A modular network for legged locomotion

被引:159
作者
Golubitsky, M [1 ]
Stewart, I
Buono, PL
Collins, JJ
机构
[1] Univ Houston, Dept Math, Houston, TX 77204 USA
[2] Univ Warwick, Inst Math, Coventry CV4 7AL, W Midlands, England
[3] Boston Univ, Dept Biomed Engn, Boston, MA 02215 USA
来源
PHYSICA D | 1998年 / 115卷 / 1-2期
基金
美国国家科学基金会;
关键词
Hopf bifurcation; symmetry; control pattern generator; gaits;
D O I
10.1016/S0167-2789(97)00222-4
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
In this paper we use symmetry methods to study networks of coupled cells, which are models for central pattern generators (CPGs). In these models the cells obey identical systems of differential equations and the network specifies how cells are coupled. Previously, Collins and Stewart showed that the phase relations of many of the standard gaits of quadrupeds and hexapods can be obtained naturally via Hopf bifurcation in small networks. For example, the networks they used to study quadrupeds all had four cells, with the understanding that each cell determined the phase of the motion of one leg. However, in their work it seemed necessary to employ several different four-oscillator networks to obtain all of the standard quadrupedal gaits. We show that this difficulty with four-oscillator networks is unavoidable, but that the problems can be overcome by using a larger network. Specifically, we show that the standard gaits of a quadruped, including walk, trot and pace, cannot all be realized by a single four-cell network without introducing unwanted conjugacies between trot and pace - conjugacies that imply a dynamic equivalence between these gaits that seems inconsistent with observations. In this sense a single network with four cells cannot model the CPG of a quadruped. We also introduce a single eight-cell network that can model all of the primary gaits of quadrupeds without these unwanted conjugacies. Moreover, this network is modular in that it naturally generalizes to provide models of gaits in hexapods, centipedes, and millipedes. The analysis of models for many-legged animals shows that wave-like motions, similar to those obtained by Kopell and Ermentrout, can be expected. However, our network leads to a prediction that the wavelength of the wave motion will divide twice the length of the animal. Indeed, we reproduce illustrations of wave-like motions in centipedes where the animal is approximately one-and-a-half wavelength long - motions that are consistent with this prediction. We discuss the implications of these results for the development of modular control networks for adaptive legged robots. Copyright (C) 1998 Elsevier Science B.V.
引用
收藏
页码:56 / 72
页数:17
相关论文
共 32 条
[1]  
Alexander RM., 1977, MECH ENERGETICS ANIM, P168
[2]   THE NATURE OF THE COUPLING BETWEEN SEGMENTAL OSCILLATORS OF THE LAMPREY SPINAL GENERATOR FOR LOCOMOTION - A MATHEMATICAL-MODEL [J].
COHEN, AH ;
HOLMES, PJ ;
RAND, RH .
JOURNAL OF MATHEMATICAL BIOLOGY, 1982, 13 (03) :345-369
[3]  
COLLINS JJ, 1994, BIOL CYBERN, V71, P375, DOI 10.1007/BF00198915
[4]  
COLLINS JJ, 1992, J MATH BIOL, V30, P827
[5]   COUPLED NONLINEAR OSCILLATORS AND THE SYMMETRIES OF ANIMAL GAITS [J].
COLLINS, JJ ;
STEWART, IN .
JOURNAL OF NONLINEAR SCIENCE, 1993, 3 (03) :349-392
[6]   Aperiodic stochastic resonance [J].
Collins, JJ ;
Chow, CC ;
Capela, AC ;
Imhoff, TT .
PHYSICAL REVIEW E, 1996, 54 (05) :5575-5584
[7]   HEXAPODAL GAITS AND COUPLED NONLINEAR OSCILLATOR MODELS [J].
COLLINS, JJ ;
STEWART, I .
BIOLOGICAL CYBERNETICS, 1993, 68 (04) :287-298
[8]  
COLLINS JJ, 1994, BIOL CYBERN, V71, P95, DOI 10.1007/BF00197312
[9]   Coupled cells with internal symmetry .1. Wreath products [J].
Dionne, B ;
Golubitsky, M ;
Stewart, I .
NONLINEARITY, 1996, 9 (02) :559-574
[10]   Coupled cells with internal symmetry .2. Direct products [J].
Dionne, B ;
Golubitsky, M ;
Stewart, I .
NONLINEARITY, 1996, 9 (02) :575-599