Minimum Input Walking Gait of Four-DOF Biped Model Solved by Optimal Trajectory Planning Method

Transcription

1 nu Input Walkng Gat of Four-DOF Bped odel Solved by Optal Traectory Plannng ethod Chunye Peng and Kyosuke Ono Tokyo Insttute of Technology Dept. of echancal and Control Engneerng -- Ookayaa eguro-ku Tokyo Japan Abstract In ths paper an optal traectory plannng study on a planar bped walkng echans s perfored. The bped locooton echans that has thghs shanks and sall feet was odeled as a four-degree-of-freedo (DOF) lnk syste coposed of a two-dof stance leg and a two-dof swng leg that connected drectly at the hp ont. By usng the functon approxaton ethod and nonlnear prograng ethod the nu nput walkng gats under full-actuated condton that slar to those of the huan walkng was obtaned. Also the valdty of ths ethod s confred by forward dynac sulaton.. Introducton There has long been an nterest n understandng huan walkng locooton not only fro a desre to buld bped echanss to perfor dangerous tasks nstead of huans but also to prove prosthetc devces for aputees. In recent years any studes have been publshed about bped walkng systes ether fro the vewpont of constructng huan-lke bped walkng echans by ont torque control or as to dscover walkng characterstc by easurng the huan otons durng locooton. ta and Yaaguch et al. [] Furusho and asubuch [] Kato et al. [] developed varous knds of bped echanss by usng odel-followng control ethods to sulate and realze bped locooton. Vladr [5] wrote a book descrbng the ethods techncal devces and procedures used when easurng both pathologcal and/or healthy huan locooton. owever ost of the researchers dd not consdered the energy effcency proble of the walkng locooton wth a notable excepton of Slva and achado [6] Ono and Lu [7]. Slva and achado characterzed the bped oton by a set of locooton varables and establshed the correlaton aong these locooton varables and the energy perforance. Ono and Lu bult a -degree-of-freedo (DOF) walkng odel that coposed of a knee-less stance leg and a -DOF swng leg. Dvded one swng phase nto two sectons: controlled swng oton of -DOF swng leg wth straght stance leg and free oton of straght swng leg wth straght stance one. And they obtaned an energy effcent gat wth nu nput torque of the controlled secton by usng optal traectory plannng ethod. Experental studes of huan locooton [8] support the hypothess that the choce of a gat pattern s nfluenced by energy consderatons. That s the huan body wll ntegrate the otons of the varous segents and control the actvty of the uscles so that the etabolc energy requred for a gven dstance walked s nzed. Bearng these n nd a traectory plannng study on a -DOF planar bped syste s perfored to gve a better nsght nto the echanss of huan walkng locooton whch s yet to be fully understood. The reander of the paper s organzed as follows: A -DOF bped odel and ts oton equatons of one step accopany wth the cyclc walkng condtons derved fro the pulse-oentu equatons for the toe collson are gven n secton. In secton we descrbe the algorth used to plan the optal traectores of the bped echans. Based on these ndces an effcent and natural walkng gat n the swng phase s solved by the optal traectory plannng ethod and several nuercal results and ther forward dynac confraton are presented n secton. Fnally n secton 5 we outlne the an conclusons and the perspectves towards future research.. Bped Walkng echans and Its Dynac Equatons Fgure shows the bped echans that walks wth bent knee. The new odel looks a lttle dfferent fro our precedng one. In the present study we assue that one step of walkng s dvded nto two phases as shown n Fg. : a) Swng phase n whch one leg s n contact wth the ground and the other leg swngs forward b) Stance phase n whch the legs trade ther roles. In the swng phase (posture -) the stance leg s n contact wth the ground and carres the weght of the body fro left to rght lke a -DOF nverted pendulu whle the swng leg oves forward lke a -DOF pendulu n preparaton for the next step. The te nterval of ths phase s denoted by t. The foot exchange takes place nstantly (posture-)

2 once the swng leg touches the ground. The te nterval of ths stance phase t s zero. ere the pact of the swng leg s assued to be perfectly nelastc whle ensurng that no slppage occurs and an adequate reacton force s prescrbed allowng a sooth transton of support. Fro posture the next swng phase begns. Ths cyclc pattern of walkng oveents s repeated over and over step after step wth a reasonable assupton that successve cycles are all about the sae. Fg. : One step of walkng. Fgure shows the planar bped odel wth the notaton used thought ths paper. ere we focus only on the bped locooton n the sagttal plane. The bped echans odel conssts of four lnks n order to approxate locooton characterstcs slar to those of the lower extretes of the huan body (.e. hp thgh and shank). Fg. : Planar bped odel ere we dsregarded the torso ass because t has lttle effect on walkng locooton [7]. And the two legs assued to be drectly connected to each other through a servo-actuator at the hp ont and both knee and ankle onts are drven by ndvdual servoactuators. The ankle of the stance leg s odeled as a rotatng ont fxed to the ground whle the foot of the swng leg s neglected. Notatons l a I and u n the fgure are defned as follows: l : length of lnk : ass of lnk a : dstance between the ass center of lnk and ts upper ont I : oent of nerta about the ass center of lnk : angle of lnk u : nput torque at ont Usng Lagrangan dynacs the dynac equatons for the -DOF bped odel of the swng phase are derved as follows: Sy 0 AntSy () C C C K u u 0 C C K u u = 0 C K u u 0 K u where C and K are calculated fro l a I and. In ths bped odel t s assued that the foot exchange takes place nstantly and the collson between the swng leg and the ground s perfectly nelastc for the sake of splcty. The behavor of the stance phase (posture -) s shown n Fg.. represent the angular velocty of lnk at the oent rght before the foot exchange (posture n Fg.) whereas stands for the angular velocty of lnk rght after the foot exchange (posture n Fg.). By usng the pulse-oentu equatons for translaton and rotaton the relatonshp between and s obtaned as follows:

3 p = 0 Sy p (5) Equaton () and (5) s the cyclc condton of walkng locooton and can be rewrtten n the general for: cb p p p ( ) = 0 (6) Fg. : Stance phase (Foot exchange) = () 0 Sy where a and are calculated fro l I and. Fro posture the next swng phase begns. Wth the assupton that the successve step s the sae wth current step the oton state varables of posture ust be the sae as those of posture. Ths can be expressed as: p = = p p p () Fro Fg. and Equaton () the angular relaton between posture and posture can be calculated as follows: p p p p = = = = p p p p π π π π () By pluggng Equaton () back nto Equaton () the angular veloctes of posture can be expressed by those of posture as follows:. Optal Traectory Plannng In ths paper the swng phase of walkng locooton s solved by the optal traectory plannng ethod. Based on the nzed energy hypothess the obectve functon J here s defned as the su of the ntegraton of the square nput torque for one step: t J = k u ( t) dt (7) w = 0 where k w : weghtng factor correspondng to ont here we set k w = for full actuated condton. u : nput torque at ont. t : te nterval of one step. Rearrangng Equaton () u can be expressed as: = Γ ( ) ( =~) (8) u where { } T =. Applyng the functon approxaton ethod the traectory of the ont angle s represented by a lnear cobnaton of the erte bass functons [9]. For exaple f the swng phase s dvded nto subsectons then the ont oton traectory of the -th subsecton can be expressed as: T ( p t) = h( t) p (... = ) (9) where h (t) s the erte bass functon vector and p s the boundary state varable vector. In ths study we adopt 7th-order erte bass functons and set = thus the p s of the for:

4 T T T T T T T T { } T p = b e b e b e b e (0) where b are the begnnng boundary state b varables of the swng phase whle e are the e end boundary state varables of the swng phase. Substtutng Equatons (8) and (9) nto Equaton (7) we get an obectve functon J related to p thus the proble of the calculus of varaton subects to the dfferental equaton constrants s translated nto a paraeter optzaton proble for the bass functon coeffcents. That s to fnd proper unknown boundary state vectors n Equaton (0) whch nze the obectve functon n Equaton (7) n a syste that s descrbed by Equaton () and subect to the constrant condtons descrbed by Equaton (6). Fg. shows the flowchart of the calculaton algorth.. Results and Dscussons Usng the traectory plannng ethod stated above we got the optal traectory walkng gats of the - DOF bped odel that has the sae segent paraeters of the lower lb as those of ost of us. Table shows the lnk paraeter values used n the calculaton. Table. Lnk paraeter values Paraeters st Lnk nd Lnk rd Lnk th Lnk Length l [] ass [kg] Center of ass a [] oent of nerta at ass center I [kg ] Fg. : Flowchart In Fg. 5 we gve the optzed traectory results of the bped odel walkng through a fxed step length of about 0.5 eters n dfferent te ntervals wth soe knee flexon durng stance. The lowest energy consupton occurs near T=0.60s. We note that the gat of ths te nterval s very sooth and closely slar to that of huan bengs. In fgures 6 and 7 we gve the ont nput torque and lnk angular veloctes of each te case respectvely. Coparng to other te cases (Fg. 5-7 a c and d) we can see that the case (b) have the sallest aount and varaton both of the ont nput torque and the lnk angular veloctes. So case (b) can be consdered to be the ost optal walkng gat. For the purpose of coparng we also calculated the optal traectory gats of the bped odel walkng through a sae step length of about 0.5 eters n dfferent te ntervals wthout any knee flexon durng stance. Ths te the ost energy effcent gat occurs near T=0.50s a lttle qucker than that of the walkng gat wth knee flexon. But the stck fgure (Fg. 8a) s not as sooth as the forer ones (Fg. 5). And the toe traectory of the swng leg exhbt soewhat undulaton together wth the fluctuaton of the ont nput torque (Fg. 8b) and lnk angular veloctes (Fg. 8c). The calculated obectve functons J that of the optzed walkng locooton wth and wthout knee flexon durng stance are shown n Fg. 9.

5 (a) T=0.50 s (a) T=0.50 s (b) T=0.60 s (b) T=0.60 s (c) T=0.70 s (c) T=0.70 s (d) T=0.80 s Fg. 5: Stck fgures of walkng through a dstance of about 0.5 wth knee flexon n dfferent te ntervals (d) T=0.80 s Fg. 6: Input torque of walkng wth knee flexon. Note that case (b) sees ost natural because both the nput torque and the leg undulaton of ths case are saller than those of the others.

6 (a) T=0.50 s (a) Stck fgure of walkng wthout knee flexon (b) T=0.60 s (b) Jont torque of walkng wthout knee flexon (c) T=0.70 s (d) T=0.80 s Fg. 7: Angular veloctes of walkng wth knee flexon. Case (b) sees ost natural for slar reasons as those of Fg. 6. (c) Angular velocty of walkng wthout knee flexon Fg. 8: Walkng wthout knee flexon (T=0.50 s). It sees unnatural because of large ont torque and lnk angular velocty undulatons. One of our as of ths study s to obtan optal nput torque at all onts of the bped odel as shown n Fg. to understand the way how huan beng do t and to realze bped walkng echanss wth energy effcency. ere coes a queston that s whether or not the sae bped locooton can be perfored by usng the calculated nput torque. Adopt the optzed nput torque shown n Fg. 6(b) as feed forward control we sulated the walkng oton of the bped and the results nclude stck fgure (a) and lnk angular veloctes (b) are shown n Fg. 0. It s clear that they are alost all the sae wth the

7 traectory planned ones (Fg.5b and Fg.7b). uch. And the experental studes of huan locooton [8] deonstrated the fact that there DO have soe degrees between the thgh and the shank of the supportng leg durng stance. 5. Concluson Fg. 9: Calculated obectve functons. Note that the knee flexon not only lowered the energy cost but also lengthened the energy effcent step perod. In ths paper the optal oton traectory of the - DOF bped walkng odel s nuercally calculated by usng the traectory plannng ethod. And the correctness of ths ethod has been proved by forward dynac sulaton. The calculated traectory sees closely lke that of people and confred the hypothess that the choce of a walkng gat pattern s nfluenced by energy consderatons. We can say that ths optal traectory plannng ethod s an effectve way to analyze the huan walkng locooton. We wll contnue the study on huan walkng by addng torso and feet to the bped odel to gve a better nsght nto the prncples of huan walkng locooton. References (a) Forward sulated stck fgure (b) Forward sulated angular velocty Fg. 0: Forward dynacs sulaton results (T=0.60 s). They see that alost all the sae wth the traectory planned ones (Fg. 5b and Fg. 7b). Although ost of us do not have the experence of walkng wthout any knee flexon durng stance fro the above fgures especally fro the optzed stck fgures of the gat wth knee flexon we have to say that t DO sees lke the way we walk everyday very [] tat. YaaguchT. KashwaseT.et al. Realzaton of a hgh speed bped usng odern control theory. Internatonal Journal of Control 0(): pp [] Furusho J. and ashbuch. Control of a dynacal bped locooton syste for steady walkng. Journal of Dynac Systes easureent and Control Vol.08: pp [] Furusho J. and ashbuch. A theoretcally otvated reduced order odel for the control of dynac bped locooton. Journal of Dynac Systes easureent and Control Vol.09: pp [] KatoT. TakanshA. NatoG. and KatoI. The realzaton of the qusas dynac walkng by the bped walkng achne. Proceedngs of the Internatonal Syposu on Theory and Practce and anpulators POANSY pp [5] Vladr. easureent of huan locooton. CRC Press 00 [6] Slva F. and Tenrero J. Energy analyss durng bped walkng. Internatonal Conference on echanscs Autoaton Detrot chgan 999. [7] Ono K. and Lu R. Optal bped walkng locooton solved by traectory plannng ethod. Journal of Dynac Systes easureent and Control Vol. 00. [8] Jessca R. and Jaes G. uan walkng (second edton). Waverly Copany 99. [9] Iadu A. and Ono K. Optal traectory plannng ethod for a syste that ncludes passve onts (proposal of a functon approxaton ethod). JSE Internatonal Journal () 999 pp.09-5.