Sensors & ransduers, Vol., Speal Issue, May 3, pp. 4-47 Sensors & ransduers 3 by IFSA http://www.sensorsportal.om Path Followng Control of a Spheral Robot Rollng on an Inlned Plane ao Yu, Hanxu Sun, Qngxuan Ja, We Zhao Shool of Automaton, Bejng Unversty of Posts and eleommunatons, 876, Bejng, Chna el.: +866857, fax: +866857 E-mal: yutaogzm@sohu.om Reeved: 3 Marh 3 /Aepted: 4 May 3 /Publshed: 3 May 3 Abstrat: In ths paper, the path followng ontrol of a spheral robot rollng wthout slppng on an nlned plane s dsussed. We frst study the knemat onstrants of the spheral robot and develop the dynam model of the robot through the onstraned Lagrange method. We then present a state spae realzaton of ths onstraned system through the null spae method and usng nonlnear feedbak. We nvestgate the path followng ontrol algorthms and develop a varable struture approah to the ontrol of ths nonholonom system. By hoosng approprate output equatons for path followng, we desgn the sldng surfaes as the funtons of the output trakng errors. Usng Lyapunov stablty theorem and exponental reahng law, we derve the sldng mode ontrol law. he asymptot stablty of the sldng surfaes s theoretally proved, and the valdty of the proposed path followng method s further valdated through MALAB smulatons. Copyrght 3 IFSA. Keywords: Spheral robot, Inlned plane, Path followng, Sldng mode ontrol, Nonholonom system.. Introduton Most moble robots we have today have wheels. hat s an obvous hoe as there s a onsderable amount of knowledge about ths type of loomoton. However, more and more possble applatons our wheeled robots have some flaws. Spheral robots ould be a soluton to some of these problems. As the robot s enompassed n a ball t s possble to effetvely seal everythng to enable the robot to wthstand exposure to dust, humdty, dangerous substanes and other envronmental threats. As we an understand, ths ould be very handy n suh applatons as planetary exploraton, survellane and others. he above mentoned stuatons often nvolve dealng wth dffult terran as well. Whle wheeled robots an ope wth t pretty good, the rsk of fallng over stll perssts. A spheral robot, on the other hand, an't fall over at all. Over the last few deades, there has been onsderable nterest n the development of powerful methods for moton ontrol of moble robots. he problems addressed n the lterature an be roughly lassfed nto three groups: trajetory trakng, path followng and pont stablzaton []. Wth respet to spheral robots, there have not been establshed methodologes to resolve these ontrol problems, although many studes have been made durng the past. Alves and Das [] presented a lne trakng method of a spheral robot based on knemats. Zhan and Lu et al. [3] dsussed the trajetory trakng problem of a spheral robot usng baksteppng approah. Zheng and Zhan et al. [4] nvestgated the trajetory trakng algorthm for a 4 Artle number P_SI_339
Sensors & ransduers, Vol., Speal Issue, May 3, pp. 4-47 spheral robot based on the RBF-PD ontroller. Ca and Zhan et al. [5] proposed a real-tme fuzzy gudane sheme for trajetory trakng of a spheral robot. Lu and Sun et al. [6] developed a lne followng ontroller for a spheral robot based on sldng modes. Current researhes on moton ontrol of spheral robots usually assume that the robot remans strtly on a level plane. As a result, the dynam model fals to represent the atual moton when the robot rolls up a slope. hs paper fouses on pratal solutons to trajetory trakng and path followng ontrol of a spheral robot rollng on an nlned plane. he man ontrbutons of the paper nlude two parts. Frstly, the knemats and dynams of the robot subjet to no-slp and no-spn onstrants are derved. Seondly, a sldng mode ontrol sheme for path followng of the robot s proposed. of the sphere B. We denote (,, ) to be the ZYX Euler angles from the nertal frame O to the body oordnate frame B. Fg. presents the geometral model of the rollng sphere and defnton of neessary varables to dedue the mathematal model. Here R represents the radus of the sphere, C s the ontat pont between the sphere and slope surfae wth ts oordnates (x, y ) wth respet to O, and denote the torques exerted on the sphere along the axs X b and Y b respetvely. Z b R B Y b X b. Mathematal Model.. System Desrpton Z Y C x, y ) ( BYQ-VIII s a pendulum-drven spheral robot, the mehanal struture of the robot s llustrated n Fg.. he robot s manly onssted of three parts: the spheral shell, the nternal gmbals and the pendulum 3. he robot has the nternal drvng unt mounted nsde the spheral shell. he steerng moton of the robot s aheved by tltng the pendulum, and the drvng moton s performed by swngng the pendulum ndretly through the nternal gmbal. X Fg.. Defnton of system varables for BYQ-VIII. We now derve the knemats and dynams of the spheral robot on the bass of the followng assumptons: () no-slp onstrant: the sphere rolls on the perfetly flat surfae of the nlne wthout slppng, () no-spn onstrant: rotatons of the rollng sphere around the Z axs are not allowed. Let ν B and ω B denote the veloty of the enter of mass of the sphere and ts angular veloty wth respet to the nerta frame O. hen, we have j k B x y z os os x sn os snos, y sn z (), j, k are the unt vetors of O. he no-spn onstrant an be formulated as Fg.. hree-dmensonal model of BYQ-VIII... Knemat Constrants We frst assgn two oordnate frames. Let O XYZ be a fxed nertal frame whose XY plane s anhored to the surfae of the nlne and Z s the vertal poston to the surfae. Let B X b Y b Z b be the body oordnate frame whose orgn s loated at the enter sn () he onstrant n () represents a nonholonom onstrant. he onstrants result from the requrement that the sphere rolls wthout slppng on the nlne,.e., the veloty of the ontat pont on the sphere s zero at any nstant ν C =. Now we an express ν B as vb BrBC v C, (3) 43
Sensors & ransduers, Vol., Speal Issue, May 3, pp. 4-47 r BC = -Rk represents the vetor from pont C to B. Substtutng () nto (3) gves v x y jz k (4) B x R os snos (5) y os os R sn (6) Z (7) he onstrants n (5) and (6) are nonholonom, as the onstrant n (7) s holonom and an be ntegrated to obtan Z = R. herefore the onfguraton of the robot system an be desrbed by a vetor of fve generalzed oordnates q = (x, y,,, ). Combng (), (5) and (6), the nonholonom onstrants an be wrtten as Aq Aqq (8) Ros Rsnos sn os os R R sn.3. Robot Dynams We study the moton equatons by alulatng the Lagrangan L = - P of the system, and P are the knet energy and potental energy of the system respetvely. he spheral shell s assumed to have mass m and the moment of nerta I. hen and P of the robot system an be alulated as follows mx y Ix y z P mg y sn Ros, (9) s the nlnaton angle of the slope. Usng the onstraned Lagrangan method, the moton equatons of the robot system are desrbed by, t M q q V q q E q A q () m m Mq I Isn Is Isn I sn mg V q, q I os I os I os E q t t t o elmnate the Lagrange multplers [7], we frst partton A(q) as A = [A A ], A A Let Ros Rsnos sn os os R R sn A A Cq () I k, k,,. It s straghtforward to verfy that C(q) satsfes that A(q)C(q) =. If we hoose and to be the two quas-oordnates, t and we an verfy that () s satsfed. t q C q () Dfferentatng () yelds q C C (3) Usng () and (), we have C E = I. Substtutng (3) nto (), and premultplyng both sdes by C gves t t t (4) C MC C MC C V Usng the state varable x = (q, υ ), we have C x f M t, (5) f M C MC C V. We apply the followng nonlnear feedbak [8] M C MC, t M u f, (6) u = (u, u ) represent the new ontrol nputs. 44
Sensors & ransduers, Vol., Speal Issue, May 3, pp. 4-47 hen the state equaton smplfes to the form f x f x b x u (7) x C q bx I y J. h h J h u, We defne the followng sldng surfaes s y y s y y d, (3) (4) 3. Path Followng 3.. Controller Desgn In the path followng task, the ontroller s gven a geometr desrpton of the assgned Cartesan path. For ths task, tme dependene s not relevant beause one s onerned only wth the geometr dsplaement between the robot and path. he path followng problem s rephrased as the stablzaton to zero of a sutable salar path error funton. Sne the robot system has two ontrol nputs, we may hoose two output varables. By approprately hoosng the output varables h and h we an aheve path followng ontrol. Suppose the referene path s f ( x, y )= (8) We defne the path error funton e f as e = f ( x, y ) (9) f hen h an be hosen as h q e = f f( x, y) () he other output varable h s hosen to be one of the quas-velotes of the robot system. h () Dfferentatng () one and twe respetvely gves s a real postve onstant, y d s the desred value for y. Let S be the vetor of omponents s and s. s y y Jh C y S (5) s y yd y yd Dfferentatng (5) yelds J h C g J h Let S f g u (6) Jh C C J h C f q s s sgn S sgn sgn (7) η, η are real postve onstants. We hoose the ontrol law u as follows u g sgn S KS f (8) k K onstants. 3.. Stablty Analyss, k and k are real postve k h q y= q J h q C q q y Jh C C J h q C q q J f f h x y. Dfferentatng () yelds u () heorem : Suppose that the system n (7) s ontrolled by the ontrol law gven by (8), then the sldng surfaes s, s are asymptotally stable. Proof: Substtutng (8) nto (6) yelds s sgn s ks s sgn s k s Consder the Lyapunov funton anddates V s,, (9) 45
Sensors & ransduers, Vol., Speal Issue, May 3, pp. 4-47 Dfferentatng V wth respet to tme gves V ss s k s (3) wth the ontroller parameters = 4.9, k = 4.5, k =.3, η =, η = Integratng both sdes of (3), we an obtan V t s V t s ks V lm d t (3) he performane of the robot n rular path followng s shown n Fg. 4. 3 atual desred From (3), we have s L, s L. From (3), we have s L. Consequently, aordng to Babalat s lemma we have lm s,,. t y(m) 4. Smulaton Study We developed a omputer smulaton n order to verfy the valdty of the ontrol algorthms dsussed n the prevous setons. he dmensons and nertal parameters are representatve of the spheral robot. Aordng to the notaton ntrodued before: m =.85 kg, R =.4 m, I =. kg m, = º..5.5.5 3 x (m) Fg. 3. Control performane n straght lne path followng. 4.. Bas Paths.5 Consder a straght lne path y = x, as shown n Fg. 3. he ntal values of the system onfguraton varables are suh that (x, y,,, ) = (.4,.,,, ) and the ntal veloty s zero. For the path followng algorthm, h = x - y, y d =.5 y(m).5 atual desred.5.5 x (m) wth the ontroller parameters = 4.8, k = 4.3, k = 4.5, η =.6, η =. he performane of the robot n straght lne path followng s shown n Fg. 3. he star marker ndates the startng pont of the robot. Consder a rular path (x -.8) + (y - ) =.5 as shown n Fg. 4. he ntal values of the system onfguraton varables are suh that (x, y,,, ) = (.,.4,,, ) and the ntal veloty s zero. For the path followng algorthm, h = (x -.8) +(y - ) -.5, y d =.5 Fg. 4. Performane of the robot n rular path followng. In both ases, the referene pont of the robot s able to reah the path and stay on the path. he path followng algorthm seems to exhbt a gradual merge, wth the path followng ontroller the atual path followed s smooth. 4.. Peewse Contnuous Paths An example of a omposte path s shown n Fg. 5, whh s omposed of two rular ars and a straght lne. he performane of the ontrol system s aeptable as seen from the fgure. he dsontnutes n urvature are negotated wthout any dffulty and there s almost no devaton of the atual path from the desred one. 46
Sensors & ransduers, Vol., Speal Issue, May 3, pp. 4-47 y(m).5 atual desred Fund of the Key Sentf and ehnal Innovaton Projet, Mnstry of Eduaton of Chna (No. 78) and Bejng Natural Sene Foundaton Program and Sentf Researh Key Program of Bejng Munpal Commsson of Eduaton (No. KZ85)..5.5.5.5 3 3.5 4 4.5 x (m) Fg. 5. Control performane n omposte path followng. 5. Conlusons We presented a varable struture method for path followng ontrol of a spheral robot movng on an nlned plane. We derved the knemats by mposng the onstrant ondtons of no-slp and nospn. We dedued the robot dynams usng the onstraned Lagrange formulaton. We elmnated the Lagrange multplers to obtan a state spae desrpton of the system. We devsed a sldng mode sheme to aheve output trakng. We onsdered the hoe of output varables for path followng and derved the sldng mode ontrol laws to satsfy the exstene ondton of sldng modes. Computer smulaton results were presented to llustrate the performane of the proposed ontrol algorthm. Aknowledgements he authors wsh to aknowledge the fnanal support provded by the Natonal Natural Sene Foundaton of Chna (No. 57548), the Cultvaton Referenes []. Soeanto D., Laperre L., Pasoal A., Adaptve nonsngular path-followng ontrol of dynam wheeled robots, n Proeedngs of 4 nd IEEE Conferene on Deson and Control,, 3, pp. 765-77. []. Alves J., Das J., Desgn and ontrol of a spheral moble robot, n Proeedngs of the Insttuton of Mehanal Engneers. Part I: Journal of Systems and Control Engneerng, 7, 3, No. 6, pp. 457-467. [3]. Lu Z., Zhan Q., Ca Y., Moton ontrol of a spheral moble robot for envronment exploraton, Ata Aeronauta Et Astronauta Sna, 9, 8, No. 6, pp. 673-678. [4]. Zheng M., Zhan Q., Lu J., Ca Y., rajetory trakng of a spheral robot based on an RBF neural network, n Proeedngs of the IEEE Internatonal Conferene on Eletral Engneerng and Automat Control, 7,, pp. 5-54. [5]. Ca Y., Zhan Q., X X., Path trakng ontrol of a spheral moble robot, Mehansm and Mahne heory, 5,, pp. 58-73. [6]. Lu D., Sun H., Ja Q., Stablzaton and path followng of a spheral robot, n Proeedngs of the IEEE Internatonal Conferene on Robots, Automaton and Mehatrons, 8, pp. 676-68. [7]. Bloh A. M., Reyhanoglu M., MClamroh N. H., Control and stablzaton of nonholonom dynam systems, IEEE rans. Aut. Control, 37, 99, No., pp. 746-757. [8]. Sarkar N., Yun X., Kumar V., Control of mehanal systems wth rollng onstrants: applaton to dynam ontrol of moble robots, Internatonal Journal of Robots Researh, 3, 994, No., pp. 55-69. 3 Copyrght, Internatonal Frequeny Sensor Assoaton (IFSA). All rghts reserved. (http://www.sensorsportal.om) 47