, 10), T (2, 7), T (3, 4)

T (1, 2)

T (1, 4)

T (1, 7)

T (1, 10), T (2, 10), T (3, 5)

T (1, 1)

T (1, 5)

T (1, 7)

T (1, 9), T (2, 7), T (3, 3)

T (1, 1)

T (1, 4)

T (1, 8)

T (1, 10), T (2, 10), T (3, 2)

T (1, 2)

T (1, 3)

T (1, 7)

severely handicapped people - A One Click Approach”. In: 2007 IEEE 10th International Conference on Rehabilitation Robotics, 2007, 582– 589, 10.1109/ICORR.2007.4428484. [5] A. Ghorbel, M. J. Jallouli, and L. Amouri. “A HW/SW Implementation on FPGA of Absolute Robot Localization Using Webcam Data”. In: O. Kanoun, F. Derbel, and N. Derbel, eds., Sensors, Circuits and Instrumentation Systems. De Gruyter, Berlin, Boston, January 2017. [6] M. Ghorbel, R. Kadouche, and M. Mokhtari, “User & service modelling in assistive environment to enhance accessibility of dependent people”, Hammamet, Tunisia, 2007, 6. [7] P. Jia, H. H. Hu, T. Lu, and K. Yuan, “Head gesture recognition for hands-free control of an intelligent wheelchair”, Industrial Robot, vol. 34, no. 1, 2007, 60–68, 10.1108/01439910710718469. [8] L. Lapierre, P. Lepinay, and R. Zapata, “Simultaneous Path Following and Obstacle Avoidance Control of a Unicycle-type Robot”. In: ICRA: International Conference on Robotics and Automation, Roma, Italy, 2007, 2617–2622. [9] S. P. Levine, D. A. Bell, L. A. Jaros, R. C. Simpson, Y. Koren, and J. Borenstein, “The NavChair Assistive Wheelchair Navigation System”, IEEE Transactions on Rehabilitation Engineering, vol. 7, no. 4, 1999, 443–451, 10.1109/86.808948. [10] Y. Matsumoto, T. Ino, and T. Ogasawara, “Development of intelligent wheelchair system with face and gaze based interface”. In: Proceedings 10th IEEE International Workshop on Robot and Human Interactive Communication. ROMAN 2001 (Cat. No.01TH8591), 2001, 262–267, 10.1109/ROMAN.2001.981912.

[11] A. Mihailidis, P. Elinas, J. Boger, and J. Hoey, “An Intelligent Powered Wheelchair to Enable Mobility of Cognitively Impaired Older Adults: An Anticollision System”, IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol. 15, no. 1, 2007, 136–143, 10.1109/TNSRE.2007.891385. [12] J. d. R. Millán, F. Renkens, J. Mouriño, and W. Gerstner, “Noninvasive brainactuated control of a mobile robot by human EEG”, IEEE transactions on bio-medical engineering, vol. 51, no. 6, 2004, 1026–1033, 10.1109/TBME.2004.827086. [13] R. C. Simpson, “Smart wheelchairs: A literature review”, Journal of Rehabilitation Research and Development, vol. 42, no. 4, 2005, 423–436. [14] T. Taha, J. V. Miro, and G. Dissanayake, “POMDP-based long-term user intention prediction for wheelchair navigation”. In: 2008 IEEE International Conference on Robotics and Automation, 2008, 3920–3925, 10.1109/ROBOT.2008.4543813. [15] K. Tsui, H. Yanco, D. Kontak, and L. Beliveau, “Development and evaluation of a flexible interface for a wheelchair mounted robotic arm”. In: 2008 3rd ACM/IEEE International Conference on Human-Robot Interaction (HRI), 2008, 105– 112, 10.1145/1349822.1349837. [16] K. M. Tsui, H. A. Yanco, D. J. Feil-Seifer, and M. J. Matarić, “Survey of Domain-specific Performance Measures in Assistive Robotic Technology”. In: Proceedings of the 8th Workshop on Performance Metrics for Intelligent Systems, New York, NY, USA, 2008, 10.1145/1774674.1774693.

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Heat Transfer Model of a Small Size Satellite on Geostationary Orbit Submitted: 15th August 2019; accepted: 24th September 2019

Philippe Preumont, Roman Szewczyk, Paweł Wittels, Filip Czubaczyński

DOI: 10.14313/JAMRIS/3-2019/24 Abstract: The purpose of the study is to compute an approached satellite thermal model to be able to estimate the heat transfer and the maximum temperature through the harmonic gear for small size geostationary satellite. We use finite element method using known data with Nastran In-CAD FEM software. The presented results show temperature gradients compatible with experimental information. Though the results are not correlated with dedicated tests, they seems to be in line with what can be found in literature. The paper use these information in dedicated analysis related to impact of temperature gradient in precision mechanism. The second purpose of the study is to show that simplified tools and methods allows to reach results sufficient for preliminary analysis. The space business is changing fast with growing private companies that are challenging conservative space standards. A simplified analysis allowing to reduce cost and increase competitiveness is presented. Keywords: Simplified thermal model, FEM modelling, Nastran InCAD, Harmonic gear, Heat transfer

1. Introduction Mechanisms in the space environment are usually more robust to extreme temperatures than electronics. The components driving the thermal design and thermal control of satellites are the most sensitive ones. For this reason, mechanism are susceptible to be submitted more constraining temperatures. These temperatures would not disturb the proper basic functioning of the mechanism, but still, some aspects of mechanism are impacted by temperature gradients. For example, the positioning accuracy. Literature [1–11] present relevant information to perform accurately thermal analysis of satellite. Thermal behavior of small satellite were studied in [12] and simplified model of temperature evolution in [13,14]. Many different cases and configuration were analyzed as in [15-18]. This study present estimated temperature evolution analysis for a small-size satellite. The

development of this type of satellite is growing fast and opportunities to find application of such developments seems more favorable. Finally this article present the worst case temperature distribution in position mechanism for the defined type of satellite. The paper presents innovative simplified way to determine approximately temperature gradient in satellite, hoping to initiate new possibilities for decreasing engineering costs at early stages of satellite architecture development. On the base of presented simulations, the temperature distribution on the mechanical joints may be estimated. Basing on this estimation, the temperature correction factors for mechanical elements, such as harmonic gear, can be calculated.

2. Materials and Methods To perform the study we use simplified methods with conservative approach. The program used is Nastran In-CAD (Inventor®). The Nastran module of Inventor® is limited in terms of radiation module analysis [19]. We compare the results obtained with data available in literature but also cross check with simplified analytical calculation. The purpose of this work being to estimate approximately the extreme temperature in positioning mechanism, the mesh will not be refined nor manual but automatic through Inventor – Nastran In-CAD. The considered satellite is on geostationary orbit. We estimate the worst case temperature that is seen by positioning mechanism. When the satellite is on dark side of Earth, its temperature can be regulated by heaters and thus it is not expected to be very low. We then concentrate on the hot case, when the satellite goes out of the shadow of Earth and enters the sunlight, as it is presented in figure 1. We consider two parts in our study. The first part consists in the steady state analysis. We present the temperature gradient in mechanism for the same condition. We then consider the transient case at the moment the satellite enter the sun beam to verify the thermal behavior at mechanism level and check if temperature in mechanism exceed steady state temperature.

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Electronics radiator Antenna

Solar Panel

Fig. 1. Geostationary satellite entering the sunlight from the Earth shadow

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Figure 2 presents the simplified model of the small-size satellite used in presented modelling. The modules of the satellite are connected by the aluminum shafts, as it is commonly done in the real structures. For the proposed modelling, the assumption was made, that the Sun flux angle on the satellite side equals 30°.

Fig. 2. Satellite top view – sunbeam angle of incidence Considering the angle of incidence of sun flux on satellite side is about 30° and using values presented in the Table 2, the flux from sun seen by satellite side is considered to be:

The satellite analyzed in this paper is of small size – 200 x 300 x 500 mm. The material selected for the model is Aluminum 7075, which is commonly used in space applications. We will not consider here MLI [20]. = S S ide 1368 W / m2 × sin 60° = 1185 W / m2 , (1) The Satellite is modelled as a plain structure to consider mass of electronics and instruments. Followin a similar way, we obtain the solar flux on front suring assumptions were made: face of the satellite (facing to Earth): – The mass of the satellite is approximately 40 kg. S F ront 1368 W/m2 × sin 30° = 684 W/m2 , (2) – The mesh element size is ~10 mm and the number = of elements is more than 134 000. – The initial temperature of the satellite coming Moreover, the solar panels are considered to be from dark side of Earth is set to 200 K [3]. It is the oriented orthogonally to the sun beams, so that they approximate mean temperature of a geostationary see the maximum flux from Sun. satellite in dark side of Earth. We consider in this preliminary analysis that the flux is constant in the first hour when the satellite enThe model contains important simplification. The ters the sunlight. satellite emissivity will not be considered but the temperature of main components will be fixed. This is considered as a conservative case. 3. Results The constrains placed on the model are the temperatures of satellite electronics radiator, the antenna 3.1. Steady-State Case and solar panel. The electronics radiator temperature is fixed to maximum temperature allowed for elecConsidering the mentioned boundary conditions tronics, 313 K. The solar panel and the antenna temand thermal loads, the steady state model shows that perature are also fixed temperature set to typical opmaximum temperature reached at mechanism locaerating maximum temperature allowed, respectively tion is approximately 369 K. The distribution of the 373 K and 333 K. Boundary conditions are presented temperature on the satellite is presented in figure 3, in the table 1, whereas heat fluxes from the environwhereas the distribution of temperature on the joint ment are summarized in the table 2. of modules is presented in figure 4. Tab. 1. Boundary condition - Typical temperature ranges of satellite components [1,2] Component

Typical Operating Temperature Range [K]

Electronics

253 to 313

Antenna Dish

193 to 333

Solar Panels

173 to 373

Tab. 2. Used fluxes from environment [1] From

Typical Flux value used [W/m2]

Sun

1368

Albedo

456

Earth

236

Fig. 3. Satellite temperature in steady state condition

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(b)

(a) Fig. 4. Satellite temperature in steady state condition: (a) Temperature gradient in antenna positioning mechanism; (b) Temperature gradient in solar panel positioning mechanism As a point of comparison, in steady state, temperature was obtained on the base of formula [2]:    α   Ashaft _ inc  T (K ) =  S    /σ    ε   Ashaft _ rad  

0,25

sents the mechanism temperature variation with regard to time.

(3)

Where, for the aluminum shaft of radius and length r = 12,5 mm and l=100 mm, the area of surface of Sun incidence normal to Sun, Ashaft_inc, is:

A( shaft _ inc ) = 2 × r × l = 2500 mm2 (4)

the area of surface radiating to space, Ashaft_rad, is:

Ashaft _ rad = 2π × r × l= 7850 mm2 (5)

with the total radiance, S, seen by the shaft (considering the data presented in the table 2): S = 1368 W/m2 + 456 W/m2 + 236 W/m2 = 2060 W/m2

m2 + 236 W/m2 = 2060 W/m2 (6)

and with the Boltzmann Constant, σ:

= σ 5,67 × 10−8 WK 4 /m2 (7)

Considering that the emissivity and absorptivity of the shaft surfaces, is for aluminium ε = 0,11 and α = 0,14 [2] respectively, we obtain a steady state temperature of T = 348 K. The second point of comparison, we can find in literature [1], presenting typical maximal test temperature of solar array mechanism equal 353 K.

3.2. Transient Case

Due to the fact that it is necessary to confirm the most critical temperature on the mechanism, the main analyses will concentrate on the transient case. As mentioned in [3], transient event may be the most important design driver. For this reason, the heat transfers in the satellite structure was modelled, while cold steady state is perturbed by Sun radiation. In the transient case, we verify how temperature of the satellite is changing with time and especially at positioning mechanism location. The figure 5 preArticles

Fig. 5. Satellite maximum temperature variation with time The figure 6 presents the location of the maximum temperature of 390 K on shaft/mechanism location. This temperature appears also after a time of about 20 minutes. Significant temperature differences are observed. It should be highlighted, that these temperature gradients may significantly influence on the accuracy of precision of the positioning systems, such as harmonic gear boxes.

Fig. 6. Satellite shaft positioning mechanism temperature gradient

4. Conclusion Presented results confirm the usability of proposed simplified method of estimation of temperature gradients in satellite shafts. Such temperature gradients are critical from the point of view of accuracy of positioning mechanism, especially mechanisms utilizing harmonic gear boxes. This is mainly due to the fact that temperature gradients are the source of non-symmetrical geometrical modification of gear components.

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Proposed method is in good agreement with analytical calculations presented previously in literature considering the temperature of mechanisms in steady state. Using proposed method utilizing FEM, we obtained central value of temperature 369 K compared to 348 K by rough analytical calculation and 353 K for similar application [1]. This difference is due to the fact that we wanted to stay conservative in our analysis and didn’t consider radiation in our model. The presented transient thermal analysis shows that mechanism could be subjected to higher temperature than the maximum steady state temperature as well as for additional temperature gradients. Considering presented worst case scenarios for temperature and gradients, it is possible to obtain input data for detailed mechanical analysis of harmonic gear boxes. As a result, the new method of correction of temperature drifts of such mechanism may be proposed. It should be highlighted, that proposed method presents an alternative solution for preliminary analysis, leading to the decrease of design cost at early stage of very demanding elaboration process of satellite project.

AUTHORS Philippe Preumont* – PIAP Space Sp. z o. o., Jerozolimskie 202, Warsaw 02-486, Poland, e-mail: philippe. preumont@piap-space.com.

Roman Szewczyk – Warsaw University of Technology, Faculty of Mechatronics, Poland. Paweł Wittels – PIAP Space Sp. z o. o., Jerozolimskie 202, Warsaw 02-486, Poland. Filip Czubaczyński – PIAP Space Sp. z o. o., Jerozolimskie 202, Warsaw 02-486, Poland. * Corresponding author

References

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[3] G. C. Birur, G. Siebes, and T. D. Swanson. “Spacecraft Thermal Control”. In: R. A. Meyers, ed., En cyclopedia of Physical Science and Technology (Third Edition), 485–505. Academic Press, New York, 2003.

[4] R. Gubby and J. Evans, “Space environment effects and satellite design”, Journal of Atmospher

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ic and Solar-Terrestrial Physics, vol. 64, no. 16, 2002, 1723–1733 DOI: 10.1016/S1364-6826(02)00122-0.

[5] D. Gilmore, Spacecraft Thermal Control Handbook: Fundamental Technologies, The Aerospace Press, 2002 DOI: 10.2514/4.989117. [6] R. Henderson, “Thermal control of spacecraft”. In: P. Fortescue and J. Stark, eds., Spacecraft sys tems engineering, Wiley, New York, 1995.

[7] C. J. Savage. “Thermal control of spacecraft”. In: P. Fortescue, J. Stark, and G. Swinerd, eds., Space craft Systems Engineering. 3rd edition, Wiley, New York, 2003. [8] C. J. Savage. “Thermal Control of Spacecraft”. In: Spacecraft Systems Engineering, John Wiley & Sons, 2011, 357–394 DOI: 10.1002/9781119971009.ch11.

[9] J. Meseguer, I. Pérez-Grande, and A. Sanz-Andrés, Spacecraft thermal control, Woodhead Publishing Limited, 2012 DOI: 10.1533/9780857096081. [10] J.-R. Tsai, “Overview of Satellite Thermal Analytical Model”, Journal of Spacecraft and Rockets, vol. 41, no. 1, 2004, 120–125 DOI: 10.2514/1.9273.

[11] C. A. Wingate, “Spacecraft thermal control”. In: V. Piscane and R. Moore, eds., Fundamentals of Space Systems, London, 1994, 433–466.

[12] Pérez-Grande, A. Sanz-Andrés, C. Guerra, and G. Alonso, “Analytical study of the thermal behaviour and stability of a small satellite”, Ap plied Thermal Engineering, vol. 29, no. 11, 2009, 2567–2573 DOI: 10.1016/j.applthermaleng.2008.12.038. [13] J. Gaite, A. Sanz-Andrés, and I. Pérez-Grande, “Nonlinear analysis of a simple model of temperature evolution in a satellite”, Nonlinear Dynam ics, vol. 58, no. 1, 2009, 405–415 DOI: 10.1007/s11071-009-9488-x. [14] J. Gaite, “Nonlinear analysis of spacecraft thermal models”, Nonlinear Dynamics, vol. 65, no. 3, 2011, 283–300 DOI: 10.1007/s11071-010-9890-4. [15] L. Jacques, E. Béchet, and G. Kerschen, “Finite element model reduction for space thermal analysis”, Finite Elements in Analysis and Design, vol. 127, 2017, 6–15 DOI: 10.1016/j.finel.2017.01.001.

[16] G. Fernández-Rico, I. Pérez-Grande, A. Sanz-Andres, I. Torralbo, and J. Woch, “Quasi-autonomous thermal model reduction for steady-state Articles

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problems in space systems”, Applied Thermal Engineering, vol. 105, 2016, 456–466 DOI: 10.1016/j.applthermaleng.2016.03.017.

17. M. A. Gadalla, “Prediction of temperature variation in a rotating spacecraft in space environment”, Applied Thermal Engineering, vol. 25, no. 14, 2005, 2379–2397 DOI: 10.1016/j.applthermaleng.2004.12.018. 18. L. Liu, D. Cao, H. Huang, C. Shao, and Y. Xu, “Thermal-structural analysis for an attitude maneuvering flexible spacecraft under solar radiation”, International Journal of Mechanical Sciences, vol. 126, 2017, 161–170 DOI: 10.1016/j.ijmecsci.2017.03.028.

19. W. Younis, Up and Running with Autodesk Nastran In-CAD 2019: Simulation for Designers, CreateSpace Independent Publishing Platform, 2018. 20. V. Nenarokomov, L. A. Dombrovsky, I. V. Krainova, O. M. Alifanov, and S. A. Budnik, “Identification of radiative heat transfer parameters in multilayer thermal insulation of spacecraft”, International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27, no. 3, 2017, 598–614 DOI: 10.1108/HFF-03-2016-0136.

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Preface to Special Issue on Contemporary Problems of Computer Science, Physics and Applied Mathematics

DOI: 10.14313/JAMRIS/3-2019/25 This special issue of the Journal of Automation, Mobile Robotics and Intelligent Systems is dedicated to informing the readership of selected aspects of Contemporary issues of Computer Science, Physics, Economy and Applied Mathematics. The articles contained in this issue of the journal have been introduced in preliminary version between 2-5 July, 2018, during the 3rd Conference on Information Technology, Systems Research and Computational Physics (ITSRCP’18), as well as the 6th International Symposium CompIMAGE’18 – Computational Modeling of Objects Presented in Images: Fundamentals, Methods, and Applications (CompIMAGE’18), which were organized by the Faculty of Physics and Applied Computer Science of the AGH University of Science and Technology and co-organized by the Systems Research Institute of the Polish Academy of Sciences in Warsaw, Poland. The idea behind this special edition was to create a specific volume containing a number of interesting cutting edge scientific articles. Herein, one can find contributions dealing with computational algorithms, data mining, clustering, variance analysis, trend analysis, logical connectives, orthomodular lattice, quantum logic, copulas, elliptically contoured distribution, and some aspects of tomography. This issue contains the following original papers in their special, extended versions.

The first paper is entitled Decomposition Integral without Alternatives, its Equivalence to Lebesgue Integral, and Computational Algorithms, and was authored by Adam Š� eliga. It introduces a new class of decomposition integrals called ‘collection integrals’. The paper is focused upon two special types of collection integrals, namely the chain integral and the min-max integral. Some computational algorithms are also discussed.

Jana Kalicka, Maria Minarova, Jaroslav Halvonik and Lucia Majtanova in their work entitled Statistical Analysis of Models for Punching Resistance Ensuring, provide a statistical assessment of models that ensure the safety of reinforced concrete slabs. The obtained results, hence, the best model, will become the European Union standard after 2020. Based on the statistical parameter evaluation and in accordance with engineer best practice, a new model was suggested, statistically verified and nominated as the normative.

The work entitled Global and local trend analysis and change-point analysis of selected financial and market indices, by Dominika Ballová, uses trend analysis. Herein, the author uncover the global evolution of selected indices. After evaluating the global trend in the series, local trend analysis occur, and the two sets of figures are compared. Using change-point analysis, the intent was to detect the moments in which the indices differ. By means of cluster analysis, indices that are most similar in long-term development are made to standout. In each analysis, the most appropriate methodology is uncovered and compared.

Oľga Nánásiová, Viera Č� erňanová and Ľubica Valášková authored the paper Probability Measures and logical connectives on Quantum Logics. In this paper, the authors delve into aspects for modelling of the probability of logical connectives in quantum logic via G-mapping. What is interesting is that in this article, authors show that unlike classical (Boolean) logic, probability measures of projections in quantum logic are not necessarily pure projections. They then go on to indicate how it is possible to define a probability measure of implication in quantum logic using G-mapping, and subsequently provide a study of some properties of this measure that are different from measure of implication in Boolean algebra.

The paper entitled 2D-Raman Correlation Spectroscopy as a Method to Recognize of the Interaction at the Interface of Carbon Layer and Albumin was written by the team consisting of Anna Kołodziej, Aleksandra Wesełucha-Birczyńska, Paulina Moskal, Ewa Stodolak-Zych, Maria Dużyja, Elżbieta Długoń, Julia Sacharz and Marta Błażewicz. The article is devoted to the analysis of two types of model carbon layers differing primarily in topography, and to their interactions with blood plasma proteins. Herein, the first layer was formed of pyrolytic carbon C (CVD) and the second was constructed of multi-walled carbon nanotubes obtained by electrophoretic deposition (EPD), both layers are set on a Ti support.

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Finally, Tomas Bacigal, Magdalena Komornikova and Jozef Komornik, provide a paper entitled State-of-the-art in Modelling Nonlinear Dependence Among Many Random Variables With Copulas and Application to Financial Indexes. The study focuses attention on multi-dimensional copula models for returns of the indexes of selected prominent international financial markets. In this paper, the authors dig into modelling results based on elliptic copulas, 7-dimensional hierarchical Archimedean copulas, vine copulas and factor copulas, and demonstrate the dominant role of the SPX index among the considered major stock indexes. It is noteworthy that the dominance of these models is most striking over the interval of the recent financial market crisis. At the same time, the best Student class models were providing a substantially poorer fit. We would like to thank all those who participated in, and contributed to the Conference program, as well as all the authors who had submitted their papers. We also wish to thank all our colleagues and the members of the Program Committee, both for their hard work during the review process and for their cordiality and outstanding efforts in the local organization of the Conference. Editors: Piotr A. Kowalski Systems Research Institute, Polish Academy of Sciences and Faculty of Physics and Applied Computer Science, AGH University of Science and Technology Szymon Ĺ ukasik Systems Research Institute, Polish Academy of Sciences and Faculty of Physics and Applied Computer Science, AGH University of Science and Technology Piotr Kulczycki Systems Research Institute, Polish Academy of Sciences and Faculty of Physics and Applied Computer Science, AGH University of Science and Technology

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Journal Journal of of Automation, Automation,Mobile MobileRobotics Roboticsand andIntelligent IntelligentSystems Systems

VOLUME 2019 VOLUME 13,13, NÂ°NÂ°3 3 2019

Dďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ Iďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ Aďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ Iďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ Aďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ Submďż˝ďż˝ed: 1ďż˝th June 2019; accepted: 10th September 2019

Adam Ĺ eliga DOI: 10.14313/JAMRIS/3-2019/26

2. Preliminaries

Abstract: ďż˝n this paper we present a new class of decomposiďż˝on integrals called the collecďż˝on integrals. ďż˝rom this class of integrals we take a closer look on two special types of collecďż˝on integralsďż˝ namely the chain integral and the minmax integral. ďż˝uperdecomposiďż˝on ďż˝ersion of collecďż˝on integral is also deďż˝ned and the superdecomposiďż˝on duals for the chain and the min-max integrals are presented. ďż˝lsoďż˝ the condiďż˝on on the collecďż˝on that ensures the coincidence of the collecďż˝on integral with the ďż˝eďż˝esgue integral is presented. ďż˝astlyďż˝ some computaďż˝onal algorithms are discussed.

In this paper, without loss of generality, we will consider a ďż˝ixed ďż˝inite space X = {1, 2, . . . , n} â&#x160;&#x201A; N. A chain on X is any sequence {Ai }ki=1 such that â&#x2C6;&#x2026; Ě¸= A1 Âˇ Âˇ Âˇ Ak â&#x160;&#x2020; X. A full chain on X is any chain {Ai }ki=1 such that k = n. Also, only positive functions on X will be considered, i.e., functions with domain X and co-domain [0, â&#x2C6;&#x17E;[. The class of such functions will be denoted by F. The set of not strictly increasing functions will be denoted by Fâ&#x2020;&#x2018; . A capacity is any set function Âľ : 2X â&#x2020;&#x2019; [0, â&#x2C6;&#x17E;[ that is grounded, i.e., Âľ(â&#x2C6;&#x2026;) = 0, and monotone with respect to set inclusion, i.e., A â&#x160;&#x2020; B implies Âľ(A) â&#x2030;¤ Âľ(B). The class of all capacities will be denoted by M. A measure is any additive capacity, i.e., if A, B â&#x160;&#x2020; X are two disjoint sets then Âľ(A â&#x2C6;Ş B) = Âľ(A) + Âľ(B) holds. A symbol M+ denotes the set of all measures on X. A collection, mostly denoted by D, is any nonempty subset of 2X \{â&#x2C6;&#x2026;}. A decomposition system H is X any non-empty subset of 22 \{â&#x2C6;&#x2026;} , i.e., a decomposition system consists of at least one collection.

Keywords: decomposiďż˝on integralsďż˝ nonlinear integralsďż˝ computaďż˝onal algorithms

ďż˝. ďż˝ntrodďż˝cďż˝on Theory of linear integration has found many applications throughout mathematics. In the last century some concepts of nonlinear integrals appeared and are under investigation to this day. These nonlinear integrals found many applications also outside the world of mathematics, e.g., in psychology, productivity maximization, and others. A wide class of nonlinear integrals that contains other nonlinear integrals used today, namely decomposition integrals, were presented not so long ago by Even and Lehrer [4ďż˝. In this paper we deďż˝ine a subclass of these integrals and present some results concerning them. This contribution is organized as follows. Basic building blocks of this paper are introduced in the section ďż˝. In the section ďż˝ we deďż˝ine a special class of decomposition integrals called the collection integrals and in section 4 two such integrals are closely investigated. In the ďż˝ifth section we are interested in two concepts. Firstly in characterizing a positive bases for the spaces F+ and F and, secondly, in characterizing all collection integrals that yield to the Lebesgue integral if restricted to the space of measures. In the section 6, superdecomposition version of the collection integral is presented. The last section of this paper is devoted to the discussion of some computational algorithms. Recall that, in general, decomposition systems consist of more than one collection. These collections represent some choice alternatives. In this paper, we will consider only singleton decomposition systems, i.e., those consisting of a single collection. Hence the integral with single decomposition alternative will be discussed.

ďż˝eďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ďż˝ 2.ďż˝. A decomposition integral [4, 7] with respect to a decomposition system H is a mapping IH : F Ă&#x2014; M â&#x2020;&#x2019; [0, â&#x2C6;&#x17E;[ such that IH (f, Âľ) is equal to aA Âľ(A) : aA â&#x2030;Ľ 0, aA 1A â&#x2030;¤ f . Dâ&#x2C6;&#x2C6;H

Aâ&#x2C6;&#x2C6;D

Based on the choice of H we get a different types of decomposition integrals. In the following example some of the well known decomposition integrals are presented.

Example 2.2. If H1 consists of all singleton collections, we speak about the Shilkret integral [8], i.e., Sh(f, Âľ) = Âľ(A) min f (A) : A â&#x2C6;&#x2C6; 2X \ {â&#x2C6;&#x2026;} .

Note that we use the following abbreviate notation min f (A) = â&#x2C6;§ {f (x) : x â&#x2C6;&#x2C6; A}. If H2 consists only of partitions of X then we speak about the Pan integral [9], i.e., Pan(f, Âľ) = Âľ(A) min f (A) : Ď â&#x2C6;&#x2C6; Prt(X) , Aâ&#x2C6;&#x2C6;Ď

where Prt(X) denotes the set of all partitions on X. In case that H3 is the class of all chains on X then the corresponding integral is the Choquet integral [1], i.e., â&#x2C6;&#x17E; Ch(f, Âľ) = Âľ(f â&#x2030;Ľ t) dt. 0

1 41

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VOLUME 13, NÂ° NÂ°33 VOLUME 13,

Lastly, conceding that H4 = 22 \{â&#x2C6;&#x2026;} we get the concave integral cov(f, Âľ) introduced by Lehrer [5]. Note that the choice H5 = 2X \ {â&#x2C6;&#x2026;} also yields to the concave integral.

obtain

The decomposition integrals represent the integration from below as, for example, the lower Riemann sum. The integration from above is represented by socalled superdecomposition integrals.

ďż˝eďż˝ďż˝ďż˝ďż˝ďż˝ďż˝oďż˝ 2.3. A superdecomposition integral [6] with respect to a decomposition system H is a mapping â&#x2C6;&#x2014; â&#x2C6;&#x2014; IH : F Ă&#x2014; M â&#x2020;&#x2019; [0, â&#x2C6;&#x17E;] such that IH (f, Âľ) is equal to

Dâ&#x2C6;&#x2C6;H

Aâ&#x2C6;&#x2C6;D

aA Âľ(A) : aA â&#x2030;Ľ 0,

Aâ&#x2C6;&#x2C6;D

aA 1A â&#x2030;Ľ f

Note that the decomposition integral can attain only ďż˝inite values. In the case of superdecomposition integrals also unbounded values, i.e., â&#x2C6;&#x17E;, can be attained. Take, for example, X = {1, 2}, H = {{{1}}}, and f (x) = 1.

â&#x2030;¤

xâ&#x2C6;&#x2C6;X

aA Âľ(A) :

Aâ&#x2C6;&#x2C6;D

Âľ({x})

xâ&#x2C6;&#x2C6;X

Aâ&#x2C6;&#x2C6;D

aA 1A (x) :

Aâ&#x2C6;&#x2C6;D

aA 1A â&#x2030;¤ f

Aâ&#x2C6;&#x2C6;D

Âľ({x})f (x) = Leb(f, Âľ).

Corollary 3.3. ID = Leb if and only if for every function f â&#x2C6;&#x2C6; F there exist aA â&#x2030;Ľ 0, A â&#x2C6;&#x2C6; D, such that aA 1A = f. Aâ&#x2C6;&#x2C6;D

4.1. Chain Integral

As already mentioned, the value of a superdecomposition integral might be lower than the value of the corresponding decomposition integral. If we restrict ourselves to only measures and collection integrals this is no longer the case. Theorem 3.2. Let f â&#x2C6;&#x2C6; F , Âľ â&#x2C6;&#x2C6; M+ and let D be any collection. Then â&#x2C6;&#x2014; ID (f, Âľ) â&#x2030;¤ Leb(f, Âľ) â&#x2030;¤ ID (f, Âľ).

Proof. From the deďż˝inition of the collection integral we Articles

aA 1A â&#x2030;¤ f

ďż˝e are interested in the problem of ďż˝inding such collections D which lead to the equality ID = Leb. From the proof of the previous theorem we trivially get the following corollary.

â&#x2C6;&#x2014; In general, the inequality IH (f, Âľ) â&#x2030;¤ IH (f, Âľ) does not hold and thus the superdecomposition integral can attain values lower than the corresponding decomposition integral. In this paper we will be interested also in the equivalence of a special class of decomposition integrals with Lebesgue integral. The Lebesgue integral of a function f with respect to a measure Âľ will be denoted by Leb(f, Âľ).

ďż˝eďż˝ďż˝ďż˝ďż˝ďż˝ďż˝oďż˝ 3.ďż˝. A collection integral with respect to a collection D is a mapping ID : F Ă&#x2014; M â&#x2020;&#x2019; [0, â&#x2C6;&#x17E;[ such that ID (f, Âľ) = IH (f, Âľ) where H = {D}. Analoâ&#x2C6;&#x2014; :FĂ&#x2014; gously, super-collection integral is a mapping ID â&#x2C6;&#x2014; â&#x2C6;&#x2014; M â&#x2020;&#x2019; [0, â&#x2C6;&#x17E;] such that ID (f, Âľ) = IH (f, Âľ).

â&#x2C6;&#x2014; (f, Âľ) can be proved The inequality Leb(f, Âľ) â&#x2030;¤ ID analogously and thus the theorem follows.

4. ďż˝ďż˝aďż˝ďż˝leďż˝ ďż˝ďż˝ Cďż˝lleďż˝ďż˝ďż˝n Integralďż˝

In this section we will deďż˝ine a collection integral that represents special class of decomposition integrals.

Example 2.4. For decomposition integrals mentioned in previous example there is a corresponding superdecomposition integral. Observe that in the case of the decomposition system H3 the same integral is obtained.

ďż˝. Cďż˝lleďż˝ďż˝ďż˝n Integral

ID (f, Âľ) =

2019 2019

In this section we will take a closer look to two special types of collection integrals called a chain integral and a min-max integral.

The chain integral is a collection integral with respect to a single chain.

ďż˝eďż˝ďż˝ďż˝ďż˝ďż˝ďż˝oďż˝ 4.ďż˝. Let B be a chain on X. A mapping chB = IB is called a chain integral with respect to a chain B. The following deďż˝inition will be useful in proving a recursive equation for the chain integral.

ďż˝eďż˝ďż˝ďż˝ďż˝ďż˝ďż˝oďż˝ 4.2. Let B = {Ai }ki=1 be a chain. A sequence {ai }ki=1 will be called f -B-feasible if and only if ai â&#x2030;Ľ 0, 1 â&#x2030;¤ i â&#x2030;¤ k, and k i=1

ai 1Ai â&#x2030;¤ f.

A f -B-feasible sequence {ai }ki=1 will be called min-f B-feasible if and only if ak = min f (Ak ).

Lemma 4.3. Let B = {Ai }ki=1 be a chain. For every f -B-feasible sequence {ai }ki=1 there exists min-f -Bfeasible sequence {bi }ki=1 such that k i=1

ai Âľ(Ai ) â&#x2030;¤

bi Âľ(Ai ).

i=1

Proof. The proof of this lemma will be divided into two k cases. Case 1: i=1 ai â&#x2030;¤ min f (Ak ). Then we can deďż˝ine {bi }ki=1 by min f (Ak ), if i = k, bi = 0, otherwise,

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for i = 1, 2, . . . , k. Truly, k i=1

ai Âµ(Ai ) â&#x2030;¤

ai Âµ(Ak ) = Âµ(Ak )

i=1

â&#x2030;¤ Âµ(Ak ) min f (Ak ) =

bi Âµ(Ai ).

i=1

The fact that {bi }ki=1 is min-f -B-feasible follows from the fact that {ai }ki=1 is f -B-feasible. k Case 2: i=1 ai > min f (Ak ). Then there exists iâ&#x2C6;&#x2014; â&#x2C6;&#x2C6; k {1, 2, . . . , k} such that i=iâ&#x2C6;&#x2014; +1 ai < min f (Ak ) and k k i=iâ&#x2C6;&#x2014; ai â&#x2030;¥ min f (Ak ). Then {bi }i=1 is given by ï£± if i = k, min f (Ak ), ï£´ ï£´ ï£´ ï£² k a â&#x2C6;&#x2019; min f (A ), if i = iâ&#x2C6;&#x2014; , k i=iâ&#x2C6;&#x2014; i bi = ï£´ if i = 1, . . . , iâ&#x2C6;&#x2014; â&#x2C6;&#x2019; 1, ai , ï£´ ï£´ ï£³ 0, otherwise,

for i = 1, 2, . . . , k. Indeed, k

ai Âµ(Ai ) + aiâ&#x2C6;&#x2014; Âµ(Aiâ&#x2C6;&#x2014; ) +

i=1

â&#x2030;¤

â&#x2C6;&#x2014; i â&#x2C6;&#x2019;1

ai Âµ(Ai ) +

i=1

i=iâ&#x2C6;&#x2014; +1 k

i=iâ&#x2C6;&#x2014;

ai Âµ(Ai )

ai â&#x2C6;&#x2019; min f (Ak ) Âµ(Aiâ&#x2C6;&#x2014; )

+ min f (Ak )Âµ(Ak ) =

bi Âµ(Ai ).

For the following lemma, let us denote Î&#x17E; = {ai }ki=1 : {ai }ki=1 is f -B-feasible ,

Î&#x2DC; = {ai }ki=1 : {ai }ki=1 is min-f -B-feasible .

Lemma 4.4. Let f â&#x2C6;&#x2C6; F, Âµ â&#x2C6;&#x2C6; M, and B = {Ai }ki=1 be any chain on X. Let us denote k Î¾= ai Âµ(Ai ) : {ai }ki=1 â&#x2C6;&#x2C6; Î&#x17E; i=1

and

Î¸=

k i=1

Then Î¾ = Î¸.

ai Âµ(Ai ) : {ai }ki=1 â&#x2C6;&#x2C6; Î&#x2DC; .

Proof. Note that Î&#x2DC; â&#x160;&#x2020; Î&#x17E; and thus Î¸ â&#x2030;¤ Î¾. On the other hand, based on the previous lemma, for every element {ai }ki=1 â&#x2C6;&#x2C6; Î&#x17E; there exists an element {bi }ki=1 â&#x2C6;&#x2C6; Î&#x2DC; such that k k ai Âµ(Ai ) â&#x2030;¤ bi Âµ(Ai ) i=1

Theorem 4.5. Let B = {Ai }ki=1 be a chain on X. Let Ï&#x201E; = min f (Ak ) and BÌ&#x192; = {Ai }kâ&#x2C6;&#x2019;1 i=1 . Then chB (f, Âµ) = Ï&#x201E; Âµ(Ak ) + chBÌ&#x192; (fË&#x153;, ÂµÌ&#x192;),

where fË&#x153; = f Akâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019;Ï&#x201E; and ÂµÌ&#x192; = Âµ 2Akâ&#x2C6;&#x2019;1 .

Proof. From previous two lemmas we can easily see that k ai Âµ(Ai ) : {ai }ki=1 â&#x2C6;&#x2C6; Î&#x17E; chB (f, Âµ) = =

i=1

bi Âµ(Ai ) :

i=1

{bi }ki=1

â&#x2C6;&#x2C6;Î&#x2DC;

= Âµ(Ak )Ï&#x201E; kâ&#x2C6;&#x2019;1 k + bi Âµ(Ai ) : {bi }i=1 is min-f -BÌ&#x192;-feasible i=1

Inducing the previous theorem we obtain the following formula. Corollary 4.6. Let f â&#x2C6;&#x2C6; F , Âµ â&#x2C6;&#x2C6; M, and let B = {Ai }ki=1 be any chain on X. Then chB (f, Âµ) = Âµ(Ak ) min f (Ak ) kâ&#x2C6;&#x2019;1 + Âµ(Ai ) min f (Ai ) â&#x2C6;&#x2019; min f (Ai+1 ) , i=1

i=1

Again, the fact that {bi }ki=1 is min-f -B-feasible follows directly from the fact that {ai }ki=1 is f -B-feasible and thus the lemma is proved. and

Now we can pose and prove a recursive formula for the chain integral.

which proves the theorem.

ai Âµ(Ai )

i=1

â&#x2C6;&#x2014; i â&#x2C6;&#x2019;1

VOLUME 2019 VOLUME 13,13, NÂ°NÂ°3 3 2019

i=1

and thus Î¾ â&#x2030;¤ Î¸ which implies that Î¾ = Î¸ as states the lemma.

chB (f, Âµ) =

k i=1

Âµ(Ai ) min f (Ai ) â&#x2C6;&#x2019; min f (Ai+1 )

with convention that min f (Ak+1 ) = 0.

Also from the previous formulae we can ï¿½ind a lower bound on the chain integral as follows. Corollary 4.7. Let f â&#x2C6;&#x2C6; F , Âµ â&#x2C6;&#x2C6; M, and let B = {Ai }ki=1 be any chain. Then chB (f, Âµ) â&#x2030;¥ Âµ(Ak ) min f (Ak ).

From the theory of Choquet integration it is known that m Ch(f, Âµ) = fi â&#x2C6;&#x2019; fiâ&#x2C6;&#x2019;1 Âµ(Ai ) i=1

where {fi }m i=1 is the increasing enumeration of Im(f ) â&#x2C6;ª {0} and Ai = {x â&#x2C6;&#x2C6; X : f (x) > fiâ&#x2C6;&#x2019;1 }

for i = 1, 2, . . . , m. Then it can be seen that Ch(f, Âµ) =

m i=1

Âµ(Ai ) min f (Ai ) â&#x2C6;&#x2019; min f (Ai+1 )

with convention that min f (Am+1 ) = 0. In other words, for every function f â&#x2C6;&#x2C6; F there exists a chain B such that chB (f, Âµ) = Ch(f, Âµ). This chain is called Ch-maximizing chain. Articles

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�e��o� 4.�. A chain B = {Ai }ki=1 is called Chmaximizing for function f if and only if {min f (Ai ) : i = 1, 2, . . . , k} \ {0} = Im(f ) \ {0}.

Then the following theorem follows from the theory of Choquet integration. Theorem 4.9. A chain B is Ch-maximizing for a function f if and only if chB (f, µ) = Ch(f, µ).

Example 4.10. Following the original example with workers of Lehrer in [5], let us assume that X = {1, 2, 3, 4} represents the set of workers and let f : X → [0, ∞[, f (i) = 5 − i, denote a non-negative function where f (i) represents the maximum number of working hours for worker i ∈ X. Let us choose a chain B = {Ai }3i=1 where A1 = {3}, A2 = {2, 3}, A3 = {1, 2, 3, 4}. This chain can represent the following situation: at the same moment only the all workers, only workers labeled by 2 and 3, and only worker labeled 3 can work at any moment. Let µ represent the number of articles made per hour: µ(A1 ) = 3, µ(A2 ) = 4 and µ(A3 ) = 6. Then chB (f, µ) represents the maximum number of articles made in this situation. From previous formulae it follows that chB (f, µ) = 13. 4.2. Min-max Integral

Note that the de�inition of the Choquet integral can be rewritten to the form Ch(f, µ) = chB (f, µ) B={Ai }n i=1

where the supremum operator runs over all full chains B on X. The motivation behind the min-max integral is to replace the �irst supremum operator by in�imum operator. �e��o� 4.11. A min-max integral of a non-negative function f ∈ F with respect to a capacity µ ∈ M is de�ined by I ∧∨ (f, µ) = chB (f, µ) B={Ai }n i=1

where the in�imum operator runs over all full chains B on X.

From the previous discussion on the chain integral we get a lower bound on the min-max integral.

Lemma 4.12. Let f ∈ F and µ ∈ M. Then the inequality I ∧∨ (f, µ) ≥ µ(X) min f (X) holds.

Proof. Let B = {Ai }ni=1 be any full chain which implies that An = X. Then by Corollary 4.7 we obtain that chB (f, µ) ≥ µ(X) min f (X) for any full chain B on X which implies that I

∧∨

(f, µ) ≥ µ(X) min f (X)

and thus the result follows. Articles

VOLUME 13, N° N°33 VOLUME 13,

2019 2019

Now we need to prove that this value is not only the lower bound but also the value of the min-max integral. Theorem 4.13. I ∧∨ (f, µ) = µ(X) min f (X).

Proof. Following the previous lemma it is enough to �ind a full chain B = {Ai }ni=1 such that chB (f, µ) = µ(X) min f (X). Let x∗ ∈ X be such that f (x∗ ) = min f (X). Then let B be any chain such that A1 = {x∗ } which implies that x∗ ∈ Ai for all i ∈ X. Then trivially chB (f, µ) = µ(X) min f (X) and thus the theorem follows. To this moment we could not really see that the min-max integral belongs to the class of collection integrals. Knowing the formula to compute the value of the min-max integral we can easily see that this integral is indeed the collection integral. Theorem 4.14. The min-max integral belongs to the class of the collection integrals, I ∧∨ = I{{X}} .

Example 4.15. Let f and µ be as in Example 4.10. The value of the min-max integral I ∧∨ (f, µ) represents the maximum number of articles made if only all workers can work together. In this setting, I ∧∨ (f, µ) = 6. Remark 4.16. Observe that the min-max integral is the smallest decomposition integral related to decomposition systems H dealing with X as an element of some collection from H.

�. ��i�alen�e �� lle��n an� �e�esg�e Integrals In this section we start by characterization of a positive basis for the space of non-negative functions F starting with �inding a basis for the space of increasing non-negative functions F↑ . This discussion will yield to an easy characterisation of such collections D that yield to the Lebesgue integral if we restrict ourselves to the class of measures. Note that both spaces, F and F↑ , are of dimension n and thus the positive basis will consist of at least n elements. �e��o� 5.1. A positive basis of a function space S is any sequence {Ei }m i=1 ⊆ S such that for every element f ∈ S there are non-negative real numbers {αi }m i=1 such that m αi 1Ei = f. i=1

For the set of increasing functions F↑ we have the following characterization of a positive basis.

�e��o� 5.2. A set B = {Ex }x∈X ⊆ 2X \ {∅} is called a ↑-compatible basis if and only if 1) for every x ∈ X we have min Ex = x; and

2) if there exists z ∈ Ax ∩ Ay where z > max{x, y} then z ∈ Ax ∩ Ay .

Remark 5.3. �ote that the second condition of �e�inition 5.2 can be stated as follows: if there exists z ∈ Ax ∩ Ay where z > x > y then x ∈ Ay , or, equivalently, if x ̸∈ Ay where x > y then z ̸∈ Ay for all z > x.

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Theorem 5.4. A sequence B = {Ex }xâ&#x2C6;&#x2C6;X â&#x160;&#x2020; Fâ&#x2020;&#x2018; is a positive basis of Fâ&#x2020;&#x2018; if and only if B is â&#x2020;&#x2018;-compatible basis. Proof. Let B = {Ex }xâ&#x2C6;&#x2C6;X be a positive basis and let f â&#x2C6;&#x2C6; Fâ&#x2020;&#x2018; . We will ďż˝ind a positive real numbers {Îąx }xâ&#x2C6;&#x2C6;X such that Îą x 1E x . f= xâ&#x2C6;&#x2C6;X

Let us recursively deďż˝ine

Îąx = f (x) â&#x2C6;&#x2019;

xâ&#x2C6;&#x2019;1 y=1

Firstly, we need to prove that Îąx â&#x2030;Ľ 0 for all x = 1, 2, . . . , n. We will use the proof by induction. For x = 1 we obtain that Îą1 = f (1) â&#x2030;Ľ 0 and thus Îą1 is nonnegative. Now, let us assume that Îą1 , Îą2 , . . . , Îąx â&#x2030;Ľ 0. We want to prove that Îąx+1 â&#x2030;Ľ 0. It follows that = Îąx+1 + = Îąx+1 â&#x2C6;&#x2019;

y=1 x y=1

Îąy 1Ey (x + 1) â&#x2C6;&#x2019; Îąx â&#x2C6;&#x2019;

xâ&#x2C6;&#x2019;1

x y=1

Îąy 1Ey (x)

y=1

Îąy 1Ey (x) â&#x2C6;&#x2019; 1Ey (x + 1)

which implies that Îąx+1 â&#x2030;Ľ

Îąy 1Ey (x) â&#x2C6;&#x2019; 1Ey (x + 1) .

Now it is sufďż˝icient to prove that 1Ey (x) â&#x2030;Ľ 1Ey (x + 1) for all y â&#x2030;¤ x. If y = x then the claim holds trivially. Now for y < x let us assume that the claim does not hold, i.e., 1Ey (x) = 0 and 1Ey (x + 1) = 1, or, equivalently, x Ě¸â&#x2C6;&#x2C6; Ey and (x+1) â&#x2C6;&#x2C6; Ey . We have x Ě¸â&#x2C6;&#x2C6; Ay where x > y and thus based on the Remark 5.3 it follows that z Ě¸â&#x2C6;&#x2C6; Ay for all z > x. Choose z = x + 1 which contradicts that (x + 1) â&#x2C6;&#x2C6; Ay and thus 1Ey (x) â&#x2030;Ľ 1Ey (x + 1) for all y â&#x2030;¤ x. This proves that Îąx+1 â&#x2030;Ľ 0 implying that Îąx are non-negative for all x â&#x2C6;&#x2C6; X. Now it is easy to see that

xâ&#x2C6;&#x2C6;X

Îąx 1Ex (y) =

yâ&#x2C6;&#x2019;1

Îąx 1Ex (y) + Îąy

x=1

f (y) n

If xâ&#x2C6;&#x2014; â&#x2C6;&#x2C6; Ey then Îąy = 0. On the other hand, if xâ&#x2C6;&#x2014; Ě¸â&#x2C6;&#x2C6; Ey then 1Ey (xâ&#x2C6;&#x2014; ) = 0. This implies that

yâ&#x2C6;&#x2C6;X

Îąy 1Ey (xâ&#x2C6;&#x2014; ) = 0 Ě¸= 1 = f (xâ&#x2C6;&#x2014; )

and thus f is not decomposeable by B. If the second condition of Theorem 5.4 is omitted then the function f can be constructed analogously.

Example 5.5. Let X = {1, 2, 3, 4}. The sequences {Ex }xâ&#x2C6;&#x2C6;X given by - E1 = {1}, E2 = {2}, E3 = {3};

Îąy 1Ey (x).

0 â&#x2030;¤ f (x + 1) â&#x2C6;&#x2019; f (x)

VOLUME 2019 VOLUME 13,13, NÂ°NÂ°3 3 2019

Îąx 1Ex (y) = f (y). x=y+1 0

Now we need to prove that the reversed claim holds, i.e., if any of the conditions of Theorem 5.4 is omitted then there exist a function that is not decomposeable by B. Let us thus assume that the condition 1 of the positive basis of Fâ&#x2020;&#x2018; does not hold, i.e., there exists an element x â&#x2C6;&#x2C6; X such that min Ex Ě¸= x. Let xâ&#x2C6;&#x2014; be the smallest such element and deďż˝ine 1, if x â&#x2030;Ľ xâ&#x2C6;&#x2014; , f (x) = 0, otherwise.

- E1 = {1, 2, 3, 4}, E2 = {2, 3, 4}, E3 = {3, 4};

- E1 = {1, 2}, E2 = {2, 3}, E3 = {3, 4}; and E4 = {4} form positive bases of the space consisting of increasing non-negative functions Fâ&#x2020;&#x2018; . On the other hand, sequences - E1 = {1, 2, 3, 4}, E2 = {1, 3, 4}, E3 = {3, 4}; - E1 = {1, 4}, E2 = {2, 4}, E3 = {3, 4}; and E4 = {4} do not form such bases.

Remark 5.6. Note that the set {4} is always part of â&#x2020;&#x2018;compatible basis.

Following the results in theory of positive linear dependence [2] we get that every positive basis of Fâ&#x2020;&#x2018; in spite of Theorem 5.4 is minimal and thus the following result follows. Theorem 5.7. Let D be any collection on X. Then there exist coefďż˝icients ÎąA â&#x2030;Ľ 0, A â&#x2C6;&#x2C6; D, such that ÎąA 1A = f Aâ&#x2C6;&#x2C6;D

for all f â&#x2C6;&#x2C6; Fâ&#x2020;&#x2018; if and only if there exist a B â&#x160;&#x2020; D such that B is â&#x2020;&#x2018;-compatible basis. Note that for every non-negative function f â&#x2C6;&#x2C6; F there exists a permutation Ď&#x192; : X â&#x2020;&#x2019; X such that f â&#x2014;Ś Ď&#x192; belongs to Fâ&#x2020;&#x2018; . Thus we can deďż˝ine sets FĎ&#x192; = {f â&#x2C6;&#x2C6; F : f â&#x2014;Ś Ď&#x192; â&#x2C6;&#x2C6; Fâ&#x2020;&#x2018; } .

For these sets it is easy to characterize bases.

Theorem 5.8. Let Ď&#x192; be any permutation of X. A collection B is a basis of FĎ&#x192; if and only if Ď&#x192;(B) = {Ď&#x192;(A) : A â&#x2C6;&#x2C6; B} is â&#x2020;&#x2018;-compatible basis.

Proof. Let fĎ&#x192; â&#x2C6;&#x2C6; FĎ&#x192; and let B be a collection such that Ď&#x192;(B) is a basis in spite of Theorem 5.4. Note that f â&#x2014;ŚĎ&#x192; â&#x2C6;&#x2C6; Fâ&#x2020;&#x2018; and there are coefďż˝icients aA â&#x2030;Ľ 0, A â&#x2C6;&#x2C6; Ď&#x192;(B), such that aA 1A = f â&#x2014;Ś Ď&#x192;. Aâ&#x2C6;&#x2C6;Ď&#x192;(B)

Now apply Ď&#x192; â&#x2C6;&#x2019;1 on the right and obtain that aA 1A â&#x2014;Ś Ď&#x192; â&#x2C6;&#x2019;1 = f. Aâ&#x2C6;&#x2C6;Ď&#x192;(B)

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Note that

A∈σ(B)

aA 1A ◦ σ −1 =

aA 1σ−1 (A)

σ −1 (A)∈B

aσ−1 (A) 1A

A∈B

and thus

aσ−1 (A) 1A = fσ

A∈B

which implies that for every function fσ ∈ Fσ there are coef�icients bA = aσ−1 (A) ≥ 0, A ∈ B, such that bA 1A = fσ . A∈B

Also note that |B| = |σ(B)| = n and thus B is the minimal basis of Fσ which completes the proof.

�e��o� 5.�. Let σ be any permutation of X. A set B is called σ-compatible basis if and only if σ(B) is ↑compatible basis.

Again, based on the theory of positive linear dependence, we obtain the following result. Theorem 5.10. Let D be any collection on X and let σ be any permutation on X. Then there exist coef�icients aA ≥ 0, A ∈ D, such that a A 1A = f A∈D

for all f ∈ Fσ if and only if there exist B ⊆ D such that B is σ-compatible basis.

Remark 5.11. Note that the set σ −1 ({4}) is always part of σ-compatible basis. Now it is trivial to see that Fσ F= σ

where the union operator runs through all permutations σ on X. Finally, we can formulate the theorem that characterizes all collections D that decompose all functions from F.

Theorem 5.12. Let D be any collection on X. Then there exist coef�icients aA ≥ 0, A ∈ D, such that a A 1A = f A∈D

for all f ∈ F if and only if there exists a subset Bσ ⊆ D such that Bσ is a σ-compatible basis for every permutation σ on X. �e��o� 5.13. A collection D is called Lebcompatible if and only if there exist Bσ ⊆ D such that Bσ is σ-compatible basis for every permutation σ on X.

This de�inition of Leb-compatible collections might seem hard to imagine. The following theorem gives an easy property that characterizes such collections. Articles

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2019 2019

Theorem 5.14. A collection D is Leb-compatible if and only if {{x} : x ∈ X} ⊆ D. Proof. Let us denote P = {{x} : x ∈ X}. Firstly, let us assume that D is Leb-compatible. Then we know that σ −1 ({n}) ∈ D for every permutation σ on X which implies that P ⊆ D. On the other hand, let us assume that P ⊆ D. Then we want to prove that every function is decomposable by D, i.e., there exist non-negative numbers aA ≥ 0, A ∈ D, such that aA 1A = f A∈D

for every f ∈ F. The choice f (x), if A = {x}, aA = 0, otherwise, yields the desired decomposition.

To this moment we characterised all collections D that are Leb-compatible, i.e., every function can be decomposed to some non-negative linear combination of elements in D. Now we can formulate the main theorem of this section and characterise all collections D such that ID , restricted to the class of measures, yields to the Lebesgue integral.

Theorem 5.15. Let D be any collection on X and let ID be a collection integral with respect to the collection D. Then ID F ×M+ = Leb if and only if {{x} : x ∈ X} ⊆ D.

Proof. Follows directly from Corollary 3.3 and Theorem 5.14.

ϲ͘ ^ƵƉĞƌͲĐŽůůĞĐƟŽŶ /ŶƚĞŐƌĂů

�n this section we provide the de�inition of the super-collection integral and discuss superdecomposition duals of the chain and the min-max integral.

�e��o� 6.1. A super-collection integral with re∗ spect to a collection D is a mapping ID :F ×M → ∗ ∗ [0, ∞] such that ID = I{D} .

The superdecomposition duals of integrals discussed in Section 4 are presented in the following examples. Example 6.2. A super-chain integral of a function f ∈ F with respect to a capacity µ ∈ M is de�ined by ∗ ch∗B (f, µ) = IB (f, µ).

Example 6.3. A max-min integral of a function f ∈ F with respect to a capacity µ ∈ M is de�ined by I ∨∧ (f, µ) = µ(X) max f (X).

Remark 6.4. Analogously to Remark 4.16, the value of max-min integral is the upper bound to the values of those decomposition integrals IH that contain X in at least one collection.

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∗ What conditions must D satisfy to ensure that ID is equivalent to the Lebesgue integral? From the Theorem 3.2 it follows that the conditions are the same as in the case of ID . ∗ Theorem 6.5. Let D be any collection on X and let ID be a super-collection integral with respect to the collection D. Then ∗ ID F ×M+ = Leb

if and only if {{x} : x ∈ X} ⊆ D.

7. Com�uta�onal Algorithms The last section is devoted to the discussion of known computational algorithms for special types of decomposition integrals. 7.�. Conca�e Integra�on as �inear ��mi�a�on �roblem Note that the problem of the concave integration can be rewritten to the following optimization problem: maximize

subject to n

n 2 −1

ai µ(Pi )

i=1

Aa ≤ f and a ≥ 0

2 −1 where {Pi }i=1 is any enumeration of 2X \ {∅}, A n is n × (2 − 1) matrix with Ai,j = 1Pj (xi ), f is ndimensional vector whose ith element is f (xi ) and a is unknown (2n − 1)-dimensional vector. The following result concerning this optimization problem was proved [3].

Theorem 7.1. The problem of the concave integration posed as a linear optimization problem is harder than NP. 7.2. Choquet and Chain Integrals

The Choquet integral can be computed using the ordered values of Im(f ). Ordering of n elements can be done in O(n log n) steps which yield O(n log n) algorithm. Similar approach can be taken for the chain integral. This again yields to O(n log n) algorithm. 7.3. Min-Max Integral

The computation of the min-max integral is straightforward, i.e., I ∧∨ (f, µ) = µ(X) min f (X).

The only unknown value is the value of min f (X). This can be done using only O(n) steps. Thus the algorithm computing the value of the min-max integral will take at most O(n) steps. 7.4. Brute Force Algorithms

With other types of decomposition integrals, e.g., the Shilkret and the Pan integrals, the situation is not so easy. Brute force algorithms, i.e., algorithms that check all possible combinations, must be used.

VOLUME 2019 VOLUME 13,13, N°N°3 3 2019

Theorem 7.2. Computation of the Shilkret and the Pan integrals belong to at most NP class. Proof. To prove this claim it is enough to �ind polynomial veri�iers for both integrals. The solution of Shilkret integral is identi�ied with a set from 2X \ {∅}. Given such set A, the minimum of f (A) can be computed in polynomial time and also the product µ(A) min f (A). This gives polynomial veri�ier for the Shilkret integral. For the Pan integral, the solution is identi�ied with a partition {Ai }i∈J of X. Such partition has at most n elements and thus min f (Ai ) for i ∈ J can be computed using polynomial time algorithm. Also, the sum µ(Ai ) min f (Ai ) i∈J

can be computed in polynomial time yielding to a polynomial veri�ier. Thus the computation of the Shilkret and the Pan integrals belong to at most NP class of computational problems.

Brute force algorithm for computing the Shilkret integral goes as follows. For every set A ∈ 2X \ {∅} compute min f (A) and �ind a minimum of µ(A) min f (A). The computation of µ(A) min f (A) for any A takes at most O(n) operations. The set 2X \ {∅} has exactly (2n − 1) elements which yield to O(2n n) algorithm. For the Pan integral we need to check all partitions. The number of partitions of a set win n elements is bounded by Catalan numbers, i.e., to generate all partitions we need O(3n ) operations. For each partition we need to compute at most n minimums which yield to O(n2 ) operations per partition and thus the brute force algorithm for the Pan integral takes at least O(3n n2 ) operations. 7.�. ��ecial Classes o� Ca�aci�es

If we restrict ourselves to a special class of capacities then the computation of decomposition integrals might be simpli�ied. The �irst such considered class is the class of all measures, i.e., all additive capacities. Theorem 7.3. If µ is a measure then

Ch(f, µ) = Pan(f, µ) = cov(f, µ) = Leb(f, µ).

The same theorem holds if µ is a sub-additive capacity, i.e., µ(A ∪ B) ≤ µ(A) + µ(B) for all disjoint sets A, B ∈ 2X . For the super-additive capacities, i.e., set functions µ such that µ(A ∪ B) ≥ µ(A) + µ(B) for all disjoint sets A, B ∈ 2X , the situation is more complicated. A capacity is super-modular if and only if µ(A ∪ B) + µ(A ∩ B) ≥ µ(A) + µ(B)

holds for all A, B ∈ 2X .

Theorem 7.4. If µ is a super-modular capacity then cov(f, µ) = Ch(f, µ). Articles

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8. Conclusion In this contribution we constructed a special class of decomposition integrals called the collection integral. From this class we took a closer look on two special integrals, namely to the chain integral and the minmax integral and we closely investigated their properties. Superdecomposition dual of the collection integral, the super-collection integral, was also de�ined and brief discussion of superdecomposition duals for the chain and the min-max integrals, namely the super-chain and the max-min integrals, is presented. The main result is in characterizing all collections such that if we restrict ourselves to the class of all measures we obtain the collection integral that coincides with the Lebesgue integral. An interesting question is what conditions must a decomposition system H ful�ill to ensure that the decomposition integral coincides with the Lebesgue integral. Open problem. Let H be a decomposition system. What conditions must H ful�ill to ensure that IH F ×M+ = Leb?

Lastly, basic computational algorithms for computing the value of some decomposition integrals, i.e., the Choquet, the Shilkret and the Pan integrals, are examined. Also algorithms for computing the chain and the min-max integrals are discussed. Nevertheless, the computational complexity of such algorithms is analyzed.

ACKNOWLEDGEMENTS

This work was supported by the Slovak Research and Development Agency under the contract no. APVV-170066. Also the support of the grant VEGA 1/0682/16 is kindly announced. AUTHOR Adam Šeliga – Slovak University of Technology, Radlinské ho 11, 810 05 Bratislava, Slovakia, e-mail: adam.seliga@stuba.sk, www: www.math.sk/seliga.

REFERENCES

[1] G. Choquet, “Theory of capacities”, Annales de l’Institut Fourier, vol. 5, 1954, 131–295.

[2] C. Davis, “Theory of Positive Linear Dependence”, American Journal of Mathematics, vol. 76, no. 4, 1954, 733–746, 10.2307/2372648.

[3] A. S� eliga, “A note on the computational complexity of Lehrer integral”. In: Advances in Architectural, Civil and Environmental Engineering: 27th Annual PhD Student Conference on Applied Mathematics, Applied Mechanics, Geodesy and Cartography, Landscaping, Building Technology, Theory and Structures of Buildings, Theory and Structures of Civil Engineering Works, Theory and Environmental Technology of Buildings, Water Resources Engineering, 2017, 62–65. Articles

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2019 2019

[4] Y. Even and E. Lehrer, “Decomposition-integral: unifying Choquet and the concave integrals”, Economic Theory, vol. 56, no. 1, 2014, 33–58, 10.1007/s00199-013-0780-0.

[5] E. Lehrer, “A new integral for capacities”, Economic Theory, vol. 39, no. 1, 2009, 157–176, 10.1007/s00199-007-0302-z. [6] R. Mesiar, J. Li, and E. Pap, “Superdecomposition integrals”, Fuzzy Sets and Systems, vol. 259, 2015, 3–11, 10.1016/j.fss.2014.05.003.

[7] R. Mesiar and A. Stupň anová , “Decomposition integrals”, International Journal of Approximate Reasoning, vol. 54, no. 8, 2013, 1252–1259, 10.1016/j.ijar.2013.02.001.

[8] N. Shilkret, “Maxitive measure and integration”, Indagationes Mathematicae (Proceedings), vol. 74, 1971, 109–116, 10.1016/S1385-7258(71)800173. [9] Q. Yang, “The pan-integral on the fuzzy measure space”, Fuzzy Mathematics, vol. 3, 1985, 107–114.

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VOLUME 2019 VOLUME 13,13, N°N°3 3 2019

STATISTICAL ANALYSIS OF MODELS FOR PUNCHING RESISTANCE ENSURING Submitted: 13th June 2019; accepted: 10th September 2019

Jana Kalická, Mária Minárová, Jaroslav Halvoník, Lucia Majtánová DOI: 10.14313/JAMRIS/3‐2019/27 Abstract: In this paper the statistical assessment of the models that ensure the safety of the reinforced concrete slabs is fo‐ cused. The investigation results in choosing the best mo‐ del as the one that can aspire to become normative for European Union after 2020. Authors dispose with a suffi‐ cient number of data yielded from experimental tests. Ha‐ ving input geometrical and physical parameters of each experiment at hand, the corresponding theoretical value is computed by using three formulas provided by three models involving the same inputs. Case by case, the ratio between measured and theoretical value reveals the sa‐ fety immediately. This ratio stands as the one parametric dimensionless statistical variable which is analysed after‐ wards. Due to statistical parameters evaluation and in accordance with engineers a new model was suggested, statistically verified and nominated as the normative one. Keywords: data mining, reinforcement structures pun‐ ching reliability, Shapiro‐Wilk test, Tuckey’s fence, quarti‐ les, coefficient of variance

1. Introduction There are various type of column system invol‑ ved in constructions. In Fig. 1 there are three types of columns‑slab junction performed, �lat slabs, down‑ stand beam support and locally supported slabs in the place of column ‑ ceiling junction. Even though there exist some drawbacks as softness of the system re‑ sulting in great de�lections of the slab, heavy shear load of the slab nearby the column often resulting in punching; the �lat slabs are very often used right be‑ cause the nice �lat lower ceiling, because of simple ca‑ sing, reinforcement by nets enabled, uni�ied concrete mixture and short building time of the construction. Concrete reinforced �lat slabs are used very fre‑ quently in civil engineering. Punching is the most often failure of them. Unfortunately, due to its brittle charac‑ ter it is very sudden and very dangerous. Brittle cha‑ racter means that the failure spreads from the initial crack place very quickly towards all directions causing a progressive collapse. Indeed, the reinforced concrete �lat slabs have to be built up with respect to prior investigation based signi�icantly on experimental re‑ sults. The investigation aiming to the constructions failure avoiding, together with economical reasoning involved, resulted in normative prescriptions ensu‑ ring the their further safety. Several models were built up, introduced and implemented in various countries of Europe. Recently there exists an effort to involve the

prescription for building up the �lat slabs as to ensure the maximal safety of the designed constructions. No‑ wadays there were three models introduced that com‑ pete to become normative. 1.1. Flat Slab Reliability Models

As detected from experiments, each failure of �lat slab involves so called critical crack. That is why so called Critical shear crack theory was developed invol‑ ving all physical parameters of concrete as well as the geometry of the structure. Some models ensuring sa‑ fety of these constructions employ this theory. �e have veri�ied three models their match with ex‑ perimental data set and safety: The �irst model is fully empirical. It was set up in 1990 in Model Code, [7]. Only the statistical observa‑ tion of the experimental data were taken into account. The formula (1) performs the shear stress dependence on the reinforcement ratio ρ and on the concrete com‑ pressive stress fck , formula (1). Later the nonlinear elasticity theory was employed in the investigation and the Critical shear crack theory was developed. The theory was introduced by Muttoni and Schwartz in [6] and upgraded by Mutttoni in [3], resulting in Model Code 2010 normative form (2), [4]. The model is more complex as it inter alia includes gre‑ ater number of geometrical and physical parameters, e.g. longitudinal shear reinforcement, and magnitude of the aggregate. Lately, due to some simpli�ication ef‑ fort, the third model EC (2017) was developed, repre‑ sented by (5) [5]. Afterwards it was included to the se‑ cond generation of Euro Code of the second genera‑ tion, EC2. Each model is represented by unique formula quantifying the shear stress resistance VRd,c ; for bet‑ ter clarity sake, some auxiliary forms are provided. Model EC2 (2004)

VRd,c =

1 CRk,c k(100ρfck ) 3 u1 d γc

with ‑ CRk,c [M P a] empirical factor ‑ γc [−] partial safety factor

(1)

‑ k[−] size effect factor, k = 1 + (200[mm]/d[mm]) 2

‑ d[m] effective depth of the slab, i.e. the vertical dis‑ tance from the bottom of the slab up to the reinfor‑ cement placement ‑ u1 [m] basic control perimeter at the distance 2d from the axis of the column

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Fig. 1. Columns without (left), with downâ&#x20AC;?stand beam (middle) with a local upper support (right), [2] â&#x20AC;&#x2018; Ď x and Ď y [â&#x2C6;&#x2019;] reinforcement ratios in x and y diâ&#x20AC;&#x2018; rection respectively: Asx Ď x = dx b Fig. 2. Critical crack of flat slab, [2]

Ď y =

Asy dy b

with â&#x20AC;&#x2018; Asx and Asy [m2 ] areas of reinforcement in x and y direction respectively â&#x20AC;&#x2018; b[m] longitude of specimen Model MC (2010)

VRd,c = kĎ&#x2C6;

â&#x2C6;&#x161;

fck b0 d v Îłc

1 1.5 + 0.9kdg Ď&#x2C6;d rs fy d mSd 32 Ď&#x2C6;= d Es mRd

kĎ&#x2C6; =

(3) (4)

with â&#x20AC;&#x2018; dv [m] effective depth of the slab, usually dv = d b0 [m] the length of control perimeter at the distance dv /2

Fig. 3. Theory of Critical Shear Crack, [1]

â&#x20AC;&#x2018; kdg [m] factor involving maximal aggregate magniâ&#x20AC;&#x2018; 32 tude dg [mm]: kdg = 16+d g

â&#x20AC;&#x2018; rs [m] distance from axis of the column to the line of contraďż˝lexure of radial bending momentums

â&#x20AC;&#x2018; fyd [M P a] yielded strength of principal reinforceâ&#x20AC;&#x2018; ment â&#x20AC;&#x2018; mSd [mâ&#x2C6;&#x2019;1 ] average design bending capacity per unit length

â&#x20AC;&#x2018; mRd [mâ&#x2C6;&#x2019;1 ] average design bending capacity per unit length

Model EC2 (2017)

Fig. 4. Graphical performance of failure criterion as stipulated by different models

VRd,c = 1

Ď = (Ď x Ď y ) 2

with Articles

(2)

( )1 b0 dv â&#x2C6;&#x161; min kb 100Ď fck adgv 3 ,0.6 fck Îłc

kb = max 1,

d 8Âľ b0

(5) (6)

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with av [m] being shear span (≥ 2.5d) , geometric average of shear spans in both orthogonal directions, µ parameter accounting shear force and bending mo‑ mentums in the shear region, in case of indoor column without unbalanced momentum it is set to 8. Accor‑ dingly, for normal weight concrete it is taken dg = 32mm. These three models performed by formulas for VRd,c computation, compete for becoming normative in the on‑coming uni�ied European norm that is inten‑ ded to be valid from 2020.

2. Statistical Analysis

2.1. Description of Data Set The focused data set was withdrawn form a large database with more than 600 experimental results gathered through the decades, initiated in 1938 in RTWH Aachen and collected by Carsten Siburg. The data of shear resistance were recorded together with several input information, i.e. geometrical and physi‑ cal characteristics of tested samples. As time lapsed, the set of input parameters had been standardized. However, some of older data are not usable in our in‑ vestigation as they lack some essential input informa‑ tion. We have used as much data as possible, different number of data for particular model, as the models re‑ quired different input parameters. The database is still being enhanced by new experimental data from Euro‑ pean countries. Such tests are realized in our univer‑ sity, as well. Primary idea was to compare three models men‑ tioned above. Comparison of match between theoreti‑ cal and experimental values and the level of safety was decisive. For this sake the statistical analysis has been carried out. As reasoned above, we disposed with 404 items of EC2004, 385 items of Model Code 2010 and 385 items of Euro Code 2017 within the statistical ana‑ lysis. Several characteristics within the data can be ana‑ lysed within the investigation. From the engineering point of view, three in�luencing input parameters were selected and focused more precisely: ‑ effective depth d[m], with values d ∈ [50, 660.9]

‑ reinforcement ratio ρ[−], with values ρ [0.0025, 0.0702]

2.2. Primary Statistical Analysis Primary statistical analysis involves graphical ana‑ lysis and computation of basic characteristics of lo‑ cation and variation. The aim of graphical analysis is a synoptic comparison of the three models perfor‑ med. Graphical analysis involves histograms depicting the probability distribution of ratio values, box‑and‑ whiskers diagrams demonstrating the overall distri‑ bution of data set, quartile distribution as well and de‑ tecting the outstanding data. We have found out that all of three data sets corresponding to the three mo‑ dels include some outstanding data. Afterwards, the Tuckey’s fence test (7) detects and omits these out‑ standing values (outliers) from the further considera‑ tion. [Q1 − k(Q3 − Q1 ), Q3 + k(Q3 − Q1 ))]

(7)

with ‑ Q1 , Q3 the �irst (lower) and the third (upper) quar‑ tile respectively

‑ k coef�icient of outlying, usually k = 1.5 for outliers, k = 3 for far out values. From Figs. 5, 6 it is evident that model EC2 (2017) is not suf�iciently safe. The Tab. 1 af�irms that even the mean value is below one. Histograms refer to the nor‑ mality of data distribution in all three cases both be‑ fore and after excluding the outstanding data. From the point of view of civil engineering practice it is in‑ teresting to trace the outstanding data with regard to the particular input parameters. In Fig. 5 we provide an example of such an approach. It is apparent that the most outstanding data are situated in the interval of the most used values of fck , i.e. [20, 40]. The normality is af�irmed by Shapiro‑Wilk test, see Chapter 2.3.

∈

‑ cylindrical stiffness of concrete fck , with values fck ∈ [9.282, 119] In this paper we provide the more detailed analysis with regard to cylindrical concrete compressive stress fck . All values (3 theoretical and 1 experimental) are included in the database. In accordance with civil en‑ gineering practice, not differences, but the ratios bet‑ ween experimental value Vtest and corresponding the‑ oretical value VRd,c (Vmodel ) will be treated, namely in three cases of VRd,c : that yielded form (1), from (3) or from (5), respectively. Thus the ratio Vtest = xi VRd,c

stands as the statistical variable to be handled. test enhances the �ive‑dimensional vec‑ The item VVRd,c tor (d, ρ, fck , VRd,c , Vtest ) belonging to each model, by one. Moreover, it is worth to note that the ratio above 1 means safety, under 1 means failure. The statistical investigation and data mining is exerted.

Fig. 5. Box plot of ratio experimental/theoretical value of punching resistance EC2 (2004), EC2(2017), Model Code (2010), original data Statistical characteristics ‑ mean

µ̂ = x̄ =

1 xi n i=1 Articles

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VOLUME N°33 2019 2019 VOLUME 13,13, N°

Fig. 6. Histograms of ratio experimental/theoretical value of punching resistance EC2 (2004), EC2 (2017), Model Code (2010), outstanding data excluded model

median

average

variance

variation coe��icient

5 0.05 quan‑ tile

P(xi ≤ 1)

EC2(2004)

1.12054

1.12842

0.186157

0.164971

0.809558

0.238845

MC(2010)

1.15752

1.16523

0.160081

0.137381

0.884772

0.137466

EC2(2017)

0.958613

0.96237

0.124694

Tab. 1. Basic statistical characteristics of three focused models

‑ standard deviation

σ̂ = sx =

‑ variation coef�icient ‑ k quantile qk where

1 (xi − x̄)2 n − 1 i=1

Vx =

x̄ sx

P (X ≤ qk ) = k xi =

Vtest Vmodel

with n number of measurements. For the civil engineers, 0.05 quantile is interesting, as well as safe and unsafe zone split of data set. He‑ rein safe zone involves all data xi ≥ 1. Even from the primary statistical analysis it is evident that although EC2 (2017) is least safe (more than 62% of data falls to unsafe zone), low value of standard deviation refers to low degree of data set variability. Indeed, large num‑ ber of data fall below zero very narrowly. That indi‑ cates a new proposal to modify perspective normative model EC2 (2017) in sense of increasing mean i.e. to ensure more safeness.

2.3. Subinterval Analysis In engineering practice, absolutely prevalently used concrete is the one of concrete compressive stress value fck within the interval [20, 40]. That is why we have paid the special attention to data within this interval. That is why we carried out a subinterval analysis and compare the three models piecewisely, as well. By Articles

0.12957

0.760439

0.627072

such an analysis also the suggestion can be reasoned, how EC2 (2017) should be eventually improved. On each of the subinterval it is performed the com‑ ple� primary statics, normality of data veri�ication in‑ cluded. Moreover, on each subinterval test have been carried out, detecting whether samples originate from the same distribution. In case of normally distributed data sets and equality of variance (in statistical sense) the parametric test ANOVA was used, otherwise non‑ parametric Kruskal‑Wallis test was employed. The re‑ sults are gathered in Table 2. Graphical performance of statistical characteristics are performed in Figs. 7‑9.

Fig. 7. Subinterval analysis. Global average (dashed line) and subinterval (continuous line) average on particular subintervals It is apparent from Table 2 and from Fig. 7 and 8 that the subintervals are different as far as distribu‑ tion concerned. Even in the most important subinter‑ val of fck , i.e. the most frequently used qualitative type of concrete the difference is statistically meaningful. The average of model EC2(2004) and MC (2010) are on the side of safety, but overestimated generally and in each subinterval, as well. On the other hand, the mo‑ del EC2 (2017) generally falls into the unsafe region,

Journal of Journal of Automation, Automation,Mobile MobileRobotics Roboticsand andIntelligent IntelligentSystems Systems

model

median

average

EC2(2004)

1.12054

1.12842

MC(2010)

1.15752

1.16523

EC2(2017)

0.958613

0.96237

VOLUME 2019 VOLUME 13,13, N°N°3 3 2019

variance

variation coef�icient

0.05 quantile

P(xi ≤ 1)

0.124694

0.164971 0.12957

0.809558

0.137381

0.760439

0.238845

0.884772

0.137466

0.186157 0.160081

Tab. 2. Basic statistical characteristics of three focused models

0.627072

subinterval

normality

variance

test

distribution

≤ 20

yes

equal

nonparametric

different

yes

not equal

nonparametric

different

20 − 40 40 − 60 60 − 80 80 − 120

yes yes yes

not equal equal equal

Tab. 3. Subintervals statistical comparison of three models

Fig. 8. Subinterval analysis. Global standard deviation (dashed line) and subinterval (continuous line) variance on particular subintervals

Fig. 9. Subinterval analysis. Global median (dashed line) and subinterval (continuous line) median on particular subintervals

slightly underestimated, therefore, roughly spoken, it should be refused. The underestimation was additio‑ nally veri�ied by recent experiment in the laboratory of our university.

nonparametric parametric parametric

different different different

Fig. 10. Subinterval analysis. Box whisker plots

Fig. 11. Amount of data below 1, i.e. on the side of unsafety, in particular three cases Partial Conclusion Regarding the lower variance both in global and interval‑wise sense, see Fig. 10 we have suggested an improvement of model EC2 (2017) represented by (5). The suggestion consists in certain modi�ication of the normative formula that increases the ratio of experimental‑to‑model value up to the safety side with keeping variance almost unchanged. After such modi‑ Articles

Journal Journal of of Automation, Automation,Mobile MobileRobotics Roboticsand andIntelligent IntelligentSystems Systems

�ication the newly arisen model will be again subjected to the statistical analysis targeting in stipulating the best of all four models as the normative for the future. 2.4. Time Dependent Analysis

VOLUME N°33 2019 2019 VOLUME 13,13, N°

results the data should be cut from the further analy‑ sis. Validation of potential normative models is in�lu‑ encing factor of future normative formulas utilization in civil building practice. It contributes to the higher safety of construction having economical optimality in mind.

ACKNOWLEDGEMENTS

This work was supported by grants APVV‑14‑0013, VEGA 1/0810/16 and VEGA 1/0420/15.

AUTHORS Fig. 12. Time split of entire data set to two sets: until 1980(left), after 1980 (right). Outliers emphasised Due to the change in technologies and lab measu‑ rement equipment and instruments and with regard the fact that the time and location item is involved in the global database, a time dependent statistical analysis is justi�ied and enabled. We carried out the time dependent analysis. Then in accordance with en‑ gineers’ request we have split the data to two time sub‑ intervals for future additional observation. We have found out a peculiar fact. In contrary to our presump‑ tion, and it can be seen in Fig. 12, there are higher number outstanding data and quite a higher number of far outliers in newer (after 1980) group of measure‑ ments. After more detailed look we could see this sur‑ prising fact is probably caused by a systematic error in one of the lab in 1984. Namely, in case of EC2 (2004) more than one half of outliers came from unique lab yielded in 1984, EC2 (2017) more than one third of the outliers come from the same lab and the same year, for MC (2010) almost one half of the outliers came the same lab and the same year, see Fig 13. A glance to the Table 4 gives us the idea of data qua‑ lity within the two time dependent sets ‑ though the mean value is greater, in case of the set after 1980, the dispersion is greater, too. Accordingly, the data was distributed more narrowly before than after 1980.

3. Conclusion

The importance of the statistical analysis of data and data mining in engineering practice is indisputa‑ ble. In case of reliability of punching resistance investi‑ gation including three models initially, inspired us to a completely new solution. As the result, instead of plain choosing the best of three provided models, an idea of new model creation arose which is still in progress. Another fact contributing to the improvement of �inal result was detected during the analysis. Techni‑ cal problems in one of the laboratories involved in the database caused mistaken measurements in one year. The wrong data were disclosed within the time depen‑ dent analysis. In order to increase the preciseness of Articles

Jana Kalická – Slovak University of Technology, Radlinské ho 11, 810 05 Bratislava, Slovakia, e‑mail: jana.kalicka@stuba.sk. Mária Minárová∗ – Slovak University of Technology, Radlinské ho 11, 810 05 Bratislava, Slovakia, e‑mail: maria.minarova@stuba.sk. Jaroslav Halvoník – Slovak University of Technology, Radlinské ho 11, 810 05 Bratislava, Slovakia, e‑mail: ja‑ roslav.halvonik@stuba.sk. Lucia Majtánová – Slovak University of Technology, Radlinské ho 11, 810 05 Bratislava, Slovakia, e‑mail: lu‑ cia.majtanova@stuba.sk.

∗

Corresponding author

REFERENCES

[1] “EN1992‑1‑1 Design of Concrete Structures, Part 1‑1 General Rules and Rules for Buildings”, Euro‑ pean Committee for Standardisation, 2004. [2] J. Hanzel, L. Majtá nová , and J. Halvonı́k, “Punching Resistance of Flat Slabs without Shear Reinforce‑ ment”. In: M. Kostelecka, ed., Proceedings from 21st Czech Concrete Day 2014, Hradec Kralove, 2014.

[3] A. Muttoni and M. Ferná ndez Ruiz, “Shear Strength of Members without Transverse Reinforcement as Function of Critical Shear Crack Width”, ACI Structural Journal, vol. 105, no. 2, 2008, 163–172, 10.14359/19731. [4] A. Muttoni and M. Ferná ndez Ruiz, “The levels‑ of‑approximation approach in MC 2010: ap‑ plication to punching shear provisions”, Struc‑ tural Concrete, vol. 13, no. 1, 2012, 32–41, 10.1002/suco.201100032.

[5] A. Muttoni and M. Ferná ndez Ruiz, “The Critical Shear Crack Theory for punching design: From a mechanical model to closed‑form design expressi‑ ons”. In: SP‑�1� ACI��ib International Symposium on Punching Shear in Structural Concrete Slabs: Honoring Neil M. Hawkins, vol. 315, 2017, 237– 252. [6] A. Muttoni and J. Schwartz, “Behaviour of be‑ ams and punching in slabs without shear reinfor‑ cement”, IABSE reports, vol. 62, 1991, 703–708, 10.5169/seals‑47705.

Journal of Journal of Automation, Automation,Mobile MobileRobotics Roboticsand andIntelligent IntelligentSystems Systems

VOLUME 2019 VOLUME 13,13, N°N°3 3 2019

Fig. 13. Particular models outliers on the data set collected after 1980 EC2(2004)

median

average

variance

variation coe��icient

0.05 quantile

P(xi ≤ 1)

until 1980

1.2043

0.228571

variance

0.176507

0.817335

average

0.201077

0.158658

median

1.1392

0.179722

EC2(2017)

1.13008

1.3277

variation coe��icient

0.05 quantile

P(xi ≤ 1)

until 1980

0.94732

0.734848

variance

0.139664

0.761974

average

0.136169

0.114922

median

0.974977

0.108319

MC(2010)

0.972726

0.942543

variation coe��icient

0.05 quantile

P(xi ≤ 1)

until 1980

1.11726

1.12244

0.124926

0.111298

from 1980

1.19188

1.18848

0.168826

Tab. 4. Basic statistical characteristics of three focused models

0.142052

0.7794788

0.7551

0.906012 0.884772

0.232877

0.57561

0.164063 0.123223

[7] T. C. Zsutty, “Beam shear strength prediction by analysis of existing data”, American Concrete Insti‑ tute Journal, vol. 65, no. 11, 1968, 943–951.

Articles

Journal of Automation, Mobile Robotics and Intelligent Systems

VOLUME 13,

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2019

Global and Local Trend Analysis and Change-Point Analysis of Selected Financial and Market Indices Submitted: 13th June 2019; accepted 10th September 2019

Dominika Ballová

DOI: 10.14313/JAMRIS/3-2019/28 Abstract From the macroeconomic point of view, the stock index is the best indicator of the behavior of the stock market. Stock indices fulfill different functions. One of their most important functions is to observe developments of the stock market situation. Therefore, it is crucial to describe the long-term development of indices and also to find moments of abrupt changes. Another interesting aspect is to find those indices that have evolved in a similar way over time. In this article, using trend analysis, we will uncover the global evolution of selected indices. After evaluating the global trend in the series we compare the results with local trend analysis. Other goal is to detect the moments in which this development suddenly changed using the change-point analysis. By means of cluster analysis, we find those indices that are most similar in long-term development. In each analysis, we select the most appropriate methods and compare their results. Keywords: Trend Analysis, Change-Point Analysis, Cluster Analysis

1. Introduction

Stock market indices express the value of a section of the stock market. By observing the historical development of market indices, we can determine the trend of their long term development. It can be useful for construing predictions of the future development of the valuation process. Locating change-points is also essential factor in the analysis of the development of indices. Understanding the long-term development and abrupt changes in the prices of indices is a key factor for the investor in the decision making about where to invest. Therefore, our aim is to reveal the presence of the trend and identify its nature in the time series of 11 selected indices. We use non-parametric tests based on the fact that the data follows non-normal distribution. Comparing the results of the Cox-Stuart test, Mann-Kendall test, and Spearman’s rho test we seek to find trend and its character. The power of the trend will be expressed by Sen’s slope. This will be compared with the values of Kendal’s tau and Spearman’s rho. Another important goal is to find change-points in the series. First we obtain single change-points for each time series using the Pettitt’s test. Next we use multiple

change-point analysis using divisive and agglomerative estimation. We compare the results of the three procedures and we will look for common change-points. In the last part indices, which long-term development are similar, and which development are the most different from the others will be found. For this purpose agglomerative techniques of cluster analysis will be used.

2. Statistical Methods 2.1. Trend Analysis The trend analysis in the time series of stock market indices has been evaluated using the following nonparametric tests: Cox-Stuart test, Mann-Kendall test and Spearman’s rho test. We will denote X1, X2, ..., Xn as a sample of n independent variables. The above tests are testing the null hypothesis that there is no trend in the data, against the alternative hypothesis that there is a statistically significant increasing/decreasing trend. Positive/negative values of the statistics implies increasing/decreasing trend. Cox-Stuart test. The Cox-Stuart test is based on the signs of the differences

= y1 x1+ d − x1 = y2 x2+ d − x2 

y= x n − x n− d d

(1)

where d = n/2, if the size n is odd, otherwise d = (n + 1)/2. Assign y1, y2, ..., ym the sample of positive differences. The test statistic of Cox-Stuart test is

T = ∑ i =1sign ( yi ) m

(2)

On the significance level α we reject the null hypothesis if |T| > tα, where tα is the quantile of binomial distribution. For m > 20, we can approximate tα with the α-quantile wα of the standard normal distribution

= tα

1 m + w (α )  m . [1] 2

(3)

Mann-Kendall test. The Mann-Kendall test statistic is defined as

Journal of Automation, Mobile Robotics and Intelligent Systems

∑ ∑

i −1

(

)

sign x i − x j

=i 1=j 1

VOLUME 13,

(4)

For n > 8, S can be approximated by normal distribution, thus the standardized test statistic is given:

 S −1 S >0   D( S )  Z = 0, S 0 =  S +1  S <0  D ( S )

(5)

We reject the null hypothesis, if Z > Z1–α/2, and that means there is increasing trend in the series, or if Z < –Zα/2 what means decreasing trend. Z1–α/2 and Zα/2 are the critical values of the standard normal distribution. [2], [3]. Spearman’s rho test. The test statistic of Spearman’s rho test is given

6∑ i =1 ( Ri − i ) n

D= 1 −

(

)

n n2 − 1

(6)

where Ri is the rank of the i-th observation in time series. The standardized statistic is given

Z SR = D

n−2 1 − D2

(7)

If |ZSR| > t(n – 2,1 – α/2), the trend exists. t(n – 2,1 – α/2) is the critical value of Student’s t distribution [4].

2.2. Change-Point Analysis First we checked the homogeneity of our time series using Wald-Wolfowitz test and then we used Pettitt’s test for single change-point detection, then we detected multiple change-points. Wald-Wolfowitz test. It is a nonparametric test for verifying homogeneity in time series. The null hypothesis says that a time series is homogenous between two given times. The test statistic is given

= R

∑

n −1

x x i + 1 + x1 x n

i =1 i

(8)

where x1, x2, ..., xn are the sampled data. For n > 10 we can make an approximation

R − E ( R) D ( R)

. [5]

(9)

Pettitt’s test. The null hypothesis of this test is that there is no change in the series against the alternative hypothesis there is change. The test statistic of Pettitt’s test is Û = max|Uk|

(10)

U= 2∑ i =1ri − k ( n + 1) k k

N° 3

2019

(11)

where k = 1, 2, ..., n and ri are the ranks of Xi. The most probably change-point is located where Û reaches maximum value [6]. Hierarchical divisive estimation E-divisive. This method applies single change-point detection iteratively. Details on the estimation of these changepoint’s locations can be found in [7].

Hierarchical agglomerative estimation E-agglo. This method assumes an initial segmentation of the data. If there are no initial segmentations defined, then each observation can be considered as a separated segment. In this method bordering segments are sequentially merged to maximize the goodness-of-fit statistic. The estimated locations of change points are assigned by the iteration which maximized the penalized goodness-of-fit statistic. More details about this method can be found in [7].

2.3. Cluster Analysis

Cluster analysis belongs to multidimensional statistical methods used to seek out similar objects and grouping them into clusters. Clusters contain objects with the highest degree of similarity, while high dissimilarity among each cluster is desirable. Results of cluster analysis can be the best shown by dendrogram which represents each object and the linkage distance of these objects. It is a figure which arranges the analyzed objects so that individual joining of objects to clusters can be observed. Since there are several aggregation methods, each of which generally yields different results, it is necessary to determine the most appropriate method of aggregation. Such measure is the cophonetic correlation coefficient CC. The cophonetic correlation coefficient is defined as the Pearson coefficient of correlation between actual and predicted distance. For the most suitable agglomeration method, we choose the one for which the cophonetic correlation coefficient is the highest [8]. In hierarchical clustering, we can choose the appropriate number of clusters from the dendrogram by cutting through its branches at the selected distance level on the corresponding axis. For this several indices has been developed as a criteria. Detailed criteria used to select the number of clusters can be found in [9].

3. Analysis of the Development of Selected Stock Indices In this paper we analyzed 11 stock market indices: SPX (measure performance of the broad US economy), CCMP (a broad-based capitalization-weighted index of stocks in all three NASDAQ tiers: Global Select, Global Market and Capital Market), INDU (a price-weighted average of 30 blue-chip stocks that are generally the Articles

Journal of Automation, Mobile Robotics and Intelligent Systems

leaders in their industry), DAX (The German Stock Index), UKX (index of the 100 highest capitalized companies of the London Stock Exchange), CAC (the most widely-used indicator of the Paris market), NKY (a price-weighted average of 225 top-rated Japanese companies of the Tokyo Stock Exchange), HSI (weighted index of a selection of companies from the Stock Exchange from Hong Kong), LEGATRUU (a measure of global investment grade debt from twenty-four local currency markets), SPGSCITR (the leading measure of general commodity price movements in the world economy), CCI (an equal-weighted geometric average of commodity price levels relative to the base year average price). We analyzed monthly time series from January 1995 to January 2018. To understand the basic relationship between the indices we calculated Kendall’s correlation coefficient. The correlations are illustrated in Figure 1. We can see that most of the correlation coefficients are positive. That indicates similar nature of development (increasing/decreasing) of the series through time. The highest correlation with the value of 0.87 is observed between indices SPX and CCMP and also between SPX and INDU indices. Other high level of correlation is between CCMP and INDU, DAX and SPX, CCMP and INDU. From the indices of the European market there is strong positive correlation between the indices UKX and DAX. These indices also strongly correlate with the indices SPX, CCMP and INDU. Weak correlation is between SPGSCITR, NKY and the rest of the indices. Weak negative correlation is only between NKY and the indices LEGATRUU, SPGSCITR and CCI.

Fig. 1. Kendall’s correlation coefficient

After observing the correlation during the whole time period, we were interested in the development of the local correlations. For this purpose we chose seven intervals with length of 72 months overlapping by 36 months with the neighbouring intervals. The results of this analysis can be found in Figure 2. Correlations greater than 0.7 are highlighted. In the legend of the figure we can see the couples of indices with significant positive correlations. These indices have very similar development through the intervals and also during the whole observed time period. SPX Articles

VOLUME 13,

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2019

is strongly correlated with indices CCMP, INDU, DAX, UKX and CAC during all intervals. The German stock market index DAX also gain positive correlation with more of the indices, i.e. CCMP, INDU, UKX and CAC. UKX is beside above mentioned indices also highly correlated with INDU and CAC indices. Significant positive correlation is also between indices CCMP and INDU.

Fig. 2. Local Kendall’s correlation coefficients

3.1. Global Trend Analysis Trend analysis plays vital role in various fields of study since researcher are often interested in the long term development of processes. Describing the long term character of the stock indices can reflect the progress of market efficiency. For this purpose we analyzed the presence and the character of the trend of 11 stock market indices: SPX, CCMP, INDU, DAX, UKX, CAC, NKY, HSI, LEGATRUU, SPGSCITR and CCI index. We analyzed monthly time series from January 1995 to January 2018. Since the multidimensional normality was rejected by testing, for this purpose Mann-Kendall test, Cox-Stuart test and Spearman’s rho test was used. These tests were evaluated in software R, using the packages: trend [10] for the Mann Kendall test and Cox-Stuart test and pastecs [11] for Spearman’s rho test. The results of the three trend tests are listed in Table 1. From this table it is found that the Mann-Kendall test indicates the presence of a statistically significant monotonic trend for all the observed indices. Cox and Stuart test and Spearman’s rho test rejected the presence of trend at the level 0.05 only for the NKY index. The character (increasing/decreasing) of the trend was obtained by Sen’s slope using the R package TTAinterfaceTrendAnalysis [12]. All of the indices except JPY have a long-term increasing tendency. JPY index indicated decreasing trend. According to the magnitude of Sen’s slope 61.1 the HSI index has the highest rising tendency among all the analyzed indices. Other high level of increase was observed in the INDU Index with the value of 45.05 and DAX Index with the value of slope 29.4. The only decreasing trend in NKY Index reached the value of -6.73. P-values of Sen’s slope magnitudes indicate that all of the magnitudes

Journal of Automation, Mobile Robotics and Intelligent Systems

VOLUME 13,

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2019

Tab. 1. Results of Trend Analysis Index SPX

CCMP INDU

Cox-Stuart test

p-value

9.75

< 10–6

9.75

< 10

15.30

< 10

17.97

< 10–6

CAC

4.75

< 10

NKY

0.38

p-value

< 10–6 0.70

6.95

< 10

–6

< 10–4

4.17

< 10–4

13.67

< 10–6

29.40

0.48 0.28

< 10–4

0.64 0.17 0.55

< 10

4.98

–6

< 10

45.05

–6

0.81 0.62

< 10–6

0.35 0.98

3.30

< 10–6

0.66

0.90

SPGSCITR

9.75

< 10–6

CCI

12.25

-0.11

16.01 22.37

< 10–6

-0.08

< 10–6

0.67

0.04

9.75

0.78 0.86

HSI

LEGATRUU

p-value

0.72

< 10–6

-2.06

Magnitude

< 10

< 10–6

12.02

–6

p-value

0.75

–6

Sen’s slope

Rho

0.62

< 10–6

16.44

Tau

Spearman’s test

< 10

–6

16.58

–6

9.75

8.29

–6

DAX

UKX

Mann-Kendall test

< 10

< 10–6

6.51

0.06

< 10–6

-6.74

0.84

< 10–6

61.10

0.20

< 10–3

6.43

< 10–6

0.79

< 10–6

9.91

–6

< 10–6

0.04

< 10–6

1.30

< 10–6

1.26

< 10–6

< 10–4

are statistically significant at the 0.05 level. We obtained the same results considering the signs of the statistics of Mann-Kendall test. Other aspect can be the value of Mann-Kendall’s tau and Spearman’s rho, which all indicate decreasing trend for NKY and increasing trend for all the other indices. Remarkably, according to these last three criteria, LEGATRUU index shows the highest level of increasing trend and on the other hand SPGSCITR and CAC the lowest increasing level.

Tab. 3. Local trend test results for LEGATRUU index

Our next goal was to illustrate the partial development of the trend in the series. For this purpose we tested the presence of the trend over chosen intervals. We chose the same intervals as we chose for the local correlation. We evaluated the above mentioned trend tests for each index for all seven interval. Our aim was to compare the results of the global analysis with the local trend analysis results. Similarly to the previous section, the decision making criterion whether to reject the null hypothesis or not, was the p-value. If we rejected the null hypothesis; that means there is statistically significant trend in the selected interval; then using Sen’s slope we evaluated the character of the trend. The results of these tests for the SPX index are organized in Table 2.

The null hypothesis is rejected for intervals except the one from 2004 to 2012. It means that there is statistically significant monotonic trend for all of the other intervals. In the interval from 2004 to 2012 the development of the SPX index price was constant. This time period overlaps with the duration of the Great Recession. Where it appears a significant trend the prices were increasing, except the period from 1998 to 2003, when the prices were decreasing. The overall trend test for this index led to a significant increasing trend in the series. We evaluated the local trends for the other indices as well. The following patterns were observed. For most of the indices there is a period with a constant trend in the series. This constant trend was from 2004 to 2012. Here belong all of the indices except the LEGATRUU index. The development of this index is increasing for all of the intervals. The result for this index is organized in Table 3. Other common period occurred for the indices SPX, CCMP, DAX, UKX and CAC from 1998 to 2003. The trend for these indices was decreasing during this interval. They were decreasing for no other interval. INDU and HSI indices are significant because except the period with constant trend they were increasing for all of the intervals. Constant trend was observed for indices DAX and CAC also during the period from 2001 to 2006. INDU index revealed constant trend from 1998 to 2003. Two intervals with decreasing trend and one with constant

3.2. Local Trend Analysis

Tab. 2. Local trend test results for SPX index SPX Interval

CS test

MK test

SP test

Trend

1998–2003

0.0002

< 10–6

↘

2004–2009

0.22

0.11

0.46

1995–2000

2001–2006 2007–2012

2010–2015

2013–2018

< 10

–6

0.004 0.41

< 10–6 < 10–6

< 10

–6

0.003 0.32

< 10–6 < 10–6

< 10

–6

< 10–6 0.39

< 10–6 < 10–6

↗ ↗ −

−

↗ ↗

LEGATRUU Interval

CS test

MK test

SP test

Trend

1998–2003

< 10–6

↗

1995–2000

2001–2006 2004–2009

2007–2012

2010–2015 2013–2018

< 10

–6

< 10–6 < 10

–6

< 10

–6

0.004

< 10–6

< 10

–6

< 10–6 < 10

–6

< 10

–6

< 10–6 0.002

< 10

–6

< 10–6 < 10

–6

< 10

–6

< 10–6 0.001

↗

↗ ↗

↗

Articles

Journal of Automation, Mobile Robotics and Intelligent Systems

VOLUME 13,

N° 3

2019

Tab. 4. Results of Change-point detection Index SPX

CCMP INDU

Pettitt’s test

Position 191

188

130

p-value < 10–6

< 10

–6

< 10

–6

DAX

140

< 10–6

CAC

< 10

UKX NKY

191 69

< 10–6 –6

< 10–6

HSI

138

< 10–6

SPGSCITR

< 10–6

LEGATRUU CCI

139 135

< 10

–6

< 10–6

Divisive estimation

Agglomerative estimation

Position

p-value

37,74,127,159,190,223

0.01

47,74,119,225

0.01

60,80,131,227

34,80,133,163,223

0.01

Position 31,226

37,75,106,136,166,218,248

0.01

33,90,126,162,218,248

0.01

25,91,117,165,175,217

34,72,129,165,197,227

0.01

79,129,157,217

40,97,155,188,229

0.01

37,90,129,162,218,248 38,90,126,165,225 51,81,140,175,216

63,111,166,196,240 40,102,132,189,238

0.01 0.01 0.01 0.01

230

39,91,125,165,230 144

91,192

110,239 136

Fig. 3. Results of the change-point detection trend was obtained for the NKY index, which from the global point of view was the only index with decreasing trend. Even if the overall trend test indicated increasing trend in the series, analyzing the local trend we also received intervals with constant and decreasing trend for all of the series except LEGATRUU index.

3.3. Change-Point Analysis

Presence of a change-point in time series is a vital question in the development of various processes. Our aim was to find abrupt changes in the time series of each index. First we used Pettitt’s test from package trend for single change-point detection. After finding the single change-points we carried out multiple change-point analysis using divisive estimation and agglomerative estimation of changepoints from the ecp [13] package. Results of this analysis are in Table 4 and Figure 3. According to Pettitt’s test three indices- SPX, CCMP and UKX have a significant change-point in 2010. Other common significant change was detected in INDU, DAX, HSI, LEGATRUU 60

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and CCI indices from October 2005 to August 2006. CAC index has a significant change in December 1998. NKY index in September 2000, SPGSCITR in December 2012. Next, the results of multiple change-point analysis were compared. As we can see in general we obtained more results using the divisive estimation. Some of the results are similar to the agglomerative estimation, although there are differences. All the detected change-points obtained by divisive estimation are statistically significant. We can see some pattern in the positions of the change-points. Most of the indices have the first abrupt change from September 1997 to April 1998. Other significant period can be considered from December 2000 to September 2001. DAX, UKX and CAC changed abruptly in June 2002. Further common changes were observed in the period from June 2005 to August 2006. Another significant changes in most of the indices was from March 2008 to July 2009. LEGATRUU, SPGSCITR, CCI and CCMP has significant changes from August 2010 to April 2011. Other changes was found from the end of 2012 to the

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beginning of 2014. For most of the indices it was the last change-point. For INDU, DAX and UKX, the last change-points were detected in August of 2015. For CCI and SPGSCITR indices it was in October and December 2014. As we can see in Figure 3, for most of the indices, divisive estimation allocated one of the multiple change-points near to the ones found by Pettitt’s test. Also in most case the results of agglomerative estimation are close to the ones gained by divisive estimation.

3.4. Cluster Analysis

In practice it is common to seek for similar objects and explain the relationship between these objects. Thus our other goal was to identify the indices which development are similar in time and can be separated into clusters. We used hierarchical methods from the stats package [14], because of the low number of the clustering objects. Since indices are measured in various currencies, first we standardized the values of the series. To determine the best cluster analysis method we calculated the cophonetic correlation coefficient from package stats, for each method using Euclidean distance. We chose the method which contains the highest cophonetic correlation coefficient. According to this coefficient the best clustering methods to use are the average linkage method and Ward´s method. In average linkage method the average distance between objects of each cluster is used as distance between clusters. Ward´s method is based on minimization of the within-cluster variance. The number of clusters was determined using the NbClust package based on 30 indices as criteria. Most of the indices proposed the three cluster solution. The results of this analysis was creating using dendextended package [15] and can be found on the dedndrogram in Figure 4.

Fig. 4. Results of cluster analysis- average linkage method and Ward’s method As we can see the results have something in common. On the lowest linkage distance SPX and INDU indices are joined into a cluster. DAX and CCMP are very similar to them. The most different from other indices are NKY index and SPGSCITR index. NKY index was the only index showing decreasing trend. It also

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has a unique change-point found with Pettitt’s test. In SPGSCITR some different change-point can be found from the change-points of other indices. We can also see that correlation analysis gave similar results. We can also observe the basic behavior of Ward’s clustering method, that objects are joined into existing clusters and new cluster is created just in case of high dissimilarity.

4. Conclusion The aim of this paper was to analyze the development of 11 market indices. Indices are very important indicator of market development. Our first goal was to reveal the trend in time series of indices. We compared the results of three nonparametric methods to determine the trend: Cox-Stuart test, Mann-Kendall test and Spearman’s rho test. The Cox-Stuart test showed a significant increasing trend for all of the indices except NKY index. Mann-Kendall test and Spearman’s test showed a statistically significant trend for all of the indices. Except NKY index it was a statistically significant increasing trend. The magnitude of the trend was calculated by Sen’s slope. According to this statistics HSI index has the highest increasing tendency. All magnitudes are statistically significant on the level of 0.05. Next we analyzed the trend locally to see its development. We chose seven intervals with length of 72 months overlapping by 36 months with the neighbouring intervals. Testing each interval using the previous tests we obtained the local trends. For most of the indices there was a period with a constant trend in the series. This constant trend was from 2004 to 2012. LEGATRUU index had all of the local trends increasing. In the development of other indices there was also at least a constant trend. Other important point of view on the development of indices is finding change-points. Single change-point detection was carried out by Pettitt’s test. Single change-points was found in 1998, 2000, from 2005 to 2006, in 2010 and 2012. Multiple change-point analysis was performed by using divisive and agglomerative estimation. The results of these methods are similar although a little biased. Also the agglomerative estimation proposes less change-points then divisive methods. The results of divisive estimation are statistically significant on the level of 0.05. Also the change-points found by Pettitt’s test are located near to the ones obtained by divisive estimation. These change-points can be caused by changes in the components of the indices or by economic depression. In the third part of this paper we found indices which development is similar in time. Results of average linkage method and Ward’s method basicly give very similar clusters. The most similar development has INDU and SPX indices. Very similar to them are DAX, CCMP and UKX. The lowest level of similarity is between NKY and the other indices. We determined the three cluster solution as the most appropriate. Articles

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Acknowledgements The support of the grant VEGA 1/0420/15 is kindly announced.

AUTHOR

Dominika Ballová – Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Bratislava, 810 05, Slovakia, E-mail: dominika.ballova@stuba.sk.

References

[1] E. Lehtinen, U. Pulkkinen, and K. Pörn, “Statistical Trend Analysis Methods for Temporal Phenomena”, SKi, 1997.

[2] M. G. Kendall, Rank correlation methods, Griffin: London, 1975. [3] H. B. Mann, “Nonparametric Tests Against Trend”, Econometrica, vol. 13, no. 3, 1945, 245–259 DOI: 10.2307/1907187.

[4] R. Sneyers, On the statistical analysis of series of observations, Secretariat of the World Meteorological Organization: Geneva, 1990, Technical Note no. 143. [5] A. Wald and J. Wolfowitz, “On a Test Whether Two Samples are from the Same Population”, The Annals of Mathematical Statistics, vol. 11, no. 2, 1940, 147–162.

[6] A. N. Pettitt, “A Non‑Parametric Approach to the Change‑Point Problem”, Journal of the Royal Statistical Society. Series C (Applied Statistics), vol. 28, no. 2, 1979, 126–135 DOI: 10.2307/2346729.

[7] D. S. Matteson and N. A. James, “A Nonparametric Approach for Multiple Change Point Analysis of Multivariate Data”, Journal of the Ame rican Statistical Association, vol. 109, no. 505, 2014, 334– 345 DOI: 10.1080/01621459.2013.849605. [8] Gilmore, Dynamic Time and Price Analysis of Market Trends, Bryce Gilmore & associates, 1999.

[9] R. D. Edwards, J. Magee, and W. H. C. Bassetti, Technical Analysis of Stock Trends, CRC Press, 2001.

[10] E. Brodsky, Change‑Point Analysis in Nonstatio nary Stochastic Models, CRC Press, 2017.

[11] B. S. Everitt, S. Landau, M. Leese, and D. Stahl, Cluster Analysis, John Wiley & Sons, Ltd: Chichester, UK, 2011 DOI: 10.1002/9780470977811. 62

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[12] A. Kassambara, Practical Guide to Cluster Analy sis in R: Unsupervised Machine Learning, STHDA, 2017.

[13] J. Chen and A. K. Gupta, Parametric Statistical Change Point Analysis with Applications to Ge netics, Medicine, and Finance, Birkhäuser Basel, 2012 DOI: 10.1007/978‑0‑8176‑4801‑5.

[14] J. S. Racine, “Nonparametric Econometrics: A Primer”, Foundations and T DOI: 10.1561/0800000009. [15] T. Mills, Modelling Trends and Cycles in Economic Time Series, Palgrave Macmillan, 2003 DOI: 10.1057/9780230595521.

[16] W. Palma, Time Series Analysis, John Wiley & Sons, 2016. [17] “Cophenetic correlation coefficient – MATLAB cophenet”. https://www.mathworks.com/help/ stats/cophenet.html. Accessed on: 2019-11-07. [18] M. Charrad, N. Ghazzali, V. Boiteau, and A. Niknafs, “NbClust: An R Package for Determining the Relevant Number of Clusters in a Data Set”, Journal of Statistical Software, vol. 61, no. 1, 2014, 1– 36 DOI: 10.18637/jss.v061.i06. 59

[19] T. Pohlert, “trend: Non‑Parametric Trend Tests and Change‑Point Detection”. c2018, https:// CRAN.Rproject.org/package=trend. Accessed on: 2019-11-07. [20] P. Grosjean, F. Ibanez, and M. Etienne, “pastecs: Package for Analysis of Space‑Time Ecological Series”. c2018, https://CRAN.R-project.org/package=pastecs, Accessed on: 2019-11-07.

[21] Devreker and A. Lefebvre, “TTAinterface – TrendAnalysis: Temporal Trend Analysis Graphical Interface”. c2018, https://CRAN.R-project.org/ package=TTAinterfaceTrendAnalysis. Accessed on: 2019-11-07. [22] N. A. James and D. S. Matteson, “ecp: An R Package for Nonparametric Multiple Change Point Analysis of Multivariate Data”, Journal of Statistical Software, vol. 62, no. 1, 2015, 1–25, 10.18637/jss.v062.i07.

[23] “R: The R Stats Package”. https://stat.ethz.ch/ R-manual/R-devel/library/stats/html/00Index. html. Accessed on: 2019-11-07.

[24] T. Galili et al, “dendextend: Extending ’dendrogram’ Functionality in R”. c2019, https://CRAN. Rproject. org/package=dendextend. Accessed on: 2019-11-07.

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[25] “R: Documentation”. https://www.r-project. org/other-docs.html. Accessed on: 2019-11-07. [26] T. Wei, V. Simko, M. Levy, Y. Xie, Y. Jin, and J. Zemla, “corrplot: Visualization of a Correlation Matrix”. c2017, https://CRAN.R-project.org/package=corrplot. Accessed on: 2019-11-07.

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VOLUME N°33 2019 2019 VOLUME 13,13, N°

PROBABILITY MEASURES AND LOGICAL CONNECTIVES ON QUANTUM LOGICS Submitted: 13th June 2019; accepted: 10th September 2019

Oľga Nánásiová, Ľubica Valášková, Viera Čerňanová DOI: 10.14313/JAMRIS/3‐2019/29

2. Basic Definitions and Properties

Abstract: The present paper is devoted to modelling of a probabi‐ lity measure of logical connectives on a quantum logic via a G‐map, which is a special map on it. We follow the work in which the probability of logical conjunction (AND), dis‐ junction (OR), symmetric difference (XOR) and their nega‐ tions for non‐compatible propositions are studied. Now we study all remaining cases of G‐maps on quantum lo‐ gic, namely a probability measure of projections, of impli‐ cations, and of their negations. We show that unlike clas‐ sical (Boolean) logic, probability measures of projections on a quantum logic are not necessarilly pure projections. We indicate how it is possible to define a probability me‐ asure of implication using a G‐map in the quantum logic, and then we study some properties of this measure which are different from a measure of implication in a Boolean algebra. Finally, we compare the properties of a G‐map with the properties of a probability measure related to logical connectives on a Boolean algebra.

In the �irst part of this section, we recall fundamen‑ tal notions: orthomodular lattice, compatibility, ortho‑ gonality, state, and their basic properties. For more de‑ tails, see [6, 24]. In the second subsection, we recall some situations with two‑dimensional states allowing to model a probability measure of logical connectives in the case of non‑compatible events [16], [15]‑ [11], [26].

Keywords: logical connectives, orthomodular lattice, quantum logic, probability measure, state

1. Introduction The problem of modelling of probability measu‑ res for logical connectives of non‑compatible proposi‑ tions started by publishing the paper Birkhoff, G., von Neumann, J. [2]. Quantum logic allows to model situa‑ tions with non‑compatible events (events that are not simultaneously measurable). Methods of quantum lo‑ gic appear in data processing, economic models, and in other domains of application e.g. [2, 28, 9, 19, 27]. Calculus for non‑compatible observables has been described in [16], while modelling of logical connecti‑ ves in terms of their algebraic properties and algebraic structures can be found in [7, 8, 21]. The present paper follows up the work [13], where the authors studied logical connectives: conjuction, disjunction, and symmetric difference together with their negations, from the perspective of a probability measure. An overview of various insights into this is‑ sue is provided in [25]. The paper is organized as follows. Section 2 re‑ minds some basic notions and their properties. A spe‑ cial function that associates a probability measure to some logical connectives on a quantum logic is de�ined and studied in Section 3 – Section 5. In the last Section 6 properties of a G‑map are compared with properties of a probability measure related to logical connectives on a Boolean algebra. 64

2.1. Quantum logic ϐ ʹǤͳ An orthomodular lattice (OML) is a lat‑ tice L with 0L and 1L as the smallest and the greatest element, respectively, endowed with a unary operation a → a′ that satis�ies� (i) a′′ := (a′ )′ = a; (ii) a ≤ b implies b′ ≤ a′ ; (iii) a ∨ a′ = 1L ; (iv) a ≤ b implies b = a ∨ (a′ ∧ b) (the orthomodular law). ϐ ʹǤʹ Elements a, b of an orthomodular lat‑ tice L are called – orthogonal if a ≤ b′ ; (notation a ⊥ b ); – compatible if a = (a ∧ b) ∨ (a ∧ b′ ); (notation a ↔ b). ϐ ʹǤ͵ A state on an OML L is a function m : L → [ 0 , 1 ] such that (i) m(1L ) = 1; (ii) a ⊥ b implies m(a ∨ b) = m(a) + m(b). Note that the notions state and probability measure are closely tied, and it is clear that m(0L ) = 0. There exist three kinds of OMLs: without any state, with exactly one state and with in�inite number of sta‑ tes (see e.g. [20]). The �irst and the second type of OLMs as a basic structure are not suitable to build a generalized probability theory. The last type of OMLs, which has in�inite number of states is considered in the present paper.

ϐ ʹǤͶ An OML L with in�inite number of sta‑ tes is called a quantum logic (QL).

When studying states on a quantum logic, one can meet some problems, that do not exist on a Bool‑ ean algebra. It means, that some of basic properties

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of probability measures are not necessarilly satis�ied for non‑compatible random events. Here are some of them: Bell‑type inequalities (e.g. [9,10,23,26]), Jauch‑ Piron state, (e.g. [4, 22]), problems of pseudometric (see [13]).

2.2. Probability Measures of Logical Connectives on QLs

In [14], the notion of a map for simultaneous me‑ asurements (an s‑map) on a QL has been introduced. This function is a measure of conjunction even for non‑ compatible propositions, see [25].

A map p : L × L → [0, 1] is called a map for simul‑ taneous measurements (abbr. s‑map) if the following conditions hold: (s1) p(1L , 1L ) = 1; (s2) if a ⊥ b then p(a, b) = 0;

(s3) if a ⊥ b then for any c ∈ L:

p(a ∨ b, c) = p(a, c) + p(b, c), p(c, a ∨ b) = p(c, a) + p(c, b).

The following properties of s‑map have been proved: Let p : L×L → [0, 1] be an s‑map and a, b, c ∈ L. Then 1) if a ↔ b then p(a, b) = p(a ∧ b, a ∧ b) = p(b, a);

2) if a ≤ b then p(a, b) = p(a, a);

3) if a ≤ b then

for any c ∈ L;

p(a, c) ≤ p(b, c) p(c, a) ≤ p(c, b)

4) p(a, b) ≤ min{p(a, a), p(b, b)};

5) the map mp : L → [0, 1] de�ined as mp (a) = p(a, a) is a state on L, induced by p.

The property 1. shows that s‑maps can be seen as pro‑ viding probabilities of ‘virtual’ conjunctions of pro‑ positions, even non‑compatible ones, for in the case of compatible propositions the value p(a, b) coincides with the value that a state mp generated by p takes on the meet a∧b, which in this case really represents con‑ junction of a and b [25]. On the other hand, the identity p(a, b) = p(b, a) may not be true in general. So an s‑map can be used for describing of stochastic causality [16–18]. Moreover, for any a ∈ L: mp (a) = p(a, a) = p(1L , a) = p(a, 1L ). Logical connectives disjunction (j‑map) and syme‑ tric difference (d‑map) are studied on a QL [13, 5].

Let L be a QL. A map q : L × L → [0, 1] is called a join map (j‑map) if the following conditions hold: (j1) q(0L , 0L ) = 0, q(1L , 1L ) = 1;

(j2) if a ⊥ b then q(a, b) = q(a, a) + q(b, b);

(j3) if a ⊥ b then for any c ∈ L:

q(a ∨ b, c) = q(a, c) + q(b, c) − q(c, c) q(c, a ∨ b) = q(c, a) + q(c, b) − q(c, c).

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If p is an s‑map on a QL, mp is a state induced by p and qp : L × L → [0, 1] such that for any a, b ∈ L qp (a, b) = mp (a) + mp (b) − p(a, b),

then qp is a j‑map. It is easy to see that if a ↔ b, then qp (a, b) = mp (a) + mp (b) − mp (a ∧ b) = mp (a ∨ b)

which explains its name.

Let L be a QL. A map d : L × L → [0, 1] is called a difference map (d‑map), if the following conditions hold: (d1) d(1L , 1L ) = d(0L , 0L )

d(1L , 0L ) = d(0L , 1L )

(d2) if a ⊥ b then d(a, b) = d(a, 0L ) + d(0L , b); (d3) if a ⊥ b then for any c ∈ L: d(a ∨ b, c)

d(c, a ∨ b)

d(a, c) + d(b, c) − d(0L , c)

d(c, a) + d(c, b) − d(c, 0L ).

If a ↔ b, then

d(a, b) = md (a b) = md (a ∧ b′ ) + md (a′ ∧ b),

where md is a state induced by d.

3. Special Bivariables Maps on QLs 3.1. Measures and Boolean Functions Let B be a Boolean algebra and f : B n → B be a Boolean function. It means, that f is such n‑ary opera‑ tion on B, which is composed of binary operations ∨, ∧, a unary operation complement ′ , and brackets ().

�or the sake of simpli�ication, the expressions of the type (x1 , · · · , xi−1 , ai , xi+1 , · · · , xn )

will be written as (y 1 , ai , y 2 )

Proposition 3.1 Let B be a Boolean algebra, f : B n → B a Boolean function and m : B → [0, 1] be a probability measure on B. Then the composition of functions m ◦ f : B n → [0, 1], (m ◦ f )(x1 , · · · , xn ) = m(f (x1 , · · · , xn )) satis�ies follo�ing properties� (G1) Let x1 , · · · , xn ∈ {0B , 1B }n . Then m(f (x1 , · · · , xn )) ∈ {0, 1}. (G2) Let ai , bj ∈ B, ai ⊥ bj . Then m(f (y 1 , ai , y 2 , bj , y 3 )) =

m(f (y 1 , 0B , y 2 , bj , y 3 )) +m(f (y 1 , ai , y 2 , 0B , y 3 ))

−m(f (y 1 , 0B , y 2 , 0B , y 3 )). Articles

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VOLUME N°33 2019 2019 VOLUME 13,13, N°

(G3) Let ai , bi ∈ B, ai ⊥ bi . Then m(f (y 1 , ai ∨ bi , y 2 )) =

Consider a, b ∈ B, a ⊥ b, and put xi = a ∨ b. Then

m(f (y 1 , ai , y 2 )) +m(f (y 1 , bi , y 2 ))

−m(f (y 1 , 0B , y 2 )). Proof. (G1) Let f : B n → B be a Boolean function. Let x1 , · · · , xn ∈ {0B , 1B }n . Then f (x1 , · · · , xn ) ∈ {0B , 1B }

and then

f (y 1 , a, y 2 , b, y 3 ) = f (x1 , · · · , xn ) ∧ U,

(1)

(a ∧ b′ ∧ Q1 ) ∨ (a′ ∧ b ∧ Q2 ) ∨ ∨(a′ ∧ b′ ∧ Q3 ) ∨ (a ∧ b ∧ Q4 ),

where Qi , i = 1, 2, 3, 4, are boolean expressions that do not contain any of the elements a, a′ , b, b′ . Assume that a ⊥ b. Then f (y 1 , a, y 2 , b, y 3 ) = (a∧Q1 )∨(b∧Q2 )∨(a′ ∧b′ ∧Q3 ). If we put m(f (y 1 , a, y 2 , b, y 3 )) = µ, then

µ = m(a ∧ Q1 ) + m(b ∧ Q2 ) + m(a′ ∧ b′ ∧ Q3 ). (2)

Since m is a probability measure, it follows that =

= =

m(a ∧ Q1 ) + m(b ∧ Q2 ) + m(Q3 ) −m((a ∨ b) ∧ Q3 ) m(a ∧ Q1 ) + m(b ∧ Q2 ) + m(Q3 )

−m(a ∧ Q3 ) − m(b ∧ Q3 ) m(a ∧ Q1 ) + m(a′ ∧ Q3 ) + m(b ∧ Q2 ) +m(b′ ∧ Q3 ) − m(Q3 ).

On the other side, from (2) we obtain

m(f (y 1 , a, y 2 , 0B , y 3 )) = m(a ∧ Q1 ) + m(a′ ∧ Q3 ), m(f (y 1 , 0B , y 2 , b, y 3 )) = m(b ∧ Q2 ) + m(b′ ∧ Q3 ), m(f (y 1 , 0B , y 2 , 0B , y 3 )) = m(Q3 ).

Thus (G2) is satis�ied.

(G3) Similarly, any Boolean function f : B → B can be written as f (x1 , . . . , xn ) = (xi ∧ Q) ∨

(x′i

∧ P ),

where the Boolean expressions Q, P do not contain xi , x′i . Thus

m (f (x1 , . . . , xn )) = m(xi ∧ Q) + m(x′i ∧ P ). (3)

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+m(b′ ∧ P ) − m(P ).

On the other side, from (3) we obtain m(f (y 1 , 0B , y 2 )) = m(P ).

where U = (a ∧ b′ ) ∨ (a′ ∧ b) ∨ (a′ ∧ b′ ) ∨ (a ∧ b)). This can be rewritten as

m(a ∧ Q) + m(a′ ∧ P ) + m(b ∧ Q)

m(f (y 1 , b, y 2 )) = m(b ∧ Q) + m(b′ ∧ P )

(G2) Let f : B n → B be a Boolean function. Then for any a, b ∈ {x1 , . . . , xn }

m(a ∧ Q) + m(b ∧ Q) + m(P ) −m(a ∧ P ) − m(b ∧ P )

m(f (y 1 , a, y 2 )) = m(a ∧ Q) + m(a′ ∧ P )

m(f (x1 , · · · , xn )) ∈ {0, 1}.

f (y 1 , a, y 2 , b, y 3 )

m (f (y 1 , a ∨ b, y 2 )) m((a ∨ b) ∧ Q) + m((a ∨ b)′ ∧ P )

Thus (G3) is satis�ied.

(Q.�.�.)

It follows from the previous proposition that each pro‑ bability measure of any boolean function has the pro‑ perties (G1) – (G3). Then it should be interesing to study a function G : B n → [0, 1] which is endowed with properties (G1) – (G3). It is easy to see, that for n = 1 a function G is a classical measure (G(1B ) = 1 and G(0B ) = 0) or a negative measure (G(1B ) = 0 and G(0B ) = 1) on B. This article is devoted to functions G on a QL for n = 2. 3.2. Bivariable G‐Maps on QLs

A special bivariable map G satisfying G(0L , 1L ) = G(1L , 0L )

has been introduced in [13]. The following de�inition brings an extended version of this G‑map. �e��o� 3.� Let L be a QL. A map G : L × L → [0, 1] is called a G‑map if the following holds: (G1) if a, b ∈ {0L , 1L } then G(a, b) ∈ {0, 1}; (G2) if a ⊥ b then

G(a, b) = G(a, 0L ) + G(0L , b) − G(0L , 0L ); (G3) if a ⊥ b then for any c ∈ L: G(a ∨ b, c) = G(a, c) + G(b, c) − G(0L , c) G(c, a ∨ b) = G(c, a) + G(c, a) − G(c, 0L ). A G‑map enables modelling of probability of logical connectives even for non‑compatible propositions.

Lemma 3.3 Let G : L × L → [0, 1] be a G‑map, where L is a QL. Then for a ↔ b it holds G(a, b)

G(a ∧ b, a ∧ b) + G(a ∧ b′ , 0L ) +G(0L , a′ ∧ b) − 2G(0L , 0L ).

Proof. See in [12].

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Proposition 3.4 Let G : L × L → [0, 1] be a G‑map, where L is a QL. Then the map G′ = 1 − G is a G‑map. Proof. See in [12].

There are sixteen families Γi , (i = 1, ..., 16) of maps G according to values in vertices (1L , 1L ), (1L , 0L ), (0L , 1L ), (0L , 0L ).

Eight of them with G(1L , 0L ) = G(0L , 1L ) are studied in [13]. More details can be found in Table 5, section 6.

Family Γ2 is the set of all s‑maps (measures of con‑ juntion), Γ3 the set of all j‑maps (measures of dis‑ junction), and Γ4 is that of all d‑maps (measures of symmetric difference) on a QL (see [13] for more de‑ tails). In the present paper, the remaining cases Γi (i = 9, ..., 16) with G(1L , 0L ) = G(0L , 1L )

are focused on.

4. Probability Measures of Projections on QLs

This part is devoted to Γ9 − Γ12 with values in the vertices shown in the Table 1. As G ∈ Γ11 iff 1 − G ∈ Γ9 , and G ∈ Γ12 iff 1 − G ∈ Γ10 (Proposi‑ tion 3.4 and Table 1), and moreover, Γ9 and Γ10 are analogical cases (Γ11 and Γ12 as well), only Γ9 is stu‑ died in detail.

Lemma 4.1 Let L be a QL and G ∈ Γ9 . Then for any a, b ∈ L it holds 1) G(1L , a) = 1, G(0L , a) = 0; 2) G(a, 0L ) = G(a, a) = G(a, 1L ); 3) G(a, 0L ) =

1 2 (G(a, b)

′

+ G(a, b ));

4) G(a, 0L ) =

1 n

n ∑

G(a, bi ),

i=1

Proof. See in [12].

Proposition 4.2 Let L be a QL, and G ∈ Γ9 . Then for any a, b ∈ L it holds 1) If a ↔ b then G(a, b) = G(a, 0L ). 2) For any choice of b, the map mb : L → [0, 1]: mb (a) = G(a, b) is a state on L.

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If L is a Boolean algebra, then for any G ∈ Γ9 it holds G(a, b) = G(a, 0L ) for all a, b ∈ L. Analogously for any G ∈ Γ10 it holds G(a, b) = G(0L , b) for all a, b ∈ L. If L is a QL but not a Boolean algebra, then the iden‑ tity does not hold in general, as illustrates the follo‑ wing example.

Example 4.3 Consider L = {0L , 1L , a, a′ , b, b′ }, a ho‑ rizontal sum of Boolean algebras Ba = {0L , 1L , a, a′ }, Bb = {0L , 1L , b, b′ }.

Consider r1 , r2 , u1 , u2 ∈ [0, 1]. Every G ∈ Γ9 can be fully de�ined by Table 2, where α=

1 (r1 + r2 ), 2

1 (u1 + u2 ) 2 according to Lemma 4.1. If r1 = r2 then β=

G(a, b) = G(a, 0L ). From Table 2, one can extract all states on L, related to the choice of r1 , r2 , u1 , u2 . Each column in the Table 2 represents a state on L. As example, mb and m0 are in Table 3. �e�inition 4.4 Let G ∈ Γ9 . The map G is called a me‑ asure of pure projection (a pure projection) if G(a, b) = G(a, 0L ) for any a, b ∈ L. �n a Boolean algebra, the projection onto the �irst coordinate may be expressed by a Boolean function f (a, b) = (a ∧ b) ∨ (a ∧ b′ ) = (a ∧ b) ∨ (b′ ∧ a) = a,

where b1 , · · · , bn is an orthogonal partition of unity 1L .

Proof. See in [12].

VOLUME VOLUME 13,13, N°N°33

From Proposition 4.2 it follows that any G ∈ Γ9 is a probability measure of the projection onto the �irst coordinate. Analogical properties are full�iled for any G ∈ Γ10 , which is a probability measure of the pro‑ jection onto the second coordinate.

what motivates us to de�ine on a QL L four G‑maps with the use of p ∈ Γ2 : G1 (a, b) G2 (a, b)

p(a, b) + p(a, b′ ),

p(b, a) + p(b′ , a),

G3 (a, b)

p(a, b) + p(b′ , a),

G4 (a, b)

p(b, a) + p(a, b′ ).

Maps Gi are measures of projection onto the �irst coordinate, i.e. Gi ∈ Γ9 what we prove below. If p is a commutative s‑map, all Gi coincide, Gi (a, b) = p(a, a)

what is a pure projection. If p is a non‑commutative s‑ map, then G1 (a, b) = G2 (a, b) = p(a, a)

is a pure projection, while G3 and G4 are not pure pro‑ jections since: G3 (a, b)

p(a, b) + p(a, a) − p(b, a),

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Tab. 1. Γ9 ‐ Γ16 values in vertices Γ9 0 0 1 1

G(0L , 0L ) G(0L , 1L ) G(1L , 0L ) G(1L , 1L )

Γ10 0 1 0 1

Γ11 1 1 0 0

Tab. 2. G‐maps from Γ9 on a hor izontal sum of Boolean algebras a a′ b b′ 0L 1L

a α 1−α u1 1 − u1 0 1

a′ α 1−α u2 1 − u2 0 1

a r1 α

a′ 1 − r1 1−α

Γ12 1 0 1 0

Γ13 1 1 0 1

b r1 1 − r1 β 1−β 0 1

Γ14 1 0 1 1

b′ r2 1 − r2 β 1−β 0 1

0L α 1−α β 1−β 0 1

Tab. 3. States on L mb m0

and

G3 (a, 0L ) =

0 ≤

p(a, a),

′

G3 (a, b) = p(a, b) + p(b , a) ′

≤

′

p(b, b) + p(b , b ) = 1.

(2)

Values in vertices:

(3)

If a ⊥ b, i.e. a ≤ b′ then

G3 (0L , 0L ) = G3 (0L , 1L ) = 0, G3 (1L , 0L ) = G3 (1L , 1L ) = 1.

G3 (a, b)

p(a, b) + p(b′ , a) = 0 + p(a, a).

= p(a, 0L ) + p(1L , a) + p(0L , b) + p(b′ , 0L ) − 0 = p(a, a).

G3 (a ∨ b, c)

= =

′

p(a ∨ b, c) + p(c , a ∨ b) p(a, c) + p(b, c) + p(c′ , a) + p(c′ , b).

G3 (a, c) + G3 (b, c) − G3 (0L , c) p(a, c) + p(c′ , a) + p(b, c)

+p(c′ , b) + p(0L , c) + p(c′ , 0L ). 68

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= = = = =

G3 (c, a ∨ b) p(c, a ∨ b) + p((a ∨ b)′ , c)

p(c, a) + p(c, b) + p(1L , c) − p(a ∨ b, c)

p(c, a) + p(c, b) + p(1L , c) − p(a, c) − p(b, c)

p(c, a) + p(a′ , c) + p(c, b) + p(b′ , c) − p(1L , c) G3 (c, a) + G3 (c, b) − G3 (c, 0L ).

Proposition 4.5 For every s‑map p there exists a G– map Gp ∈ Γ9 such that Gp (a, b) = Gp (a, 0L ). Proof. Let

where p is an arbitrary s‑map. Then Gp ∈ Γ9 and for any b ∈ L.

(Q.E.D.)

The results for Γ9 −Γ12 are summarized in Table 4.

Tab. 4. Results for Γ9 − Γ12

From the other side =

1L 1 1

Gp (a, b) = Gp (a, 0L )

G3 (a, 0L ) + G3 (0L , b) − G3 (0L , 0L )

If a ⊥ b and c ∈ L then

0L 0 0

1L α 1−α β 1−β 0 1

Gp (a, b) = p(a, b) + p(a, b′ ) = p(a, a),

From the other side

(4)

b′ 1−β 1−β

Γ16 0 1 0 0

The second identity:

and if p(a, b) = p(b, a) then G3 (a, b) = G3 (a, 0L ). Now we prove that G3 is a projection (case G4 is analogical).

(1) G3 (a, b) ∈ [0, 1]

b β β

Γ15 0 0 1 0

probability of

Γ9 a

Γ10 b

Γ11 a′

Γ12 b′

5. Probability Measures of Implications on QLs Values in vertices for families Γ13 − Γ16 are in the Table 1. Similarly to the relations between Γ9 ‑ Γ12 , for families Γ13 − Γ16 hold

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G ∈ Γ13 iff 1 − G ∈ Γ15 ,

2) If a ⊥ b, then mG (a ∨ b)

G ∈ Γ14 iff 1 − G ∈ Γ16 .

Γ15 and Γ16 are analogical cases. For these reasons only one of the famillies, Γ15 , will be focused on. Lemma 5.1 Let L be a QL and G ∈ Γ15 . Then for any a, b ∈ L it holds 1) G(a, a) = G(a, 1L ) = G(0L , a) = 0; 3) If a ↔ b then G(a, b) = G(a ∧ b , 0L ). ′

4) If a ≤ b then G(a, b) = 0.

Proof. 1) Let G ∈ Γ15 and a ∈ L, then = =

G(1L , 1L )

G(a, 1L ) + G(a′ , 1L ) − G(0L , 1L ) G(a, 1L ) + G(a′ , 1L ).

Taking into account that G(a, b) ∈ [0, 1], one concludes that G(a, 1L ) = 0 for any a ∈ L. Furt‑ her 0 =

= =

′

G(a, a) + G(a, 0L ) + G(0L , a ) − G(0L , 0L , )

−G(a, 0L )

Thus G(a, a) = G(0L , a) = 0.

2) Let G ∈ Γ15 and a ∈ L, then with the use of what preceeds, =

G(a, a) + G(a′ , a) − G(0L , a) G(a′ , 0L ) + G(0L , a) − G(0L , 0L )

1 =

Proposition 5.3 Let L be a QL. The famillies Γ2 and Γ15 are isomor�i�.

ii) If p is an s‑map on L and Gp (a, b) = p(a, b′ ), then Gp ∈ Γ15 . i) Let G ∈ Γ15 and pG (a, b) = G(a, b′ ). The proper‑ ties (s1) – (s3) of s‑map are veri�ied bellow. (s1) pG (1L , 1L ) = G(1L , 0L ) = 1 (s2) If a ⊥ b, then pG (a, b) = G(a, b′ ) = 0. It im‑ plies from Lemma 5.1 as a ≤ b′ . (s3) If a ⊥ b and c ∈ L, then pG (a ∨ b, c)

G(a ∨ b, c′ ) G(a, c′ ) + G(b, c′ ) − G(0L , c′ ) pG (a, c) + pG (b, c).

The second identity:

It suf�ices to show that G(c, a′ ) + G(c, b′ ) = G(c, a′ ∧ b′ ). From the orthomodular law it follows that a′ = b ∨ (b′ ∧ a′ ) and b′ = a ∨ (a′ ∧ b′ ). = =

Consequently,

′

G(1L , a) = 1 − G(a, 0L ) = G(a , 0L ).

3) If a ↔ b then G(a, b) = G(a ∧ b , 0L ) follows di‑ rectly from Lemma 3.3. ′

4) a ≤ b is a particular case of a ↔ b, where a ∧ b′ = 0L . This leads immediatelly to ′

G(a, b) = G(a ∧ b , 0L ) = G(0L , 0L ) = 0.

Lemma 5.2 Let L be a QL and G ∈ Γ15 . Then the map mG : L → [0, 1] de�ined as mG (a) = G(a, 0L ) is a state on L.

G(c, a′ ) + G(c, b′ ) G(c, b) + G(c, a′ ∧ b′ ) − G(c, 0L )

+G(c, a′ ∧ b′ ) + G(c, a) − G(c, 0L )

(G(c, b) + G(c, a) − G(c, 0L )) +G(c, a′ ∧ b′ ) − G(c, 0L )

G(1L , 0L ) = G(a, 0L ) + G(a′ , 0L ).

Proof. 1) mG (1L ) = G(1L , 0L ) = 1

= =

G(a , 0L ).

(Q.E.D.)

mG (a) + mG (b).

(Q.E.D.)

′

From the other side,

pG (c, a ∨ b) = G(c, (a ∨ b)′ ) = G(c, a′ ∧ b′ ) pG (c, a) + pG (c, b) = G(c, a′ ) + G(c, b′ ).

G(a, a) + G(0L , a′ ).

= =

G(a ∨ b, 0L ) G(a, 0L ) + G(b, 0L ) − G(0L , 0L )

= =

G(a, 1L ) = G(a, a) + G(a, a′ ) − G(a, 0L )

G(1L , a)

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Proof. Since Γ2 is the set of all s‑maps on L, it suf�ices to prove: i) If G ∈ Γ15 and pG (a, b) = G(a, b′ ), then pG is an s‑map on L.

2) G(1L , a) = 1 − G(a, 0L ) = G(a′ , 0L );

0 =

VOLUME VOLUME 13,13, N°N°33

= =

+G(c, a′ ∧ b′ )

G(c, a ∨ b) + G(c, (a ∨ b)′ )

−G(c, 0L ) + G(c, a′ ∧ b′ ) G(c, 1L ) + G(c, a′ ∧ b′ ) G(c, a′ ∧ b′ ).

Consequently

pG (c, a ∨ b) = pG (c, a) + pG (c, b).

ii) Let p be an s‑map and Gp (a, b) = p(a, b′ ). We want to prove G ∈ Γ15 .

‑ It is clear that the values of Gp in vertices match the maps of Γ15 . ‑ Let a ⊥ b. Then Gp (a, b) = p(a, b′ ) = p(a, a) as a ≤ b′ . On the other hand = =

Gp (a, 0L ) + Gp (0L , b) − Gp (0L , 0L ) p(a, 1L ) + p(0L , b′ ) − p(0L , 1L ) p(a, a) = Gp (a, b).

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‑ Let a, b, c ∈ L and a ⊥ b. Then Gp (a ∨ b, c) = = =

p(a ∨ b, c′ ) p(a, c′ ) + p(b, c′ ) Gp (a, c) + Gp (b, c) − Gp (0L , c).

The second identity:

Gp (c, a ∨ b) = p(c, a′ ∧ b′ )

Gp (c, a) + Gp (c, b) − Gp (c, 0L ) p(c, a′ ) + p(c, b′ ) − p(c, 1L ).

It suf�ices to show that

p(c, a′ ∧ b′ ) = p(c, a′ ) + p(c, b′ ) − p(c, 1L ).

Since

p(c, a ∨ b) = p(c, a) + p(c, b)

p(c, a ∨ b) = p(c, 1L ) − p(c, a′ ∧ b′ )

p(c, 1L ) − p(c, a′ ∧ b′ )

= p(c, 1L ) − p(c, a′ ) + p(c, 1L ) − p(c, b′ )

thus

p(c, a′ ∧ b′ )

p(c, a′ ) + p(c, b′ ) − p(c, 1L ).

(Q.E.D.)

In a classical Boolean logic it holds (principle of a proof by contraposition) a⇒b

⇔

b′ ⇒ a′ .

On a Boolean algebra is any measure of both the left and the right hand side the same. Quantum logics and some measures of implication G ∈ Γ13 (induced by a non‑commutative s‑map) enable to model a situa‑ tion where these measures are not equal. First look at basic properties of the class of implications, Γ13 . Lemma 5.4 Let L be a QL and G ∈ Γ13 . Then for any a, b ∈ L it holds 1) G(a, a) = G (a, 1L ) = G (0L , a) = 1; 2) G(1L , a) = 1 − G(a, 0L ) = G (a′ , 0L ); 3) If a ↔ b then G(a, b) = G (a′ ∨ b, 0L );

4) If a ≤ b then G(a, b) = 1.

Proposition 5.5 Let L be a QL and G ∈ Γ13 . Then the map mG : L → [0, 1] de�ined as mG (a) = G(1L , a) is a state on L. Proposition 5.6 Let L be a QL. The families Γ2 and Γ13 are isomor�i�. Proof. The statement follows immediately from: i) p ∈ Γ2 iff Gp ∈ Γ15 , where Gp (a, b) = p (a, b′ ) .

ii) G ∈ Γ15 iff 1 − G ∈ Γ13 . From the above it is clear that p ∈ Γ2 iff Gp ∈ Γ13 , where Gp (a, b) = 1 − p(a, b′ )

The measure of implication Gp is called a measure in‑ duced by s‑map p. (Q.E.D.) Let us return to the tautology

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VOLUME N°33 2019 2019 VOLUME 13,13, N°

a ⇒ b iff b′ ⇒ a′ .

We would expect an equal measure of propositions a ⇒ b & b′ ⇒ a′ ,

or equivalently: for any G ∈ Γ13 it holds G(a, b) = G (b′ , a′ ) . As already noted, this is true on a Boolean algebra, but not necessarilly on a quantum logic. In‑ deed, if a measure of implication Gp is induced by a non‑commutative s‑map p, and the events a, b are not compatible, one can obtain G(a, b) = 1 − p (a, b′ )

different of

G (b′ , a′ ) = 1 − p (b′ , a) .

Note that, if a measure of implication is induced by a commutative s‑map p, we have a classical situation.

6. Conclusion

An overview of all classes is in Table 5 and in Table 6. It is clear from these tables that on a Boolean alge‑ bra, a value of a G‑map is a probability measure of a Boolean expression, according to the known table for the propositional logic. This leads to the interpretation of values of a function G on a quantum logic. 6.1. Relations between Classes Γ1 ‐ Γ16 .

On a Boolean algebra classes Γi and Γj are isomor‑ phic for i, j = 1, 8. Another situation occurs in the case of non‑compatible random events, that is, in the case of a quantum logic: ‑ Γ4 and Γ7 are isomorphic.

‑ Γi and Γj are isomorphic for

i, j ∈ {2, 3, 5, 6, 13 − 16}.

‑ In [13] it is shown that for any p ∈ Γ2 there exists a Gp ∈ Γ4 induced by p. On the other side, there exists G ∈ Γ4 such that the map pG induced by G is not in Γ2 (pG is not an s‑map).

‑ Γ9 ‑ Γ12 are mutually isomorphic, but their relation to other classes is not quite clear. Nevertheless, for any s‑map there exists a projection, as it follows from Proposition 4.5.

6.2. Problem of Existence of G‐maps on QLs.

Two principal questions related to G‑maps arise in a quantum logic: existence of such map and its proper‑ ties. From the foregoing considerations it follows that the existence of a probability measure of conjunction (s‑map) guarantees the existence of a probability mea‑ sure of all other logical connectives. Therefore, the key question, listed as an open problem Q3 in [25], is the existence of an s‑map on any quantum logic. The existence of an s‑map in the case of a separable quantum logic and additive states has been solved in [15] and [14].

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Tab. 5. Results from the paper [13] G(0L , 0L ) G(1L , 0L ) G(0L , 1L ) G(1L , 1L ) probability of

Γ1 0 0 0 0 0L

Γ2 0 0 0 1 a∧b a↔b

Γ3 0 1 1 1 a∨b a↔b

Γ4 0 1 1 0 (a ⇔ b)′ a↔b

Γ5 1 1 1 0 a ′ ∨ b′ a↔b

Γ6 1 0 0 0 a ′ ∧ b′ a↔b

Γ7 1 0 0 1 a⇔b a↔b

Γ8 1 1 1 1 1L

Tab. 6. Γ9 ‐ Γ16 , G(1L , 0L ) ̸= G(0L , 1L ). For a ↔ b: a ⇒ b = a′ ∨ b. G(0L , 0L ) G(0L , 1L ) G(1L , 0L ) G(1L , 1L ) probability of

Γ9 0 0 1 1 a

Γ10 0 1 0 1 b

Γ11 1 1 0 0 a′

Γ12 1 0 1 0 b′

Proposition 6.1 ( [15], Proposition 1.1.) Let L be an OML, let {ai }ni=1 ∈ L, n ∈ N where ai ⊥ aj , for i = j. If for any i there exists a state αi , such that αi (ai ) = 1, then there exists σ‑CS such that for any where ki ∈ [0, 1] for i ∈ {1, · · · , n} k = (k1 , · · · , kn ), ∑ with the property i ki = 1, there exists a conditional state fk : L × Lc → [0, 1], such that for any i and each d∈L fk (d, ai ) = αi (d); and for each aj fk (aj , ∨i ai ) = ki . Proposition 6.2 ( [14] Proposition 2.2.) Let L be an OML, let there be an s‑map p. Then there exists a con‑ ditional state fp such that p(a, b) = fp (a, b)fp (b, 1L ). Let L be a QL and let Lc = L − {0L }. If f : L × Lc → [0, 1] is a conditional state, then there exists an s‑map pf : L × L → [0, 1]. s‑maps, whose existence is guaranteed by the above cited propositions, can be constructed using techniques similar to those known for the con‑ struction of copulas. ( [1, 3] ).

6.3. Some Differences Between G‐maps on a Boolean algebra and G‐maps on a QL.

1) Each probability measure on B induces a pseudo‑ metric. It means, that for any probability measure m, the map dm : dm (a, b) = m(a∧b′ )+m(a′ ∧b) is a pseudometric on B induced by m. On a quantum lo‑ gic, if p ∈ Γ2 and dp (a, b) = p(a, b′ ) + p(a′ , b), then dp ∈ Γ4 but it can happen that dp is not a pseudo‑ metric.

Γ13 1 1 0 1 a⇒b a↔b

Γ14 1 0 1 1 a⇐b a↔b

Γ15 0 0 1 0 (a ⇒ b)′ a↔b

Γ16 0 1 0 0 (a ⇐ b)′ a↔b

2) Let L be a QL, m be a state on L and p be an s‑map on L. The �irst Bell‑type inequality (4) is not ne‑ cessarily ful�illed for all values a, b ∈ L while its version (5), via an s‑map p is always satis�ied. m(a) + m(b) − m(a ∧ b)

p(a, a) + p(b, b) − p(a, b)

≤

(4)

(5)

The second Bell‑type innequality (6) is not neces‑ sarily ful�illed for all values a, b, c ∈ L while its ver‑ sion (7) is ful�illed for every s‑map, which induces a pseudometric on L [26].

m(a)+m(b)+m(c)−m(a∧b)−m(a∧c)−m(c∧b) ≤ 1 (6) p(a, a)+p(b, b)+p(c, c)−p(a, b)−p(a, c)−p(c, b) ≤ 1 (7)

3) Analogically, implication (8) (Jauch‑Piron state, see e.g. [4,22]) can be violated on L but implication (9) is always valid m(a) = m(b) = 1 ⇒ m(a ∧ b) = 1

p(a, a) = p(b, b) = 1 ⇒ p(a, b) = 1,

and moreover for any c ∈ L

(8)

(9)

p(a, c) = p(c, a) = p(c, c).

4) On a Boolean algebra, every projection is a pure projection. On a quantum logic, a G‑map G (G ∈ Γi , i ∈ {9, 10, 11, 12} ) is not necessarilly a pure projection, see Example 4.3.

5) Quantum logics and G‑maps enable to model situ‑ ations that can not occur in a Boolean algebra. The use of G‑maps to model these situations on QLs is illustrated by the following considerations: a) Quantum logics and non‑commutative s‑maps (class Γ2 ) enable to model stochastic causality. Articles

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Tab. 7. d‐map not satisfying triangle inequality if k > 0 a b c a′ b′ c′ 0 1

a 0 k 0 1 1−k 1 α 1−α

b k 0 0 1−k 1 1 β 1−β

c 0 0 0 1 1 1 γ 1−γ

a′ 1 1−k 1 0 k 0 1−α α

b′ 1−k 1 1 k 0 0 1−β β

Let L be a quantum logic, p an s‑map on L, and a, b ∈ L. The conditional probability of some event a, given the occurrence of some other event b is p(a, b) P (a|b) = . p(b, b) Assume that p is a non‑commutative s‑map. Then there are non‑compatible events a, b, for which p(a, b) = p(b, a). This situation models a stochas‑ tic causality using a non‑commutative measure of conjuction p. In this case Bayes’s theorem is vi‑ olated ( [16, 17]). Assume moreover that the event a is indepen‑ dent of b, i.e. it holds P (a|b) =

p(a, b) = p(a, a). p(b, b)

On the other side, the event b is not independent of a, as P (b|a) =

p(b, a) p(b, a)p(b, b) = = p(b, b). p(a, a) p(a, b)

Using a commutative s‑map, we have a classical situation. A commutative s‑map ps can be obtai‑ ned from an arbitrary s‑map p e.g. as ps (x, y) =

1 (p(x, y) + p(y, x)) . 2

Whether an event a is independent of b or not is determined by the measure of conjunction. The‑ refore it is suitable to say that a is independent of b with respect to a measure (s‑map p). b) Quantum logics and some d‑maps (class Γ4 ) ena‑ ble to distinguish elements that are not distin‑ guishable on a Boolean algebra. Symmetric difference (d‑map) on a Boolean alge‑ bra ful�ills the triangle inequality d(a, b) ≤ d(a, c) + d(c, b).

Consequently, if a, c and b, c are indistinguisha‑ ble, then a, b are also, because d(a, c) = d(c, b) = 0 ⇒ d(a, b) = 0.

On a quantum logic exists a set of symmetric dif‑ ferencies (subclass of Γ4 ), that do not ful�ill the 72

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c′ 1 1 1 0 0 0 1−γ γ

0L α β γ 1−α 1−β 1−γ 0 1

1L 1−α 1−β 1−γ α β γ 1 0

triangle inequality. Table 7 gives an example of such symmetric difference under condition k > 0. For elements a, b, c it holds: d(a, c) = d(c, b) = 0,

but d(a, b) = k > 0.

ACKNOWLEDGEMENTS Oľga Ná ná siová would like to thank for the support of the VEGA grant agency by means of grant VEGA 1/0710/15 and the author Ľubica Valá š ková would like to thank for the support of VEGA 1/0420/15.

AUTHORS

Oľga Nánásiová∗ – Inst. of Computer Science and Mathematics, Slovak University of Techno‑ logy in Bratislava, Ilkovič ova 3, 812 19 Bratislava, Slovakia, e‑mail: nanasiova@stuba.sk, www: ma‑ tika.elf.stuba.sk/KMAT/OlgaNanasiova. Ľubica Valášková – Department of Mathematics and Descriptive Geometry, Slovak University of Techno‑ logy , Radlinské ho 11, 810 05 Bratislava, Slovakia, e‑mail: valaskova@stuba.sk, www: math.sk. Viera Čerňanová – Department of Mathematics and Computer Science, Faculty of Education, Trna‑ va University, Priemyselná 4, 918 43 Trnava, Slo‑ vakia, e‑mail: vieracernanova@hotmail.com, www: pdf.truni.sk/katedry/kmi/pracovnici. ∗

Corresponding author

REFERENCES

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[11] O. l. Ná ná siová and L. Valá š ková , “Marginality and Triangle Inequality”, International Journal of The‑ oretical Physics, vol. 49, no. 12, 2010, 3199–3208, 10.1007/s10773‑010‑0414‑2. [12] O. Ná ná siová , V. C� erň anová , and L. Valá š ková , “Probability Measures and Projections on Quan‑ tum Logics”. In: P. Kulczycki, J. Kacprzyk, L. T. Kó czy, R. Mesiar, and R. Wisniewski, eds., Information Technology, Systems Research, and Computational Physics, Cham, 2020, 321–330, 10.1007/978‑3‑030‑18058‑4_25. [13] O. Ná ná siová and L. Valá š ková , “Maps on a quan‑ tum logic”, Soft Computing, vol. 14, no. 10, 2010, 1047–1052, 10.1007/s00500‑009‑0483‑4. [14] O. Ná ná siová , “Map for Simultaneous Measure‑ ments for a Quantum Logic”, International Jour‑ nal of Theoretical Physics, vol. 42, no. 9, 2003, 1889–1903, 10.1023/A:1027384132753.

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[19] M. Pavicic and N. D. Megill, “Is Quantum Logic a Logic?”, arXiv:0812.2698 [quant‑ph], 2008, arXiv: 0812.2698.

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[25] J. Pykacz and P. Frąckiewicz, “The Problem of Conjunction and Disjunction in Quantum Logics”, International Journal of Theoretical Physics, vol. 56, no. 12, 2017, 3963–3970, 10.1007/s10773‑ 017‑3402‑y.

[26] J. Pykacz, L. Valá š ková , and O. Ná ná siová , “Bell‑ Type Inequalities for Bivariate Maps on Ortho‑ modular Lattices”, Foundations of Physics, vol. 45, no. 8, 2015, 900–913, 10.1007/s10701‑015‑ 9906‑5. [27] G. Sergioli, G. M. Bosyk, E. Santucci, and R. Giun‑ tini, “A Quantum‑inspired Version of the Classi‑ �ication Problem”, International Journal of Theo‑ retical Physics, vol. 56, no. 12, 2017, 3880–3888, 10.1007/s10773‑017‑3371‑1. [28] S. Sozzo, “Conjunction and negation of natural concepts: A quantum‑theoretic modeling”, Jour‑ nal of Mathematical Psychology, vol. 66, 2015, 83–102, 10.1016/j.jmp.2015.01.005.

[16] O. Ná ná siová and M. Kalina, “Calculus for Non‑ Compatible Observables, Construction Through Conditional States”, International Journal of The‑ oretical Physics, vol. 54, no. 2, 2015, 506–518, 10.1007/s10773‑014‑2243‑1.

[17] O. Ná ná siová and A. Khrennikov, “Representation Theorem of Observables on a Quantum System”, International Journal of Theoretical Physics, vol. 45, no. 3, 2006, 469–482, 10.1007/s10773‑006‑ 9030‑6. Articles

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2D-Raman Correlation Spectroscopy as a Method to Recognize of the Interaction at the Interface of Carbon Layer and Albumin Submitted: 13th June 2019; accepted 10th September 2019

Anna Kołodziej1, Aleksandra Wesełucha-Birczyńska1*, Paulina Moskal1, Ewa Stodolak-Zych2, Maria Dużyja3, Elżbieta Długoń2, Julia Sacharz1, Marta Błażewicz2 DOI: 10.14313/ JAMRIS/3-2019/30 Abstract: In modern nanomaterial production, including those for medical purposes, carbon based materials are important, due to their inert nature and interesting properties. The essential attribute for biomaterials is their biocompatibility, which indicates way of interactions with host cells and body fluids. The aim of our work was to analyze two types of model carbon layers differing primarily in topography, and developing their interactions with blood plasma proteins. The first layer was formed of pyrolytic carbon C (CVD) and the second was constructed of multi-walled carbon nanotubes obtained by electrophoretic deposition (EPD), both set on a Ti support. The performed complex studies of carbon layers demonstrate significant dissimilarities regarding their interaction with chosen blood proteins, and points to the differences related to the origin of a protein: whether it is animal or human. However the basic examinations, such as: wettability test and nano sctatch tests were not sufficient to explain the material properties. In contrast, Raman microspectroscopy thoroughly decodes the phenomena occurring at the carbon structures in contact with the selected blood proteins. The 2D correlation method selects the most intense interaction and points out the different mechanism of interactions of proteins with the nanocarbon surfaces and differentiation due to the nature of the protein and its source: animal or human. The 2D correlation of the Raman spectra of the MWCNT layer+HSA interphase proves an increase in albumin β-conformation. The presented results explain the unique properties of the Clayers (CVD) in contact with human albumin. Keywords: Multi-Walled Carbon Nanotubes; Pyrolytic Carbon; Carbon Layers; Raman Microspectroscopy; Plasma Blood Proteins, 2D Correlation.

1. Introduction

The diverse and growing needs of modern societies can be met by the extraordinary development of nanotechnology methods that represent a unique position for varies application areas and importance for economic reasons [1]. The presented research topic concerns materials for regenerative medicine due to their specificity included in

the area of nanomedicine. Size of nano-additives may lead to their direct involvement in the processes and even biological structures [1-7]. Innovative biomaterials have contributed to the field of controlled drug delivery applications [8-10], cardiovascular diseases [11-13] and orthopedics [3,14], while novel materials with carbon nanotubes coatings are desirable for application as sensors and neural electrodes or as a platform for Central Nervous System diseases [15-17]. Selecting appropriate nanofiller can cause that structural, mechanical and electronic properties of nanocomposite material are shaped in a manner that would induce the desired characteristics of interaction with the biological environment [18]. So the nanoparticle by modern nanotechnology methods may be adjusted to modify, as intended, the polymer matrix [19-21], or to alter the surface of the synthetic material that comes into contact with the tissue of the living organism [11, 22-26]. Carbon materials are attractive due to their unique properties and large variety of carbon structures and nanostructures, so they can be used as a modifying particles [27]. Two types of carbon structures, pyrolytic carbon and multiwall carbon nanotubes (MWCNTs), have been chosen to identify their properties. Pyrolytic carbon can occur with different microstructures, that depend on the various forming conditions, belongs to a group of turbostratic carbons, with a structure related to graphite. However, graphite consists of carbon atoms that are covalently bonded in hexagonal arrays. These arrays are stacked and held together by weak interlayer binding, while pyrolytic carbon and other turbostratic carbons present a lack of order in the neighboring graphitic layers, so that occur wrinkles or distortions within layers. This ensures that the pyrolytic carbon has an improved durability compared to graphite [28]. Although this type of structure has been the most popular material available for artificial mechanical heart valves for about half a century there are some requirements to consider. The second type of the analyzed forms of carbon are MWCNTs, that are fibrous nanostructures [29], recently used are synthesized [27,29]. They can be also found naturally or as a byproduct of industrial processes [30,31]. The Ijima discovery caused a great interest for CNTs because of their characteristics: small size and mass, high strength, and high electrical conductivity [32].

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For biomaterials the most important feature is their biofunctionality and biocompatibility, which refers to mechanical characteristic and to the interaction with host cells and fluids, respectively [33]. The biological response to the synthetic material is determined at the interface of the surface of a biomaterial and cell. One must be aware that the interface is complex, as biomolecules and synthetic material are composed of three-dimensional entities [34]. Therefore, the number of surface parameters determining the interfacial area such as: surface morphology, roughness (micro and nano), wettability, and the degree of crystallinity are parameters that result from the surface properties and significantly affect the biological properties of the material [25,26, 35-37]. Research on the adsorption of protein to synthetic material is considered as method to assess biomaterials quality, not only that in blood-contacting applications. It is considered that the adsorption of selected proteins, may reveal the specific biological properties of the examined nanomaterials. Usually albumin is selected to study its interaction with the surface of the nanomaterial. Albumin is the blood plasma protein of the next highest abundance after hemoglobin, which is involved in transport and storage and regarded as an inhibitor of blood clotting [38]. The commonly used are proteins of animal origin, which are considered to be equivalents of human protein. However, our earlier studies have shown that animal equivalents differently affect the surface of a carbon material than human proteins [25,26]. Therefore, modeling interactions with proteins of animal origin can be questioned in some cases. As target proteins, albumin from chicken egg white (Alb), bovine serum albumin (BSA) and human serum albumin (HSA), were employed to check their influence on the surface during performed research. Conventional methods of material engineering such as wetting angle measurement and electron microscopic techniques (SEM) were employed to obtain the characteristics of the materials surface. The mechanical properties of the protein layers on the materials were tested with the Nano Scratch Tester. This method was applied to study the adhesion and scratch resistance of model coatings incubated with selected proteins. Then, materials were tested using Raman spectroscopy. The Raman spectroscopy was selected as an important spectroscopic technique to test short-range ordering. It has been used to describe the structure of two reference carbon layers and then the effect of adsorption on these layers of selected plasma proteins. Finally, a 2D Raman correlation spectroscopy was applied to the collected Raman spectra and allowed for the resolution of the phenomena occurring at the interphase, the carbon layer surface in contact with albumin, the most abundant blood proteins. In this mathematical analysis as an external perturbation the spatial position, in which the spectrum was measured, was taken into account.

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2. Materials and Methods 2.1. Preparation of Carbon Layers on Titanium Titanium plates (Grade 2 according to ASTM B265) in the form of discs with a diameter of 12 mm and a thickness of 0.5 mm were chosen as appropriate substrates for the deposition of carbon layers, which were: pyrolytic carbon (C (CVD)) and carbon nanotubes (MWCNTs) layer. The first, pyrolytic C-layer was obtained in the Plasma-enhanced chemical vapor deposition process (PECVD; Elettrorava, Italy). All depositions were performed by the RF PECVD method, in which the plasma was generated by radio frequency waves of 13.56 MHz and of power 60W. The formation of the layers was preceded by ion-etching in argon plasma during 10 minutes to eliminate of the TiOx surface layer. Then the C-layer layer was deposited at room temperature (25°C) during 30 minutes from a methane (gas flow 10 cm³/min) and argon (gas flow 75 cm³/min) mixture, while the chamber pressure was kept constant (53 Pa). Argon was used as inert gas. The second layer of MWCNTs (#1213NMGS, Nanostructured & Amorphous Materials, Inc., USA; outside diameter 10-30nm; core diameter 5-10nm, length of 1-2 μm and purity >95%.) was produced in the electrophoretic deposition (EPD). Details on the oxidizing procedure, preparation of the suspension, titanium handling and EPD set-up are presented in our previous studies [24,25, 26]. The albumin from chicken egg white (Alb), bovine serum albumin (BSA), and human serum albumin (HSA), were purchased from Sigma-Aldrich (Poland). Both carbon coatings were incubated in 1% albumin solution for 15 minutes.

2.2 SEM Images Analysis

The morphology and chemical composition of the obtained coatings were examined using a scanning electron microscope Nova Nanosem (FEI) equipped with an adapter for EDS X-ray microanalysis (EDAX). The system was operated with 10–15 kV accelerating voltage, high vacuum mode.

2.3 Wettability Measurements

The contact angle [θ] for a liquid droplet on a tested, solid surface, was specified between the surface of the liquid and the outline of the contact surface, in the point where three phases meet: solid, liquid and gas. The contact angle measurements were taken using a direct method (DSA 10 Kruss goniometer). The tests were performed at room temperature applying the sitting drop technique (the drop of deionized water of 0.15-0.25 μl in volume). The measurements were taken five times in order to found variability and the standard deviation (SD) (that was estimated as ± 2.5%). All experiments were performed for reference and both tested surfaces and also for the selected blood proteins conditioned with tested surfaces. All tests were performed under ambient conditions. Articles

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2.4. The Nano Scratch Test All tests were carried on a platform with the NST (Nano Scratch Tester) head made by CSM Instruments SA (currently Anton Paar TriTec) (Switzerland). The parameters, while testing, were set as follows: the load Fn increased linearly from 0,1 to 5 mN on the 3 mm distance, the speed of loading was set for 10 mm/ min and the Rockwell certified indenter radius was equal to 2 μm. Two nanocomposite layers were analyzed: the C-layer grown in the CVD process and the MWCNTs deposited in the Ti support, after incubation with chicken egg white albumin and human serum albumin (HSA).

2.5. Raman microspectroscopy

A Renishaw inVia spectrometer, connected to a Leica microscope, was used for the measurements of the Raman spectra. The beam from a 514.5 nm Ar+ ion Modu Laser was focused on the samples by 100x magnifying, a high numerical aperture (NA = 0.9) topclass Leica objective for standard applications. Laser power was kept sufficient low, c.a. 1-3 mW at the sample, to ensure minimum disturbances to the samples.

2.6. 2D Raman Correlation

The generalized 2D correlation analysis based on the Noda method [39-41] was performed using Raman spectra as an input data for generating the correlation maps. The spatial position was regarded as an external perturbation [42]. The five points, morphologically similar were measured for each sample, and they were regarded as a dynamical spectra in the 2D correlation. 2Dshige, v.1.3 software was employed [43].

3. Results and Discussion

3.1. SEM Images of Carbon Layers SEM investigations indicate dissimilarities in the topography of both materials. Two types of carbon structures are different primarily in the topography. The first coating was a layer formed of pyrolytic carbon (CVD) (Fig.1A) and the second was constructed of multi-walled carbon nanotubes obtained by electrophoretic deposition (EPD) (Fig. 1B), both set on a Ti support. A slight, intentional difference in the magnification of both images allows to see the specificity of the structure of micro- and nano-layer of C (CVD) and MWCNTs, respectively.

Fig. 1. SEM image of: (A) C (CVD) layer 10000x magnification, (B) MWCNTs (EPD) layer 20000x magnification.

3.2. Wettability of Carbon Layers The surfaces wettability was analyzed by the static sessile drop method. The contact angles of water droplets on the top face of the reference C (CVD) layer are 82.2 ± 2.8°, respectively (Figure 2A). The C (CVD) layer does not change the specificity of the surface with respect to the titanium substrate [26]. The difference in the contact angles for C (CVD) incubated with Alb (78.4 ± 2.3°) and HSA (84.5 ± 1.9°) with comparison to a reference the C (CVD) layer is not so significant, it fits within the limit of 5% (Figure 2A). However, the C (CVD) incubation in BSA leads to the creation of a film with hydrophobic characteristics and contact angle of 112.0 ± 6.7°. The contact angles on the top face of the MWCNTs nanocomposite layer is 25.0 ± 0.9° (Figure 2B). This value implies a hydrophilic character of the surface of the MWCNTs coating. The Alb and HSA form a layer having a wettability 62.6 ± 1.3° and 57.3 ± 0.3°, respectively. The BSA conditioned MWCNTs nanocarbon layer reaches the highest contact angle of 77.1 ± 1.3°.

Fig. 2. Wettability of studied reference carbon samples (A) C (CVD) layer; (B) MWCNTs (EPD) layer, and after incubation with selected proteins.

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3.3. Nano Scratch Test The performed test consisted of three steps. In the first step the profile of the sample (Pre Scan) was collected. In the second phase the indenter was pressed into the sample with linearly increasing load and moving along the sample at a defined length to scratch off the coating (test phase). During the third step the profile inside scratch (Post Scan) was accumulated. After the performed scratch the panoramic photograph is taken and the obtained features are analyzed. HSA has the best adhesion to the C (CVD) layer (Table 1). This coating did not break during the performed test and the Ti substrate is not visible. It can be observed that the C (CVD) layer with a thin HSA sheet looks like a “pressed and smeared” during the test and therefore presents the best adhesion to the Ti base. In addition, violation of coating, chipping or cracking up to the 5mN load is not observed. Tab. 1. Summary of the Scratch test Parameters (linear scrach; load range 0.1-5 mN; loading rate 10 mN/min). Sample

coating failed at 2.16 ± 0.20

MWCNTs + Alb

coating failed at 1.63 ± 0.10

MWCNTs + HSA

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Tab. 2. Observed Raman bands [cm-1] and their assignments for carbon coatings excited with 514.5 nm laser line. Sample C (CVD)

MWCNTs/ (EPD) C (CVD)

Peak position [cm-1] 1349 1364 1594

MWCNTs (EPD)

1595

MWCNTs (EPD)

2713

MWCNTs (EPD)

1633

2956

Assignment [20-28, 44-47] D-mode; breathing mode that requires a defect for its activation

G-mode; E2g mode at the Brillouin zone center D’-mode; effect of double resonance as an intravalley process

2D (G’); second order of the D peak

D+D’; combination of phonons with different momenta, requires a defect for theirs activation

critical load [mN]

C ( CVD) + Alb

C ( CVD) + HSA

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coating is not destroyed

coating failed at 2.17 ± 0.16

For the MWCNTs nanolayer the HSA film is de-laminated with a load of about 2-2.5 mN. It is observed that the entire carbon nanolayer along with the thin film of protein is detached from the Ti substrate. Adhesion of the Alb to the carbon C (CVD) layer is better, but the difference is small, the surface is scratched with a load of about 2.16 mN. The surface of the Alb on the MWCNTs nanolayer is scratched off easily, the visible surface of the Ti substrate is noticeable from the start of the test which means that the critical load for this layer is below 1.63 mN.

3.4. Raman Microspectroscopy Characterization of Carbon Layers and their Interaction with Selected Blood Proteins

The Raman band positions and assignments of the reference carbon layers excited by the 514.5 nm laser line are collected in Table 2. The first, the C (CVD) layer is formed of pyrolytic carbon, which is an anisotropic material. Therefore, only the main G- and D-bands are observed, what proves the graphite-type arrangement in this coating (Figure 3A). For the second, MWCNTs nano-layer, in addition to G- and D- also characteristic 2D, D’ and D+D’ bands are visible confirming more complex organization in this coating (Figure 3B) .

Fig. 3. Raman spectra for reference carbon layers (A) C (CVD), and (B) MWCNTs (EPD); 514.5 nm excitation line. The G-band at ca.1580 cm-1 is typical for the sp2 carbon materials and is assigned to the high frequency E2g optical phonon [44]. The G-band position for the C (CVD) differs from that for the MWCNTs (EPD) layer, which is more heterogenous [45, 46]. Due to the excellent sensing properties of the carbon particle, the interaction with proteins can be observed and determined. The interaction occurs in the interphase region, between the two phases, and depends on both of them, on the type of carbon material and on protein (Fig. 4). The noticed shift of the G-band position shows the identifiable type of interaction in relation, first of all, to the reference coating. Furthermore, the positions of the G-band for the animal albumins, Alb and BSA, differ from that of HSA, what confirms that interaction occurs between the thin protein layer and the carbon coating as well as their specificity for different types of albumin. Therefore, you need to consider what properties of the carbon layers are crucial in the application you are working on.

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Fig. 4. Position of the G- band for the reference carbon layers (A) C-(CVD), and (B) MWCNTs (EPD), and after incubation with the selected proteins; 514.5 nm excitation line.

3.5. 2D Raman Correlation Method Two-dimensional correlation uses mathematical formalism to obtain two-dimensional correlation spectra from any transient or time-resolved spectra of an arbitrary waveform [39, 40]. The experimental approach takes into account that an external perturbation, applied to a studied system, selectively excites various chemical constituents of the system. A 2D experiment in optical spectroscopy is performed by introducing a relatively slow external perturbation applied to the system of interest [41]. Spectral changes that are noticed under certain dynamic perturbation are the variation of intensities, shifts of spectral band positions, and alteration in the shape of peaks. These fluctuation of spectral signals are transformed into two-dimensional spectra by using a correlation method formalism. The type of physical information enclosed in a dynamic spectrum, is determined by the selection of perturbation method and electromagnetic probe. Therefore, 2D spectra obtained by 2D correlation method can highlight valuable information often hidden in the original time-resolved spectra.

3.6. 2D Raman Correlation Spectroscopy Characterization of Carbon Layers Interaction with Selected Blood Proteins

Carbon materials are characterized with a very high polarizability, therefore Raman spectroscopy is the method of choice for their analysis. However, the studied interaction occurs primarily in the interphase area, so not only carbon material, but also the complex biological structures of albumin are involved in the interaction. The most interesting would be to analyze the spectra in the vibration range of amide I, which is also the region of appearance of the D- and G-band of carbon layers. However, the Raman signal from the carbon component on the phase boundary might cover the spectrum of the protein, due to the high polarizability of sp2 system. Hence, a two-dimensional correlation analysis was used to control Articles

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changes in the structure of the proteins on the interaction with the carbon layers in order to decode the relations hidden in the Raman spectra. [40, 42]. In the performed 2D correlation spectroscopy analysis the variable intensities were linked with a location on a sample characterizing the respective protein film carbon coating interactions. The 2D spectrum indicates the clear differentiation between the origins of the Raman spectral signals [40]. Synchronous signal fluctuations indicate a common chemical constituent at ca. 1588 and 1345 cm-1 and also at ca. 1613 and 1313 cm-1 originating from the graphite G and D- band components of the studied coating for the C (CVD) and MWCNTs layer, respectively. The nonsynchronous signal fluctuations indicate chemically dissimilar components, thus the map pattern differs for the two carbon coatings and additionally animal albumins are clearly distinguishable from that of human (Figs. 5 & 6). The asynchronous correlation map for the C (CVD) + Alb layer in the 1720–1530 cm-1 range shows intensive negative cross-peaks originating from the amide I component bands at 1650 cm-1 (α-helix conf.), 1665 cm-1 (β-sheet conf.) and 1682 cm-1 (β-turn conf.) with the 1597 cm-1 band due to the G-band of the carbon layer. Otherwise it is for the C (CVD)+ BSA layer, there is an intensive positive band in a different location at (1600, 1552 cm-1) that arises from the carbon G-band and Glu vibrations. The C (CVD)+ HSA layer presents a positive asynchronous peak derived of 1635 cm-1 (Trp, Arg, His) and 1652 cm-1 (α-helix conf.) and 1661 cm-1 (β-sheet conf.) of amide I and at ca. 1594 cm-1 of the carbon layer G-band.

Fig. 5. Asynchronous 2D correlation Raman spectra of the C (CVD) sample incubated with: (A) chicken egg white albumin (Alb); (B) bovine serum albumin (BSA); (C) human serum albumin (HSA); in the wavenumber range of 1720-1530 cm-1; the red and blue color represent positive and negative cross peaks, respectively. A different pattern is observed for the 2D asynchronous maps for the second type carbon layer. The MWCNTs + Alb present a positive asynchronous correlation cross-peak at 1600 cm-1 of protein aromatic ring vibrations and 1622 cm-1 of Tyr with 1578 cm-1 of carbon nanotubes G- vibration. Another tested system MWCNTs +BSA shows intensive negative cross peak –(1630,1602 cm-1) owed to His, Tyr and (G+) carbon band and –(1582,1603 cm-1) originating from the G+band of the MWCNTs and the Phe albumin vibrations.

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Tab. 3. Observed significant asynchronous 2D correlation cross-peaks and their assignments for the C (CVD) and MWCNTs coating incubated in Alb, BSA, HSA in the wavenumber ranges of 1750–1500 cm-1 [20-28, 44-47].

assignment

amide I, α-helix conf.

amide I, β- sheet conf. amide I, β-turn conf. G-band

D’-band D’-band

D’-band

amide I, α-helix conf.

C (CVD) + Alb

2D asynchronous cross-peaks

cross-peaks

assignment

-(1665,1597)

G-band

Tyr

-(1650,1597)

-(1682,1597)

C (CVD)+BSA

+(1598,1552)

G-band

Phe, His

+(1613,1558)

Glu

Asp

Phe

+(1635,1601)

G-band

Tyr

C (CVD)+HSA

+(1635,1594)

+(1652,1601)

G-band

The MWCNTs+ HSA layer gives a negative asynchronous peak -(1618,1602) due to the Tyr and G+ carbon band. The other cross-peak originated of the amide I band of 1659 cm-1 (β-sheet conf.) and 1687 cm-1 (β-turn conf.) with 1601 cm-1 (G+) carbon nanotube vibrations.

Fig. 6. Asynchronous 2D correlation Raman spectra of the MWCNTs (EPD) sample incubated with: (A) chicken egg white albumin (Alb); (B) bovine serum albumin (BSA); (C) human serum albumin (HSA); in the wavenumber range of 1720-1530 cm-1. The red and blue color represent positive and negative cross peaks, respectively. The calculated asynchronous cross–peaks are enclosed in Table 3. The 2D correlation spectroscopy allows to differentiate the adhesion specificity of the selected blood protein to the studied model, carbon layers.

4. Summary The results of the performed complex studies of the two types of model carbon coatings display significant dissimilarities regarding their interaction with the chosen blood proteins but also the difference is related to the origin of a protein: whether it is animal or human. Both of the studied carbon layers were incubated with the selected albumins, and the interaction

His

amide I, α-helix conf. amide I, β-turn conf.

MWCNTs +Alb cross-peaks

assignment

+(1622,1578)

G–- band

+(1600,1578)

MWCNTs +BSA

+(1603,1583)

+(1630,1603)

MWCNTs +HSA

G–- band

G+- band G+- band

-(1618,1602)

G+- band

-(1687,1601)

G+- band

-(1660,1602)

G+- band

between these two materials is visible by the variation of the contact angle (Fig. 2). They substantially differ in their surface image, the nanotubes form an isotropic fibrous system with a characteristic nano topography while the C (CVD) pyrolytic surface is smoother (Fig. 2B). For both types of the studied carbon layers a similar sequence of changes reflecting the occurring interaction with the albumins, which can be estimated by measuring contact angle. For each of the albumin, these differences are clearly marked, and a remarkably different contact angle is noticed for BSA (Fig. 2 A and B). The CVD layer, which is known to be antithrombogenic, is characterized by high adhesion of protein to the surface while in other cases, the protein layer weakly adheres to the substrate (Table 1). The results of Raman spectroscopy indicate that carbon layers interact differently with the selected blood proteins, as indicated by the parameters determined from the Raman spectra, e.g. the position of the characteristic G-bands (Fig. 3). This parameter allows to uncover the type of interactions and their extent. The 2D asynchronous maps offer the possibility of determining the type of protein interaction with the surface of the carbon layer. Considering only the most intense cross-peaks you will notice that Alb adopts the structure with a comparable contribution of the α-helix, β-sheet and a sizable portion of β-turn conformation while signals from the individual amino acids Phe, His, Tyr are observed for the MWCNTs layer (Fig. 5A & 6A) [48-50]. For the BSA there is a correlation signal from the individual amino acids: Asp, Glu and Phe, His for C (CVD) and MWCNTs layer, respectively (Fig. 5B & 6B ). [47-49]. The 2D correlation spectroscopy does not provide evidence that the secondary structure is mainly α-helix (50-60% in its native state) [49,51,52]. HSA acts in an exceptional way with a synthetic nanomaterial, showing equal participation conformation α-helix and also α-helix and Articles

Journal of Automation, Mobile Robotics and Intelligent Systems

b-turn in addition to the Tyr signal for the C (CVD) and MWCNTs layer, respectively [49,53]. But, the vibrations of the amide I are ahead of the changes in the carbon nanolayer for the C (CVD). The MWCNTs layer shows the opposite order of events (Fig. 5C & 6C). The 2D analysis shows that the amide I bond is modified by the aromatic MWCNTs structure while the interaction on the interphase occurs, what was also noticed for the SWCNTs [54].

5. Conclusion The albumin conformation is different for the studied surfaces, the amide I band maximum observed for the C (CVD) pyrolytic carbon layer shifts toward the higher vibrations for the MWCNT coating confirming an increase in the amide I of β-conformation. The conducted research and spectroscopic characteristics of the studied surfaces topography as a key element in the synthetic surface with blood protein interaction and allow for the explanation of the nature of this process in relation to the type of protein. The phenomena occurring on the surface of the C-pyrolytic carbon with contact with human albumin have a different character than those observed in other cases. It can therefore be assumed that these phenomena, leading to the conformational changes in HSA and strong adhesion, indicates non-thrombogenic characteristic of this type surface coatings.

ACKNOWLEDGEMENTS

This project was financed from the National Science Centre (NCN, Poland) granted on the decision number DEC-2013/09/B/ST8/00146 and UMO-2014/13/B/ ST8/01195. AK has been partly supported by the EU Project POWR.03.02.00-00-I004/16.

AUTHORS

Anna Kołodziej – Faculty of Chemistry, Jagiellonian University, Kraków, Poland.

Aleksandra Wesełucha-Birczyńska* – Faculty of Chemistry, Jagiellonian University, Kraków, Poland, email: birczyns@chemia.uj.edu.pl. Paulina Moskal – Faculty of Chemistry, Jagiellonian University, Kraków, Poland.

Ewa Stodolak-Zych – Faculty of Materials Science and Ceramics, AGH-University of Science and Technology, Kraków, Poland. Maria Dużyja – Technolutions, Łowicz, Poland.

Elżbieta Długoń – Faculty of Materials Science and Ceramics, AGH-University of Science and Technology, Kraków, Poland.

Julia Sacharz – Faculty of Chemistry, Jagiellonian University, Kraków, Poland. 80

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Marta Błażewicz – Faculty of Materials Science and Ceramics, AGH-University of Science and Technology, Kraków, Poland. *Corresponding author

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VOLUME 2019 VOLUME 13,13, N°N°33 2019

STATE‐OF‐THE‐ART IN MODELING NONLINEAR DEPENDENCE AMONG MANY RANDOM VARIABLES WITH COPULAS AND APPLICATION TO FINANCIAL INDEXES Submitted: 13th June 2019; accepted: 10th September 2019

Tomáš Bacigál, Magdaléna Komorníková, Jozef Komorník DOI: 10.14313/JAMRIS/3‐2019/31 Abstract: In this paper, we focus our attention on multi– dimensional copula models for returns of the indexes of selected prominent international financial markets. Our modeling results, based on elliptic copulas, 7‐ dimensional hierarchical Archimedean copulas, vine co‐ pulas and factor copulas demonstrate a dominant role of the SPX index among the considered major stock indexes (mainly at the first tree of the optimal vine copulas). Some interesting weaker conditional dependencies can be de‐ tected at it’s highest trees. Interestingly, while global op‐ timal model (for the whole period of 277 months) belong to the Factor FDG copulas class, the optimal local models can be found (with very minor differences in the values of GoF test statistic) in the classes of Factor FDG and hier‐ archical Archimedean copulas. The dominance of these models is most striking over the interval of the financial market crisis, where the quality of the best Student class model was providing a substantially poorer fit. Keywords: dependence, copula, elliptically contoured distribution, vine copula, factor copula, hierarchical Ar‐ chimedean copula, international financial market indexes

1. Introduction In this paper we apply multi–dimensional copula to model dependence among returns of selected pro‑ minent indexes of international �inancial markets. The following indexes were considered (with months’ va‑ lues from the time interval 31.1.1995 – 31.1.2018): SPX (Standard and Poor’s Index is designed to mea‑ sure performance of the broad US economy through the aggregate market value of 500 stocks represen‑ ting all major industries), DAX (The German Stock In‑ dex is a total return index of 30 selected German blue chip stocks traded on the Frankfurt Stock Exchange), UKX (The FTSE 100 Index is a capitalization‑weighted index of the 100 most highly capitalized companies traded on the London Stock Exchange), NKY (The Nikkei‑225 Stock Average is a price‑weighted average of 225 top‑rated Japanese companies listed in the First Section of the Tokyo Stock Exchange), HSI (The Hang Seng Index is a free‑�loat capitalization‑weighted in‑ dex of a selection of companies from the Stock Ex‑ change of Hong Kong), LEGATRUU (The Bloomberg Barclays Global Aggregate Bond Index is a �lagship me‑ asure of global investment grade debt from twenty‑ four local currency markets), SPGSCITR (The S&P GSCI Total Return Index in USD is widely recognized as the leading measure of general commodity price mo‑ 84

vements and in�lation in the world economy). The paper is organized as follows. The second section is devoted to a brief overview of the theory of hierarchical Archimedean copulas, vine copulas, factor copulas and methodology of copula �itting to multi–dimensional time series. The third section con‑ tains application to real data modeling. Finally we dis‑ cuss results and conclude.

2. Theory

Copula represents a multivariate distribution that captures the dependence structure between/among random variables leaving alone their marginal distri‑ butions. Due to Sklar [25] F (x1 , ..., xn ) = C [F1 (x1 ), ..., Fn (xn )] ,

where F is joint cumulative distribution function of random vector (X1 , ..., Xn ), Fi is marginal cumulative distribution function of Xi , and C : [0, 1]n → [0, 1] is a copula which is a n‑increasing function with 1 as neu‑ tral element and 0 as annihilator, see e.g. monograph Nelsen (2006) [20]. Besides three fundamental copu‑ las M (x1 , ..., xn ) = min(x1 , ..., xn ) W (x1 , x2 ) = max(x1 + x2 − 1, 0) n ∏ Π(x1 , ..., xn ) = xi i=1

which model perfect positive dependence, perfect ne‑ gative dependence (not applicable for n > 2) and in‑ dependence, respectively, there exist numerous para‑ metric classes, such as Archimedean, Extreme‑Value and elliptical copulas. Within the last one there belong such important parametric families as Gaussian copu‑ las [ ] −1 CG (x1 , ..., xn ) = Φ Φ−1 1 (x1 ), ..., Φn (xn )

and Student t‑copulas

[ ] −1 Ct (x1 , ..., xn ) = t t−1 1 (x1 ), ..., tn (xn ) ,

(where Φ and t are joint distribution functions of mul‑ tivariate normal and Student t distributions, similarly Φ−1 and t−1 i i , i = 1, ..., n are univariate quantile functions related to Xi ), able to �lexibly describe de‑ pendence in multidimensional random vector.

Journal of Journal of Automation, Automation,Mobile MobileRobotics Roboticsand andIntelligent IntelligentSystems Systems

2.1. Hierarchical Archimedean Copulas Archimedean copulas are easy to handle, however in more than two dimensions their dependence struc‑ ture is too simplistic. Nevertheless, they are used as building blocks in other, more �lexible classes of copu‑ las. One such class, that is suitable for modeling mul‑ tidimensional stochastic dependence, are hierarchical Archimedean copulas. Let us recall fundamentals. Ar‑ chimedean copulas are de�ined (for any dimension n) by formula C(x1 , . . . , xn ) = ϕ(ϕ

−1

(x1 ) + . . . + ϕ

−1

(xn ))

where the so‑called generator ϕ : [0, ∞) ↘ [0, 1] satis‑ �ies boundary conditions ϕ(0) = 1, ϕ(∞) = 0 (strict Archimedean copulas) and absolute monotonicity (for further details see [17]). Such a construction is analy‑ tically convenient and very �lexible in bivariate setting, however it is too restrictive in higher dimensions since the whole dependence structure is rendered by a sin‑ gle univariate function, and – moreover – it is exchan‑ geable. Hierarchical Archimedean copulas (HAC) over‑ come this problem by nesting simple Archimedean co‑ pulas. Since the general multivariate structure is nota‑ tionally too complex, we illustrate the principle in four dimensions. For example, fully nested HAC (Fig. 1, left) can be given by ( ( ) ) C(s) (x1 , . . . , x4 ) = C3 C2 C1 (x1 , x2 ), x3 , x4 = ( ( ) −1 −1 −1 = ϕ3 ϕ−1 3 ◦ ϕ2 ϕ2 ◦ ϕ1 (ϕ1 (x1 ) + ϕ1 (x2 ) + ) ) −1 ϕ−1 (u ) + ϕ (u ) , (1) 3 4 2 1

where Cj , j = 1, . . . , n − 1 are Archimedean copu‑ las with their corresponding generators ϕj and s = (((1, 2), 3), 4) the nesting structure. An example of partially nested Archimedean copula (Fig. 1, right) is given by ) ( C(s) (x1 , . . . , x4 ) = C3 C1 (x1 , x2 ), C2 (x3 , x4 ) , (2)

where s = ((1, 2), (3, 4)). Fully and partially nested Archimedean copulas form a class of hierarchical Ar‑ chimedean copulas which can adopt arbitrarily com‑ plex structure s, generally s = (. . . , (ia , ib ), ic , . . .), where i· ∈ {1, . . . , n} is reordering of the indices of variables with a, b, c ∈ {1, . . . , n | a ̸= b ̸= c}, see, e.g., [9, 12, 22]. This makes it a very �lexible yet par‑ simonious distribution model. The generators within a single HAC can come either from a single genera‑ tor family or from different families. In the �irst case there is required complete monotonicity of composi‑ tion ϕ−1 i ◦ ϕj , (i ̸= j), which imposes some constraints on their parameters, see suf�icient conditions given by [16]. For majority of generators HAC requires decrea‑ sing parameters from top to bottom in its hierarchy. In the case of different generator families, the condi‑ tion of complete monotonicity is not always ful�illed. The software implementation in R, the HAC package [22] which we use in our study, considers only single‑ parameter generators from the same family. Then the

VOLUME 2019 VOLUME 13,13, N°N°3 3 2019

Fig. 1. Fully nested and partially nested Archimedean copulas structure. whole distribution is speci�ied with at most n − 1 pa‑ rameters which can be seen as an alternative to co‑ variance driven models, as remarked in [22], nevert‑ heless, besides the parameters also structure s needs to be estimated. As there are already n!/k! possibili‑ ties of combining n variables to fully nested HAC with k‑dimensional AC on its lowest level, the greedy ap‑ proach to structure estimation would be unreasonable even in moderate dimensions, therefore HAC package offers computationally ef�icient recursive procedure suggested by [21] 2.2. Vine Copulas Another class that can use bivariate Archimedean copulas as building blocks are Vine copulas. Howe‑ ver, they are not restricted to that class and can com‑ bine copulas of arbitrary kind via a vine tree structure, which can be estimated (by default following the cor‑ relation strength ordering), visualized and interpre‑ ted, see [1, 4, 24]. Formally, an n–dimensional regular vine tree structure S = {T1 , ..., Tn } is a sequence of n−1 linked trees with properties (see [3, 4]): ‑ Tree T1 is a tree on nodes 1 to n; ‑ Tree Tj has n + 1 − j nodes and n − j edges;

‑ Edges in tree Tj become nodes in tree Tj+1 ;

‑ Two nodes in tree Tj+1 can be joined by an edge only if the corresponding edges in tree Tj share a node. In the following, we outline the construction of three‑dimensional probability density function f

f (x1 , x2 , x3 ) = f1 (x1 )·f2|1 (x1 , x2 )·f3|12 (x1 , x2 , x3 ) = = f1 (x1 ) · c12 [F1 (x1 ), F2 (x2 )] · f2 (x2 )· [ ] · c31|2 Fx3 |x2 (x2 , x3 ), Fx1 |x2 (x1 , x2 ) ·

· c23 [F2 (x2 ), F3 (x3 )] · f3 (x3 )

(3)

where fi is a (marginal) probability density function of Xi , i = 1, 2, 3, fi|j (xi , xj ) =

f (xi , xj ) fj (xj )

is conditional density function of Xi given Xj . A co‑ pula density cij couples Xi and Xj while cij|k cou‑ ples bivariate conditional distributions of Xi |Xk and Xj |Xk , i, j, k ∈ {1, 2, 3} , i ̸= j ̸= k ̸= i. Finally, Fxi |xj =

∂Cij [Fi (xi ), Fj (xj )] ∂Fj (xj )

is a conditional cumulative distribution function of Xi given Xj . Articles

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VOLUME N°33 2019 2019 VOLUME 13,13, N°

Fig. 4. 1‐factor copula graphical model of dependence corresponding to construction (4)

Fig. 2. Vine trees corresponding to (3)

Fig. 3. C‐Vine tree (left) and D‐Vine tree (right) The construction (3) represented by two vine trees shown in Fig. 2 is one of the three possible pair–copula decompositions. Since n = 3, its vine structure coinci‑ des both with ‑ canonical (C–) vines: each tree has a unique node connected to n − j edges (use only star like tree ‑ useful for ordering by importance); and

‑ drawable (D–) vines: no node is connected to more than 2 edges (use only path like trees ‑ useful for temporal ordering of variables) and these differ for n ≥ 4 as illustrated by Fig. 3, see [5]. However in higher dimensions, C–vines and D–vines are just small subsets of a more general class ‑ regular vines, see [4, 3]. Besides the well‑known 2–dimensional product copula and elliptical copulas (Gaussian and Student), as construction blocks of vine copula we utilized also numerous 2–dimensional families of Archime‑ dean and Extreme‑value copulas, as well as their ro‑ tations, described below in the section Methods. 2.3. Factor Copulas

Yet another subclass within pair‑copula con‑ struction approach is getting considerable attention: factor copulas. According to [13] factor copula models are conditional independence models where observed variables (U1 , . . . , Un ) are conditionally independent given one or more latent variables (V1 , . . . , Vp ). These models extend the multivariate Gaussian model with factor correlation structure. They can be also viewed as p‑truncated C‑vine copulas rooted at the latent va‑ riables, one just needs to integrate out latent variables in the joint copula density to get density of observa‑ bles. The most popular are 1‑factor copulas de�ined as C(u1 , . . . , un ) =

with the density

c(u1 , . . . , un ) =

Articles

∫

n 1∏

0 i=1

∫

n 1∏

0 i=1

Ci|V1 (ui , v1 )dv1

ciV1 (ui , v1 )dv1

(4)

thus the dependence among Ui s is induced by the so‑ called linking copulas CiV1 , i = 1, . . . , n, and there are no constraints among the bivariate copulas. The de‑ pendence structure of n‑dimensional 1‑facor copula is graphically illustrated in Fig. 4. It is interesting to note, that the Archimedean co‑ pulas generated by universal generators (those ba‑ sed on Laplace transform, see e.g. [2] for examples) are special case of 1‑factor copulas, with exceptionally simple form. A main advantage of factor copula mo‑ dels comparing to Archimedean and Gaussian copulas is that it allows for asymmetric dependence structure (both re�lection asymmetry and non‑exchangeability) among observables. Later we will see that they are �lexible enough to compete with more complex class of vine copulas while keeping relative parsimony and interpretability. The main drawback nowadays, howe‑ ver, is the lack of software implementation. Commer‑ cial programs are rather conservative in bringing new statistical methods and from the open source tools, only in R, the most popular environment for statistical calculations and visualizations [23], we found single package related to factor copula: FDGcopulas. In this package a Durante class of bivariate copulas de�ined by C(u, v) = min(u, v)f (max(u, v)) ,

are used as linking copulas, where the generator f : [0, 1] → [0, 1] is differentiable and increasing function such that f (1) = 1 and t → f (t)/t is decre‑ asing. There may be chosen four different parametric families, such as Cuadras‑Augé f (t) = t1−θ , θ ∈ [0, 1], Fré chet f (t) = (1 − θ)t ) θ ∈ [0, 1], Durante‑ ( θ+ θ, t −1 , θ > 0, and Durante‑ exponential f (t) = exp θ

sin(θt sinus f (t) = sin(θ) with parameter θ ∈ (0, π/2], ple‑ ase refer to [15] for �iner details. The downside of this class is a singular component present in the model, which is not natural for most economic, hydrologic or other frequently analyzed phenomenons. However, as the authors argue, it is not of that much importance in higher dimensions, where just certain features of dis‑ tribution is preferred (such as critical levels or tail be‑ havior) instead of its overall shape. On the other hand, the class of 1‑factor copulas with Durante generators reduce the computational burden of general 1‑factor copulas while giving a good �it to observed data, as we will see in the results.

2.4. Methods

Within the considered classes of 2–dimensional copulas as well as n‑dimensional elliptical copulas, the optimal models were selected using the Maximum li‑

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VOLUME 2019 VOLUME 13,13, N°N°3 3 2019

kelihood estimation (MLE) method. Recall that for gi‑ ven m observations {Xj,i }i=1,...,m of j‑th random va‑ riable Xj , j = 1, ..., n, the parameters θ of all co‑ pulas under consideration were estimated by maximi‑ zing the likelihood function L(θ) =

m ∑ i=1

log [cθ (U1,i , . . . , Un,i )] ,

(5)

where cθ denotes density of a parametric copula fa‑ mily Cθ , and m

Uj,i =

1 ∑ 1(Xj,k ≤ Xj,i ), m+1

i = 1, ..., m,

k=1

are so‑called pseudo‑observations. The higher dimen‑ sional structures of HAC, vine and factor copulas were estimated as described in [22], [6] and [15], respecti‑ vely. Goodness‑of‑�it was performed by a test proposed by Genest et al. [8] and based on empirical copula pro‑ cess using Cramer‑von Misses test statistic SCM =

m ∑ i=1

[Cθ (U1,i , . . . , Un,i ) − Cm (U1,i , . . . , Un,i )]

with empirical copula

m n 1 ∑∏ Cm (x) = 1(Xj,i ≤ xj ) m i=1 j=1

(6)

and indicator function 1(A) = 1 whenever A is true, otherwise 1(A) = 0. All calculations were done in R [23] with the speci‑ �ic help of packages copula [10], HAC [22], VineCopula [19], and FDGcopulas [14]. Because their goodness‑of‑ �it methods, including the Cramer‑von Misses metric, are not directly comparable and computing the values of vine copula cumulative distribution function invol‑ ves integration over 7‑dimensional space, we rather approximated Cθ from (6) by empirical copula of the random samples generated from the corresponding copulas each counting 100 000 realizations. Besides the usual parametric families of Archime‑ dean class such as Gumbel, Clayton, Frank, Joe, copu‑ las BB1, BB6, BB7, BB8 and Tawn copulas (see e.g. [11, 18, 20, 26]) in bivariate case, to build vine copu‑ las we used also their rotations Cα by angle α de�ined C90 (x1 , x2 ) = x2 − C(1 − x1 , x2 ), C180 (x1 , x2 ) = x1 + x2 − 1 + C(1 − x1 , 1 − x2 ),

C270 (x1 , x2 ) = x1 − C(x1 , 1 − x2 ),

that are implemented in the package VineCopula. As a preliminary analysis of dependence between random variables, we employ classical measures of dependence such as Pearson’s and Kendall’s correla‑ tion coef�icients, moreover to inspect the conditional (in)dependence (which is further investigated para‑ metrically with vines) the partial correlation matrix comes handy. Given a Pearson’s correlation matrix Σ,

the partial correlation between variables Xi , Xj con‑ ditional on all the other pairs in vector (X1 , . . . , Xn ) can be computed −pij ρij|−ij = √ pii pjj

where pij (i, j = 1, . . . , n) are elements of the matrix P = Σ−1 . Recall that partial correlation is a measure of the strength and direction of a linear relationship between two continuous random variables that takes into account (removes) the in�luence of some other continuous random variables. Partial correlations are important, e.g., a) when building (Gaussian) graphical models, where insigni‑ �icant connections are removed to obtain more parsi‑ monious model, as well as b) to better understand the structure of estimated vine copula. The R source script used for calculations can be obtained from the corresponding author upon request or on his web page.

3. Results

All indexes are computed in terms of returns

returni =

indexi − indexi−1 , indexi−1

i = 2, 3, ..., n.

Before further analysis, we �iltered all considered time series of returns by ARIMA‑GARCH �ilters ( [7]). For all investigated series of returns, the best �ilters were identi�ied (by the system Mathematica, �ersion 11) in the class GARCH(1,1). The obtained residuals have pairwise Kendall cor‑ relation coef�icients τ in the interval (−0.08, 0.42), maximal value was achieved for the couples SPX–DAX and SPX–UKX, see Fig. 5. Fig. 6 reveals partial correlations, showing that the relations of (�iltered) returns of SPX–UKX, SPX–DAX, SPX–HSI and DAX–UKX attain the largest values, which is in accordance with corresponding strongest depen‑ dencies between couples in the �irst tree of the optimal global vine copula in Table 1. Subsequently, the residuals were transformed by their respective empirical distribution function into so‑called pseudo‑observations uniformly distributed over unit interval (see diagonal of Fig. 5). Results ser‑ ved as inputs to calculations of 7–dimensional copula models. We extended our analysis by examining evolu‑ tion of the Kendall’s correlations. We have chosen fre‑ quency of calculations of Kendall’s correlation coef‑ �icients over the intervals of 72 months overlapping by 36 months with the neighboring intervals, the last time period spans only 60 months. For each of the cou‑ ples of considered indexes, we calculated a sequence of 7 local Kendall correlation coef�icients on individual local time intervals. We can see (in Fig. 7) that just 6 out of all 21 couples of indexes have Kendall corre‑ lation coef�icients in most of the periods signi�icantly positive. In the following subsections, �irst, an overall de‑ pendence structure is modeled by means of Elliptical, Articles

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VOLUME 13, N° N°33 VOLUME 13,

SPX

0.088

0.27

0.061

0.018

0.36

0.12

0.24

−0.003

−0.0047

0.072

0.23

0.035

0.001

0.076

0.058

−0.088

0.086

0.03

z2 z2

DAX−HSI UKX−HSI

confidence band less significant

−0.2

0.0

NKY

HSI z1

0.4

Kendall's tau

z2 z2 z2

UKX z1

SPX−HSI DAX−UKX

0.2

DAX

SPX−DAX SPX−UKX

0.8

0.42

0.6

0.42

2019 2019

1995 − 2000

1998 − 2003

2001 − 2006

2004 − 2009

2007 − 2012

2010 − 2015

2013 − 2018

period

−0.081

LEG z1

Fig. 7. Evolution of Kendall’s τ for all couples of the (filtered) returns

SPG

Fig. 5. Pairwise scatter plots with Kendall’s tau (upper triangle), bivariate density contour plots with standard normal margins (lower) and marginal density (diagonal) of pseudo‐observations.

SPX 0.27

0.36

SPG

DAX

−0.18

−0.02

0.37

0.04

−0.04

−0.09

0.04 0.26

0.12

LEG

0.3

−0.01

0.01

0.13

HSI

0.35

−0.01

−0.06

0.19

0.06

UKX

0.08

NKY

Fig. 6. Partial correlation coefficients for the residuals (conditioned on the remaining elements of the considered group of residuals). nested Archimedean, vine and 1‑factor copulas. Then, with the same classes of models we examine depen‑ dence in subsequent periods. 3.1. Global Models of Dependence

The best 7–dimensional vine copula (based on for‑ ward selection of trees and AIC criterion for pair‑ copulas, see [19]) is summarized in Table 1, Fig. 8 and . We observe that at the lowest tree there are modeled stronger links between SPX with the triple UKX, DAX and HSI. It illustrates a very strong international po‑ 88

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Tab. 1. The summary of the best 7–dimensional vine copula (I1=SPX, I2=DAX, I3=UKX, I4=NKY, I5=HSI, I6=LEGATRUU, I7=SPGSCITR) tree 1

edge family par1 par2 I4 ‑ I7 t ‑0.13 6.04 I2 ‑ I4 SC 0.32 0.00 I1 ‑ I2 t 0.61 2.80 I1 ‑ I3 t 0.62 2.75 I5 ‑ I1 t 0.43 3.19 I6 ‑ I5 G 1.09 0.00 2 I2 ‑ I7; I4 t ‑0.01 5.83 I1 ‑ I4; I2 J 1.11 ‑ I3 ‑ I2; I1 t 0.26 3.87 I5 ‑ I3; I1 t 0.12 3.76 I6 ‑ I1; I5 Tawn 17.67 0.01 3 I1 ‑ I7; I2 ‑ I4 SJ 1.08 0.00 I3 ‑ I4; I1 ‑ I2 t 0.07 7.19 I5 ‑ I2; I3 ‑ I1 t 0.09 5.18 I6 ‑ I3; I5 ‑ I1 I ‑ ‑ 4 I3 ‑ I7; I1 ‑ I2 ‑ I4 I ‑ ‑ I5 ‑ I4; I3 ‑ I1 ‑ I2 Tawn90 ‑9.33 0.00 I6 ‑ I2; I5 ‑ I3 ‑ I1 t ‑0.10 6.14 5 I5 ‑ I7; I3 ‑ I1 ‑ I2 ‑ I4 I ‑ ‑ I6 ‑ I4; I5 ‑ I3 ‑ I1 ‑ I2 G 1.06 0.00 6 I6 ‑ I7; I5 ‑ I3 ‑ I1 ‑ I2 ‑ I4 F ‑0.81 0.00 type: R‑vine logLik: 257.52 AIC: ‑455.03 BIC: ‑346.42

τ ‑0.08 0.14 0.42 0.43 0.28 0.08 ‑0.01 0.06 0.17 0.08 0.01 0.05 0.05 0.05 0.00 0.00 0.00 ‑0.06 0.00 0.06 ‑0.09

sition of the US economy. (It is also interesting to re‑ alize that there exist historically strong ties of HSI to the increasingly in�luential Chinese economy.) At the second tree, we can clearly observe a modest depen‑ dence between UKX and DAX, conditioned on SPX. All other elements of the second tree are clearly weaker. Interestingly, at the very last tree, a slight dependence (negative) between LEGATRUU with SPGSCITR, condi‑ tioned on all considered stock indexes can be obser‑ ved. The best HAC is shown in Fig. 9. According to the GoF test statistics, see Tab. 2, the best models for the investigated data are in the class of 1–factor copula with Durante generators followed by Student t–copula and HAC with bivariate Gumbel Archimedean copula. For comparison, the distance of product copula to empirical copula is equal to 0.167. There is no interesting structure to be illustrated about both 1‑factor copulas, Fig. 10 shows scatter‑plot of 300 simulated observations, one may observe a sin‑ gular component present in the FDG copula depen‑ dence model. Although not clearly visible, bivariate

Journal Systems Journalof ofAutomation, Automation,Mobile MobileRobotics Roboticsand andIntelligent Intelligent Systems

Tree 1

Tree 2

τ = 0.08

2,4

τ = 0.02

SPG

τ = 0.12

LEG

6,5

τ = 0.24

1,2

NKY

τ = 0.42

5,1

1,3

τ = 0.44

HSI

DAX

Tree 4

Tree 3

SPX

2019 2019

4,7

VOLUME VOLUME 13,13, N°N°33

1,4 ; 2

5,2 ; 3,1

3,2 ; 1

2,7 ; 4

UKX

6,3 ; 5,1

I 3,4 ; 1,2

Fig. 9. The best global Gumbel HAC copula

5,3 ; 1

Tab. 2. GoF test statistics for all four global multi–dimensional models

1,7 ; 2,4

6,1 ; 5

Tree 6

Tree 5

class elliptical

5,7 ; 3,1,2,4

6,2 ; 5,3,1

5,4 ; 3,1,2

HAC

3,7 ; 1,2,4

Gaussian 0.022 Gumbel 0.019 (Fig. 8) 0.027 Fré chet 0.016

Vine

Factor FDG

6,4 ; 5,3,1,2

6,7 ; 5,3,1,2,4

5,7 ; 3,1,2,4

6,4 ; 5,3,1,2

3,7 ; 1,2,4

5,4 ; 3,1,2

family Student 0.017 Clayton 0.052

Frank 0.026

Joe 0.023

Cuadras‑Auge 0.014

SPG

LEG 6,2 ; 5,3,1

1,7 ; 2,4

3,4 ; 1,2

5,2 ; 3,1

Cuadras−Auge

HSI

6,3 ; 5,1

2,7 ; 4

1,4 ; 2

3,2 ; 1

5,3 ; 1

6,1 ; 5

4,7

2,4

1,2

1,3

5,1

NKY

UKX

DAX 6,5

SPX

Fig. 8. The best global Vine copula: trees with pair copula family indicated on edges (up) and density contour plots (down), see Tab. 1 for a legend margins of 1‑factor copula with generator of Frechet family are radially symmetric (equal tail dependence), those Cuadras‑Auge family generator based have zero lower tail dependence. 3.2. Modeling Evolution of Dependence by Means of Lo‐ cal Models

We continued by searching models for the 7 time intervals described above (for which sequence of Ken‑ dall�s correlation coef�icient was calculated). A best

Frechet

Fig. 10. Simulated observations from bivariate margins of the two best 1‐factor copulas with Durante generators vine copula was identi�ied (tree structure) for each in‑ terval but estimated (in the same structure) also for all the other intervals. This way we got the selection of 7 different, locally best �itting vine copula structures and their corresponding sequences of estimated vine co‑ pulas. Through almost whole considered time period, the best vine copula was that one from period 2007‑ 2012 (V5). Similarly we estimated a sequence of 7 el‑ liptic, HAC and factor copulas. Among the elliptic co‑ Articles

Journal Journal of of Automation, Automation,Mobile MobileRobotics Roboticsand andIntelligent IntelligentSystems Systems

pulas, t‑copula was mostly better (except for the last 2 subintervals). We have selected HAC from the clas‑ ses Gumbel, Clayton, Frank, Joe and Ali–Mikhail–Haq. Throughout all considered time intervals, the best mo‑ del among them was Gumbel HAC (H3, H4) with the same hierarchical structure as the global HAC but with DAX and SPX swaped. Factor copulas performs simi‑ larly except for the �irst period, when Cuadras‑Auge family �itted better. The corresponding GoF test sta‑ tistic (for the best copulas in each class) is displayed in Fig. 11 and it shows slightly superior performance of hierarchical Archimedean copula over elliptical, fac‑ tor and vine copulas throughout the whole analyzed period (except for the �irst 3 subintervals). Here come two interesting observations. First, the breath‑taking performance of HAC considering its par‑ simony: for illustration take now only bivariate co‑ pulas used in vines, factor and HAC copulas, then number of parameters needed for construction of n‑ dimensional normal copula are n(n − 1)/2, vine copu‑ las n + (n − 1) + . . . + 1, 1‑factor copulas n and HAC copulas only n − 1. It is true that when (conditional) independence takes place in the random vector, vine copula gets signi�icantly reduced, however in our par‑ ticular case as for global copula 20 parameters are in‑ volved comparing to 6 of HAC, and as for the optimal evolving copula the vine structure contains 8 parame‑ ters. Second, unlike elliptical copulas, the best HAC, vine and factor copulas reveal some asymmetry with re‑ spect to tail behavior, and while vines are better for directly displaying conditional relationships, hierar‑ chical Archimedean copulas shows clusters of random variables in somewhat clearer way.

4. Conclusion and Future Work

Modeling dependencies between international �i‑ nancial market indexes is interesting and important for investors, risk managers and policy makers. Appli‑ cation of more dimensional copulas is bringing a new insight and experience for modeling activities. Interestingly, while global optimal model (for the whole period of 277 months) belong to the Factor FDG copulas class, the optimal local models can be found (with very minor differences in the values of GoF test statistic) in the class of HAC. The dominance of this model is most striking over the interval of the �inan‑ cial market crisis, where the quality of the best Student class model was providing a substantially poorer �it.

ACKNOWLEDGEMENTS

This work was supported by APVV‑14‑0013.

AUTHORS

Tomáš Bacigál∗ – Slovak University of Technology in Bratislava, Radlinské ho 11, Bratislava, Slovak republic, e‑mail: tomas.bacigal@stuba.sk. Magdaléna Komorníková – Slovak Univer‑ sity of Technology in Bratislava, Radlinské ho 90

Articles

VOLUME N°33 2019 2019 VOLUME 13,13, N°

11, Bratislava, Slovak republic, e‑mail: magda‑ lena.komornikova@stuba.sk. Jozef Komorník – Faculty of Management, Co‑ menius University, Odbojá rov 10, P.O.BOX 95, 820 05 Bratislava, Slovak republic, e‑mail: Jo‑ zef.Komornik@fm.uniba.sk. ∗

Corresponding author

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VOLUME 13, N° 3, 2019 www.jamris.org (ONLINE)

Journal of Automation, Mobile Robotics and Intelligent Systems pISSN 1897-8649 (PRINT) / eISSN 2080-2145

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