Use of PSO in Parameter Estimation of Robot Dynamics; Part Two: Robustness

Transcription

1 Ue of PSO in Paraeter Etiation of Robot Dynaic; Part Two: Robutne Hoein Jahandideh, Mehrzad Navar Abtract In thi paper, we analyze the robutne of the PSO-baed approach to paraeter etiation of robot dynaic preented in Part One. We have ade attept to ake the PSO ethod ore robut by experienting with potential cot function. The iulated yte i a cylindrical robot; through iulation, the robot i excited, aple are taken, error i added to the aple, and the noiy aple are ued for etiating the robot paraeter through the preented ethod. Coparion are ade with the leat quare, total leat quare, and robut leat quare ethod of etiation. I. INTRODUCTION Invere dynaic of a robot i obtained in the following for [1]: τ = D ( qq ) + Cqqq (, ) + gq ( ) + Fignq ( ) + Fq (1) where τ denote the vector of force/torque applied to the robot joint, D i the anipulator inertia atrix, C i the corioli/centripetal atrix, g i the gravity vector, F c i the coulob friction and F v i the vicou friction. The vector q contain all the joint variable- length for priatic joint, and angle for revolute joint. Without knowledge of the ae, center of a, inertia paraeter, and frictional paraeter of the robot link, (1) i incoplete. Conventional ethod of robot paraeter etiation, factorize (1) into the for: c τ = Y ( q, q, q) α () where Y i the linear regreor and α i the et of bae paraeter. The bae paraeter are the iniu nuber of paraeter that influence the dynaic behavior of the robot. They ay be cobination of the a, inertia, friction, and gravity paraeter. Finding the et of bae paraeter i called paraeterization. In part one [], particle war optiization (PSO) wa ued to etiate the robot paraeter without the tep of paraeterization. In thi paper, we wih to exaine the perforance of the PSO approach to etiation, on the condition where the aple which are to be ued for etiation have relatively large error. Thi analyi will be perfored by coparion with the leat quare (LS), total leat quare (TLS), and robut leat quare (RLS) ethod. Mehrzad Navar i a faculty eber of the control yte group in the Electrical Engineering departent at Sharif Univerity of Technology, Tehran, Iran. He received hi PhD in Control yte in 001 fro the Grenoble Intitute of Technology (INPG) in France. navar@harif.ir Hoein Jahandideh i a tudent in the Electrical Engineering departent at Sharif Univerity of Technology, Tehran, Iran ( ) h.jahan@gail.co v All ethod are iulated for the etiation of the paraeter of a cylindrical robot. The cylindrical robot ha only four identifiable (influential) paraeter. The four paraeter are etiated by the four ethod and the etiated value are copared to the real paraeter value given to the iulated robot for excitation and apling. The LS, TLS, and RLS ethod require paraeterization of the invere dynaic, which we have thu provided and ued for thi purpoe. We have ade effort to ake iproveent to the PSO approach. A relevant reearch to thi paper i [3], which ha experiented with robut cot function for the PSO, to identify coplex nonlinear yte (not particularly robot). Siilarly, in thi paper we have experiented with potential cot function, though not the one teted in [3]. Unlike [3], in which all copared ethod are PSO-baed, we have ade coparion to the RLS ethod to achieve jutifiable reult. Previou reearch on reducing the effect of eaureent noie on robot paraeter etiation i found in [4] and [5]. The LS-baed ethod in [4] depend on data filtering, and data filtering depend on a very high apling frequency. Siilarly, a weighted leat quare (WLS) approach wa propoed in [5], which relie on data filtering and knowledge of the exact propertie of the noie. On the contrary, the TLS and RLS olution [6-9] are deigned for noiy condition; they do not rely on data filtering, and thu can be obtained by a liited nuber of aple (a long a the nuber of aple exceed the nuber of identifiable paraeter). Likewie, the PSO doe not rely on a high apling rate; hence in thi paper the PSO approach i copared to the TLS and RLS ethod rather than ethod baed on data filtering uch a in [4] and [5]. In thi paper we take a liited nuber of aple for our etiation and the propertie of noie are not conidered. The ret of thi paper i organized a follow: In ection, PSO i introduced. The leat quare ethod are introduced in ection 3. The iulated experient (including the introduction of the cylindrical robot) i explained in detail in ection 4. In ection 5, the reult of the iulation are uarized. Concluion are drawn in ection 6. II. PARTICLE SWARM OPTIMIZATION Particle Swar Optiization (PSO) i a war intelligence optiization algorith inpired by iulating bird flocking or fih chooling. Exaple of war and evolutionary algorith and their application are explained in an orderly anner in [10]. PSO wa firt introduced by Kennedy and Eberhart in [11]. The atheatical analye behind PSO were explained by Clerk and Kennedy in [1].

2 PSO ha been utilized in a wide range of cientific field (including robotic, control, and yte identification), exaple of which can be found in [3] and [13-15]. Let ƒ : R n R be the function to be optiized. Without lo of generality, we'll take our objective to be iniization. Objective: iniize ƒ(x) ubject to: xϵχ The contraint xϵχ can be efficiently erged with the function ƒ(x) [0]. PSO algorith ue a war of k particle a agent to earch for the optial olution in an n- dienional pace. The tarting poition of a particle i randoly et within the range of poible olution to the proble. The range i deterined baed on an intuitive gue of the axiu and iniu poible value of each coponent of x, but doen t need be accurate. Each particle analyze the function value (ƒ(p)) of it current poition (p), and ha a eory of it own bet experience (Pbet), which i copared to p in each iteration, and i replaced by p if ƒ(p)<ƒ(pbet). Aide fro it own bet experience, each particle ha knowledge of the bet experience achieved by the entire war (the global bet experience denoted by Gbet). Baed on the data each agent ha, it oveent in the i-th iteration i deterined by the following forula: V = w. V + C r ( Pbet p ) i i i i 1 + Cr( Gbet p ) i 1 where V i, P i, Pbet, and Gbet are n-vector (or iilar object, uch a atrice with n coponent), r 1 and r are rando nuber between 0 and 1, re-generated at each iteration. C 1 and C are contant poitive nuber, C 1 i the cognitive learning rate and C i the ocial learning rate. w i i the inertia weight, the iportance of which i coprehenively dicued in [16]. The new poition of each particle at the i-th iteration i updated by: i i 1 i (3) p = p + V (4) After certain condition are et, the iteration top and the Gbet at the latet iteration i taken a the optial olution to the proble. In thi paper, we let the PSO algorith end when the nuber of iteration reache a certain nuber. In [], we ued PSO for etiating robot paraeter by uing the invere dynaic function defined in robot iulation oftware (in our cae, [17]). Each aple we have of the robot dynaic contain the following data: τ ( ), q ( ), q ( ), q ( ), where τ i the n-vector i i i i of force/torque, and q i the tate of the n joint variable (n i equal to the degree of freedo of the robot). The index (i) i ued for the i-th aple. For each aple, baed on the etiated paraeter and the invere dynaic odel, a vector ˆ τ can be calculated by the cited oftware. Define E (i) for the i-th aple: E = τ ˆ τ ( i ) ( i ) ( i ) (5) Now define the error atrix E for which the i-th colun i E (i) : E = [ E E... E ] (1) () ( N ) where N i the nuber of aple available. The cot function for the PSO algorith i defined for the atrix E; for exaple (7). f ( E) = E The objective of the PSO algorith in our etiation tak i to find the et of paraeter that iniize the cot function ƒ. In thi paper, we wih to iprove the perforance of the PSO algorith in ter of robutne. Many contribution have been ade to odifying and iproving the PSO algorith (e.g. [18-0]). However, thee odification have been ade to the algorith itelf, while in our application, it i een that due to the aple error, the cot function of the real paraeter value i higher than that of the etiated value. Thi iplie that regardle of the odification ade to the algorith, the etiated value will not iprove unle the cot function i odified. Thu in thi paper we have experiented with different candidate for the cot function. The advantage of the PSO algorith over analytical ethod uch a LS i the flexibility of the cot function. The cot function ay be non-linear, non-convex, and/or non-differentiable. The cot function ay even be dicontinuou at certain point. III. THE LEAST SQUARES METHODS A. Leat Square The conventional ethod for robot paraeter etiation i the leat quare ethod and it derivative. [1] i a very old reference which introduce LS a a yte identification tool. Exaple of reearch which have baed robot paraeter etiation on LS are found in [4] and [-5]. In the LS approach, the invere dynaic are paraeterized a in (). The regreor Y i coputed for the data fro each aple to create aple of the regreor. If the aple-regreor for the i-th aple i denoted by Y (i), the obervation atrix W i defined by: (6) (7) T T T T... (1) () ( N ) (8) W = Y Y Y where N i the nuber of aple taken. τ i defined by: T T T T... (1) () ( N ) (9) τ = τ τ τ where τ (i) i an n 1 atrix containing the torque(force) of the i-th aple and N i the total nuber of aple. The LS etiation of α iniize ƒ LS defined by: f α = τ W α (10) ( ) LS It i known fro [4, 11, 1, 16, 1-5] that the olution to thi proble i given by:

3 1 ( T T α = W W ) W τ LS B. Total Leat Square The TLS olution [8, 9] ( the following forulae: αˆ TLS (11) ) iniize ρ defined by ( W +Δ W ) α = τ +Δ τ (1) LS [ W τ ] Δ= Δ Δ (13) ρ = Δ (14) F where the atrice W and τ are defined uch that there exit a unique vector of paraeter value α that olve (1).. F denote the Frobeniu nor of a vector or atrix (in thi cae a vector). It ha been hown in [9] that the TLS olution can be obtained by uing ingular value decopoition, which reult in the following olution: T 1 T α = ( W W σ I) W τ (15) TLS where σ denote the allet ingular value of the atrix [W τ ]. σ i zero if τ i linearly dependant on the colun of W (i.e. if there exit an α that olve Wα=τ ). C. Robut Leat Square The RLS ethod wa introduced in [6, 7]; it i known to be le accurate, while being ore robut (i.e. le riky) than the TLS ethod a it take into account the wort ituation that our eaureent data could have. The RLS olution ( α ) iniize r defined by the ˆRLS following forulae: Subject to: r = ax ( W +ΔW ) α ( τ +Δ τ) (16) [ ] F ΔW Δτ ρ (17) where ρ i the perturbation bound. If ρ i unknown (uch a conidered in our proble), the ρ obtained fro (14) by olving the TLS proble ay be ued in the RLS proble. The RLS proble can be forulated a a econd-order cone proble [7]: iniize λ ubject to: W τ α α λ γ, γ ρ ρ 1 (18) In thi paper the olution to thi proble i obtained with the help of CVX, a package for pecifying and olving convex progra [6, 7]. IV. SIMULATED EXPERIMENT The robot ued in our iulation i a cylindrical robot. The cylindrical robot i depicted in fig. 1. The link paraeter of a cylindrical robot, following the traditional notation, are given in table 1 []. Figure 1. A iple depiction of the cylindrical robot TABLE I. THE LINK PARAMETERS OF THE CYLINDRICAL ROBOT link nuber a (i)() α (i)(rad) d (i)() q (i) (i) θ 1 0 -π/ 0 d d 3 The invere dynaic equation of the cylindrical robot are given by []: τ = [( I + I + I ) + ( + d ) ] θ 1 1zz yy 3yy 3 3z 3 1 (19) + [ ( + d )] d θ 3 3z τ = ( + )( d + g) (0) τ 3 = d ( + d ) θ z 3 1 (1) where 3z denote the coponent of the center of a of the third link along it own z-axi; θ 1, d, and d 3 are the joint variable, i i the a of the i-th link, g i the gravity force, and I abc i the bc coponent of the oent of inertia of link a about it center of a. The frictional factor have been oitted for iplicity. (19-1) have been derived with the auption that link 1 i zero dienional and link and 3 each are one dienional figure. We have choen thi iple exaple to be able to eaily copare the etiation obtained by the different ethod. There are only four paraeter to be etiated;, 3, 3z, and Ι. Ι i defined a: I = I + I + I () 1 3 zz yy yy The invere dynaic of the cylindrical robot are paraeterized into the for () a: θ 0 dd θ + d θ d θ + d θ Y = 0 d + g d + g d d θ θ (3)

4 I + I + I + α = 3 3 3z 1zz yy 3yy 3 3z (4) Once α i etiated by a leat quare ethod, in order to copare the etiated paraeter to thoe etiated by PSO, the value of the four paraeter, 3, 3z, and Ι ut be extracted fro α. There are countle potential cot function that can be defined for the PSO algorith; here we will conider 16 of thee. It i poible that replacing the error atrix E (fro (6)) by E rel defined by the following equation ight yield better reult: Τ= τ τ... τ (5) (1) () ( N ) [ ] [ τ τ... τ (1) () ( N ) ] Τ= ˆ ˆ ˆ ˆ (6) E rel : E = ( Τ Τˆ ) Τ rel. The eight cot function conidered are: (7) f ( E ) = E (8) 1 F f ( E) = E (9) f ( E ) = E (30) 3 1 f ( E ) = E (31) 4 f ( E ) E E 5 1 = (3) f ( E ) E E E 6 1 = (33) f E = E (34) ( ) ax( ) 7 i j f ( E ) = ax( E ) (35) 8 i, j The ae eight cot function are defined for E rel a well (ƒ 9 to ƒ 16 repectively). It i notable that the CVX can be ued to olve the LS proble for the relative error, ubtituting the relative error vector for the abolute error vector in (10). TLS and RLS can alo be defined for relative error by replacing W and τ by: W Wr: Wr = (36) τ r( i ) ( i ) τ : τ = 1 (37) r Hence, we have alo obtained the relative LS, TLS, and RLS olution in our experient, denoted repectively by LS-rel, TLS-rel, and RLS-rel in table to 5. It ay be thought that ince the -nor, 1-nor, and infinity-nor are convex function, CVX can be ued to obtain the optial value for f, f3, and f4 (f10, f11, and f1). Thi i not true, due to that the LS ethod ue the error vector, wherea the PSO ethod ue the error atrix. If the error atrix i defined a a function of the error vector, and f, f3, or f4 (f10, f11, or 1) are ued on the reulting atrix, the total function i a non-convex function of the paraeter in α. For the PSO algorith, the learning rate of C 1 and C in (3) are both et to 1.3, and w i decreae linearly fro 0.9 to 0.4 through the firt 100 iteration and i et at 0.4 for the next 00 iteration (the total nuber of iteration are et at 300). The war population i et to 0. The reference trajectory ued for apling i planned according to the PSO-baed ethod preented in []. Saple are taken, rando error i given to the aple, and the ae aple are ued for all 11 ethod of etiation (i.e. the LS, TLS, RLS, and the 8 PSO ethod); the reult are copared. The phyical boundarie oberved by the planned trajectory are a follow: π θ ( ) π, 4 θ ( ) 4, 3 θ ( ) 3 rad rad rad d ( ) 1, d ( ), d ( ) 0 d ( ) 1, 1.5 d ( ) 1.5, 1 d ( ) 1 The PSO-planned trajectory i a follow: θ = 0.43in(. t) + 0.3in(1.8 t) 3.4in(0.06 t) ( t ) d = in(0.1 t) 0.3in(0.07 t) in(1.3 t) ( t ) d = 0.1in(0.1 t) 0.1in(.7 t) in(0.14 t) ( t ) (38) 0 t 10 (39) Table -5 how the etiated paraeter baed on different et of aple. In table, ten aple are ued; each tate eaureent (i.e. q, q, q ) of all aple i given a rando error of up to 0%. In table 3, 10 aple are ued and error of up to 0% are placed upon on all force/torque eaureent (i.e.τ ). Table 4 how the etiated value baed on 0 aple with error of up to 0% on all eaureent (both tate eaureent and force/torque eaureent). Table 5 i baed on aple which have up to 70% error on ten aple coponent (of total 10 aple coponent, i.e. 10(aple) 1(tate and force/torque)) and up to 5% error on the ret. The value tated a the PSO etiate are the ean etiate of ten PSO run, oitting all etiate with cot function that are relatively too large. The real value tated in the table are the value dictated to the iulated robot fro which the aple are obtained. V. DATA ANALYSIS It i een in table that for the cae where the eaureent error are on the tate aple only, uing the abolute error atrix (vector, for the LS ethod) rather

5 than the relative error atrix (vector) yield ore accurate reult. However, in table 3, where the error are on the force/torque eaureent only, table 4, where the error are on all data, and table 5, where few, very large error exit, for ot ethod uing relative error ee to reult in ore accurate etiation. Thu we conclude that uing relative error i ore reliable unle it i known that only the tate eaureent have notable error, and that all error are relatively all. TABLE II. ESTIMATED VALUES BASED ON THE FIRST SET OF SAMPLES 3-3z Ι coputation tie real value LS µ TLS µ RLS LS-rel µ TLS-rel µ RLS-rel PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ TABLE III. ESTIMATED VALUES BASED ON THE SECOND SET OF SAMPLES 3-3z Ι coputation tie real value LS µ TLS µ RLS LS-rel µ TLS-rel µ RLS-rel PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ TABLE IV: ESTIMATED VALUES BASED ON THE THIRD SET OF SAMPLES 3-3z Ι coputation tie real value LS µ TLS µ RLS LS-rel µ TLS-rel µ RLS-rel PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ TABLE V: ESTIMATED VALUES BASED ON THE FOURTH SET OF SAMPLES 3-3z Ι coputation tie real value LS µ TLS µ RLS LS-rel µ TLS-rel µ RLS-rel PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ PSO-ƒ The TLS ethod i deigned to be the ot accurate in the cae where the error are very all and alot negligible. A i een in table to 5, TLS i an untable ethod when the error grow; table 4 and 5 are epecially notable for their original TLS etiation being copletely off-liit. The RLS etiation, both the original and rel verion, are een to give better reult than their LS counterpart in nearly all cae. It i predictable and verified that PSO-f1 and PSO-f9 would reult in approxiately the ae etiate a LS and LS-rel repectively. In table, any of the PSO ethod have reulted in better etiation than RLS ethod, and in each table, there are one or ore PSO etiate which urpa the RLS-

6 etiate in ter of accuracy. However, a it i uncertain which cae of error correpond to our data, we ut chooe a PSO cot function that urpae the RLS-rel etiate in alot all cae. f13 ee to be the function we eek. f13 conider both the infinity-nor and the one-nor, and wa thu hypotheized and verified to be a robut ethod. f13 i a non-convex function, thu analytical ethod baed on convex optiization cannot be ued to iniize the function (particularly with the help of the CVX oftware); thi i one of the advantage of the PSO approach. The -nor by itelf (PSO-f10) doe not ee to be a robut ethod (ee table and 4 where the reult of RLS-rel greatly urpae that of PSO-10) however, adding the - nor to f13 to build f14, in oe cae ee to cancel out the deficiencie of f13. An average of the reult of PSO-f13 and PSO-f14 can be taken a the final etiation, which i een to be ore accurate than the RLS-rel ethod in all four cae of error type. All the idea ipleented were ade poible thank to the flexibility of PSO, which i another advantage of the PSO approach. VI. CONCLUSION In thi paper, we have analyzed the perforance of the nonlinear particle war optiization (PSO) approach preented in the previou paper, in ter of robutne toward error on eaureent aple. 8 PSO ethod (differed by their cot function) a well a the leat quare (LS), total leat quare (TLS) and robut leat quare (RLS) ethod were ued to etiate the paraeter of a iulated cylindrical robot for 4 type of error condition. Both the relative force/torque error and the abolute force/torque error were conidered and copared. It wa een that the conideration of the relative error, rather than the abolute error, yield ore reliable etiation. The etiation reult how that aide fro the iplicity of the ipleentation of the PSO ethod a decribed in [], the PSO approach ha the advantage of being flexible and able to iniize non-convex cot function, which allow the uer to tune the PSO ethod to ake it ore robut than the RLS ethod. In particular, it wa een that f13, which utually conider the 1-nor and the infinite-nor of the relative error atrix, i a robut ethod, ade ore robut by uing it in conjunction with f-14, which conider alo the -nor of the relative error atrix. There i uch pace for experienting and changing the PSO to ake it ore robut, experienting with idea which have not been ipleented in thi paper. REFERENCES [1] De Luca A. and Ferrajoli L., A Modified Newton-Euler Method for Dynaic Coputation in Robot Fault Detection and Control, IEEE Intern. Conf. Robotic and Autoation, Kobe, Japan, May 009. 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