Solving position-posture deviation problem of multi-legged walking robots with semi-round rigid feet by closed-loop control

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1 J. Cent. South Unv. (214) 21: DOI: 1.17/s Solvng poston-posture devaton problem of mult-legged walkng robots wth sem-round rgd feet by closed-loop control CHN Gang( 陈刚 ), JIN o( 金波 ), CHN Yng( 陈鹰 ) State Key Laboratory of Flud Power ransmsson and Control (Zhejang Unversty), Hangzhou 3127, Chna Central South Unversty Press and Sprnger-Verlag erln Hedelberg 214 bstract: he sem-round rgd feet would cause poston-posture devaton problem because the actual foothold poston s hardly known due to the rollng effect of the sem-round rgd feet durng the robot walkng. he poston-posture devaton problem may harm to the stablty and the harmony of the robot, or even makes the robot tp over and fal to walk forward. Focused on the poston-posture devaton problem of mult-legged walkng robots wth sem-round rgd feet, a new method of poston-posture closed-loop control s proposed to solve the poston-posture devaton problem caused by sem-round rgd feet, based on the nverse velocty knematcs of the mult-legged walkng robots. he poston-posture closed-loop control s dvded nto two parts: the poston closed-loop control and the posture closed-loop control. hus, the poston-posture control for the robot whch s a tght couplng and nonlnear system s decoupled. Co-smulatons of poston-posture open-loop control and poston-posture closed-loop control by ML and DMS are mplemented, respectvely. he co-smulaton results verfy that the poston-posture closed-loop control performs well n solvng the poston-posture devaton problem caused by sem-round rgd feet. Key words: poston-posture devaton; sem-round rgd feet; closed-loop control; mult-legged walkng robots 1 Introducton Mult-legged walkng robots have many advantages compared wth wheeled robots and tracked robots, such as good adaptablty to the rough terran, and nearly no damages to the passed ground. herefore, more and more researchers have studed on the mult-legged walkng robots. nd poston-posture control whch decdes the stablty and harmony of the robot s an mportant research topc. GONZLZ D SNOS et al [1 3] used the ntermttent correcton method to adjust the poston-posture durng the RIMHO walkng. L and INGR [4], and OODN et al [5] appled force control to keep gdog balance so that the poston-posture was controlled ndrectly. GRND et al [6], and JRRUL et al [7] adopted the closed-loop control method to control the wheeled-legged robot called Hylos. In the mult-legged walkng robots, sem-round rgd feet were generally employed [8 1], though feet wth flat soles were sometmes used, such as SILO4 [11], IN [12], and COM [13]. he sem-round rgd feet have some advantages compared wth feet wth flat soles. Frstly, the structure of sem-round rgd feet s smpler than that of feet wth flat soles, because feet wth flat soles should be connected wth the legs by actve or passve jont ankles, whch also rases ts expense. Secondly, the robot may be stumbled by passve jont ankles durng walkng, or complex control s needed for feet wth actve jont ankles. he rollng effect of sem-round rgd feet may cause poston-posture devaton problem when multlegged walkng robots move forward. GURDRZO et al [14] frstly concerned about the body msplacement problem caused by sem-round rgd feet n walkng robots. hey employed knematcs correcton algorthm to correct the foot trajectory. he correcton method worked well n the flat terran. ut the contact ponts between feet and the ground were hardly known n the rough terran so that the approach to solve the msplacement problem was unavalable. In ths work, poston-posture closed-loop control was proposed to solve the poston-posture devaton problem caused by sem-round rgd feet thoroughly. he poston-posture was regarded as an objectve varable to follow the poston-posture value planned n advance through adjustng jont angles of the mult-legged walkng robot usng poston-posture closed-loop control wthout payng attenton to the actual contact ponts between feet and the ground. herefore, ths method s sutable for any terran and can acheve accuracy results n the postonposture control of the mult-legged walkng robot. Foundaton tem: Project(512214) supported by the Scence Fund for Creatve Research Groups of Natonal Natural Scence Foundaton of Chna; Project supported by the Program for Zhejang Leadng eam of S& Innovaton, Chna Receved date: ; ccepted date: Correspondng author: JIN o, Professor; el: ; -mal: bjn@zju.edu.cn

2 Poston-posture closed-loop control algorthm to solve devaton problem 2.1 Velocty knematcs of sngle leg wth three degrees of freedom leg whch can move n the three-dmensonal space should have three degrees of freedom at least. he sngle leg wth a sem-round rgd foot shown n Fg. 1 has three jonts: the root jont, hp jont and knee jont. he jont frames are formed accordng to D-H method, as shown n Fg. 1, where the frame {} s a word coordnate system and the frame {} s a body coordnate system. he D-H parameters of the sngle leg are gven n able 1. J. Cent. South Unv. (214) 21: Consderng the word coordnate system {} as the reference frame, the poston of the foot end pont can be expressed by P P R P (2) where P s the poston of the foot end pont expressed n the word coordnate system {}, P s the poston of the center of the robot body relatve to the word coordnate system {}, and P s the poston of the foot end pont wth respect to the body coordnate system {}. R s the rotaton matrx between frame {} and {}. Dfferentatng q. (2) yelds P P R P R P (3) P may be gven by dfferentatng q. (1) as P J θ (4) where Fg. 1 D-H model of sngle leg wth sem-round rgd foot able 1 D-H parameters of sngle leg Lnk Lnk twst, α 1 /( ) Lnk length, α 1 /m Lnk offset, d /m l2 Jont angle θ,1 (From 3 to 12 ) θ,2 (From 6 to 12 ) θ,3 (From to 12 ) th respect to the body frame {} and gven the orgn coordnate of frame { O } ndcated by O ( x, O O y, z ), the thgh length l 2, the shank length l 3, the root angle θ,1, the hp angle θ,2, and the knee angle θ,3, the poston of the foot end pont s as follows: x x l3 cos,1 cos(,2,3) O l cos cos 2,1,2 y y l3 sn(,2,3) 2 sn l O,2 z z l3 sn,1 cos(,2,3) O l2 sn,1 cos,2 O (1) l3 sn,1 cos(,2,3) l 2 sn,1 cos,2 J l3 cos,1 cos(,2,3) l2 cos,1 cos,2 l cos sn(,2,3) l2 cos,1 sn,2 l cos( 3,3) l2 cos l sn sn(,2,3) l sn sn 2 3 3,2,1,1,1 s the Jacoban matrx of the leg. θ θ θ θ [, 1,2,3],2,2 l3 cos,1 sn(,2,3) l3 cos(,2,3) l3 sn,1 sn( ),2,3 s the jont angular velocty vector of the leg. herefore, the relaton between the velocty of the foot end pont and the jont angular velocty of leg s formed by P P R RJ θ (5) P 2.2 Inverse velocty knematcs of mult-legged walkng robot based on velocty knematcs of sngle leg he dervatve of the rotaton matrx s as follows [15]: z y R ( t) z x R( t) y x S( ) R( t) (6) z where ω. P x y Substtutng q. (6) nto q. (5) yelds G P P S( ) R RJ θ (7) Furthermore, we have

3 J. Cent. South Unv. (214) 21: θ ( RJ ) 1 ( P P ( ) S R P 1 J ( R) ( P P S( ) R P ) (8) ecause the poston-posture of the mult-legged walkng robot s determned by the support legs, we focus on the knematcs of the support legs. herefore, the nverse velocty knematcs of the mult-legged walkng robot whch descrbes the relaton between jont angular velocty vector θ of support legs and body angular velocty vector ω, body translaton velocty V s formed as follows: 1 θ J R ( V V SR P) (9) θ [ J1 J = R= ] 1 2 n J 2 J n 3n3n R R V = V = R3n 3n P 1 P 2 P n P P P S( ) ( ) = S S 3n3n 3n3n S( ) 3n3n P P P P P where θ s the jont angular velocty vector of support legs, J s the Jacoban matrx of the mult-legged walkng robot, R s the rotaton matrx of the mult-legged walkng robot, V s the velocty matrx of support legs expressed n the frame {}, V s the body velocty matrx of the mult-legged walkng robot wth respect to the frame {}, S s the skew-symmetrc matrx of the mult-legged walkng robot, P s the velocty vector of support legs expressed n the frame {}, and n s the ) n number of support legs. 2.3 Poston-posture closed-loop control algorthm based on nverse velocty knematcs of multlegged walkng robot o mplement the accurate poston-posture control, a closed-loop control strategy s needed. In q. (9), V s not a zero matrx and ts value s not easy to be calculated because of the rollng effect of round rgd feet. hle V can be consdered as a zero matrx and the devaton caused by the approxmaton s corrected by the closed-loop control. hus, q. (9) can be wrtten as 1 θ J R ( V SR P ) (1) ccordng to q. (1), a closed-loop control structure on the poston-posture of the mult-legged walkng robot s constructed, as shown n Fg. 2. he poston-posture closed-loop control s dvded nto two parts: the poston closed-loop control and the posture closed-loop control. he poston closed-loop control ams to make the poston of the robot body accurately trace the planned poston. nd the posture closed-loop control s appled n order to mplement the closed-loop control on posture of the robot body accordng to the planned posture. herefore, the poston-posture control s decoupled and becomes easy to be mplemented through constructng the two closed-loops. Fnally, the jont angles of the support legs are obtaned by the nverse velocty knematcs of the mult-legged walkng robot. V can be calculated by a proportonal control law n the poston closed-loop as V = P P P P =K ( P P ) P = P x y z I I I I I = P x y z K P a a a a k p1 k p 2 k p3 a 3n3n (11) where P s computed by a proportonal control law, P 1 s the object poston of the mult-legged walkng robot s body planned n advance, P a s the actual poston of the mult-legged walkng robot s body measured by the locaton sensor, and K p s a proportonal

4 4136 J. Cent. South Unv. (214) 21: Fg. 2 Structure of poston-posture closed-loop control for mult-legged walkng robot coeffcent matrx. nd S s calculated by a proportonal control law n the posture closed-loop as S( ) ( ) = S S S( ) n n 3 3 x y z 1 sna cosa sna cos a sna cosa cosa K K k 1 k 2 ( 1 a ) k I [ I I I ] a [ a a a ] 3 (12) where ω s computed by a proportonal control law, K s a proportonal coeffcent matrx, I s the object posture of the mult-legged walkng robot s body descrbed by uler angle vector and planned n advance, and a s the actual posture of the mult-legged walkng robot s body descrbed by uler angle vector and measured by the gyroscope. hen, the jont angles of the support legs can be calculated as tt t 1 θ J R ( V SR P )dt (13) ccordng to the jont angles calculated by q. (13), the support legs of the mult-legged walkng robot can move to adjust the poston-posture of the robot. 3 Verfcaton results Co-smulatons based on ML and DMS are mplemented to verfy whether the poston-posture closed-loop control algorthm s avalable to solve the poston-posture devaton problem caused by the sem-round rgd feet for mult-legged walkng robots. sx-legged walkng robot shown n Fg. 3 s used as the prototype for the co-smulatons. he geometrcal parameters of the sx-legged walkng robot are lsted n able 2. hs robot has sx legs whch are nstalled symmetrcally on the two sdes of the body wth the same knematcal structure. ach leg has three degrees of freedom: root jont, hp jont and knee jont, whch are powered by three Dynamxel RX-64 servomotors [16]. In order to smulate the rough terran, the ground wth heght dfference of 4 mm s adopted, and the sxlegged walkng robot stands on the ground, as depcted n Fg. 4. he ground and the sx-legged walkng robot model are constructed n DMS software, as shown n Fg. 4, where the world frame {} s located on the lower ground and the body frame {} s located on the center of the robot body. Meanwhle, the control strategy s mplemented n ML software. Co-smulatons of poston-posture open-loop control and closed-loop control are carred out so as to vvdly demonstrate the valdty of the poston-posture closed-loop control

5 J. Cent. South Unv. (214) 21: part. Meanwhle, the actual poston-posture trajectory can be obtaned. sne-wave trackng co-smulaton of the postonposture open-loop control s conducted by ML and DMS. he planned poston trajectory as shown n Fg. 6 s formed as Fg. 3 Sx-legged walkng robot platform able 2 Geometrcal parameters of sx-legged walkng robot Length, L /mm dth, /mm Length of thgh, l 2 /mm Length of shank, l 3 /mm Radus of semround rgd foot, R/mm x y z 5sn t 4sn t 27 15sn t (14) nd the planned posture trajectory as shown n Fg. 7 s expressed as 2sn t 1.5sn t sn t (15) Fg. 4 Ground and sx-legged walkng robot n DMS method, through comparng the results. nd n the cosmulatons, all legs are set to support the robot. 3.1 Sne-wave trackng co-smulaton of postonposture open-loop control he co-smulaton for the open-loop control on poston-posture of the sx-legged walkng robot has two parts: the ML part and the DMS part, as shown n Fg. 5. he ML part s used to calculate the jont angles based on the nverse knematcs of the robot wth the poston-posture trajectory gven and provde these jont angle values to the DMS part [15]. he DMS part constructs the three-dmensonal model of the ground and the sx-legged walkng robot, as shown n Fg. 4, and the sx-legged walkng robot moves accordng to the jont angle values from the ML Fg. 5 Relaton between ML and DMS n cosmulaton for open-loop control on poston-posture Fg. 6 Planned and actual poston trajectores of sx-legged walkng robot n sne-wave trackng co-smulaton of postonposture open-loop control: (a) X component of poston and devaton; (b) Y component of poston and devaton; (c) Z component of poston and devaton

6 4138 Fg. 7 Planned and actual posture trajectores of sx-legged walkng robot n sne-wave trackng co-smulaton of postonposture open-loop control and devatons between them: (a) Yaw angle of posture and devaton; (b) Roll angle of posture and devaton; (c) Ptch angle of posture and devaton where α, β, γ are yaw angle, roll angle and ptch angle of the robot posture, as shown n Fg. 4. he co-smulaton results are presented n Fg. 6 and Fg. 7. It s shown that there exst devatons between the actual poston-posture and the planned poston-posture of the robot. s mentoned above, the sem-round rgd feet can cause the nterference among legs. he actual poston of the foothold always changes by the roll effect of the sem-round foot, and ths further breaks the harmony movement of the robot. Under the smulaton envronment, the nterference among legs can make the J. Cent. South Unv. (214) 21: robot slp on the plane ground when the nterference force s more than the statc frcton force between feet and ground. herefore, devatons of the poston along x axs and y axs are larger and the yaw angle devaton s larger n poston-posture open-loop control. ut poston devaton along z axs, the roll angle devaton and the ptch angle devaton are relatvely small wth the ground constran. From the sne-wave trackng co-smulaton results, we can also conclude that the nterference among legs and slppage caused by the sem-round rgd feet exert great nfluence on the devatons n postonposture. 3.2 Sne-wave trackng co-smulaton of postonposture closed-loop control In order to verfy that poston-posture closed-loop control algorthm for mult-legged walkng robots proposed can work well to solve the poston-posture problem caused by sem-round rgd feet, co-smulatons of closed-loop control on poston-posture of the sxlegged walkng robot by ML and DMS are carred out. s shown n Fg. 8, dfferent modules of the co-smulaton are completed respectvely n ML and DMS. he jont angles of the robot are calculated n ML accordng to the planned poston-posture trajectory and the actual poston-posture got from DMS, based on the proportonal control law and the nverse velocty knematcs model of the robot. nd n DMS, the sx-legged walkng robot moves n accordance wth the jont angle values computed n ML and provdes the nformaton of the actual poston-posture of the robot to ML. sne-wave trackng co-smulaton of the poston-posture closed-loop control s carred out. he planned poston-posture trajectores shown n Fg. 9 and Fg. 1 are the same forms as those n the sne-wave trackng co-smulaton of the poston-posture open-loop control. In the sne-wave trackng co-smulaton, the proportonal coeffcent matrx K P s set as K P and K s set as K he sne-wave trackng co-smulaton results of the poston-posture closed-loop control are shown n Fg. 9 and Fg. 1. he poston-posture closed-loop control method works well n the sne-wave trackng. he

7 J. Cent. South Unv. (214) 21: Fg. 8 Relaton between ML and DMS n co-smulaton for closed-loop control on poston-posture Fg. 9 Planned and actual poston trajectores of sx-legged walkng robot n sne-wave trackng co-smulaton of poston-posture closed-loop control: (a) X component of poston and devaton; (b) Y component of poston and devaton; (c) Z component of poston and devaton poston-posture devatons n closed-loop control are much smaller than those n open-loop control shown n Fg. 6 and Fg. 7. It s worth notng that the actual poston-posture n closed-loop control lags behnd the planned poston-posture, as shown n Fg. 9 and Fg. 1, but the phenomenon does not exst n the open-loop control of the poston-posture, as depcted n Fg. 6 and Fg. 7. hrough the co-smulatons n closed-loop control and open-loop control on poston-posture for multlegged robots wth sem-round rgd feet, t s obvous that sem-round feet can cause large devatons n poston-posture of the robot, especally about the poston, along x and y, and the yaw angle, when the poston-posture adjusts wth the open-loop control. On the contrary, the poston-posture devaton problem caused by sem-round feet s successfully solved by the closed-loop control on the poston-posture of the robot. lthough the closed-loop control on the poston-posture has another ssue that the actual poston-posture lags behnd the planned one n control, the delay tme s so short that t has lttle nfluence on the control accuracy.

8 414 J. Cent. South Unv. (214) 21: jont angular veloctes of the robot s establshed based on the nverse velocty knematcs model of the multlegged walkng robot, whch s the bass of the postonposture closed-loop control algorthm. 3) Poston-posture closed-loop control s dvded nto two parts: the poston closed-loop control and the posture closed-loop control, based on the nverse velocty knematcs of the mult-legged walkng robot, so that the poston-posture of the robot can be controlled accurately. he poston-posture closed-loop control method presented can solve the poston-posture devaton problem caused by sem-round rgd feet successfully. 4) Co-smulatons by ML and DMS are mplemented to verfy the poston-posture closed-loop control method for solvng the poston-posture devaton problem. he results show that the poston-posture closed-loop algorthm can accurately control the poston and posture of the robot, wthout carng how the semround rgd feet move. 5) In the near future, the poston-posture closedloop control method for solvng the poston-posture devaton problem wll be used n the sx-legged walkng robot platform to verfy the valdty. Furthermore, the poston-posture control wth closed-loop control method wll be appled n the sx-legged walkng robot platform to mprove the adaptaton of the robot to the rough terran. References Fg. 1 Planned and actual posture trajectores of sx-legged walkng robot n sne-wave trackng co-smulaton of postonposture closed-loop control and devatons between them: (a) Yaw angle of posture and devaton; (b) Roll angle of posture and devaton; (c) Ptch angle of posture and devaton 4 Conclusons 1) Focused on the poston-posture devaton problem for the mult-legged walkng robots wth sem-round rgd feet, a poston-posture closed-loop control method s presented. he method does not concern about the poston of the actual contact pont between the foot and the ground so that t can solve the poston-posture devaton thoroughly. 2) he mathematc relaton between velocty, angular velocty of the mult-legged walkng robot and [1] GONZLZ D SNOS P, JIMNZ M, RVIJO J, R J. tttude and alttude control usng dscontnuous gats for walkng machnes [C]// Internatonal Conference on Systems ngneerng n the Servce of Humans. Le ouquet, France, 1993: [2] NONI JIMNZ M, GONZLZ D SNOS P. tttude and poston control for non-rgd walkng machnes [J]. Mechansm and Machne heory, 1998, 33(7): [3] JIMNZ M, GONZLZ D SNOS P. tttude and poston control method for realstc legged vehcles [J]. Robotcs and utonomous Systems, 1996, 18(3): [4] L D V, INR. gdog-nspred studes n the locomoton of goats and dogs [J]. Integratve and Comparatve ology, 211, 51(1): [5] OODN D, MLCHNO M, LNKSPOOR K, HORD, RIZZI, RIR M. utonomous navgaton for gdog [C]// I Internatonal Conference on Robotcs and utomaton. laska, US, 21: [6] GRND C, NMR F, PLUM F. Moton knematcs analyss of wheeled-legged rover over 3D surface wth posture adaptaton [J]. Mechansm and Machne heory, 21, 45(3): [7] JRRUL P, GRND C, IDUD P. Robust obstacle crossng of a wheel-legged moble robot usng mnmax force dstrbuton and self-reconfguraton [C]// I Internatonal Conference on Intellgent Robots and Systems, Calforna, US, 211: [8] GONZLZ D SNOS P, SRMR J, GRCI. Optmzng leg dstrbuton around the body n walkng robots [C]//

9 J. Cent. South Unv. (214) 21: I Internatonal Conference on Robotcs and utomaton, arcelona, Span, 25: [9] GONZLZ D SNOS P, CONO J, GRCI, SRMR J, RMD M. sx-legged robot-based system for humantaran demnng mssons [J]. Mechatroncs, 27, 17(8): [1] KLKRISHNN M, UCHLI J, PSOR P, MISRY M, SCHL S. Learnng, plannng, and control for quadruped locomoton over challengng terran [J]. he Internatonal Journal of Robotcs Research, 211, 3(2): [11] SRMR J, GONZLZ D SNOS P. Free gats for quadruped robots over rregular terran [J]. he Internatonal Journal of Robotcs Research, 22, 21(2): [12] HIROS S, YOKO S, ORII, OG M, SUGNUM S, KI K, KO K. Quadruped walkng robot centered demnng system-development of ttan-ix and ts operaton [C]// I Internatonal Conference on Robotcs and utomaton. arcelona, Span, 25: [13] IRN, NONMI K, OHROKU H, KUSU Y, IMMUR S. daptve mpedance control wth complant body balance for hydraulcally drven hexapod robot [J]. Journal of System Desgn and Dynamcs, 211, 5(5): [14] GURDRZO, JIMNZ M, GONZLZ D SNOS P. nalysng and solvng body msplacement problems n walkng robots wth round rgd feet [J]. Robotcs and utonomous Systems, 26, 54(3): [15] CRIG J J. Introducton to robotcs: mechancs and control [M]. US: Prentce Hall, 24. [16] JIN o, CHN Cheng, LI e, LI Xang-yun. Desgn and confguraton of a hexapod walkng robot [C]// Proceedngs of ICMM. Shangha, Chna, 211: (dted by YNG ng)