Optimal Path Planning for Flexible Redundant Robot Manipulators

Transcription

1 25 WSEAS In. Conf. on DYNAMICAL SYSEMS and CONROL, Venice, Ialy, November 2-4, 25 (pp ) Opimal Pah Planning for Flexible Redundan Robo Manipulaors H. HOMAEI, M. KESHMIRI Deparmen of Mechanical Engineering Isfahan Universiy of echnology IRAN Absrac: - Vibraion is one of he mos imporan resources of error in moion of ip of flexible robo manipulaors. Alhough much work has been done in he design of conrollers for flexible manipulaors o follow a specified ip rajecory, i has been done a lile work in rajecory planning iself. For redundan robo manipulaors, rajecory planning can be accomplished wih he aim of opimizing some objecive funcions while racing along a given ip rajecory. In he presen work, he opimal rajecory-planning problem for a flexible redundan manipulaor will be formulaed as a wo poin boundary value problem (PBV) by globally minimizing he elasic deformaion of flexible links during he moion. Key-Words: Manipulaor - Redundan - Flexible - Vibraion - Opimizaion - Pah Planning 1 Inroducion Precise conrol of a robo manipulaor is an imporan aspec of robo performance. In he pas, his has led o he design of heavy and siff links and large moors. Hence mos exising manipulaors have very low payload o oal weigh raio. However, o increase produciviy by fas moion and o have less energy consumpion, robo arms are required o have ligh and consequenly flexible srucures. he main difficuly which arises from flexible srucures is vibraion. his vibraion leads o an error in moion of he manipulaor ip along a given rajecory and especially afer reaching he manipulaor o he desired end goal poin. Since his vibraion needs addiional seling ime before any ask can be begun, an effecive mehod should be employed o reduce he vibraion and srain energy of he robo manipulaor. Alhough here has been much work in he area of conrolling flexible manipulaors o follow a desired ip rajecory [1], lile work has been done in choosing he rajecory iself. Such a choice is necessary eiher for poin o poin moion or for resolving he redundancy of robos wih more degrees of freedom han necessary o follow he given ip rajecory. Meirowich and Chen used LQR conrol o reduce he elasic disurbance for a flexible redundan robo [2]. Eisler e al. considered join orque bounds and deermined minimum ime rajecories o minimize ip racking error for a flexible redundan robo [3]. However hese works have been based on minimizing a cos funcion independen of elasic deflecions and velociies. Redundancy, which is widely used in robo manipulaors o achieve addiional performance while racing a given end-effecor rajecory, can be used for vibraion reducion of flexible robo manipulaors. Shigang analyzed he effecs of iniial configuraion in vibraion reducion of flexible robos wih kinemaic redundancy [4]. Alhough he problem of opimizing he join moion for rigid, redundan manipulaors has already received much aenion from researchers, here has been a lile work concerning flexible redundan manipulaors. Opimal pah generaion by minimizing an objecive funcion evaluaed locally, "local opimizaion", involves less complexiy and requires less compuaional effor, while minimizing an objecive funcion evaluaed along he enire pah, "global opimizaion", resuls in more complexiy, bu more desirable manipulaor performance. In he previous works, some mehods have been developed on opimal pah generaion for flexible redundan manipulaors by local opimizaion of some objecive funcions [5], and in he presen work we will propose a mehod for opimal pah generaion by global minimizaion of elasic deformaion along a specified pah. he approach developed in his paper formulaes he rajecory planning problem for a flexible redundan manipulaor by a wo poin boundary value problem (PBV) where globally minimizes he elasic deformaion during he moion. he remainder of he paper is organized as follows. Firs, he equaions of moion for a general flexible robo manipulaor are briefly presened. he formulaion of opimal rajecory planning will be developed in he hird secion. hen differen numerical schemes for solving wo-poin boundary value problems will be defined and finally a numerical example will be solved by he presened formulaion a he end.

2 25 WSEAS In. Conf. on DYNAMICAL SYSEMS and CONROL, Venice, Ialy, November 2-4, 25 (pp ) 2 Dynamic Modeling Dynamic equaions of a general flexible robo can be derived using differen mehods such as finie elemen or assumed mode mehods. I can be shown ha dynamic equaions for a flexible robo manipulaor can be wrien as: M ( θ,φ) & θ + M ( θ,φ) Φ&& + d ( θ,θ,φ & , Φ& ) = τ (1) M ( θ,φ) && θ + M ( θ,φ) Φ&& + d ( θ,θ,φ & , Φ& ) = (2) Where θ, θ & and & θ are vecors of join angles, velociies and acceleraion, Φ, Φ & and Φ & are vecors of generalized coordinaes, velociies and acceleraions describing deformaion behavior of he flexible links and τ is he acuaor orque vecor. d 1 and d 2 are erms involving damping, siffness, cenrifugal, corilois of rigid and flexible coupling, and graviaional effecs. 3 rajecory Opimizaion For redundan manipulaors he number of joins is more han he number of degrees of freedom ha is required o follow a desired ip rajecory. In opimal pah generaion approaches, he rajecory of joins is designed o opimize some objecive funcion along he enire rajecory. In his paper we will find he opimal rajecory of a redundan, flexible robo by globally minimizing a funcion of flexible coordinaes in he form of: g ( Φ) = Φ CΦ (3) where C is a posiive definie weigh marix. o minimize a funcion evaluaed along he enire pah, he sandard mehods of calculus of variaions such as Hamilonian approach for opimal conrol problems may be employed. Opimal rajecory of he manipulaor is deermined by globally minimizing g (Φ), subjec o some consrains. he firs consrain is he moion of he ip of he manipulaor along a specified rajecory: X e = X() (4) Also we can see ha (2) represens anoher consrain. Essenially his is a differenial consrain relaing rigid acceleraion & θ and elasic acceleraion Φ &. I should be noed ha for general manipulaors his consrain is non inegrable. In moion planning problems, (1) is no a consrain because he orque τ is no considered in hese problems, bu only he moion of joins is deermined for a specified ip rajecory and once he acceleraions and he curren sae are known, (1) may be employed o obain acuaor orques. I is apparen from (2) ha elasic coordinaes are no conrolled direcly. hus he problem of moion planning is o calculae join acceleraion vecor & θ which globally opimizes some objecive funcions such as (3), subjec o he differenial consrain (2), ip rajecory consrain (4) and suiable boundary condiions: s. 12 ( θ,φ) && θ + M 22( θ,φ) Φ&& + h2( θ,θ,φ &, Φ& ) = min J = Φ CΦ M, (5) X e = X() I should be noed ha he ip posiion of he manipulaor is a funcion of join angles; In oher words he consrain (4) can be wrien as: X e = h( θ) = X( ) (6) he sae of he manipulaor can be defined as: x = [ θ, θ&, Φ, Φ& ] (7) As can be seen, (2) is a differenial consrain of degree 2. We can sae he consrains (2) and (6) as a differenial consrain of degree 1 wih a suiable ransformaion. hen he consrains of he problem can be summarized by he sae dynamics equaion: x & = f ( x, (8) where (2) and (6) have been used o obain & θ and Φ & as funcions of x and a conrol vecor u. If he number of join angles is n, he number of elasic coordinaes is r and X e is a vecor of dimension s, (2) and (6) represen r + s consrain equaions. We can differeniae (6) wice o have: J ( θ) & θ = X&& ( ) J& ( θ ) θ& (9) where J is he Jacobian marix: h( θ) J( θ) = (1) θ I is clear ha (2) and (9) are linear wih respec o & θ and Φ &, and may be used o obain hem as funcions of x. I is eviden ha we have n + r unknowns bu r + s equaions, so if we define: u = [ && θ, && 1 θ2,..., & θ n s ] (11) where n s is he number of degrees of robo redundancy, by his definiion i will be possible o sae & θ and Φ & as funcions of x and u, and as a consequence he derivaive of he sae vecor x can be saed as a funcion of x and u as in (8). Anoher imporan problem ha we should consider is boundary condiions. In he presen problem we wan o obain he opimal join rajecory from known iniial condiions. In oher word he vecor x is assumed o be known a ime. herefore we can sae he problem by: f min J = F( x, d s. x & = f ( x,, x ( ) = x (12)

3 25 WSEAS In. Conf. on DYNAMICAL SYSEMS and CONROL, Venice, Ialy, November 2-4, 25 (pp ) I should be noed ha in he presen case he final ime f is fixed. Following he LQR heory developed for opimal conrol [6], we can choose a general form for F: F ( x, = x Qx + u Ru (13) Where Q and R are consan weighing marices, Q is a posiive semi-definie and R is a posiive definie marix. If our aim is minimizaion of some objecive funcion in he form of (5), we should define Q as he followings: Q = diag(,, C, ) (14) While if our aim is minimizing he join velociies along he enire pah we can choose: Q = diag(, C,, ) (15) o obain differenial equaions governing he opimal rajecory, he differenial consrain is incorporaed ino he problem formulaion by augmening F using he Lagrange muliplier mehod. So we can sae (12) by min J = f [ F( x, + λ ( f ( x, x& )] d (16) Where λ is he vecor of Lagrange mulipliers or cosaes. For an opimal rajecory we should have: δ J = (17) δ J can be derived from (16) and o obain differenial equaions and associaed boundary condiions we should have: f F (, ) = f x u δ J [( + λ + λ& ) δx + δλ x x F (, ) f x u f + ( + λ ) δu] d λ δx = ( f ( x, x& ) (18) hus he equaions governing he opimal rajecory will be developed as: f λ& F = λ x x (19) x & = f ( x, (2) F f ( x, u = λ f ( x, 2u R = λ (21) Conrol vecor u can be obained as a funcion of x and λ from he las equaion because R is a nonsingular marix and f is a linear funcion of u. So we can inegrae (19) and (2) o obain saes and cosaes from iniial ime o final ime f. Also he boundary condiions saisfy he expression: f λ δ x = (22) I should be noed ha he sae vecor x is unknown a final ime f, so (22) leads o: x ( ) = x, λ = (23) f As can be seen he problem of opimal rajecory planning for a flexible redundan robo manipulaor racing along a specified ip rajecory wih minimum elasic deflecions leaded o a wo-poin boundary value problem. he soluion of he wopoin boundary value problem defined by (19) and (2) associaed wih he given iniial and final condiions for saes and cosaes (23) will yield he opimal rajecory of he manipulaor. Noe ha by choosing R o be a posiive definie marix, we will have he posiive definie Hessian and consequenly he soluion derived by he aforemenioned scheme will be a unique global minimum. Also i should be noed ha when he aim is no minimizaion of u, as for he presen problem, R should no be made very large o affec he soluion. 4 Numerical Mehods here are some mehods for solving wo-poin boundary value problems. he finie-difference mehod, shooing mehod, Rayleigh-Riz mehod, collocaion mehod and dynamic programming [7] are some of hese mehods. he imporan difference beween hese mehods is in how well he differenial equaions and he boundary condiions of he problem can be saisfied. In finie-difference mehods he derivaives in he differenial equaions of he problem are approximaed by finie differences beween he soluion a discree imes [8]. In some problems as for he presen case o have an accurae soluion wih finie-difference mehod a very high number of discree imes may be required. In shooing mehods he differenial equaions of he problem are solved accuraely and he soluion is obained wihin inegraion olerances. his is done by iniially approximaing some of he boundary condiions and an ieraive mehod is used o updae he iniial approximaions o converge o a soluion which saisfies he given boundary condiions. he error in he known boundary condiions expressed as a funcion of he iniial approximaions can be used o modify he iniial approximaions of he boundary condiions and a nonlinear opimizaion mehod is used o updae he iniial guesses o reduce he error. In Rayleigh-Riz mehod he soluion of a problem is approximaed by a linear combinaion of some basis funcions [8], so a good experience in choosing he appropriae basis funcions is very essenial o obain an accurae soluion by his mehod. Collocaion mehod is a mehod similar o he Rayleigh-Riz mehod in which dynamics consrains are saisfied a discree imes. Comparing hese mehods, we employed collocaion mehod for solving he wo poin

4 25 WSEAS In. Conf. on DYNAMICAL SYSEMS and CONROL, Venice, Ialy, November 2-4, 25 (pp ) boundary value problem described in he previous secion. 5 Numerical Example As an example we consider opimal pah generaion for a wo degree of freedom planar manipulaor (Fig. 1) wih he aim of minimizaion of he elasic deflecions of he flexible links along he enire pah. he ip of he manipulaor is o move along a rajecory consrained by he equaion: x = 2cos() (24) he above consrain can be wrien in he form of (6) as: x = l1 cos( θ 1) + l2 cos( θ1 + θ2) = 2cos( ) (25) We have a wo degree of freedom manipulaor ha should move along a rajecory consrained by only one consrain, so we have a robo wih 1 degree of redundancy. Link 1 is a rigid link of lengh 2 m and link 2 is a flexible link of lengh 4 m, Young's modulus E=71 GPa and momen of ineria of he cross secion I= m 4. wo links have he same densiy of 271 kg/m 3. A dynamic model for his manipulaor can be developed using finie elemen or assumed mode mehod. Here, only ransverse displacemen of he flexible link 2 is considered in he model. In order o make a comparison beween he resuls, we have accomplished rajecory planning for wo differen problems. In he firs problem, an objecive cos funcion of elasic deflecions is used o minimize elasic vibraions along he enire pah, while in he second problem we have used a cos funcion of join velociies in he form of: f 2 2 min J = ( & θ & + ) d (26) y 1 θ2 θ 2 For he firs problem marix Q is obained from (14) and for he second problem i is obained from (15) and ideniy marix wih appropriae dimension may be subsiued for C in (14) and (15). In he presen example, vecor u is a one dimensional vecor, because we have one degree of redundancy. So he one dimensional weigh marix R is made posiive definie bu no large enough o affec he soluion. Here we consider: R= 1-6 (26) Now we should specify he iniial condiions x from which he opimal rajecory will be iniiaed, we consider: θ1() = π / 3 θ2() =.27 (27) θ & = Φ = Φ& = I is apparen ha he iniial condiions for join angles saisfy he consrain (25). Also he iniial join velociies, elasic deflecions and elasic raes are considered o be zero. For he firs problem he error in racking he specified pah (24) is shown in Fig. 2. I is seen ha he error is very small and he ip of he manipulaor moves along he specified pah wih an accepable accuracy. he elasic deflecion of he end poin of link 2 is shown in Fig. 3 and as is shown, he maximum ip deflecion is less han.4% of he lengh of he flexible link 2. Also he configuraions of he manipulaor as racing he specified rajecory are shown in Fig. 4. he amoun of he cos funcion of elasic deflecions (J 1 ) and he cos funcion of join velociies (J 2 ) calculaed in his problem are: J 1 = e-11 J 2 =.99 For he second problem he error in racking he specified pah (24) is shown in Fig. 5. I is seen ha he ip of he manipulaor moves along he specified pah wih a small error. he elasic deflecion of he end poin of link 2 is shown in Fig. 6. he maximum ip deflecion is more han 1% of he lengh of he link 2. he amoun of he cos funcion of elasic deflecions (J 1 ) and he cos funcion of join velociies (J 2 ) calculaed in his problem are: J 1 =.15 J 2 =.959 θ 1 Fig. 1 A wo link flexible manipulaor x

5 25 WSEAS In. Conf. on DYNAMICAL SYSEMS and CONROL, Venice, Ialy, November 2-4, 25 (pp ) 4 x x 1-8 racking error (m) racking error (m) ime (s) Fig. 2 Error in racking he desired pah (firs problem) ime (s) Fig. 5 Error in racking he desired pah (second problem) 5 x Elasic deflecion (m) -5-1 Elasic deflecion (m) ime (s) Fig. 3 Elasic deflecion of he end poin of link 2 (firs problem) y ime (s) Fig. 6 Elasic deflecion of he end poin of link 2 (second problem) I is apparen ha as we expeced J 1 calculaed in he firs problem is less han J 1 obained in he second problem, Also he amoun of he cos funcion J 2 in he second problem is less han J 2 obained in he firs problem x Fig. 4 Manipulaor configuraions (firs problem) 6 Conclusion An approach for solving he opimal pah planning problem for a flexible redundan robo manipulaor has been presened. he opimal pah has been generaed by globally minimizing an objecive cos funcion dependen on he elasic deformaions of he flexible links of he manipulaor. he presened mehodology leaded o a wo-poin boundary value problem and he collocaion mehod has been employed for solving his problem. he efficiency of he proposed approach has been shown by a simple numerical example of opimal pah planning for a wo degree of freedom manipulaor wih link flexi-

6 25 WSEAS In. Conf. on DYNAMICAL SYSEMS and CONROL, Venice, Ialy, November 2-4, 25 (pp ) biliy. his approach is applicable o a general flexible redundan manipulaor for minimizaion of differen objecive funcions. References: [1] W. J. Book. Srucural flexibiliy of moion sysems in he space environmen, IEEE ransacions on Roboics and Auomaion, vol. 9, no. 5, pp , Ocober [2] L. Meirowich and Y. Chen. rajecory and conrol opimizaion for flexible space robos, Journal of Guidance, Conrol, and Dynamics, vol. 18, no. 3, pp , May-June [3] G. R. Eisler, R. D. Robine, D. J. Segalman, and J. D. Feddema. Approximae opimal rajecories for flexible-link manipulaor slewing using recursive quadraic programming, ransacions of he ASME, Journal of Dynamic Sysems, Measuremen and conrol, vol. 115, pp , Sepember [4] Y. Shigang. Weak-vibraion configuraions for flexible robo manipulaors wih kinemaic redundancy, Journal of Mechanism and Machine heory, vol. 35, pp , 2. [5] X. Zhang, Y. Q. Yu. Moion conrol of flexible robo manipulaors via opimizing redundan configuraions. Journal of Mechanism and Machine heory, vol. 36, pp , 21. [6] S. M. Shinners. Modern Conrol Sysem heory and Design. John Wiley & Sons, 2nd ediion, [7] D. P. Bersekas. Dynamic Programming and Opimal Conrol. Ahena Scienific, 2nd ediion, [8] R. L. Burden and J. D. Faires. Numerical analysis. PWS-Ken Publishing, Boson, 6h ediion, 1997.