Use of PSO in Parameter Estimation of Robot Dynamics; Part One: No Need for Parameterization

Transcription

1 Use of PSO in Paraeter Estiation of Robot Dynaics; Part One: No Need for Paraeterization Hossein Jahandideh, Mehrzad Navar Abstract Offline procedures for estiating paraeters of robot dynaics are practically based on the paraeterized inverse dynaic odel. In this paper, we present a novel approach to paraeter estiation of robot dynaics which reoves the necessity of paraeterization (i.e. finding the iniu nuber of paraeters fro which the dynaics can be calculated through a linear odel with respect to these paraeters). This offline approach is based on a siple and powerful swar intelligence tool: the particle swar optiization (PSO). In this paper, we discuss and validate the ethod through siulated experients. In Part Two we analyze our ethod in ters of robustness and copare it to robust analytical ethods of estiation. I. INTRODUCTION The inertial and friction paraeters of a robot are norally needed for exact coputed torque control of the robot. There is a wealth of literature on offline ethods of estiating the paraeters of robots' dynaical odel (e.g. [1-4]). However, ost existing research on this subject is based on paraeterization. Exaples of ethods for obtaining the dynaics of a robot can be found in [5]. The basic forulation of a robot s inverse dynaics has the following for [5]: τ = D ( q) q + C( q, q ) q + g( q) + Fsign( q ) + Fq (1) where τ denotes the vector of forces/torques applied to the robot s joints, D is the anipulator inertia atrix, C is the coriolis/centripetal atrix, g is the gravity vector, F c is the coulob friction and F v is the viscous friction. The vector q contains all configuration variables (displaceent for prisatic joints and joint angles for revolute joints). Paraeterization refers to the linear factorization of (1) as: c τ = Y( qqq,, ) α (2) where Y is the linear regressor and α is the set of base paraeters. The base paraeters are the iniu nuber of paraeters that influence the dynaic behavior of the robot. They ay be cobinations of the ass, inertia, friction, and gravity paraeters. An efficient procedure for reaching this factorization is clearly explained in [6]. Such paraeterization akes it possible for analytical ethods to the paraeters. The Least Squares ethod [2, 4], for exaple, which is the ost widely used Mehrzad Navar is a faculty eber of the control systes group in the Electrical Engineering departent at Sharif University of Technology, Tehran, Iran. He received PhD in Control systes in 2001 fro the Grenoble Institute of Technology (INPG) in France. navar@sharif.ir Hossein Jahandideh is a student in the Electrical Engineering departent at Sharif University of Technology, Tehran, Iran ( ) hs.jahan@gail.co v estiation ethod in the literature, depends on linear paraeterization. The goal of this paper is to present a ethod which can the values of the actual physical paraeters (i.e. center-of-ass, inertia paraeters, etc.) rather than the cobinations of the, without getting involved with the paraeterization procedure. This ethod is based on the particle swar optiization (PSO). Swar and evolutionary algoriths are optiization tools that are inspired by natural phenoena. Exaples of swar and evolutionary algoriths and their various applications have been discussed in an orderly anner in [7]. The PSO, in particular, was otivated by the siulation of bird flocking or fish schooling. PSO was first introduced by Kennedy and Eberhart in [8], and analyzed thoroughly by Clerk and Kennedy in [9]. As an optiization technique, PSO found way into nuerous applications throughout the years. Exaples of the applications of PSO in robotics are discussed in [3, 7]. PSO quickly found way into syste identification probles. [10], for exaple, presented a ethod based on artificial neural networks and PSO for the identification of a general dynaical syste. However, in [10] and siilar papers, identification of a syste requires building a odel to predict the input-output behavior of the syste, while in robots, the exact for of the dynaical odel is obtained fro the kineatics of the robot according to [5]. If the kineatics is uncertain, there are ethods to the kineatics; [11] for exaple, presented a PSO approach to the kineatics estiation. This being said, the exact for of a robot s dynaical odel is known, and strong analytical ethods exist that can the values of the paraeters in the odel. Hence, PSO has rarely been used in estiating the dynaical odel paraeters of a robot. However, [3] applies PSO to estiating the dynaical odel paraeters of the first three links of a Staubli RX-60 Robot and copares the ethod to the conventional least squares ethod. In [3], paraeterization is used as a preliinary step to estiating the paraeters through PSO. In this paper, this preliinary step (paraeterization) is reoved, and siulation results are shown to support our ethod. We have siulated the estiation of all paraeters of the three links of a cylindrical robot to deonstrate the siplicity and efficiency of our approach. It is iportant to note that all offline ethods of dynaic odel paraeter estiation utilize experiental saples, thus the way we obtain these saples (i.e. the way we excite our syste to obtain these saples) plays an iportant role in the accuracy and reliability of our estiation. In this paper, we have used PSO not only in the estiation procedure, but also in our trajectory planning.

2 The rest of this paper is arranged as follows: In section II, the PSO algorith is explained. Section III describes how PSO can be used without the preliinary step of paraeterization and section IV notes the advantages of using this ethod. Section V introduces the cylindrical robot and the paraeters to be d. Section VI explains the application of PSO in planning an acceptable excitation trajectory. Section VII is dedicated to a siulated experient. Finally, section VIII concludes the paper. II. PARTICLE SWARM OPTIMIZATION Let ƒ: R n R be the function to be optiized. Without loss of generality, we ll take our objective to be iniization. Objective: iniize ƒ(x) The PSO algorith assigns a swar of k particles to search for the optial solution in an n-diensional space. The starting position of a particle is randoly set within the range of possible solutions to the proble. The range is deterined based on an intuitive guess of the axiu and iniu possible values of each coponent of x, but doesn t need be accurate. Each particle analyzes the function value (ƒ(p)) of its current position (p), and has a eory of its own best experience (Pbest), which is copared to p in each iteration, and is replaced by p if ƒ(p)<ƒ(pbest). Besides its own best experience, each particle has knowledge of the best experience achieved by the entire swar (the global best experience denoted by Gbest). Based on the data each agent has, its oveent in the i-th iteration is deterined by the following forula: V = w. V + C r( Pbest p ) i i i i 1 + Cr( Gbest p ) 2 2 i 1 where V i, P i, Pbest, and Gbest are n-vectors (or siilar objects, such as atrices with n coponents), r 1 and r 2 are rando nubers between 0 and 1, re-generated at each iteration. C 1 and C 2 are constant positive nubers, C 1 is the cognitive learning rate and C 2 is the social learning rate. w i is the inertia weight. The new position of each particle at the i-th iteration is updated by: i i 1 i (3) p = p + V (4) After certain conditions are et, the iterations stop and the Gbest at the latest iteration is taken as the optial solution to the proble. In this paper, we let the PSO algorith end when the nuber of iterations reaches a certain nuber III. PSO IN PARAMETER ESTIMATION OF ROBOT DYNAMICS Each saple we have of the robot dynaics contains the following data: τ, q, q, q, where τ is the n-vector of ( i ) ( i ) ( i ) ( i ) forces/torques, and q is the state of the n joint variables (n is equal to the degrees of freedo of the robot). The index (i) is used for the i-th saple. For each saple, based on the d paraeters and the inverse dynaics odel, a vector ˆ τ can be calculated. Referring to (1), we have: ˆ τ = D ( q ) q + C ( q, q ) q ( i ) est ( i ) ( i ) est ( i ) ( i ) ( i ) + gq ( ) + F signq ( ) + F q c v ( i ) est ( i ) est ( i ) where D est, C est, F vest, and F cest are calculated according to the d paraeters. Thus, for every set of potential s of the paraeters, we have a τ and a ˆ τ. Define e (i) for the i-th saple: ( i ) ( i ) ( i ) (5) e = τ ˆ τ (6) Now define the atrix e for which the i-th colun is e (i) : [... (1) (2) ( N ) ] E = e e e (7) where N is the nuber of saples available. The cost function for the PSO algorith is defined for the atrix E. An analysis of what function to define in this stage is presented in Part Two. In this paper we define the cost function as: f = E (8) ( e ) 2 The objective of the PSO algorith in our estiation task is to find the set of paraeters that iniizes the cost function ƒ. Software for siulating robot dynaics (both sybolic and nueric) are introduced in [12]. The software used in our siulation, is Robotics Toolbox for Matlab [13]. This software, just as any other robotics software, can nuerically and efficiently calculate the inverse dynaics of a robot; eaning that given the kineatics of the robot links, and a set of potential s for the ass, center of ass, inertia, and frictional paraeters, a odel of the robot is siulated which can perfor the inverse dynaics function of the robot. Thus for every set of potential s of the paraeters, a cost function can be calculated via (6-8). If the position of each particle in the PSO swar is defined as an of the robot paraeters, the PSO algorith can find the which iniizes (8). The algorith was prograed in Matlab software: For each particle, the toolbox siulates the inverse dynaics of the robot and uses it on all saples to calculate ˆ τ ( i ); the function value of the particle is then calculated through (6-8); Pbest and Gbest are updated and the next ove is obtained fro (3). As can be seen, this ethod does not require the paraeterization step. IV. ADVANTAGES When the need for paraeterization is oitted, each particle of the PSO swar can directly represent the ass, center of ass, inertia, and frictional paraeters, rather than cobinations of the. This way, a physical insight is given of the characteristics of the robot. When paraeterization is required, the paraeters becoe very coplicated as the degrees of freedo increase. Even if we use software such as ScrewCalculus

3 [14] to accoplish the sybolic paraeterization task, the user ust organize the sybolic paraeters to prepare the for estiation via nueric algoriths, which can be a frustrating task, considering the coplicated sybolic analysis. To the coponents of α (defined in (2)) via a conventional ethod of estiation, the atrix Y ust be encoded. For ore coplicated robots and higher degrees of freedo, Y and α becoe too coplicated for analysis. After the estiation procedure, the robot odel is ready to be used for control purposes, for either odel based or non-odel based control ethods. For control ethods which require only the prediction of the input-output behavior of the odel, a robot can be siulated in the sae software used for estiation, which siulates the input-output behavior of the robot. The identification procedure results in a siulation odel of the robot fro which the D, C, and g atrices defined in (2) can also be calculated by the robot siulation software, given the state of the syste. Thus, control ethods which require calculation of the entioned atrices can also be ipleented after this siple paraeter estiation procedure is carried out. V. THE CYLINDRICAL ROBOT We have used the cylindrical robot in our siulations for siplicity of presentation. The link paraeters of a cylindrical robot are shown in table 1 [4]. TABLE I. THE LINK PARAMETERS OF THE CYLINDRICAL ROBOT link nuber a (i)() α (i)(rad) d (i)() q (i) (i) θ π/2 0 d d 3 The paraeters that constitute the robot dynaics are as follows: s ia is the coponent of the center of ass of the i-th link along its own a-axis (a could be x, y, or z); i is the ass of the i-th link, and I iab is the ab coponent of the oent of inertia of the i-th link about its center of ass. f c and f v are the coulob and viscous frictions respectively. If the links are treated as one diensional figures and the frictional factors are oitted (as they are in [4], and in section VI of this paper), this robot has only 4 identifiable paraeters ( 2, 3, s 3z, and I 1zz +I 2yy +I 3yy ). For a ore realistic siulation, such assuptions have not been ade in section VII of this paper. VI. EXCITATION TRAJECTORY Defining a proper trajectory by which to excite a dynaical syste for sapling purposes, plays an iportant role in the accuracy of the estiation based on the obtained saples. If part of a syste is not excited, the paraeters pertaining to the dynaics of that part will not be identifiable. Siilarly, if part of a syste is insufficiently excited, sall errors on the easureent saples can cause large errors in the d values of the paraeters pertaining to the insufficiently excited syste variables. The excitation trajectory ust be planned in a way that the physical constraints on the syste are observed (if they are not, the planned trajectory will not be executable) and all state variables are properly excited. Much research has been carried out on planning excitation trajectories for robots. Particularly, [15] uses the genetic algorith to obtain an optial trajectory for excitation. In [15], it is atheatically proven that the larger the deterinant of the square atrix W T R v -1 W, the less is the error of our least squares estiation; where R v is the autocorrelation atrix of the error vector τ sa -τ est, and W is built by cobining all the Y (i) obtained fro easureent saples (Y is the regressor defined in (2)). Y Y W = Y ( q1, q 1, q 1) ( q2, q 2, q 2)... ( qn, qn, qn ) When R v is unknown, it can be replaced by the identity atrix, thus the objective of the genetic algorith becoes axiizing the deterinant of the square atrix W T W. H= W T W (10) Objective: Maxiize H Subject to physical constraints on the joint variables A perceivable explanation of this theore is that in order to axiize H, the coponents of the regressor ust be large as well as properly varied, thus the syste states are well excited. Thus, a cognitive hypothesis is that if W is replaced by Q sa as defined below, the obtained trajectory will reain an efficient excitation. If the hypothesis is correct, we can obtain an efficient trajectory without paraeterization. Q sa q q q q q q... q q q q q q q q q... q q q =... q q q q q q... q q n 1n 1n n 2n 2n N 1 N 1 N 1 N 2 N 2 N 2 Nn Nn (9) (11) where q ij is the j-th joint variable of the i-th saple. We replace the genetic algorith by PSO and ust follow the following procedure for obtaining the desired trajectory: 1- Define: q j =a 1j sin(ω 1j t)+ a 2j sin(ω 2j t)+ a 3j sin(ω 3j t) j=1,2,,n (12) (n is the degrees of freedo of the robot.) 2- Define a kj and ω kj as the paraeters deterined by PSO. (6n variables in total)

4 3- Calculate the saples (defined by (12)) at the ties T i ( i = 1,..., N ) and place the inside Q sa. N (T is the sapling tie and N is the nuber of desires saples.) 4- Define : T 40 = ( ).10 (13) Q sa sa H abs Q Q a where a is a binary value, such that it is 0 if all saples eet the physical constraints, and is 1 if any coponent of any saple fails to eet the physical constraints. The function abs denotes the absolute value function. Note that it is possible that none of the saples break the constraints, while the trajectory does break the constraints at soe point in tie. Thus, we ust progra the constraints slightly above their lower liits and slightly below their higher liits. 5- Use the PSO algorith to axiize H Q and deterine the values of a kj and ω kj. (k=1,2,3 j=1,2,,n) We will test our hypothesis through a siulated experient. Consider the exaple of the cylindrical robot. Consider the following constraints to siplify the exaple: 0 q 1 1 q, q 1 j j j ( j = 1,2,3) (14) The starting point for the joint variables is considered to be: (q 1, q 2, q 3 ) = (0.5, 0.5, 0.5) The PSO paraeters of the trajectory planned by the above procedure and two rando trajectories (which eet the constraints) are given in table 2. These trajectories were used as reference trajectories to obtain easureent saples for each. The three sets of saples were affected by the sae noise disturbance. The PSO algorith for paraeter estiation of the robot s dynaics was run 5 ties for each set of saples and the results are copared in table 3. It is seen in table 3 that the estiation derived fro the saples of the obtained trajectory are very close to the real values, even though the saples had errors. The algorith reached the exact sae s in all its 5 runs. For the rando trajectories, we see a large variance in the d values, and the perturbations have caused relatively large errors on the average d values. Our hypothesis is validated by this siulation. This ethod of trajectory planning is used in the next section of this paper and throughout "part two". TABLE II. THE PLANNED AND RANDOM TRAJECTORY PARAMETERS PSO planned trajectory Rando trajectory 1 Rando trajectory 2 a ω a ω a ω a ω a ω a ω a ω a ω a ω TABLE III. ESTIMATED VALUES BASED ON ONE PLANNED TRAJECTORY AND TWO RANDOM TRAJECTORIES Planned Rando 1 Rando 2 Real values average ax-in average ax-in average -s 3z ax-in - s 3z average I 1zz+I 2yy+I 3yy in-ax I 1zz+I 2yy+I 3yy average cost function VII. SIMULATION RESULTS In order to achieve a realistic siulation, all physical paraeters have been considered and d. The trajectory suggested by the PSO algorith (16) is taken as the reference trajectory in our sapling operation. All sapling data have been given a rando error of up to 10% of their total value. The PSO paraeters used in our siulation and the average coputational tie for each run are suarized in table 4. The algorith was run 10 ties; the results are shown in tables 5-9, and the unidentifiable paraeters have been disclosed. A paraeter is classified as unidentifiable if there is too uch variation in its d value. Note that all real values of the robot paraeters and the physical constraints are dictated to the siulated robot; the siulated robot in our experient is not eant to odel a specific counterpart in the real world. TABLE IV. PSO PARAMETERS USED IN SIMULATION AND AVERAGE CPU TIME swar population nuber of c 1, c 2 w i average CPU tie iterations seconds

5 The physical constraints to be observed by the trajectory planning algorith are assued to be as follows: π θ ( ) π, 4 θ ( ) 4, 3 θ ( ) 3 rad rad rad 1 1 s 1 s 2 2 s 2 s 3 3 s 3 2 s 0 d ( ) 1, 2 d ( ) 2, 2 d ( ) 2 0 d ( ) 1, 1.5 d ( ) 1.5, 1 d ( ) 1 The following trajectory was planned for the experient: 1( t ) 2( t ) 3( t ) (15) θ = 0.97 sin(1.15 t) sin(1.1 t) sin(0.42 t) 2.63 d = 0.96sin(0.57 t) sin(2.05 t)-1.1sin(0.12 t) d = -2.3sin(0.07 t) sin(1.5 t) sin(0.38 t) t 10 (16) It is iportant to note that once the unidentifiable paraeters are recognized, in siulating the robot for control or siilar purposes, we are not free to set the values of the unidentifiable paraeters to arbitrary values. As an exaple, in the cylindrical robot, I 1zz, I 2yy, and I 3yy, all are unidentifiable paraeters. The suation of these paraeters, however, is crucial to the robot dynaics. We will define such paraeters as sei-identifiable (SI). Soe paraeters, such as 1 in this exaple, are not copletely absent in a robot s dynaics, but have such little effect that renders it unidentifiable. We will define such paraeters as nearly unidentifiable (NUI). For siulation purposes after paraeter estiation, all SI and NUI paraeters ust be set in accordance with the suggestion of the PSO algorith. In tables 5-7, SI or NUI paraeters have been recognized by the following procedure: A paraeter classified as not identifiable by the PSO database is classified as SI/NUI if the total cost function is affected by the variation of that specific paraeter while all other paraeters are kept unchanged; otherwise the paraeter is classified as unidentifiable. It is difficult to identify whether a paraeter is SI or NUI, but it can usually be said that a relative variation in the d values for the SI paraeters cause larger variation in the cost function than does the sae relative variation in the d value of an NUI paraeter. For further verification, the ean d values (including of the SI, NUI, and UI paraeters) have been used to siulate a second robot (aside fro the siulation representing the real robot) and the following reference trajectory was given to both robots to copare the resulting force/torques and the results are shown in figures 1 to 3: θ 1( t ) = 1.97sin(0.5 t ) sin(2.2t) sin(0.9t)-2.7 d2( t ) = 0.6sin(1.7 t) 0.3sin(1.45 t) sin(0.7 t)-0.06 d3( t ) = 0.4sin(0.3 t) + 0.4sin(1.3 t) sin(1.2 t) t 10 (17) paraeter TABLE V. SIMULATION RESULTS FOR LINK 1 real ean Coefficient ax-in value of status (I, UI, SI) Variation M NUI -s x NUI -s y NUI -s z UI I xx UI I yy UI I zz SI I xy UI I yz UI I xz UI f c SI/NUI f v SI/NUI TABLE VI. SIMULATION RESULTS FOR LINK 2 paraeter real value ean Coefficient of ax-in status (I, UI, SI) Variation M I -s x NUI -s y NUI -s z NUI I xx UI I yy SI I zz UI I xy UI I yz UI I xz UI f c SI/NUI f v SI/NUI TABLE VII. SIMULATION RESULTS FOR LINK 3 paraeter real value ean Coefficient of ax-in status (I, UI, SI) Variation M I -s x I -s y NUI -s z I I xx UI I yy SI I zz UI I xy UI I yz UI I xz UI f c SI/NUI f v I As seen in tables 5-7, unidentifiable paraeters have been recognized and acceptable s of the identifiable paraeters have been given. The d paraeters are verified for being acceptable by the result of a verification siulation (figures 1 to 3). The sae algorith ay be used to the paraeters of any industrial robot, given the robot s link paraeters and saples obtained fro an acceptable (though not necessarily perfect) excitation trajectory. The perforance of the d robot is verified by figures 1 to 3. It is seen that even though soe of the ean estiations in tables 5-7 differ greatly fro their real values, the behavior of the d robot follows closely the behavior of the real robot. This confirs that the paraeters classified as SI/NUI or UI have been correctly discovered.

6 Figure 1. Torque coparison for the verification trajectory; Joint 1. solid line: real robot; dashed line: d robot VIII. CONCLUSION In this paper, a tie-efficient, cost-effective, easily ipleented, and flexible ethod based on particle swar optiization was applied to the paraeter estiation of robot dynaics. As shown through siulation on a cylindrical robot, this ethod is easily executable on any industrial robot with any nuber of degrees of freedo. With this ethod, any user with access to robot siulation software can identify a robot and prepare it for control or other purposes without getting involved with paraeterization. In order to copletely avoid paraeterization, the excitation trajectory was also planned based on a PSO approach which requires only the link paraeters and the physical constraints of the joint variables. In part 2, this ethod of robot paraeter estiation is copared to least squares, total least squares, and robust least squares ethods in ters of robustness toward relatively large errors in the saple data. Figure 2. Torque coparison for the verification trajectory; Joint 2. solid line: real robot; dashed line: d robot Figure 3. Torque coparison for the verification trajectory; Joint 3. solid line: real robot; dashed line: d robot REFERENCES [1] Basilio B. and Aldo C., Identification of Industrial Robot Paraeters for Advanced Model-Based Controllers Design, IEEE International Conference on Robotics and Autoation, p , April [2] Chan S., An Efficient Algorith for Identification of Robot Paraeters Including Drive Characteristics, Journal of Intelligent and Robotic Systes 32: , [3] Bingul Z. and Karahan O., Dynaic identification of Staubli RX-60 robot using PSO and LS ethods, Expert Systes with Applications, Volue 38, [4] Khosla P. and Kanade T., Paraeter Identification of Robot Dynaics, Proc. 24th Conf. Decision and Control, Deceber [5] De Luca A. and Ferrajoli L., A Modified Newton-Euler Method for Dynaic Coputations in Robot Fault Detection and Control, IEEE Intern. Conf. Robotics and Autoation, Kobe, Japan, May 12-17, [6] Gabiccini M., Bracci A., Artoni A., Direct Derivation of the Dynaic Regressors for Serial Manipulators with Mixed Rigid/Elastic Joints XX Congresso AIMETA, Bologna, Italy (2011). [7] Mostajabi T., Poshtan J., Control and Syste Identification via Swar and Evolutionary Algoriths, International Journal of Scientific and Engineering Research Volue 2, Issue 10, October [8] Kennedy J. and Eberhart R., Particle Swar Optiization, Proceedings of IEEE International Conference on Neural Networks, Piscataway, NJ. pp , [9] Clerc M. and Kennedy J. The Particle Swar: Explosion, Stability, and Convergence in a Multi-Diensional Coplex Space, IEEE Transactions on Evolution Coputer, 6(1): 58-73, [10] Okar S. and Mudigere D. Non-Linear Dynaical Syste Identification Using Particle Swar Optiization, Proc. Int. Conf. Advances in Control and Optiization of Dynaical Systes, [11] Barati M., Khoogar A., and Nasirian M., Estiation and Calibration of Robot Link Paraeters with Intelligent Techniques, Iranian J. Electrical and Electronic Eng., Volue 7, No. 4, Dec [12] Toz M., Kucuk S., Dynaics Siulation Toolbox for Industrial Robot Manipulators, Coputer Applications in Engineering Education, Volue 18, Issue 2, pages , June [13] Corke P., Robotics TOOLBOX for MATLAB, CSIRO Manufacturing Science and Technology, [14] Gabiccini M., ScrewCalculus: a Matheatica Package for Robotics, DIMNP, University of Pisa, [15] Calafiore G., Indri M., and Bona B., Robot Dynaic Calibration: Optial Excitation Trajectories and Experiental Paraeter Estiation, J. Robotic Systes, Vol. 18, Issue 2, pp , Feb