JAMRIS 2012 Vol 6 No 1 by JAMRIS

JOURNAL of AUTOMATION, MOBILE ROBOTICS & INTELLIGENT SYSTEMS

Editor-in-Chief Janusz Kacprzyk

Executive Editor: Anna Ładan aladan@piap.pl

(Systems Research Institute, Polish Academy of Sciences; PIAP, Poland)

Associate Editors: Jacek Salach (Warsaw University of Technology, Poland) Maciej Trojnacki (Warsaw University of Technology and PIAP, Poland)

Co-Editors: Oscar Castillo (Tijuana Institute of Technology, Mexico)

Statistical Editor: Małgorzata Kaliczyńska (PIAP, Poland)

Dimitar Filev (Research & Advanced Engineering, Ford Motor Company, USA)

Kaoru Hirota Editorial Office: Industrial Research Institute for Automation and Measurements PIAP Al. Jerozolimskie 202, 02-486 Warsaw, POLAND Tel. +48-22-8740109, office@jamris.org

(Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, Japan)

Witold Pedrycz (ECERF, University of Alberta, Canada)

Roman Szewczyk (PIAP, Warsaw University of Technology, Poland)

Editorial Board: Chairman: Janusz Kacprzyk (Polish Academy of Sciences; PIAP, Poland) Mariusz Andrzejczak (BUMAR, Poland) Plamen Angelov (Lancaster University, UK) Zenn Bien (Korea Advanced Institute of Science and Technology, Korea) Adam Borkowski (Polish Academy of Sciences, Poland) Wolfgang Borutzky (Fachhochschule Bonn-Rhein-Sieg, Germany) Chin Chen Chang (Feng Chia University, Taiwan) Jorge Manuel Miranda Dias (University of Coimbra, Portugal) Bogdan Gabryś (Bournemouth University, UK) Jan Jabłkowski (PIAP, Poland) Stanisław Kaczanowski (PIAP, Poland) Tadeusz Kaczorek (Warsaw University of Technology, Poland) Marian P. Kaźmierkowski (Warsaw University of Technology, Poland) Józef Korbicz (University of Zielona Góra, Poland) Krzysztof Kozłowski (Poznań University of Technology, Poland) Eckart Kramer (Fachhochschule Eberswalde, Germany) Piotr Kulczycki (Cracow University of Technology, Poland) Andrew Kusiak (University of Iowa, USA) Mark Last (Ben–Gurion University of the Negev, Israel) Anthony Maciejewski (Colorado State University, USA) Krzysztof Malinowski (Warsaw University of Technology, Poland) Andrzej Masłowski (Warsaw University of Technology, Poland)

Patricia Melin (Tijuana Institute of Technology, Mexico) Tadeusz Missala (PIAP, Poland) Fazel Naghdy (University of Wollongong, Australia) Zbigniew Nahorski (Polish Academy of Science, Poland) Antoni Niederliński (Silesian University of Technology, Poland) Witold Pedrycz (University of Alberta, Canada) Duc Truong Pham (Cardiff University, UK) Lech Polkowski (Polish-Japanese Institute of Information Technology, Poland) Alain Pruski (University of Metz, France) Leszek Rutkowski (Częstochowa University of Technology, Poland) Klaus Schilling (Julius-Maximilians-University Würzburg, Germany) Ryszard Tadeusiewicz (AGH University of Science and Technology in Cracow, Poland)

Stanisław Tarasiewicz (University of Laval, Canada) Piotr Tatjewski (Warsaw University of Technology, Poland) Władysław Torbicz (Polish Academy of Sciences, Poland) Leszek Trybus (Rzeszów University of Technology, Poland) René Wamkeue (University of Québec, Canada) Janusz Zalewski (Florida Gulf Coast University, USA) Marek Zaremba (University of Québec, Canada) Teresa Zielińska (Warsaw University of Technology, Poland)

Publisher: Industrial Research Institute for Automation and Measurements PIAP

If in doubt about the proper edition of contributions, please contact the Executive Editor. Articles are reviewed, excluding advertisements and descriptions of products. The Editor does not take the responsibility for contents of advertisements, inserts etc. The Editor reserves the right to make relevant revisions, abbreviations and adjustments to the articles.

JOURNAL of AUTOMATION, MOBILE ROBOTICS & INTELLIGENT SYSTEMS VOLUME 6, N° 1, 2012

CONTENTS 3

Linear Speed Control for Multi-Machine system Using Fuzzy-Sliding Mode Bousmaha Bouchiba, Abdeldjebar Hazzab, Hachemi Glaoui, Fellah Med-Karim, Ismaïl Khalil Bousserhane, Pierre Sicard

Chaotic Mobile Robot Workspace Coverage Enhancement Ashraf Anwar Fahmy 39

Accelerometer-based Measurements of Axial Tilt Sergiusz Łuczak

Effective Measyrand Estimators for Samples of Trapezoidal PDFs Zygmunt Lech Warsza

Increased Performance of a Hybrid Optimizer for Simulation Based Controller Parameterization Reimund Neugebauer, Kevin Hipp, Arvid Hellmich, Holger Schlegel

Automated Map Comparison using Non-invariant Fourier Descriptors Robert Ouellette, Kotaro Hirasawa

Postural Equilibrium Criteria Concerning Feet Properties for Biped Robots Alejandro González de Alba, Teresa Zielinska

Measuring of the Basic Parameters of LCD Displays Roman Barczyk, Błazej Kabzinski, Danuta JasinskaChoromanska, Agnieszka Stienss

Pole-placement Adaptive Control for a Plant with Unknown Structure and Parameters – a Simulation Study Dariusz Horla

Intelligent BRT in Tehran Peyman Parvizi, Sasan Mohammadi, Farzad Norouzi Fard

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Linear Speed Control for Multi-Machine system Using Fuzzy-Sliding Mode Submitted 16th March 2010; accepted 1st June 2010

Bousmaha Bouchiba, Abdeldjebar Hazzab, Hachemi Glaoui, Fellah Med-Karim, Ismaïl Khalil Bousserhane and Pierre Sicard

Abstract: In this contribution, a control scheme based on multi input multi output Fuzzy Sliding Mode control (MIMOFSMC) for linear speed regulation of multi-motors system is proposed. Once the decoupled model of the multi-motors system is obtained, a smooth control function with a threshold was chosen to indicate how far the state from to the sliding surface is. However, the magnitude of this control function depends closely on the upper bound of uncertainties, and this generates chattering. So, this magnitude has to be chosen with great care to obtain high performances. Usually the upper bound of uncertainties is difficult to known before motor operation, so, a Fuzzy Sliding Mode controller is investigated to solve this difficulty; a simple Fuzzy inference mechanism is used to reduce the chattering phenomenon by simple adjustments. A simulation study is carried out and shows that the proposed controller has great potential for use as an alternative to the conventional sliding mode contro. Keywords: multi-motors system, fuzzy logic, sliding mode control, MIMO

1. Introduction

The systems handling web material such as textile, paper, polymer or metal are very common in the industry. The modelling and the control of multi-motors systems have been studied already for several decades [1]. The increasing requirement on control performance, however, and the handling of thinner web material led us to search for more sophisticated control strategies. One of the objectives in such systems is to increase web velocity as much as possible, while controlling web tension over the entire production line. This requires decoupling between web tension and speed, so that a constant tension can be maintained during speed changes [2, 3]. The decoupled adaptive fuzzy Sliding Mode Control (SMC) for robotic manipulators. This controller is proposed for a class of Multiple-Input Multiple-Output (MIMO) systems with unknown non-linear dynamics. Indeed, an online fuzzy adaptation scheme is suggested to approximate unknown non-linear functions to design SMC [4]. The stable adaptive fuzzy sliding-mode controller is developed in [5] for nonlinear multivariable systems with unavailable states. When the system states are not available, the estimated states from a semi high gain observer are used to construct the output feedback fuzzy controller by incorporating the dynamic sliding mode. It is proved that uniformly asymptotic output feedback stabilization can be achieved with the tracking error approaching to zero. The design and application of an adaptive fuzzy total sliding-mode

controller (AFTSC) are addressed in [6]. The proposed control scheme comprises a special fuzzy sliding-mode controller and an adaptive tuner. The former is a main controller, which is designed without reaching phase to retain the merits of a total sliding-mode control approach. In this work the design of fuzzy sliding-mode (FSMC) to control a multi-motors system are proposed in order to improve the performances of the control system, which are coupled mechanically, and synthesis of the robust control and their application to synchronize the five sequences and to maintain a constant mechanical tension between the rollers of the system [7]. In this contribution, based on fuzzy variable structure control concept, the authors introduced a control scheme for the design and the tuning of fuzzy logic controllers with an application to winding system. To show the benefits of the MIMO-FSMC, simulation results comparing the performance of the proposed controller with that of Single-Input Single-Output Sliding Mode Controller (SISO-SMC) and with that of the conventional Multi-Input Multi-Output Proportional and Integral (MIMO-PI) controller are presented. The results obtained confirm that the proposed control structure improves the performance and the robustness of the drive system. The model of the winding system and in particular the model of the mechanical coupling are developed and presented in Section II. Section III shows the development of sliding mode controllers design for winding system. The proposed MIMO fuzzy sliding mode control is given in the section IV. Section V shows the Simulation results using Matlab Simulink of different studied cases. Finally, the conclusion is drawn in Section VI.

2. System model’s In this system, the motor M1 carries out unreeling, M3 drives the fabric by friction and M5 is used to carry out winding, each one of the motors M2 and M4 drives two rollers via gears “to grip” the band (Fig.1). Each one of M2 and M4 could be replaced by two motors, which each one would drive a roller of the stages of pinching off. The elements of control of pressure between the rollers are not represented and not even considered in the study. The stage of pinching off can make it possible to isolate two zones and to create a buffer zone [8, 9]. The objective of these systems is to maintain the tape speed constant and to control the tension in the band. The used motors are three phase induction motors type which each one is supplied by an inverter voltage controlled with Pulse Modulation Width (PWM) techniques. Articles

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Part 2 :Roll velocity model d (J 1(t )Ω1 ) = R1(t )T2 + C em 1 − f1(t )Ω1  dt  d (J 2 (t )Ω2 ) = R2 (t )(T3 − T2 ) + C em 2 − f2 (t )Ω2  dt  d (J 3 (t )Ω3 ) = R3 (t )(T4 − T3 ) + C em 3 − f3 (t )Ω3  dt  d (J 4 (t )Ω4 ) = R4 (t )(T5 − T4 ) + C em 4 − f4 (t )Ω4  dt  d (J 5 (t )Ω5 ) = R (t )(−T ) + C − f (t )Ω em 5 5 5 5 5  dt 

Five motors websystem transport system Fig. 1. FiveFig.1. motors web transport

A model based on circuit equivalent equations is generally sufficient in order to make control synthesis. The electrical dynamic model of three-phase Y-connected induction motor can be expressed in the d-q synchronously rotating frame as [13]: 2     dids = 1  −  R +  Lm  ⋅ R  ⋅ i + σ L ω i +   r s e qs  dt σ ⋅ Ls   s  Lr   ds     Lm ⋅ Rr L  ⋅ ϕdr + m ⋅ ϕqr ⋅ ωr + Vds   L2r Lr   2  di      qs = 1  −σ L ω i −  R + Lm ⋅ R  ⋅ i −   s e ds r  dt σ ⋅ Ls   s  Lr   qs     Lm L ⋅R  ⋅ ϕdr ⋅ ωr + m 2 r ⋅ ϕqr + Vqs   L L   r r  dϕ L R R ⋅  dr = m r ⋅ i − r ⋅ ϕ + ω − ω ⋅ ϕ dr e r dr ds  dt Lr Lr  R dϕqr Lm ⋅ Rr ⋅ iqs − ωe − ωr ⋅ ϕdr − r ⋅ ϕqr  dt = L Lr r  2 d ω P .Lm f P . iqs .ϕdr − ids .ϕqr − c .ωr − .Tl  r = dt J J L J .  r   (1)

(

Where by:

)

is the coefficient of dispersion and is given L2m LsLr

(2)

By using of Hooke’s law, Coulomb’s law, mass conservation law and the laws of motion for each rotating roll the mechanical part model of the system is given by the two following parts equations [8, 9]: Part 1: Tension model between two consecutive rolls  dT2 L1  dt  dT3 L2 dt  L dT4  3 dt  dT 5 L  4 dt

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(

= ES (V2 − V1 ) − TV 2 2 = ES (V3 − V2 ) + TV − T3V3 2 2 = ES (V4 − V3 ) + T3V3 − T4V4 = ES (V5 − V4 ) + T4V4 − TV 5 5

(3)

)

3. Sliding mode control

The sliding mode control consists in moving the state trajectory of the system toward a predetermined surface called sliding or switching surface and in maintaining it around this latter with an appropriate switching logic. In the case of the nth-order system, the sliding surface could be defined as [12]:  ∂  S (x ) =  + λ   ∂t 

)

σ = 1−

(4) k = 2, 3, 4, 5. Lk−1 is the web length between roll k−1 and roll k; Tk is the tension on the web between roll k−1 and roll k; Vk is the linear velocity of the web on roll k; Wk = wk ⋅ P : The rotational speed of roll k; Rk is the radius of roll k; E is the Young modulus; S is the web section; C emk = iqsk .ϕdrk − idsk .ϕqrk is the k-th motor torque, fk is the friction torque of k-th motor torque.

n −1

⋅ e(x )

(5)

Where λ > 0 The control law is divided into two parts, the equivalent control Ueq and the attractivity or reachability control Un. The equivalent control is determined off-line with a model that represents the plant as accurately as possible. If the plant is exactly identical to the model used for determining Ueq and there are no disturbances, there would be no need to apply an additional control Un. However, in practice there are a lot of differences between the model and the actual plant. Therefore, the control component Un is necessary which will always guarantee that the state is attracted to the switching surface by satisfying the condition [12]: .

S (x ) ⋅ S (x ) < 0

Therefore, the basic switching law is of the form: U = U eq + U n

(6)

With U n = −M (⋅) ⋅ sgn(S (⋅)) M(S): the magnitude of the attractivity control law Un, and sgn is the sign function In a conventional variable structure control, Un generates a high control activity. It was first taken as constant, a relay function, which is very harmful to the actuators and may excite the unmodeled dynamics of the System. This is known as a chattering phenomenon. Ideally, to reach the sliding surface, the chattering phenomenon

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( )

 S U n − f x , t = −α i sat  1 _ f ϕ   1

( )

Consider the class of nonlinear time varying systems described by the equations:

yj = xj

(8)

(12)

U eq x , t = −D(t , x )−1 ⋅ F (t , x )

4. MIMO fuzzy sliding mode control (MIMO-FSMC)

(7)

2012

U = U eq + U n − f

should be eliminated [12, 13]. However, in practice, chattering can only be reduced.

x (jn ) = f j (X 1, , X m ) + bj (X 1, , X m )u j + d j (t )

N° 1

Si _ f

ϕi

Sn _ f   (13)  ϕ5   

The Fig. 2 shows the SMC control strategy scheme for each induction motor Rectifier L Filtre

X j = x j , x j , , x (jn −1)  : the j-th components of the state

Pont

vector.

INV.

PWM

X = [X 1 ,....., X m ]

u j is the j-th control input and y j is the j-th system output

In (7) the function fj , the control gain bj, and the disturbance dj are assumed to be unknown. The dynamics of (7) describe a large number of nonlinear systems encountered in practice, including a vast class of controllable nonlinear systems that could be converted into (7) by using appropriate transformations. Then we can write a state space representation of (7) in terms of e j = x *j − x j ( x *j = constant ), and its derivatives: e j 1 = e j 2  e j 2 = e j 3   e = e jn  jn −1 e jn = − f j (X ) − bj (X ) u j − d j (t ) 

(9)

Rule j: IF ei ∈ Ri j THEN λi = λij

Where j = 1,… , ri Where ei is the traking error for the i th system variable, and ri is the total number of rules for the i th system variable. In (10), Rij is the j th fuzzy set on the i th universe of discourse, Characterized by membership function u (e ) . Therefore, each tracking error ei , a fuzzy system is built such that each rule j has a specific control bandwidth in the consequent part. The aggregate control bandwidth λi _ f is obtained by center average defuzzification and can be viewed as a nonlinear interpolation between linear mappings: i

λi _ f =

∑u j =1

Vref + -

FSMC

* Cem

IFOC

* Vds * Vqs

PARK

PAR K-1

w*s ids

iqs

Fig. 2. Block diagram for each motor with FSMC control

5. Simulation results

Where: e j 1 = e j ,… ,e jn = e(jn −1) and x j 1 = x j ,… , x jn = x (jn −1) The fuzzy system rule base for control bandwidths λi is defined as follows Rule 1: IF ei ∈ Ri1 THEN λi = λi1 Rule2: IF ei ∈ Ri2 THEN λi = λi2 (10)

j i

φr*

j i

λij

∑ uij

(11)

The winding system we modeled is simulated using MATLAB SIMULINK software and the simulation is carried out on 10s. To evaluate system performance we carried out numerical simulations under the following conditions: Start with the linear velocity of the web of 5m / s. The motor M1 has the role of Unwinder a roll radius R1 (R1 = 2.25 m). The motors M2, M3, M4 are the role is to pinch the tape. The motor M5 has the role of winding a roll of radius R5. The aims of the STOP block is to stop at the same time the different motors of the system when a radius adjust to a desired value (for example R5 = 0.8 m), by injecting a reference speed zero. The comparison between the two controllers SISOFSMC and MIMO-FSMC is achieved in the two cases: Comparison of the control performances: it has been made by the comparison of the average speeds of the five motors Vavg, for each controller this average is expressed by the equation (14). Comparison of synchronism between the speeds of the five motors: in this point one makes a comparison between the deviation standard of speeds of five motors Vstd, for each controller this average is expressed by the equation (15). Vavg =

j =1

Based on the result from (11), the resulting sliding surface is represented as: Si _ f = e + λi _ f . ei

Finally the proposed MIMO Fuzzy Sliding Mode control (MIMO-FSMC) law is

Vstd = (

1 n

∑V

∑ (V −V i

i =1

(14)

i =1

avg

1 2 2

) )

(15)

As shown in Figs. (3-5). An improvement of the linear speed is registered, and has follows the reference speed for both PI controller and FSMC control, but in case of Articles

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PI controller, the overshoot in linear speed of Unwinder is 25%. Fig (4) and Fig (5) show that with the MIMOFSMC controller the system follows the reference speed after 0.3 sec, in all motors, however, in the SISO-FSMC and PI controller the system follows after 1.3 sec and 2 sec, respectively. From the Figures (3-5), we can say that: the effect of the disturbance is neglected in the case of the MIMO-FSMC controller. It appears clearly that the classical control with PI controller is easy to apply. However, the control with MIMO-FSMC offers better performances in both of the overshoot control and the tracking error. Fig. 6 and Fig. 7 show the comparison between the MIMO-FSMC controller, the SISO-FSMC controller and the MIMO-PI controller. After this comparison we can judge that the MIMO-FSMC controller presents a clean improvement to the level of the performances of control, compared to the MIMO-PI controller, the synchronism between the five motors is improved with MIMO-FSMC controller compared to SISO-FSMC controller.

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Fig. 5. The linear speed of winder M5

6. Conclusion The sliding mode control of the field oriented induction motor was proposed. To show the effectiveness and performances of the developed control scheme, simulation study was carried out. Good results were obtained despite the simplicity of the chosen sliding surfaces. The robustness and the tracking qualities of the proposed control system using sliding mode controllers appear clearly.

Fig. 6. comparison between the MIMO-FSMC, SISOFSMC and PI MIMO with average speeds of five motors

Fig. 3. The linear speed of unwinder M1 Fig. 7. Comparison between the MIMO-FSMC, SISOFSMC and PI MIMO with the deviation standard of speeds of five motors

Fig 4. The linear speed of motor M2 6

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Furthermore, in order to reduce the chattering, due to the discontinuous nature of the controller, fuzzy logic controllers were added to the sliding mode controllers. These gave good results as well and simplicity with regards to the adjustment of parameters. The simulations results show the efficiency of the FSMC-MIMO controller technique, however the strategy of FSMC-MIMO Controller brings good performances, and she is more efficient than the FSMC-SISO controller and classical PI-MIMO controller.

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Authors: Bousmaha Bouchiba* – Laboratory of command Analysis and Optimization of the Electro-Energizing Systems, Faculty of Sciences and technology, BECHAR University B.P. 417 BECHAR, 08000 ALGERIA (e-mail: bouchiba_bousmaha@yahoo.fr). Hachemi Glaoui, Abdeldejbar Hazzab and Ismaïl Khalil Bousserhane – Laboratory of command Analysis and Optimization of the Electro-Energizing Systems, Faculty of Sciences and technology, BECHAR University B.P. 417 BECHAR, 08000 Algeria (e-mail: glaouih@ yahoo.fr, a_hazzab@yahoo.fr and bou_isma@yahoo.fr) Fellah Med-Karim – Laboratory of ICEPS Intelligent Control and Electrical power Systems University of sidi bel Abbes, ALGERIA( e-mail: mkfellah@yahoo.fr). *Corresponding author

References: 1.

Christian Thiffault Pierre Sicard Alain Bouscayrol, “Tension Control Loop Using A Linear Actuator Based On The Energetic Macroscopic Representation”, CCECE 2004–CCGEI 2004, Niagara Falls, May 2004. S. Charlemagne, A. Bouscayrol, Slama Belkhodja, J.P. Hautier, “Flatness based control of non-linear textile multimachine process”. In: Proc. of EPE’03, CD-ROM, Toulouse (France), September 2003. Adlane Benlatreche Dominique Knittel “State Feedback Control with Full or Partial Integral Action for Large Scale Winding Systems”. In: Industry Applications Conference, Oct. 2005, vol. 2, pp. 973-978. Hanène Medhaffar, Nabil Derbel ,Tarak Damak “A decoupled fuzzy indirect adaptive sliding mode controller with application to robot manipulator”, Int. J. Modelling, Identification and Control, vol. 1, no. 1, 2006.

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Shaocheng Tong and Han-Xiong Li “Fuzzy Adaptive Sliding-Mode Control for MIMO Nonlinear Systems”, IEEE Transactions on Fuzzy Systems, vol. 11, no. 3, June 2003, pp. 354-360. 6. Chung-Chun Kung, Kuo-Ho Su and Lun-Ping Hung “Adaptive Fuzzy Total Sliding-Mode Controller Design and Its Application to Tension Control of a Winding process”. In: 2006 IEEE International Conference on Systems, Man, and Cybernetics, 8th –11th October, 2006, Taipei, Taiwan, pp. 3659-3664. 7. Rahmi Guclu “Sliding mode and PID control of a structural system against earthquake”, Mathematical and Computer Modeling, vol. 44, issues 1–2, 2006, pp. 210-217. 8. Hakan Koç, Dominique Knittel, Michel de Mathelin, and Gabriel Abba, “Modeling and Robust Control of Winding Systems for Elastic Webs”, IEEE Transactions on Control Systems Technology, vol. 10, no. 2, March 2002, pp. 197-208. 9. H. Koc, “Modélisation et commande robuste d’un system d’entrainement de bande flexible”, Ph.D. thesis, Université Louis Pasteur (Strasbourg I University), 2000. 10. Zhongze Chen Changhong Shan Huiling Zhu “Adaptive Fuzzy Sliding Mode Control Algorithm for a Non-Affine Nonlinear System ”, IEEE Transactions on Energy Conversion, vol. 19, issue 2, 2004 , pp. 362-368. 11. Yujie Zhao, Qingli Wang, Jinxue Xu, Chengyuan Wang, ”A Fuzzy Sliding Mode Control Based on Model Reference Adaptive Control for Permanent Magnet Synchronous Linear Motor”. In: 2nd IEEE Conference on Industrial Electronics and Applications – ICIEA 2007, pp. 980-984. 12. Chien-An Chen, Huann-Keng Chiang, Chih-Huang Tseng, “The Novel Fuzzy Sliding Mode Control of Synchronous Reluctance Motor”. In: 8th International Conference on Intelligent Systems Design and Applications – ISDA’08, vol. 1, 2008, pp. 576-581.

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Effective Measyrand Estimators for Samples of Trapezoidal PDFs Submitted 5th December 2010; accepted 22nd March 2011

Zygmunt Lech Warsza

Abstract: This paper is final overview of investigations on the accuracy of basic estimators of trapezoidal probability distribution samples of the measured data. For symmetrical trapezoidal PDF of straight as well concaved sides, using Monte-Carlo method of simulation, the standard deviation (SD) of linear 1- and 2-component estimators are evaluated. Approaches for theirs evaluation are proposed. It is established that in the ratio of upper and bottom bases of trapezoidal PDF in the range from 1 to 0,35 the mid-range value has smaller standard deviation (SD) than the mean value and median. It is find then for the whole family of the symmetric linear trapezoidal PDF more accurate than above single element estimators are two-component (2C) estimators as the linear form of the mean and mid-range values of the sample. Their coefficients are found, properties discussed and formulas of SD are given. The new simplified 2C-estimator of equal coefficients is also proposed. These estimators successfully extend estimation of the measurand value as the sample mean and description of its accuracy by the uncertainty type A recommended by the international guides of uncertainty evaluation in measurement GUM-2008 [1], EA4/02 [2] and by Handbook NASA [3]. Approaches of described below investigations could be effectively applied also for other models of convoluted PDF-s. Keywords: estimators of probability density function, trapezoidal PDF, mid-range, uncertainty evaluation

1. Introduction

Random components of measurement data can be in many cases more accurately modelled by non-Gaussian probability density distribution function (PDF) than by Normal distribution as the range of data random dispersion is commonly limited in reality. The mean value as the most effective measurand estimator of the n-element sample of Normal distribution is also used for other distributions. Its standard deviation (SD) is defined in GUM [1] as the uncertainty type A. For data processing it is very important to choose an effective estimator of the centre coordinate of PDF, i.e. estimator of the smallest SD, as not proper evaluation entails incorrect assessment of the measurement accuracy. For samples modelled by Normal, Uniform and Laplace (double-exponential) PDF distributions, it is presented in the paper [4] of 15th IMEKO TC4 Symposium in Iasi Romania, how to regard the data autocorrelation and which estimator has the smallest standard deviation (SD) to be chosen as the better accurate for any of them. 8

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E. g. more effective estimator then mean value of measurand of Uniform samples is mid-range and for Laplace sample – median, respectively. Using one of goodnessof-fit tests (Kolmogorov–Smirnov, Cram´er–von Mises, Chi-Square and other tests) we make decision about the estimation choice. The main purpose of this work is the expansion of opportunities for choosing the best single or a few component estimators of empirical data modelled by more complex non-Gaussian distributions than the above models. It is assumed that treated measurement data do not contain unknown systematic errors and are not selfcorrelated. The estimator of the distribution parameter should meet also requirements of solvency, sufficiency, efficiency and be unbiased. First of all, efficiency of estimators is researched.

2. Single component estimators

Let’s check up which one of single-component estimators of PDF of particular samples: mean X, mid-range qV / 2 or median X med , satisfies the requirement of efficiency, i.e. has the least-possible sum of the square dispersion, denotes a minimum standard deviation in comparison with other estimators. Similarly, it is possible to receive results for other basic non-Gaussian distributions. In columns 3–5 of Tab. 1 values of standard deviations of three estimators of a few basic distribution models of empirical data (for demonstration of difference order only) are presented. Standard deviation of the best single component estimator of the particular non-Gaussian distribution is significantly less then of other estimators even if difference between their values, e.g. between midrange and mean, is small. This is the cause to search for estimators better then the sample mean.

3. The best single component estimators of trapeze distributions 3.1. Linear trapeze

It is important to consider the problem of choice of an effective estimator for composition of simple distributions. In the measurement systems practically all analogue signals now are digitalised, and then uniform distributions are very common in these systems. So, with convolution of two different uniform distributions we get PDF as a symmetrical trapezoid of linear sides, from triangular to the uniform distribution as its boundary cases. The effective single component estimators of the centre of the triangular and uniform distributions are the sample mean and the mid-range respectively – see again Table 1.

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Тable 1. Comparison of sufficiency of different estimators and expression of the standard uncertainty Standard deviations of sample estimators Distribution

S mean

Smidrange

S med

The most effective estimator

Normal

0,010

0,220

0,013

sample mean

Uniform

0,006

1,4·10-4

0,010

mid-range

Double-exponential

0,007

0,870

7·10-5

median

Triangular

0,0040

0,0045

0,0049

sample mean

Arcsine

0,067

5·10-5

0,146

mid-range

The aim of the following research is to find a position of border separating trapezoids of better mid-range or mean values. There are two ways to obtain the trapezoid in MC simulations : – to generate two uniform distributions and theirs sum [9]; – to use the inverse function method (derived in [9] for trapezoid). Both techniques were tested. Samples from population with trapezoidal distribution with β = a/b ratio of their shorter upper a and longer bottom b basis are simulated and stable results are obtained. Obviously β ∈ (0; 1) was taken. Fig. 1 shows how standard deviations of mean and mid-range are changed with a ratio β and number of oba)

Standard uncertainty of the most effective estimator uA = Sx / n

[1]

V n +1 2 (n − 1) n + 2 [4]

S x / 2n Sx / n

[4]

[3] - [5]

S x ⋅ 5π 4 / n 2

[5]

servations n [11]. Median SD is significantly larger and is not shown on fig 1. Border value α of ratio β has been found for us also analytically by young mathematician P. Endovitskyi from TU Kiev. He obtained (1) Novitzky and Zograph in their original book [6] show dependence of estimator (mid-range) efficiency on a type of distribution. Topographical classification of distributions is also offered and dependence of the estimator efficiency on the counter-kurtosis æ is presented. Variances of estimators are equal when æ =0,675. This value of æ corresponds to kurtosis E=1/æ2= -0,805. For Normal PDF E=3. Dependence of kurtosis differences E-3 from Normal PDF on ratio of trapezium bases β are given on Fig. 2. Then we can find that E = -0,805 corresponds to β = 0,35.

3.2. Curvilinear trapeze

Fig. 1. Efficiency of single component sample estimators of Trap(a,b) distributions [10]: a. Dependences of sample mean and midrange standard deviations S on ratio β of linear trapeze bases and of sample size n, b. cut-set of S surfaces for n=const.= 400

In Table 1 of GUM Supplement 1 [2] the curvilinear trapezoidal of concave sides is given. This PDF model has the symbol CTrap(a,b,d). It is proposed to be used when limits of upper a and lower b sides are inexactly given, i.e. a ± d and b ± d, where a, b and d, with d > 0 and a +d < b − d, are specified. Histogram of these type simulated data is given on Fig. 3. Fig. 4a, b shows how standard deviations of main estimators depend on the number n of observations in the sample and a ratio βc=(a2–a1–2d)/(b2–b1+2d) of curvilin-

Fig. 2. Kurtosis differences (E -3) of trapezoid and Normal PDFs as function of ratio β of trapeze bases Articles

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of best values, or by known analytical equations. These equations are derived by numeric methods too and they based on shape coefficients or parameters of the distribution model.

4. Multi-component estimators of trapeze distribution

4.1. Three- and two-component estimators based on kurtosis E value

Fig. 3. Example of curvilinear trapezoid PDF ear trapezoid basis. It is shown that median here is the best single component estimator if 0 < βc< 0,08; mean – if 0,08 < βc< 0,5 and mid-range if 0,5 < βc< 1. But, it should be taken into account, that in practice, uncertainty of uncertainty may be limited up to even 20 – 30 %, then: d=

(b − a ) (1 − β c ) (b − a ) ⋅ = 0,3 ⇒ β = 0,54 2 (1 + β c ) 2

(2)

and could be decided that the mid-range may be applied as the most effective estimator to the border drawn in Fig. 4. To increase accuracy of the measurement result other types of estimators, which contain a few components, may be also considered. According to considered approaches, ratio of these components could be found by modelling and selection a)

Zakharov and Stephen in [7, 8] considered for nonGaussian symmetrical PDF the linear 3-component (3C) estimator of measurand value: Xˆ = k1 X + k2 qV /2 + k3 X med

(3)

as the efficient estimate of the expectation. Coefficients k1, k2 and k3 depend on the kurtosis E of the distribution of observation results. For linear trapezoids of E ∈ (−1,15; − 0, 2) only two such coefficients are enough [7]: k1 = −1, 05 E + 1, 22, k2 = −0, 05 E − 0, 22 , k3 = 0 (4)

Modelling shows that such proposed estimator is biased [10] and it is not consistent with requirements of the effective estimator. For unbiased estimator the sum of all three coefficients must be equal to 1. From MC investigations [10, 11] k1 = −1, 05 E + 1, 22,

, k3 = 0 (5)

Standard deviations of X , qV/2 and 2C estimator Xˆ corrected due (2) for linear trapezes of different β are given in Fig. 5.

Fig. 5. Dependences of standard deviations for different statistics on a ratio of bases (linear trapeze) In case of the curvilinear trapeze its kurtosis E ∈ (−1, 2;0, 2) and following coefficients have been obtained

Fig. 4. Efficiency of single component estimators of CTrap(a,b,d) distribution: a) Dependences of SD on a ratio of bases β and on sample size n of the curvilinear trapezoidal PDF, b) visualization of crossing points for n=const=400 10

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4.2. Two-component (2C)estimators of trapeze distributions based on ratio β of their bases

We investigated broadly [11] a whole family of linear trapezoids and found the most effective unbiased twocomponent estimator. It could be expressed by X eff = k1 X + (1 − k1 )qV / 2 ,

(7)

This estimator was analyzed by changing k1 in (7) for different trapezoids from rectangular to triangular shapes and values of k1 corresponding for min {S [ X eff ]} and with negligible correlation of X and qV/2 (large n), were obtained. Results with the uncertainty under 10% are as follows:

Fig. 7. Dependences standard deviations of the different statistics’ on a ratio of bases mator for a full range. From analyzes of some cases of β > 0,54 it is recommend to use in practice two below formulas of the best estimator:

(8)

1 1  ⋅ qV /2 + X , if 0,54 < β < 0,8; = 2 2 qV /2 if β > 0,8.

X eff

, Results are stable even when n is changed from 10 up to 10000 for trapezoids with different ratio of their bases. Results of application (3) after converting k1, k2 coefficients from E to βC are given in Fig. 6. One can see, that in a short interval β c ∈ [0; 0,8] the best estimator is median. That does not appear in approach proposed in [7] and [8].

(10)

Application of (7) gives the dependence on β as in Fig. 7. The difference between analytical results and modeling is less than 5%. And it is natural that the triangular distribution is not exactly like Normal PDF, but an intermediate one, between Uniform and Normal. So its the best estimator consisting also of both components.

5. Uncertainty evaluation

5.1. Theoretical background

Because of correlation between X min and X max conjoint density function has to be found here. Standard deviation SD of two-component estimator (6) is u A2 = S [ X eff ] = k12 S 2 [ X ] + (1 − k1 ) 2 S 2 [qV /2 ] + 2ρk1 (1 − k1 ) S [ X ]S [qV /2 ]

Fig. 6. Dependences of standard deviations for different statistics on a ratio of bases (curvilinear trapeze) where:

4.3. Simplified 2C-estimator of trapeze distributions

Let us analyze simplified 2C-estimator based on two equal components ~ X = 0,5 X + 0,5 qV / 2 .

(9)

The results of its MC modelling for linear trapezium are given also in Fig. 6. From these results one can see that simple 2C-estimator (9) is the best for a wide range of trapezoids ( 0 < β < 0, 75 ). If trapezoid is considered as the convolution of two uniform distributions, the range for one of them is 2(1 + 2 β ) times larger than for the other one. Values β > 0,75 correspond to the ratio of ranges over 8. It means that one of uniform distributions is dominant. For this range the mid-range is the best estimator (see Fig. 8). The results of investigations: how far formula (9) is applicable also for curvilinear trapezoid, are clear from Fig. 6. One can see that it is not the most effective esti-

S[ X ] = S 2 [qV /2 ] =

(11)

Sx n;

V 2 (1 − β 2 ) n ⋅ 16 (n + 1)(n + 2) .

The recommendations on correlation coefficient values obtained by MC simulation are given in Table 2. Table 2. The values of correlation coefficients n

[100; 200) [200; 300) [300; 500) 0,25

0,2

0,15

n→∞

ρ →0

Variance of the best estimator Xeff should be minimum. Let’s try to find analytically its value of k1. For large n last component in (11) from correlation between X and qV/2 is negligible. Then S 2 [ X eff ] = ( k1 ⋅ S [ x ]) + ( (1 − k1 ) ⋅ S [qV /2 ]) , (12) 2

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where:

VOLUME 6,

is the SD of whole population.

u A = S [ X eff ] =

Coefficient k1 could be find from

∂k1

S V (1 − β 2 ) n 1 S x2 V 2 (1 − β 2 ) + ⋅ + 2ρ x . 2 n 16 (n + 1)(n + 2) (n + 1)(n + 2) (14) If ρ → 0

S 2 [qV /2 ] S 2 [ x ] + S 2 [qV /2 ]

u A = S [ X eff ] =

(13)

For triangular distribution (β =0) [5], [6]:

k1 =

3 ⋅ (4 − π) 2 σX , 2n

k ( P) =

3(4 − π ) ≈ 0, 56 . 2 + 3(4 − π )

2σ2X = 0,5 . n

It coincides with (10). For rectangular distribution (β =1):

k1 =

3σ2X 2(n + 1)(n + 2) σ2X n

3σ2X

3n 2

2n + 9n + 2

, lim

n →∞

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(1 −

(1 − P ) ⋅ (1 − β 2 )

= 0.

2(n + 1)(n + 2)

Fig. 8. Dependences of standard deviations on k1 12

2n + 9n + 2

If n → ∞, k1 → 0. For n = 30 , k1 = 0, 04 and n=10: k1= 0, so these results are not very far from the above k1= 0 for n → ∞. Dependences of SD on k1 for boundary cases of trapezium shape (triangular and rectangular PDF) are shown in Fig 8. It is natural that the triangular distribution is not exactly like Normal PDF, but an intermediate one, between Uniform and Normal. So it’s the best estimator consisting also of both components.

(14a)

)

(15)

Considerations has to be illustrated below by the numerical example of measurand value and uncertainty calculations. Data values of the sample size n=200 obtained in simulated experiment are shown in Fig. 9. As no other information is available then should be presume that this observations are not autocorrelated and cleaned before from systematic errors. Let’s find the measurement result as the best estimator of measurand value, its standard and expanded uncertainties. The proper PDF model of this sample has to be chosen. Sample observations are arranged into 15 groups (Fig. 10).

For trapezoid with β = 0,35 we find that, and from (6): σ2X n

6 1+ β 2

6. Numerical Example

It coincides with results of the earlier MC simulation.

k1 =

1 S 2 [ X ] + S 2 [qV /2 ] 2

As standard deviation of the proposed estimator is used the standard uncertainty, we should give expressions for coverage factor k(P) to expanded uncertainty calculation. The equation for the large sample size is [11, 12]:

5.2. Particular cases

S 2 [qV /2 ] =

1 S 2 [ X ] + S 2 [qV /2 ] + 2ρ S [ X ] S [qV /2 ] = 2

After calculations: k1 =

2012

For simplified two-component estimator of (6) The standard uncertainty (equivalent to uA in GUM) is:

S[ X ] = σ x / n , σ x

∂S 2 [ X eff ]

N° 1

Fig. 9. Values of sample observations.

Fig. 10. Histogram of data relative frequencies

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Hypothesis about compliance with three different theoretical distributions are verified by χ 2 test. Number of freedom is 11. Compliance with Uniform and Normal distributions is not fulfilled, but with linear trapezoidal distribution is accepted at significance level 0,05, because: χ 2 = 17,3 < χ 211, 0,05 = 19,7 .

X = 22,873 , qV /2 = 23, 010 , X = 22,942 . Sample standard deviation: S X = 1,309 .

Standard deviation of the mean: S [ X ] = S x / n = 0, 0926 .

Standard deviation of the mid-range: V n ⋅ (1 − β 2 ) ⋅ = 0, 089. 4 (n + 1)(n + 2)

Standard uncertainty of the 2-component estimator is uA =

1 S 2 [ X ] + S 2 [qV /2 ] + 2 ⋅ 0, 2 ⋅ S [ X ] ⋅ S [qV /2 ] = 0, 0703. 2

2 ⋅ S [ X ] ⋅ S [qV /2 ] = 0, 0703. The value of uncertainty for estimator (5) does not differ significantly from above. Distributions of qV /2 and X for trapeze pdf are unknown but expected to be smoother than Normal one. For these estimators is taken the same coverage factor as for normal pdf, i.e.: K(P = 0,95) = 1,96. For coverage probability P expanded uncertainty is: U ( P) = K ( P) ⋅ u A

(16)

Results are put together in Table 3. Table 3. Representations of the measurement result and accuracy By standard uncertainty

X = 22,87; u A = 0,09

qV /2 X = 23,01; u A = 0,09

X = 22,94; u A = 0,07

2012

The most accurate is the last one – simplified 2C estimator X . Values of each estimator are lying in the expanded uncertainty ranges of two others.

7. Final conclusions It is very important to choose the most accurate, i.e. effective estimator at data processing for correct estimation of the measurand uncertainty corresponding to u A (type A). – For samples of distributions modelled by trapezoid, the best single-component estimator depends on its shape. If it is nearer to rectangular (1≥ β ≥0,35) then the best effective estimator of measurand is the midrange. Below β=0,35 up to β=0 of the triangle distribution, the sample mean is better. – The 2-component estimator as the linear form of above two estimators is better for samples of trapezium PDF. – For the broad range of trapezium shapes (0,75 ≥ β ≥0) the simplified form of this double component estimator of equal both coefficients k1= k2 = 0,5 is proposed and may be used with sufficiently good accuracy acceptable in practice. – For a number of sample observations n ≥ 10 all coefficients are practically independent from n. For smaller size n<10 individual modelling is needed for trapezium PDF. – All conclusions are positively tested by MC simulations and also by several numerical examples. – Estimators of trapezoidal distributions given in this work could be applied not only in measurement practice and for extending of GUM, NIST and NASA recommendations [1] – [3], [9] but also in the statistics, when trapezoidal models are also used [8]. One could forecast that way to obtain two-component measurand estimators for samples modelled by convolution of other two distributions such as Uniform and Normal, Uniform and arcsine, etc. may be interesting. –

The trapezoid PDF model of 5.38 and 1.79 bases are found. Its parameter β=1/3. As the best estimator of the measurand value is used (4). Values of distribution parameters are:

S [qV /2 ] =

N° 1

by expanded uncertainty X = (22,87 ± 0,19), P = 0,95 X ∈ (22,68; 23,06), P = 0,95

Aknowledgments

Author wishes to express his many of thanks to Maryna Galovska M.Sc., now a scientific assistant at the Institute of Manufacturing Metrology (Institut für Produktionsmesstechnik), Technical University of Braunschweig Germany, which in the years 2007 - 2009 as PhD student in Ukrainian National Technical University „Kiev Polytechnic Institute“ on their own choice cooperated with the author on the above issues and gave a lot of her own initiative for obtaining the results of this work by Monte Carlo simulation [10-12]. Thanks to that a number of intuitive ideas of the author about trapezoidal distributions are checked and new one- and two-component estimators for these distributions are established.

X = (23,01 ± 0,18), P = 0,95 X ∈ (22,83; 23,19), P = 0,95 X = (23,01 ± 0,14), P = 0,95 X ∈ (22,87; 23,15), P = 0,95

Author

Zygmunt Lech Warsza – Industrial Institute of Control and Measurement PIAP, Warsaw, Poland, E-mail: zlw@wp.pl. Articles

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References [1]

[2] [3]

[4]

[5]

[6]

Evaluation of measurement data - Guide to the expression of uncertainty in measurement (GUM), BIPM, JCGM 100, (Ed. 1993 –2008), and Supplement 1 Propagation of distributions using a Monte Carlo method. Guide OIML G1-101, 2007. EA-4/02 • Expression of the Uncertainty of Measurement in Calibration, EA European Cooperation for Accreditation, December 1999, pp. 63-65. Measurement Uncertainty Analysis Principles and Methods, NASA Measurement Quality Assurance Handbook –Annex 3, HDBK-8739.19-3, July 2010 Washington DC. Dorozhovets M., Warsza Z., “Methods of upgrading the uncertainty of type A evaluation (2). Elimination of the influence of autocorrelation of observations and choosing the adequate distribution”. In: Proceedings of 15th IMEKO TC4 Symposium, Iasi, pp. 199-204. Johnson N. L., Leone F. C., Statistics and experimental design in engineering and physical sciences, vol.1, 2nd ed., John Wiley & Sons, New-York, 1977. Novickij P.V., Zograf I.A., Оcenka pogreshnostiej resultatov izmierenii (Estimation of the meas-

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urement result errors), Energoatomizdat, Leningrad,1985 (in Russian only). [7] Zakharov I.P., Shtefan N.V., ”Algorithms for reliable and effective estimation of type A uncertainty”, Measurement Techniques, vol. 48, 5, 2005, pp.427-437, www. Springer com. (transl. from Izmieritelnaja Tekhnika no 2, 2005 p. 9-15) [8] Van Dorp J.R., Kotz S., “Generalized Trapezoidal Distributions”, Metrika, vol. 58, Issue 1, July 2003. [9] Kacker R. N., Lawrence J. F., “Trapezoidal and triangular distributions for Type B evaluation of standard uncertainty” Metrologia, no. 44, 2007, pp. 117–127. [10] Warsza Z. L., Galovska M., “About the best measurand estimators of trapezoidal probability distributions”, Przegląd Elektrotechniki (Electrical Review), no. 5, 2009, pp. 86–91. [11] Warsza Z. L., Galovska M., “The best measurand estimators of trapezoidal PDF”. In: Proceedings of IMEKO World Congress Fundamental and Applied Metrology, 2009, Lisbon, CD, pp. 2405–2410. [12] Galovska M., Warsza Z. L., The ways of effective estimation of measurand”, PAKgoś (Pomiary Automatyka Komputery w gospodarce i ochronie środowiska), no.1, 2010, pp. 18-20.

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Automated Map Comparison using Non-invariant Fourier Descriptors Submitted 5th April 2011; accepted 5th July 2011

Robert Ouellette, Kotaro Hirasawa

Abstract: In this paper, we use non-invariant Fourier descriptors to derive transformation variables which allow us to optimally localize and reorient robot-generated maps based on the map shapes in order to determine, in an automated way, the accuracy of the generated maps. Our method uses only 4 simple calculations for alignment, therefore is extremely fast and gives a very good optimization for data maps that contain consistent, high frequency noise. A drawback to this method is occlusions in the map which affect the low frequency Fourier descriptors and cause localization and orientation errors. Preprocessing and optimization can help minimize these drawbacks. This application can be easily adapted to other areas such as image comparison or fault detection. Keywords: Fourier descriptors, robot mapping, SLAM, map comparison, image comparison

1. Introduction

Simultaneous localization and mapping (SLAM) provides an effective way of helping robots localize themselves within a known environment or during an ongoing mapping operation [1]-[3]. One way of determining how well generated maps (a.k.a. “data maps”) reflect the area they map is to compare them to a model map by digitally superimposing the data map with the model map in such an orientation that the optimal amount of black pixels gets covered. In this paper, we present a new technique for automating this type of alignment. Until now, map alignment has been done by manually manipulating the maps such that superimpositioning of the maps gives the best visual match. Validating maps this way is painstaking and time consuming [4] therefore it would be advantageous to automate the process. To do so, some problems must first be overcome. One problem is that simply maximizing the number of black pixels covered (in this paper, both data and model maps are assumed to be black and white images – the map lined in black against a white background) does not guarantee that the map has been oriented optimally. Another issue to consider is that when a map is automatically generated such as that done by mapping robots, no specific orientation or reference point gets defined which in turn presents a lack of a common reference point from which the software would be able to relate maps to one another. Instead, we propose that the shape of the map’s boundary is enough to quantify the map’s orientation, position, and scale, which provides us with a way to relate maps to one another and allows us to calculate the differences between them.

The technique we use to quantify map shapes is done in the frequency domain using Fourier Descriptors [5]. Fourier descriptors have been used extensively in shapebased image recognition [5]-[12]. They allow a simple method of image normalization [12] and, when doing so, if the Euclidean distance between the Fourier descriptors of the two image borders falls within a given value, the images are considered the same. In this research however, we pre-assume that the maps themselves are already the “same” regardless of the quality of the data map and, rather than normalize the images, we adapt the normalization techniques in order to determine the values of the transformation variables needed to reorient the images for superimposition. At that point we can go about measuring how accurate the data maps are – not how similar they are to the model map.

2. Fourier descriptors

The concept of Fourier descriptors were first introduced by Zahn and Roskies [5] in 1972 and are simply the result of taking the discrete Fourier transform (DFT) [13] of a closed boundary. Zahn and Roskies [5] used the Cartesian version of the DFT but the method we use is based on the complex interpretation as used by Granlund [6]. This is done by first expressing the points on the boundary as two parametric equations: x ( n ) = xn , n = 0, 1, 2, … , N − 1

(2.1)

y ( n) = yn , n = 0, 1, 2, … , N − 1.

(2.2)

Now, if we consider the points to lie in the complex plane, we can combine the two parametric equations above into a single equation: s ( n) = x ( n) + iy ( n) for n = 0, 1, 2,..., N -1.

(2.3)

Using a parametric description in this way allows us to reduce the dimensionality of the boundary without loss of information. Using a complex interpretation also simplifies some of the math needed to solve for the transformation variables. Once the data is expressed in this form, we can use the complex version of the 1D DFT on (2.3): N −1

a (u ) = ∑ s ( n)e − i 2π un / N for u = 0, 1, 2,..., N -1,

(2.4)

n=0

The complex coefficients a(u) are the Fourier descriptors of the boundary and are exactly the same as the Fourier coefficients produced by a DFT, with the Articles

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only stipulation being that the DFT be done on a closed boundary. The dual of (2.4) is:

s ( n) =

N −1

∑ a (u ) e N

i 2 π un / N

for n = 0, 1, 2,..., N -1,

u =0

(2.5)

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(3.2)

| a (u ) | | a (u ) |

(3.3)

 a (k )a (m)  −iN ln 2π (m − k )  a (m)a (k ) 

(3.4)

α=

n0 =

∆ xy =

1 (a (0) − α eiθ a (0)) N

(3.5)

where a indicates the FDs of the original border, the FDs of the transformed border, Δx=Re(Δxy), Δy=Im(Δxy), and k, m, p, q, and u are FD indices. Choosing which FDs to use in the equations can be done almost completely arbitrarily; however, derivation of the equations required that: q=2p; p, q ≠ 0 α is real & positive k ≠m; k, m ≠ 0. While data maps and model maps refer to the same physical area, they are technically different, particularly from the software’s point of view. In order to be able to relate the maps in software, we stipulate that data maps are copies of the model maps that have been transformed by some unknown amount and have been infused with noise. In the case that the data map were a perfect copy of the model map with the inclusion of random, high frequency noise, the FDs of the borders of both maps would be the same for all the FDs except the highest order FDs. It follows that we could determine the transformation values between the data and model map borders using only the lower-ordered FDs. This makes sense because, Table I. Spatial-Frequency Transformation Relationship Type

Spatial Domain (boundary)

Frequency Domain (Fourier Descriptors)

Identity

si ( n)

Rotation

s r (u ) = s ( n ) e

Scaling

ss ( n) = α s ( n)

a s (u ) = α a (u )

Start point

ssp ( n) = s ( n − n0 )

asp (u ) = a (u )e

ai (u ) iθ

a r (u ) = a (u ) e

iθ

− i 2 π un0 / N

Translation st ( n) = s ( n) + ∆ xy † at (u ) = a (u ) + N ∆ xyδ (u ) ‡

Transformation types: rotation (θ), scaling (α), change in the starting point (n0), and translation (Δxy).

(3.1) †

 [a( p)]2 a (q) | a ( p) |   [a ( p)]2 a( q)|a( p)| 

θ = i ln 

3. FD-based map alignment

a (u ) = eiθ α e − i 2π un0 / N a (u ) + N ∆ xyδ (u ).

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where are the FDs of the transformed boundary. By exploiting the relationships in Table I, we can algebraically manipulate the FDs of a map boundary and a transformed version of itself in such a way that we can find the values of the transformation variables of (3.1) using the equations summarized in (3.2)-(3.5).

which can be used to regenerate the original sequence of points.

In image processing, a variety of transformations exist that transform an image from one form to another [14], [15]. Since image boundaries are part of the images themselves, transformations of an image cause the boundaries to be transformed the same way. Conversely, using the same transformation values that were used to transform a border, when applied to the image body, will cause the image to be transformed in the same way as well. In this paper we will be realigning the data map to match the model map by first finding the transformation values that allow us to align their borders. Once these transformation values are found, we can use the same transformation values that we used to align the borders to align the maps themselves. Certain transformations on a boundary produce known relationships on their respective Fourier descriptors [14], [15]. Table I includes four of these: rotation (θ), scaling (α), translation (Δxy), and start point (n0). Spatially, the point from which boundary point trace begins is irrelevant; eventually the border will be completely recreated once the trace returns to that first point at the end of the trace cycle. However, in the frequency domain, since the sinusoids that make up the border are all dependent on the point that they are referenced from, starting from a different point will cause a phase shift in the sinusoids. Therefore, for boundary-related transformations, the first point that the trace begins from (i.e. the “start point”) is critical to a correct border reconstruction, thus we also include it in this paper. Notably, non-uniform affine transformations (i.e. “shear”) also have a chance of occurring during the mapping process but they would likely only occur due to severe problems with the hardware and/or software thus we have omitted them from this discussion. Any transformation from one location/orientation to another can be minimized to a combination of one of each of the transformations listed in Table I. The amount of each transformation is dependent on the transformation sequence taken [reference analytic, image processing, robotics], but in this paper we set the sequence to be: rotation→scaling→start point→translation. Our reasoning for using this sequence is that it gives us an easy way to determine the respective transformation values for a transformed image. Combining each of the transformation equations in Table I in the above sequence we get,

N° 1

∆ xy = ∆x + i∆y

‡

0, u ≠ 0 1, u = 0

δ (u ) = 

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as indicated above, the lower-ordered FDs capture the location, orientation, and shape. Higher-ordered FDs capture things like noise. So, in this paper, rather than choosing the FD indices mostly arbitrarily, we select the first four FDs, FD(0)-FD(3) instead. Our reasoning for choosing is based in the geometric interpretation of the FDs. References [7] and [16] gives some insight into this reasoning. Given that we can calculate the transformation values from the FDs of a transformed boundary, and given the relationships between transformations on FDs and on the boundaries themselves, we can transform the border back to its original location by: s (n) =

( s (n) − N ∆ xy )ei 2π nn0 / N e − iθ .

(3.6)

Since we are comparing the maps themselves and not the borders, we can ignore the start point part of the equation and use simply the inverse transform, s (n) =

(3.7)

( s (n) − N ∆ xy )e − iθ ,

and use the non-complex version, (3.8)

x = Re[ s (n)], y = Im[ s (n)],

in order to recreate the whole map.

4. Map comparison automation

preprocess

extract boundary

find FDs

All Test Maps

Model Map (1x)

In Fig. 1 we outline the steps taken during map comparison. The potential exists for there to be breaks in continuity in the model boundary as well as “islands” such as objects within the map so preprocessing the model and test maps entails making sure that the map outline has no breaks. We could automate the closing of gaps, however, considering that we only needed a few maps for validation, we found it was sufficient to fill in any gaps by hand. Islands are eliminated by using

Determine transformation variables “Inverse transform” the test map Superimpose test and model maps Do pixel-wise comparison

Figure 1. Procedure used in map comparison automation

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only the outermost boundary of the map and ignoring all others. We use the Moore boundary tracking algorithm [18] as described by [14] for finding the boundary of the model map and test maps. In practice, we should only have to do this with the model map since our sensor of choice, the laser scanner, by its very nature, gives us a discretized version of the boundary for the map. Below we summarize the Moore tracking algorithm. 1. Start in the upper left-hand corner of the map. Scan the map from left to right, top to bottom until you reach the first black pixel in the map. Denote the location of that pixel as the start point, b0. 2. Define c0 as the “west” neighbor of b0 (the pixel to the immediate left of b0). Starting at c0, examine the neighbors of b0 while moving in a clockwise direction. Record the position of the first non-zero pixel as b1 and register this location as the second point on the boundary. Let c1 be the point immediately preceding b1 in the sequence. Store b1and b0 for later reference. If no neighbor is found, the pixel is a singular “pixel island”. 3. Let b = b1, c = c1. 4. Starting at c, examine the neighbors of b in a clockwise direction, like above until the first black pixel is found. Label the location for this pixel as b and the pixel just before it in the search sequence as c. 5. Register b as the next point on the boundary and use this as the next evaluation pixel. 6. Repeat steps 4 and 5 until b=b0 and the next boundary point found is b1 Unsupervised automation of border extraction does not guarantee that the first black pixel you find will be on the border you want to compare. Objects within or outside the map outline itself such as simple noise or a map legend could theoretically cause the software to find an unintended boundary. To minimize this potential problem, we found that finding all the borders in the map and choosing the border that contains the largest area to be that which worked best. This method may or may not solve all problems, but worked fine in our limited case. Once the border is found, the remaining steps are mostly straightforward. Finding the Fourier descriptors of the maps, extracting the transformation variables, and performing the inverse transform on the test maps are as described in the previous sections. It is important to reiterate that we are not “inverse-transforming” the test map border, instead we are reorienting the whole data map using the extracted transform variables in order to compare all portions of the maps.

5. Results

Validation of our approach was done in two steps. In the first step, we transformed the boundary of the model map shown in Fig. 2 using known transformation values. Each type of transformation was applied individually as well as a rotation→ scaling→ start point→ translation transformation. The different transformations are superimposed with the model boundary in Fig. 3. Since we know by how much each of the transformed boundaries were transformed, when we extract the variables from Articles

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the transformed boundaries, the resultant extracted variables should be the same as the original transformation. The results, shown in Table II, verify that there is no error from the extracted variables using this “fixed” data method. We confirmed this visually by superimposing the inverse-transformed data boundary with the model boundary as shown in Fig. 4, showing that the boundaries match overall. Fig. 5 gives a zoom-in of the upper left-hand corner of Fig. 4, verifying that the boundaries match at the smallest scale as well. The second step in validating our method involved applying our method to hand-generated, “raw” data. The data was acquired by hand-tracing a scanned image of the model map. Doing so ensured a reasonably well matching data map with enough regular noise to simulate a robot-produced closed-loop data map. During the scanning phase, the image was simultaneously given a random orientation and was scaled down to 75 percent (reference) of its original size. The hand-traced map and its pre-processed boundary are shown in Fig. 6 and the hand-traced boundary superimposed with the model map is shown in Fig. 7. Since the data map is a different size (smaller in this case) than the model map, Moore-tracing the boundary will give a different amount of data points, therefore it is necessary to resample either the model boundary or the data boundary so that the number of data points are the same. We chose to resample the model map since it was larger than the data map. Resampling the data map would have entailed splitting the distance between neighboring pixels which can be done easily mathematically, but is not so easy to imagine visually. We used a “quick and dirty” resampling method to resample the border as given in (7.1): where

s [n] = s[n ] + (nα - n )( s[n + 1] - s[n ])

α=

match between the hand-generated data boundary and the model map boundary. There is, of course, some deviation as shown in the close up in Fig. 9, but this is simply due to the original freehand-generated error. This type of error will be present in any real-world environment. Even though we could clearly see that the map borders match, our goal was to compare the actual maps. Therefore, we applied the transformation variables extracted from the border of the hand-traced map to the actual data map and superimposed the data map on top of the model map as shown in Fig. 10. As expected, we see that, like the border comparison above, the data map (red pixels) and the model map (black pixels) match well with errors only due to tracing as shown in Fig. 11. All image transformations and variable extractions were done using Matlab. The Fourier descriptors themselves were generated using the DIPUM 1.1.3 package for Matlab.

(5.1)

6. Discussion

and is after being resampled. This gives a regular sampling period with only the distance from the last point to the first point not necessarily matching the sampling distance. This method was sufficient for our purposes. Following the algorithm outlined in Fig. 1, we did an inverse transformation using the transformation variables extracted from the hand-traced test data. The results are shown in Fig. 8 which shows a very good

As shown in the results above, this method has the potential to work very well. The maps we used were convenient in that the data maps were, in essence, exactly the same map as the model map with differences only in regular, high frequency noise which has little effect on the transformation variables themselves. In reality, a robot mapping a building has to deal with open or closed doors, obstacles, etc, which, in turn, if taken by themselves, would affect the low frequency Error after Inverse Transform

Table II. Individual and complete transformation validation error results

Δxy

Δxy = 0

0.00+0.00i

Δxy = 0

0.00+0.00i

0.00-0.00i

0.00+0.00i

Scaling

α =2.2

n0 = 00

Shifted Start Point

α =0

n0 = 4575 θ = 0

Rotation

α =0

n0 = 00

θ = 1.14 Δxy = 0

0.00+0.00i

Translation

α =0

n0 = 0

θ=0

Δxy = 440+560i

0.00+0.00i

n0 = 4575 θ = 1.14 Δxy = 440+560i

0.00+0.00i

0.00-0.00i

0.00+0.00i

Rot.-Scaling-S.P.-Translation α =2.2 18

Transformation Amount

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Figure 2. Model map (top) and its boundary (bottom)

size( s (n)) , n = floor(α n), size( s (n))

Transformation Type

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Figure 7. Boundary of hand-traced map in its raw state superimposed over the model map boundary Figure 3. Original boundary (in bold) and the boundary after undergoing various transformations

Figure 4. Model map boundary (bold, hyphenated) and the data boundary after being inverse-transformed (solid line)

Figure 5. The top-left corner of Figure 4 zoomed to show that there is no discernible variation from the original boundary and the inversed-transformed boundary

Figure 8. The data map boundary (solid) after undergoing an inverse transformation and superimposed against the model map boundary (bold, hyphenated)

Figure 9. The top-left corner of Figure 8 zoomed in to show the slight deviation from the model boundary due to the error given by drawing by hand

Figure 10. The data map inverse-transformed (red) from the transformation variables extracted from the handtraced border shown in Figure 6

Figure 6. Hand-tracing of model map (above) and its border (below)

Figure 11. The top-left corner of Figure 10 zoomed in to show the slight deviation from the model boundary due to the error given by tracing the map by freehand Articles

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descriptors and would most certainly cause the center of mass of the map border to be off, and, almost as certainly, cause the orientation to be off as well. This issue can be minimized in practice because in order to compare a data map with a model map (manually or automatically), we already have to preprocess both the data map and model map to some degree. During that preprocessing, it is easy to ensure that no extra rooms exist due to opened doors, etc. An issue also exists with the derivation of the transformation variables. We use only the first four Fourier descriptors to determine the transformation variables. Doing so greatly reduces the number of computations to roughly 4N where N is the number of boundary points. This method is much faster than a regular Fourier transform or even the Fast Fourier Transform however it only allows the capture of information in the very lowest frequencies. It does capture general shape and orientation, but it does not consider information from the higher frequencies. Thus, a better method might be to calculate a few values of orientation, scaling, etc, based on more, or even all, Fourier descriptors and take the Euclidean distance which gives the best matching value. We have considered this, but our primary objective with this paper is to show that this approach will give very good results using even the simplest approach. Furthermore, it gives a good balance between speed and accuracy. Any optimization beyond this needs to consider preprocessing, the number of boundary points (i.e. the sampling frequency), the number of Fourier descriptors used to calculate the transformation variables, the number of calculations involved, and the number of maps to be compared, among other things and should be addressed as a topic by itself. As mentioned above, any noise in this approach should be regular and high in frequency. Figure 9 presents a good example of such noise as well as the addition of low frequency noise. Although the high frequency noise is relatively low in amplitude with respect to the size of the map, the noise is clearly seen in the “roughness” of the data points superimposed about the model map. The key is that regardless whether the noise is high in amplitude or not, the algorithm will always average the high frequency noise out through the low frequency-based transformation calculations. Lastly, it is important to mention that in this paper we assume that the data maps and the model maps are maps of the same location. If, for some reason (e.g. by mistake), the data map and the model map are maps of two different places, our method will still force the maps to be localized and orientated in the optimal position for comparison, even though if it the result is nonsensical. So while this method will align two maps regardless if they are different or not, it will not tell us if the maps are actually the same. In order to do so, we could take the Euclidean distance between the normalized Fourier descriptors of the data map and model map where the smaller the error would indicate the better the match. The details for similarity matching are outside the scope of this paper, but many of the other references listed in this paper such as [5]-[12] discuss this topic in depth. 20

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7. Conclusion and Future Work

In this paper we extracted transformation variables based on information derived from the shape of model and data map boundaries then used those variables to align the data and model maps so that the data map can be checked for accuracy. This method is very fast as it only needs four simple calculations to determine the values of the transformation variables in the most basic alignment approach. Optimization needs to be addressed for accuracy-constrained applications, but overall, the methods we developed here are very useful for automating map comparison. While we limited the scope of our discussion to map alignment, it is easy to realize the many different areas that this can be applied to such as image comparison, defect detection, or object tracking, to name a few. In future work, we will apply these methods in a piecewise sense to perform robot localization and mapping.

Authors: R. P. Ouellette* is a robotics consultant at Open Thoughts Research. He has held positions in embedded hardware & software, signal processing and robotics research in various companies such as Raytheon, GMDJapan (now Fraunhofer FhG), The Foxboro Company, among others. He has a BS in Engineering Physics from the University of Maine, an MS in Electrical & Computer Engineering from Kyushu Institute of Technology, and is currently pursuing his PhD in Robotics at Waseda University. (phone: +81-90-1191-1514; e-mail: rpo@ openthoughts.com). Kotaro Hirasawa received the B.S. and M.S. degrees from the Kyushu University, Japan, in 1964 and 1966, respectively. From 1966 to 1992, he worked at Hitachi Ltd. at the Hitachi Research Laboratory. From December 1992 to August 2002, he was a Professor at the Graduate School of Information Science and Electrical Engineering of Kyushu University. Since September 2002, he has been a Professor at the Graduate School of Information, Production and Systems, Waseda University. Dr. Hirasawa is a member of the Society of Instrument and Control Engineers, the Institute of Electrical Engineers of Japan and IEEE. *Corresponding author.

References

1. A. Eliazar, R. Parr, “DP-SLAM: Fast, robust simultaneous localization and mapping without predetermined landmarks”. In: Proc. 18th Int. Joint Conf. on Artificial Intelligence (IJCAI’03). 2. A. Nüchter, H. Surmann, K. Lingemann, J. Hertzberg, S. Thrun, “6D SLAM with an Application in autonomous mine mapping”. In: Proc. of the IEEE International Conference on Robotics and Automation, New Orleans, USA, April 2004, pp. 1998-2003.

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3. W.Y. Jeong, K.M. Lee, “Visual SLAM with Line and Corner Features”. In: Proc. IEEE/RSJ International Conference on Intelligent Robots and Systems, Oct. 2006, pp. 2570-2575. 4. R. Ouellette, K. Hirasawa, “A comparison of SLAM Implementations for Indoor Mobile Robots”. In: Proc. IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS’2007), Oct. 2007, pp. 1479-1484. 5. C. T. Zahn, R. Z. Roskies, “Fourier Descriptors for Plane Closed Curves”, IEEE Transactions on Computers, vol. 21, 1972, pp. 269-281. 6. G. H. Granlund, “Fourier Preprocessing for Hand Print Character Recognition”, IEEE Transactions on Computers, vol. 21, 1972, pp. 195-201. 7. O. Bertrand, R. Queval, H. Maitre, “Shape Interpolation using Fourier Descriptors with Application to Animation Graphics”, Signal Processing, vol. 4, 1982, pp. 53-58. 8. D. S. Zhang, G. Lu. “A Comparative Study on Shape Retrieval Using Fourier Descriptors with Different Shape Signatures”. In: Proc. of the International Conference on Intelligent Multimedia and Distance Education (ICIMADE’01), Fargo, ND, USA, June 1-3, 2001, pp.1-9. 9. L. Keyes, A.C. Winstanley, “Fourier Descriptors as a General Classification Tool for Topographic Shapes”. In: Proc. of the Irish Machine Vision and Image Processing Conference (IMVIP ‘99), Dublin City University, 1999, pp. 193-203. 10. Y. Rui, A. C. She, T. Huang, “A Modified Fourier

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12.

13. 14. 15. 16. 17. 18.

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Descriptor for Shape Matching in MARS”. In: S. K. Chang (ed.), Image Databases and Multimedia Search, Series of Software Engineering and Knowledge Engineering, World Scientific Publishing, 1998, pp. 165-180. D.J. Lee, S. Antani, L. Rodney Long, “Similarity Measurement Using Polygon Curve Representation and Fourier Descriptors for Shape-based Vertebral Image Retrieval”, Journal of Visual Communication and Image Representation, vol. 15, no. 3, Sept. 2004, pp. 285-302. C. S. Lin, C. L. Hwang, “New Forms of Shape Invariants from Elliptic Fourier Descriptors”, Pattern Recognition, vol. 20, no. 5, 1987, pp. 535-545. J. W. Cooley, J. W. Tukey, “An algorithm for the machine calculation of complex Fourier series”, Math. Comput., 19, 1965, pp. 297-301. R. C. Gonzalez, R. E. Woods, Digital Image Processing, 3rd ed., Pearson Education, Inc., 2008. A. K. Jain, Fundamentals of Digital Image Processing, Pearson Education, Inc., 1989. B. Jahne, Digital Image Processing, 6th ed., Springer, 2005. A. Oppenheim, R. Schafer, J. Buck, Discrete-time Signal Processing, 2nd ed., Prentice Hall, 1998. G. A Moore, “Automatic Scanning and Computer Processes for the Quantitative Analysis of Micrographs and Equivalent Subjects”. In: C. G. Cheng et al. (ed.), Pictorial Pattern Recognition, 1968, pp. 275-326.

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Postural equilibrium criteria concerning feet Properties for biped robots Submitted 7th July 2011; accepted 29th September 2011

Alejandro González de Alba, Teresa Zielinska

Abstract: This article presents a study on the postural equilibrium conditions for biped robots. Criteria for dynamic walking, such as ZMP and CoP are introduced and their similarities discussed. We also introduce the effects of a compliant foot and take them into consideration during the evaluation of the criteria. A model of a planar biped is used to imitate the movements of a human subject as recorded by the VICON motion capture system. In order to estimate the criteria, body segments accelerations and ground reaction forces are needed. ZMP and CoP are analyzed during both single and double support phases for the model’s motion. A linear shift function is used to transport the load of the biped between the supporting legs during the double support phase. We compare simulated CoP trajectories obtained using a rigid foot and a compliant, deformable spring-damper system located between the ankle joint and the sole of the foot. It is seen that the foot’s deformation smoothes the CoP trajectory and improves the biped’s stability. Keywords: biped robots, motion analysis

1.1. Robotic Feet

The study of feet in humanoid robotics has an ample field of application. In 2002 the humanoid robot H6 and its successor H7 were fitted with an actuated toe-jointed foot (see Fig. 1a) to increase their range of motion [11]. These articulated feet made H6 capable of kneeling while maintaining contact of the soles with the ground, walk faster and climb up higher steps. Speed was increased from 160 mm∙s-1 to 270 mm∙s-1, while gait cycle characteristics remained unchanged. Also, the increased height of affordable obstacles (such as stairs) required a smaller torque at the ankle when compared to the un-jointed foot. On the other hand, passive toe-joints have been proposed [1, 6]. They rely on the use of springs-damper systems to support the motion and improve the energy consumption. Compliant limbs have been shown to adjust better to difficult terrains [9]. Robot WABIAN-2R [6] was fitted with a foot capable of recreating the role of a human foot’s longitudinal arch a)

1. Introduction

The role of the ankle and toe joints during bipedal locomotion has been discussed for a long time. It has been observed that they are able to store energy at heel strike and release it during push-off with a spring like action [2, 4, 10]. This energy release decreases the total energy consumption for the gait cycle [2]. It is only natural that this advantage is sought to be replicated in both walking robots and prosthetic devices [1, 2, 6, 11]. Two main approaches exist. The first actively reproduces the motion of the ankle, and in some cases of the toes. The second stores energy dissipated at heel-strike, by means of a variety of springs, releasing it at the appropriate time 2. This second, compliant approach can reduce the overall weight of the foot. It is also able to closely mimic human gait in regards to the distribution of ground reaction forces at key events of the gait cycle [2, 6]. The work presented here focuses on biped locomotion and the calculation of equilibrium criteria ZMP and CoP. These criteria are widely used in the field of walking robots [3, 14, 17]. Following sections will describe the method used for the evaluation of these criteria, and a brief discussion over the obtained results. 22

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Fig. 1. Robotic feet. a) shows the actuated ankle of the H7 robot [12]; while b) presents an schematic diagram of WABIAN-2R’s compliant foot [6]

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(see Fig. 1b). WABIAN-2R’s foot also has two toes and is divided along its sagital plane; this creates a transversal arch. It has a soft, elastic material limiting the motion of the toe joint and features a steel wire which recreates the function of the plantar aponeurosis ligament1. This wire is run from the heel to the toes as shown in Fig. 1b. The pull force created by the wire increases the push-off forces and gives the gait a natural look. In order to implement suitable control strategies, the feet proposed by Borovac and Slavnic [1] and Nishiwaki et al. [11] are equipped with force sensors in the sole for real time calculation of ZMP. Hashimoto et al. [6] make no mention of the control strategy used in WABIAN-2R but the effectiveness of their foot in imitating that of a human was validated by comparing ground reaction forces during push-off and overall motion of the foot.

1.2. Biped model

We define a walker as an 8 element linkage (see Fig. 2). The knees are represented by revolute joints, while the joints between hip and thighs and those at the ankles are considered to be spherical. This allows motion both along the motion direction and a direction normal to it. The foot consists of a spring-damper system, placed perpendicular to the ground. In Fig. 2 the spring-damper

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system is shown only as a spring, for the sake of readability. No toe joint is considered. Since we will focus on the lower limbs, the upper part of the body is treated as single link with equivalent mass sometimes referred to as HAT (Head And Trunk). Global reference frame was defined with the y-axis along the motion direction and the z-axis normal to the ground surface pointing upwards. The set of parameters taken into consideration can be found in Table 1 [8]. It is important to note that the values for moment of inertia (Ji) are presented as measured with the local frame place at the center of mass of the corresponding segment. Local frames attached to the model’s links are not shown as their definition is not definitive for the method followed here. For analysis, we consider the subject to fall within the values of a 5 percentile U.S. male crew member. Limbs will be represented by point masses located at a distance measured from the joint of the link closest to the hip. Table 1. Geometric and dynamic parameters for the 5% U.S. male crew member. [8] Shin

Thigh

Trunk

Length [m]

0.47

0.43

0.80

Mass [kg]

3.30

10.60

47.20

Jx [kgm2]

4.37e-2

12.25e-2

21.53e-2

Jy [kgm2]

4.30e-2

11.63e-2

25.56e-2

Jz [kgm2]

0.51e-2

3.16e-2

107.31e-2

1.3. Stability criteria

1.3.1. Zero Moment Point [ZMP] It is impossible to mention ZMP without referring to M. Vukobratović’s works, such as [17, 18], and their importance in the control of biped walking robots. ZMP is defined as the point on the ground where the tipping moment due to gravitational, inertial and reaction forces acting on the biped equals zero. For case of the planar walker, the ZMP coordinates can be determined from equations (1). n n y y n ∑i mi ( z i + g )⋅xi −∑i mi x i zi −∑i Ii ω i x ZMP = n ∑i mi ( z i + g ) (1) n n n ∑i mi ( z i + g )⋅ yi −∑i mi y i zi + ∑i Iixω ix y ZMP = n ∑i mi ( z i + g )

Fig. 2. Biped walker model. A spring is shown in place of a spring-damper system for the sake of readability

1 The planar aponeurosis ligament is the main component of the plantar fascia, a group of ligaments and connective tissue located on the sole of the foot. It is capable of maintaining the shape of the longitudinal arch by pulling on its ends, resembling a taut bow.

Where: xZMP , y ZMP – position of the ZMP along the corresponding axis; mi – mass of link i; xi , yi , zi – position of center of mass of link i along appropriate axis; g – acceleration due to gravity ( g = −9.81 m ⋅ s −2 ); I i j – moment of inertia of link i along axis j expressed on the global reference frame; ωi j – angular velocity of link i along axis j. Articles

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These use the notation found in [19] (interested readers may refer to [cite{Sardain}, cite{Vukobratovic2004} for further information) and hold for both single and double support phase. Owing to its definition, the coordinate along z-axis belongs in the ground plane.

Knowing the shape of the polygon is useful for generating a ZMP/CoP based gait. Erbatur et al. [3] prescribe the desired location of ZMP point to be located in this polygon and determine the motion of the legs in order to realize it.

1.3.2. Center of Pressure [CoP]

2. Reconstruction of the gait and equilibrium conditions.

CoP is a point in the ground surface. It is located in such a way that the sum of moments caused by ground reaction forces is equal to zero. In other words; CoP is defined as a function of the measured ground reaction forces, while ZMP is determined via the kinematic behavior of the walker. It can be proved that these two points, though differently defined, are in fact equivalent [14]. As such, they can be used somewhat interchangeably. CoP position can be found by means of (2) which are modified from the work of Sardain and Bessonnet [14] and are similar to those presented by Schepers et al. [15]. M = OR × Fr + OL × Fl CoP =

n× M n ⋅ (Fr + Fl )

Where: Fi – pressure force on the corresponding foot (l for left and r for right); n – vector normal to the ground surface (for as model defined as in Fig. 2, n = [0 0 1]T) OR, OL – distance from the frame’s origin to the application point of the corresponding pressure force (see Fig. 2). Relation (2) will hold for both single and double support phases; and with an appropriate choice of vector n it does not restrict the ground to a horizontal surface.

A human subject was recorded while walking, by means of a VICON motion capture system. After processing, the angles between body segments (hip, knees and shins) are stored and used to analyze the motion. The procedures used are to be detailed in the following.

2.1. Kinematics

To reconstruct the individual’s gait, we must determine the biped’s pose at each time instant. For each set of measured angles, we think of the biped’s structure as a serial linkage fixed at the hip. This will allow us to determine the position of the trunk, knees and ankles in the hip’s frame. Motion is then recreated by placing correctly the supporting foot/feet in the global reference frame. Joint positions at each time instant are calculated and stored. In order to make the supporting foot appear stationary, it is necessary to offset the position of the model for each time instant. This offset is represented by the vector d shift (see Fig. 4) and may be used to set an initial condition for the walk.

1.3.3. Usage of stability criteria

While using a ZMP/COP based control, the corresponding point is required to stay inside the supporting polygon in order for the gait to be considered stable [3, 17]. During single support phase, this support polygon is limited by the footprint (see Fig. 3a). For the double support phase, it can be considered to span the area connecting both feet (see Fig. 3b). a) Fig. 4. Shifting of the biped to give the impression of motion After finding the trajectory of each body segment, the location of the walker’s center of mass (PCM) may be calculated as given by equation (3).

PCM =

b) 24

Fig. 3. Support polygon for a) single support and b) double support phase Articles

Σmi PCMi Σmi

Where: PCMi – corresponds to the global position link’s i center of mass; PCM – is the position of the biped’s center of mass.

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In order to find the ZMP of the biped, as defined by (1), we must first find the angular velocity and acceleration of its links, as well as the linear acceleration of their centers of mass. Numerical differentiation can be used to find these values. Determining reaction forces during single support offers no difficulty. On the contrary, during the double support phase we find the system does not have a unique solution. In order to obtain one, we assume the forces experienced are transferred between the supporting legs by means of a shift function [7, 16] denoted by f. In order to do this, the time instants for the beginning and ending of the single support phases must be known. A function fi is created with a value of one for the single support phase, and a value of zero during transfer. Transition between both states is taken to be linear and takes place during the double support phase. Different shift functions have been proposed. In her work, Ruiz Garate [16] points to the work of Lei Ren et al. [13], who use an exponential function for the analysis of 3D data with good results in the sagital plane. A linear function (4) as used here yields good results while remaining simple to implement.

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2.2. Ground reaction force’s point of application

CoP is the point where the sum of moments caused by the ground reaction force’s vectors is equal to zero. With this in mind, it is necessary to locate the point of application of the reaction forces along the sole of the foot (we denote it here as ‘C’).

Fig. 6. Compliant foot model with spring-damper system In an analogous manner, we propose to find point ‘C’ where the sum of moments caused by the forces acting on the foot is equal to zero (see Fig. 6). When assuming a foot with no mass, the location of point ‘C’, measured from the ankle ( d c ) is given by (6).

(4) Function fi is defined for the left and right legs. This function changes between 0 and 1 during the double support phase. Here th and tp refer to the time instant of heelstrike and push-off for the corresponding leg; subindex i refers to the first supporting leg, j to the other one during one complete gait cycle (see Fig. 5).

dc = −

Fay Faz

(6)

2.3. Compliant foot model

We propose to model the foot’s compliance by means of a spring-damper system located between the ankle and the sole (see Fig. 6). We assume that deformation will occur only along the z-axis and that this segment is always perpendicular to the ground. The walker’s motion may then be modeled as shown in Fig. 7. In order to solve the ordinary differential equa-

Fig. 5. Shift function defined for transfer of ground reaction forces during walk Following this, ground reaction forces on each leg are given by relation (5).

Fr = − f r ⋅ FCM Fl = − f l ⋅ FCM

(5)

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tion (7) and find the deformation of the spring element, we take the vertical acceleration of the walker’s center of mass as input to the system. With the vertical position of the center of mass being denoted as h; we find its acceleration to be h . Relation (7) is the classic definition of the springdamper system, where z s = 0 corresponds to the equilibrium position. According to Geyer et al. a stable walking gait may be obtained with a spring constant (‘K’) close to 18 kN∙m-1. This value nears the highest stiffness registered for running [5] and was used here for calculations. The value of the damper constant (‘B’) is taken to be 100 N∙s∙m-1. This value was chosen in order to see the effect of the spring damper system while offering a naturallooking gait.

1   z s   0   z s   0  z  = − K / m − B / m  z  + − 1 h  s     s 

(7)

In order to find the deformation in each foot (zr and zl) during double support phase, we propose to make use of the shift function introduced in Section 2.2 and the resulting deformation (zs), as obtained by solving (7). The largest deformation should be measured on the spring bearing the highest load. For this we can suggest relations (8).

Fig. 9. Spatial motion of the CoP. a) shows a stiff foot, while b) shows the effect of the spring damper-system

zr = f r ⋅ zs zl = f l ⋅ z s

(8)

3. Results

After reconstructing the gait in simulation, we compare the position of point ‘C’ (application point of ground reaction forces) in the reference frame of the supporting foot (with its origin under the ankle, as given in Fig. 6). Fig. 8 shows the location of point ‘C’ in the supporting foot’s local frame; the solid line represents double support phase, while the dashed line indicates single support. When no deformation is allowed, we see large

Fig. 8. Location of point ‘C’ expressed in the supporting foot’s frame 26

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peaks for the location of ‘C’. This is due to the changes in forward velocity of the walker’s center of mass when entering double support phase. In order to respect the equilibrium condition set by (6); ‘C’ must sometimes be located outside of the sole of the foot. This is of course, not desirable. After allowing for deformation of the foot structure the moments caused by the forces at the ankle are diminished and ‘C’ remains closer to the ankle (see Fig. 8). Although not shown here, it is interesting to note that large vertical accelerations of the biped’s center of mass (P CM ) are registered at these time instants. This is brought about by the change of trajectory followed by the center of mass, as indicated in the traditional inverted pendulum model for biped walking gait. Fig. 9 shows the trajectory of the CoP point, projected on the ground plane. As a reference for both the non compliant and compliant foot (Fig. 9a and Fig. 9b respectively) we use the CoP as calculated for a single point contact located under the ankle for the corresponding foot (shown with a solid line). CoP point moves abruptly during the double support phase while using an un-deformable foot structure. This motion corresponds to the peaks observed in the position of ‘C’. After modeling the foot as a compliant element, we observe a smother trajectory; closer to that of the straight line, which would be expected under the assumed conditions. By looking at the behavior of ‘C’ and its effect on the CoP criteria, we note that CoP travels to the heel of the foot before changing the support leg (see Fig. 9); in the same way as ‘C’ does (see Fig. 8).

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4. Future work

Future work should focus on: i) the analysis of the evolution of CoP for different motion situations; ii) testing the proposed method by comparing CoP as obtained using direct force measurement; iii) studying the effect of toe joints and different spring-damper parameters whose respective actuation and values are linked to the desired motion and environment. These investigations may lead to a better estimation of ground reaction forces, such as those obtained by Geyer et al. [5] and Ruiz Gárate [16]. They may also redefine the duration of the single and double support phases for the proposed model.

5. Conclusions

Criteria for stability evaluation, as given by Vukobratović [17, 18], were presented and evaluated for the single and double support phases. It is seen that for the reconstructions presented here, a compliant foot structure is preferred; since it keeps the ZMP/CoP within the support polygon (see Fig. 3 and Fig. 9). Such a compliant structure is proposed to be modeled by a spring-damper system with viscous friction. Due to the characteristics of the biped model used here, the compliancy experienced on the foot may also be thought of as belonging to the ground. Acknowledgments This research was in part supported by the Ministry of Scientific Research and Technology Grant N N514. We would like to thank Prof. M. Inaba from Johou Systems Kougaku (JSK) Laboratory for allowing the use of Fig. 1a, and Prof. K. Hashimoto at Waseda University for permitting the use of Fig. 1b.

Authors

Alejandro González de Alba* – Aeronautics and Applied Mechanics, Warsaw University of Technology, Warsaw, 00-665, Poland. E-mail: jandro.glez@gmail.com Teresa Zielińska – Aeronautics and Applied Mechanics, Warsaw University of Technology, Warsaw, 00-665, Poland. *Corresponding author

[5]

[6]

[7]

[8]

[9] [10] [11]

[12]

[13]

[14]

[15]

References

[1] B. Borovac, S. Slavnic, “Design of multisegment humanoid robot foot”, Research and Education in Robotics EUROBOT 2008, vol. 33, Communications in Computer and Information Science. [2] S.H. Collins, A.D. Kuo, “Recycling energy to restore impaired ankle function during human walking”, PLoS ONE, 5(2):e9307, 02/2010. [3] K. Erbatur, Ö. Koca, E. Taşkıran et al. “ZMP based reference generation for biped walking robots”, World Academy of Science and Technology, no. 58, 2009, pp. 943–950 [4] D.P. Ferris, M. Louie, C.T. Farley, “Running in the real world: adjusting leg stiffness for different sur-

[16]

[17] [18] [19]

N° 1

2012

faces”, Proceeding of The Royal Society, 1998, pp. 989–994. H. Geyer, A. Seyfarth, and R. Blickhan, “Compliant leg behavior explains basic dynamics of walking and running”, Proceeding of The Royal Society, 2006. K. Hashimoto, Y. Takezaki, K. Hattori et al., “A study of function of foot’s medial longitudinal arch using biped humanoid robot”. In: 2010 IEEE/ RSJ International Conference on Intelligent Robots and Systems (IROS), Oct. 2010, pp. 2206–2211. B. Koopman, H. Grootenboer, H. de Young, “An inverse dynamics model for the analysis, reconstruction and prediction of bipedal walking”, Journal of Biomechanics, vol. 28, no. 11, pp. 1369–1376. J.T. McConville, T.D. Churchill, I. Kaleps et al., “Anthropometric relationships of body and body segment moments of inertia”. In: Report No. AFAMRL-TR-80-119, Wright-Patterson Air Force Base, OH, 1980. L.A. Miller, D.S. Childress, “Analysis of a vertical compliance prosthetic foot”, Rehabilitation Research and Development, no. 34, 1997, pp. 54–57. N. Milne, The ankle and foot in locomotion. 2010. Course material at UWA. Website: http://www.lab. anhb.uwa.edu.au/hfa213/week5/lec5afoot.pdf K. Nishiwaki, Y. Murakami, S. Kagami et al., “A Six-axis Force Sensor with Parallel Support Mechanism to Measure the Ground Reaction Force of Humanoid Robot”. In: Proceedings of IEEE International Conference on Robotics and Automation. ICRA 2002, vol. 3, pp. 2277–2282. K. Nishiwaki, S. Kagami, Y. Kuniyoshi et al., “Toe joints that enhance bipedal and fullbody motion of humanoid robots”, Proceedings of IEEE International Conference on Robotics and Automation. ICRA 2002, vol. 3, pp. 3105-3110. L. Ren, R.K. Jones, D. Howard, “Whole body inverse dynamics over a complete gait cycle based only on measured kinematics”, Journal of Biomechanics, 41, 2008, pp. 2750-2759. P. Sardain, G. Bessonnet, “Forces acting on a biped robot. Center of Pressure Zero Moment Point”, IEEE Transactions on Systems, Man and Cybernetics, Part A: Systems and Humans, vol. 34, no. 5, Sep. 2004, pp. 630–637. H.M. Schepers, E. van Asseldonk, J.H. Buurke, P.H. Veltink, “Ambulatory Estimation of Center of Mass Displacement During Walking”, IEEE Transactions on Biomedical Engineering, vol. 56, no. 4, Apr. 2009, pp. 1189–1195. V. Ruiz Gárate, “Inverse Dynamic problem of human gait – Investigation for robotic application”, Master diploma thesis, Warsaw University of Technology, 2010. M. Vukobratović, “Dynamic models, control synthesis and stability of biped robots gait”, CISM Courses and Lectures, no. 375, 1997, pp. 153–190. M. Vukobratović, B, Borovac “Zero Moment Point – thirty five years of its life”, International Journal of Humanoid Robotics, no. 1, 2004, pp. 157–173. T. Zielińska, Postural equilibrium in two-legged locomotion, Working material. Warsaw, Poland, 2010. Articles

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2012 2012

Pole-placement Adaptive Control for a Plant with Pole-placement adaptive control for a –plant with unknown Unknown Structure and Parameters a Simulation Study structure and parameters – a simulation study Submitted 4 march 2011; accepted 23 November 2011 th

Submitted 4th March 2011; accepted 23th November 2011.

Dariusz Horla

Abstract: Adaptive pole-placement control for the plant with unknown orders and coefficients of its model is presented in the paper, in an on-line approach. In order to adapt to the plant, the considered controller changes its structure and parameters, along with the identification process. In order to combine structural and parametric identification, the approach presented in [5] has been used, with the simulation runs performed for continuous plant and a discrete-time controller and identification algorithms. Keywords: adaptive control, identification

Full knowledge of the plant is required to design the classical controller with parameters computed on the basis of well-known tuning rules, such as of Ziegler-Nichols for PID controllers. A good choice of controller parameters assures achieving expected performance of the control system. Because the tuning is performed for a specific plant which structure (e.g., order) or parameters may change with time, the computed controller parameters may turn out to be inappropriate. In such a case, one uses adaptive control, tuning the controller to improve the performance by using information about current polynomial orders and estimates of plant parameters. The paper presents the topic of adaptive control with estimation of parameters and orders of the plant polynomials in the reference signal tracking task in a fully discretetime control system. The control algorithm combined with estimation yields time- and structure-varying controller with parameters tuned on-line to the current structure and properties of the plant.

2. General model of the plant Let G(q −1 ) and H(q −1 ) be certain transfer functions and q −1 be a one-sample shift operator, q −1 yt = yt−1 . The general structure of the model [3, 4] = =

G(q −1 )ut + H(q −1 )ξt = C(q −1 ) B(q −1 ) u ξt , + t−d A(q −1 ) A(q −1 )

(1)

where G(q −1 ) is a transfer function of control circuit, and H(q −1 ) of disturbance circuit, can be put in the form A(q −1 )yt = B(q −1 )ut−d + C(q −1 )ξt , 28

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A(q −1 ) B(q

−1

C(q

−1

)

1 + a1 q −1 + · · · + anA q −nA ,

b0 + b1 q 1 + c1 q

−1

+ · · · + bnB q

+ · · · + cnC q

−nB

−nC

(3) (4) (5)

The basic model considered in the paper is the autoregressive model with moving-average with auxogenous input (ARMAX) as in (2). A special form of ARMAX will be of interest here, namely CARMA model.

3. Pole-placement controller

1. Introduction

where d is a dead-time and polynomials from (2) are given as:

(2)

The considered controller is to assure appropriate placement of poles of discrete-time system (with discrete-time controller and discrete-time model of the plant). Such an adaptive controller can be put in RST form with control signal ˆ −1 ) ut − S(q ˆ −1 )yt + Tˆ(q −1 )rt , (6) ut = 1 − R(q

where rt is the reference signal tracked by yt . Having omitted estimate symbols, the controller polynomials: R(q −1 )

S(q −1 )

1+r1 q −1 +· · ·+rnB+d−1 q −nB−d+1 , (7)

s0 + s1 q −1 + · · · + snA−1 q −nA+1 . (8)

Using the knowledge about plant polynomials A(q −1 ) and B(q −1 ), known delay d and choosing closed-loop characteristic polynomial AM (q −1 ) = 1 + aM,1 q −1 + · · · + aM,nAM q −nAM (9) one can introduce Diophantine equation A(q −1 )R(q −1 ) + q −d B(q −1 )S(q −1 ) = AM (q −1 ) , (10) with AM (1) . (11) T (q −1 ) = B(1) By solving this equation one obtains controller parameters, what in turn allows the current control sample to be computed. Having substituted control law to the plant equation and using the Diophantine equation one can obtain closed-loop transfer function yt rt

q −d B(q −1 )T (q −1 ) , AM (q −1 )

from which it can be seen that closed-loop poles are in prescribed locations.

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

VOLUME 6, VOLUME 6,

w RS =

Let the regression vector

, ϕot

where: 

W AB

0 ···

0 .. .

  a1 1   a2 a1   .. ..  . a2 . 0   .. . .. 1  . =  ..  anA . a1    0 a a2 nA   0   . . . . ..  .. .. . . 0 0 · · · anA

w RS

w AM A

∈R

··· .. . ···

0 .. .

b0 0 · · ·

0 .. .

0 .. . 0

b 1 b0 .. . b1 . bnB .. 0 bnB .. . 0

(nR+nS+1)×(nR+nS+1)

2012 2012

5.2. Recursive estimation method

The general solution of the Diophantine equation for given nA and nB is W −1 AB w AM A

N° 1 N◦ 1

.. ..

. 0

. b0 b1 . . .. . .

···

, T [r1 , · · · , rnR , s0 , · · · , snS ] ,

bnB



          ∈          

(12) (13)

[aM,1 −a1 , aM,2 −a2 , · · · , 0, · · · , 0] . (14) T

o yt − θˆ t−1 ϕot , t ≥ 0, εt = 0 (t < 0) ,(16)

εt

ˆθo t

o = ˆθ t−1 + k ot εt , P ot−1 ϕot = , 1 + ϕot T P ot−1 ϕot

k ot P ot

ˆθ t

(19)

[−ˆ a1,t , −ˆ a2,t , . . . , −ˆ anAo ,t , ˆb0,t , ˆb1,t , . . . , ˆbnB o ,t , (20)

cˆ1,t , cˆ2,t , . . . , cˆnC o ,t ]T

is obtained from ELSRO [4] algorithm (16)–(19) (extended least-squares in reducing orders) for overparametrised model. The ELSRO estimate ˆ

(nA, nB, nC) θˆ t

[−ˆ a1,t , −ˆ a2,t , . . . , −ˆ anA,t ˆ ,

ˆb0,t , ˆb1,t , . . . , ˆb ˆ , nB,t

(21)

T cˆ1,t , cˆ2,t , . . . , cˆnC,t ˆ ]

of the vector of plant parameters at time t is given by ˆ

(nA, nB, nC) θˆ t = t −1 ˆ ˆ ˆ ˆ nB, ˆ nC) ˆ T 1 (nA, nB, nC) (nA, ϕi ϕi + I × h i=1 t

(22)

ϕ(inA, nB, nC) yi ,

i=1

– positive-real condition is satisfied,

where nAo , nB o , nC o are maximal assumed orders.

(18)

The arbitrary vector ˆθ 0 (for t = 0) is of size h = nAo + nB + nC o + 1 and P o0 = hI. Vector of estimates

5. Estimation of orders and parameters for linear plants in ARMAX form [5]

– true polynomial orders (nA, nB, nC) belong to known and finite set ˆ nB, ˆ nC) ˆ : 0 ≤ nA ˆ ≤ nAo , M = (nA, ˆ ≤ nC o , ˆ ≤ nB o , 0 ≤ nC 0 ≤ nB

(17)

P ot−1 − k ot ϕot T P ot−1 .

4. Formulation of the problem

5.1. Preliminaries The algorithm has been designed for ARMAX-type plants but by omitting the information about C(q −1 ) polynomial one can use the information concerning A(q −1 ) and B(q −1 ). The method will be cited in brief, the complete algorithm can be found in [5]. It requires the following assumptions: – polynomials A(q −1 ), B(q −1 ), C(q −1 ) are co-prime and their leading coefficients are non-zero,

(15)

correspond to the information gathered for maximal polynomial orders and εt is computed as in RELS algorithm based on equations [4]:

The parameters of A(q −1 ) are put in the first nR columns of W AB , and the zero matrix in right upper corner has the (d − 1) × (nS + 1) dimension.

ˆ nB, ˆ nC) ˆ The main problem is given the triple (nA, of model orders and the current estimate (21), to find the improved model for which no improvement in polynomial orders are unnecessary. Subsequently, for such a model the adaptive pole-placement discrete-time controller of the form (6) has to assure tracking properties for the discretetime model of the plant (2).

[yt−1 , yt−2 , . . . , yt−nAo , ut−d , ut−d−1 , . . . , ut−d−nB o , εt−1 , εt−2 , . . . , εt−nC o ]T

and ˆ

ϕ(tnA, nB, nC)

[yt−1 , yt−2 , . . . , yt−nA ˆ , ut−d , ut−d−1 , . . . , ut−d−nB ˆ , ˆ nB, ˆ nC) ˆ (nA,

ˆ nB, ˆ nC) ˆ (nA,

εt−1

, εt−2

...,

ˆ nC) ˆ nb, ˆ T (nA, εt−nC ] ˆ

, (23)

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VOLUME 6, VOLUME 6,

The estimate (22) can be recursively computed from the algorithm: Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

Îľt

yt â&#x2C6;&#x2019;

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; T (nA, Ë&#x2020; Ë&#x2020; Ë&#x2020; Î¸Ë&#x2020; tâ&#x2C6;&#x2019;1 Ď&#x2022;(tnA, nB, nC)

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

(24)

,t â&#x2030;Ľ 0,

, (25)

Ë&#x2020;

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA, Ë&#x2020;Î¸(nA, nB, nC) + k (nA, Îľt , t tâ&#x2C6;&#x2019;1 Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA, = P tâ&#x2C6;&#x2019;1 â&#x2C6;&#x2019; Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; Ë&#x2020; Ë&#x2020; Ë&#x2020; T (nA, (nA, Ď&#x2022;(tnA, nB, nC) P tâ&#x2C6;&#x2019;1 kt

(26)

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA, Pt

(27)

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

with the initial condition Ë&#x2020;Î¸ 0

as a part of Ë&#x2020;Î¸ 0 (see Ë&#x2020; + nB Ë&#x2020; + = mI, where m = nA

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA, P0

(17)) and Ë&#x2020; + 1 â&#x2030;¤ h. nC Plant-model mismatching residual Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

Ď&#x192;t

t i=1

Ë&#x2020;

(nA, nB, nC) Ë&#x2020; Ë&#x2020; Ë&#x2020; Ď&#x2022;(inA, nB, nC) yi â&#x2C6;&#x2019; Î¸Ë&#x2020;i Ë&#x2020;

Ë&#x2020;

yt â&#x2C6;&#x2019;

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA, Î¸Ë&#x2020; t

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

+â&#x2C6;&#x2020;Î¸Ë&#x2020; t

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

2h t with:

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

â&#x2C6;&#x2020;Î¸Ë&#x2020; t

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

(28)

+ Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

â&#x2C6;&#x2020;Î¸Ë&#x2020; t

Ë&#x2020; t , nB Ë&#x2020; t , nC Ë&#x2020; t , nA Ë&#x2020; t , nB Ë&#x2020; t â&#x2C6;&#x2019; 1, nC Ë&#x2020; t ) < 0, then if St (nA then drop the last parameter of polynomial B;

â&#x2C6;&#x2019;

(nA, nB, nC) (nA, nB, nC) Î¸Ë&#x2020; t â&#x2C6;&#x2019; Î¸Ë&#x2020; tâ&#x2C6;&#x2019;1 , (30)

N tâ&#x2C6;&#x2019;1

Ë&#x2020;

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA, Ë&#x2020;

Ë&#x2020;

Ď&#x2022;(tnA, nB, nC) Ď&#x2022;(tnA, nB, nC) ,(31) Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA, ht

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA, h0

= =

(32)

0, Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA, h tâ&#x2C6;&#x2019;1 + Ë&#x2020; Ë&#x2020; Ë&#x2020; Ď&#x2022;(tnA, nB, nC) yt ,

(33)

(34)

Ytâ&#x2C6;&#x2019;1 +

yt2

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

Ë&#x2020; , nB Ë&#x2020; , nC Ë&#x2020; ) (nA

As a convention N t are, respectively, submatrix 30

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â&#x20AC;&#x201C; step 5 Ë&#x2020; t , nB Ë&#x2020; t , nC Ë&#x2020; t , nA Ë&#x2020; t , nB Ë&#x2020; t , nC Ë&#x2020; t â&#x2C6;&#x2019; 1) < 0, then if St (nA then drop the last parameter of polynomial C; â&#x20AC;&#x201C; step 6 go to step 2. If the initial assumptions hold and initial orders have Ë&#x2020; t , nB Ë&#x2020; t , nC Ë&#x2020; t) been set as maximal, then estimates (nA form monotonically non-increasing trains.

6. Simulation results

6.1. Discrete-time model of the plant The plant can be described as first-order inertia with discrete-time transfer function

, Y0 = 0 ,

Ď&#x192;0

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

model on the basis of (24)â&#x20AC;&#x201C;(27); compute Ď&#x192;t Ë&#x2020; t , nB Ë&#x2020; t, from (28) on the basis of (29)â&#x20AC;&#x201C;(35), with (nA Ë&#x2020; Ë&#x2020; Ë&#x2020; Ë&#x2020; nC t ) substituted for (nA, nB, nC);

â&#x20AC;&#x201C; step 4

Ë&#x2020;

Ë&#x2020; t , nB Ë&#x2020; t , nC Ë&#x2020; t , nA Ë&#x2020; t â&#x2C6;&#x2019; 1, nB Ë&#x2020; t , nC Ë&#x2020; t ) < 0, then if St (nA drop the last parameter of polynomial A;

(29)

Ë&#x2020;

5.3. Recursive algorithm of simultaneous estimation of orders and parameters The shortened algorithm for ARMAX model is cited from [5]: â&#x20AC;&#x201C; step 0 (initialisation)

Ë&#x2020;

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

ence between residuals Ď&#x192;t â&#x2C6;&#x2019; Ď&#x192;t â&#x2030;Ľ Ë&#x2020; Ë&#x2020; Ë&#x2020; Ë&#x2020; Ë&#x2020; Ë&#x2020; 0, that is St (nA, nB, nC, nA , nB , nC ) < 0, Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; should be modified to thus the orders (nA, Ë&#x2020; , nB Ë&#x2020; , nC Ë&#x2020; ). (nA

â&#x20AC;&#x201C; step 3

Ë&#x2020; Ë&#x2020; Ë&#x2020; Ď&#x2022;(tnA, nB, nC)

where Î˛n = Îą(log(t))2 and Îą > 0. Ë&#x2020; , nB Ë&#x2020; , nC Ë&#x2020; ) â&#x2030;¤ (nA, Ë&#x2020; nB, Ë&#x2020; nC), Ë&#x2020; then the differIf (nA

â&#x20AC;&#x201C; step 2

can be recursively evaluated as = Ď&#x192;tâ&#x2C6;&#x2019;1

(36)

â&#x2C6;&#x2019; Î˛t ,

stop if t â&#x2030;Ľ tk , otherwise substitute t := t + 1; evaluate Ë&#x2020; t , nB Ë&#x2020; t , nC Ë&#x2020; t) (nA Ë&#x2020; , nB Ë&#x2020; t , nC Ë&#x2020; t) (nA Îľ t and Ë&#x2020;Î¸ for current plant

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; T Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA, (nA, Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA, Î¸Ë&#x2020; t Nt Î¸Ë&#x2020; t

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

â&#x2C6;&#x2019;

set initial values of appropriate variables;

(nA, nB, nC) Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA, Yt â&#x2C6;&#x2019; 2Î¸Ë&#x2020; t ht +

Ď&#x192;t

Ë&#x2020; ) , nC

â&#x20AC;&#x201C; step 1

Ë&#x2020;

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA, Ď&#x192;t

set the values of start t = 0 and stop time t = nk (nk > 0);

= Ë&#x2020;

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

ht be obtained directly from without additional computations. Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; â&#x2C6;&#x2C6; M the information function Let for (nA, be given

Ë&#x2020; , nB Ë&#x2020; , nC Ë&#x2020; ) (nA

2012 2012

and their values can

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA, Nt ,

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; Ë&#x2020; Ë&#x2020; Ë&#x2020; (nA, Ď&#x2022;(tnA, nB, nC) P tâ&#x2C6;&#x2019;1 Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; T (nA, Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA, (nA, P tâ&#x2C6;&#x2019;1 Ď&#x2022;t 1+Ď&#x2022; t Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

Ë&#x2020;Î¸ t

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

and h t

Ë&#x2020; , nB Ë&#x2020; Ë&#x2020; nB, Ë&#x2020; nC, Ë&#x2020; nA Ë&#x2020; , nB Ë&#x2020; , nC Ë&#x2020; ) = Ď&#x192;t(nA St (nA,

Ë&#x2020; nB, Ë&#x2020; nC) Ë&#x2020; (nA,

NÂ° 1 Nâ&#x2014;Ś 1

(35)

= 0.

Ë&#x2020; , nB Ë&#x2020; , nC Ë&#x2020; ) (nA

and h t and subvector

from which:

yt 16.5q â&#x2C6;&#x2019;1 = , ut 1 â&#x2C6;&#x2019; 0.9q â&#x2C6;&#x2019;1

A(q â&#x2C6;&#x2019;1 ) = 1 â&#x2C6;&#x2019; 0.9q â&#x2C6;&#x2019;1 ,

and d = 1.

B(q â&#x2C6;&#x2019;1 ) = 16.5

(37)

(38)

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

VOLUME 6, 0,N° 1 N◦ 2012 VOLUME 1 2012

Journal of Automation, Mobile Robotics & Intelligent Systems

nA Journal of Automation, Mobile Robotics & Intelligent Systems 5 nB 4 nC 3 nA 2 5 nB 1 Journal of Automation, Mobile Robotics & Intelligent Systems 4 nC 0 3 0 50 100 150 200 250 300 Journal of Automation, Mobile Robotics & Intelligent Systems 2 nA 1 5 nB 0 4 nC 2 50 100 150 200 250 300 3 0 1 2 a ˆ2 , a ˆ3 nA 1 0 5 nB 0 2 50nC 100 150 200 250 300 −1 4 0 50 100 150 200 250 300 a ˆ nA 1 1 3 a ˆ2 , nB a ˆ3 5 2 0 4 nC 1 50 100 150 200 250 300 −1 3 2 ˆb0 0 a ˆ1 2 15 50 100 150 200 250 300 1 0 1 a ˆ2 , a ˆ3 10 0 ˆb0 50 ˆ 100 200 250 300 5 0 −1 ˆ 150 2 a ˆ1 b1 , . . . , b3 15 0 1 10 a ˆ250 ,a ˆ3 100 150 200 250 300 −5 2 0 5 ˆ ˆ ˆ .b.0. , b3 150 −10 50 b1 , 100 200 250 300 1 −1 a ˆ2 , a ˆ3 0 15 a ˆ 1 −15 0 50 100 150 200 250 300 −5 10 100 150 200 250 300 −1 rt , yt 50 150 −10 a ˆ1 ˆ 5 b3 b1 , .ˆb. . , ˆ 0 −15 0 100 15 rt , yt 50 100 150 200 250 300 −5 ˆb0 10 50 150 −10 15 5 ˆ b3 b1 , . . . , ˆ 0 100 −15 10 0 50 100 150 200 250 300 −50 50 100 150 200 250 300 5 rt , yt 50 ˆ ˆ −5 b1 , . . . , b3 150 0 −100 −10 100 50 100 150 200 250 300 −5 −15 −150 −50 50 rt , yt −10 −100 150 −15 0 ut −150 1.0 rt , yt 50 100 100 150 200 250 300 −50 150 0.5 50 −100 100 ut 0 1.0 −150 50 50 100 150 200 250 300 −0.5 −50 0.5 0 −1.0 −100 0 ut 50 100 150 200 250 300 −50 1.0 50 100 150 200 250 300 −150 −0.5 −100 0.5 Fig. 1. Simulation I −1.0 −150 0 ut 1.0 50 100 150 200 250 300 −0.5 Fig. 1. Simulation I 0.5 ut −1.0 1.0 0 0.5 50 100 150 200 250 300 −0.5 Fig. 1. Simulation I 0 −1.0 50 100 150 200 250 300 −0.5

t t t t t t t t t

t t tt

t t t

t t

5 4 3 2 5 1 4 0 3 2 1 5 0 4 1 3 0 2 1 −1 5 1 0 −2 4 0 3 5 2 −1 4 1 15 −2 3 1 0 10 2 0 5 1 −1 0 15 −5 −2 10 1 −10 5 0 0 1 −1 15 −5 0 10 −2 −10 150 5 −1 0 100 −2 15 −5 50 150 10 −10 5 15 0 100 0 10 −50 50 −5 5 150 −10 −100 0 100 −5 −150 −50 −10 50 −100 150 0 −150 100 1.0 −50 150 50 0.5 −100 100 0 0 1.0 −150 50 −50 −0.5 0.5 0 −100 −1.0 0 −50 1.0 −150 −0.5 −100 0.5 −1.0 −150 0 1.0 −0.5 0.5 −1.0 1.0 0 0.5 −0.5 0 −1.0 −0.5

nA nB nC nA nB 50nC

nA nB a ˆ 2 50nC

0 0

a ˆ a ˆ1 50 3 nA a ˆ 2 50nB nC nAa ˆ a ˆ1 50nB 3 a ˆ nC 2 ˆ b0 50 ˆb1 , ˆ b2a ˆ a ˆ1 50 3 50 ˆ b0 a ˆ2 ˆb1 , ˆ b2

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ˆ3 100 a ˆ2 a a ˆ1 50 50 100 rt , ˆ y b0t ˆ3 100 ,ˆ b2a a ˆ1ˆb150

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31 5

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

6.2. Simulations In order to verify the performance of simultaneous order and parameter estimation algorithm, the following simulations have been carried out in discrete-time adaptive control system by stipulating the triplets (nA, nB, nC) or plant orders: I) (3, 4, 1), T

θ 0 = [−0.5, −0.5, −0.5, −15, 10, −10, 5, −3, 2] , II) (3, 2, 3), θ 0 = [−0.05, 0.8, 0.3, −15, 10, 1.2, 0.2, 0.2, 0.2]

with attaching discrete-time transfer function in series with plant 0.4 , 1 − 0.6q −1 and resulting transfer function of a additional inertiaplant connection yt ut

16.5q −1 0.4 · = 1 − 0.9q −1 1 − 0.6q −1 6.6q −1 . 1 − 1.5q −1 + 0.54q −2

(39)

The reference has the same shape in all simulations and aM 1 = −0.908. The estimates of C(q −1 ) during computations have chosen the values close to zero (or equal to zero), and as it has already been mentioned, the polynomial has been ignored here. As it can be seen in Fig. 1 for Simulation I, the algorithm has caused overparametrisation of polynomial B with the last three parameters almost equal to zero. In fact, one can assume that the order or B is zero. The orders of plant model changed their values twice (by this changing the structure of the controller twice). Having fixed the orders, it has taken a few sample times to find improved estimates’ values in the initial stage. Large changes in the reference signal caused the control input to saturate on a certain level, leading to insufficient excitation and poor tracking. Nevertheless, as it can be seen, the tracking is of satisfactory performance past the adaptation period (say, 50th sample) and the values of parameters do not change. In the case of Simulation II (Fig. 2), by initially choosing high order values one could obtain acceptable tracking performance in comparable time with respect to the previous simulation, but with appropriate order of B. The structure of the adaptive controller changed three times (model of the plant has changed). As in the previous simulation, the model has been initially inaccurate, what has caused control signals to saturate.

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VOLUME 6, VOLUME 6,

N° 1 N◦ 1

2012 2012

The performance of tracking is the same and with the same prescribed dynamics as in the previous simulation – the pole placement has been performed correctly.

7. Summary

The presented algorithm, as shown in the simulations, allows one to estimate the orders of plant model polynomials in a closed-loop system. On the basis of numerous simulations and the first simulation included in the paper, it can be said that structural identification is not flawless. A common case was that estimation stopped with orders higher than real orders, leading to overparametrised models, as in Simullation I. Nevertheless, control performance in such a case does not suffer from overparametrisation as much as in the case of underparametrisation which leads to poor tracking. It has been verified that the controller assures good performance in a case of an unknown plant, assuring full adaptivity features. Since discrete-time system analysis allows one to draw conclusions about behaviour of sampled-data systems, one can expect that a sampled-data pole-placement control for a real plant could have similar performance. The conclusions drawn from this paper, have been applied to a real-time control of a servo drive with minimumvariance controller (not a subject of this paper) and the same identification algorithms, leading to comparable results in real-world experiment (for reference see [2]). AUTHOR Dariusz Horla∗ – Poznan University of Technology, Institute of Control and Information Engineering, ul. Piotrowo 3a, 60-965 Poznan, Poland, e-mail: Dariusz.Horla@put.poznan.pl ∗

Corrresponding author

References

[1] Horla D., Adaptive Control. Laboratory classes (in Polish), Publishing House of Poznan University of Technology, Poznan 2010. [2] Horla D., Minimum Variance Adaptive Control of a Servo Drive with Unknown Structure and Parameters, Asian Journal of Control, 2011, doi: 10.1002/asjc.475. [3] Isermann R., Lachmann K-H., Matko D., Adaptive Control Systems, UK, Prentice Hall International 1992. [4] Ljung L., System Identification. Theory for the User, 2nd ed., Prentice Hall, New York, 1999. [5] Ruan R., Chang-Li Y., Huixin C., Bin L., On-line Order Estimation and Parameter Identification for Linear Stochastic Feedback Control Systems (CARMA Model), Automatica 39(2), pp. 243–253, 2003.

Articles

Journal of Automation, Mobile Robotics & Intelligent Systems

VOLUME 6,

N° 1

2012

Chaotic Mobile Robot Workspace Coverage Enhancement Submitted 24th May 2011; accepted 10th January 2012

Ashraf Anwar Fahmy

Abstract: The chaotic mobile robot implies a mobile robot with a controller that ensures chaotic motions. Chaotic motion is characterized by the topological transitivity and the sensitive dependence on initial conditions. Due to the topological transitivity, the chaotic mobile robot is guaranteed to scan the whole connected workspace. The Chua’s circuit, which is low cost and easy to construct for trajectory generators, exhibits a rich variety of bifurcation and chaotic behaviors. This particular circuit is analyzed. It is shown how to generate a sequence of chaotic behaviors by varying the value of a linear resistor of the Chua’s circuit. According to the simulation, with best adjusting parameters, and mapping the appropriate chaotic variables to robot’s kinematic variables, the chaotic behavior is enhanced in the sense of wide area coverage and evenness index.

the integration of Chua’s circuit equations to mobile robot model, and the performance criteria to be applied. Section 5 is reserved to the simulation results. Finally section 6 concludes this paper.

2. Chua’s Circuit Analysis

The circuit components are two capacitors, one inductance and two resistors – one linear and another one nonlinear. The non-linear resistor is known as Chua’s diode and it’s actually made from several components (as seen in the dashed rectangle of Fig.1). The diodes are nonlinear components, and the opamp together with R1, R2 and R3 represent a negative resistor [13].

Keywords:chaos, mobile robot, chaotic motion, chaotic mobile robot, Chua’s circuit

1. Introduction

Chaos is a typical behavior of nonlinear dynamical systems, and has been studied deeply in different fields such as mathematics, physics, engineering, economics, and sociology. In robotics, the first chaotic mobile robot that can navigate following a chaotic pattern was proposed by Nakamura and Sekikuchi [1, 2], where the Anorld’s equation was used to generate the desired motions. Further investigations on chaotic trajectories of the same type of the robot using other equations were carried out in [3-12]. The main objective in exploiting chaotic signals for an autonomous mobile robot is to increase and to take advantage of coverage areas resulting from its travelling paths. Large coverage areas are desirable for many applications such as robots designed for scanning of unknown workspaces with borders and barriers of unknown shape, as in patrol or cleaning purposes. The Chua’s pattern is the most interesting one among other candidate’s patterns due to its largest coverage areas, low cost, and ease for implementation [12]. The aim of this paper is to seek the best combinations of Chua’s circuit parameters and the appropriate mapping of the state variables to robot’s kinematic variables in order to fulfill the requirements of high coverage coefficient and high evenness index of a specified area crossed far and wide erratically by the mobile robot. The paper structure is as follows: The next section presents the analysis of the Chua’s circuit. The mobile robot model is illustrated in section 3. Section 4 is dedicated to describe the methodology which includes

Fig. 1. Chua’s circuit The equivalent input resistor is actually a negative resistor, accordingly; the equivalent Chua’s circuit is as shown in Fig. 2.

Fig. 2. Equivalent Chua’s circuit Using kirchoff laws: dVc1 V V i = − c1 + c 2 − NL dt RC 1 RC 1 C 1 dVc 2 Vc1 V i = − c2 + L dt RC 2 RC 2 C 2

(1) (2) Articles

Journal of Automation, Mobile Robotics & Intelligent Systems

VOLUME 6,

di L V = − c2 dt L

(3)

Knowing the Chua’s diode I-V piecewise-linear characteristic given by Fig. 3, the non-linear part of the circuit can be solved by the following system of equations:

X n +1 = X n + h f (x n , t n )

(9) Where Xn+1 is the solution of the ODE at time tn+1, Xn is the current value, and h is the step-size. Using (9) to solve (5) we obtain:

)

(

)

−g2Vc1 + g1 − g2 BP1, Vc1 < −BP1  = −g1Vc1 − BP1 ≤Vc1 ≤ BP  −g2Vc1 + g2 − g1 BP1, Vc1 < −BP1

(4)

This set of equations can be put in the form of statespace:   Vc 1  Vc 1      Vc 2  = A Vc 2  + b   I  I   L L  

(5)

From (1) to (4), the state space equations of the Chua’s circuit can be written as in (6-8):     −1 + g2 Vc 1   RC 1 C 1    1 Vc 2  =     RC 2 I   0  L   

 1  (g2 − g1  0  RC 1  Vc 1   C 1   1    −1 Vc 2  +  0    RC 2 C 2 I   0   L   −1 0     L  VC 1<−BP

    −1 + g1 Vc 1   RC 1 C 1    1 Vc 2  =  2 RC    I   0  L   

    −1 + g2 Vc 1   RC 1 C 1    1 Vc 2  =     RC 2 I   0  L   

(10)

3. The Mobile Robot Model

Fig. 3. Chua’s diode I-V characteristic

(

2012

This set of ordinary differential equations (ODE) can be solved using a numerical method such as Forward Euler (FE). The FE method states that [13]:

X n +1 = X n + h (Ax n + b)

I NL

N° 1

The mobile robot considered in this work is of two degrees-of-freedom, including two active, parallel and independent wheels, a third passive wheel with exclusively equilibrium functions, and proximity sensors capable of obstacles detection. The active wheels are independently controlled on velocity and rotation sense. The sensors provide short-range distances to obstacles. For instance, these sensors can be infrared devices commonly used in mobile robots, with adequate accuracy. Additionally, the robot is supposed to be equipped with specific sensors for detection and recognition of searched objects. The robot is supposed to operate on a horizontal plane with a motion described in terms of linear velocity u(t) and direction q(t) . The geometry of this motion scheme is shown in Fig. 4 [7]. The mathematical model, of this kinematic problem considers two control variables, velocity u(t) and rotational velocity w(t) and three state variables, the robot position and orientation (xr(t), yr(t), qr(t)), is as follows:

(11) The discrete form of the equation (11): (6)

 1  (g2 − g1  0  RC 1  Vc 1   C 1   1    −1 V + 0  RC 2 C 2   c 2    I   L  0  −1  0    L  −BP1 ≤ VC 1 ≤ Bp1

x r(n +1) ≈ x r(n) +t ncosq r(n) y r(n +1) ≈ y r(n) + t nsinq r(n) q r(n +1 ) ≈ q r(n) + t ω r(n)

(12)

Where t is sufficiently small step

(7)  1  (g1 − g 2  0  RC 1  Vc 1   C1   1    −1 Vc 2  +  0    RC 2 C 2 I   0   L   −1 0     L  VC1>BP 1

(8) 34

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Fig. 4. Geometry of the robot motion on Cartesian plane

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4. Methodology

2012

by the following algorithm: 1. We divide the specified area into (NxN) pixels. 2. Initially, assign the value 0 for all pixels. 3. We get the x-coordinate and y-coordinate of the pixels which passes through the trajectory of the robot. 4. We assign the value 1 for each pixel passes through the trajectory. 5. We count the number of ones (pixels which passes through the trajectory) which is Au 6. We count the number of zeroes Z (pixels which don’t passes through the trajectory). 7. The total area At is the sum of Au and Z. Similarly, let us consider a rectangular shape area, Fig. 5. The total area can be partitioned into four quarter, denoted Q =1, 2, 3, 4. The quantitative measurement of the trajectory can be evaluated by using the following equation:

4.1. Integrated system.

In order to integrate the Chua’s circuit into the controller of the mobile robot, we name the following states x1 ≡ Vc1, x 2 ≡ Vc2 , x 3 ≡ i L,

N° 1

(13)

From equation (5) the set of equations of Chua’s circuit become:

(14) Consequently, the state equation of the chaotic mobile robot after integrating the set of equations of Chua’s circuit with the mathematical model of the mobile robot equation (11), we obtain the following system of equations:

KQ =

AuQ

(17)

AtQ

Where KQ is the performance index of the Qth quadrant, AuQ is the area used by the trajectory in the Qth quadrant. In our case, we have:

(15)

At 4

AtQ =

(18)

Equations (16)-(18) will be used as performance indices in section 5.

The corresponding mapping parameters from 3-D chaotic circuit into 2-D one is as follows: Table 1. Mapping chaotic variables to robot’s kinematic variables System

Case 1

Case 2

Case 3

Fig. 5. Partition of the specified area

Where q corresponds to the orientation angle of the mobile robot.

b) An evenness index E refers to how close in numbers each species in an environment are. The evenness index can be represented in our situation by [14]:

4.2. Evaluation criteria

The evaluation criteria are set according to the application purpose. Since we would like to use the robot in wandering around area in the area of no maps, the chaotic trajectory should cover the entire areas of patrolling as much as possible. The following two performances criteria are to be considered to evaluate the coverage rate of the chaotic mobile robot, namely the performance index k and the evenness index E a) A performance index K representing a ratio of areas that the trajectory passes through or used space (Au), over the total working area (At) K =

∑ E = 1−

C2 1.5nF

L 1mH

Vc2(0) 0V

ln(s)

(19)

5. Simulation Results

The system of equations (15), introduced in section 4 is simulated through Matlab, including circuit simulation of the Chua’s circuit, introduced in section 2 and mobile robot model introduced in section 3. The values used for the Chua’s circuit simulation were the following [13]:

Au At

Vc1(0) 10-5

KQ ln(kQ )

Q =1

Where s: No of species = 4 Quarters in our case. E is constrained between 0 and 1. The less variation in covering the areas between the species, the higher E is.

(16) The used area Au and the total area At can be calculated C1 450 PF

IL(0) 0A

g1 1/1358

g2 1/2464

BP1 0.114

h 10-7

R 1-2 KΩ Articles

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VOLUME 6,

The values of all the components of the circuit are constant except the resistor R which is varied from (10002000) Ω to get the best value of performance index K and the evenness index E, introduced in section 4 as a performance measure of the chaotic trajectory of the Chua’s circuit for the integrated system, described in section 4. For simplicity, an area of 20m x 20m is used as a workspace coverage trajectory in computer simulation. The robot moves as if is reflected by the boundary “mirror mapping”. The initial conditions were established and the program was set to run n iterations. The resultant Chua’s pattern in three dimensions at iteration n=3000, is depicted in Fig. 6

Resistance R[Ω]

% of K

% of Q=1

% of Q=2

% of Q=3

% of Q=4

% of E

1000

82.10

93.69

87.85

71.07

75.80

54.74

1100

80.47

88.84

90.97

71.88

70.19

51.17

1200

85.00

86.98

80.81

82.94

89.28

60.34

1300

84.08

94.86

84.28

73.96

83.24

58.89

1400

81.76

70.89

77.63

93.31

85.21

53.74

1500

88.07

86.66

86.48

89.48

89.67

67.77

1600

83.99

82.08

77.76

85.68

90.44

58.09

1700

86.39

92.15

84.12

80.78

88.49

63.84

1800

83.90

82.40

88.34

85.29

79.56

57.69

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84.78

77.31

78.30

92.33

91.16

60.44

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74.91

90.42

64.86

60.30

84.07

40.66

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Table 3. Case 2 and run time for n= 3000

x 10

0.2

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0.05 0

-5 -0.5

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0.1

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Fig. 6. Chua’s pattern in three dimensions Table 4. Case 3 and run time for n=3000 The simulation results for case 1, case 2, and case 3, for number of iterations n= 3000 and by varying the value of the resistance R is shown in Table 2, Table 3, Table 4, respectively. From the results we can deduce the following: a) In case 1, the performance index k and the evenness index E are maximum when R=1100. b) In case 2, the performance index k and the evenness index E are maximum when R=1500. c) In case 3, the performance index k and the evenness index E are maximum when R= 1625. The resultant chaotic trajectories of the robot in case 1, case 2, and case 3 at the specified values of the resistor R are illustrated in Fig. 7, Fig. 8, and Fig. 9, respectively.

Resistance R[Ω]

% of K

% of Q=1

% of Q=2

% of Q=3

% of Q=4

% of E

1000

79.97

84.62

68.67

74.63

91.96

49.88

1100

77.57

89.80

66.19

65.48

88.82

45.74

1200

74.74

87.04

93.88

61.24

56.74

42.17

1300

75.12

94.46

80.09

57.79

68.16

41.59

1400

74.84

65.49

69.52

83.88

79.02

37.72

1500

80.34

83.90

84.63

76.74

76.07

49.53

1625

80.86

84.23

68.88

76.63

93.70

51.94

1700

76.35

78.32

80.63

74.92

72.16

40.76

1800

46.64

72.41

57.49

25.08

31.59

8.90

1900

76.48

85.73

73.35

67.72

79.15

41.69

2000

45.01

54.29

29.17

33.75

62.81

2.64

Table 2. Case 1 and run time for n=3000 % of K

% of Q=1

% of Q=2

% of Q=3

% of Q=4

% of E

1000

85.82

86.06

88.15

85.58

83.52

62.19

1100

85.93

89.22

80.04

82.49

91.95

62.79

1200

85.17

92.35

82.00

78.23

88.10

61.06

1300

76.01

63.89

59.71

87.18

93.28

43.84

1400

83.07

77.70

79.40

88.54

86.64

55.91

1500

80.52

84.14

90.24

76.43

71.26

50.61

1600

82.36

88.13

80.28

76.76

84.27

54.20

1700

67.34

95.64

46.65

41.66

85.41

35.24

1800

81.71

83.32

74.64

79.80

89.08

52.88

1900

77.85

74.12

70.10

81.27

85.93

44.47

2000

67.70

95.84

58.56

44.16

72.27

31.49

Trajectory of the chaotic mobile robot "case 1"

5 0 -5 yr[m]

Resistance R[Ω]

-10 -15

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-20 -25 -25

-20

-15

-10 xr[m]

-5

Fig. 7. The chaotic trajectory of the robot in case 1 To investigate the relation between the value of the resistance R of the Chua’s circuit and the performance indices K and E, the bar plots of this relation for the case 1, case 2, and case 3, are illustrated in Fig. 10, Fig. 11, and Fig. 12, respectively.

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Case 3 90 K% E%

80 70 60 50 40 30 20 10 0

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

Resistance [ohm]

Fig. 8. The chaotic trajectory of the robot in case 2

Fig. 12. Bar plot of K and E vs. R for case 3 The relation of the performance indices K and E for the specified values of the resistance R as n changed from 1000 to 10000, for case 1, case 2, and case 3 are illustrated in Fig. 13, Fig. 14, and Fig. 15, respectively. Case 1 100 K% E%

90 80 70 60 50 40 30 20 10 1000

Fig. 9. The chaotic trajectory of the robot in case 3 Case 1

2000

3000

4000

5000 6000 Iterations "n"

7000

8000

9000

10000

Fig. 13. The performance indices K and E vs. iterations in case1

Case 2

100

K% E%

90 80

60 70

60 50

30 20

10 0 1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Iterations "n" 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

Resistance R [ohm]

Fig. 14. The performance indices K and E vs. iterations in case 2

Fig. 10. Bar plot of K and E vs. R for case 1 Case 2 90 K% E%

Case 3 100 K% E%

80 70

40 30

20 10

10 0

0 1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Iterations "n" 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000

Resistance [ohm]

Fig. 11. Bar plot of K and E vs. R for case 2

Fig.15. The performance indices K and E vs. iterations in case 3 Articles

Journal of Automation, Mobile Robotics & Intelligent Systems

6. Conclusion

This article introduced an enhancement to the chaotic mobile robot, using Chua’s circuit as a controller. The Chua’s circuit, which is low cost and easy to construct for trajectory generators, exhibits a rich variety of bifurcation and chaotic behaviors. First, we studied the effect of the “resistor” value of the Chua’s circuit on the resultant robot kinematics for enhancing the robot trajectory in the sense of wide area coverage and evenness index. Second, we studied the effect of the setting of mapping between circuit variables and robot angle of rotation. We deduce from simulation results that, we can get the highest value of the covering coefficient and evenness index at R= 1500 Ω, conditionally that we map Vc2 of the Chua’s circuit to the angle of rotation (θ), of the robot. Third, we deduced that: by increasing iterations of simulations (time), the covering coefficient and evenness index are enhanced with different rate depending on appropriate mapping, is used. In case 1 and case 2, the covering coefficient and evenness index reach above 90% in 5000 iterations (time unit), but case 3 realizes that in 6000 iterations. We can deduce that, by setting optimal values of the parameters of the Chua’s circuit and the appropriate mapping of chaotic variables to robot's kinematic variables, is suitable for generating navigating signal to implement the robot target-searching tasks due to its excellent trajectory throughout a given work-area.

Author:

Ashraf Anwar Fahmy – Assistant Professor, Department of Computer Engineering, College of Computers and Information Technology, Taif University, Taif, Saudia Arabia. His research interests are mainly in the area of tracking system, control, and robotics. E-mails: ashraaf@tu.edu.sa; ashrafmanwar90@yahoo.com

References

[1] Y. Nakamura and A. Sekiguchi, “Chaotic mobile robot,” IEEE Transaction on Robotics and Automation, vol. 17, no. 6, 2001, pp. 898–904. [2] A. Sekiguchi and Y. Nakamura, “The Chaotic mobile robot”, in: Proc. IEEE/RSJ. Int. Conf. Intelligent Robots and Systems, vol. 1, 1999, pp. 172-178.

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[3] A. Jansri, K. Klomkarn, and P. Sooraksa, “Further investigation on trajectory of chaotic guiding signals for robotics system”. In: Proc. Int. Symp. Communication and Information Technology, 2004, pp. 1166–1170. [4] A. Jansri, K. Klomkarn, and P. Sooraksa, “On comparison of attractors for chaotic mobile robots”. In: Proc. 30th Annu. Conf. IEEE Industrial Electronics Society, IECON, vol. 3, 2004, pp. 2536–2541. [5] C. Chanvech, K. Klomkarn, and P. Sooraksa, “Combined chaotic attractors mobile robots”. In: Proc. SICE-ICASE Int. Joint Conf., 2006, pp. 3079–3082. [6] L. S. Martins-Filho, R. F. Machado, R. Rocha, and V. S. Vale, “Commanding mobile robots with chaos”. In: J. C. Adamowski, E. H. Tamai, E. Villani, and P. E. Miyagi (Eds.) ABCM Symposium Series in Mechatronics, vol. 1, ABCM, Rio de Janeiro, Brazil, 2004, pp. 40–46. [7] S. Martins et al.,” Kinematic control of mobile robots to produce chaotic trajectories”. In: ABCM Symposium Series in Mechatronics, vol. 2, 2006, pp. 258–264. [8] S. Martins et al., „Patrol Mobile Robots and Chaotic Trajectories”. In: Mathematical problems in engineering, vol. 2007, Article ID61543, 13 pages, 2007. [9] J. Palacin, J. A. Salse, I. Valganon, and X. Clua, “Building a mobile robot for a floor-cleaning operation in domestic environments”, IEEE Transactions on Instrumentation and Measurement, vol. 53, 2004, no. 5, pp. 1418–1424. [10] M. Islam and K. Murase, “Chaotic dynamics of a behavior-based miniature mobile robot: effects of environment and control structure”, Neural Networks, vol. 18, no. 2, 2005, pp. 123–144. [11] U. Nehmzow, “Quantitative analysis of robot-environment interaction-towards scientific mobile robotics”, Robotics and Autonomous Systems, vol. 44, no. 1, 2003, pp. 55–68. [12] P. Sooraksa and K. Klomkarn, „No-CPU chaotic robots from classroom to commerce”. In: IEEE circuits and systems magazine, 10.1109/MCAS, 2010, pp. 46–53. [13] V. Araujo, „Circuit analysis and matlab simulation of chua oscillator”, ARISER, vol. 5, No. 1, pp. 49– 57, 2009. [14] J. Nicolas et al., „A comparative analysis of evenness index sensitivity”, Int. Review Hydrobiology, vol. 88, no. 1, 2003, pp. 3–15.

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ACCELEROMETER-BASED MEASUREMENTS OF AXIAL TILT Submitted 27th June 2011; accepted 1st September 2011

Sergiusz Łuczak

Abstract:

The paper deals with a specific type of tilt measurements, where an axial tilt is to be determined. The measurements are realized by means of accelerometers – MEMS devices most preferably. Various mathematical relations between the axial tilt and the Cartesian components of the gravitational acceleration are presented. Each relation is described in detail, especially in the terms of the resultant uncertainty of the measurement, as well as the requirements regarding the employed accelerometers. Results of experimental studies realized by means of commercial MEMS accelerometers are presented and discussed, especially with regard to the measurement accuracy that has been evaluated for each mathematical relation. Scope of application of each relation is proposed. Keywords: MEMS, accelerometer, tilt, measurements

Designations used in Fig. 1, have the following meaning: g – gravitational acceleration, gx, gy, gz – components of the gravitational acceleration, ϕ – tilt angle. Let us assume that we use a triaxial accelerometer whose sensitive axes are x, y, z, and we want to determine the tilt of axis z (note that even though rotation about this axis results in a change of gx and gy, it should not result in a change of angle ϕ being determined).

2. Computations of the tilt angle and the resultant uncertainty Whereas the pitch is primarily related to component gx and the roll to gy, angle ϕ is primarily related to gz: φ1 = arccos

1. Introduction In the case of tilt measurements, usually two component angles are determined: pitch and roll [1]. However, there are some cases when such approach is not convenient. An example can be here e.g. directional drilling [2], where the tilt angle between the rotation axis of a drill bit and the gravitational acceleration must be carefully observed and kept at constant value. Another instance may be monitoring an object against losing its stability, as far as its vertical position is concerned. In the mentioned cases both pitch and roll occur, as if it were a dual axis tilt measurement. However, in fact it is a single axis measurement, where orientation of the rotation axis is not important. So, an unconventional approach should be employed here. Its idea is illustrated in Fig. 1, where the considered tilt angle ϕ is presented against the Cartesian components of the gravitational acceleration. As measurements of tilt realized by means of accelerometers are very advantageous in many aspects, the further considerations are limited to this measurement method.

Fig. 1. Tilt angle and Cartesian components of the gravitational acceleration

gz g

(1)

Making use of the fact that: (2)

g = g x2 + g y2 + g z2

further formulas for determining angle ϕn can be derived (the subscript n=1…5 has been introduced in order to distinguish between particular formulas later in the text): f2 = arcsin

f3 = arctan

g x2 + g y2 g g +g 2 x

(3) 2 y

(4)

Even though the nominal values of angles ϕn (n = 1…3) are the same, the uncertainty of their determination differs significantly. As suggested by the International Organization for Standardization [3], the uncertainty can be calculated according to a general formula: 2

uc ( fn ) =

  ∂fn   ∂fn  ∂g u ( g x ) +  ∂g u g y  +  y  x 2

( )

 ∂fn   ∂fn   ∂g u ( g z ) +  ∂g u ( g ) z

(5)

Assuming that:

( )

u ( g x ) = u g y = u ( g z ) = u ( g x... z ) >> u ( g )

(6) Articles

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the uncertainty for Eq. (1), (3) and(4) will be respectively: uc ( f1 ) =

uc ( f2 ) =

uc ( f3 ) =

u ( g x... z ) g u ( g x... z ) g u ( g x... z ) g

1 sin f1

(7)

1 cos f2

(8)

= const

(9)

Courses of the Eq. (7)-(9), as well as the following Eq. (10) and (12), are presented in Fig. 2. It was assumed that the illustrated uncertainties are related to a unit relative standard uncertainty of the accelerometer: u ( g x... z ) g

(10)

In the case of Eq. (7)-(8) (courses u2 and u1), their maximal values (approaching infinity) have been limited in the chart.

Analyzing the courses of uncertainties uc(ϕ1) and uc(ϕ2) presented in Fig. 2, one may easily conclude that it is worthwhile to introduce still another way of determining angle ϕ, defined as follows [4]:  f1 , f2 < 45° ⇒ f4 = f2     f1 , f2 > 45° ⇒ f4 = f1 

2012

Then, the considered uncertainty approximately equals [5]: uc ( f5 ) ≈

u ( g x... z ) g

≈ const

(14)

If the uncertainties featured by particular sensitive axes of the applied accelerometer, related to measurements of the respective component accelerations, are not equal, as expressed by Eq. (6) and assumed in Fig. 2 (and thus in Eq. (11) and (13), consequently), the relevant formulas should be rearranged. Eq. (11) should feature an angle other than 45°, and the variable weight coefficients in Eq. (13) should be expressed by functions other than sin2(x) or cos2(x), otherwise the respective uncertainty would increase.

3. The experimental studies The presented theoretical considerations have been fully confirmed in an experimental way. In order to carry out appropriate tests a special test station was used. It has been described by the author in [6], whereas the observed methodology of performing the experiments minutely discussed in [7]-[8]. The measurements were realized by means of a tilt sensor built of two dual-axis MEMS accelerometers ADXL 202E from Analog Devices Inc. [9], whose sensitive axes were arranged into a Cartesian coordinate system. Results of the tests are illustrated in Fig. 3, which presents variations of errors corresponding to each kind of the considered formulas for determining the tilt angle. The error en (n=1…5) has been defined as [3]: en = q − fn

Fig. 2. Uncertainty of tilt measurements for various mathematical formulas

N° 1

(15)

where θ is the real tilt angle applied by means of the test station and ϕn is the tilt angle calculated according to the respective formula, on the basis of accelerometer indications. The graphical form of each error is consistent with its corresponding uncertainty in Fig. 2.

(11)

As results from Eq. (11), uncertainty uc(ϕ4) is a combination of uc(ϕ1) and uc(ϕ2), expressed in the following way:  f1 , f2 < 45° ⇒ uc ( f4 ) = uc ( f2 )      f f f f > ° ⇒ = , 45 u u  1 2  c ( 4) c ( 1)  

(12)

Additionally, the tilt can be computed using a principle of a weighted average having variable weight coefficients, as follows [5]: f5 = f1 sin 2 f4 + f2 cos 2 f4

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(13)

Fig. 3. The sensitivity of tilt measurements for various mathematical formulas

Journal of Automation, Mobile Robotics & Intelligent Systems

4. Conclusions As results from the courses of the uncertainties illustrated in Fig. 2, as well as variations of the measurement errors in Fig. 3, the most accurate way of determining the considered axial tilt is to employ Eq. (4) or Eq. (13) – uncertainties u3 and u5, errors e3 and e5. The later is much more complicated, however while using triaxial MEMS accelerometers it may turn out that it is more advantageous, as the manufacturing technologies of MEMS devices are in fact usually semi-three-dimensional [10]. As a result, accuracy in the third axis of a triaxial accelerometer is often lower than in the other two. Then Eq. (13) may be constructed in such a way as to regard this fact. In the case of using the most simple formulas: Eq. (7)-(8), the resultant uncertainty significantly increases for tilt angles of 80°÷90° and 0°÷10° respectively – uncertainties u2 and u1, what corresponds to clearly higher errors e2 and e1. This way of determining the tilt is acceptable rather in the case of using only the remaining part of the measuring range, or using both formulas interchangeably, as proposed by Eq. (11). In conclusion, while comparing the respective courses of uncertainties and the corresponding errors, presented in Fig. 2 and Fig. 3, it can be stated that the experimental studies proved the reasoning presented in section 2 to be true.

5. Summary While determining tilt of an axis with application of MEMS accelerometers, one must first decide what kind of formula will be applied for this purpose. In Tab.1 there are gathered the most important features of determining tilt according to particular kind of the mathematical formula. Tab. 1. Characteristics of the mathematical formulas. A

Eq. (1)

∞÷1

³ ±1g

Eq. (3)

1÷∞

³ ±1g

Eq. (4)

³ ±1g

Eq. (11)

1÷1.41

³ ±0,71g

Eq. (13)

³ ±1g

A – no. of the formula B – uncertainty (defined as in Fig. 2) C – necessary no. of sensitive axes of the applied accelerometer D – measurement range of the applied accelerometer (g = 9.81 m/s2)

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It must not be neglected that application of particular formula is connected with a different complication of the related data processing, as it results from the structure of each formula. Application of Eq. (1) is advantageous, because a singleaxis accelerometer can be applied. However, if the measured tilt angles are small, the errors have considerable values. It is just contrary in the case of Eq. (2). On the other hand, application of Eq. (3) and (5) ensures the lowest uncertainty of measurements (so, the highest accuracy), whereas application of Eq. (11) makes it possible to use an accelerometer with a smaller measuring range.

Author Sergiusz Łuczak – Division of Design of Precision Devices, Institute of Micromechanics and Photonics, Faculty of Mechatronics, Warsaw University of Technology, ul. Boboli 8, 02-525 Warsaw, Poland; tel. (+48)(22) 2348315, fax (+48)(22) 2348601. E-mail: s.luczak@mchtr.pw.edu.pl.

References: [1] S. Popowski, “Determining pitch and roll in inexpensive land navigation systems”, J. Aeronautica Integra, vol. 3, no. 1, 2008, pp. 93-97 (in Polish). [2] J. Qian, B. Fang, W. Yang, X. Luan, H. Nan, “Accurate Tilt Sensing with Linear Model”, IEEE Sensors J., (in press). [3] Wyrażanie niepewności pomiaru. Przewodnik, Główny Urząd Miar: Warsaw, 1999 (in Polish). [4] S. Łuczak, W. Oleksiuk, M. Bodnicki, “Sensing Tilt with MEMS Accelerometers”, IEEE Sensors J., vol. 6, no. 6, 2006, pp. 1669-1675. [5] S. Łuczak, W. Oleksiuk, “Increasing Accuracy of Tilt Measurements”, Eng. Mech., vol. 14, no. 3, 2007, pp. 143-154. [6] S. Łuczak, “Dual-Axis Test Station for MEMS Tilt Sensors”, Metrology and Measurement Sys., (in press). [7] S. Łuczak, “Doświadczalne wyznaczanie wybranych parametrów akcelerometrów MEMS – metodyka badań doświadczalnych”, Pomiary Automatyka Kontrola, (in press), (in Polish). [8] S. Łuczak, “Doświadczalne wyznaczanie wybranych parametrów akcelerometrów MEMS – wyniki badań doświadczalnych”, Pomiary Automatyka Kontrola, (in press), (in Polish). [9] Low Cost ±2g Dual Axis Accelerometer with Duty Cycle Output, ADXL 202E, Analog Devices Inc.: Norwood, 2000. [10] Kaajakari V., Practical MEMS, Small Gear Publishing: Las Vegas, 2009, p. 5.

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Increased Performance of a Hybrid Optimizer for Simulation Based Controller Parameterization Submitted 27th June 2011; accepted 12th October 2011

Reimund Neugebauer, Kevin Hipp, Arvid Hellmich, Holger Schlegel

Abstract:

The controller parameterization is often carried out by applying basic empirical formulas within an integrated automatic design. Hence, the determined settings are often insufficiently verified by the resulting system behavior. In this paper an approach for the controller parameterization by using methods of simulation based optimization is presented. This enables the user to define specific restrictions e.g. the complementary sensitivity function (CSF) to influence the dynamic behavior of the control loop. Furthermore it is possible to choose alternative optimization criteria. A main influence factor for practical offline as well as controller internal optimization methods is the execution time, which can be reduced by applying a hybrid optimization strategy. Thus, the paper presents a performance comparison between the straight global Particle-Swarm-Optimization (PSO) algorithm and the combination of the global PSO with the local optimization algorithm of Nelder-Mead (NM) to a hybrid optimizer (HO) based on examples. Keywords: controller parameterization, simulation based optimization, particle swarm optimization, Nelder-Mead

1. Introduction

In the field of operations research a large number of methods were developed to support decision-making processes. It has been proven, that there is a wide field of application. In this paper a brief introduction using these methods for mechatronic controller parameterization is given with the goal of increased speed using a hybrid optimizer. In section 2 the basics of simulation optimization as well as the used optimization algorithms are stated. Subsequently in section 3 the application for controller parameterization is briefly introduced for two examples. The structure and functionality of the hybrid optimizer is the subject of section 4. A performance evaluation of the hybrid optimizer is done in section 5. The paper closes with a comparison and conclusions given in section 6.

2. Simulation based controller parameterization

Generally, simulation based optimization is a methodology of searching for the global extremum of an objective function by the coupling of a simulator with an optimizer [1]. It results in a cyclic sequence between the optimizer and the simulator. The optimizer determines a possible solution and passes it to the simulator for evaluation. According to the 42

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result of the simulator the optimizer calculates a possible better solution. The core of the simulator is a model of the entire system which is examined. Therefore, an optimization problem (eq. 1) has to be solved [1].

F (θ ) → min( F (θ )

(1)

θ ∈Θ

F(), called fitness function [2], is a real-valued function which represents the evaluation of the actual solution. In general, the implementation of constraints is realized by using punishment values. If a constraint is violated, a punishment value is added to the evaluation of the actual solution. Therewith it is depreciated and avoided by the optimizer. The evaluation of a solution is calculated in accordance to equation 2. F ( xn ) = Main_Criterion ( xn ) +

∑ Punishment_Constraint

( xn )

(2)

Optimization techniques are divided into global and local algorithms [3]. The objective of global optimization is to find the global extremum over the entire function space. In contrast, local methods start from a defined point in the search space and try to determine a better solution. According to [4] simulation based optimization could be used to adjust controller parameters considering definable constraints. It exits a large number of optimization algorithms for different application fields. Hereafter the PSO and the NM algorithm are described.

1.2. Particle-Swarm-Optimization

PSO is a common heuristic technique [5], which is based on the simulation of the movement of herds or swarms. An individual of a swarm is called particle. The trajectory of each particle depends on the movement of the other individuals of the swarm and random influences. The advantages of the algorithm are among other things its simple structure, no need for gradient information and its performance. The position of every particle in the -th step is described by the vector . The position of each particle in the ()-th step is update according to equations 3 and xik +1 = x ki + ∆xik +1

(3)

∆xik +1 = ω∆xik + c1r1,ik ( xibest ,k − xik ) + best ,k c2 r2,ik ( xswarm − xik )

(4)

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Where c1, c2 and w are positive constants, r 1,k i , r 2,k i , and are two random values in the range [0, 1]. The term xibest, k represents the best previous position of particle i till step k and as the best known position among all particles in the population. Therefore, xibest, k is called “simple nostalgia” because the individual tends to return to the place that most satisfied it in the past. The term realizes the publicized knowledge, which also every individual tends to [6].

2.2. Nelder-Mead

The NM algorithm (or simplex method), which was originally presented in [7], uses a geometric structure, the simplex, with points in the search space with n +1 the dimension, e.g. for n = 2 the simplex is a triangle. At the beginning the simplex is constructed around a committed start point. The edges of the simplex are called vertex and have to be arranged equidistant from each other. The basic principle of the algorithm is the modification of the simplex towards the extremum. In general this is achieved by replacing the worst vertex by a better one using four functions: reflect(), expand(), contract() and shrink(). A detailed description of the algorithm can be found in [8].

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To reduce the overshoot of the system the CSF T(s) could be used [10]. The mathematical structure is: T (s) =

G R ( s ) * GS ( s ) 1 + G R ( s ) * GS ( s )

(7)

It allows an evaluation of the influence of changes in the command signal. By specifying the CSF it is possible to affect the dynamic of the control loop. Therefore, by setting T (s) =

1.1 0.1s + 1

(8)

the permissible amplification is limited. The new results are KR= 12.327, KI = 17.936 and KD = 2.07. As expected the overshoot is reduced (Figure 2) while the rise time increases.

3. Optimization Problem 3.1. Control Loop Dynamic

Assuming the stated closed loop system structure [9], the system behavior is described with a transfer function GS (eq. 5). GS ( s ) = K *

1 + b1s + b2 s 2 + ... + bm s m 1 + a1s + a2 s 2 + ... + an s n

(5)

It is supposed to use a PID controller G R in the additive structure (eq. 6). GR ( s ) = K R +

KI + KDs s

(6)

Fig. 2. Comparison of parameter settings

3.2. Energy Consumption

As an alternative to the optimization based on the CSF (section 3.1), the approach can also be used to minimize other criteria, such as the energy consumption of a servo drive. To illustrate this, a PI velocity controller according to equation 9 is used.  1   GPI ( s) = K P ⋅ 1 + T N ⋅s 

(9)

The model of the controlled system includes the closed current loop and the total moment of inertia. The parameter identification was carried out in [11] and leads to the following first order integral plus dead time system (FOIPD) (eq. 10). GS ( s ) =

Fig. 1. Comparison of parameter settings In Figure 1 possible attainable transition functions of a PT3 plant (K=0.6, a1= 0.92, a2= 0.234, a3=0.018) with a PID controller are shown. The main optimization criterion is the control area. No constraints were defined. The results of the optimization process are KR= 12.327, KI = 17.936 and KD = 2.07.

746 ⋅ e −s⋅0.00025 (1 + 0,0004 ⋅ s ) ⋅ s

(10)

To achieve a higher flexibility in formulating the quantification factors (control effort, control area, disturbance area) and the penalty (max. overshoot = 15%), the simulator was realized in MATLAB® Simulink®. Focusing on the energy consumption, the control effort was chosen as main criterion. The resulting controller parameterization as well as the results of the automatic controller tuning system, included in the servo drive system, are listed in Table 1. Articles

Journal of Automation, Mobile Robotics & Intelligent Systems

Table 1. Controller parameters Method

KP [Nm s/rad]

TN [ms]

Automatic Tuning (SIEMENS)

1.309

8.73

Optimizer

0.9

5.2

Fig. 3. Control effort step response Figure 3 shows the resulting step responses and the integrated control effort. Notice that the overshoot reaches the predefined limit without exceeding it. Furthermore the reduction of the energy consumption is visible in the time plot, while the settling times of both variants are comparable.

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Fig. 4. Structure of hybrid optimizer

5. Performance comparison

The performance tests have been carried out using a self developed modular optimization application written in C# using VisualStudio 2010 supporting different optimization algorithms. The simulation models were implemented in MATLAB®. 44

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Four different transfer functions (table 2) were utilized to compare the required number of calculations to determine the global extremum with a defined tolerance to the best known solution. For every transfer function the optimization was performed thirty times. As the results in table 1 show the HO only requires 6% to 69% invocations of the simulator in contrast to the standalone PSO. This reduces the execution time of an optimization run tremendous. Furthermore the HO was always able to detect the global extremum. The reason for the high number of necessary calculations for system three can be justified with the complex shape of the search space. The gradients around the global extremum are very high and therefore the location of the extremum is very small. Table 2. Overview of chosen test functions Calculations PSO Hybrid

Controlled System

Transfer function

1 (PT3)

K=1, a1 = 2, a2 = 2, a3 = 1

1737

342

2 (PT3)

K=1, a1 = 3.1, a2 = 2.3, a3 = 0.2

2007

459

3 (PT3)

K=1, a1 = 2, a2 = 2, a3 = 3

23459

1464

4 (PT2)

K=1, a1 = 3, a2 = 2

542

374

4. Hybrid optimizer

In this paper the HO is a combination of the global PSO and the local NM algorithm with the objective of better performance in comparison to a standalone optimization approach. The operation of the algorithm is the following: First the PSO is performed with a small number of calculations and then terminated. Hence, the PSO is only used for global exploration of the search space. Subsequently three instances from NM algorithm start from the best, the 3rd best and the 5th best point examined by the PSO realizing a local search. The solution of the optimization is the best result of the three NM instances (Figure 4). The reason behind starting the local search from different points is the robustness against local extremes. It has been investigated, that if only one instance is used, the hit rate of the HO to find the extremum is reduced [12].

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6. Conclusion

The combination of the PSO and NM to a hybrid optimizer increased the performance dramatically in comparison to the standalone PSO algorithm. The advantage of the HO is the switch from the well performing global optimization technique of the PSO to the NM, which is more effective in local exploration. Even with the three instances of the NM the required number of the calculations is still smaller. This is essential to enable online and real time applications. But furthermore investigations in adjusting the tuning parameter of the algorithms concerning the problem of controller parameterization must be carried out. Moreover even different optimization techniques e.g. the Newton’s method, genetic algorithms and different combinations to a hybrid optimizer must be investigated. Furthermore it is conceivable to use the methodology of simulation based optimization for tuning more complex systems like a controller cascade with filters.

Acknowledgements Funded by the European Union (European Social Fund) and the Free State of Saxony.

Journal of Automation, Mobile Robotics & Intelligent Systems

Authors

Reimund Neugebauer, Kevin Hipp*, Arvid Hellmich, Holger Schlegel – Chemnitz University of Technology, Faculty of Mechanical Engineering, Institute for Machine Tools and Production Processes, Reichenhainer Str. 70, 09126 Chemnitz, Germany; e-mails: wzm@mb.tu-chemnitz.de; kevin.hipp@mb.tu-chemnitz.de, arvid.hellmich@mb.tu-chemnitz.de, holger.schlegel@mb.tu-chemnitz.de *Corresponding Author

References [1] [2]

[3]

[4]

P. Köchel, “Simulation Optimisation: Approaches, Examples and Experiences”, TU Chemnitz, 2009, ISSN 0947–5125. Y. Carson, A. Maria, “Simulation Optimization: Methods and Applications“. In: Proceedings of the 1997 Winter Simulation Conference, 1997, pp. 118–126. E. Tekin, I. Sabuncuoglu, “Simulation optimization: A comprehensive review on theory and applications“, IIE - Transactions, vol. 36, no. 11, 2004, p. 1067. R. Neugebauer, K. Hipp, S. Hofmann, H. Schlegel, “Application of simulation based optimization methods for the controller parameterization con-

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sidering definable constraints”, Mechatronik 2011, 2011, pp. 247–252. [5] R. Eberhart, J. Kennedy, “A new optimizer using particle swarm theory“. In: MHS’95, Proceedings of the Sixth International Symposium, 1995, pp. 39–43. [6] R. Eberhart, J. Kennedy, “Particle Swarm Optimization“. In: IEEE International Conference on Neural Networks Proceedings, 1995, 1942–1948. [7] J. A. Nelder, R. Mead, “A Simplex Method for Function Minimization“, The Computer Journal, vol. 7, no. 4, Jan. 1965, pp. 308–313. [8] H. Schwefel, Evolution and Optimum Seeking, Wiley VCH, 1995, ISBN 0471571482. [9] J. Lunze, Regelungstechnik 1: Systemtheoretische Grundlagen, Analyse und Entwurf einschleifiger Regelungen, 8th Edition, Springer, Berlin, 2010, ISBN 9783642138072. [10] K. Aström, T. Hägglund, Advanced PID Control, ISA – The Instrumentation, Systems and Automation Society, 2006, ISBN 1556179421. [11] S. Hofmann, “Time-Based Parameter Identification and Controller Design for Motion Control Systems“. In: Conference Proceedings 55. IWK Ilmenau, 2010, pp. 404–415 [12] K. Hipp, Entwurf, Implementierung und Test eines Software-Werkzeuges zur Bestimmung optimaler und robuster Regler mittels Verfahren der simulationsbasierten Optimierung, Diploma Thesis, Chemnitz, 2010.

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Measuring of the Basic Parameters of LCD Displays Submitted 27th June 2011; accepted 2nd September 2011

Roman Barczyk, Błazej Kabzinski, Danuta Jasinska-Choromanska, Agnieszka Stienss

Abstract:

This article is about realization of the laboratory stand used for measuring parameters of LCD displays. The stand together with additional accessories (spectrophotometer and measuring probes) as well as software allow to measure a wide range of parameters (colour gamut, response time, contrast coefficient and its irregularity, luminance of black colour and its irregularity, luminance of white colour and its irregularity, changes of luminance depending on the viewing angle). This paper also show method and results of measurements of differences in contrast ratio and luminance uniformity between particular pieces of the same model of a display screen (research has been carried out on 5 pieces of the same model of a display). The presented results show that the measured value of the viewing angle may significantly differ from the value provided by producers of displays. Keywords: LCD, parameter, contrast ratio, view angle, luminance, measurement

1. Introduction

This article is about realization of the laboratory stand used for measuring viewing angles of LCD displays. There is a need to build such stands in order to check whether the parameters of displays reflect the ones provided by their producers. ISO norms guidelines as regards tests of displays (ISO 9241-303:2008 “Ergonomics of human-system interaction – Part 303: Requirements for electronic visual displays” and ISO 9241-305:2008 “Ergonomics of human-system interaction – Part 305: Optical laboratory test methods for electronic visual displays“) have been used to make sure the results of such research are reliable. The stand together with additional accessories (spectro-photometer and measuring probes) as well as software allow to measure a wide range of parameters (colour gamut, response time, contrast coefficient and its irregularity, luminance of black colour and its irregularity, luminance of white colour and its irregularity, changes of luminance depending on the viewing angle). In this article the construction of a stand has been described, as well as the method of measuring changes of luminance depending on the viewing angles, which is one of the most important parameters in the situation when there are several people looking at a certain screen. This paper also shows the method and the results of measurements of differences in contrast ratio and luminance uniformity between particular pieces of the same model of a display screen (research has been carried out on 5 pieces of the same model of a display). The presented results show that the measured value of the viewing angle may 46

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significantly differ from the value provided by producers of displays.

2. Reasons for independent measurement methods

Theoretically all LCD monitors seem to have parameters allowing to use them almost in any application. According to manufacturers specifications LCD monitors have wide view angles (over160o), contrast ratio at least 500:1, and 16 million colour, so average user should not be able to observe any difference between view angle of TN matrix and PVA or MVA matrix. Measurements used in technical specifications are usually made by LCD matrix manufacturers, not final monitor manufacturers. During such measurement procedure LCD matrices are placed on special platforms with selected light source powered by high quality power supply. Final result is when two different monitors equipped with the same kind of LCD matrix, but different light source and electronics, have different technical parameters.

3. View angle – luminance changes depending on view angle

According to standards maximal view angle is defined for a picture where contrast ratio in central point of monitor drops to value 10:1. However, the picture is distorted while contrast ratio drops to value 100:1, therefore real loss of picture quality is visible at smaller angles. Moreover, some producers accepted 5:1 as maximal contrast ratio to allow them expand theoretical view angle by approximately 20 degrees.

Measurement method

In order to get information about changes in observed picture while moving observer around the monitor, a measurement stand was designed and build. Measurement of view angle is based on contrast ratio factor:

CR =

Lmax Lmin

(1)

where: Lmax – maximal luminance display point, Lmin – minimal luminance display point.

Compatibility of measurement stand with ISO standard

To make our results comparable with values given by manufacturers, an assumptions according to ISO standards was made: •Optimal distance of monitor observation according to

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ISO 9241-303:2008 [1] is between 300 and 750 mm from central point of the screen. • According to ISO 9241-305:2008 [2]: − monitor which is being tested should be isolated from external light sources, − measurement device should be placed perpendicular to screen axis, going through centre of LCD matrix, measurements are made every 10 degrees or less, − monitor displays 100% white plane, − when measurements are done, luminance uniformity of measurement points is calculated using equation: uniformity =

Lmin ⋅ 100% Lmax

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Measurements

Analysis of acquired data is based on finding points of minimal and maximal luminance, but most important is average luminance of whole matrix (or selected region). Measurements of horizontal and vertical angles were made for 5 monitors (same manufacturer and model), to reveal differences between theoretically identical copies of the same monitor. Charts below show average values of whole monitor matrix.

(2)

Measurement stand

Digital SLR camera is mounted on a tripod, which allows to change height of camera from lower to upper edge of monitor. Camera with tripod is able to move around the monitor on previously set radius (range ±90˚from perpendicular position to matrix); motion is realized manually. Before measurements, according to ISO-9241-305:2008, few conditions should be fulfilled: • remove all dust from monitor matrix, • warm up monitor – luminance is stable after approximately 20 minutes, • preliminary parallel position of monitor and camera matrices • monitor calibration with spectrophotometer, • setting monitor to native resolution (physical matrix resolution), • external condition in closed premises, humidity 25– 85%, temperature 20–50oC, atmospheric pressure 860–1060 hPa.

Fig. 3. Chart of the dependence of gray-scale level on horizontal (up) and vertical (down) viewing angle

4. Uniformity of Contrast Ratio

In contrast ratio measurement procedure and its uniformity a set of two checkerboard. In place of white rectangles on one board, black ones are on the second.

Fig. 1. Measuring stand – schematic diagram

Fig. 2. Measuring stand – measuring angle’s range

Fig. 4. Test board Articles

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Measurements were made using electronic circuit OPT101 consisting of photodiode with trans-impedance amplifier in one structure. Electric current is proportional to light stream. Procedure is simple, but needs accuracy. First step is to display board on the left, and measure with probe each white field (measurement on centre of rectangle), next step is to display the second board and repeat measurement on the black fields. Results are measurements in 13 points for each board, and measurements were repeated five times to make estimation more accurate and certain.

Significant difference in contrast uniformity is observed, but average value of difference in uniformity is about 50%. It cannot be discussed without looking at results of contrast ratio for whole surface of matrix.

Measurements

5. Conclusions

Results in percents show distortion in uniformity of contrast ratio while setting backlight for 100% and 50% for group of 5 monitors (M1-M5). Tab.1. Contrast Ratio and contrast uniformity difference measurement results M1

Param [%]

100

Min CR

68,9

67,8

63,8

63,7

58,0

Max CR

127,9

105,5

113,5

115,3

110,3

∆CR

59,0

37,7

49,7

51,6

52,3

diff. CU

53,8

64,3

56,2

55,2

52,6

Param [%]

Min CR

57,9

69,9

64,3

63,8

50,0

Max CR

112,3

100,0

105,4

122,2

100,6

∆CR

54,4

30,1

41,1

58,4

50,6

diff. CU

51,5

70,0

61,0

52,2

49,7

Contrast uniformity: CU =

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Tab. 2. Contrast Ratio values

40:1

43:1

46:1

38:1

Measurement stand allow to measure LCD monitor parameters representative for image reproduction quality and comparing to values given by manufacturers. Our research shows that our results of viewing angle, as well as contrast ratio strongly differs from that published by manufacturers in technical specifications. Contrast uniformity and luminance uniformity are parameters that seem to be relevant in describing quality of monitor and are not included in official specifications. Significant differences in uniformity of white luminance are main source of differences in contrast ratio.

Authors:

Roman Barczyk – Warsaw University of Technology, Faculty of Mechatronics, 8 Św. A. Boboli Street, Warsaw, 02-525, Poland, r.barczyk@mchtr.pw.edu.pl Błażej Kabziński* – Warsaw University of Technology, Faculty of Mechatronics, 8 Sw. A. Boboli Street, Warsaw, 02-525, Poland, b.kabzinski@mchtr.pw.edu.pl Danuta Jasińska-Choromańska – Warsaw University of Technology, Faculty of Mechatronics, 8 Sw. A. Boboli Street, Warsaw, 02-525, Poland, danuta@mchtr.pw.edu.pl Agnieszka Stienss – Interactive Education Institute estakada.pl 80/82 Grzybowska Street Warsaw, 00-844, Poland, a.stienss@gmail.com *Corresponding author

References C min ⋅ 100% C max

(3)

where Cmin i Cmax are minimal and maximal measured value of CR.

N° 1

[1] ISO 9241-303:2008: Ergonomics of human-system interaction – Part 303: Requirements for electronic visual displays [2] ISO 9241-305:2008: Ergonomics of human-system interaction – Part 305: Optical laboratory test methods for electronic visual displays.

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Intelligent BRT in Tehran Submitted 27th June 2011; accepted 2nd September 2011

Peyman Parvizi, Sasan Mohammadi, Farzad Norouzi Fard

Abstract: An intelligent BRT system is necessary when communities looking for new ways to use high capacity rapid transit at a reduced cost. This paper will describe the intelligent control system that works with Datacenter. With the help of GPS system, the data center can monitor the situation of each bus and bus station. Through RFID technology, bus station and traffic light can transfer data with bus and by Wimax communication technology all of parts can talk together; data center learns all information about the location of bus, the arrival of bus in each station and the number of passengers in station and bus. Finally, the paper presents the case study of those theories in Tehran BRT.

2.2. Tehran BRT

In Iran, the BRT system has been implemented in Tehran. Tehran Bus Rapid Transit has been officially inaugurated by Tehranâ&#x20AC;&#x2122;s mayor in order to facilitate the motor traffic in Tehran on January 14, 2008. Tehran has five BRT lines. The first stretch of Tehran BRT corridor from Azadi square to Tehran-pars has been operational since Jan (2008). Table I. Teheran BRT Lines Line Status

Start point

End point

Stations

Distance (KM)

18.7

Azadi Terminal Tehran-Pars

Azadi Terminal

Khavaran Terminal

18.7

Elm-O- Sanat Terminal

Khavaran Terminal

14.3

Under Parkway construction

South Terminal

21.5

Under study

Dehkadeh Olympic

Under Babayee construction Highway

Under Railway station Tajrish construction

Under study Besat Highway Basij

2. Background

Under study Besat Highway

Babayee Highway

2.1. What is BRT

Under construction Ponak

Azadi Terminal

Keywords: Tehran BRT, RFID, intelligent transportation

1. Introduction

This paper presents a design of an intelligent BRT system to improve BRT service in Tehran, such as minimizing the delay of BRT buses by changing management on traffic lights and BRT stations that increasing speed and reducing travel time; that minimizes the count of active BRT buses that causes reducing the cost [1], [2]. In addition, a brief background of Tehran BRT can be found in section 2, the tools used are described in section 3, the methodology employed and the idea for improvement described in section 4 and conclusion can be found in section 5.

BRT is a high quality, high capacity rapid transit system that in many ways improves upon tradition rail transit systems. Vehicles travel in exclusive lanes, thus avoiding traffic. Passengers walk to comfortable stations, pay their fare in the station, and board through multiple doors like a train. Service is very frequent and often passengers can choose between express and local routes, an option not available on most train systems.

Fig. 1. BRT

Elm-O-Sanat Terminal

Ponak

19 27

17.5 6.2

The total length of BRT in Tehran is about 100 km that will be increased to 300 km in future.

Fig. 2. Tehran BRT Articles

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3. Tool used

power up and transmit a response. Most passive tags signal by backscattering the carrier signal from the reader. This means that the antenna has to be designed to both collect powers from the incoming signal and to transmit the outbound backscatter signal.

3.1. Automatic Passenger Count System (APC)

3.2.2. Semi-passive RFID tags Semi-passive RFID tags are very similar to passive tags except for the addition of a small battery. This battery allows the tag IC to be constantly powered. This removes necessity the aerial to collect power from the incoming signal. Therefore, Aerials can be optimized for the backscattering signal. Semi-passive RFID tags are faster in response and therefore stronger in reading ratio compared to passive tags.

The BRT buses and bus stations will be equipped with; APC (Automatic Passenger Count System), RFID (Radio Frequency Identification) and GPS (Global Positioning System) [3], [4]. APC is an advanced technology solution that helps transit improves. APC technology puts infrared sensor at the bus doors and bus station gates to count passengers as they board and leave.

Fig. 3. APC APC collects ridership data on every bus, per door per every bus station, and per every gate basis. Counted passengers are stored on the bus and bus station and downloaded into a data center where the information can be readily correlated to scheduled runs, routs, stops, time (arrive – leave ), date and destination.

3.2.3. Active RFID tags Active RFID tags or beacons, on the other hand, have their own internal power source, which are used to power any ICs and to generate the outgoing signal. They may have longer range and larger memories than passive tags, as well as the ability to store additional information sent by the transceiver. To economize power consumption, many beacon concepts are operated at fixed intervals. A RFID-Reader is also required on bus stations and traffic lights to receive the signal from the BRT buses. The RFID-Reader requires the exact position of the intersection to make decisions; the code in each RFID-Reader is unique [6], [7].

Fig. 4. Infrared sensor

3.2. Radio Frequency Identiﬁcation (RFID)

RFID is an electronic method of exchanging data over radio frequency waves. There are three major components for a RFID system: Transponder (Tag), Antenna and a Controller. RFID tags can be active, semi-passive (=semi-active) or passive [5].

Fig. 6. RFID

3.2. Global Positioning System (GPS)

RFID

Passive RFID tags have no internal power supply

Semi-passive RFID tags are very similar to passive tags except for the addition of a small battery

Fig. 5. RFID Types 3.2.1. Passive RFID tags Passive RFID tags have no internal power supply. The minute electrical current induced in the antenna by the incoming radio frequency signal provides just enough power for the CMOS integrated circuit (IC) in the tag to 50

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Active RFID tags have their own internal power source , which are used to power any ICs and to generate the outgoing signal

Fig. 7. GPS GPS has been developed by the United States’ Department of Defense. It uses between 24 and 32 Medium Earth Orbit satellites that transmit precise microwave signals. This enables GPS receivers to determine their current location, time and velocity.

Journal of Automation, Mobile Robotics & Intelligent Systems

WiFi

GPS

Power Usage Low to Medium

High

Medium

Data rate

Low to Medium

High

Not Applicable

Coverage

Medium

High

Very High

Security

Medium

High

Not Applicable 1. Station

Act signal

BUS

Station

Signal

This algorithm used with buses, bus stations and traffic lights, which have the hardware as described in Tools used. Modes of operation: the algorithm runs in different modes. First data center call a bus and the intelligent cycle will be started [8], [9].

Act signal

Act signal signal

2. traffic light

Act signal

BUS

Data entry

Traffic light

Signal

Bus Antenna

Fig. 10. A ﬂowchart illustrating the ﬂow of algorithm

4.2. Trafﬁc light mode

Light time Bus timing Emergency cars Number of passengers Bus numbers

Actuator system

signal

Data Center

Start

Traffic Light Antenna

2012

When the distance between bus and bus station is less than 50 meters, bus sends a signal to the bus station and bus station listens for BRT interrupts – capacity, direction, speed and number of passengers that bus sends then bus listens for BRT bus station interrupts – free gate number, capacity and number of passengers in station [10], [11].

4. Methodology employed

Bus Station Antenna

N° 1

4.1. Bus station mode

Table II. Compare (RFID&WiFi&GPS) RFID-active

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Bus Bus Station Traffic Light

When the distance between bus and traffic light is less than 50 meter, bus sends a signal to the traffic light and traffic light listens for BRT interrupts – capacity, direction, number of passengers then the bus listens for traffic light interrupts emergency mode, stop or keep riding [12], [13]. Li

Fig. 8. A ﬂowchart of system 1) Traffic light Act: •Green mode •Red mode •Idle mode 2) Bus Act: •Speed change •Stop •Acceleration mode •Deceleration mode •Passenger count 3) Bus Station Act: •Choice terminal for bus •Choice terminal for passenger •Passenger count

Bus passengers (BPL1,BPL2)// bus passenger line 1,2 BPL1= BP1+…+BPn BPL2= BP1+…+BPn BRT lines total (L1,L2)// BRT line1,2 Traffic light time (TLT2,TLT3,TLT4) Emergency cars lines (ECL1,ECL2,ECL3,ECL4) Street passengers (SPL1,SPL2,SPL3,SPL4) Traffic in rout (TL1,TL2,TL3,TL4) Traffic light color (green , yellow , red)// TLC=G,Y,R EC= emergency cars L1,2,3,4 = Line1,2,3,4

TLT=2

BLT=1

BPL1+ X > Y

YES

L1= Bus & EC L2 - EC

YES

TLC = Green for L 1 and after cross the last bus , it will send a signal to the traffic light and TLC=G Green for line 2 (L2)

L1= Bus L2 = EC

YES

YES TLC = Green for L 1 and after cross the last EC TLC=G Green for line2 (L2)

Yes NO

Fig. 9. Bus signals

L1= EC L2 = EC

TLC = Green for L 2 TL2 = Max

TLC = Green for L1 and after cross the last bus , it will send a signal to the traffic light and TLC=G Green for EC in line2 (L2)

TLC = Green for L 2 and after X min TLC = Green for line1 (L1) and Bus can move

YES

TLC = Green for L 1 and after cross the last bus , it will send a signal to the traffic light and TLC = Red for bus rout (L1)

YES

TL2 = Max

YES

L1=Bus & EC

B NO

EC<>0

TL2 = Max

No TLC = Green for EC in line2 (L2) then after cross it , TLC= Green for Bus & EC in line1 (L1)

Bus wait for X min

YES

TL2 = Max

TLC = Green for L 2 and after cross the last EC TLC=G Green for line 1 (L1)

TLC = Green for L 1 and after cross the last bus , it will send a signal to the traffic light and TLC =G Green for line 2 (L2)

Fig. 11. Trafﬁc Light Algorithm Articles

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5. Conclusion

With this research, this data will be accessible for Datacenter: 1) All of passengers in each station equal to all of passengers in waiting, 2) All of passengers in each bus in equal to of passengers in transfer, 3) Ability to analyze the critical stations, 4) Ability to Schedule traffic lights information, 5) Ability to record diary. To get a good result we need data that are without defects in determination centers. With given information above two important points are enforceable: 1) Just in time determination, 2) Future forecasting with diary data. Benefits: 1) Speed will be increased in BRT, 2) Cost of transformation will be decreased, 3) Time of delay to be decreased in stations, 4) Time of delay to be decreased behind traffic lights, 5) Passengers will be distributed among the buses.

Authors:

Peyman Parvizi* – Department of Mechatronics, Islamic Azad University South Tehran Branch, Tehran, Iran, St_p_parvizi@azad.ac.ir Sasan Mohamadi – Department of Mechatronics, Islamic Azad University South Tehran Branch, Tehran, Iran, s_mohammadi@azad.ac.ir Farzad Norouzifard – Department of Mechatronics, Islamic Azad University South Tehran Branch, Tehran, Iran, St_f_norouzifard@azad.ac.ir *Corresponding author

[4]

[5] [6]

[7]

[8]

[9]

[10]

[11] [12]

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Yin Zhu, “Study on Intelligent Traffic Control Based BRT”. In: Conf. Intelligent Systems and Applications (ISA), Beijing, China, May 2010. T. Gang, “The Application Study of Model and Algorithm on Bus Scheduling”, Journal of Qingdao University of Science and Technology, China, 2004. L. Da-ming, J. Xiao-jing, “General Introduction of Bus Rapicl Transport Intelligent System of Beijing

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South-Center Corridor”. In: Conf. Communication and Transportation Systems Engineering and Information, China, 2005. W.H. Hsieh, C.J. Ho, G.J. Jong, “Vehicle Information Communication Safety Combined with Mobile RFID”. In: Conf. Intelligent Information Hiding and Multimedia Signal Processing, China, 15th–17th August 2008. S. Inoue, H.Yasuura, “RFID privacy using usercontrollable uniqueness (Privacy Workshop)”, Nov. 2003. B. Jirong, S. Liya, L. Jinyuan, “Design of transportation safety management system for dangerous goods based on RFID”, CNKI:SUN:NXGJ.0.2010-03-017. X.H. Fan, Zhang, Y.L.A, “Design of bi-verification vehicle access intelligent control system based on RFID”. In: Conf. International Conference on Electronic Measurement & Instruments, Beijing, China, 16th–19th August, 2009. B.C. da Silva, R. Junges, D. de Oliveira, Bazzan, “An intelligent transportation system for urban mobility (Autonomous agents and multiagent systems)”, AAMAS’06, 2006, pp. 1471–1472. G. Kotusevski, K.A. Hawick, “A Review of Traffc Simulation Software”. In: Conf. Information and Mathematical Sciences, New Zealand, 23rd July, 2009. S. Wenchao X. Jianmin Y. Feng, “An Intelligent Transportation Simulation System Based on RFID Orientation Technology” , CNKI:SUN:JTJS.0.2009-01-031. J.-P. Hubaux, S. Capkun, J. Luo, “The security and privacy of smart vehicles”, IEEE Security and Privacy, 2004, pp. 49–55. Y. Sato, K. Makane, “Development and Evaluation of In-Vechicle Signing System Utilizing RFID Tags as Digital Traffic Signals”, Int. J. ITS, 2006, pp. 53–58. W. Hongjian, T. Yuelin, L. Zhi, “ RFID Technology Applied in Highway Traffic Management (Periodical style)”. In: Conference on Optoelectronics and Image Processing, vol. 2, November 2010, pp. 348–351.