TRAJECTORY CONTROL OF A THREE DOF DUAL ARM SPACE ROBOT WITH SMALL ATTITUDE DISTURBANCE

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1 RAJECORY CONROL OF A HREE DOF DUAL ARM SPACE ROBO WIH SMALL AIUDE DISURBANCE Haresh Patola (a), P. M. Pathak (b), S. C. Jan (c) (a) (b) Robotcs and Control Laboratory, Mechancal and Industral Engneerng Department, Indan Insttute of echnology, Roorkee, Inda, (c) Mangalayatan Unversty, Algarth, Inda (a) haresh_patola@yahoo.co.n, (b) pushpfme@tr.ernet.n, (c) sjanfme@yahoo.com ABSRAC hs work presents a trajectory plannng strategy for a three degree of freedom (DOF) dual arm space robot n work space. he strategy s based on the prncple of dynamc couplng between the tp moton and the vehcle moton of the space robot. he strategy uses the two arms of three lnk seral manpulator. One arm called msson arm acheves the trajectory control task whle the other arm called balance arm moves n a way to reduce the atttude of the vehcle. A robust overwhelmng controller s used for trajectory control of tp of the msson arm. he frst and second jont rotatons of balance arm are based on the frst and second jont rotatons of msson arm whereas the thrd jont rotaton of balance arm s carred out such that small atttude dsturbance of vehcle takes place. An eample of three DOF dual arm space robot s consdered to llustrate the methodology. Bond graph has been adopted as modelng tool as t facltates the system modelng from the physcal paradgm tself and t s easy to develop varous control strateges by modfyng the physcal paradgm. Keywords: dual arm space robot, robust overwhelmng controller, atttude control, bond graph modelng 1. INRODUCION In a large number of space applcatons lke mantenance of components, capture of floatng objects, removal of debrs or old satelltes, orbtal replacement unt operatons, refuelng etc., the manpulator moton s requred to be controlled precsely. In a free-floatng space robotc system, the spacecraft poston and the atttude are not actvely controlled usng eternal jets or thrusters durng manpulator actvty. Due to the conserved lnear and angular momenta, the spacecraft moves freely n response to the dynamc dsturbances caused by the manpulator moton. Such dsturbances may result n sgnfcant devaton of the end-effector from the desred trajectory, and are partcularly sgnfcant when the manpulator s holdng a heavy payload. Moreover, the angular momentum conservaton constrants are non-ntegrable renderng the system non-holonomc. Hence, t s not possble to cancel ths base dsturbance usng statc state feedback. hs complcates the moton plannng and control ssue further (Nakamura and Mukherjee 1991). he dynamc couplng between space vehcle and manpulators has been a subject of ntense nvestgaton snce the Space Shuttle RMS went nto servce n 198. Due to dynamc couplng, the resultng moton at the shuttle base caused the end-effector to mss ts target. hs type of problem can be partcularly acute durng satellte rendezvous n whch errors of even a few centmeters could result n a faled capture attempt or even damage to the satellte. Also vehcle atttude dsturbance assumes prme mportance because any dsorentaton would affect the communcaton lnk of the satellte wth the control staton. One of the key ssues n space robotcs s usng arms to contact workste elements safely, quckly, and accurately wthout accdentally contactng unntended objects or mpartng ecessve forces beyond those needed for the task. For ths purpose trajectory plannng assumes mportance. A well planned trajectory, takng care of sngulartes of manpulator arm, s essental for effcent and smooth controlled moton of arm tps. Control of space robot has been studed by many authors but control studes on mult arm space robot are only few. Dubowsky and Papadopoulos (199a, 199b) dscussed concepts of vrtual Manpulator (VM) model wth an applcaton to workspace analyss, path plannng usng Enhanced Dsturbance Map (EDM) as well as effects of dynamc sngulartes on workspace of free flyng manpulator. Moosavan and Papadopoulos (1998) proved the advantage of Drect Path Method over Barycentrc vector approach whle developng knematc modelng of rgd mult arm space robots. Papadopoulos and Moosavan (1994) compared by smulaton study, the performance of Euler angle and Euler parameter base control law to that of a transposed Jacoban algorthm and showed that the latter gves reasonably good performance wth reduced computatonal burden. Huang, Xu, and Lang (5) developed a space robot system consstng of two arms, wth one arm (msson arm) for accomplshng the capture msson, and the other one (balance arm) compensatng for the dsturbance of the base. Marches Page 1

2 Fgure 1: Schematc Dagram of hree DOF Dual Arm Space Robot and Angroll (1997) focused on control strategy for a free-flyng space manpulator and descrbed the methods used to solve the nverse knematc problem of a non-redundant robotc arm mounted on a free floatng spacecraft for mnmal dsturbance of spacecraft atttude. Yokokohj, oyoshma, and Yoshkawa (199) proposed an effcent computaton algorthm for the trajectory control of mult arm free-flyng space robot usng the generalzed Jacoban matr technque. Chen and ang (6) presented the optmal non-holonomc moton plannng of free-floatng space robot system wth dual arms. Based on the nsghts developed from bond graph modelng, Ghosh (199) developed a robust overwhelmng controller for the robotc manpulator, whch does not requre the knowledge of the robot parameters and the payload. Pathak, Kumar, Mukherjee, and Dasgupta (8) presented a scheme for robust trajectory control of free-floatng space robots. hs work presents a strategy for trajectory control of msson arm wth small atttude dsturbance of vehcle. rajectory control of msson arm s acheved by usng robust overwhelmng controller (Ghosh 199; Pathak, Kumar, Mukherjee, and Dasgupta 8; Mukherjee, Karmakar, and Samantray 6). Due to msson arm tp moton, vehcle atttude gets dsturbed. A control law s proposed for jont control of balance arm such that when t works wth trajectory control of msson arm small vehcle atttude dsturbance s obtaned. It s assumed that the space vehcle s equpped wth three small reacton wheels. A numercal eample of free-floatng space vehcle carryng two manpulators, each of havng three lnks s presented to demonstrate the effcacy of the proposed control strategy. he complete dynamcs s modeled usng bond graphs (Karnopp, Margols, and Rosenberg 6; Mukherjee, Karmakar, and Samantray 6; Breedveld and Dauphn-anguy 199). For the purpose of modelng and smulaton, the bond graph package SYMBOLS Shakt (Users Manual) has been used.. MODELING OF HREE DOF DUAL ARM SPACE ROBO o llustrate the control strategy for the space robot, a three DOF free-floatng space robot consstng of two seral manpulators each of havng three lnks mounted on a space vehcle s consdered and shown n Fgure 1. Jont rotaton ( ) of manpulator and Euler angle of the base (,, ) n --1 conventon are used as generalzed coordnates. All jonts are assumed to be revolute. he jont between lnks () and (+1) s numbered as (+1). In Fgure 1, {A} s the absolute frame and {V} s the vehcle frame located at the centre of mass (CM) of the vehcle. he frames {} and {5} are attached on the vehcle to represent the msson arm and balance arm base locaton on the vehcle at a poston (r, r y, r z ) and (-r, -r y, -r z ) from the orgn of frame {V}, respectvely. he frames {1} and {6} are attached on the frst lnks of msson arm and balance arm at ther bases, respectvely. he frames {1} and {6} have relatve moton wth respect to frames {} and {5} about Y-as. Frames {} and {} are attached at the Page

3 Fgure : Mult Bond Graph Model of hree DOF Dual Arm Space Robot base of second and thrd lnks of the msson arm whereas frames {7} and {8} are attached at the base of second and thrd lnks of the balance arm, respectvely. hey represent the jont aes for the second and thrd jonts of both the arms and have rotaton about ther Z- aes. he moments of nerta are defned about the body-fed prncpal coordnate system. It s assumed that all the lnks are to be rgd and the specfed task space trajectory s wthn the reachable workspace of the manpulators for smplcty. he frames {4} and {9} represent the tps of msson and balance arm, respectvely. he angular relatve veloctes of lnks of the msson arm are defned as, 1 1 [ 1 ] [ ] (1) ] [ Here, 1 and are the jont veloctes of frst, second and thrd jonts of the msson arm, respectvely. Modelng of dual arm space robot nvolves modelng of the lnear and angular dynamcs of the manpulators as well as the space vehcle on whch they are mounted. he complete bond graph model of three DOF dual arm space robot s created and shown n Fgure. Varous sub-models requred for drawng bond graph model are brefly presented here..1. Knematc Relatons A sgnal structure has been made for transformng the vehcle angular velocty to Euler angle rates usng the relaton, 1 ts c s / c tc s y c / c z where, and are the Euler angle rates;, () y and z are the vehcle angular veloctes. Also t tan, c cos, c cos and s sn. he angular velocty of (+1)th lnk wth respect to the 1 A absolute frame {A}, ( 1) can be computed recursvely usng the followng relaton (Crag, 1986), 1 A 1 A 1 ( ˆ 1 ) R ( ) 1 U 1 () Page

4 1 where R s the rotaton matr of drecton cosnes A relatng frame {} wth respect to frame {+1}, ( ) s the angular velocty of ()th lnk wth respect to the absolute frame {A}, 1 s the velocty of (+1)th jont 1 and Uˆ 1 s a unt vector representng the drecton of the jont as. hs computaton s mplemented through angular velocty propagaton (AVP) sub-model of lnk as shown n Fgure. hs sub-model takes the angular velocty of prevous lnk and the jont velocty between the prevous and current lnk as nput and the angular velocty of the current lnk s gven out after calculaton. he jont velocty 1 s relatve between two adjacent lnks, can be about any one of the aes of the jont coordnate frame. Based on the as of the jont rotaton, one of the transfer modul (Mt[], Mt[1] or Mt[]) connected to the jont velocty juncton wll be unty, and the remanng wll be zero. hs s decded by the 1 vector Uˆ 1. he relaton for lnear velocty of the tp of the lnk can be computed as, N N y z I YY y ( I XX I ZZ ) z (6) I ( I I ) (7) ZZ z YY XX here N, N y and N z represent the moments actng on the vehcle epressed n vehcle frame n X, Y, and Z drecton, respectvely. I XX, I YY and I ZZ are the moment of nerta of the rgd body about X, Y and Z drectons, respectvely. he created sub-model EJS s shown n Fgure.. CONROLLER DESIGN OF SPACE ROBO.1. rajectory Control of Msson Arm For the msson arm tp trajectory control, robust overwhelmng controller (ROC) presented elsewhere, s used. he bond graph sub-model for ROC s shown n Fgure. A reference flow nput for plant s suppled to sub-model ROC from man bond graph whch s shown n Fgure. hs sub-model s also provded wth msson arm tp velocty as nput from man bond graph y A A ( 1 1 A A A A V ) ( V ) R[ ( ) ( P )] (4) l where ( P 1 ) l, [ ( P 1 )], and l l = length of ()th lnk. he lnear velocty of the CM of the lnk can be ( P ) l n place of obtaned by usng G ( P 1 ) n Eq. (4). he bond graph sub-model for lnear velocty propagaton (LVP) of lnk s created and s shown n Fgure. hs sub-model takes the angular velocty of current lnk and tp velocty of prevous lnk as nput and gves velocty of CM of current lnk and tp velocty of current lnk as output. he CM veloctes of the space vehcle and lnks of the manpulators are obtaned from the lnear nerta of the space vehcle and lnks, respectvely. In Fgure, junctons represent 1 the jont relatve veloctes and the ntegrator submodels used as dsplacement detectors... Euler Juncton Structure A rgd body has translaton and angular moton. hus ts dynamcs has two parts. he translaton dynamcs s modeled usng Euler s frst law and the angular dynamcs usng the Euler s second law. In order to model the Euler s equatons, the gyrator loop commonly known as an Euler Juncton Structure (EJS) s used and a sub-model EJS s created to represent the rotatonal dynamcs of the space vehcle and that of the lnks. he followng Euler equatons are used to create the sub-model EJS, N I ( I I ) (5) XX ZZ YY y z G Fgure : Robust Overwhelmng Controller (ROC) Sub-model and t gves correcton efforts as output to sub-model CRL. In Fgure, pad sub-model s used to avod dfferental causalty. Pads are lumped flebltes wth hgh values of stffness and dampng. hree ROC submodels (one for each X, Y and Z drecton) are used to control the space robot as shown n Fgure. A sgnal structure CRL can be created usng forward knematc relatons to convert correcton efforts from ROC nto correcton torques to be suppled to dfferent jonts of the msson arm. For creaton of ths sgnal structure CRL, the tp velocty of the msson arm wth respect to ts base frame {} epressed n absolute frame {A} can be evaluated as, A ( V4 4 A ) R ( V ) (8) k j here, ( V ) denotes the velocty of the orgn of {}th frame as observed from {j}th frame and epressed n {k}th frame, represents the orentaton of a frame A R {} wth respect to a frame {A}. he term ( V4 ) n Eq. (8) can be derved as, ( s1sl s1s ( V 4) ( c1s l c1s l ) l ) ( c c l 1 ( s l ( s c l 1 c c s s c 1 l ) l ) 1 l ) c 1cl 1 sl s 1cl) (9) Page 4

5 where sj c cos, s sn, c cos( ) and sn( ). One can wrte Eq. (9) n a compact form as, j ( V 4 ) [ Q][ 1 ] (1) where [Q] s the transformaton matr n Eq. (9). he transformaton A A A V R s computed as R R where, V R cc css sc csc ss A V R sc sss cc ssc cs (11) s cs cc V and R s dentty matr, c cos, s sn, c cos, s sn, c cos and s sn. hus Eq. (8) can be wrtten as, A A ( V 4 ) [ R][ Q][ 1 ] (1) hus, the relatonshp between jont correcton torques and correcton efforts from controller can be epressed as, [ 1 1 ] [ Q ] [ A R][ F Fy Fz ] (1) where s the torques eerted on lnk () by lnk (-1), epressed n terms of frame {}. F, Fy and Fz are correctve efforts n X, Y and Z drectons, respectvely. he created sub-model CRL for sgnal structure for converson of correcton efforts to correcton torques s shown n Fgure. j j proportonal gan parameter and parameter. K v v s dervatve gan () he control law for the actuator at thrd jont of balance arm can be wrtten as, B K p( d a) Kv( d a) K p( d a) Kv( d a) K ( ) K ( ) p d a d a (15) where B s the torque at thrd jont of balance arm;, and are the Euler angles;, and are the Euler angle rates; suff d stands for desred whereas suff a stands for actual. hs control law has been devsed based on the observaton that the body atttude s controllable from the jonts of the balance arm (Dauphn-anguy, Rahman, and Sueur, 1999). 4. SIMULAION AND RESULS o valdate the control strategy dscussed n Secton and, a bond graph model as shown n Fgure has been smulated. he parameters used for the smulaton are gven n Append A. It s assumed that ntally the absolute frame and vehcle frame are concdent. For the smulaton, the reference velocty command for msson arm n absolute frame s taken as, X ref R sn( t) n X drecton (16) Y ref Asn( t) n Y drecton (17) R cos ( t) n Z drecton (18) Z ref where R s the radus of the reference crcle n X-Z plane; A s the ampltude of velocty n Y drecton. It s a crcular trajectory n X-Z plane and parabolc.. Atttude Control of Space Vehcle he proposed control strategy for atttude control of space vehcle can be worked out as, () Assumng that the motons of actuators at frst and second jonts of msson arm and balance arm are equal and opposte drecton to each other. he control law for the actuators at frst and second jonts of the balance arm can be gven as, K p ( M j ) K v ( M j ) for j = 1,. (14) here s the torque at (j)th jont of balance a rm; M j s the poston of (j)th jont of msson arm; s the poston of (j)th jont of balance arm; s the M j (j)th jont angular velocty of msson a rm; s the (j)th jont angular velocty of balance arm; K p s Fgure 4: Intal Confguraton of hree DOF Dual Arm Space Robot Page 5

6 Fgure 5: -D Plot of Reference and Actual p rajectory of Msson Arm Fgure 6: Plot of Reference and Actual p rajectory of Msson Arm n X-Z Plane Fgure 7: Plot of Reference and Actual p rajectory of Msson Arm n X-Z and Z-Y Plane Fgure 8: Plot of Msson Arm p Poston Error versus me Fgure 9: -D Plot of Actual p rajectory of Balance Arm trajectory n X-Y plane and Y-Z plane. he jont torques for balance arm s gven as per control law shown n Eq. 14 and 15. At the begnnng of the smulaton, the Fgure 1: Plot of Actual p rajectory of Balance Arm n X-Z Plane overwhelmer ntal poston s set to msson arm tp poston n order to keep the ntal error to be zero. Page 6

7 Fgure 11: Plot of Varaton of Euler Angle for Sngle and Dual Arm Space Vehcle versus me Fgure 1: Plot of Varaton of Euler Angle for Sngle and Dual Arm Space Vehcle versus me only follow the atttude of the vehcle base. However reacton wheels can be actuated by nput source of voltage as and when requred. he ntal confguraton of the space robot s shown n Fgure 4. he smulaton s carred out for 1 seconds. A three-dmensonal plot of reference and actual trajectores of tp of the msson arm are shown n Fgure 5. he reference and actual tp trajectores of the msson arm n X-Z plane are shown n Fgure 6, whereas n Fgure 7, the reference and actual tp trajectores of the msson arm n X-Y and Z-Y planes are shown. Fgure 8 shows the plots of error between the reference and actual tp postons wth respect to tme for msson arm. Here for msson arm, X tp poston error vares between -.59 to +.5 mm, Y tp poston error vares between -.8 to +.4 mm and that of Z tp, t vares between -.6 to +.1 mm, ecept the ntal phase and t s due to that the tp acqures a velocty from the rest poston. From Fgure 5 to 8, t s observed that msson arm tp s effectvely dragged along the reference trajectory. Fgure 9 shows a three-dmensonal plot of the balance arm tp trajectory. he trajectory followed by tp of the balance arm n X-Z plane s presented n Fgure 1. From Fgure 9 and 1, t s seen how the balance arm tp moved whle controllng the space vehcle atttude dsturbance. he varatons n Euler angles,, and for the space vehcle carryng two arms as well as the space vehcle carryng sngle arm (msson arm) only wth respect to tme are shown n Fgure 11, 1 and 1, respectvely. From these fgures, t s observed that for a same set of parameters and reference trajectory that of the dual arm, the space vehcle carryng sngle arm havng a contnuous drft n Euler angle from to -.55 radans. he space vehcle carryng two arms, the varaton n Euler angle s reduced sgnfcantly and s between -.8 to +.77 radans only. he varaton n Euler angle for sngle arm space vehcle s. to +.65 radans and t s further mproved for dual arm space vehcle whch vares from to -.8 radans. However, there s no any mprovement n Euler angle but t vares between -.75 to +.75 radans for dual arm space vehcle. 5. CONCLUSION hs work contrbutes a control strategy for small atttude dsturbance of space vehcle whle msson arm follows the desred trajectory. Here our objectve s that the msson arm should follow the desred trajectory but vehcle atttude dsturbance should be small. hs objectve has been acheved wth the help of the proposed control laws of balance arm. Fgure 1: Plot of Varaton of Euler Angle for Sngle and Dual Arm Space Vehcle versus me As the reacton wheels are provded to ntroduce the redundancy n the system, the nput source of voltage to them s kept zero. In ths condton, they wll ACKNOWLEDGMENS We are thankful to Indan Space Research Organzaton (ISRO) for fundng the project vdes Grant No. ISRO/RES//596/9-1. Page 7

8 APPENDIX A Parameters used n smulaton Lnk Parameters Mass (kg) Length l (m) Length l G (m) I XX (kg m ) I YY (kg m ) I ZZ (kg m ) Intal Jont angle (rad) For msson arm Lnk Lnk / Lnk / For balance arm Lnk Lnk / Lnk / Space vehcle Intal Arm Base Poston from Vehcle CM Msson Arm Balance Arm r. m -. m r y. m -. m r z. m. m Overwhelmng Controller Parameters Parameter value Mass (M c ) 1. kg Stffnesss (K c ) 1e5 N/m Rasstance (R c ) 1e Ns/m Feed forward gan ( H ) 1. Atttude Controller Gan Parameters Proportonal gan of controller (Kp) 1 Dervatve gan of controller (Kv) 5 Reference rajectory Parameters Radus of reference crcle (R). m Ampltude of velocty n Y-drecton (A).1 m Angular velocty () 1. rad/s Actuator and reacton wheel Parameters Inductance of the armature (I m ).1 H Resstance of the armature (R m ). ohm Inerta of reacton wheel (I RW ).5 kg m Bearng resstance for reacton wheel (R b ). Nm/rad/s Pad Parameters Stffness of hard sprng (K h ) e5 N / m Stffness of soft sprng (K s ) 1e5 N/m Dampng resstance (R d ) 1e Ns/m Jont Resstance (R j ).1 Nm/rad/s Page 8

9 REFERENCES Dauphn-anguy, G., Rahman, A., and Sueur, C., Bond graph Aded Desgn of Controlled Systems. Smulaton Practce and heory, 7(5-6), Dubowsky, S. and Papadopoulos, E., 199a. he Knematcs, Dynamcs, and Control of Free-Flyng and Free-Floatng Space Robotc Systems. IEEE ransactons on Robotcs and Automaton, 9(5), Dubowsky, S. and Papadopoulos, E., 199b. Dynamc Sngulartes n the Control of Free-Floatng Space Manpulators. ASME Journal of Dynamc Systems, Measurement and Control, 115(1), Huang, P., Xu, Y., and Lang, B., 5. Dynamc Balance Control of Mult-arm Free-Floatng Space Robots. Internatonal Journal of Advanced Robotc Systems, (), Moosavan, S. A. A. and Papadopoulos, E., On the Knematcs of Multple Manpulator Space Free-Flyers and ther Computaton. Journal of Robotc Systems, 15(4), Nakamura, Y. and Mukherjee R., Non-holonomc Path Plannng of Space Robots va a Bdrectonal Approach. IEEE ransactons on Robotcs and Automaton, 7(4), Pathak, P. M., Kumar, R. P., Mukherjee, A., and Dasgupta, A., 8. A Scheme for Robust rajectory Control of Space Robots. Smulaton Modelng Practce and heory, 16(9), Yokokohj, Y., oyoshma,., and Yoshkawa,., 199. Effcent Computatonal Algorthms for rajectory Control of Free-Flyng Space Robots wth Multple Arms. IEEE ransactons on Robotcs and Automaton, 9(5), Chen, L. and ang, X., 6. Optmal Moton Plannng of Atttude Control for Space Robot System wth Dual-arms, Proceedngs of 6th World Congress on Intellgent Control and Automaton, pp June 1-, Dalan, Chna. Marches, M. and Angrll, F., Control Strategy for a Free-flyng Space Manpulator. Proceedngs of 8th Internatonal Conference on Advanced Robotcs, pp July 7-9, Monterey, CA. Papadopoulos, E. and Moosavan, S. A. A., Dynamcs and Control of Mult-arm Space Robots durng Chase and Capture Operatons. Proceedngs of the Internatonal Conference on Intellgent Robots and Systems, pp September 1-16, Munch, Germany. Breedveld, P. C. and Dauphn-anguy, G., 199. Bond Graphs for Engneers, Amsterdam North-Holland, Elsever. Crag, J. J., Introducton to Robotcs Mechancs and Control, Addson-Wesley Publshng Co. Karnopp, D. C., Margols, D. L., and Rosenberg, R. C., 6. System Dynamcs: Modelng and Smulaton of Mechatroncs Systems, New Jersey, John Wley and Sons Inc. Mukherjee, A., Karmarkar, R., and Samantray, A. K., 6. Bond graph n Modelng, Smulaton and Fault Identfcaton, Inda, I.K. Internatonal Publshng House Pvt. Ltd. Ghosh, A. K., 199. Dynamcs and robust control of robotc system: a bond graph approach. hess (PhD). Department of Mechancal Engneerng, Indan Insttute of echnology, Kharagpur, Inda. Users Manual of SYMBOLS Shakt, 6. Hgh-ech Consultants, S..E.P., Indan Insttute of echnology, Kharagpur. AUHORS BIOGRAPHY Haresh Patola receved the B. E. degree n 1991and M. E. degree n 1, both n Mechancal Engneerng from B. V. Mahavdyalaya, V. V. Nagar, Inda. He joned the Department of Mechancal Engneerng at B. V. Mahavdyalaya, V. V. Nagar, Inda n Currently he s a Ph. D. canddate at the Department of Mechancal and Industral Engneerng, Indan Insttute of echnology, Roorkee, Inda. P. M. Pathak receved the B. ech. degree n 1988 from R. E. College, Calcut, Inda and M. ech. degree n 1998 from II, Kanpur, Inda both n Mechancal Engneerng. He receved Ph. D. degree n 4 from II, Kharagpur, Inda. He s an Assstant Professor at the Department of Mechancal and Industral Engneerng, Indan Insttute of echnology, Roorkee, Inda. Hs areas of nterest are space robotcs, walkng robots, dynamcs and control. S. C. Jan receved the B. Sc. Engg. degree n 1969 from AMU, Algarh, Inda and M. E. degree n 1971 from Unversty of Roorkee, Inda, both n Mechancal Engneerng. He receved Ph. D. degree n 1985 from Unversty of Roorkee, Inda. He s a Vce-Chancellor at Mangalayatan Unversty, Algarth, Inda. Hs areas of nterest are trbology, vbraton and nose control, CAD, and robotcs. Page 9

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