Coordination in CPG and its Application on Bipedal Walking

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1 Coordination in CPG and it Application on Bipedal Walking Weiwei Huang Department of Mechanical Engineering National Univerity of Singapore Singapore 96 Chee-Meng Chew Department of Mechanical Engineering National Univerity of Singapore Singapore 96 Geok-Soon Hong Department of Mechanical Engineering National Univerity of Singapore Singapore 96 Abtract Reearch of bipedal walking ha been motivated by it great flexible in rough terrain walking, a well a it benefit to prothei development. Thi paper aim to achieve coordination of central pattern generator (CPG) and aymptotically table walking behavior. CPG i alway referred a everal ocillator with coupled mutual inhibition. In thi paper, the ocillator model propoed by Matuoka wa ued. We ue output difference between two ocillator a the feedback to adjut the output of the target ocillator. Further exploration achieve to coordinate multiple ocillator with different frequencie. To verify the method, we demontrate a 3D robut walking controlled by a imple CPG tructure with everal ocillator. Index Term CPG, Neural Ocillator, Coordination, Bipedal Walking. I. INTRODUCTION CPG reearch i greatly motivated by the rapid growth of interet in biological inpired reearch. Biped walking of humanoid robot i a complex and challenging tak. However, human being walk gracefully and with good efficiency. CPG reearch will help to undertand the baic trategy of how human or animal perform walking or running. Walking eem baic and imple for human and animal, while it i daunting for legged robot. Well undertanding of the reearch problem could lead to an advance in humanoid robotic reearch and breakthrough in prothei development. With an effective CPG tructure, robot could achieve more natural, efficient and robut locomotion. CPG alway refer to a tructure which combine multiple coupled ocillator to generate walking gait. Ocillator i a baic component of CPG tructure. Variou model of ocillator have been propoed to explain the mechanim of the automatic ocillatory activitie [] [] [3]. Among thee model, neural ocillator i mot commonly ued which ha many good propertie uch a bio-inpired, entrainment and adaptation propertie. Beide ocillator model deign, CPG tructure need to well deigned to control the motion [4] [5] [6]. By properly deigning the tructure of the ocillator, a biped robot can achieve mooth walking in both imulation and real implementation [7] [8]. In thee work, deigning a coordination control tructure i the primary goal. To do thi, manipulation of phae relationhip between ocillator i very important, ince it determine the equence of each ub-motion controlled by ocillator and therefore the behavior of the robot. Thee relationhip will hift becaue of the external enory feedback. Klavin et al. propoed a general analyi of coordination between ocillator by phae regulation [9]. Other intereting work include the ynchronization of Kuramoto ocillator [] and coordination of a group of mobile robot by CPG []. In thi paper, we focu on the coordination between ocillator to achieve the mooth walking. The ocillator model we ue i neural ocillator model propoed by Matuoka. We preent a novel method to coordinate ocillator with feedback. An intereting apect of our approach i that an arbitrary phae relationhip could be adjuted and maintained between ocillator with the ame or different frequency. It could be alo applied to the cae of multiple ocillator. The preent tudy cloely connect the ocillator in CPG which ynchronize by the main ocillator. Thi paper i organized a follow. In ection II, we introduce the neural ocillator model and it entrainment propertie. The propoed method i preented in the ection III. We further explore the method to the cae of different frequency and multiple ocillator. Numerical reult are given. In Section IV, we verify the method by implementing it on a 3D bipedal robot to achieve walking behavior. In Section V, we preent dicuion of our reult and future work. II. NEURAL OSCILLATOR DESCRIPTION A. Neural Ocillator Model Neural ocillator i the mot commonly ued ocillator model in CPG reearch.it wa inpired from the behavior of biological neuron. It can output rhythmic ignal without external input and ha limit cycle property. The mathematical model i decribed by the following equation []. τ u = c u βv a[u ] + h j [g j ] + () τ v = [u ] + v () τ u = c u βv a[u ] + + h j [g j ] (3) τ v = [u ] + v (4) [u ] + = max(,u ) [u ] =min(,u ) (5) Y = [u ] + [u ] + (6) /8/$5. c 8 IEEE RAM 8

2 Here u () i the tate of the neuron; v () i the degree of neural adaptation; c i the contant timuli; τ and τ are the time contant; β i the parameter that indicate the effect of adaptation; a repreent the trength of inhibition connection between neuron; g j i the external input which uually could be the feedback pathway to ocillator; h j i a factor to adjut the input; Y i the ocillator output. The ocillator model conit of two imulated neuron arranged in mutual inhibition; each neuron ha an inner adaptation variable v i, a hown in Fig.. The tonic excitation c determine the amplitude of the ocillator, with amplitude proportional to c. The two time contant τ and τ determine the frequency of the ocillator and the parameter β affect the hape of ocillator output Fig.. Propertie of neural ocillator; top: entrainment property, bottom: uppreion of ocillator by a large contant input Fig. 3. New tructure where the error between ocillator output and enory input i conidered a an external input Fig.. Schematic of Matuoka neural ocillator model; white cycle mean excitation and black cycle mean inhibition B. Entrainment Property Ocillator ha the entrainment property. Williamon found that when an ocillatory input i applied, the ocillator can entrain the input, lock onto the input frequency [6]. When the external input ignal i large and contant, the ocillator output could be uppreed by the ignal. Fig.(top) how an example that the ocillator output lock to a inuoidal input, and Fig.(bottom) illutrate an example that the ocillator output i uppreed by a large contant external input. Entrainment lock the ocillator output with the external input, which hope to deign the natural repone of ocillator. Reearch work include operate a Silink toy to coordinate the motion of two arm by the entrainment property. Although the frequency could be locked to the external input, their phae relationhip i unclear. Here, we ue the error between ocillator output and enory input a the new external input to the neural ocillator. Fig. 3 how the tructure where K i the caling parameter. We urpriedly found that the ocillator could be converged to the external input. Fig.4 how the example; the ocillator output trie to follow the external input. For demontration, here parameter K only ha value when time i between to 6 econd; K equal in the other time. Many other different type of enory input are given to verify the congregation of ocillator. In Fig.5, three different type of enory input are given: triangular wave, quare wave and mall contant value. The ocillator can track thee different type of enory input. Here the contant could not be, becaue i one of the table point of ocillator. If the ocillator tate value go to, it will never ocillator again. Therefore, we et a mall value which i.. III. METHODOLOGY A. Propoed Method Ocillator ha the entrainment property. Thi property could be ued to coordinate ocillator in CPG. A we have dicued in the previou ubection, when the external input of the ocillator become the error between the ocillator output and Fig. 4. Example of ocillator converging to enory input with new tructure where K only ha value between -6 econd

3 O O Fig. 7. Coordination example between ocillator with ame frequency and different initial phae Fig. 5. Example of ocillator congregation with different type of enory input where K only ha value between -6 econd the enory input, the ocillator could be converged to the enory input. Thi bring about the potential olution for the coordination between ocillator. If the reference ocillator output i one of the enory input of the target ocillator, the target ocillator could be ynchronized by the reference ocillator. maintained after the effect of external input. A imple way to olve the problem i to give a reference trajectory for each ocillator and thee reference trajectorie are generated by one main ocillator. Fig. 8. phae The model to coordinate ocillator with different frequencie and Fig. 6. The model to coordinate two neural ocillator with ame frequency Fig.6 how the propoed method to adjut the output between two ocillator. Here g and g are election function. g = K max(e, ), g = K min(e, ) (7) where e i the error between O and O; K i the caling factor to modify the peed of adjutment. In thi tructure, O offer the reference output to O. The output error will feedback to O and O will try to match the match of O. Fig. 7 how an example of ocillator adjutment. In Fig.7, O and O have the ame frequency but different initial phae. O i able to ynchronize O and make O converge to O. Fig. 7 alo how that ometime amplitude may be affected in the coordination. Although it i not the deired one, it may play a poitive effect on the walking. For example, when the robot body i moving ahead while the wing leg motion i late and behind, a larger and fater wing tep may help to balance the walking cycle. The influence of thi effect will be further explored in the future reearch. B. Applied in Coordination Baed on previou method, the target ocillator output will converge to reference ocillator output. However, in the mot cae of CPG, we do not want ocillator have exactly ame output. We want the ocillator with different initial phae and frequency to ynchronize together. Thee different could be The propoed method i to get the phae value of main ocillator and then convert it to the reference phae value of target ocillator whoe frequency and initial value i different from main ocillator. The reference phae i changed to reference input to target ocillator by Inv function which ha a unique mapping form phae to output value. Fig.8 how the propoed method tructure. O and O are two ocillator which have different frequencie and different initial phae. Function f i ued to calculate the approximate phae value of the ocillator output. For each cycle of ocillator output, we divide it phae from to π. Φ = f(g(x)) (8) π x< A mp g(x) = arcin( x A mp ) A mp <= x<= A mp (9) π x<a mp { mod(y, π) = f(y) = () π y = where Φ i the approximate phae value; x i the output of ocillator; A mp i the amplitude of the ocillator. The output range of arcin i [ π, π ]. To convert the phae range to [, π], we define a parameter. The parameter indicate the direction change of the ocillator: =indicate an increae of the ocillator value while = indicate a decreae of ocillator value. The function f i ued to get the reference phae of O. Φ ref = f (Φ )=mod( F Φ, π)+φ d () F where F i the frequency of the th ocillator; Φ d i the deired phae different between O and O. For example,

4 two ocillator O and O need to maintain phae difference Φ d. The reference phae Φ r will be equal to Φ +Φ d.since O will converge to Φ r becaue of the coordination, the phae difference i maintained. When two ocillator frequencie are different, function mod( F F Φ, π) will help to make the reference phae change a fat a the phae of target ocillator. Since each phae ha only one ocillator output, we build a unique mapping from phae value to ocillator output. Inv function give the ocillator output according to reference phae. In thi cae, the reference input i the ame type of ocillator output a the target ocillator External O O External O+O Fig.. No coordination between ocillator when external input wa added External Fig. 9. The model to coordinate between multiple ocillator External O O The propoed method could alo be applied to the coordinate between multiple ocillation. Fig.9 how the model which could be ued to adjut multiple ocillator. Here we give an example of coordination between two ocillator which have different frequencie and initial phae value. The frequency of one neuron i twice of the other. The target output trajectory i the um of the output of the two ocillator. When there i an external input through g j which i alway the enory input, the hape of trajectory change a hown in Fig.. The reaon i that the phae relationhip between two ocillator i changed, a indicated by the red cycle. Thi i becaue two ocillator repone differently to the external input. When CPG i applied to control certain motion tak, change to enory input i neceary. It help balance walking according to environment change. However, the deired trajectory need to be recovered after the affection of external input. When there i no coordination between ocillator, thi can not be achieved. When the coordination i connected between ocillator, the deired ocillator output could be recovered. Fig. how the performance of the two ocillator with coordination. When there i external input, the ocillator are able to adjut the phae and recover to original target output. IV. DYNAMIC SIMULATION To further verify our propoed method, we have teted it on a 3D biped for walking. Firtly, we will preent our control Fig O+O Coordination between ocillator when external input wa added architecture and control trategy. Then, the dynamic walking imulation on level ground i implemented. A. Control Architecture In thi 3D imulation, the imulation model i built baed on a real 3D robot NUSBIP-II (http : //guppy.mpe.nu.edu.g/ legged group/). Each leg ha ix degree of freedom: three on hip, one on knee and two on ankle. Our imulation i carried out in Yobotic environment. Yobotic i a Java package for imulating multibody dynamic ytem (http : // Fig. how the imulation model. The dimenion and ma are hown in Table I. TABLE I THE SPECIFICATION OF SIMULATION MODEL link ma(kg) Ixx(kgm ) Iyy Izz length(m) Body Thigh Shank Foot

5 In thi approach, we propoe a CPG arrangement with repect to the poition of the leg in the Carteian coordinate pace (ee in Fig.). The ocillator are arranged to control X and Y direction of tance leg hip(hip x and Hip y )andx and Z direction of wing foot(f oot x,f oot z ). Since the target i traight level ground walking, the deired motion of hip Z direction and foot Y direction are et to be contant. The reference joint angle are then calculated by invere kinematic. We employed poition baed control in joint pace. Compared to joint pace implementation, thi arrangement ignificantly reduce the total number of ocillator parameter and provide a imple way to find effective feedback pathway. A hown in Fig., we aign the reference frame on the ankle joint of the tance leg. Baed on the reference frame, during walking Hip x trajectory alway move forward. The trajectory of Hip x i hown in Fig.3(dot line). The trajectory are not continuou when the upport leg witche. To obtain continuou trajectory, we invere Hip x every time when a particular upport leg touche the ground. The reulting trajectory i hown in Fig.3. correponding to the nd half period but negatived. We adopt the ame trategy for the wing foot horizontal poition F oot x reference to the tance foot. The hip Y direction trajectory i deigned to be cloe a inuoidal wave []. We aume that the walking height i contant which mean Hip z i a contant. The foot vertical trajectory F oot z i generated by an ocillator. To achieve a double upport phae, we et a threhold for ocillator output. The value val determine the double upport period. { O4 val O4 val F oot z = () O4 <val Fig.4 how the reference trajectory of tance hip and wing foot. O and O give the Hip x and Hip y trajectorie; O3 give the F oot x trajectory; F oot y i generated by O4. Coordination adjutment i connected between thee ocillator. O give the reference information to other ocillator(fig.) Reference Hip y Reference Hip x Reference Foot x Reference Foot z Fig. 4. Reference trajectory of tance hip and wing foot Fig.. L R Ocillator arrangement of the biped robot Fig. 3. Modified Hip x Original Hip x L L L L R R R R Reference Hip x trajectory. For the tance hip, a neural ocillator generate a periodic trajectory. Auming that the robot tand with left upport, the reference trajectory for the left upporting hip i correponding to the t half period of the neural ocillator trajectory and the reference trajectory for the right upporting hip i B. Senory Feedback Two imple enory feedback are ued to adjut the ocillator output: body pitch and roll(fig.5). They are given through g j. The roll angle i ued to adjut Hip Y direction motion while the pitch angle i connected to all the ocillator to balance the walking. The caling factor of thee feedback are optimized by GA method. C. Simulation Reult In the imulation, the ocillator parameter are derived uing the following rule [3]: ) To implify the frequency calculation, we let β = a. The frequency formula become F = πτ b where b = τ τ ; c ) Let +β+a = A, where A i the deired amplitude; 3) The value of b i choen to make a b mall; By thi procedure, we can roughly obtain all the value of the ocillator parameter which atify our requirement. In the imulation, the walking tep length i.4m and walking period i econd. The walking peed i approximate.4 m/. When there i no enory feedback and robot i controlled by

6 Pitch&Roll. Pitch Roll With Feedback Hip y No Feedback Fig. 5. Feedback pathway for robot motion the planed CPG trajectory, the CPG fail to balance the walking and the robot fall(fig.6). However, when proper feedback add on the CPG, the trajectory will be modified according to the environment and robot achieve a robut walking behavior. In Fig.8, we plot the trajectorie generated without and with feedback. It i obviou on thee graph that the pitch and roll feedback modified the trajectorie to balance the walking. Fig.7 how the naphot of normal level ground walking for the biped. Fig. 6. Robot fall in the level ground walking when when no feedback add on the CPG Fig. 7. CPG Robut level ground walking when proper feedback add on the V. CONCLUSION Thi paper preent a method to coordinate ocillator in CPG to achieve robut walking behavior. Numerical imulation demontrate that the method could adjut the phae of ocillator. We have alo teted the approach on a 3D biped to achieve table walking. An appropriate enory feedback i elected to help the ocillator repone correctly to the environment change. With the coordination between ocillator, the robot how a robut walking behavior. In thi paper, proper enory feedback help balance the walking behavior. More efficient feedback pathway will be tudied. Different way to coupling ocillator will be explored in the future reearch. Hip x Foot x Foot z Fig. 8. Effect of feedback on the generation of trajectory REFERENCES [] K. Matuoka, Sutained ocillation generated by mutually inhibiting neuron with adaptation, Biological Cybernetic, vol. 5, pp , 985. [] T. Zieliinka, Coupled ocillator utilied a gait rhythm generator of a two-leggeeed walking maachine, Biological Cybernetic, vol. 74, pp , 996. [3] J. Nakanihi, J. Morimoto, G. Endo, G. Cheng, S.Schaal, and M. Kawato, A framework for learning biped locomotion with dynamical movement primitive, Humanoid, pp , 4. [4] L. Righetti and A. J. Ijpeert, Programmable central pattern generator: an application to biped locomotion control, pp , IEEE International Conference on Robotic and Automation, may 6. [5] G. Taga, Y. Yamaguchi, and H. Shimizu, Self-organized control of bipedal locomotion by neural ocillator in unpredictable environment, Biological Cybernetic, vol. 65, pp , 99. [6] M. M. Williamon, Neural control of rhythmic arm movement, Neural Network, pp , 998. [7] G. Endo, J. Morimoto, J. Nakanihi, and G. Cheng, An empirical exploration of a neural ocillator for biped locomotion control, pp , IEEE International Conference on Robotic and Automation, April 4. [8] G. Endo, J. Morimoto, J. Nakanihi, and G. Cheng, Experimental tudie of a neural ocillator for biped locomotion with qrio, pp , IEEE International Conference on Robotic and Automation, April 5. [9] E. Klavin and D. E. Koditchek, Phae regulation of decentralized cyclic robotic ytem, International Journal of Robotic Reearch, vol., pp ,. [] N. Chopra and M. W. Spong, On ynchronization of kuramoto ocillator, 44th IEEE Conference on Deciion and Control, and the European Control Conference, 5. [] S. Wichmann, M. Hle, J. F. Knabe, and F. Paemann, Synchronization of internal neural rhythm in multi-robotic ytem, Adaptive Behavior, vol. 4, pp. 7 9, 6. [] J. Roe and J. G.Gamble, Human Walking. LIPPINCOTT WILLIAMS and WILKINS, 6. [3] W. W. Huang, C. M. Chew, and G. S. Hong, Coordination between ocillator: An important feature for robut biped walking, International Conference on Robotic and Automation, 8. Accepted.