Journal of Automation, Mobile Robotics and Intelligent Systems, vol. 15, no. 3 (2021)

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Articles

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Journal of Automation, Mobile Robotics and Intelligent Systems Volume 15, N° 3, 2021 DOI: 10.14313/JAMRIS/3-2021

Contents 3

System Identification and Heuristic Control of Segmented Ailerons for Enhanced Stability of Fixed Wing UAVs Abdul Sattar, Liuping Wang, Abdulghani Mohamed, Alex Fisher DOI: 10.14313/JAMRIS/3-2021/14 15

An Approach Towards Parametric Optimisation of Construction Frames for Cartesian Industrial Robots Filip Gwardecki, Piotr Falkowski DOI: 10.14313/JAMRIS/3-2021/15 29

Software for the Control and Monitoring of Work of a Collaborative Robot Wojciech Łabuński, Andrzej Burghardt DOI: 10.14313/JAMRIS/3-2021/16 37

FDA*: A Focused Single‐Query Grid-Based Path Planning Algorithm Mouad Boumediene, Lamine Mehennaoui, Abderazzak Lachouri DOI: 10.14313/JAMRIS/3-2021/17

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Articles

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Risk Analysis Method by the Extreme Data of Dependent Exogenous Variables Ihor Tereshchenko, Anton Tereshchenko, Nataliya Bilous, Svetlana Shtangey, Zygmunt L. Warsza DOI: 10.14313/JAMRIS/3-2021/18 54

Unified Model of Disturbances Acting Upon Gimbal Seeker in Anti-Tank Guided Missile Radosław Nawrocki DOI: 10.14313/JAMRIS/3-2021/19 70

3D Maps Integration based on Overlapping Regions Matching Michał Drwięga DOI: 10.14313/JAMRIS/3-2021/20 81

Structurally R-Controllable and Structurally R-Observable Descriptor Linear Electrical Circuits Tadeusz Kaczorek, Kamil Borawski DOI: 10.14313/JAMRIS/3-2021/21

and Intelligent Systems Journal of Automation, Automation,Mobile MobileRobotics Robotics and Intelligent Systems

2021 VOLUMEVOLUME 15, 15, N° 3N° 3 2021

SYSTEM IDENTIFICATION AND HEURISTIC CONTROL OF SEGMENTED AILERONS FOR ENHANCED STABILITY OF FIXED WING UAVS Submitted: 7th June 2021; accepted: 10th December 2021

Abdul Sattar, Liuping Wang, Abdulghani Mohamed, Alex Fisher DOI: 10.14313/JAMRIS/3‐2021/14 Abstract: Different from a conventional aircraft, an investigation on system identification and control design has been car‐ ried out on a small fixed‐wing unmanned aerial vehicle (UAV) with segmented ailerons. The multiple aileron se‐ tup is configured as a multi‐input and single‐output sy‐ stem, and each segment is modeled as a control input. Ex‐ periments are conducted in the wind tunnel to determine the frequency responses of the system and the correspon‐ ding transfer functions. Multiple PID controllers are de‐ signed and implemented in a cascaded form for each con‐ trol surface. Furthermore, a heuristic switching control strategy is implemented for the aircraft where the multi‐ ple aileron segments perform as a single aileron pair in a normal flight condition and adapt to multi‐segment con‐ trol when encountering severe turbulence or significant angle reference changes. Experimental results reveal that although each control surface can stabilize the aircraft, the proposed control strategy by combining the multiple actuation surfaces reduces the mean squared errors for the roll angle up to 38 percent in the highly turbulent en‐ vironment providing superior disturbance rejection pro‐ perties.

Keywords: Fixed‐wing UAV, PID Control, Segmented Con‐ trol Surfaces, System Identification

1. Introduction With their small size and portability, small un‑ manned aerial vehicles (UAVs) have proven their ef‑ �icacy in a number of applications, including surveil‑ lance and mapping in complex terrain [1, 2] data gat‑ hering around large infrastructures with modern col‑ lision avoidance strategies [3–5], environmental mo‑ nitoring and telecommunication relaying in urban en‑ vironments [6–8]. While the small size has many ad‑ vantages, there is a distinct drawback that associated with it that is susceptibility and sensitivity to envi‑ ronmental disturbances [9]. In the same regard, the stability of the aircraft is severely degraded, especi‑ ally concerning the disturbances in roll axis [10]. More speci�ically, small aircraft have dif�iculties at maintai‑ ning straight and level �light when encountering se‑ vere turbulence. When a certain turbulence intensity limit is crossed, the aircraft may head to signi�icant de‑ viation from the designated �light path and exhibit at‑ titude �luctuations with disastrous consequences. The performance of these vehicles is often limited by ac‑ tuators’ bandwidth, given that the turbulence band‑ width can quickly saturate the actuator bandwidth

resulting in compromised stability and overall per‑ formance. This might be the case even when imple‑ menting latest sensing techniques [11, 12]. Improving the actuation of �ixed‑wing UAVs with respect to cont‑ rol authority and rapidity constitutes an important re‑ search area, bound to improve the small aircraft’s per‑ formance.

Considering an aircraft’s structure, various de‑ signs have been conceptualized and analyzed to achieve a higher degree of performance. Recent ad‑ vances in �ixed‑wing UAV design include the free wing design in [13], segmented control surfaces [14–16], split aileron wing [17], �lexible wing and wing mor‑ phing [18–22] blended wing body design [23]. Seg‑ mented control surfaces are the focus of this paper, gi‑ ven their capability to upgrade the control response and rapidity of the UAV. Prior work in [14, 15] demon‑ strates the practicality of multiple control surface de‑ sign and discussing various advantages linked to it. For example, segmented surfaces can act as a supplement to pitch and rudder controls, minimize induced drag, and contribute in active wing lift distribution [14].

In 2018, work reported by [24] under the NASA project utilized multiple aileron segments for active in‑�light load redistribution. The method uses optical �iber to sense strain on the wing and then actuating various segments to distribute the load evenly. Simi‑ lar work, in its early stages, has also been reported in patents [25,26] using tabbed and multiple ailerons. In modern bigger aircraft, utilization of multiple ailerons can be common practice to reduce �luttering of the wing and increase passenger comfort [27]. For exam‑ ple, in Airbus 380, multiple ailerons have been incor‑ porated and can be actuated in various ways based on the aircraft’s speed [28]. Although the physical designs have been accomplished, a control system architec‑ ture has not been developed and analyzed for a small �ixed‑wing aircraft having multiple aileron segments.

However, for ordinary aircraft with single single aile‑ ron per side of the main wing, the literature reports various developments and optimization, which inclu‑ des PID controllers [29–31], sliding mode controller [32] [33], model predictive controller (MPC) based in [34, 35], fuzzy control based [36] & back‑stepping ba‑ sed control in [37, 38].

This paper investigates the dynamic model and control system design of a segmented aileron based small �ixed‑wing unmanned aerial vehicle. Since the roll axis is the most sensitive axis to the atmospheric disturbances hence it is the only axis analyzed throug‑ hout this work. This �ixed‑wing UAV is designed by in‑

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corporating multiple aileron segments on each side of the wing. During the control system design, each ai‑ leron pair is considered to be an independently mani‑ pulated variable. Hence, the multi‑segment �ixed‑wing UAV control system is con�igured as multi‑input & sin‑ gle output system for the dynamics of the roll‑axis.

System identi�ication experiments are carried out in the wind tunnel to obtain the frequency response data & the transfer function models. Cascade PID control‑ lers are designed on the basis of the transfer function models for the multi‑segment �ixed‑wing UAV. Con‑ trol strategy is devised to manage the operation of the multiple actuators relative to the tracking errors of the roll‑axis. The remainder of this paper includes Section 2 and 3, which describe the hardware of a multi‑segment �ixed‑wing UAV and the system identi‑ �ication experiments as well as the transfer function models obtained for the two inputs and one output sy‑ stem. Section 4 presents the cascade PID controller de‑ sign using the frequency response data and the cont‑ rol strategy to deal with severe turbulence. In Section 5 the cascade control system for the multi‑segment �ixed‑wing UAV is validated with the control surfa‑ ces from the inner segment and outer segment in a wind tunnel to demonstrate their individual capacity for maintaining closed‑loop stability. Section 6 pro‑ poses the use of an error threshold to activate multi‑ ple control surfaces when encountering a large refe‑ rence change or severe air turbulence and demonstra‑ tes that the proposed strategy has signi�icantly impro‑ ved the closed‑loop performance. Section 7 concludes the research �inding.

2. Multi‐segment Fixed‐wing UAV and Experi‐ mental Environment

This section describes the hardware and software of the multi‑segment �ixed‑wing UAV along with expe‑

VOLUME 15, N° 3 2021 VOLUME 15, N° 3 2021

rimental procedure used to identify roll‑axis dynamic models.

2.1. Aircraft Specifications and Control Hardware Unlike the conventional �ixed‑wing UAV, in the multi‑segment con�iguration, each aileron control sur‑ face is segmented into two, as a result a total of four segments are obtained. A separate high‑speed servo is attached to each aileron segment. Because the servo motors operate independently, the aileron cont‑ rol surfaces are de�ined as δa_o for the actuation using the outer segments and δa_i for the actuation using the inner segments. Figure 1 illustrates the aircraft’s seg‑ mented control surfaces for inner and outer compo‑ nents. A more detailed illustration of the experimental model is given by Figure 2, which shows the exact me‑ asurements of wingspan, size of aileron segments, and distribution of segments alongside the main wing.

A �lat plate type airfoil is selected for this UAV. Table 1 presents the properties of this airfoil. A high speed microcontroller namely, Cortex M4 pro‑ cessor (32‑bit) is deployed to analyze the incoming roll attitude data and prepare proper output signal for each control surface. The roll attitude estimation is accomplished through the combination of digital‑ motion‑processor or DMP documented in [39] and IMU (inertial‑motion‑unit). The dedicated DMP out‑ puts a noise‑free attitude signal without involving the main processor, which saves the main control loop’s execution time. Table 2 presents the details of compo‑ nents utilized to develop the roll attitude control sy‑ stem. Very high speed servos are deployed to achieve swift control system response. Table 3 gives the spe‑ ci�ications of servos. These servos are made up of set of durable metallic gears and can handle the pulse fre‑ quencies up to 333 Hz to make sure the system exhibits minimal delay.

[ ] 1 bp br + C ℓr + Cℓδa δa + Cℓδr δr ṗ = Γ1 pq − Γ2 qr + ρVa2 Sb Cℓ0 + CℓβC βC + Cℓp 2 2Va 2Va [ ] 2 Sc cq ρV q̇ = Γ5 pr − Γ6 (p2 − r2 ) + a + Cmδe δe Cm0 + C mα + C mq 2Iy 2Va ] [ 1 bp br ṙ = Γ7 pq − Γ1 qr + ρVa2 Sb Cn0 + CnβC βC + Cnp + C nr + Cnδa δa + Cnδr δr 2 2Va 2Va Tab. 1. Features of UAV Features Airfoil Leading Edge Wing length (per side) Chord Camber Aileron segment size Cruise speed Fig. 1. The fixed‐wing UAV with multiple aileron segments. Key: (a) inner segments, (b) outer segments 4

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Details Flat plate Ellipsoid 290.0 mm 115 mm 4.0 mm 145 x 45 mm 10.0 m/s

(1)

Journal and Intelligent Systems Journal of of Automation, Automation,Mobile MobileRobotics Robotics and Intelligent Systems

Tab. 2. Parts used in UAV Parts Microcontroller IMU

Servo Data Logger Voltage Regulator

Details MK64FX512VMD12 MPU6050 (built‑in DMP) RJX‑FS0435HV Teensy 3.5 SD‑logger Step‑down DC‑DC con‑ verter

Tab. 3. Features of high speed servo Parts Operating Voltage Speed Pulse frequency Torque Gear Type Weight

2.2. Experimental Setup

Details 4.8 ‑ 7.4 V 0.04 sec/60◦@ 7.4 V 333 Hz 3.4 kg/cm @ 7.4 V All Metal Gear 20 g

All the experiments have been carried out in RMIT’s Aerodynamics Wind Tunnel (AWT), which is con�igured as a closed‑circuit wind tunnel. A 380 kilo‑ watt DC motor precisely controls the air pressure in‑ side the tunnel. The sensors measure the pressure in‑ side the wind tunnel’s hexagonal test section measu‑ ring 1.37 × 1.08 × 2 meters (WxHxL), which is then converted into relevant wind speed. A detailed study of the characteristics and environments of AWT is out‑ lined in [40].

The �irst stage of the experiments is focused on system identi�ication with segmented aileron control surfaces in relatively smooth air�low (turbulence in‑ tensity < 0.1%). A special roll rig, developed and tes‑ ted in [41] is utilized to conduct the experiments. This rig only allows motion along roll axis, prohibiting any coupling from other axis. It helps to speci�ically study and analyze the effect of ailerons since they primarily affect the roll motion. The experimental setup inside AWT test section is shown by Figure 3. The wind speed through all the experiments has been kept to normal �lying speed of the small �ixed wing UAV, which is 10 m/s [9].

3. System Identification for Multi‐segment Fixed‐wing UAV

In order to understand the dynamic model for the multi‑segment �ixed‑wing UAV, we will �irst present the mathematical models for the conventional case.

3.1. Dynamic Model for Single Segment UAV

The dynamic models for a conventional �ixed‑wing with single segment are described by the set of diffe‑ rential equations as given by (1) [42, 43] where p, q and r are the roll, pitch and yaw rates in the body frame, the control manipulated variables are the aile‑ rons, elevator and rudder de�lections, de�ined as vari‑ ables δa , δe and δr . Among the remaining parameters,

2021 VOLUMEVOLUME 15, 15, N° N° 3 3 2021

Cxy is the aerodynamics derivative coef�icients corre‑ sponding to their respective variables, ρ is the air den‑ sity, Va is the airspeed, S is the wing platform area, b is the wingspan of the airframe, c is the mean chord of the wing, and βC is the course angle.

For attitude control of the aircraft, the system out‑ puts are the roll, pitch Euler angles, and yaw angular velocity, de�ined as variables ϕ, θ and r respectively.

The relationships between the body frame angular ra‑ tes and the Euler angular rates are captured by the dif‑ ferential equations (2).    ϕ̇ 1 sin(ϕ)tan(θ)  θ̇  = 0 cos(ϕ) 0 sin(ϕ)sec(θ) ψ̇

  cos(ϕ)tan(θ) p −sin(ϕ)  q  (2) cos(ϕ)sec(θ) r

The control objective is that for the given reference signals ϕ∗ , θ∗ and r∗ , the roll, pitch Euler angles, and yaw rate will follow their respective reference signals and reject air turbulence disturbances and the payload of the �ixed‑wing UAV. It can be seen from (1)‑(2) that the mathematical models for the conventional �ixed‑ wing UAV are nonlinear and contain many unknown physical parameters. The control system design pro‑ blems were tackled more ef�iciently by the direct iden‑ ti�ication of linear models.

3.2. System Identification of Multi‐segment Fixed‐wing UAV

Due of the spatial difference between the aileron control surfaces, the nonlinear model described by (1) requires a rigorous modi�ication to render useful for the multi‑segment �ixed‑wing UAV, which can be a lengthy and cumbersome process. Alternatively, the system is considered to have two input variables i.e. two ailerons δa_i (t) and δa_o (t) and one output vari‑ able roll rate p(t). With two ailerons functioning in‑ dependently, the multi‑segment UAV is regarded as a two‑input and one output system. This paper treats the UAV as an unknown system and attempts to iden‑ tify system dynamics from scratch. To begin with, a relay feedback experiment is performed to determine the important frequency region for the control system design [44]. Afterward, roll dynamics are found in de‑ tail by estimating the frequency responses of the two inputs and one output system with a series of sinusoi‑ dal input signals working as the excitation signals and converting the estimated frequency response points to transfer functions to reveal the dynamics of the sy‑ stem.

The relay with hysteresis experiment is depicted in Figure 4. The aim of the relay experiment is to excite the system around a certain frequency. This frequency ( or period of oscillations) relies on three major vari‑ ables such as the value of hysteresis, relay amplitude and the nature of system’s dynamics [45]. Apart from the system’s dynamics, which are unknown, the user can specify the relay’s hysteresis and amplitude va‑ lues. The value of hysteresis is chosen to be 11 deg/sec, which prevents the false switching of relay in case of measurement noise. An amplitude of 20 deg/sec was Articles

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Fig. 2. The UAV model with dimensions

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Roll Rate Relay out

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Fig. 3. Aircraft inside the test section of Aerodynamics Wind Tunnel

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Fig. 5. Relay experiment: Response of inner segments 30

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Fig. 4. The setup of relay experiment

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selected for the relay, keeping in mind the stable ope‑ ration and maximum working range of aileron seg‑ ments. Figure 5 depicts the response of relay experi‑ ment when only inner segments are active. It must be noticed that with exactly the same characteristics of the relay, roll rate response of the outer segments is different as shown by Figure 6 from the response of inner segments as shown by Figure 5. This is a good indicator of non‑linearity between inner and outer ai‑ leron segments.

With the relay experiments, the cross‑over fre‑ quency regions for both inner and outer segments are identi�ied, which are used in the selections of sinus‑ oidal testing signals in order to reveal the entire fre‑ quency response of the system. The sinusoidal excita‑ 6

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Fig. 6. Relay experiment: Response of outer segments tion signal u(t) has the general form: u(t) = 20 sin ωk t

where the frequency ωk varies from experiment to

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experiment, sweeping through higher and lower fre‑ quency regions based on the cross‑over frequency identi�ied by the relay experiments. Figure 7 shows the frequency regions for which sinusoidal signals are used as the excitation inputs for the aileron control surfaces δa_i and δa_o . The minimum frequencies are determined when the measured roll rate p(t) can not produce sustained sinusoidal responses and the max‑ imum frequencies are determined when the measu‑ red roll rate p(t) is buried in the measurement noise.

From the input and output sinusoidal testing signals, Fourier analysis is used to estimate the frequency re‑ sponse Gi (jωk ) and Go (jωk ) such that Y (jωk ) Gi (jωk ) = Ui (jωk )

where Y (jωk ) is Fourier transform of the measured roll rate signal at the frequency ωk and Ui (jωk ) is the Fourier transform of the inner segment control sur‑ face signal. The same procedure is applied to the outer segment testing data to obtain the frequency response estimates.

There are 25 sinusoidal experiments conducted for each inner and outer aileron control surface to cover the entire frequency region. Figure 8 compares these estimated frequency points for the inner and outer ai‑ leron segments. It clearly shows the existence of non‑ linearity, as the frequency responses of the two sys‑ tems are very different.

3.3. Estimation of Transfer Functions The frequency responses obtained from frequency injection experiments are utilized to estimate transfer functions for inner and outer segments. The following structure of transfer function is assumed:

N (s) b1 sm + b2 sm−1 + b3 sm−2 + · · · + bm+1 = M (s) a1 sn + a2 sn−1 + a3 sn−2 + · · · + an+1 (3) where m and n are the orders for the numerator and denominator of the transfer function and the parame‑ ters ai (i = 1, . . . , n + 1)and bi (i = 1, . . . , m + 1) are to be estimated from the frequency response data.

MATLAB function ‘invfreqs.m’ is used to convert the estimated phase and magnitude data to transfer G(s) =

Fig. 8. Comparison of frequency responses between inner and outer control surfaces functions. A detailed description of transfer function estimation algorithm is given in Levy [46]. The propo‑ sed algorithm solves system of linear equations deve‑ loped by minimizing the objective function (4) for the coef�icients ai (i = 1, . . . , n+1) & bi (i = 1, . . . , m+1) as follows, J = min{bi , ai }

M ∑

k=1

wg(k)|h(k)A(w(k)) − B(w(k))|2

(4) where the wg(k) denotes the set of weights used to diminish the effect of high frequency components and M denotes the total number of frequency points.

The estimation is further re�ined by deploying dam‑ ped Gauss‑Newton method for iterative search as gi‑ ven in [47]. The aforementioned methodology assists in minimizing the SSE (sum of squared errors) i.e. difference between desired and the actual response data acquired via weighted optimization. The transfer function for the inner segment is determined as a third order system with the following form: −74.15s + 8892 (5) s3 + 59.11s2 + 1599s + 7936 and the transfer function for the outer segment deter‑ mined as a fourth order model has the following struc‑ ture: G(s) =

4.746s3 − 392.5s2 + 2.443 × 104 s + 2.064 × 105 + 80.26s3 + 3026s2 + 2.829 × 104 s + 1.21 × 105 (6) The diagrams in Figure 9 compare the actual fre‑ quency response of segments to that of estimated transfer functions. It can also be deduced that the ou‑ ter segment has higher gain than inner ones as shown by in Figure 10. An additional phase lag can also be noticed at higher frequencies for outer segments in the given Bode plot. This is another indicator of higher gain of outer segments in terms of roll moment when compared to inner segments. It can be seen that the outer ailerons are moving too fast for UAV’s roll mo‑ tion to react to. Moreover, the given response can also be used to prevent over‑actuation of the servos.

G(s) =

s4

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VOLUME N° 33 VOLUME 15,15, N°

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Tab. 4. PID parameters for inner segments 2

Transfer Function Response of inner segments Experimental freq points of inner segments Transfer Function Response of outer segments Experimental freq points of outer segments

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Fig. 9. Transfer function frequency responses vs. experimental data

see [45, 48]. Assuming that G(jω) is the frequency re‑ sponse of the transfer function model, the basic idea in the PID controller design is to use two frequency response points G(jω1 ) and G(jω2 ) for the controller design, where ω1 is chosen to be the frequency when the frequency response G(jω) across the imaginary axis �irst time (− π2 ), and ω2 is chosen to be the cross‑ over frequency (−π). Then the PID controller para‑ meters are calculated through linear curve �itting in the frequency domain [49]. �ore speci�ically, a PID controlled system will exhibit open loop frequency re‑ sponse at frequency ω1 as, L(jω1 ) =

and ω2 as,

L(jω2 ) =

c2 (jω1 )2 + c1 (jω1 ) + c0 (jω1 ) G(jω1 ) (7) jω1 c2 (jω2 )2 + c1 (jω2 ) + c0 (jω2 ) G(jω2 ) (8) jω2

By equating the actual open‑loop frequency response to the desired open‑loop frequency response Ld (jω) at the two frequency points: L(jω1 ) L(jω2 )

Fig. 10. Bode plot of inner and outer control surfaces

4. Control System Design Figure 11 shows the cascaded closed‑loop control system con�iguration. The two manipulated variables are the inputs to the inner segments δa_i and outer seg‑ ments δa_o . For the secondary control system, the out‑ put is the roll angle rate p in body reference frame and for the primary control system, the output is the Eu‑ ler angle ϕ. Because the coef�icients for the mathema‑ tical model (2) are given, the primary system transfer function is simply an integrator obtained from linea‑ rization of nonlinear model at the zero angle, together with an estimated delay from the inner‑loop dynamics.

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To design PID controllers for higher order transfer functions given by the equations (5) and (6), model or‑ der reduction techniques may be used to reduce the higher order transfer functions to second order trans‑ fer functions so that model based PID controllers can be designed. Instead, a simpler approach is to directly use two frequency points for the PID controller design 8

Articles

=

Ld (jω1 )

=

Ld (jω2 )

(9)

(10)

the PID controller parameters c0 ,c1 and c2 are calcula‑ ted. These parameters are then converted into the pro‑ portional gain Kc , integral time constant τI and deri‑ vative time constant τD using the following relations: Kc = c1 ; τI =

c1 c2 ; τD = . c0 c1

The PID controller design method, mentioned above, is used to calculate parameters for all the controllers in the cascade control structure. Tables 4 and 5 present the proportional controller gain, integral and deriva‑ tive time constant. It is seen from these two tables that there are large differences between the inner‑loop PID controllers. However, small differences are noticed be‑ tween the outer‑loop PID controllers. This is due to the dynamics from the outer‑loop system, which is being dominated by integrator with time delay obtai‑ ned from the linearization of the nonlinear model (2) at small angles of ϕ and θ.

5. Experimental Validations for Independent Actuation Experimental validation is performed in two diffe‑ rent environments in the wind tunnel. In the �irst envi‑ ronment, laminar air�low is used to mimic the normal

Journal of Automation, Mobile Robotics and Intelligent Systems Journal of Automation, Mobile Robotics and Intelligent Systems

VOLUME 15, N° 3 2021 VOLUME 15, N° 3 2021

Fig. 11. Cascade control system for the UAV with multiple aileron segments Tab. 5. PID parameters for outer segments 100

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Fig. 12. Inner segments response. In the laminar flow (a) shows roll angle and (b) shows roll rate while in turbulent flow (c) shows roll angle and (d) shows roll rate wind conditions for the multi‑segment aircraft. In con‑ trast, the second environment uses turbulent air�low to capture the situation where the multi‑segment air‑ craft faces severe challenges such as, in a storm. The cascade control structure is utilized in which all the controllers are PID, receiving both roll rate and angle feedback signals from IMU. 5.1. Inner Segment Control Performance

The �irst set of experimental evaluations will focus on the control performance of the inner segments of the wings. When performing this set of assessments, the high‑speed servo motors for the outer segments are disabled. Therefore, no electrical energy is consu‑ med by outer segments.

The reference signal to the roll angle is chosen to mimic the pilot who is transmitting a challenging refe‑ rence signal.

A challenging roll reference signal is selected which mimics the pilot operating in a harsh environ‑ ment, sweeping through both positive and negative roll reference changes. It starts from 0 degree, chan‑ ges to −30 degree, and then to +30◦ . This incorpora‑ tes an amplitude change of 60 degree, which puts the �light control system to a hard test. Figure 12a shows the roll angle step response using the inner set of seg‑ ments and Figure 12b shows the roll rate response in the cascade control system. It is seen that there is an overshoot in the roll angle step response, and the sett‑ ling time is about 5 seconds. Both closed‑loop respon‑ ses demonstrate that the inner set of segments has the capacity to become an independent actuator for the ai‑ rcraft when the roll angle takes step changes. The ex‑ perimental evaluation is repeated in a turbulent envi‑ ronment. Figure 12c shows the step response of the roll angle and Figure 12d shows the roll rate response in presence of external disturbance i.e. turbulence. In comparison to the experimental results from the la‑ Articles

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Journal of Automation, Mobile Robotics and Intelligent Systems Journal of Automation, Mobile Robotics and Intelligent Systems

minar �low environment, the roll angle response is no longer smooth. However, it is able to maintain closed‑ loop stability.

5.2. Outer Segment Control Performance

The experimental evaluations are performed for the outer aileron segments in identical experiments.

Figures 13a and 13b show the closed‑loop step re‑ sponses of the roll angle and roll rate in the laminar �low environment. The results are identical to those obtained using the inner segment actuation. When the aircraft encounters the turbulent �low, the outer seg‑ ment control actuation can also overcome the turbu‑ lence and maintain the closed‑loop stability as shown in Figure 13c for the closed‑loop step response of the roll angle and in Figure 13d for the closed‑loop re‑ sponse of roll rate.

6. A Heuristic Approach Towards Turbulence Mitigation Using Combined Actuation

The adverse effect of a turbulent environment is widely known on a normal �ixed‑wing UAV. �onse‑ quently, to handle severe turbulence increasing aile‑ ron size would require bigger actuators compromising on its response time and energy consumption. In ad‑ dition, resulting challenges such as the need for high‑ speed computational devices and sensors will have to be addressed. To address this problem, a heuris‑ tic switching strategy of aileron segments has been exercised in this aircraft. In order to save the battery energy, at a routine operation, only one of the seg‑ ments is recommended to be active for reference fol‑ lowing and disturbance rejection. The second pair of control surfaces is used as a redundant actuator in the case of actuator faulty or is used to improve reference following and overcome severe air turbulence.

The control strategy to determine when the redun‑ dant pair of control surfaces should be switched on is based on the amplitude of the feedback error between the desired roll angle ϕ∗ (t) and the measured roll an‑ gle ϕ(t). With this in mind, the control signal becomes δa (t) = δa_o (t) + λδa_i (t)

where the outer segment control surface is chosen to be the main actuator. For a pre‑de�ined threshold ϵ, if the error |ϕ∗ (t) − ϕ(t)| ≤ ϵ

then λ = 0, leading to the inner segment control sur‑ face to become inactive. On the other hand, if the error |ϕ∗ (t) − ϕ(t)| > ϵ

10

10

then λ = 1, resulting in the inner segment control sur‑ face to become active. Figure 14 shows the control sy‑ stem con�iguration for the multi‑segment �ixed‑wing aircraft.

This new control system con�iguration is evalua‑ ted for both reference following and turbulence mi‑ tigation in the wind tunnel. For comparison purpo‑ ses, the experimental conditions are identical to those Articles

VOLUME 15, N° 3 2021 VOLUME 15, N° 3 2021

Tab. 6. Performance comparison using mean squared error (MSE) Ailerons con�iguration MSE in laminar MSE in turbu‑ �low lent �low Inner segments only 82.8340 111.2609 Outer segments only 94.8809 110.9734 Both inner & outer 55.4334 68.5341 segments

used in Section 5. The threshold ϵ for all the experi‑ mental evaluation presented in this paper is chosen to be 5 degrees. This means that if |ϕ∗ (t) − ϕ(t)| ≤ 5, then the inner segment control surface is inactive and if |ϕ∗ (t) − ϕ(t)| > 5, then both inner and outer seg‑ ments will become active.

Figures 15a and 15b show the roll angle response and roll rate response for the step reference signal du‑ ring the laminar air�low. In comparison to the respon‑ ses from the single control surface demonstrated in Section 5, it is seen that the roll angle follows the re‑ ference signal more quickly without a large overshoot.

Figure 16 presents servos reaction in degrees ver‑ sus roll angle error. It can be seen that whenever er‑ ror crosses 5 degrees threshold, inner servos are acti‑ vated otherwise they remain inactive. Simultaneously, outer servos are working all the time to correct for all roll angle errors. �uring the turbulent air�low, the advantage of the multi‑segment control surfaces be‑ come more apparent. Figures 15c and 15d show the substantial improvement in roll attitude stabilization.

It is seen that not only the overshoots are eliminated but the effect of the turbulence on the closed‑loop an‑ gle response is also decreased, making it hardly noti‑ ceable in real �light.

To further quantify the improvement of the closed‑ loop performance, the mean squared error is calcula‑ ted, which is de�ined as E=

M −1 1 ∑ ∗ (ϕ (ti ) − ϕ(ti ))2 M i=0

where M is the number of samples. Table 6 shows the mean squared errors for the three different control system con�igurations. It is seen that the closed‑loop performance from the independent actuation using either the inner segment control surfaces or the outer segment control surfaces are comparable. However, with the heuristic switching control, the mean squa‑ red error has reduction of ≈ 42 percent for the lami‑ nar �low and upto 38 percent reduction in the turbu‑ lence air�low. Figure 17 shows servos behavior against measured roll angle error. It can be seen that only at events of reference change, the inner servos are active because error recorded is greater that 5 degrees thres‑ hold.

7. Conclusion

This work investigates the control system design for the small unmanned aerial vehicle (UAV) with mul‑ tiple aileron segments. The system is con�igured as a

Journal of and Intelligent Systems Journal of Automation, Automation,Mobile MobileRobotics Robotics and Intelligent Systems

2021 VOLUMEVOLUME 15, 15, N° 3N° 3 2021

Fig. 14. Cascade control system for the UAV with multiple aileron segments utilizing roll angle deviation

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Fig. 17. Servos reacting to magnitude of error in turbulence

multi‑input and single‑output system, and each seg‑ ment is modeled as a control input. At �irst, the dy‑ namics of the inner and outer segments are identi‑ �ied and then cascaded control systems are designed.

A heuristic switching control strategy is proposed for the multi‑segment �ixed‑wing aircraft. �y selecting a threshold (±5◦ ) for the feedback error, the UAV is operated in a single segment and multi‑segment cont‑ rol surface con�iguration. �xperimental results reveal that each control surface has the capability for the sta‑ bilization of the aircraft. Whereas the combined actu‑ ation signi�icantly improves (up to 38%) the closed‑

loop performance in a turbulent environment. Depen‑ ding upon the intensity of external disturbances, the value of threshold can be easily programmed from 0◦ (all segments active, all the time) to 10◦ (selected seg‑ ments active only to reject severe disturbances). Mo‑ reover, the proposed heuristic switching strategy car‑ ries the innate ability to avoid unnecessary switching of selected actuators, preventing the wastage of limi‑ ted onboard battery energy.

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Journal of Automation, Mobile Robotics and Intelligent Systems Journal of Automation, Mobile Robotics and Intelligent Systems

AUTHORS

Abdul Sattar∗ – RMIT University, Melbourne VIC 3000, e‑mail: abdul.sattar@rmit.edu.au, www: www.rmit.edu.au.

Liuping Wang – RMIT University, Melbourne VIC 3000, e‑mail: liuping.wang@rmit.edu.au, www: www.rmit.edu.au.

Abdulghani Mohamed – RMIT University, Melbourne VIC 3000, e‑mail: abdulghani.mohamed@rmit.edu.au, www: www.rmit.edu.au.

Alex Fisher – RMIT University, Melbourne VIC 3000, e‑mail: alex.�isher@rmit.edu.au, www: www.rmit.edu.au. ∗

Corresponding author

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[48] L. Wang, T. J. D. Barnes, and W. R. Cluett, “New frequency‑domain design method for PID con‑ trollers”, IEE Proceedings ‑ Control Theory and Applications, vol. 142, no. 4, 1995, 265–271, 10.1049/ip‑cta:19951859.

[49] L. Wang, PID control system design and automatic tuning using MATLAB/Simulink, Wiley: Hoboken, NJ Chichester, West Sussex, 2020.

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Journal of and Intelligent Systems Journal of Automation, Automation,Mobile MobileRobotics Robotics and Intelligent Systems

2021 VOLUMEVOLUME 15, 15, N° 3N° 3 2021

AN APPROACH TOWARDS PARAMETRIC OPTIMIZATION OF CONSTRUCTION FRAMES FOR CARTESIAN INDUSTRIAL ROBOTS Submitted: 1st October 2021; accepted: 9th February 2022

Filip Gwardecki, Piotr Falkowski DOI: 10.14313/JAMRIS/3‐2021/15 Abstract: The paper presents an approach to parametric optimi‐ zation with response surface methodology. This process was performed based on the design of a construction frame for a Cartesian industrial robot. The presented installation is dedicated to the real industrial pick‐and‐ place application. Firstly, the case study was described with relevant information about the components invol‐ ved. Then, the finite element model with constraints and loads, as well as the settings of the response surface op‐ timization were discussed. The simulation was presented to the reader within all the stages with necessary details.

Into consideration were taken six methods of creating re‐ sponse surfaces. Influence on the final optimization result and prediction accuracy of each one was presented. In the end, to validate the outcomes of the process, the sta‐ tic structural analysis of the setup was computed. The pa‐ per compares the impact of applying different methods of response surface generation on the results of parametric optimization. Moreover, it indicates the most vulnerable fragments of dynamically loaded elements made of con‐ struction profiles. Its results may be used to select appro‐ priate settings in similar applications, mainly for frame structures.

Keywords: Cartesian robot, FEM, Industry 4.0, Optimiza‐ tion, Response surface

1. Introduction Over the past decades, the engineering design pro‑ cess has changed signi�icantly. Computer simulations and three‑dimensional modelling have been gaining more and more interest since they became commer‑ cially available. In industrial mechanical engineering, one of the most essential processes is structural ana‑ lysis. It enables determining effects such as stresses, strains, and deformations of constructions caused by the loads [1]. The modern approach to solving such a problem is often based on the �inite element met‑ hod (FEM), which approximates a real solution [2] [3].

Considered frame structures may be discretized with one‑dimensional elements with an associated speci�ic cross‑section [4] [5].

Engineers optimize their designs regarding vari‑ ous criteria, minimal mass among others. FEM allows tracking an impact of single structural parameters, but this approach requires many iterations carried out by a designer, to give satisfying results. Distribu‑ tion and maximum value of stress depend not only on the particular cross‑sections but also on the over‑

all geometry of the structure. Moreover, the correla‑ tion between these is sometimes dif�icult to predict.

Therefore, an effective optimization process for fra‑ mes requires prior determination of all the parame‑ ters with a non‑neglectable impact on the criterium.

Afterwards, parametric optimization, a process of mi‑ nimizing de�ined outputs by searching for the corre‑ sponding values of the inputs, may be carried.

It is proposed to use the response surface metho‑ dology (RSM) for optimization, so to replace the origi‑ nal inputs‑outputs correlation model with an approx‑ imated one. This approach reduces computation time and enables the assessment of relationships between input and output parameters. However, at the same time, it reduces the precision of simulations’ results.

Available response surface generation algorithms are compared in terms of their approximation accuracy as well as their impact on the optimization outcomes.

The following paper aims in introducing the rea‑ der to the application of the RSM in structural design. �articularly, as more ef�icient use of materials is a key aspect of sustainable manufacturing and Industry 4.0.

1.1. Related Works

The RSM is widely used for the planning of che‑ mical processes, where the course of reaction may be unknown [6] [7] [8] [9] [10]. However, the applicati‑ ons of this method go beyond cases where the original model is insuf�iciently well known. The substitute mo‑ del can support the design process from an early stage by visualising the in�luence of design variables on key deliverables and ensuring their optimal selection [11].

The use of RSM‑based adaptive models is aimed at im‑ proving accuracy [12] [13]. Whereas these generally require more computational power and are more dif‑ �icult to implement, the basic RSM module is built into ANSYS Workbench 2021.

1.2. Case Description

Industrial frames are often used as support con‑ structions for various mechanisms. In the paper, it is attached to the Cartesian manipulator presented in Fig. 1. It consists of four linear units driven by three stepper motors. Every unit converts torque from a mo‑ tor into the linear force through a belt‑driven mecha‑ nism. Two of them are responsible for the motion al‑ ong with the same axis. This enables operation un‑ der higher loads. However, such an application requi‑ res assuring synchronous movement, realised by con‑ necting the drives through shafts to the common en‑ gine. The robot’s tasks contain transporting objects up to 100 kg from a conveyor to boxes on EUR‑pallets. The

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analysed frame was designed in a Computer‑Aided De‑ sign (CAD) environment.

1.3. FEM and RSM

The static structural analysis of the frame was performed for the loading states resulting from the components’ weights, dynamics of movements and weights of the transported objects. Dynamic reacti‑ ons were traced within the multibody (MBD) analy‑ sis module during simulation operation. Their highest values, occurring while accelerating, braking, and lif‑ ting, were converted into static loads described in the next sections. Afterwards, they were applied as vertex and edge loads. Conducted analyses provide informa‑ tion on stress, strain and deformation distributions in a construction.

To perform optimization, selected dimensions of the frame geometry and cross‑sections of the con‑ struction pro�iles are parametrised. Their various combinations are considered and the optimal solution is searched according to a speci�ic criterion within the set ranges of parameters’ values.

Initially, for RSM, FEM analyses are performed for the design points ‑ sets of parameters with diffe‑ rent values, chosen to �ill the whole range of conside‑ red hyperspace. Thanks to this, an original, complex, unknown relationship between input and output pa‑ rameters may be substituted with approximated hy‑ perplanes [14]. Hence, the analytical gradients may be used for the computations to accelerate them. The estimated function is used in the optimization pro‑ cess providing results for applied objectives and con‑ straints [15]. However, due to the approximation, the accuracy of the result may differ from the optimal.

Fig. 1. Visual of the Cartesian robot model 1.4. Optimization

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[17] [18] [19] [20] [21] [22] [23]. They empower more effective usage of resources; thus cost reduction. In the paper, Multi‑Objective Genetic Algorithm (MOGA) [24] [25] is used within the process of designing a sup‑ porting frame for a cartesian robot shown in Fig. 1. Op‑ timization algorithms are continuously developed for a wide range of applications [26] [27] [28] [29] [30] [31] [32]. In general, they enable systematic search for the candidate points minimizing the loss function.

In RSM, algorithms perform computations based on samples from the model. Alternatively, a direct opti‑ mization that performs calculations on the model it‑ self, may be used. Each sample is obtained by the ex‑ periment, not a prediction in contrast to the RSM al‑ gorithms. Thus, it requires signi�icantly more compu‑ tational power but typically results in more accurate outcomes.

1.5. Considered Methods The optimization process is based on static struc‑ tural stress analysis. It is aimed at �inding the most vulnerable fragments and further mass minimization with constraints involving a safety factor and total de‑ formation. The safety factor is calculated as a ratio of the yield strength to the equivalent stress according to the Huber‑Mises‑Hencky hypothesis. Six methods of constructing response surfaces were taken into con‑ sideration: 1) genetic aggregation 2) full 2nd order polynomials 3) kriging

4) non‑parametric regression 5) neural network

6) sparse grid Genetic aggregation is initialized with several re‑ sponse surface methods with different settings. Every method has its parameters selected through genetic operations such as crossover or a mutation. Additio‑ nally, the weights for them are calculated in an analo‑ gue way; so to get the �inal weighted model [33].

The full second‑order polynomials method ap‑ proximates the real model with quadratic functions. It calculates the surfaces’ parameters so as to minimize an error with respect to the real design points [34].

Kriging method obtains each surface as a combi‑ nation of a global function (usually a quadratic poly‑ nomial) and local functions. The global function is de‑ termined to �it best in its whole domain, while the lo‑ cal functions improve accuracy only in the neighbour‑ hood of few design points and have no effect for more distanced ones [35] [36].

Non‑parametric regression comes to the use of Support Vector Machine (SVM) for regression [37].

The response surface s(x1 , x2 , ...) is estimated in a way that a majority of the design points are within the e space around the surface. This space is limited with two hyperplanes, called margins and described by the minimal ϵ, as presented in Eq. 1. ∀(x1 , x2 , ..., y) ∈ e : s − ϵ ≥ y ≥ s + ϵ

(1)

Journal of Automation, Mobile Robotics and Intelligent Systems Journal of Automation, Mobile Robotics and Intelligent Systems

Neural network estimated output parameters are calculated within a multiple‑layer neural network structure. The method calculates the weights and bi‑ ases so as to minimize the error between estimations and real values [38].

Sparse grid is an adaptive response surface met‑ hod. It self‑corrects by creating design points in the directions expected to contain inaccurate approxima‑ tion. The sparse grid algorithm works well for almost all problems. It is even able to estimate a response sur‑ face with multiple discontinuities [39] [40]. However, this algorithm requires much more design points than other methods, to obtain accurate results (in this study it used around 2.5 times more points). As it requires much more computation time, it is not recommended for simpler applications.

Each of the mentioned above requires the design of the experiment phase (DoE). It is a process of de�ining a set of design points covering the considered ranges of input parameters. A bottleneck of the computations is calculating output parameters for these. Therefore, it is preferable to initially exclude the input parame‑ ters that have a marginal impact on the loss function in their considered ranges. For this purpose, Spearman’s rank correlation matrix is being used, as it contains the linear relationship between the parameters [41]. Va‑ lues close to zero indicate poor correlations and may be excluded from the optimization process.

2. Plan of the Experiment 2.1. Loads

The end effector carries an object from the con‑ veyor to a box and then returns to its base position, over the conveyor; all during a cycle of 7.64 seconds.

Within this time object needs to be gripped, transpor‑ ted and released. While gripping or releasing the ob‑ ject, the end effector should stop for 0.5 seconds. The maximum velocity is 1 m/s for the vertical linear drive and 6 m/s for the other drives. A motion along the path was planned according to all these conditions, and then, internal loads of the frame were computed with Autodesk Inventor’s multi‑body dynamic simula‑ tion module. The resultant forces at the three potenti‑ ally most dangerous stages of motion were used after‑ wards for static structural analysis. These include: 1) lifting the object from the conveyor (load state 1);

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2.2. Preparation of the Model A model of the frame in a non‑native format was imported to be embedded in the Ansys SpaceClaim, where the construction pro�iles were replaced by beam elements. The model was parametrised with three variables represented in Fig. 2 and additionally with construction pro�ile cross‑sections’ properties: thickness t and size a. �nly the pro�ile, connected with the engine and shafts, has been constrained with spe‑ ci�ic dimensions (150x140 mm). All the oblique beams were identically constrained with the same parame‑ ter describing their bases’ distance from the corner of the frame. The angles between the oblique and verti‑ cal beams were taken as to 45 ◦ . The input parame‑ ters were considered within certain scopes being a re‑ sult of available space (e.g. front beam could not be placed too low, as to allow replacement of the storage boxes on pallets) and normalization standards of com‑ mercial parts. As construction pro�iles are available in standardized series, their typical combinations of di‑ mensions were used. The corners of the pro�iles are rounded; however, for simulation purposes, they were simpli�ied for the Ansys beam tool. The comparison of the simpli�ied and real cross‑sections is presen‑ ted in Fig. 3. As proved for the initial simulation, this simpli�ication does not signi�icantly affect the pro�iles’ moments of inertia and thus, calculated stresses and strains.

The frame’s mesh consists of three‑node one‑ dimensional elements of a default size of 20 mm. The element size was forced to 0.1 mm at the ends of every beam (see Fig. 5). The statistics of the mesh are pre‑ sented in Tab. 3. Stainless steel (from General Materi‑ als in Ansys Workbench 2021) was assigned as the ma‑ terial of the construction pro�iles. Its strength parame‑ ters correspond to those of commonly used industrial materials. They are presented in Tab. 4 in comparison with steel used for such construction pro�iles. The cho‑ sen material, X5CrNiMo17‑12‑2, is compliant with the EN 1.4401 standard, which is the equivalent of the AISI 316 standard. [42].

2) vertical acceleration of the system after gripping the object (load state 2);

3) vertical braking of the system before releasing the object (load state 3).

Vectors of applied acceleration �ield, forces and torque are presented in Fig. 4. Force A (610 N) is the weight of stationary drives, while Force B (173 N) is the weight of the engine and shafts, Vector C repre‑ sents earth gravity, and loads D‑G are dynamic reacti‑ ons of the system. Their values are gathered in Tab. 1.

The frame is relatively large compared to the moun‑ ting feet, which are to be anchored. Therefore, each leg’s connection to the ground is modelled as �ixed support (H).

Fig. 2. Model of the frame geometry

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Load State 1 2 3

Remote Force D [N] 0 3448.9 0

Tab. 1. Values of loads

Remote Force E [N] 0 0 3448.9

Stress distribution across beams is not constant due to the internal changes in the bending moment.

Therefore, the distributions of maximum and mini‑ mum stress along a beam were analysed. Safety factor (FS) was calculated based on the calculated stress va‑

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Remote Force F [N] 3605 0 0

Torque G [Nm] 0 26.25 ‑26.25

lues, according to the formula Eq. 2.

FS =

Re max(|σmax |, |σmin |)

(2)

Where the used symbols represent particular parame‑ ters: Re ‑ yield strength, σmax ‑ maximum combined stress, σmin ‑ minimum combined stress.

Geometry mass, safety factor, and maximum total deformation were set as the output parameters. FEM analysis was performed for each load state described in the previous sections. Afterwards, parameters cor‑ relation was checked with the analysis of the correla‑ tion matrix.

2.3. Response Surface Optimization

Fig. 3. Comparison of profiles’ cross‐sections

Response surfaces were computed with enhanced Face‑Centred Central Composite Design (FCCCD) DoE type [43] [44] (except sparse grid response surface, where the sparse grid initialization was used) and ve‑ ri�ied with 3 points not used before to create the surfa‑ ces. The DoE generally needed 53 design points, while the sparse grid generation took 131 design points.

The results of optimization with the different appro‑ aches towards the generation of response surfaces were compared to choose the most suitable algorithm for similar frame constructions.

Steel Density [kg/m3 ] Young’s Modulus [GPa] Poisson’s Ratio Yield Strength [MPa] Ultimate Strength [MPa]

Fig. 4. Loads and boundary conditions applied to the frame Parameter db [mm] df [mm] do [mm] a [mm] t [mm]

Tab. 2. Considered scopes for the parameters Parametrised 16368 32751 Beam188 axaxt

Tab. 3. Mesh statistics 18 18

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Engine mounting 1119 2240 Beam188 150 x 140 x t

EN 1.4401 7950 200 0.31 205 515

Tab. 4. Material properties for the Ansys model and the industrial steel

Scope/Discrete values ⟨100; 400⟩ ⟨400; 700⟩ ⟨200; 820⟩ [60, 70, 75, 80] [2, 3, 4]

Beams Elements Nodes Element type Cross‑section

Model 7750 193 0.31 207 586

Fig. 5. Frame mesh

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In the end, the MOGA optimization algorithm was used for each response surface model to minimize the mass of the frame. Additionally, the safety factor was constrained to remain greater or equal to 3, while the maximum total deformation was constrained not to exceed 2 mm. The settings of the algorithm are presen‑ ted in Tab. 5. Physical properties and strength results obtained with the optimal candidate points for all the methods were compared with the initial ones.

Number of initial samples Number of samples per iteration Maximum allowable pareto percentage Convergance stability percentage Maximum number of iterations Type of discrete crossover

Tab. 5. MOGA settings

5000 1000 70 2 20 one point

Fig. 7. Stress distribution at the load state 2

3. Results First of all, the equivalent stress analyses were con‑ ducted for the initial frame design. Their results are illustrated in terms of distribution in Fig. 6‑8 with highlighted places of the maximum stresses occurring.

Their values are gathered in Tab. 6.

State Lifting Acceleration Braking

σmax [MPa] 8.1634 17.824 18.211

Tab. 6. Maximum equivalent stress Correlation matrix is presented in Fig. 9. The �irst �ive parameters are inputs, and the last three are out‑ puts of optimization. The correlation coef�icients be‑ tween the df and the output parameters are almost equal to zero. Linear and quadratic trends between the df and the safety factor are illustrated in Fig. 10. They are used to investigate the in�luence of this parameter on the loss function.

Fit statistics and a response surface of safety factor as a function of db and do are presented in Fig. 11‑16 for every method. The �it statistics provide accuracy

Fig. 8. Stress distribution at the load state 3 metrics of the response surface. These affect the reli‑ ability and the quality of the �inal results. The coef�i‑ cient of determination (R2 , see Eq. 3) is a parameter that denotes how well the response surface represents the variability of the output parameter. n (yi − ŷi )2 2 R = 1 − i=1 (3) n 2 i=1 (yi − ȳ)

The other metrics describe errors of the approx‑ imation. These are: Root mean square error (σ, see Eq. 4);

n 1 σ= (yi − ŷi )2 (4) n i=1 Relative maximum absolute error (σr,max , see Eq. 5) σr,max =

1 max(|yi − ŷi |) σ

Relative average absolute error (σr,avg , see Eq. 6) σr,avg =

n 1 (|yi − ŷi |) nσ i=1

(5)

(6)

Fig. 6. Stress distribution at the load state 1

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The response surface function is a three‑ dimensional vector function of �ive variables. As it is impossible to represent on a single surface chart, the single response charts of one output parameter (safety factor) are presented dependent on two in‑ put parameters (db and do selected for this case, see Fig. 12‑16). These speci�ic variables were chosen due to the curvature of the surface, discussed in the follo‑ wing section.

The optimization results for every response sur‑ face method (named as in the �irst section) are compa‑ red with each other and with the ones obtained for the initial frame design (see Tab.7). Additionally, the out‑ put parameters were veri�ied as regular design points.

The safety factor and deformations were calculated for

Fig. 9. Linear correlation matrix

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all the three load conditions. Their extreme values (mi‑ nimum for safety factor and maximum for deformati‑ ons) in the whole structure were used for further ana‑ lysis. As may be observed, these tend to occur under load condition 3.

The �igures Fig. 17‑22 illustrate stress and defor‑ mation distributions for the frames optimized with every method. As may be observed, the maxima occur at approximately the same locations for every case, while the global minima are variable for stress distri‑ butions.

Fig. 10. Trend between the safety factor and the df

Fig. 11. The goodness of fit and the plot of response surface of safety factor dependent on db and do for genetic aggregation method

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Fig. 12. The goodness of fit and the plot of response surface of safety factor dependent on db and do for second order polynomials method

Fig. 13. The goodness of fit and the plot of response surface of safety factor dependent on db and do for kriging method

Fig. 14. The goodness of fit and the plot of response surface of safety factor dependent on db and do for non‐parametric regression method

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Fig. 15. The goodness of fit and the plot of response surface of safety factor dependent on db and do for neural network method

Fig. 16. The goodness of fit and the plot of response surface of safety factor dependent on db and do for sparse grid method

Method (number) db [mm] df [mm] do [mm] �ro�i�e si�e [mm] �ro�i�e thickness [mm] Geometry mass [kg] Safety factor Maximum deformation [mm] Geometry mass (�eri�ied) [kg] Safety factor (�eri�ied) Maximum deformation (�eri�ied)[mm]

1 179.69 429.75 200.43 80 2 131.94 6.9842 1.7331 131.95 6.7984 1.7812

Tab. 7. Optimization results for different methods

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2 364.25 456.55 203.82 80 2 130.39 6.6271 1.9063 132.06 6.7957 1.7814

3 113.33 409.66 256.25 75 2 126.91 5.7762 1.9924 126.79 6.0372 2.0904

4 120.67 417.48 203.61 75 2 131.71 6.6254 1.9664 124.88 6.0982 2.0906

5 298.31 662.74 200.64 75 2 125.25 6.7461 1.9981 124.77 6.1023 2.0861

6 130.04 419.34 200.20 80 2 130.79 6.8452 1.7334 131.95 6.7981 1.7798

Initial Frame 200.00 600.00 450.00 80 4 ‑ ‑ ‑ 276.42 11.367 0.8722

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Fig. 17. The equivalent stress distribution and the deformation at the load state 3 for genetic aggregation configuration

Fig. 18. The equivalent stress distribution and the deformation at the load state 3 for 2nd order polynomials configuration

Fig. 19. The equivalent stress distribution and the deformation at the load state 3 for kriging configuration

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Fig. 20. The equivalent stress distribution and the deformation at the load state 3 for non‐parametric regression configuration

Fig. 21. The equivalent stress distribution and the deformation at the load state 3 for neural network configuration

Fig. 22. The equivalent stress distribution and the deformation at the load state 3 for sparse grid configuration 24 24

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4. Discussion of the Results According to the FEM analysis of the frame, the parts connected to the ground are the ones with the highest values of internal stress. They are especially exposed to damage within load cases 1 and 3. It is the result of the greatest local internal torque caused by the horizontal component of the dynamic response.

Regarding this, frame structures are typically more lo‑ aded during acceleration and deceleration of the Car‑ tesian robot’s horizontal units. Moreover, the surroun‑ ding of the load application point is more stressed than the other fragments of the frame. As the design is overly stiff, the entire load is distributed among the supports. Such a dependence shall occur for the frame parts connecting the force application points with the constraint points for the constructions with similar stiffness properties. The longer the beam, the more �lexible it is, and thus, transfers less load to the sup‑ port.

The correlation coef�icients of the df with the out‑ puts indicate that this parameter has almost no in�lu‑ ence on the loss function within the considered scope.

Also, an analysis of the linear and quadratic trend graphs con�irm this theory. Therefore, it could have been excluded from the optimization process. As ex‑ pected, the response surfaces involving this parame‑ ter indicated almost no relationship with outputs.

Regarding different methods of response surface generation, the sparse grid obtained the best me‑ trics of quality. Genetic aggregation and kriging met‑ hods estimated geometry mass correctly. However, their stress and deformation estimations remained less accurate. The second‑order polynomials algo‑ rithm provided decent geometry mass estimations but failed in determining stress and deformation respon‑ ses. Finally, non‑parametric regression and the appro‑ ach based on neural networks provided the worst �it‑ ting response surfaces.

The included charts of safety factors may be used to de�ine the cause of the insuf�icient �it for response surface models. As observed, the stress changes ra‑ pidly for the upper range of input parameters associ‑ ated with oblique beams. The tendency is similar for the correlation between these parameters and the to‑ tal deformation.

The kriging and genetic aggregation algorithms provide wavy response surfaces, while others, except the sparse grid, provide smooth ones. The response surface for the sparse grid algorithm is irregular with a sharp geometry. It is especially visible for the hig‑ her values of the parameter related to oblique beams and the extremal values of the parameter related to the bottom beams. This can be observed in Fig. 12‑16, as well as for other response surfaces not presented in this paper.

Genetic aggregation and kriging provide accepta‑ ble metrics for learning points. However, the veri�ica‑ tions points indicate that the response surface models are less accurate. This situation may be caused by a lack of design points especially for such a high varia‑ bility of the safety factor.

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The second‑order polynomials method resulted in poor metrics for the safety factor and deforma‑ tion estimations. This means that quadratic functions are insuf�icient to estimate curvatures of the real re‑ sponses. Similar outcomes may be observed for non‑ parametric regression and neural networks approach.

Even though some algorithms had poor quality metrics, all methods provided similar results of opti‑ mization. Some of them underestimated displacement and while verifying, it turned out to exceed the boun‑ dary limit. However, as the RSM is a numerical esti‑ mation, some minor inaccuracies should be conside‑ red and acceptable. The genetic aggregation involves signi�icantly fewer points and leads to almost the same outcomes as the sparse grid, for relatively simple con‑ struction of a frame. Therefore, it is not recommended to use such a complex method if there are no extraordi‑ nary circumstances, such as expected discontinuities in the inputs‑outputs function.

The stress distributions in the optimized structu‑ res are similar to those for the initial design under the same loading condition. Also, the maxima for both, stress and deformation, occur at the same locations for all the methods.

5. Conclusion

The rising importance of digital technologies and particular simulations in the design process brings new possibilities for engineers. This arises a need to understand the basics of numerical algorithms laying behind these. Related knowledge and experience are useful to select appropriate methods from their wide range. This is especially important for FEM, as the de‑ cision on modelling the system and setting the compu‑ tations may signi�icantly affect their results.

In frame structures, it is advisable to inspect the connections and joints in the nearby surroundings of the frame‑ground interface. This is necessary due to the higher values of the stress appearing there. All the simulations shall be run for the extremal load cases, hence, for the greatest moment occurring nearby the supports.

Response surface optimization is a powerful tool to reduce mass at the early stage of design. Additio‑ nally, dependencies between input and output para‑ meters may be analysed by the engineer to empower suboptimal manual design. Within response surface optimization, the key aspect is to select an adequate method for an expected characteristic of the response.

However, this may be dif�icult for multi‑output sys‑ tems with numerous variables. The previous sections may be used as a base for the optimization of similar frame constructions. Nevertheless, the quality metrics of the models obtained with a particular method shall be controlled at all stages. The amount of necessary design points increases with the number of input pa‑ rameters, the span of their ranges and the complex‑ ity of the response. However, it depends mainly on the number of variables, as the domain of the estimated response grows linearly with the number of their com‑ binations.

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All the considered methods provided appropriate optimization results. However, genetic aggregation, kriging and the sparse grid had noticeably better accu‑ racy metrics. These methods are expected to give cor‑ rect results for similar frame constructions; i.e. a frame on a rectangular plan without a base and with additi‑ onal diagonal reinforcing beams �ixed in the corners, particularly for the similar cross‑sections and overall dimensions.

The wavy response surfaces cause a better local �it of kriging, genetic aggregation and sparse grid met‑ hods to the design points. Therefore, they are bet‑ ter suited for nonlinear and dynamically changing re‑ sponses. However, they are less effective if the data is noisy, e.g. for the sets with a measurement error.

The investigation may be continued in three diffe‑ rent directions. First of all, it is possible to test the ap‑ plication of RSM for more complex frames as well as the structures with two‑ and three‑dimensional �inite elements. The results of such an experiment should enable forming a benchmark of geometries connected with the best‑�itted RSM methods. Second of all, it is possible to perform direct optimization and compare the accuracy gain, considering the additional compu‑ tational cost. Also, the use of adaptive models for the design of similar industrial frames can be tested. The original research is planned to be followed up within the �irst approach, and possibly broadened for a vari‑ ety of materials as well as manufacturing techniques.

AUTHORS

Filip Gwardecki∗ – Warsaw University of Techno‑ logy, Plac Politechniki 1, Warsaw, 00‑661, e‑mail: �i‑ lip.gwardecki.stud@pw.edu.pl.

Piotr Falkowski∗ – ŁUKASIEWICZ Research Net‑ work – Industrial Research Institute for Automa‑ tion and Measurements PIAP, Al. Jerozolimskie 202, Warsaw, Warsaw University of Technology, 02‑486, Plac Politechniki 1, Warsaw, 00‑661, e‑mail: pi‑ otr.falkowski@piap.lukasiewicz.gov.pl. ∗

Corresponding author

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Software for the Control and Monitoring of Work of a Collaborative Robot Submitted: 15th July 2021; accepted: 10th December 2021

Wojciech Łabuński, Andrzej Burghardt DOI: 10.14313/JAMRIS/3-2021/16 Abstract: The development of robotic systems is correlated with the development of their software. Expanding robot implementation areas and attempts to replace more and more groups of activities carried out by people requires increasing the degrees of freedom, introducing robot interaction with the environment, and preparing software that manages over six degrees of freedom in a friendly, understandable, ergonomic, and functional manner.

The authors proposed a method of programming a collaborative robot with the use of a joystick, created the necessary software, constructed elements of the system, obtaining an original, flexible, and intuitive solution. As part of the work, the proposed solution has been simulated and verified. Verification of the proposed solution was carried out on a real bench equipped with a cobot, Kawasaki duAro. Keywords: robotics, collaborative robot, python, joystick, cobot, Kawasaki, duAro

1. Introduction As the idea of Industry 4.0 develops, the concept of human-robot collaboration in a production environment is becoming more and more popular. The first and basic criterion that must be met for a robot to be allowed to work with a human is having safety systems that will allow reduction or limitation of the number and degree of dangers that may threaten the operator during cooperation with the machine. A mechanical unit designed for direct interaction with humans in a common workspace can be called a cooperating robot [17]. In the literature we can find the terms “collaborative robot”, “ cooperating robot” [17, 22, 23] or “cobot” [10]. All these terms are synonyms or abbreviations of the term Collaborative Robot. In this publication, the above terms will be used interchangeably. The growing expansion of collaborative robots in industry creates a growing need to provide appropriate systems that allow planning of the paths of a robot tool and implementation of its movement without hindrance. Allowing robots to work closely with people without using external safety systems, such as fencing, motion sensors, safety mats, magnetic bolts, carries risks for operators and programmers. Unlike working with conventional robots, it is not necessary for the collaborative robot to be located in a space with limited access for the operator. (Fig. 1)

In this work the Kawasaki collaborative robot was used because it was available to the authors and high level of safety can be easily used in teaching programming of industrial robots. The proposed solution is dedicated mainly for educational and presentational purpose.

a)

b)

Fig. 1. Ways of working with robots: a) conventional robot, b) collaborative robot According to the ISO/TS 15066 technical specification, it should be possible to limit the velocity and power of a robot’s drives. Also, the shape and material of the machine have an impact on work safety, some manufacturers install special foam on the robot arms to reduce the impact of the machine on the environment in the event of a collision. Five levels of robot-human cooperation can be distinguished [5, 17]: monitored stop (the robot does not move when the operator is in the cooperation space), manual guidance (the operator manually guides the robot tool), monitoring velocity and distance (the robot moves only when the operator is at a certain distance from it), limiting the power and forces of machine move-

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ments (system restrictions introduced in accordance with applicable guidelines). Fig. 2 shows an operator (3) and a collaborative robot (1) working together in a workplace (4) in a space described as (2).

Fig. 2. Collaborative robot cooperation space

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An example of cooperation between people and robots in the context of the stability of performed operations and the impact of interaction on stability has been described in [1]. Factors affecting robot-human interaction as well as problems and challenges related to it are described in [6, 7, 8, 10, 11, 14, 23, 27]. Papers [6, 7] describe issues and ways of implementing collaborative robots s in Czech Republic and African industry. Paper [8] presents the results of research on key factors that affect the organization of cooperation between a robot and a human being. One example of using the capabilities of a collaborative robot to facilitate human work is a robotic station for assembling homokinetic joints [10]. An analysis of universally understood physical interaction in tasks requiring cooperation between a robot and an employee was presented in [11]. Paper [14] concerns the problem of ethics in manufacturing stations equipped with industrial robots and collaborative robots. In contrast, paper [23] presents an analysis of robotization problems of small and medium enterprises. The safety of the operator working together with a collaborative robot is analysed in [4, 17, 18, 22, 26, 35]. In paper [17], the author analyses the issue of safety in systems equipped with cooperating robots in the context of industrial standards, among others PN-EN ISO 10218-1, PN-EN ISO 10218-2. Paper [35] presents the provisions of the technical specification ISO/TS 15066. The strategy for the correct assessment of the safety of a robotic system with a cooperating robot, as well as guidelines for the construction of such workstations are included in paper [18]. Analysis of current trends in industry in the context of the safety of robot-human cooperation was carried in paper [22]. The authors of [26] presented two risk analysis paths regarding safety issues for stations with cooperating robots. The use of collaborative robots in an industrial environment is discussed in papers [2, 3, 7, 12, 13, 16, 20, 21, 25, 30, 31, 32, 37, 38]. An example of the use of a robot cooperating with a triangulation sensor for deterArticles

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mining the TCP of the robot and the calibration of the tool was presented in [12]. In paper [20], an example system was proposed in which human cooperation with many robots and an algorithm for controlling such a station would occur. Examples of collaborative robots used for assembly lines were presented and analysed in paper [21]. A review of applications for monitoring velocity and separation of the robot from humans in the light of the technical specification ISO/ TS 15066 was described in paper [25]. In paper [30], an approach was proposed to estimate the confidence interval for the duration of a robot movement for situations in which space sharing between human and robot is required. In paper [31], the authors analysed practical possibilities and limitations of introducing collaborative robots in various industrial environments. The authors of paper [36] discussed the possibility of introducing collaborative robots into welding applications. An example of symbiotic cooperation between a robot and a human in an environment with a proprietary interface was presented in paper [38].

The issues of employee confidence in industrial robots, especially collaborative robots, were described in paper [9]. In papers [19, 29] authors discussed problems and possibilities of analysing and monitoring parameters and operations of industrial robots – in standard [19] and virtual environment [29]. Trust has been identified as one of the key factors affecting proper robot-human cooperation. Research has been carried out to determine how and when employees and machine operators build trust in industrial robots. A mathematical model of robot-human cooperation can be found in paper [28], the use of Python language in robotic applications is presented in papers [24, 33], and paper [34] describes the use of the pygame module for communication with a joystick.

There are many solutions that allow programming of industrial robots in online mode. One of the most popular of these is the manual programmer that acts as a peripheral device. It allows control of movements as well as creation of programs and monitoring of the robot’s work parameters. The manual programmer is available in the product range of most robot producers: ABB, Kuka, Fanuc. Alternative solutions that are worth paying attention to are, for example, a touch screen tablet from the company Universal Robots.

Some manufacturers allow control of the movement of a robot from the level of their programming and simulation environments, for example ABB and Kawasaki. However, most often such control is unwieldy and inconvenient for the programmer. Kawasaki software enables this option from the HyperTerminal level. It only allows the setting of a defined point in space or a specific angle of rotation of the axis. The Kawasaki duAro robot does not have in its basic equipment any peripheral devices enabling manual control over the movement of the robot during manual steering.

Hence the need to prepare a system that could act as a manual controller with simultaneous monitoring of some robot operating parameters. The article proposes a solution using a two-axis joystick and a window application to communicate with the joystick with a robot controller. Analysis of the available literature

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has shown that similar proprietary solutions used to control the movement of robots, especially cooperating robots, are not found, and the subject matter brings a new view on manual steering of industrial robots.

2. Description of Workstation

The workstation with a collaborative robot (Fig. 3) used in the work is intended for educational demonstrations and development of robot programming skills. To facilitate control of the robot and memorizing points, and thus enable simpler planning of algorithms and carrying out pick and place operations, it was decided to add a joystick to the station. It plays the role of a manual robot controller, enabling steering and control of the movement of the individual arms of the collaborative robot.

Fig. 3. Elements of a workstation with a collaborative robot The workstation with a collaborative robot (Fig. 3) consists of a Kawasaki duAro robot and a test platform in the manipulator workspace. It is a two-armed robot designed to work with humans in industrial conditions. It consists of two arms (left - (1) and right - (5)) designed based on a SCARA robot. The parallel structure of the arms allows for high velocities of the mechanism. Each arm is equipped with a pneumatic gripper (2). The test platform (3) has been designed in such a way that it is connected to the robot base and rigid enough to maintain a constant level. The robot is controlled by means of a controller (6) equipped with the Cubic S option – a unit monitoring the movement of the robot and responsible for ensuring safety. The manufacturer equipped the robot with an external panel (4) that allows the brakes to be deactivated, which enables manual movement of the third axis of each arm.

Identical tools were mounted on both arms (Fig. 4).

Each of them consists of a mounting (1), pneumatic gripper (2) and gripper fingers (3). Both the mounts and fingers were manufactured by the incremental method of ABS material.

Fig. 4. Gripper

Fig. 5. Joystick A joystick was used to control individual movements of the selected robot arm and to switch between selected motion control options (Fig. 5).

Fig. 6 presents kinematic diagrams and the way in which the movements of the robot in the appropriate coordinate systems were mapped using the joystick movement. The movement of the M point associated with the robot tool was implemented in the robot’s base system - xyz. The coordinate system in the base of the joystick was adopted analogously and marked with xjy¬¬jzj. Tilting the controller in the direction of the selected axis resulted in the movement of point M along the corresponding axis of the robot system. For example, movement of the joystick by angle α in the positive direction of the y axis corresponded to the movement of point M in the positive direction of the y axis. Movement of point M in the direction of the z axis was transmitted using the slider (3) shown in Fig. 5.

Movement in the basic Cartesian system was implemented by means of tilts along two joy-stick axes (4) – axes xj and yj (Fig. 6b) and two buttons (1) responsible for the tool configuration (two-way rotaArticles

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tion). In addition, it provides the ability to control the movement of the TCP point along the z axis (3) and arm selection (2). To operate the gripper, the trigger button on the back of the joystick (6) was used. It is also possible to select the currently used arm (2).

Closing the connection between the joystick and the robot controller is ensured by means of a button (5).

Thanks to this, the operator has full control over the movement of each of the robot arms.

a)

b)

Fig. 6. Kinematic diagram: a) duAro robot arm, b) joystick

Fig. 7. Information flow in the cooperating robot control system Tab. 1. Data frame elements No. 1.

2.

3.

4.

5.

6.

7.

8.

9.

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Name of the variable robot_arm interpolation horizontal_direction vertical_direction string_direction_z rotation velocity tool S

Number of characters 2 1 2 2 3 2 3 1 1

Value “A1” or “A2” “J” or “B” “-1” or “+1” or “+0” “-1” or “+1” or “+0” from 000 to 100 “-1” or “+1” or “+0” from 000 to 100 “0” or “1” “S”

Purpose arm number junction and linear interpolation horizontal joystick deflection vertical joystick deflection tool vertical axis direction tool orientation robot TCP velocity in the base system gripper state – open / closed end of frame sign

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3. Description Using an existing robotized station equipped with a Kawasaki duAro collaborative robot, a work control system was prepared using the joystick to manipulate. The window application controlled and monitored the robot’s operation, simultaneously recording information about the status of the joystick. The processed data were sent to the robot controller using the TCP/IP protocol, where, after interpretation by the robot software, they ensured movement of the selected robot arm. Despite the limited number of joystick buttons, control over the collaborative robot was obtained.

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of prepared code. It can also help to create motion programs and robot points in a workspace, define motion and process parameters and manage program variables, databases, and input/output signals. The software helps also with monitoring performance of a machine and saving and analysing error logs.

3.1. Communication

Communication between the joystick and the collaborative robot was prepared using the TCP/IP protocol and a PC. Data flow is from the joystick to the application, which forwards it to the robot controller.

There, individual data is interpreted and translated into the appropriate robot motion. Part of the information is also passed the other way: from the robot to the application. The user can monitor and change the velocity of the TCP point of the currently operating arm (Fig. 7). The information was sent in the form of a data frame. This was a nine-element vector consisting of characters depicted in ASCII code. Each element of this structure had a clear interpretation. They are presented in Tab. 1. together with the function they perform, the values they can take, and the number of characters reserved for a given variable. The total number of characters sent from the application to the robot controller is 17. The data frame was transferred to the robot controller during the robot’s operation.

Fig. 7 schematically illustrates the flow of information in an industrial robot control system.

a)

b)

Fig. 8. The robot program was prepared on the workstation (a) using the KIDE application (b)

3.2. Control of Robot Arm – Kawasaki Program

Kawasaki the manufacturer of duAro robot enables programming of its devices using the AS language dedicated for this purpose. It is robot programming language consisting of two groups of instructions – one for motion of the arms and another for other purposes like operating signals, for and while loops etc. All commands and functions are entered using the terminal from a computer (Fig. 8a). It allows, among others, for access to the robot controller memory and data stored on it, as well as for direct control and monitoring of arm movement. Robot communication with the terminal is carried out using the TCP/ IP protocol using commands in the ASCII code. The application that facilitates writing programs for Kawasaki robots is KIDE (Kawasaki Robotics Integrated Development Environment) (Fig. 8b). It is additional application which can help with controlling a robot and its programs. It is not necessary for creating and realizing robot programs, but it is helpful when comes to management of larger portions of code and more complex schemes of action.

The graphic interface allows for simpler management of programs, variables in the controller’s memory, and a text editor adapted to the AS language syntax allows faster writing of programs and debugging

Fig. 9. Application for robot control using a joystick

3.3. Application for Joystick – Robot and Communication To solve the problem of control over robot movements, a joystick connected to a PC was used. A window application was prepared to act as a joystick controller and a server that supports data exchange between the joystick and the collaborative robot. Full use of the joystick functionality was possible thanks Articles

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to the Python language and the pygame library available to it.

To improve the programming process implemented in Python, the option of creating function definitions and saving them in a separate file was used. Such a file is called a module. Its content can be loaded into another module or into a program. One of the useful Python modules is pygame. It allows the programmer to access peripheral devices (such as joystick, pad, speakers etc.) from the Python language, which can be used, for example, as a computer game control.

The pygame module is based on the cross-platform SDL (Simple DirectMedia Layer) library, which gives access to hardware thanks to OpenGL and DirectX technologies. The graphic interface of the collaborative robot control application was prepared using Qt libraries available for the Python language. These are libraries for designing graphical user interfaces. They were created for C++ and Java, but it is possible to create applications using other popular languages such as Ruby, C# or Python.

The application window was divided into four parts (Fig. 9). In the first part (1), the user has the option to choose how the application will communicate with external devices. Real workstation control is enabled by the “Robot” option, while the “Computer” option using a local connection can be used for offline testing. The limited number of buttons and joystick sliders caused the necessity to add a robot velocity control option in a second area (2). Due to the assumption that the robot arm will be moved in linear interpolation, in the base coordinate system the robot velocity is the TCP velocity of the selected arm expressed in the range from 0% to 100%. The current percentage value was displayed below the dial. Part three (3) contains user information and operating instructions for the joystick-robot system. In contrast, part 4 provided information on which robot arm is currently being controlled.

The use of the pygame library made it easy to implement joystick control and combine its individual elements with specific robot features. The socket library made it easy to use the TCP/IP protocol to receive and transfer data between the computer and the robot. Appropriate variables in the program were assigned to individual degrees of freedom of the controller and its buttons.

4. Validation Tests

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Validation tests (Fig. 10) were carried out on a real object confirming the correct operation of the system.

The workstation was prepared by combining a collaborative robot, a computer, and a joystick. As expected, the TCP/IP protocol proved to be a sufficient means of communication for exchanging information between individual station elements. The software of individual robot functions and enabling them to be associated with joystick movements and its buttons significantly facilitated the programming of robot movements. In further work, the proposed solution can be expanded with the option of remembering the position of the tool.

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In addition, the presented solution is universal and easily adaptable to the robots of other manufacturers. After applying additional safety measures, such as fencing, they can also be used in work with industrial robots not adapted to work with humans.

Fig. 10. Verification of the prepared application

5. Conclusion The paper presents a proposal for software intended for controlling a collaborative robot implemented using a joystick. Analysis of the available literature allows us to state that similar solutions are not used. As part of the work, a Kawasaki duAro collaborative robot controller was connect-ed to a joystick using the TCP/IP protocol. Python language and available modules were used, which greatly facilitated the operation of individual joystick functions. A window application was developed using the Qt library. The graphical user interface allows easy connection of the joystick and robot with simultaneous monitoring of operating parameters. The solution prepared in this way fulfilled its task and allowed control of the robot’s movements with the help of a joystick. The next stage of the work will be connecting an external vision system to carry out an assembly operation.

AUTHORS

Wojciech Łabuński* – Faculty of Mechanical Engineering and Aeronautics, Rzeszow University of Technology, Poland, e-mail: w.labunski@prz.edu.pl.

Andrzej Burghardt* – Faculty of Mechanical Engineering and Aeronautics, Rzeszow University of Technology, Poland, e-mail: andrzejb@prz.edu.pl. *Corresponding author

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2021 VOLUMEVOLUME 15, 15, N° 3N° 3 2021

FDA*: A FOCUSED SINGLE‐QUERY GRID BASED PATH PLANNING ALGORITHM Submitted: 4th October 2021; accepted: 9th February 2022

Mouad Boumediene, Lamine Mehennaoui, Abderazzak Lachouri DOI: 10.14313/JAMRIS/3‐2021/17 Abstract: Square grid representations of the state‐space are a com‐ monly used tool in path planning. With applications in a variety of disciplines, including robotics, computatio‐ nal biology, game development, and beyond. However, in large‐scale and/or high dimensional environments the creation and manipulation of such structures become too expensive, especially in applications when an accurate re‐ presentation is needed.

In this paper, we present a method for reducing the cost of single‐query grid‐based path planning, by focu‐ sing the search to a smaller subset, that contains the optimal solution. This subset is represented by a hyper‐ rectangle, the location, and dimensions of which are cal‐ culated departing from an initial feasible path found by a fast search using the RRT* algorithm. We also present an implementation of this focused discretization method cal‐ led FDA*, a resolution optimal algorithm, where the A* algorithm is employed in searching the resulting graph for an optimal solution. We also demonstrate through si‐ mulation results, that the FDA* algorithm uses less me‐ mory and has a shorter run‐time compared to the classic A* and thus other graph‐based planning algorithms, and at the same time, the resulting path cost is less than that of regular RRT based algorithms.

Keywords: motion planning, grid‐based, path planning, mobile robots

1. Introduction Motion planning for robots has received a sub‑ stantial amount of attention in the last two decades, due to the increasing integration of robots in modern industry and even in many tasks of our daily lives.

In addition to this, motion planning plays a key role in autonomous robot navigation, and numerous other disciplines such as computational structural biology [1, 18], crowd simulation [14, 19], and video game development [2].

The fundamental problem to be solved is how to compute a path that allows the robot to move from an initial to a target location in the state‑space while avoiding the surrounding obstacles. One of the most common approaches to solving this problem is to discretize the continuous state space, through a de‑ terministic grid with a pre‑de�ined resolution. From there the implicit graph within this grid is searched using graph search algorithms, such as A* [6] and Dijkstra’s [3], to obtain a resolution‑optimal solution to the problem.

Fig. 1. Example of FDA* in a 2D Euclidean space with random obstacles, the new state‐space represented by the blue borders is discretized and searched by the A* algorithm and finally the resolution optimal path is generated between the start and goal states

This is known as grid‑based path planning, however, the number of grid cells required for this process grows exponentially with the number of the state space dimensions. in addition to this, the grid reso‑ lution needed for an accurate representation often leads to extremely large search spaces which cause the use of grids to be costly in terms of time and memory resources.

Furthermore, in large‑scale environments, only a small subset of grid cells will contribute to �inding the optimal solution especially if the distance between the start and goal states is relatively small, therefore cells outside of this subset will only consume memory space that could otherwise be allocated for other purposes. �owever, �inding the optimal path is not always a requirement. In some instances of the path planning problem, only an obstacle‑free path is needed. In such a case many cheaper solutions can be applied.

For example, various stochastic algorithms avoid creating expensive grids altogether, and instead, generate random samples in the planning domain to incrementally grow a search tree in the free space.

The Rapidly‑exploring Random Trees (RRTs) pre‑ sented by Laval [12] is one of these algorithms, it ef�iciently provides an obstacle‑free path, making it useful, especially in high dimensional environments.

We also mention the RRT* presented by Karaman and Frazzoli [8], which converges to the optimal solution asymptotically, as the number of iterations approaches in�inity.

While discretizing the full state‑space using a grid can get expensive in large or/and high dimensional en‑ vironments, one possible solution is to discretize only a smaller subset of the planning domain, under the a Articles

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condition that the optimal solution belongs with cer‑ tainty to this subset. The states outside this subset are therefore not necessary for the computation of the op‑ timal path and they will only constitute a burden on the available resources. This subset as in [5]and [15], can be described by a prolate hyper‑ellipsoid, and the calculation of its dimensions is carried out departing from the cost of an initial feasible path, which can be obtained ef�iciently by conducting a low‑cost search using stochastic algorithms such as those mentioned above.

In this paper, we present an ef�icient method for grid discretization of the state‑space to be used by grid‑ based planning algorithms. In this method, we consi‑ der discretizing only a rectangular subset, that tightly bounds the ellipsoidal subset where the optimal path is guaranteed to be found. For that purpose, RRT* is used to rapidly obtain an initial path, that can then be used for calculating the boundaries of this infor‑ med subset. We also present an implementation of this method, we call it FDA*( focused discretization A*), where the A* algorithm is used to plan the optimal path in the obtained grid .

FDA* uses FD(focused discretization) to improve upon classic grid‑based algorithms in terms of storage‑ space and execution time, exploiting the fact that a lo‑ wer number of states have to be handled using FD. The rest of the paper is organized as follows: Section 2 provides a formulation of the path plan‑ ning problem and a review of the related work in‑ cluding a comprehensive overview of the RRT* algo‑ rithm. Section 3 presents the focused discretization (FD), a method for building size‑ef�icient grids for the use of single query grid‑based planning algorithms. In section 4, an implementation of the focused discreti‑ zation method called FDA* is presented. As for section 5, it is devoted to the presentation of the results of several experiments conducted on FDA* and A* al‑ gorithms, to compare the performance of our algo‑ rithm with the classical grid‑based planners. Section 6 provides interpretations and presents drawn conclu‑ sions regarding results presented in section 5. Finally, section 7 concludes the paper with thoughts of future work.

2. Background

2.1. Problem Definition In our proposed algorithm, we tackle two variati‑ ons of the path planning problem, namely the feasible and the optimal planning. Therefore, in this section and similarly to [7] we will formally de�ine these two variants and provide some notations that will be used in the rest of this paper.

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Let X ⊆ Rn be the open set that represents the state‑space of the planning problem, where n ≥ 2.

Let Xobs X be the obstacle region, that is con‑ sisted of the states in collision with obstacles, and Xf ree = X/Xobs be the free space. Let xstart ∈ Xf ree be the initial state, and xgoal ⊆ Xf ree be the goal re‑ gion. Since we are dealing with a 2‑Dimensional state Articles

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space, let d2 (x1 , x2 ) be the 2‑D Euclidean distance between two arbitrary states x1 , x2 ∈ X 2 .

Feasible path planning is concerned with generating π : [0, 1] → Xf ree , an obstacle‑free path, where π(0) = xstart and π(1) ∈ Xgoal if such a path exists, otherwise it returns failure.

The optimal planning in the other hand deals with the generation of a path π ∗ such that : (i) π ∗ : [0, 1] → Xf ree , where π ∗ (0) = xstart and π ∗ (1) ∈ Xgoal . (ii) C(π ∗ ) = min ∑C(π). π∈ ∑ where C is the cost function, and is the set of all nontrivial paths from xstart to xgoal , however if no such path exists then it returns failure.

2.2. Related Work The early work on path planning was generally dominated by classical methods such as: arti�icial potential �ields (APF) [9], roadmaps, and cell decom‑ position [13].

The APF method proposed by Khatib in 1986 deals with the environment as a continuous state space, and use it to generate a scalar �ield, where the robot is attracted to the goal and repulsed from the obstacles.

Cell decomposition methods on the other hand build a graph by discretizing the continuous state‑space using cell decomposition techniques, and converting the problem to a graph search problem .

Approximate cell decomposition is a class of cell de‑ composition, that overlays a grid with a deterministic resolution over the state space, and then generates a route in this grid by searching the embedded graph within it. Hence, this is usually known as grid‑based path planning .

Grid‐based path planning Among the graph search al‑ gorithms used in this approach, A* is one of the most famous methods, it �inds the optimal path in a graph using a heuristic function upon which its ef�iciency de‑ pends heavily. Later, many variations of the A* algo‑ rithm were developed to deal with its limitations.

D*(dynamic A*) [17] allows for ef�icient online re‑ planning in partially known as well as dynamic envi‑ ronments. LPA*(life long A*) [11] uses heuristics to fo‑ cus the replanning process when the implicit graph’s topology or its edges costs change. However, these two algorithms come with high memory costs [17].

D* Lite [10] builds on LPA* but avoids the reordering of the priority queue, achieving a lower computatio‑ nal cost than LPA*, and guarantees an ef�iciency equal or greater than that of D*with a lower memory con‑ sumption than both of them. Field D* [4] uses linear interpolation during re‑planning to calculate accurate path cost estimates for arbitrary positions within each grid cell, generating paths with a continuous range of headings as opposed to other A* variants mentioned above.

Journal of Automation, Mobile Robotics and Intelligent Systems Journal of Automation, Mobile Robotics and Intelligent Systems

Stochastic methods The high memory and computa‑ tion time requirements needed for grid‑based plan‑ ners in large search spaces inspired the development of incremental stochastic planning methods such as RRT, where random samples are drawn from a uni‑ form distribution over the planning domain, and then used to build a search tree in the free space. Howe‑ ver, since RRT produces only a feasible sub‑optimal path, it was necessary to develop an optimal random sampling‑based algorithm, and so RRT* was proposed

by Karaman and Frazzoli in 2011 [8], where the con‑ cept of tree rewiring was used in order to enhance the paths extended by the tree using new samples. This al‑ gorithm is therefore considered to be asymptotically optimal since the cost of the generated paths approa‑ ches the optimum as the number of iterations approa‑ ches in�inity, however, the rate of convergence to that optimal solution remained an issue.

In 2014 Gamel, Srinvansa, and Barfoot [5] presented a focused version of RRT* called informed‑RRT*, they �irst use RRT* to �ind an initial path between the start pose and the goal region, this initial path is then used to calculate an ellipsoidal subset from which the new samples will be drawn instead of the full state space, focusing the search to only include this subset, which contains the states that may improve the initial path, they also proved that the informed RRT* outperforms RRT* in terms of convergence rate, �inal cost, and the ability to deal with narrow passages.

Learning‐Based Methods recently, new learning‑ based algorithms were widely applied to solve path planning problems.

MPN(motion planning Networks) [16] developed by Qureshi et al, uses a neural motion planner called MPNet to plan a path from a start to a goal position directly from a provided point cloud and proved to be more ef�icient and consistent than the state of the art BIT* algorithm. in [20], the authors implemented a policy‑based search method that can improve planning times by learning implicit sampling distributions for particular environments, however, this method doesn’t offer a solution for all types of problems, environments with narrow passages for example are still a serious challenge. [7] also proposes a method for sampling biasing, they use a conditional variational autoen‑ coder (CVAE) to construct a nonuniform sampling distribution, but the need for a lot of preprocessed conditional information, and the fact that it is a multi‑process generative model demands a great deal of time and effort for predicting the whole sampling distribution [21].

2.3. RRT*

Algorithm 1 describes the operation of the RRT* algorithm: It builds a tree rooted at the initial state xstart . At each iteration, a sample is drawn from a uni‑ form distribution over X. If the new sample is loca‑ ted in an obstacle free location, the algorithm proceeds

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Algorithm 1: RRT ∗ (X, xstart , xgoal ) 1: 2: 3: 4: 5: 6: 7: 8: 9:

10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24: 25: 26: 27: 28: 29: 30: 31: 32: 33: 34: 35: 36: 37: 38: 39: 40:

V ← {xstart },E ← ∅ T ← (V, E) cinit ← ∞ for k ∈ {1, ..., K} do xrand ← Sample(X) Xnearest ← N earest(T, xrand ) xnew ← Steer(xnearest , xrand ) if ObstacleF ree(Xnearest , Xnew ) then V ← V ∪ {xnew } Xnear ← N ear(T, xnew , rRRT ∗ ) xmin ← xnearest cmin ← cost(xmin ) + d2 (xnearest , xnew ) for all xnear ∈ Xnear do cnew ← cost(xnear ) + d2 (xnear , xnew ) if cnew < cmin then if CollisionF ree(xnear , xnew ) then xmin ← xnear cmin ← cnew end if end if end for E ← E ∪ {(xmin , xnew )} for all xnear ∈ Xnear do cnear ← cost(xnear ) cnew ← cost(xnew ) + d2 (xnew , xnear ) if cnew < cnear then if CollisionF ree(xnew , xnear ) then xparent ← P arent(xnear ) E ← E {(xparent , xnear )} E ← E ∪ {(xnew , xnear )} end if end if end for if IsGoalState(xnew ) then return Cinit ← cost(xnew ) end if end if end for return Cinit

by extracting its nearest neighbor in the tree (line 6) xnearest . The function steer then chooses an obsta‑ cle free con�iguration between these two labeled xnew so that it minimizes the distance d2 (xnearest − xnew ) while maintaining d2 (xnew − xrand ) ≤ n where n is a prede�ined value .

If the path from xnearest to xnew does not cross any ob‑ stacles, xnew is added to the RRT* vertex list (line 9).

The algorithm then loops through the set of the near neighbors of xnew in the tree, and �ind the best parent for xnew from that set, a parent that provides the mi‑ nimum cost to come to xnew from xstart through the tree. After that, the algorithm rewires the tree using xnew , by assigning it as the new parent of any node in its neighborhood, if the cost to come to that node through xnew is better than the cost to come through its old parent. The algorithm then keeps iterating until a termination condition is met.

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In our case and at this point, we are only concerned with �inding a feasible path, this means that when the RRT* tree reaches the goal region ( line 34 ), the algo‑ rithm should be terminated. Also A maximum number of iterations K can be assigned to the RRT* loop, after which we consider the algorithm to have failed in �in‑ ding the solution, and thus we return Cinit , which in this case still have its initial value ∞ [8] [5].

3. Focused Discretization

Fig. 2. This figure shows the ellipsoidal subset Xs in the case of a 2‐D path planning problem calculated departing from the initial path between xstart and xgoal of cost Cinit inside Xr which is the rectangular subset that tightly bounds Xs where its width and height are equal to Xs ’s diameters it is used to mark the new boundaries of the state‐space to be discretized In this paper, we are applying the concept of focu‑ sed search from [8], in the context of grid‑based plan‑ ning, to decrease the high consumption of resources, that generally accompanies the use of grids in state‑ space discretization.

First, let Cinit be the cost of a path π that connects xstart and xgoal , and let Cx = f (x) + g(x) be the cost of the optimal path π∗, that is constrained to in‑ clude the state x ∈ X, where f (x) = d2 (xstart , x) and g(x) = d2 (x, xgoal ). Therefore, we can de�ine a sub‑ set Xs ⊆ X that consists exclusively of the states that ful�ill the following inequality, Cx ≤ Cinit , allowing these states to be components of a path with a lower cost than Cinit . And since the objective here is to �ind the optimal path, Xs can be a good basis for focusing the search process.

According to the condition above, Xs is an n‑ dimensional prolate hyper‑ellipsoid, in the 2D case it is represented as an ellipse with xstart and xgoal as √ its focal points and D1 = Cinit and D2 = 2 2 Cinit − Cmin as its transverse and conjugate diame‑ ters, where Cmin = d2 (xstart , xgoal ) [5]. In order to successfully use a regular square grid for the discre‑ tization procedure we need to de�ine another subset Xr ⊆ X, a hyper‑rectangle that tightly bounds Xs and represents the new boundaries of the state‑space as shown in �igure 2. And �inally Xr can be used instead of X in grid‑based path planning, saving a substan‑ tial amount of resources that can be allocated to other tasks.

4. FDA*

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FDA* is an implementation of the focused discre‑ tization concept discussed in the preceding section, Articles

VOLUME 15, N° 3 2021 VOLUME 15, N° 3 2021

which results in a more cost‑effective grid‑based plan‑ ning as shown by the results presented in the next section. The pseudo‑code for FDA* is shown in algo‑ rithm 2.

The procedure followed by grid‑based methods can often be divided into two phases. The discretization phase where the original state‑space is subdivided using a grid with an initial resolution then comes the planning phase where a graph search algorithm is used to obtain the optimal path within this grid. If no path was returned, the planner can increment the re‑ solution and start again from phase 1. This procedure continues until either a solution is found or some ot‑ her termination condition is triggered.

The only difference between FDA* and classic grid‑ based planning using A*is the introduction of lines 2 and 3 in algorithm 2. Where the RRT* algorithm is executed �irst for a few iterations until an initial path is found(line2). Its cost Cinit is then passed to the function new_bounds as an argument, along with the original planning domain X and the xstart and xgoal con�igurations. This function will subsequently return the vertices of the rectangle Xr that tightly bounds the ellipsoidal subset Xs .

Following this, the new state‑space is discretized (line 4), generating a square grid Gr that can be �inally se‑ arched by the A* algorithm in order to compute the resolution optimal path between xstart and xgoal .

Algorithm 2: F DA∗ (X, Cinit , xstart , xgoal ) 1: Lr ← initialize to the original domain 2: Cinit ← RRT ∗ (X, xstart , xgoal ) 3: Lr ← new_bounds(X, Cinit , xstart , xgoal ) 4: Gr ← discretiz(X, Lr ) 5: A∗ (Gr )

5. Results and Simulations Initially in this section, we will present the results of an experiment designed for demonstrating the per‑ formance of our method regarding memory consump‑ tion, and then we will display the outcomes of an expe‑ rimental comparison between FDA* and the classical A*, aimed for highlighting the difference in execution time between these two algorithms.

Each one of these experiments is conducted by executing a large number of Monte‑Carlo simulations in environments populated with randomly generated (30x30) pixels obstacles. In addition to this, the obsta‑ cle density will be varied to create environments with different complexities (examples are shown in �igure 3). It is important to note that the density levels used in our experiments do not exceed 32% since most real‑ life applications operate within these limits.

It should also be noted that both of these experi‑ ments are implemented with the same un‑optimized code which eliminates any effects code‑ef�iciency might have on the results of the comparison, also all experiments were run on the same 2.3GHz Intel i5 8th gen processor, 8GB memory machine.

and Intelligent Systems Journal of Automation, Automation,Mobile MobileRobotics Robotics and Intelligent Systems

(a) 3%

(b) 11%

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(c) 31%

Fig. 3. Examples depicting some the of the environments used in the conducted experiments (a) a low density environment (b) a medium density environment (c) a high density environment Fig. 6. Average execution time needed by FDA* and A* for finding the optimal path in 2‐D environments with varying obstacle densities and a constant distance between the start and goal poses ments still reach an average of 83 % reduction.

5.2. Execution Time

Fig. 4. Comparison between the average memory space consumed by FDA* and A* algorithms

Fig. 5. The average percentage of memory space saved using FDA* instead of the classic A* algorithm 5.1. Memory Use In this �irst experiment, we ran �onte‑�arlo simu‑ lations for both the new proposed (FDA*) and A* al‑ gorithms in a 1200x1200 pixels map, while keeping a constant distance between the start and goal positi‑ ons. Each time a simulation was executed, the memory space allocated for each algorithm was recorded, and at the end, the results were averaged and �inally illus‑ trated in �igure 4.

This experiment shows a marked drop in memory con‑ sumption for the FDA* algorithm compared with A*, which is also displayed in �igure 5, where their ratio is illustrated, showing an average of 92% reduction in terms of memory usage in relatively simple environ‑ ments, such as environment (a) in �igure 3. And even though this percentage decreases in more clustered spaces with narrow passages, more complex environ‑

In an experimental comparison, we ran both FDA* and the classical A* algorithm through the same num‑ ber of simulations as the �irst experiment, in an envi‑ ronment of 600x1200 pixels and recorded then avera‑ ged the execution time for both of them.

Figure 6 shows the results of this experiment where we can clearly see that our algorithm is faster in envi‑ ronments where the obstacle density is less than 15 %, which is considered to be the range where most of the real‑life applications operate in. However, beyond that our algorithm is out‑preformed by the conventional A* path planning. Similar to the second experiment, se‑ veral additional experiments were performed by va‑ rying only the size of the map. Table 1 illustrates the results where we can clearly see that the density from where our algorithm becomes slower than A* gets hig‑ her with the increase of the problem scale, giving our algorithm the upper hand in relatively large‑scale pro‑ blems.

6. Discussions and Conclusions

In the course of this paper, we discuss improving the expensive process of grid‑based state‑space dis‑ cretization in single query path planning problems, through cropping the search space to only include a small subset, represented by a hyper‑rectangle, in which the optimal solution can be found.

By doing so, we eliminate the states that are unneces‑ sary for �inding the optimal path and thus reduce the amount of resources dedicated to this process.

�e have demonstrated in the �irst experiment, that focusing the discretization process can immen‑ sely decrease the memory requirements, especially in large‑scale environments. The Focused Discretization technique bene�its from the fact that in such environ‑ ments a considerable amount of states does not con‑ tribute to solving the optimal path planning problem, and by discarding them, we save the memory space needed for storing these states.

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Journal and Intelligent Systems Journal of of Automation, Automation,Mobile MobileRobotics Robotics and Intelligent Systems

Tab. 1. Comparison of the performance of FDA* with the classic A* algorithm regarding execution time in different scale environments

env 1 2 3 4 5

dimensions (pixels)

512 x 512 700 x 700 900 x 900 1200 x 1200 1500 x 1500

start‑goal distance (pixels) 300 300 300 300 300

obstacle density from which FDA* becomes out‑performed 0% 16 % 16.50 % 19.37 % 21 %

We have further shown that this approach can reduce the overall time required to �ind the optimal solution in low and medium obstacle density environments, where most of the real‑world applications operate.

This is a result of the lower number of states (i.e.grid cells), which reduces the time needed for reading and writing the grid into memory, and thus reducing the overall execution time of grid‑based planners.

Moreover, these effects are ampli�ied in large‑scale en‑ vironments where even a greater number of states do not contribute to the optimal path planning.

This was made clear by comparing FDA*, to our im‑ plementation of Focused Discretization with the clas‑ sical approach, using A* as a graph search algorithm for both of them. The results presented in �igure 5 con�irms the superiority of our technique in medium and low dense environments, this is due to the RRT* fast planning combined with the effect of the reduced number of states to be handled, resulting in the overall FDA * computation time being better than that of the conventional A*.

As the obstacles density increases, it becomes more dif�icult for the RRT* algorithm to �ind an initial path, causing our method to have an overhead in extremely clustered environments with narrow passages. Howe‑ ver, the density in which our algorithm starts to be out‑performed seems to grow higher as the problem domain grows in size, and that is caused by the higher number of states handled by the classical approach.

The path length generated by FDA* is theoretically the same as A*, due to the fact that A* is used as a planner in the second stage of FDA* after the �irst stage (focu‑ sed discretization) is done, line 5 in algorithm 2. FDA* however, is more ef�icient in generating this solution.

Such ef�iciency is the result of focusing the discreti‑ zation process to a smaller subset of the state space, which leads to consuming less time and memory re‑ sources.

7. Future Work

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In the future, we intend to investigate the possibi‑ lity of using a learning‑based technique for estimating the focused subset instead of the stochastic approach, this would immensely save the valuable computatio‑ nal and storage resources, increasing the ef�iciency of our proposed method making it more suitable for real‑ Articles

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time applications.

AUTHORS

Mouad Boumediene – Laboratoire Automati‑ que Skikda, Road Elhadaiek, BP.26, Skikda, Alge‑ ria, e‑mail: mouadboumediene@yahoo.fr, www: http://vrpg.univ‑skikda.dz/las/.

Lamine Mehennaoui∗ – Laboratoire Automati‑ que Skikda, Road Elhadaiek, BP.26, Skikda, Algeria, e‑mail: me_lamine@yahoo.fr, www: http://vrpg.univ‑ skikda.dz/las/.

Abderazzak Lachouri – Laboratoire Automatique Skikda, Road Elhadaiek, BP.26, Skikda, Algeria, e‑mail: alachouri@yahoo.fr, www: http://vrpg.univ‑ skikda.dz/las/. ∗

Corresponding author

REFERENCES

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Risk Analysis Method by the Extreme Data of Dependent Exogenous Variables Submitted: 26th July 2021; accepted: 10th March 2022

Ihor Tereshchenko, Anton Tereshchenko, Nataliya Bilous, Svetlana Shtangey, Zygmunt L. Warsza DOI: 10.14313/JAMRIS/3-2021/18

Nomenclature:

Abstract: Many practical tasks of data multivariate statistical analysis from the standpoint of a risk-oriented process approach (in accordance with ISO 9001: 2015, 31000: 2018) requires the definition of the risk values for the dependent exogenous variables of some processes. This paper proposes the method, which consist of original stages sequence for calculating value-at-risk (VaR) or conditional-value-at-risk (CVaR) of dependent exogenous variables, presented of the extreme data frame of critical manufacture process parameters or other parameters, for example, extreme data of environmental monitoring and etc. Risk analysis method by the extreme data of dependent exogenous variables, presented of the data matrix, uses the result of solving the formalized problem of defines the tails parameters of the joint distributions of exogenous variables as components of a bivariate random variable. It can be argued that the tails parameters of the joint distributions of dependent exogenous variables make the validated corrections of the VaR and CVaR estimates for such variables. This method expands the practical application of extreme value theory for the value at risk analysis of any dependent variables as process parameters.

{ X i }i≥1 :

Keywords: exogenous variables; risk-oriented process approach; extreme value theory; tailed distribution Abbreviations: VaR: Value at Risk CVaR: Conditional Value at Risk GPD: Generalized Pareto Distribution GEVD: Generalized Extreme Value Distribution POT: Peaks-Over-Threshold EVS: Extreme Value Statistics ES: Expected Shortfall EVT: Extreme Value Theory QMS: Quality Management System EI: Extremal Index EVI: Extremal Value Index Q-Q: Quantile-Quantile CI: Confidence Interval

44

X i = ( xi1,..., xiD ) :

i = 1,..., n :

D (1 ≤ d ≤ D) :

F():

xid :

M n = ( M n1,...,M nD ) :

M nd = max { xi1 ,..., xiD }

M = {M nd } : an , bn :

G( ):

ξ d , µd , σ d :

χ: u: Z:

Introduction

The sequence of independent and identically distributed (i.i.d.) vector of random exogenous variables Vector of random exogenous variables The sequence time numbers of multivariate elements (exogenous variables) Dimension the vector of random exogenous variables The marginal distribution function of exogenous variable The exogenous variable values

The data-frame of componentwise (exogenous variables) maximums

Vector of exogenous variables maximums

The n × D -dimensional array of maximums (extremes) of n statistical D-dimensional exogenous variable observations The constants for a non-degenerate distribution function G The generalised extreme value distribution function

Extremal index, location parameter, scale parameter of marginal exogenous variables distributions respectively The probability of one variable being extreme given that the other is extreme Threshold

The standardized residuals from the fitted model of Heffernan & Tawn

By variables further mean exogenous variables whose cause is external to the model and whose role is to explain other variables or outcomes in the model [1].

The extreme values of process’s variables are the values which equal to, close to or exceed the limit values acceptable to certain requirements (for example, the

Journal of Automation, Mobile Robotics and Intelligent Systems

requirements of branch standards for product’s quality, environmental standards, etc.). Therefore mentally and functionally investigations are based on the general concept of risk-oriented management by ISO 9001: 2015/ISO 31000: 2018 and the methodology of the process approach [2-4].

A software solution reviews for the extreme value statistics [5-7] and related risk analysis [8-11] indicates to a positive trend of the researchers’ interest to these areas of statistics for the information technologies and a high relevance level of such investigations.

The evolution of the risk assessment methodology and the market success of the RiskMetrics Group [12, 13] can serve as a good example of the dynamic growth of progress and the usefulness of applied risk analysis.

Dynamic trading strategies can be considered as branch trends of risk analysis with a good analogy of application [14]. These strategies targets on a predefined level of risk as measured by volatility, Value at Risk (VaR) or Conditional Value at Risk (CVaR). Investigations have shown that targeting increases the risk-adjusted performance and heightens utility gains for mean-variance investors. As such, the target factorial direction of researches is accented.

Many solutions useful for practical application in this area of statistics are concentrated in modern software products of the R programming language.

So, it is necessary to note the “fExtremes” software package [15]. This software package contains some functions were implemented from Alec Stephenson’s R-package “evir” [16] imported from Alexander McNeil’s S library “EVIS” [17], extreme values in S programming language, some are from Alec Stephenson’s R-package “ismev” [18] based on Stuart Coles code from his book “Introduction to Statistical Modeling of Extreme Values” [19] and some were written by Diethelm Wuertz. The topics of “fExtremes” package includes: data preprocessing, explorative data analysis, peak over threshold modeling, block maxima modeling, estimation of VaR/CVaR and the computation of the extreme index.

It is necessary to note the content of the R-package “ReIns” (February 10, 2020) [20, 21] for risk analysis.

The important position of the book [20] is that in the risk analysis, a global fit that appropriately captures the body and the tail of the variables distributions is essential. The whole range modeling of the variables using a standard distribution is usually very hard and often impossible due to the specific characteristics of the body and the tail of the distributions of variables.

A possible solution is to combines two distributions in a splicing model [21]: a light-tailed distribution for the body which covers light and moderate variables, and a heavy-tailed distribution for the tail to capture large variables.

Note, that the proposed solution is an adequate alternative to the approximation of tail values by the Pareto distribution.

The R-package “texmex” [22] also has the actual content of the approaches that have been implemented. This software package for the statistical extreme value modeling of threshold excesses, maxima and

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multivariate extremes uses univariate models for threshold excesses and maxima are the Generalized Pareto Distribution (GPD) and Generalized Extreme Value Distribution (GEVD). Also, for serially dependent sequences, the intervals declustering algorithm of Ferro & Segers [23, 24] is provided, with diagnostic support to aid selection of threshold, declustering horizon and the computation of the Extremal Index (EI).

Multivariate modeling is performed via the conditional approach of Heffernan & Tawn [25, 26], with graphical tools for threshold selection and to diagnose estimation convergence.

The R-package “tsxtreme” also allows characterizing the extremes values dependence structure of time series via the Peaks-Over-Threshold (POT) methods [27]. It uses the Heffernan & Tawn conditional approach [26] which is flexible in terms of extremal and asymptotic dependence structures, and Bayesian methods improves efficiency and allow for deriving measures of uncertainty. For example, the EI, related to the size of clusters in time, can be estimated and samples from its posterior distribution obtained.

Thus, the use of the conditional Heffernan & Tawn’s approach is justified for solving the relevant problem of the risk analysis by the extreme data of dependent exogenous variables as the main goal of this article.

It is important that the Heffernan & Tawn’s method, evaluated and extrapolated the distribution of a two-dimensional random variable is developed a semi-parametric approach that overcomes the limitations to arguments when components of the two-dimensional variable become large at the same rate.

For the software packages that are considered it is important to note, that the POT methodology for Extreme Value Statistics (EVS) is applied, which uses more data and allows evaluating the behavior of extreme values above some high threshold [28, 29]. POT has an advantage because it allows to inference directly on the distribution of variables extremes [28].

So, at statistical estimates the extremes of the multivariate data the problem of the dependence (in particular, correlation) of the components (covariates of variables) of multivariate observations are inevitably arises. For this currently relevant problem, the review [30] considers the most interesting examples of strongly correlated variables for which there are very few exact of the EVS. In the paper [31] a test for detecting sequential correlations in multivariate time series was proposed.

This test uses Spearman’s rank correlation properties and the theory of extreme values.

Risk assessment can be performed using the model for maxima that can be obtained by combining the GEVD for the univariate marginal distributions with extreme-value copulas to describe their dependence structure, as justified by the theory of multivariate extreme values [32]. Here it is advisable to note the use of the copula mathematical tool [33–36] for analysis and assessments of nonlinear dependencies of variables. At present, paired copula models are mainly used, for which a mathematical tools and software methods have been developed.

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The paper [37] describes a multivariate statistical dependency model for hydrological observations, which used to estimate flood losses (i.e. risks) in a large and heterogeneous region.

So, the methodology of the extreme values theory and the subject of risk analysis are currently relevant and are considered in the overwhelming majority of cases for the financial, in particular the exchange, branches [7, 11]. Therefore, one of the article goals is to expand the application of the extreme values methodology and risk analysis to data monitoring of dependent variables for the industrial, ecological and other branches.

Note that the closest to the topic of our article are the sources [30–37], in which the relationship between the observed extreme values and the influence of exogenous factors, including on risk values, is investigated. Review of the literature allows to conclusions: – The relevance of risk analysis and the need to expand it for monitoring dependent process parameters/exogenous variables not only in the financial branches.

– The POT advantage for risk methods analyses because it uses more data, allows evaluating the behavior of extreme values above some high threshold and inferences to be made directly on the distribution of exogenous extremes as a risk values. Another advantage of the POT approach is that common risk measures like VaR and Expected Shortfall (ES/CVaR) may be computed [28].

– The copula mathematical tools provides convenient instruments for analyzing pairwise dependences of exogenous variables and calculating the parameters of asymptotic dependence.

– The conditional approach of Heffernan & Tawn is flexible in terms of extremal and asymptotic dependence structures together with Ferro & Seeger’s algorithm which helps to choose the threshold and the EI of variables extremes distribution.

It is also important that the statement of the problem and the method for solving it are performed from the perspective of an object-oriented analysis, which accords to modern tendencies of the process approach to support the product Quality Management System (QMS).

Risk Analysis Problem Statement

46

So, will be used the interpretation for multivariate sequences proposed in thesis [38]. Let { X i }i ≥1 be the sequence of independent and identically distributed (i.i.d.) vector of random exogenous variables X i = ( xi1,..., xiD ) with dimension D (1 ≤ d ≤ D) and marginal distribution function F, where i = 1,..., n is the sequence time numbers of multivariate elements (exogenous variables). Many definitions are possible for the maximum of n consecutive elements of a multivariate sequence.

For estimating the dependence of the exogenous variables values xid , will use the conditional approach proposed by Heffernan & Tawn [25, 26], when maximum/extreme of n consecutive elements is defined as Articles

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the data-frame of componentwise (i.e. for each exogenous variables) maximums: , M n = ( M n1,...,M nD )

where M nd = max { xi1,..., xiD } for each 1 ≤ d ≤ D .

Thus, will form M = {M nd } as an n × D -dimensional array of maximums (extremes) of n statistical D-dimensional observations are known for process’s variables. Such data-set covers the examples of situations in which dependence at extreme levels is a consequence of proximity in space, time or dependence on a common covariate/variable [25].

Non-degenerate limits are obtained for the distribution function  M −b  n P  n n ≤ x=  F ( an +bn ) ,  an 

x = ( x1 ,..., xD ) , bn = ( bn1 ,..., bnD ) .

where

an = ( an1 ,..., anD ) > 0

(1)

and

All operations are performed componentwise, so an > 0 is translated as an1 > 0,..., anD > 0 . The Haan and Resnick theorem [39] characterizes all of the possible limit distributions. If there exist sequences of constants an > 0 and bn such that  M −b  w P  n n ≤ x  → G ( x ) as n → ∞ ,  an 

(2)

for a non-degenerate distribution function G, then each of the D one-dimensional component distributions of G is a GEVD function Gd ( xd=)

    exp  −1+ξd       

for 1 ≤ d ≤ D

−1  xd − µd   ξd    σ d    , 

(3)

with standard Frechet components, where ξ d is a EI/EVI or shape parameter (shape), µd is a location parameter (location) and σ d is a scale parameter (scale) of marginal exogenous variables distributions.

Note, that for high threshold values that are given by high quantiles (0.95 and higher), the distribution of probabilities for random maximums of exogenous variables values is close to the generalized Gaussian or Pareto distributions [28, 40].

The problem statement as follows: for an array of observed maximum values of variables – M, taking into account their pairwise dependence, develop and investigation the method for calculating VaR/CVaR as high quantiles of the respectively joint distribution functions of maximum values of variables.

Assuming, for the moment [25, 26], that the marginal distributions of variables are identical and one natural measure of their pairwise dependence is a parameters of paired conditional distributions of dependent variables.

Proposed Method

The method proposes the sequence of statistical computational procedures of processing the observations of multivariate exogenous variables extreme values.

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Consider the content of the implementation stages for proposed method.

Stage 1: definition the dependent variables of the M extreme data array using the copula mathematical tools.

Generalized univariate limiting distribution of extreme values for describe the dependence structure of the variables can be combined with copulas of this extreme values [35].

In this case, copula (C) is the joint distribution function of variables random vectors X and Y after transformation to variables U and V, with uniform [0, 1] margins, via (U , V ) = { FX ( x), FY ( y )} [25].

Here X and Y are M nd for different components d.

Then the pairs ( ui , vi ) , i = 1,..., n are independent realizations with approximate distribution C. Copula C is used to compute function values that give an empirical measure of the type and strength of the tail dependence exhibited by the data.

In particular, was calculated the values of the functions ChiBar ( χ (u ) ) and Chi ( χ (u ) ) [22] described by Coles, Heffernan & Tawn [25]: χ is the probability of one variable being extreme given that the other is extreme , (4) = χ lim Pr(V > u | U > u ) u →1

χ

 2 log Pr(U > u )  − 1 lim  u →1 log Pr(U > u , V > u )  ,

where −1 ≤ χ (u ) ≤ 1 fo all 0 ≤ u ≤ 1 .

(5)

The both measures ChiBar and Chi are needed in order to obtain a summary that is informative for variables which may be either asymptotically independent or asymptotically dependent.

Stage 2: fitting of the conditional distributions functions of the most dependent variables extreme values to the observed data via the Heffernan & Tawn’s method in accordance with results of the stage 1.

Note that the simulated Heffernan & Town’s model data above the prediction threshold can be obtained for both point and bootstrap estimates of the dependency model parameters. Q-Q diagnostics allows comparing the simulated distributions and estimating their compliance with the Pareto distribution.

Thus the simulated values of the dependent variables are created, given that the conditioning variable is above its high quantile. This collection of values is interpreted as a tail distribution.

Stage 3: definition the parameters of the conditional distributions functions of the dependent variables extreme values: the scale parameter is sigma (σ) and the shape parameter is xi (ξ) (EVI/EI).

At this stage, the parameters are determined for the conditional distribution functions of pairwise dependent variables. The probability that one variable will be extreme (exceeds a certain threshold) is determined given that the other variable is also extreme or exceeds a certain threshold. Thus, a complete description of the dependent variables, the values of which exceed the established thresholds, is available for analysis.

Note that for further risk analysis the EI estimating is a great importance. In paper, EI values were esti-

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mated using the Ferro & Seeger’s method for dependent extremes [23, 26]. According to these methods, EI value defines for variable, above which EI is stable for higher values of this variable [40]. It is recommended to choose such values of variables as a “high enough threshold” u [41].

Stage 4: the VaR and CVaR calculations as quantile-dependent values using the parameters of the conditional distributions functions of the dependent variables extreme values based on Heffernan & Town’s model.

Thus, the VaR and CVaR were determined based on the parameters of the conditional Pareto distribution that is the shape parameter ξ and the scale parameter σ, for some high enough threshold u [28]. For the simulated tail Pareto distribution of the dependent variables values the VaR and CVaR are calculated by the formulas [28]:

σ n ⋅ (( (1 − q )) −ξ − 1) ξ r σ + ξ (VaR − u ) , CVaR = 1− ξ

VaR =u +

(6) (7)

where r is a quantity of exceedances over the threshold u, q is a quantile probability.

At this stage, note the advantage of POT analysis, which is the ability to draw conclusions directly about the all distribution of variables extreme values over a given threshold [28].

Stage 5: the VaR/CVaR analyzes for the dependent and independent exogenous variables.

Comment: the VaR/CVaR without taking into account the dependence of exogenous variables are calculated using the parameters of the marginal distributions of extreme variables values.

Thus, the above sequence of the method’s stages allows solving the problem posed in section 2. An important goal is also to test the operation of the method created for proprietary data on well-known data.

Experimental Results

The paper uses five-dimensional air quality monitoring data comprising the measurements series of ground level ozone (O3), nitrogen dioxide (NO2), nitrogen oxide (NO), sulphur dioxide (SO2) and particulate matter (PM10) in Leeds (UK) city center, during the years 1994–1998 inclusively [22, 26].

In particular, was used the daily maxima for dependent NO and NO2 variables during winter periods 1994–1998 inclusively. Note, the gases are recorded in parts per billion and the particulate matter in micrograms per cubic meter.

Researches represent a synergy of parametric and nonparametric statistical procedures as well as uses descriptive and inferential statistic tools for data.

At the first stage, for M array an estimate of the variables dependence was obtained by combining the generalized extreme value distribution for the univariate marginal distributions with extreme value copulas to describe their dependence structure, as justified by the theory of multivariate extreme values [35]. So, Fig. 1 visualizes the character of dependence Articles

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of five exogenous variables. For example, variables NO and NO2 (NO/NO2) demonstrate a clear dependence.

Fig. 1. Exploratory copula-diagram for proposed data Next, the graphs of the ChiBar ( ) and Chi ( ) functions are plotted, which give an empirical measure of the type and strength of the tail dependence demonstrated by the data NO/NO2, as shown in Fig. 2.

Parameter u is the upper limit of the general marginal distribution support and is the probability of one variable being extreme given that the other is extreme. In the case χ = 0 the variables are said to be asymptotically independent [25].

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for NO/NO2 are equivalent to those for NO2/NO) [22, 26].

At the second stage the fitting of the conditional distributions functions of the most dependent variables extreme values to the observed data via the Heffernan & Tawn’s method [26] in accordance with the results of stage 1 is produced.

Three diagnostic plots are produced for dependent variables NO/NO2, as shown in Fig. 3 [22].

Fig. 3a shows scatterplots of the residuals Z from the fitted model of Heffernan & Tawn are plotted against the quantile of the conditioning variable (NO2), with a lowess curve showing the local mean of these points.

Fig. 3b shows the absolute value of Z – mean ( Z ) is also plotted again with the lowess curve showing the local mean of these points. Any trend in the location or scatter of these variables with the conditioning variable indicates a violation of the model assumption that the residuals Z are independent of the conditioning variable [28]. This can be indicative of the dependence threshold used being too low. a)

b)

c)

Fig. 3. Diagnostic plots for dependent variables NO/NO2

Fig. 2. Graphs the ChiBar and Chi values dependencies on 100 quantiles of the random value u distribution at the confidence interval (brown background) background

48

The significance level for the confidence interval is 0.05. A limiting value of ChiBar equal to 1 indicates asymptotic dependence of NO/NO2, in which case the limiting value of Chi gives a measure of the strength of dependence in this class, as shown in Fig. 2. In the case that a limiting value of ChiBar of less than 1 is it indicates asymptotic independence in which case Chi is irrelevant and the limiting value of ChiBar gives a measure of the strength of dependence [25]. In the case of the confidence interval for ChiBar excluding the value 1 for all of the largest quantiles, the plot of the Chi function is shown in grey on Fig.2.

It is important that the ChiBar and Chi values are integral estimates of the relationship of two exogenous variables that are permutation invariant to the choice of the conditioning variable (i.e. the estimates Articles

Fig. 3c shows the original data (on the original scale) and the fitted quantiles (specified by quantiles) of the conditional distribution of dependent variable given the conditioning variable. A model that fits well will have good agreement between the distribution of the raw data (shown by the scatter plot) and the fitted quantiles. Simulated data for a collection of 5 generalized Pareto models (dashed lines) is generated above the conditioning variable threshold (vertical line).

Note that the simulated data above the prediction threshold can be obtained for both point and bootstrap estimates the parameters of the dependent variables conditional distribution model by Heffernan & Town. Despite of the Fig. 3 informativeness, this visualization is not convincing enough to assess the accordance of the simulated data to the Pareto distribution. Therefore, Q-Q diagnosis is used for simulated values are based on the point and bootstrap estimates of the dependence model parameters. Q-Q diagnostics allows you to compare the simulated distributions and assess their accordance with the Pareto distribution, as shown in Fig. 4.

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a)

b)

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the estimated EI is stable over further higher thresholds. This can be analyzed using Fig. 5, where the EI is shown for each threshold.

Graphs were plotted for the Heffernan & Tawn’s model for point and bootstrap estimates of the Pareto conditional distribution parameters, as shown in Fig.5 a and b respectively. The values are equal to ξ= –0.142 for point model and ξ= –0.164 for bootstrap model. Uncertainty in the estimation of the EI and GPD parameters where assessed by using a bootstrap scheme which accounts for uncertainty in the EI estimation, and the appropriate uncertainty in the declustering of the series [22].

At the fourth stage VaR/CVaR are calculated for point and bootstrap estimates of the Heffernan & Tawn’s dependence model parameters. VaR/CVaR were also calculated for independent exogenous variables. The VaR/CVaR values are given in Table 1, where prob. – the probabilities (quantiles levels) of the conditional and marginal distributions quantiles for the VaR/CVaR calculation. a)

Fig. 4. Graphical Q-Q-verification the accordance of the reference distribution to the simulated distributions

by Heffernan & Tawn’s dependence model for point (a) and bootstrap (b) estimates of the dependence model parameters The Q-Q-plot using the Pareto distribution as a reference distribution is graphical technique to infer the tail behavior of observed variables [28]. If the excesses over thresholds are from a thin-tailed distribution, then the GPD is exponential with ξ= 0 and the Q-Q-plot should be linear. Departures from linearity in the Q-Q-plot then indicate either fat-tailed behavior (ξ>0) or bounded tails (ξ<0). The simulated data show nearly linear convergence to the reference values for excesses over thresholds with bounded-tailed (ξ= –0.12).

Note that the simulated values of variables for the procedure of fitting the generalized Pareto models to the original data according to the Heffernan & Towne’s method based on the point estimation of the dependency model parameters demonstrate close identity with the simulated values, which additionally contain simulated repeating datasets in accordance to bootstrap-estimation of dependency model parameters.

At the third stage estimates the EI of a dependent series of observations above a given threshold. An appropriate choice of threshold is one above which

b)

Fig. 5. The EI dependence graphs on the threshold exogenous variables values of the Heffernan & Tawn’s model for the point (a) and bootstrap (b) estimates of the Pareto conditional distribution parameters

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Tab. 1. VaR/CVaR for point and bootstrap estimates of the Heffernan & Tawn dependence model parameters and for independent exogenous variable №

1

2

3

4

5

prob.

point model

0.9900

0.9950

CVaR

VaR

CVaR

VaR

CVaR

691.63

768.06

677.94

750.37

494.17

531.11

819.84

865.67

0.9995

730.07

931.32

913.09

0.9999

independent exogene NO

VaR

746.83

0.9990

bootstrap model

1015.16

799.55

843.06

975.80

906.13

888.45

1071.55

948.94

986.83

1041.74

526.04

571.03

582.41

598.47

553.13

584.22

592.08 603.18

Tab. 2. Values for 99%-th quantile level of VaR/CVaR estimates and 95% CIs VAR

CVaR

Lower CI

Estimate

Upper CI

Lower CI

Estimate

Upper CI

647.49

677.94

725.12

703.36

750.37

842.59

From Table 1 it can be seen that calculated VaR/ CVaR based on the point estimate of the Heffernan & Tawn’s dependence model parameters are fairly close to the calculated values based on the bootstrap estimate of the Heffernan & Tawn’s dependence model parameters. The calculated VaR/CVaR for independent variable distribution is smaller than the risks values based on the estimates of the Heffernan & Tawn’s dependence model parameters for all quantiles levels.

At the fifth stage makes the analysis of VaR/CVaR values for dependent and independent exogenous variables. This stage uses the advantage of POT analysis which is the ability to draw conclusions about the entire available distribution of extreme variables values over a given threshold. Estimated VaR/CVaR values and confidence intervals (CIs) are shown in Fig. 6.

Fig. 6. Tail plot for exogenous variables conditional distribution over the threshold for the bootstrap estimate of the Heffernan & Tawn’s dependence model parameters and CIs (red dotted line) with the estimates of VaR/CVaR high quantiles (vertical dotted line)

50

For further analysis a graphical representation for exogenous variables conditional distribution over the threshold (solid black curve) on Fig. 6 is added by the CIs for estimates of VaR/CVaR high quantiles for the variant of bootstrap estimate of the Heffernan & Tawn’s dependence model parameters. The CIs values Articles

and estimates for VaR/CVaR in this case are characterized by the data in Table 2.

The sensitivity of the VaR estimates to changes in the threshold u to be investigated using the functions of R package “fExtremes” is shown in Fig. 7.

Fig. 7. Estimate plot for the varies of a high quantile in the tail of the simulated dataset of variables based on the Heffernan & Tawn’s conditional model in depend on threshold or number of extremes (95%-th CI shows with dashed lines) So, for analysis together with Fig. 6 is convenient to use Fig. 7 showing how the estimate of a high quantile for the tail of a conditional data based on the Heffernan & Tawn’s dependence model varies with threshold or number of extremes. The VaR estimates for 99%-th quantile level are stable for thresholds more than 410.

It can be argued that EVT is a useful supplementary risk measure because it provides more appropriate distributions to fit extreme events [40]. This makes it possible to reduce the uncertainty of VaR/CVaR estimates, which can be seen in Fig. 7 when the VaR estimates for 99% quantile level remains stable over a wide range of thresholds.

5. Conclusion

In risk analysis, a global fit that appropriately captures the body and the tail of the distribution of exogenous

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variables is essential. At statistical estimates the extremes data of the exogenous variables the problem of the dependence of these variables as the monitoring result is inevitably arises.

The accounting of the joint exogenous variables distribution is a relevant clarification of the risk value as it provides more adequate estimates of the variables distribution fitted to the extreme events. Therefore the problem of risk-analysis by the extreme data of dependent exogenous variables as the main goal of the article is solved using the conditional Heffernan & Tawn’s approach.

The result of this solution is the sequence and content of the actions (stages) that make up the proposed method. Studies have shown that the main trends in VaR/CVaR change for proprietary data of dependent exogenous variables are generally confirmed for wellknown data.

When solving the problem, the following were used: the POT approach for analysis of extreme values, the flexibility of the Heffernan & Tawn’s method to the structure of asymptotic dependencies, the advantages of the Ferro & Segers’s algorithm for choosing a threshold and an EI for the conditional exogenous variables distribution.

Thus, the method expands the application of the risk-analysis methodology for monitoring data of dependent exogenous variables not only for the financial area, but also for industrial, environmental and other areas.

It is also important that the proposed method opens up the prospect of research on the influence of VaR/CVaR estimates on the accuracy of interval estimates of observation parameters.

Risk-adjusted dependent extreme value variables research targeting is optimizes the branches performance in terms of functional load on process elements and benefits stakeholders.

Nataliya Bilous* – Software Engineering Department, Kharkiv National University of Radio Electronics, Kharkiv, 61166, Ukraine, e-mail: nataliya.bilous@nure.ua.

Svitlana Shtangey – Infocommunication Engineering Department, Kharkiv National University of Radio Electronics, Kharkiv, 61166, Ukraine, e-mail: svitlana.shtanhei@nure.ua.

Zygmunt L. Warsza* – Łukasiewicz Research Network – Industrial Research Institute for Automation and Measurements PIAP, e-mail: zlw1936@gmail.com. ∗ Corresponding authors

This paper is submitted in a competition framework of scientific and technological developments “Science for the safety of man and society” in accordance with the Resolution of the scientific council of the National Research Foundation of Ukraine (protocol №. 7 of May 11, 2020). These studies have been extended for the topic “Study of psychophysiological functions and intelligence of students in the conditions of life in a state of epidemic limitation”.

[6]

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ACKNOWLEDGEMENTS

AUTHORS

Ihor Tereshchenko – Infocommunication Engineering Department, Kharkiv National University of Radio Electronics, Kharkiv, 61166, Ukraine, e-mail: ihor.tereshchenko@nure.ua.

Anton Tereshchenko – Management of Information and Cyber Security Department, State University of Telecommunications, Kyiv, 03680, Ukraine, e-mail: arisen001@gmail.com.

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Unified Model of Disturbances Acting Upon Gimbal Seeker in Anti-Tank Guided Missile Submitted: 7th October 2021; accepted: 9th February 2022

Radosław Nawrocki DOI: 10.14313/JAMRIS/3-2021/19

2. Models of Individual Internal Disturbances in Gimbal Seeker

Abstract: Optimisation of the dynamic response of gimbal seeker plays key role from the point of view of development of anti-tank guided missile’s systems. In this study the set of the most important internal disturbances were integrated in generalized model of two axis gimbal seeker implemented in MathWorks’ Simulink environment.

Compared to previous works on the subject, it was enhanced by replacing simple friction model with dynamic LuGre friction. Furthermore, its Coulomb component was linked to the normal force induced by missile’s lateral acceleration. Control system of gimbal seeker proposed in paper was tuned with modelled disturbances turned off and then examined with them being turned one by one.

System’s responses were assessed to be significantly deteriorated, proving need of disturbance modelling and its use in control systems’ design.

Main sources of disturbances were described in article by Masten [4]. By definition, some of them could be labelled as external, which means that their source is not directly associated with seeker, but rather its environment (vehicle motion, atmospheric disturbances [5, 6], etc.). Others are internal – inherently related to gimbal’s physical phenomena and its structure. Most notable ones were listed below.

2.1. Friction

Miss distance (m)

The most important goal of seeker’s gimbal is to keep detector pointed towards acquired target or any other predefined fragment of space. Considering gimbal which uses direct drive to actuate its payload, it is easy to imagine that in idealized system with no friction, this goal would be achieved as a result of Newton’s first law of motion [4]. No friction in bearings partially separates detector from missile Keywords: anti-tank guided missiles, gimbal seeker, dismovements that often could be stochastic in nature. turbances, dynamic LuGre friction, imbalance, cross couResearch by Lin, Hsiao [7] underlines how important pling coefficients of friction are in context of miss distance criterion, which is one of the most critical functional 1. Introduction ATGM parameters. In this sense, miss distance can Third generation anti-tank guided missiles (ATGMs) be defined as distance of missile’s closest approach are one of the most important development directo target [8]. Fig. shows how increasing friction tions in category of precision-guided munitions. They disturbance torque can quickly render missile useowe this to their high effectiveness against armour less, having in mind that commonly used HEAT warachieved mostly by top attack capability and also user heads need direct hit to be effective, and standard Fig. shows how increasing friction disturbance torque can quickly render missile useless, having in mind th safety assured by fire and forget operation, despite NATO target used for evaluation of optical systems is commonly used HEAT warheads need direct hit to be effective, and standard NATO target used for evaluation their high price [1]. Their defining feature is gimbal mm×[9].

2.3 m [9]. optical systems is 2.3 2.3 m x 2.3 seeker which, through means of tilting their line of sight, allows engaging targets from the top [2].

Third generation ATGMs, which are precise and 100 highly integrated systems, are prone to many factors degrading their performance [3]. This is why it is especially important to identify and model disturbances acting upon them – mistakes made during their devel50 opment phase are very costly.

Literature research shown significant number of papers concerning modelling gimbal seekers, taking into account not all, but some of possible disturbances. Aim of this paper is to evaluate current state of 0 0,000 0,002 0,004 0,006 0,008 0,010 0,012 0,014 0,016 0,018 knowledge regarding disturbance modelling in conFriction disturbance torque (kg*m) text of gimbal seekers in third generation ATGMs, establish mathematical models for each disturbance and propose generalized disturbance model for such Fig. 1.Fig.Relationship between coefficient 1. Relationship between friction friction coefficient and miss distanceand [7] system. miss distance [7]

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According to Yu and Shang [3] system performance under heavy influence of disturbances could be improved ther by enhancing control system parameters or minimizing their magnitude through refining mechanical syste That can be done by choosing right design solutions and improving quality of assembly process or even machini precision of single parts. Concerning friction, to alleviate its impact as a disturbance, ball bearings are commonly us to bear gimbal’s structures mainly due to lower coefficients of friction.

Frictional component of torque disturbance could be modelled using many static and dynamic models as d

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According to Yu and Shang [3] system performance under heavy influence of disturbances could be improved either by enhancing control system parameters or minimizing their magnitude through refining mechanical system. That can be done by choosing right design solutions and improving quality of assembly process or even machining precision of single parts. Concerning friction, to alleviate its impact as a disturbance, ball bearings are commonly used to bear gimbal’s structures mainly due to lower coefficients of friction.

Frictional component of torque disturbance could be modelled using many static and dynamic models as described in work of Olsson, Å� ström, Canudas de Wit, Gäfvert and Lischinsky [10]. Having in mind that in case of ATGM’s gimbal one is dealing with highly dynamic movements with stops and changes of direction, only dynamic models seem to ensure sufficient level of detail to properly imitate physical behavior of such system. On top of that there are additional effects caused by bearing lubrication that should be addressed to fully model friction in gimbal bearings.

The LuGre dynamic friction model was selected as one capturing many relevant friction aspects in missile gimbal’s simulation.

As shown in paper by Dumitriu [11], LuGre friction model can be defined by set of three following equations. σ0 v dz =v− z dt g (v )

F = σ 0 z + σ1 ż + α 2 v

g ( v ) = FC + (FS − FC )e

(1)  v −   vs 

2

(2)

(3)

where z stands for pre-sliding displacement, v for relative velocity of surfaces in friction, σ0 for stiffness of the bristle, g(v) for function describing Stribeck’s effect, F for friction force, σ1 for damping, α2 for viscous friction coefficient, FC for Coulomb force, FS for stiction force and vs for Stribeck’s velocity. Mentioned model could also be used to describe friction in bearings, but all linear parameters should be interpreted as angular (for example velocity v as angular velocity and force F as torque).

2.2. Gimbal Static Imbalance

Anti-tank guided missiles are often subjected to high accelerations following sudden movements of host vehicle induced by atmospheric disturbances or even intentional manoeuvres subsequent to deflection of missile’s control surfaces [12]. Assuming that payload’s centre of gravity (CG) is located exactly on azimuth and elevation gimbal’s pivot axes, during acceleration there should not appear any additional torque. Any CG shift from that location generates disturbance torque directly proportional to payload’s mass and offset distance. That effect is called gimbal’s static imbalance. According to Toloei, Abdo, Vali and Arvan [13] those disturbance torques could be modelled as follows:

TS−EL = mEL am R EL cos (θm + ε + θ EL ) TS−AZ = mAZ am R AZ cos(Ψm + η +θAZ )

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(4)

(5)

where TS–EL and TS–AZ stand for elevation and azimuth disturbance torque respectively, mEL for mass of elevation gimbal, mAZ for mass of azimuth gimbal, am for lateral missile acceleration, REL for centre of gravity (CG) offset distance to elevation gimbal’s pivot axis, RAZ for CG offset distance to azimuth gimbal’s pivot axis, θm for missile body angle in vertical plane, Ψm for missile body angle in horizontal plane, ε for elevation gimbal angle, η for azimuth gimbal angle, θEL for gimbal offset elevation angle, θAZ for gimbal offset azimuth angle.

The problem with the equations presented herein is that proposed model does not concern am changing with missile rates caused by external disturbances as a result of Newton’s second law of motion. Slight change was proposed and presented in equations (6) and (7).   dω   TS−EL = mEL   Pj × R MG  + g  REL cos (θm+ ε + θEL ) dt   

 dω  TS−AZ = mAZ  Pk × R MG  RAZ cos (Ψm + η + θAZ)  dt 

(6)

(7)

where ωPj stands for missile pitch rate, RMG for distance between missile’s centre of gravity (CG) and gimbal’s CG, g for standard acceleration due to gravity and ωPk for missile yaw rate. This way elevation and azimuth torque disturbances are dependent on missile movements, what closer resembles reality.

2.3. Gimbal Cross Coupling

Besides static imbalance, factual missile gimbal is characterized by dynamic imbalance which is caused by non-symmetrical mass distribution around its rotation axes. Said imbalance manifests itself by products of inertia and results in non-diagonal inertia matrix [13]. It is important to mention that even when gimbal is statically balanced it does not mean that it can’t be dynamically imbalanced [12].

As can be seen in paper by Toloei, Abdo, Vali and Arvan [13], in equations of gimbal motion derived through use of Lagrange equation, there are disturbance torques in which described products of inertia occur. What’s more, in azimuth’s gimbal equations there are products of inertia and rates of elevation gimbal and that causes phenomenon called cross coupling where movement involving only one gimbal transfers to the other. For example, equation of motion for azimuth gimbal is formulated like this [13]: J eq ωAd = TAz cos (ε ) + (T d1 +T d 2 +T d3) cos( ε ) + T d'

(8)

where Jeq stands for instantaneous moment of inertia of azimuth gimbal around k axis, ωAd for acceleration of gimbal’s payload around d axis, TAZ for azimuth gimbal’s motor torque, ε for deflection of elevation gimbal and Td1, Td2, Td3, Td' for different components Articles

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of disturbance torque. Taking only last of them into consideration we get following equations [13].   Td' = J eq ωBnsin (ε ) + ωAr (ω Ae − ω Be )   

Jeq = B k + Ar sin 2 (ε ) + A d cos2 (ε ) − A rdsin( 2ε)

(9)

(10)

where ωBn stands for acceleration of azimuth gimbal around n axis, ωAr for elevation gimbal’s rate around r axis, ωAe for elevation gimbal’s rate around e axis, ωBe for azimuth gimbal’s rate around e axis, Bk for moment of inertia of azimuth gimbal around k axis, Ar for moment of inertia of elevation gimbal around r axis, Ad for moment of inertia of elevation gimbal around d axis, Ard for elevation gimbal moment of inertia around r axis, when rotated around d axis (product of inertia).

As can be easily seen, parameters associated with elevation gimbal (rates and moments of inertia) cause additional azimuth gimbal’s disturbance torque. As result of that every elevation gimbal movement have to be addressed by control system of other gimbal.

2.4. Cable Flexure

Being highly integrated electromechanical devices, gimbal seekers often require electrical connections between their payload and host vehicle. The only exception are systems employing steering stabilization paradigm, which move optical elements to manipulate line of sight (LOS) rather than whole sensor fixed to missile fuselage [4]. In classical configuration, where at least few of the connections such as video signal transmission are necessary, designer have to

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deal with periodically changing disturbance torque originating from cable flexure [14].

According to Wang [14] currently there is no well-established spring disturbance torque model that can be used during gimbal seeker design. Previous works focus mainly on models prepared for use in marine cable installation [15, 16]. Gimbal designers have to rely on building psychical models, often when rest of the design is already completed. Wang proposed and validated model based on Kirchhoff rod theorem, which proved to be well suited for this kind of task.

Due to model’s complexity and need for precise formulation of cable harness mounting conditions, as it affects expected results, it was decided to leave cable flexure out of developed generalized model.

3. Proposed Generalized Model of Gimbal

Considering above internal disturbances, generalized model of two axis gimbal seeker was built using Simulink environment within MathWorks’ MATLAB. Many previous works touched on subject of deriving two-axis gimbal’s equations of motion [17-19]. Herein, structure is based on one introduced in paper by Toloei, Abdo, Vali and Arvan [13], although changes were made including: replacing simple friction model with dynamic LuGre friction model and also making it dependent upon missile’s angular acceleration (Coulomb friction part of it). Also, separate subsystem for generating external missile pitch rate disturbance was highlighted. In table 1 all parameters used in model are listed with their corresponding symbols.

Tab. 1. Parameters and their symbols used in model Parameter

Symbol

Comment

Input elevation rate command generated by tracking loop

ω_EL

–

Input azimuth rate command generated by tracking loop Missile roll rate Missile pitch rate Missile yaw rate

Assumed 0 – model was examined in pitch axis

ω_Pk

Assumed 0 – model was examined in pitch axis

η, eta

–

ω_Ae

–

ω_Pj

ε, epsilon

Elevation gimbal rate around d axis in relation to inertial frame

ω_Ad

Elevation gimbal rate around r axis in relation to inertial frame

ω_Ar

Elevation gimbal rate around e axis in relation to inertial frame Azimuth gimbal rate around k axis in relation to inertial frame

ω_Bk

Azimuth gimbal rate around n axis in relation to inertial frame

ω_Bn

Azimuth gimbal rate around e axis in relation to inertial frame Elevation gimbal disturbance torque

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Assumed 0 – model was examined in pitch axis

ω_Pi

Elevation gimbal deflection Azimuth gimbal deflection

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ω_AZ

ω_Be

T_D-EL

–

– –

– – – –

Result of cross coupling between azimuth and elevation gimbals

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Parameter Azimuth gimbal disturbance torque Instantaneous moment of inertia of azimuth gimbal around k axis Elevation gimbal moment of inertia around e axis Elevation gimbal moment of inertia around r axis

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Comment

T_D-AZ

Result of cross coupling between azimuth and elevation gimbals

J_eq

Changes with deflection of elevation gimbal

A_r

Assumed 0.1 kg × m2

B_k

Assumed 0.1 kg × m2

B_n

Assumed 0.1 kg × m2

A_e

A_d

Azimuth gimbal moment of inertia around e axis

B_e

Azimuth gimbal moment of inertia around n axis

Elevation gimbal moment of inertia around r axis, when rotated around e axis (product of inertia)

Assumed 0.1 kg × m2 Assumed 0.1 kg × m2

Assumed 0.1 kg × m2

A_re

Assumed 0.05 kg × m2

Elevation gimbal moment of inertia around d axis, when rotated around e axis (product of inertia)

A_rd

Assumed 0.05 kg × m2

A_de

Assumed 0.05 kg × m2

Azimuth gimbal moment of inertia around n axis, when rotated around k axis (product of inertia)

B_ne

Assumed 0.05 kg × m2

B_nk

Assumed 0.05 kg × m2

B_ke

Assumed 0.05 kg × m2

Elevation gimbal moment of inertia around r axis, when rotated around d axis (product of inertia) Azimuth gimbal moment of inertia around n axis, when rotated around e axis (product of inertia) Azimuth gimbal moment of inertia around k axis, when rotated around e axis (product of inertia) Back EMF constant of the motor Torque constant of the motor Terminal inductance Terminal resistance

Torque of the elevation gimbal’s motor Torque of the azimuth gimbal’s motor Mass of elevation gimbal

K_e

Assumed 0.85

K_TM

Assumed 0.85

L_a

R_a

T_EL

T_AZ

Coefficient of rolling friction

LuGre bristle stiffness parameter

–

Assumed 0.2 m as in [11]

R_MG

Missile vertical acceleration

–

R_EL

Gimbal offset distance from missile’s CG

Rate gyro damping coefficient

Assumed 4.5 Ω as in [11]

Assumed 50 Hz as in [11]

R_AZ

Gimbal offset azimuth angle

Assumed 0.003 H as in [11]

ω_n

Azimuth gimbal CG offset distance from its rotation axis

Gimbal offset elevation angle

Nm as in [11] A

Assumed 0.4 kg

m_AZ

Elevation gimbal CG offset distance from its rotation axis

Vs as in [11] rad

m_EL

Mass of azimuth gimbal

Rate gyro natural frequency

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Symbol

Elevation gimbal moment of inertia around d axis Azimuth gimbal moment of inertia around k axis

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Teta_EL Teta_AZ ksi

a_el mi

sigma0

Assumed 0.4 kg

Assumed 0.2 m as in [11] Assumed 0.7 m

Assumed 0 as when gimbal axis in initial position is coaxial with missile body axis Assumed 0 as when gimbal axis in initial position is coaxial with missile body axis Assumed 0.7 as in [11] –

Assumed 0.005 m

Assumed 1000

Nm rad Articles

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Parameter

Symbol

LuGre damping coefficient

sigma1

Assumed 2

alfa2

Assumed 0.01

LuGre stiction force with Coulomb friction subtracted LuGre coefficient of viscous friction Stribeck velocity

In the following figures generalized model of gimbal is shown, starting with overview of whole system (figure 2), followed by close up of elevation and azimuth gimbal (figure 3 and 4 respectively). Next, cross coupling subsystem is shown with missile movement signal generator subsystem (figure 5). Inside of the first can be seen in figure 6 and 7. Figures 8 and 9 show static imbalance and LuGre dynamic friction subsystems.

In the model overview we can see outer part of the model consisting of two inner stabilization loops (rate control), cross coupling between them and also missile pitch rate disturbance generator subsystem.

Output from each rate stabilization loops is used to close second, position tracking loop. For this purpose,

Fig. 2. Simplified overview of gimbal seeker model

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alfa1

vs

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Comment Nm × s rad

Assumed 0.5 Nm

Nm × s rad

Assumed 0.01

rad s

PI controller is used. Reference value for azimuth tracking loop is set to zero because in this paper only elevation system’s response is considered.

Closeup of elevation and azimuth gimbal disturbance rejection loops shown in figures 3 and 4 reveals their internal structure with DC motor, rate gyro, inertia block and summation node where all disturbances are added (static imbalance, friction, cross coupling disturbance). It’s important to point out dependency between elevation and azimuth gimbal – elevation gimbal deflection ε acts as one of the inputs to azimuth gimbal subsystem and is used as a multiplier to motor’s torque, azimuth gimbal’s rates and also changes its inertia.

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Fig. 3. Elevation gimbal control loops

Fig. 4. Azimuth gimbal control loops

Fig. 5. Missile disturbance block and cross coupling between gimbals As can be seen in figure 5 missile movement disturbance subsystem consists of step signal generator and oscillatory signal generator (to examine input signals used for tests see subsection 3.1). Inside of cross coupling subsystem is shown in the following figures. There are several MATLAB function blocks containing elevation and azimuth gimbal’s rates calculations (figure 6) and cross coupling disturbance torques calculation (figure 7). Used equations are synonymous to those derived

by Toloei, Abdo, Vali and Arvan [13] in their work.

Lateral acceleration of gimbal a_el is being extracted from static imbalance subsystem as can be seen in figure 8. It is later used to calculate alfa0 coefficient in LuGre dynamic friction model subsystem, which corresponds with a level of Coulomb friction. This way additional tension on bearings during high-G missile manoeuvres transfers to higher friction torques in said bearings. This nuance differentiates hereby paper from previous works of Toloei, Abdo, Vali and Arvan [13].

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Fig. 6. Inside of cross coupling between elevation and azimuth gimbal subsystem – rates calculation

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Fig. 7. Inside of cross coupling between elevation and azimuth gimbal subsystem – disturbance torques calculation

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Fig. 8. Static imbalance subsystem for elevation gimbal

Fig. 9. LuGre dynamic friction model subsystem [11]

3. Results of Modelling

3.1. External Disturbances

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To see how individual internal disturbances affect responses of modelled system, two missile pitch rate input signals were proposed. First one imitates situation where sudden gust of wind pushes missile out of its trajectory and then stops allowing it to restore its initial position (figure 10a; oscillations with decreasing amplitude). Second one is classic step input sigArticles

nal which corresponds to situation of going into very sudden missile turn and continuing it with steady rate (figure 10b).

First all internal disturbances (cross coupling disturbance torque, static imbalance and LuGre friction) had been disabled and then rate and position controllers were auto-tuned with Simulink’s PID tuning app.

Following figures show responses of system just to said external disturbances.

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(a)

(b)

(a)

(b)

(a)

(b)

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Fig. 10. Missile pitch rate disturbances: oscillatory (a) and step (b) used to visualize influence of internal disturbances

Fig. 11. System response without additional internal disturbances: missile pitch rate oscillating disturbance (a, red plot), elevation gimbal rate response (a, blue plot) and line of sight (LOS) elevation rate (b, black plot)

Fig. 12. System response without additional internal disturbances: missile pitch rate step disturbance (a, red plot), elevation gimbal rate response (a, blue plot) and LOS elevation rate (b, black plot)

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3.2. Effects of Implementing LuGre Friction Model After initial tests without internal disturbances, LuGre friction subsystem was enabled. What is important is that auto-tuned controllers settings set initially were left unchanged. This way we can assess

disturbance rejection ability of the model and also evaluate if examined disturbance significantly affects system. Following figures show system responses with LuGre friction included.

(a)

(b)

(a)

(b)

Fig. 13. System response with LuGre friction: missile pitch rate oscillating disturbance (a, red plot), elevation gimbal rate response (a, blue plot) and LOS elevation rate (b, black plot)

Fig. 14. System response with LuGre friction: missile pitch rate step disturbance (a, red plot), elevation gimbal rate response (a, blue plot) and LOS elevation rate (b, black plot)

3.3. Effects of Implementing Static Imbalance Model Next friction disturbance was substituted with static imbalance what led to system responses shown on following figures.

(a)

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(b)

Fig. 15. System response with static imbalance: missile pitch rate oscillating disturbance (a, red plot), elevation gimbal rate response (a, blue plot) and LOS elevation rate (b, black plot) Articles

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(b)

Fig. 16. System response with static imbalance: missile pitch rate step disturbance (a, red plot), elevation gimbal rate response (a, blue plot) and LOS elevation rate (b, black plot)

3.3. Effects of Implementing Cross Coupling Model Last internal disturbance to examine was that associated with cross coupling. Following figures show how it affects system response.

(a)

(b)

(a)

(b)

Fig. 17. System response with cross coupling: missile pitch rate oscillating disturbance (a, red plot), elevation gimbal rate response (a, blue plot) and LOS elevation rate (b, black plot)

Fig. 18. System response with cross coupling: missile pitch rate step disturbance (a, red plot), elevation gimbal rate response (a, blue plot) and LOS elevation rate (b, black plot) Another important fact to consider is cross coupling’s effect on gimbal’s second axis which reveals itself when examining LOS azimuth rate during missile

pitch rate disturbance. Following figures show side by side comparison of LOS elevation and azimuth rate.

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(a)

(b)

(a)

(b)

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Fig. 19. Azimuth gimbal response to elevation gimbal moving during missile pitch rate oscillating disturbance – LOS elevation rate (a) and azimuth rate (b)

Fig. 20. Azimuth gimbal response to elevation gimbal moving during missile pitch rate step disturbance – LOS elevation rate (a) and azimuth rate (b)

3.4. Generalized Model Ultimately all internal disturbances were considered, as ized internal disturbance model was shown in figures 21 can be seen on model schematics in section 3 of hereby and 22. paper. An influence on system’s response to this general-

(a)

(b)

Fig. 21. System response with all internal disturbances considered: missile pitch rate oscillating disturbance (a, red plot), elevation gimbal rate response (a, blue plot) and LOS elevation rate (b, black plot)

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(b)

Fig. 22. System response with all internal disturbances: missile pitch rate oscillating disturbance (a, red plot), elevation gimbal rate response (a, blue plot) and LOS elevation rate (b, black plot)

4. Discussion As expected, while examining disturbances, system response turned out to be deteriorated in terms of LOS rate’s peak amplitude and also change in regularity of plotted curves - each internal disturbance was assessed as significant.

LuGre friction made elevation LOS’s peak rate almost double during oscillatory external disturbance and also introduced additional peaks with zero-velocity crossing of the gimbal. Supposedly this is due to stiction – an effect which is captured by LuGre friction model. Static imbalance doesn’t necessarily change character of LOS rate’s curve, but significantly increases its amplitude during oscillations of missile.

With assumed parameters change was at least order of magnitude greater than without static imbalance.

Lastly, cross coupling not only changed LOS rate of elevation gimbal in terms of amplitude and characteristic, but also induced additional disturbance torques in

azimuth gimbal, which had to be addressed by control system to keep LOS still.

All internal disturbances combined seem to affect LOS elevation rate accordingly – with changes in amplitude and plot characteristic, but not necessarily generating worst responses. In fact, some of the disturbance’s effects cancel out, so the amplitude peaks are not higher than when single disturbances were examined.

Gimbal’s inner rate control loops, sometimes called stabilization loops, are meant to keep LOS stable during flight, so seeker would never lose track of acquired target. This can happen when LOS deflection from zero position (line perpendicular to detector’s focal plane array connecting its centre with the target) surpasses system’s field of view (FOV). So, the real functional parameters of gimbal seeker are not LOS rates like discussed before, but rather LOS deflections shown in figures 20 and 21.

Fig. 23. LOS deflection in result of missile pitch oscillating disturbance and other modelled disturbances

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Fig. 24. LOS deflection in result of missile pitch step disturbance and other modelled disturbances

As can be seen in above figures, incorporating internal disturbance’s models leads to degradation in functioning of gimbal’s control system. Although shown characteristics (black – no internal disturbances and red – all internal disturbances) differ significantly, in terms of peak values they are well below typical FOV limits in anti-tank guided missiles. Nevertheless, setting different parameters of the model (and disturbances) could lead to situation where LOS peak deflection is no longer acceptable for assumed requirements. Hence including presented internal disturbances in developed model and tuning controllers for that plant has potential to greatly improve gimbal’s characteristics.

5. Conclusion

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Presented results indicate that modelling gimbal seeker’s internal disturbances such as: static imbalance, friction and cross coupling can significantly influence the quality of system responses when subjected to arbitrary inputs. This is especially important in the case of a system which controllers were initially tuned on model not considering these disturbances. Within this context, proposed, expanded unified model of disturbances is first step to significant increase of control systems disturbance rejection ability, what is critical from the point of view of system’s accuracy.

As seen in figures 23 and 24, maximum line of sight’s (LOS) deflection under circumstances of additional modelled disturbances changed by nearly 1100% and 160% respectively. Due to relatively high detection range requirements for anti-tank guided missiles, considering their restricted calibre, typical seeker’s field of view (FOV) doesn’t exceed 3 degrees [20]. Having in mind calculated increase of the LOS deflection, initial amplitude of as little as 0,15 degrees approximately could rise to unacceptable levels with the internal disturbances modelled and regulators Articles

not tuned accordingly. What is meant by “unacceptable” is losing track of a target, when it goes beyond image boundaries. Moreover, gimbal’s deflections measured with encoders are used to drive missiles autopilot subsystems during flight, so without proper filters, change recorded in figures 23 and 24 could directly affect their movement and consequently miss distance.

Therefore, proposed unified model of disturbances acting upon gimbal seeker could greatly aid early design phase, giving engineers tool to describe modelled system more accurately. Anti-tank guided missiles are very costly to manufacture, but more importantly they are expensive in terms of development.

Design that considers additional aspects of modelled physical phenomena helps in choosing correct control system’s algorithms and also in tuning process.

AUTHOR

Radosław Nawrocki – Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warsaw, Poland, email: radoslaw.nawrocki.dokt@pw.edu.pl.

ACKNOWLEDGMENTS

This research was funded by PCO S.A. The author declares no conflict of interest.

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[20] A. A. Shilin, “Обзор пассивных оптических ГСН для поражения наземных тактических целей Известия ТулГУ (Review of passive optical homing heads for destroying surface tactical targets)”, Technicheskiya Nauki, vol. 7, 2014, 202– 209, (in Russian).

[11] T. Dumitriu, “Development of a Simulink® toolbox for friction control design and compensation”, The Annals of “Dunarea de Jos“ University of Galati. Fascicle III, Electrotechnics, Electronics, Automatic Control, Informatics, vol. 28, 2005.

[12] D. R. Otlowski, K. Wiener and B. A. Rathbun, “Mass properties factors in achieving stable imagery from a gimbal mounted camera”. In: Proceedings of SPIE 6946, Airborne Intelligence, Surveillance, Reconnaissance (ISR) Systems and Applications V, 2008, 10.1117/12.778245.

[13] A. Toloei, M. Abdo, A. R. Vali and M. R. Arvan, “Research on gimbal seeker performance under variable operation conditions”. In: Proceedings of the 13th Iranian Aerospace Society Conference, Tehran, Iran, 2014.

Articles

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VOLUME 2021 VOLUME 15, 15,N° N° 3 3 2021

3D MAPS INTEGRATION BASED ON OVERLAPPING REGIONS MATCHING Submitted: 23rd September 2021; accepted: 9th February 2022

Michał Drwięga DOI: 10.14313/JAMRIS/3‐2021/20 Abstract: This paper presents a developed method of 3D maps inte‐ gration based on overlapping regions detection and mat‐ ching that works without an initial guess about transfor‐ mation between maps. The presented solution is based on a classic pipeline approach from computer vision that has been applied to the 3D maps integration with mul‐ tiple improvements related to model extraction and the descriptors matching. The process of finding transforma‐ tion between maps consists of three steps. The first one is the extraction of the model from one of the maps. Then the initial transformation is estimated between extracted model and another map based on feature extraction, des‐ cription, and matching. The assumption is that the maps have an overlapping area that can be used during the feature‐based alignment. In the last step, the initial so‐ lution is corrected using local alignment approaches, for example, ICP or NDT. The maps are stored in the octree‐ based representation (octomaps) but during transforma‐ tion estimation, a point cloud representation is used as well. In addition, the presented method was verified in various experiments: in a simulation, with wheeled ro‐ bots, and with publicly available datasets. Eventually, the solution can be applied to many robotic applications rela‐ ted to the exploration of unknown environments. Nevert‐ heless, so far it was validated with a group of wheeled robots. Furthermore, the developed method has been im‐ plemented and released as a part of the open‐source ROS package 3d_map_server.

Keywords: multi‐robot mapping, map merging, feature matching, ICP, NDT, octomaps

1. Introduction

70

The development of autonomous mobile robots has received much attention in recent years. Rapid progress in this area is stimulated by numerous possi‑ ble applications like mine exploration [11], planetary exploration, scout robots, search and rescue, recon‑ naissance, home vacuum cleaning, lawn mowing or in‑ dustrial applications, for instance, transport in ware‑ houses [3]. However, it turns out that Multi‑Robot Sy‑ stems (MRS) have several advantages over Single Ro‑ bot Systems in many of these applications. Especially, they are more time‑ef�icient because tasks execution can be parallelized. Also, the multi‑robot con�iguration can provide a higher level of reliability, for example, in case of malfunction of one of the robots.

One of the requirements for the creation of the au‑ tonomous robot is the ability to create a map of the

unknown environment and localize the robot on it.

The multi‑robot mapping of unknown environments can also be performed more ef�iciently than a single robot mapping. First of all, it can be done faster when it is executed in parallel, which is especially important during mapping larger areas. Moreover, the robots can be equipped with different types of sensors what makes it possible to create more accurate and com‑ plete world models. Nonetheless, several new pro‑ blems speci�ic to multi‑robot systems arise, like coor‑ dination of robots or communication between them.

Typically of multi‑robot mapping, each robot crea‑ tes its local map in the local coordinate frame. One of the intensively researched topics of mapping by mul‑ tiple robots is merging all of these local maps into one global map [36]. However, to create such a globally consistent world model, a few problems have to be sol‑ ved. The �irst one is �inding transformation between maps. There are a few ideas on how to use the robots’ initial poses and internal localization system to esti‑ mate transformation. But it may be not possible espe‑ cially in the case of a big drift in the pose estimated by the local localization method.

Another idea is to integrate maps only during ro‑ bots meetings. However, it needs in many cases an ad‑ ditional sensory system to detect other robots or a de‑ dicated detection method. Also, this idea is limited be‑ cause sending data between robots is not enough to estimate the orientation of robots and they have to see each other. Nevertheless, it is worth mentioning that systems that use partial information are developed as well. Such partial information could be the only dis‑ tance between robots, calculated based on signal time‑ of‑�light.

Another group of approaches depends on sending measurements from one robot to other robots and the assumption that maps have an overlapping area. With measurements from the other robot, it is possible to locate it on the map, for example, with a particle �ilter algorithm.

The last group of methods is based on the feature matching idea. The features are extracted from maps, identi�ied, and matched. �t is assumed like in the pre‑ vious group that maps have an overlapping area that can be used during the matching process.

Nonetheless, the map alignment process is more challenging than, for example, 2D laser scan matching or depth sensors measurements matching because of displacement between maps or measurements. The displacement can reach bigger values especially when robots start mapping from totally different parts of en‑

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vironments. In the case of small displacement, algo‑ rithms that are based on the local optimization can be used. However, the maps alignment process needs a global optimization that is resistant to local minima.

1.1. Contribution

This paper presents a developed global 3D maps integration method with the alignment based on the feature matching which does not require an initial transformation estimation. Most of the other approa‑ ches for 3D maps integration are based on some kind of initial information or they are sensitive to the lo‑ cal minima. In contrast, this method does not need any initial cues and it deals with the local minima pro‑ blem. The presented method is based on a classic pi‑ peline approach from computer vision that has been applied to the 3D maps integration with multiple im‑ provements related to model extraction and the des‑ criptors matching. Moreover, the introduced model di‑ vision into submodels improves the feature matching process and decreases the number of necessary calcu‑ lations in the optimistic case. To my knowledge, there is no other application of feature matching with the model subdivision‑based method for solving the 3D maps integration problem without an initial transfor‑ mation guess.

The presented approach is based on the assump‑ tion that maps have the same scale and that they have an overlapping area. The intersection area of the maps can be extracted and used in the initial alignment step.

In the initial alignment process, one of the maps is di‑ vided into regions, then each region is aligned to the second map. Finally, the best solution that consists of the transformation is selected according to the propo‑ sed quality measure, for example, a �itness score.

The method has been developed to work mainly with octree‑based maps (octomaps) but during the merging process, the point clouds representation is used as well. Dual representation makes it possible to use the advantages of two representations and ef�i‑ ciently perform speci�ic operations. But of course, the cost behind that is the increased memory usage. Mo‑ reover, the method was veri�ied in multiple test cases based on data from real robots. It was con�irmed that with some assumptions it is possible to merge large 3D maps from multiple robots.

Furthermore, the method of maps integration des‑ cribed in this paper has been implemented in C++ and released as the open‑source software. The soft‑ ware is a part of the ROS (Robot Operating System) [1] package 3d_map_server [10].

1.2. Problem Statement

74

This paper deals with the three‑dimensional (3D) feature maps integration problem. Brie�ly, the pro‑ blem can be de�ined as a data association between multiple representations of the same part of the en‑ vironment.

Let’s consider a system of N robots R = {r1 , r2 , . . . , rn , . . . , rN } in R3 space, where the each robot creates its partial map Mn in a local coordinate system Tn . The map can be de�ined as a set of nodes

2021 VOLUMEVOLUME 15, 15, N° N° 3 3 2021

Mn = {m1 , m2 , . . . , mNn } where |Mn | = Nn . In gene‑ ral, the octomaps integration can be de�ined as a crea‑ tion of one, consistent model of the world M based on the k separate models which represents regions of the environment M ′ = {M1 , . . . Mk } (�ig. 1). The problem can be narrowed down to the only two input models without loss of generality because of the assumption that more than two maps can be merged, for instance, recursively.

Fig. 1. Local maps created by different robots merged into a one consistent world model Let’s assume that exists a transformation T21 be‑ tween a pair of models that transforms M1 and M2 to the common coordinate system. The output map is a composition of the partial maps transformed to the coordinate system of the �irst map.

M = M1 ∪ T21 M2

(1)

In the presented problem the solution consists of estimated transformations between the coordinate frames of the input maps. More precisely, the goal is to �ind the transformations under which the distances between corresponding nodes in input maps are mini‑ mized. In the real world, it is dif�icult to �ind the opti‑ mal transformation T21 because of multiple inaccura‑ cies sources. Errors are introduced by similarities on the maps and not enough distinctive descriptions of fe‑ atures. Also, the input maps have different scales, and sensors used for maps creation are not ideal.

1.3. Related Work

One of the basic and intensively researched topics of multi‑robot mapping is a merging of local maps from robots into one global map [3]. Following the classi�ication proposed in [21] approaches can be clas‑ si�ied into two categories� a direct map merging and an indirect map merging.

In the direct map merging the system has additio‑ nal information about transformation between maps.

For example, this information can be acquired by vi‑ sual or range measurements during the meeting of ro‑ bots. Such an approach has been presented in [18].

It generates hypotheses by direct measurements be‑ tween robots. Then robots move to a speci�ic loca‑ tion and meet again. If a meeting happens and robots again detect each other correctly, then the hypothe‑ sis is accepted and maps are merged. In [40], also a solution based on robot‑to‑robot visual observations has been proposed. Based on the estimated transfor‑ mation, maps are transformed and in the next step, it Articles

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is checked if maps overlap. In the case of maps over‑ lapping the accuracy of robots relative poses is incre‑ ased by landmarks detection on overlapped parts of maps by matching them. Another approach has been presented in paper [20]. This probabilistic map mer‑ ging method depends on Rao‑Blackwellized particle �ilters with unknown initial poses and models inter‑ robot measurements as Gaussian processes.

The other branch of direct map merging methods includes these based on the pose of overlapping area regions or objects in multiple maps. A probabilistic on‑ line map merging approach working with an omnidi‑ rectional visual system has been described in [32]. It has been used for one robot that collects partial maps but can be adapted to a multi‑robot system as well. It uses a vision system to generate coarse transforma‑ tions between maps based on the place recognition method. Then depending on that the bounding boxes are calculated for Haar‑based place recognition that is able to discriminate new and previously visited loca‑ tions.

On the other hand, indirect map merging is ba‑ sed on �inding and matching the overlapping area of the maps which are not known a priori. These methods can be classi�ied into three groups. In the �irst group, there are mostly approaches based on point feature detection and matching. In [19] the sy‑ stem for detection of the overlapping regions of maps has been described. Maps were created with ceiling‑ vision‑based SLAM. The algorithm robustly detects the overlapping regions and estimates transformati‑ ons for map alignment.

The next group includes methods based on the scan matching. In [34] the 2D local maps were mer‑ ged by integration of Scale‑Invariant Feature Trans‑ form (SIFT) algorithm to extract, describe and match features. Also, they used a well‑known optimization technique in robotic mapping, the ICP(Iterative Clo‑ ses Point) [4, 13] that �inds a rigid transformation be‑ tween two points sets. They used that data from the 2D laser scanner and consecutive scans were matched during the map creation. In [2] the virtual robot ap‑ proach has been provided. It treats laser scans from multiple robots as range measurements to the virtual robot and generates its odometry data by detection of similar structures in local maps.

The last group includes spectral information‑ based methods. In [6,22], some approaches have been presented that utilize the spectral information on 2D maps. They consider the maps merging as a binary image matching problem, so they use the Hough trans‑ form to structure and decompose the transformation into separate operations of rotation and translation.

Paper [12] describes a method that uses geometric and topological similarities of vertices and edges to �ind a match between two maps.

With the growing demand for robotics services in complex, human environments, the need for 3D maps storing and processing methods will be essential. This kind of world representation allows robots to ope‑ rate in the interior of multi‑level buildings, inside clut‑ 72

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VOLUME 15, N° 3 2021 VOLUME 15, N° 3 2021

tered rooms, or in rough terrain. Also, 3D maps are better suited to heterogeneous multi‑robots systems [28, 30, 38], especially when robots use different sen‑ sors. Papers [9, 17] deal with the octomaps [16] mer‑ ging problem with the ICP based methods. The alter‑ native to the ICP approach for the 3D local alignment has been presented in [23,33]. It is based on NDT (Nor‑ mal Distribution Transform) and is more ef�icient than the ICP algorithm because it doesn’t require nearest neighbors search. Graph‑based merging methods are also developed [5], and they have the advantage in in‑ consistencies reduction because of a backend graph optimization. The paper [7] presents a solution for the 3D point clouds alignment based on the transforma‑ tion into the Radon/Hough domain. In [24] it was pre‑ sented the comparison of different 3D maps matching approaches.

1.4. World Representations An octomap [35,37] is a tree‑based representation, which is memory ef�icient and well suited to large en‑ vironments. It is built upon a recursive dividing of the world into eight cubic parts. One of the key features of octomap is the possibility to postpone the initializa‑ tion of nodes until a robot visits a speci�ic part of the environment. Naturally, it is not possible with �ixed re‑ presentations like voxel grids. However, the computa‑ tional complexity of a node random access is O(log d) and it depends on the depth of the octree. Another ad‑ vantage of the octomap is the possibility of optimiza‑ tion. Blocks can be divided into smaller parts as long as output blocks are distinct. If the obtained blocks are similar enough, the branch is cut and the model is con‑ sistent.

Fig. 2. Recursive space division which is the idea behind octomaps As mentioned in section 1.2, the map can be de�i‑ ned as a set of nodes which determines the probability of occupancy of represented segments of the environ‑ ment. Such probabilities are updated according to the following formula [16]: p(n | z1:t ) (2) ]−1 [ 1 − p(n | zt ) 1 − p(n | z1:t−1 ) p(n) = 1+ p(n | zt ) p(n | z1:t−1 ) 1 − p(n)

where: ‑ zt – denotes measurement at time t,

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‑ p(n) – a priori occupancy probability,

‑ p(n | z1:t−1 ) – previous probability estimation,

‑ p(n | zt ) – occupancy probability at measurement zt calculated based on sensor model.

2. Maps Integration Method

In the proposed approach a few processing steps are necessary for each of the maps. The data pipelines have been presented in the �ig. 3. On the input of the

VOLUME 15, N° 3 2021 VOLUME 15, N° 3 2021

n hypotheses H = {h1 , . . . , hn }. Then, each hypothe‑ sis is evaluated based on the selected quality measure, for example, the �itness score. As a result, the best so‑ lution is selected from the set of accepted hypotheses HA . If at least one hypothesis is accepted then the pro‑ cessing is continued and the �inal result is the transfor‑ mation that transforms one of the maps to the coordi‑ nate system of the other map. Otherwise, the proces‑ sing is stopped at this point.

In the next step, the transformation from the previ‑ ous step is corrected in the process called a local alig‑ nment. For this purpose, variants of ICP and NDT algo‑ rithms were used.

However, to generate the integrated map that con‑ sists of both parts, it is necessary to include some addi‑ tional steps that are parts of the data integration pro‑ cess. This process consists of a map conversion to the octomap, the transformation of one map to another coordinate system, and �inally the combination of data of two maps into one consistent model.

2.1. Model Extraction From One of Maps

The �irst step is to divide the �irst map into rectan‑ gular blocks that are used as models. As shown in the �ig. 4, a few cases of relations between two maps have to be considered. The maps integration can be done only in the �irst three cases when the overlapping re‑ gion exists. map 2

Fig. 3. The data pipelines in the presented maps merging approach integration method, there are two 3D maps. Nevert‑ heless, the important assumption is that maps have an overlapping area, which is necessary for correct operation. The whole maps merging process consists of two main parts: �inding of the transformation bet‑ ween maps and data integration. The output of the al‑ gorithm is an integrated map.

The �inding of the transformation could be divided into three steps: ‑ model extraction, ‑ global alignment,

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‑ and local alignment.

One of the maps �map � in the �ig. 3) is used directly and the other one is used for n models extraction. The process of models extraction is described in the follo‑ wing section.

In the global alignment, there are some common operations for both maps, like �iltration, keypoints de‑ tection, and descriptors computation. Based on the computed descriptors and the assumption that maps have an overlapping area, the descriptors from maps are matched to each other with a randomized algo‑ rithm. As a consequence of one map division into mul‑ tiple models, the result of initial alignment is a set of

map 1

map 1 I

map 1

map 2

III

map 1

map 2

map 2 II

IV

Fig. 4. Different cases of maps overlapping (I‐III) or when there is no common part between maps (IV) Moreover, it has been noticed that the process of maps integration can be speeded up in the most com‑ mon case ��ig. 4), when robots start exploration from the same place and explore different parts of the en‑ vironment. Therefore, excluding the multi‑�loor map‑ ping, kidnapped robot problem, and a case when one of the maps is entirely included in the second one, in most cases the overlapping area begins on the bor‑ ders of both maps. Because of that, processing of the map begins from the outside and proceed in a spiral towards the center ��ig. 5). To speed up the method and decrease the number of calculations, not all rec‑ tangular blocks are processed.

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n

1

1 1

1

1

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Fig. 5. A division of one map and extraction of the model ‐ 2D and 3D cases Also, assuming that robots move mostly parallel to the ground, then maps are rather limited in Z axis and wider in x and y axes. Therefore, map heights are sig‑ ni�icantly smaller than sizes in x and y axes. Then the division can be performed in two dimensions. In spe‑ ci�ic cases, for example, maps of multi‑level buildings, the model can be extracted from one of the maps based on the 3D grid (�ig. 5).

One of the stop criteria is a �itness score combined with a number of correspondences. So, if the criteria presented in the following part are satis�ied, the pro‑ cessing of the remaining blocks is stopped. If a map has a lot of nodes near the border that is in the overlap‑ ping area with another map, it should stop fast ‑ just after having processed a few blocks. However, in some cases, maps near the borders are not dense and then processing should be continued until the center of the map is reached.

θi

Fig. 6. Normal vector in specified point and the θi angle between the normal vector n and point pi ted only in selected points ‑ keypoints. The properly selected keypoints should be distinctive and repea‑ table to deal with noises or different points of view.

Commonly used approaches that deals with keypoints extraction in 3D data are NARF detector [29], ISS [39] (�ig. 7) or modi�ied Harris detector [14]. Another, less computationally demanding solution that has been applied to this work is uniform sampling. It creates a 3D voxel grid over the input point cloud and then in each voxel the present points are approximated with their centroid.

2.2. Input Data Filtration

In the process of maps merging it is necessary to correctly prepare the data. Therefore, the preproces‑ sing was done in a few steps.

The �irst one is a pass‑through �iltration that allows the rejection of points that are not inside the useful area. For example, the maps are cut in the z‑axis to re‑ move the ground and reduce the number of points to speed up further calculations.

In the next step, the point cloud is downsampled with a voxel grid �ilter. The idea is to divide space into voxels with speci�ied sizes and approximate points in each voxel.

The last �iltration step is the removal of outliers that is based on a statistical analysis of neighboring points, and points that do not meet the requirements are removed. In this work, an approach [27] that com‑ putes the distribution of the point to neighbors distan‑ ces has been applied. Assuming that resulted distribu‑ tion is �aussian, points which are outside a speci�ied range are �iltered out.

2.3. Keypoint Detection

74

After �iltration, surface normals are estimated (�ig.

6). The algorithm for each point �inds neighboring points in a speci�ied search radius and then estima‑ tes a �itting plane with the least‑square algorithm [25].

Based on the estimated plane equation ax + by + cz + d = 0, the surface normal is calculated n = [a, b, c].

The descriptors calculations is a computationally expensive step. Therefore, the descriptors are compu‑ Articles

Fig. 7. An example of keypoints (red points) detection with the ISS method 2.4. Local Features Description In order to �ind similar areas on the maps, all de‑ tected features should be described in the most con‑ cise possible way that makes it easy to compare. The‑ refore, in each keypoint selected in the previous step, the local descriptor is calculated. Local descriptors describe a local neighborhood around a query point on the surface. The widely used is FPFH method that is a local version of PFH [26] around a given keypoint. It is based on pairing the query point with neighbors and calculations of orientation differences for each pair.

After that, a weighted sum of the orientations diffe‑ rences is computed and the output vector is created by concatenation of histograms.

However, for the purpose of the maps integration method, a SHOT (Signature of Histograms of Orienta‑ tions) descriptor [31] has been used. It is based on the spherical support (neighbors points) that is divi‑ ded into spatial segments (�ig. 8). For each segment, the descriptor calculates a histogram representing the distribution of the cos θi , where θi is the angle between the surface normal in each point from the support and the surface normal vector n in the query point (�ig. 6).

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Then all local histograms are combined into one vector and the descriptor is created. One of the advantages of the SHOT descriptor is the possibility to utilize texture information like a point color received from the RGB‑D sensor.

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2.6. Local Alignment As a �inal step of transformation estimation, the lo‑ cal correction is applied. For this purpose, the scene is cropped to the size of the model in�lated by a speci�ied distance dm , as shown in the �ig. 10. dm

10

0 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 cos( i)

Fig. 8. A spatial division of the spherical support and histogram of cos θi value range

2.5. Features Matching The features matching uses the descriptors extrac‑ ted and computed in the previous steps. The �irst des‑ criptors set DM is calculated for keypoints of extrac‑ ted model M ′ = {mi | mi ∈ R3 , i = 1, . . . , NM } and the second set DS for the scene S ′ = {si | si ∈ R3 , i = 1, . . . , NS }. Those two sets of descriptors are matched to each other. It means that for each point from the set M ′ it should be found the point in S ′ with a similar descriptor that relates to the same region in the se‑ cond map. After that, one should calculate a transfor‑ mation that minimizes distances between pairs of des‑ criptors.

To match the descriptors sets, the SAC (Sample Consensus) alignment approach was used [26]. The al‑ gorithm idea is similar to the RANSAC (Random Sam‑ ple Consensus) [8] method and random matching of k‑ nearest neighbors. The method consists of three steps: ‑ Randomly select k points from the set M ′ and them do set P = {pi | pi ∈ R3 , i = 1, . . . , k}, ‑ For each pi ∈ P , �ind points with similar descrip‑ tors in S ′ and randomly select from them the one that will make a pair with the point from P ,

‑ For each pair of points, called correspondence, com‑ pute the transformation between points and the er‑ ror metric.

Repeating the above steps allows to avoid local mi‑ nima and �ind transformation which minimizes the er‑ ror metric. It is not an �inal solution but rather an ini‑ tial guess for the next step which is a local alignment.

The example of pairs of features matching between two maps has been shown in the �ig. 9.

Fig. 9. Matching of feature pairs from two maps

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model

part of a scene

Fig. 10. The scene cropped to the size of the model with the inflation distance. On the right side, there is a model matched to the cropped scene After that, the ICP based approach is used to cor‑ rect locally a transformation between the scene S = {si | si ∈ R3 , i = 1, . . . , NS } and the model M = {mi | mi ∈ R3 , i = 1, . . . , NM }. The idea behind this method depends on matching one map (scene) D to the second one M called model in such a way as to minimize distances between pairs of points (nearest neighbors) from both sets. The steps in k‑th iteration of algorithm are as follows: ‑ ∀mki ∈ M �ind the closest point (nearest neighbor) ski ∈ S, ‑ Minimize distances between corresponding points pairs with the least squares method NM 1 ∑ E(R, t) = ||Rmi + t − ski ||2 , NM i=1

(3)

‑ Transform model according to estimated rotation R and translation t M k+1 = RM k + tk ,

(4)

‑ Terminate if error value is below the threshold τ .

The �inal transformation can be estimated by repe‑ ating the above steps.

2.7. Evaluation of Estimated Transformation

To evaluate the solution, a �itness score fs has been computed. It computes the mean error between pairs of corresponding points (pi , qi ) based on the nearest neighbors calculation. The pairs of points are placed in two points sets. The �irst one is P = {pi | pi ∈ R3 , i = 1, . . . , n} and the second one is Q = {qi | qi ∈ R3 , i = 1, . . . , n}. Nevertheless, the standard version of the �it‑ ness score is not very robust, for instance, it can be cal‑ culated on the basis of a small number of pairs, which means it is not very reliable. So, there were applied a few modi�ications. One of them is the calculation of bi‑ nary weights for each pair depending on the maximum distance between corresponding points dth , so only pairs with smaller distances are considered in calcula‑ tions. Also, if the number of pairs with positive weight Articles

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nw is below the threshold value, the �itness score is marked as worthless. wi =

{

1,

0,

if ∥pi − qi ∥ ≤ dth

otherwise

nw =

n ∑

(6)

wi

i=1

 n ∑ ∥pi − qi ∥  2  wi , nw fs = i=1   +∞,

(5)

if nw ≥ nth

otherwise

(7)

where: ‑ n is a number of corresponding pairs,

‑ nth is a threshold value for the number of pairs.

The presented formula allows avoidance of false good nodes matching because of the use of a threshold. �ithout it, it is possible to get a low �itness score only based on a small number of points pairs.

Fig. 11. Model matching and transformation estimation between two maps. The first map (blue) was used as a scene. From the other map (red), a model (yellow) has been extracted and matched to the first map. The matched and transformed model was marked as a green region. The real transformation between maps was TR = (7.5, 0.2, 0.1, 1o , 1o , 5o )

3. Validation

The maps integration algorithm has been valida‑ ted in numerous experiments. The performed experi‑ ments could be divided into three parts. The �irst part is based on data from publicly available datasets from Freiburg University, released under Creative Commons Attribution License CC 3.0 [15]. The second part con‑ sists of experiments with two Turtlebot robots. The last part contains the performance evaluation for dif‑ ferent maps alignment methods.

3.1. Experiments Based on Datasets

Maps placed in dataset [15] has been created ba‑ sed on data from a SICK LMS laser scanner was placed on a pan‑tilt unit. As a result, the accuracy of maps is better in comparison to maps created with the RGB‑D sensor but they do not provide information about the color of the surface.

The selected results of the maps merging based on the datasets have been placed in the �igures 11 ‑ 13.

Other results are shown in table 1, where: ‑ n1 and n2 denote sizes (a number of nodes) of two input maps, ‑ TR is a real transformation between maps in format (x, y, z, roll, pitch, yaw),

‑ r is an approximated size of an overlapping area of both maps as a percentage of the full map,

‑ fs is a �itness score of the best solution from all ex‑ tracted models, ‑ Terr is an error value between real and estimated transformation and is calculated as Terr = ∥Test · TR−1 − I4 ∥F ,

‑ t denotes processing time in seconds.

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Fig. 12. Another example of finding transformation between maps, with different real, initial transformation TR = (9, 1.0, 0.1, 1o , 2o , 15o ) Tab. 1. Results of transformations estimation n1

n2

TR

4.0 ·105 6.5 ·105 8.9 ·105 1.0 ·106

8.3 ·105 8.3 ·105 9.9 ·105 1.0 ·106

(12, 6, 0.5, 5o , 5o , 60o ) (12, 6, 0.5, 5o , 5o , 30o ) (14, 4, 0.5, 5o , 3o , 45o ) (15, 3, 0.5, 5o , 5o , 45o )

r

fs

Terr

t[s]

16%

0.028

0.25

15

32%

0.023

0.007

26

64%

0.023

0.006

40

84%

0.023

0.009

44

3.2. Experiment With Turtlebot Robots To validate the integration algorithm on noisy data representing scenes encountered in mobile robotics applications, the experiments with two mobile ro‑ bots Turlebots (�ig. 14) were performed. The robots were equipped with an odometry system, laser scan‑ ner Hokuyo UST‑10LX and RGB‑D sensor Intel Real‑ Sense D435. The system used for the octomaps crea‑ tion was presented in the �ig. 15. Similarly like in the simulation, it was built upon the ROS framework, and it used the GMapping SLAM.

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2021 VOLUMEVOLUME 15, 15, N° 3N° 3 2021

a

b

Fig. 16. Paths of two robots during the experiment a

Fig. 13. Example of two maps integration (a) with real transformation between maps TR = (10, 1.5, 0.1, 3o , 2o , 25o ) and an integrated map (b) as a result

Fig. 14. The Turtlebot mobile robot with the sensors

b

c

Fig. 17. An experiment with a mapping of the laboratory in Wrocław University of Science and Technology. The figure shows maps before merge (a) with extracted (yellow) and matched model (green). Also it contains an output map (b) and a map of the same area created by one robot (c) 3.3. Performance Evaluation of Alignment Algorithms

Fig. 15. The high‐level control system used to create the octomap by the Turtlebot robot

80

During the experiment with Turtlebots in a robo‑ tics laboratory and corridor localized in the campus of Wrocław University of Science and Technology, robots moved following the paths shown in �ig. 16. Results from experiments have been shown in �igures 17‑19.

The performance of different maps integration method modi�ications has been evaluated. The mean value of error has been calculated for multiple testing cases which contain distinctive maps of the environ‑ ment.

The diagram (�ig. 20) contains mean errors (Terr ) for different global alignment methods. The following global alignment methods were evaluated: SAC (Sam‑ ple Consensus) and GCC (Geometry Consistency Clus‑ tering). Additionally, it was checked if a division of one map into models (model extraction process) could speed up the alignment procedure. The cases with div post�ix in name, have been using model division.

The diagram (�ig. 21) contains mean errors (Terr ) for combinations of local and global alignment met‑ hods. As a local methods, ICP [4] and NDT [23] have been compared.

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VOLUME 2021 VOLUME 15, 15,N° N° 3 3 2021

a

0.20

Terr

0.15

0.10

0.05

b

0.00

c

sac_icp

sac_ndt

gcc_icp

gcc_ndt

Fig. 21. Mean error between real and estimated transformations for combination of global and local alignment methods

Fig. 18. Integration of maps from another robots run, from the same location (a,b) but with extended area. It has been shown also the 2D map of the same area (c) b

a

Fig. 19. The last case of two maps integration from Turtlebots robots 0.200 0.175 0.150

Terr

0.125 0.100 0.075 0.050 0.025 0.000

sac_iss3d

sac_iss3d_div

gcc_iss3d

gcc_iss3d_div

Fig. 20. Mean error between real and estimated transformations for global alignment methods

4. Conclusion

78

The paper presents the approach to 3D maps inte‑ gration problem without the initial knowledge about the relative poses of robots. It is based on feature de‑ tection and matching techniques. To speed up proces‑ sing, randomized algorithms were used. Also, some optimization steps for the most common cases were introduced like processing maps from the outside part to the center.

The evaluation uses publicly available data sets Articles

and data from experiments with Turtlebots. The re‑ sults show that the approach is effective in various en‑ vironments. So, when properly tuned and maps over‑ lap enough (above 15% of common space) the method is quite robust.

However, the merging method has some weaknes‑ ses. One of them is the computational cost of the data processing pipeline, especially during the �irst robots met when initial alignment is necessary. Then pro‑ cessing can take hundreds of seconds for two 20m × 20m × 2m maps. Also, it turns out to be hard to �ind a transformation between octomaps that contain a ground plane as the ground plane is wrongly matched between maps. Of course, it can be �ixed relatively ea‑ sily, by removing the ground plane from maps or by introducing another keypoints detection method. Be‑ sides of that the open topic is a scaling of the method as it was tested only on two robots so far.

Another issue that is still not addressed is the loop closure problem. Currently, it was assumed that a local merging error is low enough and its in�luence is neg‑ ligible. However, it is not true and after multiple mer‑ ging processes it can increase to a signi�icant value and have an impact on the quality of the map. One of possi‑ ble improvements can be providing a high‑level graph‑ based approach to manage multiple partial maps from robots together with some kind of backend SLAM met‑ hod.

Further research will be also directed to the op‑ timization of algorithms, especially that many of the operations can be processed in parallel.

AUTHOR Michał Drwięga – Department of Cybernetics and Ro‑ botics, Wrocław University of Science and Technology, ul. Wybrzeż e Wyspiań skiego 27, 50‑370 Wrocław, Po‑ land, e‑mail: michal.drwiega@pwr.edu.pl.

ACKNOWLEDGEMENTS

This research was supported by the National Science Centre, Poland, under the project number 2016/23/B/ST7/01441.

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STRUCTURALLY R‐CONTROLLABLE AND STRUCTURALLY R‐OBSERVABLE DESCRIPTOR LINEAR ELECTRICAL CIRCUITS Submitted: 7th October 2021; accepted: 2nd March 2022

Tadeusz Kaczorek, Kamil Borawski DOI: 10.14313/JAMRIS/3‐2021/21 Abstract: Structurally R‐controllable and structurally R‐observable descriptor linear electrical circuits are investigated.

Sufficient conditions are given under which the R‐ controllability and R‐observability of descriptor linear electrical circuits are independent of their parameters.

Keywords: structural, R‐controllability, R‐observability, descriptor, linear, electrical circuit

1. Introduction A dynamical system is called descriptor (singular) if its mathematical model is represented by a combi‑ ned set of differential and algebraic equations. Des‑ criptor linear systems have been investigated in [3, 5– 8, 16, 18]. The computation of Kronecker’s canonical form of a singular pencil has been analyzed in [17].

The notion of controllability and observability and the decomposition of linear systems have been intro‑ duced by Kalman [11, 12]. These notions are the ba‑ sic concepts of the modern control theory [1, 2, 8, 10, 14, 15]. They have been also extended to descriptor li‑ near systems [4, 6, 8, 13, 19]. It is well‑known that the controllability and observability of linear systems are generic properties of the systems [14].

In this paper structurally R‑controllable and struc‑ turally R‑observable descriptor linear electrical cir‑ cuits will be investigated.

The paper is organized as follows. In Section 2 the basic de�initions and theorems concerning descriptor linear electrical circuits and their controllability and observability are recalled. In Section 3 structural R‑ controllability of the descriptor linear electrical cir‑ cuits is introduced and investigated. Similar results for structural R‑observability are presented in Section 4.

Concluding remarks are given in Section 5.

The following notation will be used: R ‑ the set of real numbers, Rn×m ‑ the set of n × m real matrices and Rn = Rn×1 , C ‑ the �ield of complex numbers.

2. Descriptor Linear Electrical Circuits 2.1. Preliminaries

Consider the descriptor linear electrical circuit composed of resistors, coils, capacitors and source voltages described by the equations

84

E ẋ = Ax + Bu,

(1a)

y = Cx,

(1b)

where x = x(t) ∈ Rn , u = u(t) ∈ Rm , y = y(t) ∈ Rp are the state, input and output vectors and E, A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n .

It is assumed that detE = 0, rankB = m and the pen‑ cil is regular, i.e. det[Es − A] ̸= 0 for some s ∈ C.

(2)

Usually as the state variables x1 , . . ., xn (the compo‑ nents of the state vector x) the currents in the coils and voltages on the capacitors are chosen.

It is well‑known [9] that: 1) every electrical circuit is a descriptor system if it contains at least one mesh consisting of capacitors and voltage sources only or at least one node with branches with coils;

2) every linear descriptor electrical circuit is a linear system with regular pencil (the condition (2) is sa‑ tis�ied).

2.2. Elementary Matrix Operations The following elementary operations on matrices will be used: 1) Multiplication of the i‑th row (column) by a real number c. This operation will be denoted by L[i × c] (R[i × c]).

2) Addition to the i‑th row (column) of the j‑th row (co‑ lumn) multiplied by a real number c. This operation will be denoted by L[i + j × c] (R[i + j × c]).

3) Interchange of the i‑th and j‑th rows (columns). This operation will be denoted by L[i, j] (R[i, j]).

2.3. Controllability and Observability For descriptor linear electrical circuits different notions of controllabilities and observabilities can be introduced, such as C‑controllability, R‑controllability, I‑controllability and C‑observability, R‑observability, I‑observability [6, 8].

�e������o� 1. [8] The descriptor linear electrical ci‑ rcuit (1) is called completely controllable (in short C‑ controllable) if for every initial state x0 = x(0) ∈ Rn and every �inite state xf ∈ Rn there exist a time tf > 0 and input u(t) ∈ C q (the set of q times piecewise con‑ tinuously differentiable functions) in [0, tf ] such that x(tf ) = xf .

Theorem 1. [8] The descriptor linear electrical circuit (1) is C‑controllable if and only if [ ] rank Es − A B = n for all s ∈ C (3a)

and

rank

[

E

B

]

= n.

(3b)

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�e������o� 2. [8] The descriptor linear electrical ci‑ rcuit (1) is called completely observable (in short C‑ observable� if there e�ists a �inite time tf > 0 such that for given u(t) and y(t) in [0, tf ] it is possible to �ind its unique initial condition x0 = x(0).

Theorem 2. [8] The descriptor linear electrical circuit (1) is C‑observable if and only if [ ] Es − A rank = n for all s ∈ C (4a) C and

rank

[

E C

]

= n.

(4b)

Neglecting the impulse part of the descriptor li‑ near electrical circuit we obtain the R‑controllability (R‑observability), i.e. controllability (observability) within the reachable set.

VOLUME 2021 VOLUME 15, 15,N° N° 3 3 2021

Proof. If the assumptions are satis�ied then for each row of the matrix Es−A containing entries with resis‑ tances, inductances and capacitances the correspon‑ ding rows of the matrix B have nonzero entries in‑ dependent of the circuit parameters. Using elemen‑ tary column operations it is possible to eliminate all parameter‑dependent [ ] entries in these rows of the ma‑ trix Es − A B . After this elimination procedure, which does not change the rank of the matrix, we obtain full row rank matrix with entries independent of the resistances, inductances and capacitances.

Example 1. Consider the descriptor linear electrical ci‑ rcuit shown in Figure 1 with given resistances R1 , R2 , R3 , inductances L1 , L2 , L3 , capacitance C and source voltages e1 , e2 .

Theorem 3. [8] The descriptor linear electrical circuit (1) is R‑controllable if and only if the condition (3a) is satis�ied.

Theorem 4. [8] The descriptor linear electrical circuit (1) is R‑observable if and only if the condition (4a) is satis�ied.

Similarly, neglecting the standard part of the descriptor linear electrical circuit we obtain the impulse controllability (observability), in short I‑ controllability (I‑observability).

Theorem 5. [6] The descriptor linear electrical circuit (1) is I‑controllable if and only if [ ] E 0 0 rank = n + rankE. (5) A E B

Fig. 1. Descriptor electrical circuit of Example 1 Using Kirchhoff’s laws we may write the equations

Theorem 6. [6] The descriptor linear electrical circuit (1) is I‑observable if and only if   E A rank  0 E  = n + rankE. (6) 0 C In this paper we focus on the R‑controllability and R‑ observability.

3. Structurally R‐Controllable Electrical Cir‐ cuits In this section structural R‑controllability of the descriptor linear electrical circuits will be introduced and investigated. �e������o� 3. The descriptor electrical circuit (1) is called structurally R‑controllable if its R‑controllability is independent of its resistances, inductances and capa‑ citances.

Theorem 7. The descriptor electrical circuit (1) is structurally R‑controllable if the linearly indepen‑ dent meshes contain only resistances, inductances and source voltages and the number of linearly independent meshes containing only capacitances and source volta‑ ges is equal to the number of its capacitances.

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di3 di1 + R1 i 1 + L 3 + R3 i 3 , dt dt di2 di3 + R2 i 2 − L 3 − R3 i 3 , e 2 = L2 dt dt i 3 = i1 − i 2 , u = e1 + e 2 .

e 1 = L1

The equations (7) can be written in the form     i1 i1 ] [  i2  d  i2  e1     E  , = A +B i3  e2 dt i3  u u

(7)

(8a)

where





L1  0 E=  0 0

−R1  0 A=  1 0

0 −R2 −1 0

0 L2 0 0

−R3 R3 −1 0

L3 −L3 0 0  0 0  , 0  −1

 0 0  , 0  0



1  0 B=  0 1

 0 1  .

0  1 (8b)

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Using (8b) we obtain [ ] Es − A B =  0 sL1 + R1  + R2 0 sL 2   −1 1 0 0

sL3 + R3 −sL3 − R3 1 0

0 0 0 1

1 0 0 1

 0 1  .

0  1 (9)

By Theorem 7 the R‑controllability of the electrical cir‑ cuit is independent of its resistances, inductances and capacitance. Using on the matrix (9) elementary co‑ lumn operations R[1 + 5 × (−sL1 − R1 )], R[3 + 5 × (−sL3 −R3 )], R[2+6×(−sL2 −R2 )], R[3+6×(sL3 + R3 )], R[1 + 4 × (sL1 + R1 )], R[2 + 4 × (sL2 + R2 )], or equivalently postmultiplying the matrix (9) by Q=        

1 0 0 sL1 + R1 −sL1 − R1 0

0 1 0 sL2 + R2 0 −sL2 − R2

0 0 1 0 −sL3 − R3 sL3 + R3

and taking into account (3a) we obtain [ ] rank Es − A B {[ ] } Es − A B Q = rank  0 0 0 0 1 0  0 0 0 0 0 1 = rank   −1 1 1 0 0 0 0 0 0 1 1 1 for all s ∈ C.

 0 0 0 0 0 0   0 0 0  , 1 0 0   0 1 0  0 0 1 (10)



 =4 

(11)

Example 2. Consider the descriptor linear electrical ci‑ rcuit shown in Figure 2 with given resistance R, capaci‑ tances C1 , C2 , C3 and source voltages e1 , e2 .

Fig. 2. Descriptor electrical circuit of Example 2 Using Kirchhoff’s laws we may write the equations e1 = R1 C1

86

which can be written in the form     [ ] u1 u1 d e1 E  u2  = A  u2  + B , e2 dt u3 u3

where





 0 −C3  , 0  0 −1 1 0 0 , B =  0 −1 −1 0

RC1 E =  C1 0

−1 A= 0 0

0 C2 0 

Using (3a) and (13b) we obtain [ ] rank Es − A B  sRC1 + 1 0 sC1 sC2 = rank  0 1 { 2 for s = 0, = 3 for s ̸= 0,

(13a)

 0 0 .

1

1 −sC3 1

(13b)

 1 0 0 0  0 1 (14)

By Theorem 7 the R‑controllability of the electrical circuit depends of its resistance and capacitances.

Parameter‑dependent entries of the matrix (14) can not be eliminated using elementary column operations.

Therefore, the electrical circuit is not structurally R‑ controllable.

4. Structurally R‐Observable Electrical Circuits

Therefore, the electrical circuit is structurally R‑ controllable.

du1 + u1 + u 3 , dt du1 du2 du3 + C2 = C3 , C1 dt dt dt e2 = u2 + u3 ,

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In this section structural R‑observability of the descriptor linear electrical circuits will be introduced and investigated. �e������o� �. The descriptor electrical circuit (1) is called structurally R‑observable if its R‑observability is independent of its resistances, inductances and capaci‑ tances.

Theorem 8. The descriptor electrical circuit (1) is structurally R‑observable if its outputs are linearly in‑ dependent combinations of those state variables which are expressed by ordinary differential equations in the circuit model, i.e. state variables related to linearly in‑ dependent meshes containing either resistances and in‑ ductances or resistances and capacitances.

Proof. Taking into account duality of notions of con‑ trollability and observability we can accomplish the proof in a similar way to the one of Theorem 7. If the assumptions are satis�ied then for each column of the matrix Es − A containing entries with resistances, inductances and capacitances the corresponding co‑ lumns of the matrix C have nonzero entries indepen‑ dent of the circuit parameters. Using elementary row operations it is possible to eliminate all parameter‑ dependent entries in these columns of the matrix [ ]T [Es − A]T C T . After this elimination proce‑ dure, which does not change the rank of the matrix, we obtain full column rank matrix with entries inde‑ pendent of the resistances, inductances and capacitan‑ ces.

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Example 3. (Continuation of the Example 1) The R‑ observability of the linear electrical circuit shown in Fi‑ gure 1 will be analyzed for the following two cases: [ ] Case 1. C = C1 0 ∈ ℜ2×4 , detC1 ̸= 0, sa‑ tisfying Theorem 8. [ ] Case 2. C = 0 C2 ∈ ℜ2×4 , detC2 ̸= 0, not satisfying Theorem 8.

In Case 1 we have the following results. Assuming C=

[

1 0

0 1

0 0

0 0

]

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and using (8b) and (15) we obtain [

       

Es − A C

]

sL1 + R1 0 −1 0 1 0

=

0 sL2 + R2 1 0 0 1

sL3 + R3 −sL3 − R3 1 0 0 0

0 0 0 1 0 0



  (16)  ,   

By Theorem 8 the R‑observability of the electrical cir‑ cuit is independent of its resistances, inductances and capacitance. Using on the matrix (16) elementary row operations L[1 + 3 × (−sL3 − R3 )], L[1 + 5 × (−sL1 − sL3 −R1 −R3 )], L[1+6×(sL3 +R3 )], L[2+3×(sL3 + R3 )], L[2+5×(sL3 +R3 )], L[2+6×(−sL2 −sL3 −R2 − R3 )], or equivalently premultiplying the matrix (16) by (17) and taking into account (4a) we obtain [

] Es − A rank C { [ ]} = rank P Es − AC   0 0 0 0  0 0 0 0     −1 1 1 0  =4  = rank    0 0 0 1   1 0 0 0  0 1 0 0

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for all s ∈ C.

Therefore, the electrical circuit is structurally R‑ observable.

In Case 2 we have the following results. Assuming C= 

   P =    84

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[ 1 0 0 0 0 0

0 0

0 0

1 0

0 1

]

0 −sL3 − R3 1 sL3 + R3 0 1 0 0 0 0 0 0

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and using (8b) and (19) we obtain [ ] Es − A = C  sL1 + R1 0 sL3 + R3  0 sL + R −sL 2 2 3 − R3   −1 1 1   0 0 0   0 0 1 0 0 0

0 0 0 1 0 1



  (20)  .   

By Theorem 8 the R‑observability of the electrical circuit depends of its resistances and inductances.

Parameter‑dependent entries of the matrix (20) can not be eliminated using elementary row operations.

Therefore, the electrical circuit is not structurally R‑ observable.

Example 4. (Continuation of the Example 2) The R‑ observability of the linear electrical circuit shown in Fi‑ gure 2 will be analyzed for the following three cases: Case 1. [ C= 1

Case 2. [ C= 0 1 Case 3. [ C= 0 0

0

0

0

1

]

, satisfying Theorem 8.

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]

, not satisfying Theorem 8. (22)

]

, not satisfying Theorem 8. (23)

In Case 1 we have the following results. From (8b) and (21) we obtain   0 1 sRC1 + 1 [ ]  sC2 −sC3  sC1 Es − A  . (24) =  0 1 1  C 1 0 0

By Theorem 8 the R‑observability of the electrical cir‑ cuit is independent of its resistance and capacitances.

Using on the matrix (24) elementary row operation L[1 + 4 × (−sRC1 )], neglecting its second row and ta‑ king into account (4a) we obtain   [ ] 1 0 1 Es − A rank = rank  0 1 1  = 3 C (25) 1 0 0 for all s ∈ C.

Therefore, the electrical circuit is structurally R‑ observable.

0 −s(L1 + L3 ) − R1 − R3 0 sL3 + R3 0 0 1 0 0 1 0 0

sL3 + R3 −s(L2 + R3 ) − R2 − R3 0 0 0 1



   ,   

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Journal of of Automation, Intelligent Systems Journal Automation,Mobile MobileRobotics Roboticsand and Intelligent Systems

In Case 2 we have the following results. From (8b) and (22) we obtain   0 1 sRC1 + 1 [ ]  sC2 −sC3  sC1 Es − A  . (26) =  0 1 1  C 0 1 0

By Theorem 8 the R‑observability of the electrical cir‑ cuit depends of its resistance R and capacitance C1 . The entry sRC1 of the matrix (26) can not be eliminated using elementary row operations. Therefore, the electri‑ cal circuit is not structurally R‑observable.

In Case 3 we have the following results. From (8b) and (23) we obtain   0 1 sRC1 + 1 [ ]  sC1 sC2 −sC3  Es − A  . (27) =  0 1 1  C 0 0 1 The analysis is similar to the one presented in Case 2.

By Theorem 8 the electrical circuit is not structurally R‑ observable.

5. Conclusion Structurally R‑controllable and structurally R‑ observable descriptor linear electrical circuits have been investigated. Suf�icient conditions under which the R‑controllability and R‑observability of descriptor linear electrical circuits are independent of their para‑ meters have been given (Theorems 7 and 8). The con‑ siderations have been illustrated by examples.

An open problem is an extension of presented approach to C‑controllability (C‑observability) and I‑ controllability (I‑observability).

The considerations can be also extended to des‑ criptor fractional systems with different fractional or‑ ders.

AUTHORS

Tadeusz Kaczorek – Bialystok University of Technology, Faculty of Electrical Engineering, Wiejska 45D Street, Bialystok, Poland, e‑mail: t.kaczorek@pb.edu.pl.

Kamil Borawski∗ – Bialystok University of Technology, Faculty of Electrical Engineering, Wiejska 45D Street, Bialystok, Poland, e‑mail: k.borawski@pb.edu.pl. ∗

Corresponding author

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