DESIGN OF AN ARTIFICIAL CENTRAL PATTERN GENERATOR WITH FEEDBACK CONTROLLER

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1 Intelligent Automtion nd Soft Computing, Vol., No., pp. 85-9, 4 Copyright 4, TSI Press Printed in the USA. All rights resered DESIGN OF AN ARTIFICIAL CENTRAL PATTERN GENERATOR WITH FEEDBACK CONTROLLER KAZUKI NAKADA, TETSUYA ASAI, AND YOSHIHITO AMEMIYA Deprtment of Eletril Engineering Hokkido Uniersity North 3, West 8, Spporo 6-868, Jpn nkd@spiens-ei.eng.hokudi..p ABSTRACT This pper proposes n pproh to design of n rtifiil entrl pttern genertor (CPG) with feedbk ontrol loop. CPG is the biologil neurl network tht genertes rhythmi moements for loomotion of nimls. A ruil point in designing of n rtifiil CPG ontroller is how to del with sensory informtion on surrounding enironments. Hene, we inestigted the properties of n rtifiil CPG ontroller inluding sensory feedbk. First, we nlyzed the stbility of the CPG ontroller, nd then how sensory feedbk influenes to the output of the ontroller. The results proide relisti pproh to design of n rtifiil CPG ontroller.. INTRODUCTION Key Words: Centrl pttern genertor, feedbk ontrol loop, biologilly-inspired wlking robot. Centrl pttern genertors (CPGs) re biologil neurl networks tht generte rhythmi moements for loomotion of nimls, suh s wlking, running, swimming nd flying []. The rhythmi moement generted by CPG indues oordintion of physil prts. Sine the degree of freedom relent to loomotion is ery high, this oordintion is neessry for stble loomotion. Furthermore, it is belieed tht rhythmi moement emerge s stble limit yle from globl entrinment between the neurl system tht inludes CPG nd the physil system tht interts with rying enironment [7]. As result, high utonomous dpttion to unpreditble enironments during loomotion is hieed. In reent yers, mny reserhers he pplied suh funtions of CPG to loomotion ontrol in robotis [3]-[5]. For exmple, Kimur et l. he deeloped qudruped wlking robot pble of dpting to irregulr terrin using CPG dynmis [3]. Billrd nd Ispeert he pplied CPG-bsed ontroller to n entertinment robot, AIBO [4]. Lewis et l. he designed nd fbrited ustom CPG hip for biped wlking robot [5]. In robotis, using the CPG ontroller for loomotion ontrol hs the following dntges: ) redution of the mount of lultion required for motion ontrol s result of the oordintion of physil prts indued by rhythmi moements, nd ) utonomous dpttion to unexpeted enironments used by the globl entrinment between the CPG ontroller nd the physil system suh s robot rms nd legs. In this pper, we propose n pproh to design of n rtifiil CPG ontroller inluding feedbk ontrol loop. From the point of iew of the globl entrinment, one of the key issues in the design of n rtifiil CPG ontroller is how to del with sensory informtion tht reflets the dynmis of the physil system nd surrounding enironment. Hene, we inestigte the properties of n rtifiil CPG ontroller for driing n tutor with sensory feedbk ontrol. The results proide relisti pproh to design of n rtifiil CPG ontroller. This pper is diided into fie setions. In setion, we briefly reiew the neurl bsis of the loomotion ontrol of nimls. In setion 3, we propose CPG ontroller inluding feedbk ontrol loop for driing simple tutor. Setion 4 shows the properties of the CPG ontroller through seerl omputer simultions. Finlly, our work is summrized in setion 5. 85

2 86 Intelligent Automtion nd Soft Computing. NEURAL BASIS OF LOCOMOTION CONTROL In this setion, we briefly reiew the neurl bsis of the ontrol priniples in loomotion of nimls.. Centrl Pttern Genertor Loomotion of nimls, suh s wlking, running, swimming nd flying is bsed on periodi rhythmi moements generted by CPG []. CPG onsists of sets of neurl osilltors, situted in the gnglion or the spinl ord. Indued by inputs from ommnd neurons, CPG genertes rhythmi pttern of nere tiity utomtilly, resulting in rhythmi moement for loomotion of nimls. Suh rhythmi moement indues oordintion of physil prts. Sine the degree of freedom relent to loomotion is ery high, the oordintion of physil prts is neessry for stble loomotion. Thus, CPG n be sid to ply the priniple role in loomotion of nimls.. Globl Entrinment While not neessry for generting rhythmi pttern of nere tiity, sensory feedbk plys lso importnt roles in loomotion ontrol []. One role of sensory feedbk is to regulte the frequeny nd phse of the rhythmi nere tiity depending on rying situtions. Another is to entrin between the rhythmi pttern of nere tiity nd the tul motion of the limbs interting with the externl enironment. Bsed on the biologil findings, Tg et l. he proposed tht rhythmi moement emerges s limit yle generted through globl entrinment between the neurl pttern genertor nd the physil system tht interts with the enironment [7]. As result of the globl entrinment, utonomous dpttion to the unpreditble eents is hieed. Figure shows oneptul illustrtion of the globl entrinment. Centrl Pttern Genertor Sensory signl Control signl Physil System limit yle Enironment Figure. Coneptul illustrtion of the globl entrinment. 3. CPG CONTROLLER In this setion, we propose n rtifiil CPG ontroller inluding feedbk ontrol loop. First, we re going to explin CPG model underlying the CPG ontroller. Seond, physil system drien by the CPG ontroller is desribed, nd the feedbk ontrol loop of the CPG ontroller. 3. CPG Model In the pst gret number of CPG models he been proposed [6]-[8]. Most of these he been onstruted with oupled nonliner osilltors. In the present pper, we propose CPG model bsed on the Wilson-Cown neurl osilltor [9], whih onsists of popultion of exittory neurons nd inhibitory neurons with reiprol synpti onnetions (Fig. ()). The neurl osilltor is desribed by the following equtions:

3 Design of n Artifiil Centrl Pttern Genertor with Feedbk Controller 87 du dt = u + f (u b + S ) µ u d dt = + f (u d + S ) µ () where u nd express the tiities of the popultion of the exittory nd inhibitory neurons, respetiely. The prmeters through d express the synpti strength between the popultions of neurons, S u nd S the externl inputs suh s the bis urrent nd the sensory inputs, τ u nd τ the time onstnts. The trnsfer funtion f µ (x) = tnh(µx) nd µ is its gin prmeter. Depending on ll the prmeters, the Wilson-Cown neurl osilltor shows rious osilltory behiors. For instne, Figure (b) shows limit yles for stble osilltion in the u- phse plne. Beuse of its differentible trnsfer funtion, the stbility of the Wilson-Cown neurl osilltor n be esily nlyzed. Figure 3 shows the stble region of the periodi solutions. As drien by n externl input, the neurl osilltor shows more omplex behiors suh s bursting nd dmping. Suh dynmi behiors of the Wilson-Cown neurl osilltor he been inestigted in detil []. u.5. = b u = d u exittory onnetion inhibitory onnetion () (b) Figure. The Wilson-Cown neurl osilltors. () Configurtions. (b) Phse-plne portrit. We onstruted CPG network model from the Wilson-Cown neurl osilltors. The dynmis of the network model is desribed by the following equtions: du i dt = u i + f µ ( i u b i + S ui ) d i dt = + f ( u i µ i d i + S i ) ()

4 88 Intelligent Automtion nd Soft Computing Stble Region HB Figure 3. Bifurtion digrms of the Wilson-Cown neurl osilltor, where we set prmeters s follows: = b, d =, µ =.5, τ u = τ =. nd S u = S =, nd HB represents Hopf bifurtions. where u i nd i express the tiities of the i-th popultion of the exittory nd inhibitory neurons, respetiely. Depending on the prmeters i through d i nd the externl inputs S ui nd S i, the CPG network model hs rious periodi solutions. We n utilize suh periodi solutions for ontrolling of rhythmi moements in loomotion robots. 3. Joint Atutor nd PD Controller In the following, we ssumed tht one neurl osilltor dries n tutor onneted with oint of robot. The dynmis of the oint tutor is gien s follows: Mθ + kθ + hθ = τ (3) where θ is the oint ngle, M the moment of inerti of the oint tutor, k the stiffness prmeter, h the dmping nd τ the driing fore. By using simple proportionl differentil (PD) ontroller, the driing fore is gien s follows: τ = K P (θ o θ) K D θ (4) where θ ο is the equilibrium ngle. θ nd θ n be mesured quite urtely. K P nd K D express the proportionl nd differentil prmeter, respetiely. The output of the neurl osilltor gies the equilibrium ngle expressed s follows: θ o = g(u ) (5) where g is the proportionl gin. By using the PD ontroller, the motion of the oints n be stbilized. The physil system inluding the PD ontroller is the stndrd seond-order system, whih is gien by the following eqution: G(s) = Kω n s + ςω n s + ω n (6) where ω n is the nturl ngulr frequeny of the system, ζ the dmping nd K onstnt. The dynmi hrteristis of the system depend on these prmeters. We should determine ω n nd ζ in order to get suffiient quik response of the ontroller, espeilly, ω n should be suffiiently lrge ompred with the nturl ngulr frequeny of the neurl osilltor. The other prmeters K P, K D, K nd g will be determined thorough the following equtions, one ω n nd ζ re gien:

5 Design of n Artifiil Centrl Pttern Genertor with Feedbk Controller 89 ςω n = K + h D M, ω n = K + k P M, Kω n = K g P M (7) where the onstnt K deides the dynmi rnge of the oint ngle θ. In the following setion, we ssumed tht M=., ω n =5., ζ =.5 nd K = Feedbk Controller One of the key issues in the design of CPG ontroller is how to gie sensory feedbk to the ontroller sine it might influene the stbility of the system. Hene, we should onsider the stbility of the whole system by exmining the following equtions, whih is tking into ount sensory informtion: τ u u u + f µ (u b + S u (θ,θ )) τ = + f µ (u d + S (θ,θ )) θ ω Mω K P g(u ) (K P + k)θ (K D + h)ω where S u nd S re the funtions of the sensory informtion bout the oint of the sttes θ nd θ. Sine the nonliner funtion of the Wilson-Cown neurl osilltor is differentible, the Jobin mtrix of the boe system is esily obtined. Therefore, the stbility of the system with ny resonble sensory feedbk funtion n be nlyzed. Bsed on the results, we n determine the prmeters of the CPG ontroller inluding sensory feedbk ontrol. 4. RESULTS In this setion, we show the properties of the CPG ontroller through omputer simultions. 4. Stbility Anlysis We nlyzed the stbility of the CPG ontroller inluding the feedbk ontrol (Figure 4). By omputer simultion, we inestigted bifurtions of periodi solutions of the Wilson-Cown neurl osilltor with two types of sensory feedbk suh s follows: S u (θ,θ ) = ηθ or ηθ, S (θ,θ ) = where η is the feedbk gin. The following results were obtined by using AUTO [], whih is n pplition for bifurtion nlysis of ordinry differentil equtions. Figures 4() nd (b) show the bifurtion digrms of the ontrol system, where HB represent the Hopf bifurtion. By introduing both of the sensory feedbk ηθ nd η θ, the stble region of periodi solutions is found to be extended (see Figure 3). 4. Frequeny nd mplitude modultion Willimson inestigted the frequeny entrinment between the neurl osilltor proposed by Mtsuok [6] nd the oint tutor modeled by (5) []. We lso inestigted the frequeny nd mplitude modultion of our ontroller by hnging the feedbk gin. In the following, we set prmeters s follows: =b=5.5, =.5, d=, µ=.5, nd τ u = τ =.. Figure 5() shows the osilltory frequeny of the ontroller with the sensory feedbk ηθ nd η θ. Depending on the type of the feedbk nd its gin, different responses were obsered. Figure 5(b) shows the mplitude of the periodi solution tht orresponds the oint ngle t the sme onditions. As the oint ngle θ is used for sensory feedbk, its sensory feedbk gin inreses the mplitude of the periodi solution with deresing the frequeny of the periodi solution. By using the ngulr eloity (8) (9)

6 9 Intelligent Automtion nd Soft Computing Stble Region 3 Stble Region HB HB () (b) Figure 4. Bifurtion digrms of the system inluding feedbk ontrol. () S u = ηθ nd (b) S u = η θ..... frequeny (Hz) oint ngle (rd) feedbk gin feedbk gin () (b) Figure 5. Frequeny nd mplitude modultion of the system. for sensory feedbk, we n inrese the frequeny of the ontroller with inresing the mplitude. Bsed on the results, we n mke use of the feedbk funtion gien by the following eqution for regulting the mplitude nd frequeny of the system independently (Figure 6): S u (θ,θ ) = ςθ + ηθ () where ζ is the feedbk gin. If we implement our CPG ontroller s n nlog iruit, this feedbk funtion should be useful for ontrolling rhythmi ptterns generted by the iruit. Aording to the physil system drien by the CPG ontroller, we n onstrut suitble feedbk funtion. 5. SUMMARY In the present pper, we propose n pproh to design of n rtifiil CPG ontroller inluding feedbk ontrol loop. Our ontroller is bsed on CPG model onstruted from the Wilson-Cown neurl osilltor. In order to hiee stble loomotion, we introdued the PD ontroller into the indiidul

7 Design of n Artifiil Centrl Pttern Genertor with Feedbk Controller 9.3 oint ngle (rd) time (s) Figure 6. Weforms of oint ngle, where ζ = η =. (bold line) nd ζ = 3.5, η =. (solid line). osilltor tht dries eh of the oints, nd used the sensory informtion bout the oint sttes s feedbk to the CPG ontroller. By seerl omputer simultions, we inestigted the properties of the CPG ontroller. First, we nlyzed the stbility of the periodi solutions of the ontroller. As result, it is shown tht the stble region is extended further by introduing the feedbk ontrol. Seond, we inestigted how the sensory feedbk nd its gin modulte the frequeny nd mplitude of the oint ngle drien by the ontroller. Thus, it is lso shown tht the feedbk funtion ombining the oint ngulr nd the ngulr eloity n regulte the frequeny nd the mplitude independently. These results proide useful informtion to design CPG ontroller. In future, it is hoped, we will be ble to implement the CPG ontroller for biologilly-inspired wlking robot. REFERENCES. F. Delomyn. 98. Neurl bsis of rhythmi behiour in nimls, Siene, ol., pp F. Delomyn Foundtions of Neurobiology, New York: W.H. Freemn nd Co. 3. H. Kimur, Y. Fukuok, nd K. Kong.. Adptie dynmi wlking of qudruped robot by using neurl system model, ADVANCED ROBOTICS, Vol. 5, No. 8, pp A. Billrd nd A. J. Ispeert.. Biologilly inspired neurl ontrollers for motor ontrol in qudruped wlking robot, in pro. IEEE-INNS Interntionl Joint Conferene on Neurl Networks, Itly. 5. M. A. Lewis, M. J. Hrttmnn, R. Etienne-Cummings, nd A. H. Cohen.. Control of robot leg with n dptie VLSI CPG hip, Neuroomputing, ol. 38-4, pp K. Mtsuok Mehnism of frequeny nd pttern ontrol in the neurl rhythm genertors, Biol. Cybern., ol, 56, pp G. Tg, Y. Ymguhi, nd H. Shimizu. 99. Self-orgnized ontrol of bipedl loomotion by neurl osilltors in unpreditble enironment, Biol. Cybern., ol. 65, pp H. Ngshino, Y. Nomur, nd Y. Kinouhi.. Genertion nd trnsitions of phse-loked osilltions in oupled neurl osilltors, in pro. the 4h SICE Annul Conferene, Ngoy, Jpn. 9. H. R. Wilson nd J. D. Cown. 97. Exittory nd Inhibitory intertions in lolized popultions of model neurons, Biophys., J., Vol., pp.-4.. F. C. Hoppenstedt nd E. Izhikeih Wekly Conneted Neurl Networks, Springer-Verlg.. M. Willimson Neurl ontrol of rhythmi rm moements, Neurl Networks, speil issue on neurl ontrol of moement, pp E. Doedel, X. Wng, nd T. Firgriee Auto94: Softwre for ontinution nd bifurtion problems in ODEs, Appl. Mth. Tehnil Report, Cliforni Inst. of Tehnology.