The dynamics of legged locomotion: Models, analyses, and challenges

SIAM Review

Volumn 48, Issue 2, 2006, Pages 207-304

  Holmes, Philip ^a   Full, Robert J ^b   Koditschek, Dan ^c   Guckenheimer, John ^d  

^a Princeton University   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

^c UNIVERSITY OF PENNSYLVANIA   (United States)

^d Department of Obstetrics Gynecology   (United States)

Author keywords

Animal locomotion; Biomechanics; Bursting neurons; Central pattern generators; Control systems; Hybrid dynamical systems; Insect locomotion; Lagrangians; Motoneurons; Muscles; Neural networks; Periodic gaits; Phase oscillators; Piecewise holonomic systems; Preflexes; Reflexes; Robotics; Sensory systems; Stability; Templates

Indexed keywords

BIOMECHANICS; MATHEMATICAL MODELS; MUSCLE; NEUROLOGY;

ANIMAL LOCOMOTION; GOAL-ORIENTED SENSING; NEUROBIOLOGY; NEUROMUSCULAR SUBSYSTEM; NEWTONIAN BODY-LIMB DYNAMICS;

BIOLOGY;

EID: 33744929666     PISSN: 00361445     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036144504445133     Document Type: Review

References (347)

1. L. Abbott, Single neuron dynamics, in Neural Modeling and Neural Networks, F. Ventriglia, ed., Pergamon Press, Oxford, UK, 1994, Chap. 4, pp. 57-78.
2. R. Abraham and J. Marsden, Foundations of Mechanics, Addison-Wesley, New York, 1985.
3. C. Acker, N. Kopell, and J. White, Synchronization of strongly coupled excitatory neurons: Relating network behavior to biophysics, J. Comput. Neurosci., 15 (2003), pp. 71-90.
4. A. Ahn and R. Full, A motor and a brake: Two leg extensor muscles acting at the same joint manage energy differently in a running insect, J. Exp. Biol., 205 (2002), pp. 379-389.
5. T. Akay, U. Bässler, P. Gerharz, and A. Büschges, The role of sensory signals from the insect coxa-trochanteral joint in controlling motor activity of the femur-tibia joint, J. Neurophys., 85 (2001), pp. 594-604.
6. R. Alexander, Elastic Mechanisms in Animal Movement, Cambridge University Press, Cambridge, UK, 1988.
7. R. Alexander, The mechanics of jumping by a dog (canis familiaris), J. Zool. Lond., 173 (1989), pp. 549-573.
8. R. Alexander, Principles of Animal Locomotion, Princeton University Press, Princeton, NJ, 2003.
9. R. Alexander and A. Jayes, A dynamic similarity hypothesis for the gaits of quadrupedal mammals, J. Zool. London, 201 (1983), pp. 135-152.
10. R. Alexander and A. Vernon, Mechanics of hopping by kangaroos (macropodidae), J. Zool. London, 177 (1975), pp. 265-303.
11. R. Altendorfer, D. Koditschek, and P. Holmes, Stability analysis of a clock-driven rigid-body SLIP model for RHex, Int. J. Robotics Res., 23 (2004), pp. 1001-1012.
12. R. Altendorfer, D. Koditschek, and P. Holmes, Stability analysis of legged locomotion models by symmetry-factored return maps, Int. J. Robotics Res., 23 (2004), pp. 979-999.
13. R. Altendorfer, A. Moore, H. Komsuoglu, M. Buehler, H. Brown, D. McMordie, U. Saranli, R. Full, and D. Koditschek, RHex: A biologically inspired hexapod runner, J. Autonomous Robots, 11 (2001), pp. 207-213.
14. S. Arimoto, A challenge to Bernstein's degrees-of-freedom problem in both cases of human and robotic multi-joint movements, in Proceedings of the 2004 International Symposium on Nonlinear Theory and Its Applications, Research Society of Nonlinear Theory and Its Applications, IEICE, 2004, pp. 1-15 (CD-ROM).
15. S. Arimoto, Intelligent control of multi-fingered hands, Ann. Rev. Control, 28 (2004), pp. 75-85.
16. V. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1978.
17. Y. Arshavsky, T. Deliagina and G. Orlovsky, Pattern generation, Current Opinion in Neurobiology, 7 (1997), pp. 781-789.
18. K. Autumn, M. Ryan, and D. Wake, Integrating historical and mechanistic biology enhances the study of adaptation, Quart. Rev. Biol., 77 (2002), pp. 383-408.
19. A. Back, J. Guckenheimer, and M. Myers, A dynamical simulation facility for hybrid systems, in Hybrid Systems, Lecture Notes in Comput. Sci. 736, Springer-Verlag, Berlin, 1993, pp. 255-267.
20. U. Bässler and A. Büschges, Pattern generation for stick insect walking movements-multisensory control of a locomotor program, Brain Research Rev., 27 (1998), pp. 65-88.
21. P. Beek, C. Peper, and A. Daffertshofer, Modeling rhythmic interlimb coordination: Beyond the Haken-Kelso-Bunz model, Brain and Cognition, 48 (2002), pp. 149-165.
22. R. Beer, H. Chiel, and J. Gallagher, Evolution and analysis of model CPGs for walking II. General principles and individual variability, J. Comput. Neurosci., 7 (1999), pp. 119-147.
23. R. Beer and J. Gallagher, Evolving dynamical neural networks for adaptive behavior, Adaptive Behavior, 1 (1992), pp. 91-122.
24. A. Bernstein, The Coordination and Regulation of Movements, Pergamon Press, Oxford, UK, 1967.
25. A. Bernstein, On Dexterity and Its Development, Lawrence Erlbaum Associates, Mahwah, NJ, 1996. Translated from the Russian by M. L. Latash.
26. A. Biewener and R. Baudinette, In vivo muscle force and elastic energy storage during steady-speed hopping of tammar wallabies (macropus eugenii), J. Exp. Biol., 198 (1995), pp. 1829-1841.
27. A. Biewener, D. Konieczynski, and R. Baudinette, In vivo muscle force-length behavior during steady-speed hopping in tammar wallabies, J. Exp. Biol., 201 (1998), pp. 1681-1694.
28. A. Biewener and C. Taylor, Bone strain: A determinant of gait and speed?, J. Exp. Biol., 123 (1986), pp. 383-400.
29. G. Birkhoff, Dynamical Systems, AMS, Providence, RI, 1927. Reprinted, Colloquium. Publications, Vol. 9, AMS, Providence, RI, 1966.
30. E. Bizzi, N. Hogan, F. Mussa-Ivaldi, and S. Giszter, Does the nervous system use equilibrium-point control to guide single and multiple joint movements?, Behav. Brain Sci., 15 (1992), pp. 603-613.
31. E. Bizzi, M. Tresch, P. Saltiel, and A. D'Avella, New perspectives on spinal motor systems, Nature Reviews Neurosci., 1 (2000), pp. 101-108.
32. R. Blickhan, The spring-mass model for running and hopping, J. Biomechanics, 22 (1989), pp. 1217-1227.
33. R. Blickhan and R. Full, Locomotion energetics of the ghost crab: II. Mechanics of the centre of mass during walking and running, J. Exp. Biol., 130 (1987), pp. 155-174.
34. R. Blickhan and R. Full, Similarity in multi-legged locomotion: Bouncing like a monopode, J. Comp. Physiol. A, 173 (1993), pp. 509-517.
35. R. Blickhan, R. Full, and L. Ting, Exoskeletal strain: Evidence for a trot-gallop transition in rapid running ghost crabs, J. Exp. Biol., 179 (1993), pp. 301-321.
36. A. Bloch, J. Baillieul, P. Crouch, and J. Marsden, Nonholonomic Mechanics and Control, Springer-Verlag, New York, 2003.
37. A. Bloch, H. Chang, N. Leonard, and J. Marsden, Controlled Lagrangians and the stabilization of mechanical systems: II. Potential shaping, IEEE Trans. Automat. Control, 46 (2001), pp. 1556-1571.
38. A. Bloch and P. Crouch, Nonholonomic and vakonomic control systems on Riemannian manifolds, Fields Inst. Commun., 1 (1993), pp. 25-52.
39. A. Bloch and P. Crouch, Nonholonomic control systems on Riemannian manifolds, SIAM J. Control, 33 (1995), pp. 126-148.
40. A. Bloch, P. Krishnaprasad, J. Marsden, and R. Murray, Nonholonomic mechanical systems with symmetry, Arch. Rational Mech. Anal., 136 (1996), pp. 21-99.
41. A. Bloch, N. Leonard, and J. Marsden, Controlled Lagrangians and the stabilization of mechanical systems: I. The first matching theorem, IEEE Trans. Automat. Control, 45 (2000), pp. 2253-2270.
42. V. Blondel and J. Tsitsiklis, NP-hardness of some linear control design problems, SIAM J. Control Optim., 35 (1997), pp. 2118-2127.
43. N. Borghese, L. Bianchi, and F. Lacquaniti, Kinematic determinants of human locomotion, J. Physiol., 494 (1996), pp, 863-879.
44. G. Bowtell and T. Williams, Anguilliform, body dynamics: Modelling the interaction between muscle activation and curvature, Phil. Trans. Roy. Soc. B, 334 (1991), pp. 385-390.
45. C. Bowtell and T. Williams, Anguilliform, body dynamics: A continuum model for the interaction between muscle activation and body curvature, J. Math. Biol., 32 (1994), pp. 83-91.
46. E. Brown, J. Moehlis, and P. Holmes, On the phase reduction and response dynamics of neural oscillator populations, Neural Comp., 16 (2004), pp. 673-715.
47. E. Brown, J. Moehlis, P. Holmes, E. Clayton, J. Rajkowski, and G. Aston-Jones, The influence of spike rate and stimulus duration on noradrenergic neurons, J. Comput. Neurosci., 17 (2004), pp. 13-29.
48. I. Brown and G. Loeb, A reductionist approach to creating and using neuromusculoskeletal movement, in Biomechanics and Neural Control of Movement, J. Winters and P. Crago, eds., Springer-Verlag, New York, 2000, pp. 148-163.
49. I. Brown, S. Scott, and G. Loeb, "Preflexes" - programmable, high-gain, zero-delay intrinsic responses of perturbed musculoskeletal systems, Soc. Neurosci. Abstr., 21 (1995), p. 562.9.
50. N. Brunel and X.-J. Wang, Effects of neuromodulation in a cortical network model of object working memory dominated by recurrent inhibition, J. Comput. Neurosci., 11 (2001), pp. 63-85.
51. A. Bryson and Y. Ho, Applied Optimal Control, Taylor and Francis, New York, 1975.
52. J. Buchli and A. Ijspeert, Distributed central pattern generator model for robotics application based on phase sensitivity analysis, in Biologically Inspired Approaches to Advanced Information Technology: First International Workshop, BioADIT 2004, Lecture Notes in Comput. Sci. 3141, Springer-Verlag, Berlin, 2004, pp. 333-349.
53. M. Buehler, D. Koditschek, and P. Kindlmann, A family of robot control strategies for intermittent dynamical environments, IEEE Control Systems Mag., 10 (2) (1990), pp. 16-22.
54. F. Bullo and A. Lewis, Geometric Control of Mechanical Systems, Springer-Verlag, New York, 2005.
55. R. Burke, Revisiting the notion of "motor unit types," Prog. Brain Res., 123 (1999), pp. 167-175.
56. R. Burke, Some unresolved issues in motor unit research, Adv. Exp. Med. Biol., 508 (2002), pp. 171-178.
57. M. Burrows, The control of sets of motoneurones by local interneurones in the locust, J. Physiol., 298 (1980), pp. 213-233.
58. M. Burrows and M. Siegler, Networks of local interneurons in an insect, in Neural Origin of Rhythmic Movements, A. Roberts and B. Roberts, eds., Symposia of the Society for Experimental Biology, Vol. 37, Cambridge University Press, Cambridge, UK, 1983, pp. 29-53.
59. A. Büschges and A. Manira, Sensory pathways and their modulation in the control of locomotion, Current Opinion in Neurobiology, 8 (1998), pp. 733-739.
60. R. Butera, J. Rinzel, and J. Smith, Models of respiratory rhythm generation in the pre-Bötzinger complex. I. Bursting pacemaker neurons, J. Neurophys., 81 (1999), pp. 382-397.
61. R. Butera, J. Rinzel, and J. Smith, Models of respiratory rhythm generation in the pre-Bötzinger complex. II. Populations of coupled pacemaker neurons, J. Neurophys., 81 (1999), pp. 398-415.
62. W. Calder, Size, Function, and Life History, Harvard University Press, Cambridge, MA, 1984.
63. A. Calvitti and R. Beer, Analysis of a distributed model of leg coordination. I. Individual coordination mechanisms, Biol. Cybern., 82 (2000), pp. 197-206.
64. J. Camhi and E. Johnson, High frequency steering maneuvers mediated by tactile cues: Antennal wall-following in the cockroach, J. Exp. Biol., 202 (1999), pp. 631-643.
65. C. Canavier, R. Butera, R. Dror, D. Baxter, J. Clark, and J. Byrne, Phase response characteristics of model neurons determine which patterns are expressed in a ring model of gait generation, Biol. Cybern., 77 (1997), pp. 367-380.
66. J. Carling, T. Williams, and G. Bowtell, Self-propelled anguilliform swimming: Simultaneous solution of the two-dimensional Navier-Stokes equations and Newton's laws of motion, J. Exp. Biol., 201 (1998), pp. 3143-3166.
67. S. Carver, Control of a Spring-Mass Hopper, Ph.D. thesis, Cornell University, Ithaca, NY, 2003.
68. G. Cavagna, N. Heglund, and C. Taylor, Mechanical work in terrestrial locomotion: Two basic mechanisms for minimizing energy expenditure, Amer. J. Physiol., 233 (1977), pp. R243-R261.
69. G. Cavagna, F. Saibene, and R. Margaria, Mechanical work in running, J. Appl. Physiol., 19 (1964), pp. 249-256.
70. G. Cavagna, H. Thys, and A. Zamboni, The sources of external work in level walking and running, J. Physiol. Lond., 262 (1976), pp. 639-657.
71. H. Chang, A. Bloch, N. Leonard, J. Marsden, and C. Woolsey, The equivalence of controlled Lagrangian and controlled Hamiltonian systems, ESIAM Control Optim. Calc. Vax., 8 (2002), pp. 393-422.
72. H. Chiel, R. Beer, and J. Gallagher, Evolution and analysis of model CPGs for walking I. Dynamical modules, J. Comput. Neurosci., 7 (1999), pp. 99-118.
73. H. Chiel, R. Beer, R. Quinn, K. Espenschied, and P. Larson, A distributed neural network architecture for hexapod robot locomotion, Neural Computation, 4 (1992), pp. 356-365.
74. C. Chow, J. White, J. Ritt, and N. Kopell, Frequency control in synchronized networks of inhibitory neurons, J. Comput. Neurosci., 5 (1998), pp. 407-420.
75. F. Clarac, D. Cattaert, and D. L. Ray, Central control components of a "simple" stretch reflex, Trends in Neurosci., 23 (2000), pp. 199-208.
76. A. Cohen, P. Holmes, and R. Rand, The nature of coupling between segmental oscillators of the lamprey spinal generator for locomotion: A model, J. Math. Biol., 13 (1982), pp. 345-369.
77. A. Cohen and T. Kiemel, Intersegmental coordination: lessons from modeling systems of non-linear oscillators, Amer. Zool., 33 (1993), pp. 54-65.
78. A. Cohen, R. Rand, and P. Holmes, Systems of coupled oscillators as models of central pattern generators, in Neural Control of Rhythmic Movements in Vertebrates, A. Cohen, S. Rossignol, and S. Grillner, eds., Wiley, New York, 1988, pp. 333-367.
79. A. Cohen, S. Rossignol, and S. Grillner (eds), Neural Control of Rhythmic Movements in Vertebrates, Wiley, New York, 1988.
80. A. Cohen and P. Wallén, The neuronal correlate of locomotion in fish. "Fictive swimming" induced in an in vitro preparation of the lamprey spinal cord, Exp. Brain. Res., 41 (1980), pp. 11-18.
81. M. Coleman, A. Chatterjee, and A. Ruina, Motions of a rimless spoked wheel: A simple three-dimensional system with impacts, Dynam. Stability Systems, 12 (1997), pp. 139-159.
82. M. Coleman, M. Garcia, K. Mombaur, and A. Ruina, Prediction of stable walking for a toy that cannot stand, Phys. Rev. E, 64 (2001), article 022901-1-3.
83. M. Coleman and P. Holmes, Motions and stability of a piecewise holonomic system: The discrete Chaplygin sleigh, Regular and Chaotic Dynamics, 4 (1999), pp. 1-23.
84. M. Coleman and A. Ruina, An uncontrolled walking toy that cannot stand still, Phys. Rev. Lett., 80 (1998), pp. 3658-3661.
85. J. Collins and I. Stewart, Hexapodal gaits and coupled nonlinear oscillator models, Biol. Cybern., 68 (1993), pp. 287-298.
86. S. Collins, A. Ruina, R. Tedrake, and M. Wisse, Efficient bipedal robots based on passive-dynamic walkers, Science, 305 (2005), pp. 1082-1085.
87. S. Collins, M. Wisse, and A. Ruina, A three-dimensional passive-dynamic walking robot with two legs and knees, Int. J. Robotics Res., 20 (2001), pp. 607-615.
88. J. Connor, D. Walter, and R. McKown, Neural repetitive firing: Modifications of the Hodgkin-Huxley axon suggested by experimental results from crustacean axons, Biophys. J., 18 (1977), pp. 81-102.
89. P. Constantin, C. Foias, R. Temam, and B. Nicolaenko, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer-Verlag, New York, 1989.
90. N. Cowan, J. Lee, and R. Full, Dynamical Control of Antennal Wall Following in the American Cockroach, Society for Integrative and Comparative Biology, San Diego, CA, 2005.
91. H. Cruse, The functional sense of central oscillations in walking, Biol. Cybern., 86 (2002), pp. 271-280.
92. H. Cruse, T. Kindermann, M. Schumm, J. Dean, and J. Schmitz, Walknet - a biologically inspired network to control six-legged walking, Neural Networks, 11 (1998), pp. 1435-1447.
93. G. Cymbaluk, R. Borisyuk, U. Müller-Wilm, and H. Cruse, Oscillatory network controlling six-legged locomotion: Optimization of model parameters, Neural Networks, 11 (1998), pp. 1449-1460.
94. T. Dawson and C. Taylor, Energetic cost of locomotion in kangaroos, Nature, 246 (1973), pp. 313-314.
95. P. Dayan and L. Abbott, Theoretical Neuroscience, MIT Press, Cambridge, MA, 2001.
96. C. Del Negro, S. Johnson, R. Butera, and J. Smith, Models of respiratory rhythm generation in the pre-Bötzinger complex. III. Experimental tests of model predictions, J. Neurophys., 86 (2001), pp. 59-74.
97. F. Delcomyn, Neural basis of rhythmic behaviours in animals, Science, 210 (1980), pp. 492-498.
98. F. Delcomyn, Walking and running, in Comprehensive Insect Physiology Biochemistry and Pharmacology, G. Kerkut and L. Gilbert, eds., Pergamon Press, New York, 1985, pp. 439-466.
99. F. Delcomyn, Perturbation of the motor system in freely walking cockroaches II. The timing of motor activity in leg muscles after amputation of a middle leg, J. Exp. Biol., 156 (1991), pp. 503-517.
100. F. Delcomyn, Insect walking and robotics, Ann. Rev. Entomol., 149 (2004), pp. 51-70.
101. M. Dickinson, C. Farley, R. Full, M. Koehl, R. Kram, and S. Lehman, How animals move: An integrative view, Science, 288 (2000), pp. 100-106.
102. E. Doedel, A. Champneys, T. Fairgrieve, B. S. Y. Kuznetsov, and X. Wang, Auto 97: Continuation and Bifurcation Software for Ordinary Differential Equations, Tech. Report, Concordia University, 1997. Available via FTP from directory/pub/doedel/auto at ftp.cs.concordia.ca; software available from http://indy.cs.concordia.ca/auto/ and http:// sourceforge.net/projects/auto2000.
103. J. Donelan, R. Kram, and A. Kuo, Simultaneous positive and negative external mechanical work in human walking, J. Biomech., 35 (2002), pp. 117-124.
104. A. Dorval, D. Christini, and J. White, Real-time Linux dynamic clamp: A fast and flexible way to construct virtual ion channels in living cells, Ann. Biomed. Eng., 29 (2001), pp. 897-907.
105. P. Drazin and W. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge, UK, 1981.
106. D. Dudek and R. Full, Leg function in running insects: Resilience and impedance of legs linked to a body, Amer. Zool., 41 (2001) (abstract).
107. D. Dudek and R. Full, Mechanical properties of the support tripod in running insects, in Society of Integrative and Comparative Biology Annual Meeting and Exhibition: Final Program and Abstracts, 2004.
108. V. Dürr, J. Schmitz, and H. Cruse, Behavior-based modeling of insect locomotion: Linking biology and technical application, Arthropod Structure and Development, 33 (2004), pp. 237-250.
109. J. Duysens, F. Clarac, and H. Cruse, Load-regulating mechanisms in gait and posture: Comparative aspects, Physiol. Rev., 80 (2000), pp. 83-133.
110. Ö. Ekeberg, A combined neuronal and mechanical model of fish swimming, Biol. Cybern., 69 (1993), pp. 363-374.
111. O Ekeberg, M. Blümel, and A. Büschges, Dynamic simulation of insect walking, Arthropod Structure and Development, 33 (2004), pp. 287-300.
112. Ö. Ekeberg and S. Grillner, Simulations of neuromuscular control in lamprey swimming, Phil. Trans. Roy. Soc. B, 354 (1999), pp. 895-902.
113. T. S. Eliot, in "The Hollow Men," Collected Poems, Faber and Faber, London, 1964.
114. G. B. Ermentrout, Type I membranes, phase resetting curves, and synchrony, Neural Comp., 8 (1996), pp. 979-1001.
115. G. B. Ermentrout, Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, SIAM, Philadelphia, 2002.
116. G. B. Ermentrout and N. Kopell, Symmetry and phase locking in chains of weakly coupled oscillators, Comm. Pure Appl. Math., 39 (1986), pp. 623-660.
117. G. B. Ermentrout and N. Kopell, Oscillator death in systems of coupled neural oscillators, SIAM J. Appl. Math., 50 (1990), pp. 125-146.
118. G. B. Ermentrout and N. Kopell, Multiple pulse interactions and averaging in systems of coupled neural oscillators, J. Math. Biol., 29 (1991), pp. 195-217.
119. C. Farley, J. Glasheen, and T. McMahon, Running springs: Speed and animal size, J. Exp. Biol., 185 (1993), pp. 71-86.
120. C. Farley and C. Taylor, A mechanical trigger for the trot-gallop transition in horses, Science, 253 (1991), pp. 306-308.
121. J. Felsenstein, Phylogenies and the comparative method, Amer. Nat., 125 (1985), pp. 1-15.
122. R. Fitzhugh, Thresholds and plateaus in the Hodgkin-Huxley nerve equations, J. General Physiol., 43 (1960), pp. 867-896.
123. R. Fitzhugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), pp. 455-466.
124. R. Full, Mechanics and energetics of terrestrial locomotion: From bipeds to polypeds, in Energy Transformation in Cells and Animals, W. Weiser and E. Gnaiger, eds., Georg Thieme Verlag, Stuttgart, Germany, 1989, pp. 175-182.
125. R. Full, Animal motility and gravity, Physiologist, 34 (1991), pp. S15-18.
126. R. Full, Invertebrate locomotor systems, in The Handbook of Comparative Physiology, W. Dantzler, ed., Oxford University Press, New York, 1997, pp. 853-930.
127. R. Full and A. Ahn, Static forces and moments generated in the insect leg: Comparison of a three-dimensional musculo-skeletal computer model with experimental measurements, J. Exp. Biol., 198 (1995), pp. 1285-1298.
128. R. Full, K. Autumn, J. Chung, and A. Ahn, Rapid negotiation of rough terrain by the death-head cockroach, Amer. Zool., 38 (1998), p. 81A.
129. R. Full, R. Blickhan, and L. Ting, Leg design in hexapedal runners, J. Exp. Biol., 158 (1991), pp. 369-390.
130. R. Full and C. Farley, Musculoskeletal dynamics in rhythmic systems: A comparative approach to legged locomotion, in Biomechanics and Neural Control of Movement, J. Winters and P. Crago, eds., Springer-Verlag, New York, 2000, pp. 192-205.
131. R. Full and D. Koditschek, Templates and anchors: Neuromechanical hypotheses of legged locomotion on land, J. Exp. Biol., 202 (1999), pp. 3325-3332.
132. R. Full, T. Kubow, M. Garcia, W. Schwind, and D. Koditschek, Can a Simple Neural Oscillator Generate Rapid Running in Cockroaches?, Society for Integrative and Comparative Biology, Toronto, ON, Canada, 2003.
133. R. Full, T. Kubow, J. Schmitt, P. Holmes, and D. Koditschek, Quantifying dynamic stability and maneuverability in legged locomotion, Integ. Comp. Biol., 42 (2002), pp. 149-157.
134. R. Full, D. Stokes, A. Ahn, and R. Josephson, Energy absorption during running by leg muscles in a cockroach, J. Exp. Biol., 201 (1998), pp. 997-1012.
135. R. Full and M. Tu, Mechanics of six-legged runners, J. Exp. Biol., 148 (1990), pp. 129-146.
136. R. Full and M. Tu, Mechanics of a rapid running insect: Two-, four- and six-legged locomotion, J. Exp. Biol., 156 (1991), pp. 215-231.
137. R. Full, A. Yamauchi, and D. Jindrich, Maximum single leg force production: Cockroaches righting on photoelastic gelatin, J. Exp. Biol., 198 (1995), pp. 2441-2452.
138. M. Garcia, A. Chatterjee, A. Ruina, and M. Coleman, The simplest walking model: Stability, complexity and scaling, ASME J. Biomech. Eng., 120 (1998), pp. 281-288.
139. M. Garcia, A. Kuo, A. Peattie, P. Wang, and R. Full, Damping and size: Insights and biological inspiration, International Symposium on Adaptive Motion of Animals and Machines, 2000.
140. T. Garland, Jr. and S. C. Adolph, Why not to do two-species comparative studies: Limitations on inferring adaptation, Physiol. Zool., 67 (1994), pp. 797-828.
141. T. Garland Jr., P. Harvey, and A. Ives, Procedures for the analysis of comparative data using phylogenetically independent contrasts, Sys. Biol., 41 (1992), pp. 18-32.
142. P. Getting, Comparative analysis of invertebrate central pattern generators, in Neural Control of Rhythmic Movements in Vertebrates, A. Cohen, S. Rossignol, and S. Grillner, eds., Wiley, New York, 1988, pp. 101-128.
143. H. Geyer, A. Seyfarth, and R. Blickhan, Spring mass running: simple approximate solution and application to gait stability, J. Theoret. Biol., 232 (2005), pp. 315-328.
144. R. Ghigliazza, Neuromechanical Models for Insect Locomotion, Ph.D. thesis, Princeton University, Princeton, NJ, 2004.
145. R. M. Ghigliazza, R. Altendorfer, P. Holmes, and D. Koditschek, A simply stabilized running model, SIAM J. Appl. Dynam. Systems, 2 (2003), pp. 187-218. Revised and updated version appeared in SIAM Rev., 47 (2005), pp. 519-549.
146. R. M. Ghigliazza and P. Holmes, Minimal models of bursting neurons: How multiple currents, conductances, and timescales affect bifurcation diagrams, SIAM J. Appl. Dynam. Systems, 3 (2004), pp. 636-670.
147. R. M. Ghigliazza and P. Holmes, A minimal model of a central pattern generator and motoneurons for insect locomotion, SIAM J. Appl. Dynam. Systems, 3 (2004), pp. 671-700.
148. R. Ghigliazza and P. Holmes, Towards a neuromechanical model for insect locomotion: Hybrid dynamical systems, Regul. Chaotic Dyn., 10 (2005), pp. 193-225.
149. C. Gilbert, Visual control of cursorial prey pursuit by tiger beetles (Cicindelidae), J. Comp. Physiol. A, 181 (1997), pp. 217-230.
150. M. Goldman, J. Golowasch, E. Marder, and L. Abbott, Global structure, robustness, and modulation of neuronal models, J. Neurosci., 21 (2001), pp. 5229-5238.
151. H. Goldstein, Classical Mechanics, 2nd ed., Addison Wesley, Reading, MA, 1980.
152. D. Golomb, J. Guckenheimer, and S. Gueron, Reduction of a channel based model for a stomatogastric ganglion LP neuron, Biol. Cybern., 69 (1993), pp. 129-137.
153. M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. I, Springer-Verlag, New York, 1985.
154. M. Golubitsky, I. Stewart, P. Buono, and J. Collins, A modular network for legged locomotion, Phys. D, 115 (1998), pp. 56-72.
155. M. Golubitsky, I. Stewart, P. Buono, and J. Collins, The role of symmetry in locomotor central pattern generators and animal gaits, Nature, 401 (1999), pp. 693-695.
156. M. Golubitsky, I. Stewart, and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. II, Springer-Verlag, New York, 1988.
157. P. Gove, ed., Webster's Third New International Dictionary of the English Language Unabridged, Merriam, Springfield, MA, 1985.
158. D. Graham, Pattern and control of walking in insects, in Advances in Insect Physiology, Vol. 18, Academic Press, London, UK, 1985, pp. 31-140.
159. M. Graziano, C. Taylor, and T. Moore, Complex movements evoked by microstimulation of prefrontal cortex, Neuron, 34 (2002), pp. 841-851.
160. A. Griewank, Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, SIAM, Philadelphia, 2000.
161. A. Griewank, D. Juedes, H. Mitev, J. Utke, O. Vogel, and A. Walther, Adol-c: A package for the automatic differentiation of algorithms written in C/C++, ACM Trans. Math. Software, 22 (1996), pp. 131-167, Algor. 755; see also http://www.math.tu-dresden.de/wir/project/adolc/.
162. S. Grillner, Neurobiological bases of rhythmic motor acts in vertebrates, Science, 228 (1985), pp. 143-149.
163. S. Grillner, Bridging the gap - from ion channels to networks and behaviour, Current Opinion in Neurobiol., 9 (1999), pp. 663-669.
164. S. Grillner, P. Wallén, L. Brodin, and A. Lansner, Neuronal network generating locomotor behavior in lamprey, Ann. Rev. Neurosci., 14 (1991), pp. 169-199.
165. J. Guckenheimer, Isochrons and phaseless sets, J. Math. Biol., 1 (1975), pp. 259-273.
166. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
167. J. Guckenheimer and B. Meloon, Computing periodic orbits and their bifurcations with automatic differentiation, SIAM J. Sci. Comput., 22 (2000), pp. 951-985.
168. H. Haken, J. A. S. Kelson and H. Bunz, A theoretical model of phase transitions in human hand movements, Biol. Cybern., 51 (1985), pp. 347-356.
169. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems, Springer-Verlag, New York, 1991.
170. J. Hancox and R. Pitman, Plateau potentials drive axonal impulse bursts in insect motoneurons, Proc. Roy. Soc. London Ser. B, 244 (1991), pp. 33-38.
171. D. Hansel and G. Mato, Phase dynamics for weakly coupled Hodgkin-Buxley neurons, Europhys. Letters, 23 (1993), pp. 367-372.
172. D. Hansel, G. Mato, and C. Meunier, Phase reduction and neural modeling, Concepts Neurosci., 4 (1993), pp. 192-210.
173. D. Hansel, G. Mato, and C. Meunier, Synchrony in excitatory neural networks, Neural Comp., 7 (1995), pp. 307-337.
174. H. Hatze, A myocybernetic control model of skeletal muscle, Biol. Cybernet., 25 (1977), pp. 103-119.
175. N. Heglund, G. Cavagna, and C. Taylor, Energetics and mechanics of terrestrial locomotion III: Energy changes of the centre of mass as a function of speed and body size in birds and mammals, J. Exp. Biol., 97 (1982), pp. 41-56.
176. N. Heglund and C. Taylor, Speed, stride frequency and energy cost per stride. How do they change with body size and gait?, J. Exp. Biol., 138 (1988), pp. 301-318.
177. N. Heglund, C. Taylor, and T. McMahon, Scaling stride frequency and gait to animal size: Mice to horses, Science, 186 (1974), pp. 1112-1113.
178. A. Hill, The heat of shortening and the dynamic constants of muscle, Proc. Roy. Soc. London Ser. B, 126 (1938), pp. 136-195.
179. J. Hindmarsh and R. Rose, A model of neuronal bursting using three coupled first order differential equations, Proc. Roy. Soc. London Ser. B, 221 (1984), pp. 87-102.
180. A. Hodgkin and A. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), pp. 500-544.
181. P. Holmes, Poincaré, celestial mechanics, dynamical systems theory and "chaos," Phys. Rep., 193 (3) (1990), pp. 137-163.
182. M. Höltje and R. Hustert, Rapid mechano-sensory pathways code leg impact and elicit very rapid reflexes in insects, J. Exp. Biol., 206 (2003), pp. 2715-2724.
183. F. Hoppensteadt and E. Izhikevich, Weakly Connected Neural Networks, Springer-Verlag, New York, 1997.
184. D. Hoyt and C. Taylor, Gait and the energetics of locomotion in horses, Nature, 292 (1981), pp. 239-240.
185. R. Huey, Phylogeny, history and the comparative method, in Directions in Ecological Physiology, M. Feder, A. Bennett, W. Burggren, and R. Huey, eds., Cambridge University Press, Cambridge, UK, 1988, pp. 76-97.
186. C. Hughes, The co-ordination of insect movements. I. The walking movement of insects, J. Exp. Biol., 29 (1952), pp. 267-284.
187. A. Huxley, Muscular contraction, J. Physiol., 243 (1974), pp. 1-43.
188. A. Ijspeert, A connectionist central pattern generator for the aquatic and terrestrial gaits of a simulated salamander, Biol. Cybern., 84 (2001), pp. 331-348.
189. A. Ijspeert and J. Kodjabachian, Evolution and development of a central pattern generator for the swimming of a lamprey, Artificial Life, 5 (1999), pp. 247-269.
190. Y. Ivanenko, R. Grasso, V. Macellari, and F. Lacquaniti, Two-thirds power law in human locomotion: Role of ground contact forces, Neuroreport, 13 (2002), pp. 1171-1174.
191. A. Jayes and R. Alexander, Mechanics of locomotion of dogs (Canis familiaris) and sheep (Ovis aries), J. Zool. London, 185 (1978), pp. 289-308.
192. D. Jindrich and R. Full, Dynamic stabilization of rapid hexapedal locomotion, J. Exp. Biol., 205 (2002), pp. 2803-2823.
193. D. Johnston and S.-S. Wu, Foundations of Cellular Neurophysiology, MIT Press, Cambridge, MA, 1995.
194. C. Jones, Geometric Singular Perturbation Theory, Lecture Notes in Math. 1609, C.I.M.E. Lectures, Springer-Verlag, Heidelberg, 1994.
195. S. Jones, B. Mulloney, T. Kaper, and N. Kopell, Coordination of cellular pattern-generating circuits that control limb movements: The sources of stable differences in intersegmental phases, J. Neurosci., 23 (2003), pp. 3457-3468.
196. R. Josephson, Contraction dynamics and power output of skeletal muscle, Ann. Rev. Physiol., 55 (1993), pp. 527-546.
197. R. Jung, T. Kiemel, and A. Cohen, Dynamic behavior of a neural network model of locomotor control in the lamprey, J. Neurophys., 75 (1996), pp. 1074-1086.
198. E. Kandel, J. Schwartz, and T. Jessel, Principles of Neuroscience, 3rd ed., Appleton and Lange, Norwalk, CT, 1991.
199. M. Kawato, Internal models for motor control and trajectory planning, Current Opinion in Neurobiol., 9 (1999), pp. 718-727.
200. J. Keener and J. Sneyd, Mathematical Physiology, Springer-Verlag, New York, 1998.
201. J. Kelso, P. W. Fink, C. DeLaplain, and R. Carson, Haptic information stabilizes and destabilizes coordination dynamics, Proc. Roy. Soc. London Ser. B, 268 (2001), pp. 1207-1213.
202. T. Kepler, L. Abbott, and E. Marder, Reduction of conductance-based neuron models, Biol. Cybern., 66 (1992), pp. 381-387.
203. T. Kiemel and A. Cohen, Estimation of coupling strength in regenerated lamprey spinal cords based on a stochastic phase model, J. Comput. Neurosci., 5 (1998), pp. 267-284.
204. T. Kindermann, Behavior and adaptability of a six-legged walking system with highly distributed control, Adaptive Behavior, 9 (2002), pp. 16-41.
205. E. Klavins and D. Koditschek, Phase regulation of decentralized cyclic robotic systems, Int. J. Robotics Res., 3 (2002), pp. 257-275.
206. E. Klavins, H. Komsuoglu, D. Koditschek, and R. Full, Coordination and control for locomotion, in Neurotechnology for Biomimetic Robots, A. Rudolph, ed., MIT Press, Boston, MA, 2001, pp. 351-382.
207. D. Koditschek and M. Buehler, Analysis of a simplified hopping robot, Int. J. Robotics Res., 10 (1991), pp. 587-605.
208. D. Koditschek, R. Full, and M. Buehler, Mechanical aspects of legged locomotion control, Arthropod Structure and Development, 33 (2004), pp. 251-272.
209. H. Komsuoglu, D. McMordie, U. Saranli, N. Moore, M. Buehler, and D. Koditschek, Proprioceptive-based behavioral advances in a hexapod robot, in Proceedings of the IEEE International Conference on Robotics and Automation, IEEE Press, Piscataway, NJ, 2001, pp. 3650-3655.
210. N. Kopell, Toward a theory of modelling central pattern generators, in Neural Control of Rhythmic Movements in Vertebrates, A. Cohen, S. Rossignol, and S. Grillner, eds., Wiley, New York, 1988, pp. 369-413.
211. N. Kopell and G. B. Ermentrout, Phase transitions and other phenomena in chains of coupled oscillators, SIAM J. Appl. Math., 50 (1990), pp. 1014-1052.
212. N. Kopell, G. B. Ermentrout, and T. L. Williams, On chains of oscillators forced at one end, SIAM J. Appl. Math., 51 (1991), pp.
213. N. Kopell and G. LeMasson, Rhythmogenesis, amplitude modulation, and multiplexing in a cortical architecture, Proc. Natl. Acad. Sci. USA, 91 (1994), pp. 10586-10590.
214. R. Kram, B. Wong, and R. Full, Three-dimensional kinematics and limb kinetic energy of running cockroaches, J. Exp. Biol., 200 (1997), pp. 1919-1929.
215. H. Krebs, For many problems there is an animal on which it can be most conveniently studied, J. Exp. Zool., 194 (1975), pp. 221-226.
216. T. Kubow and R. Full, The role of the mechanical system in control: A hypothesis of self-stabilization in hexapedal runners, Philos. Trans. Roy. Soc. London Ser. B, 354 (1999), pp. 849-861.
217. A. Kuo, Stabilization of lateral motion in passive dynamic walking, Int. J. Robotics Res., 18 (1999), pp. 917-930.
218. A. Kuo, The relative roles of feedforward and feedback in the control of rhythmic movements, Motor Control, 6 (2002), pp. 129-145.
219. F. Lacquaniti, R. Grasso, and M. Zago, Motor patterns in walking, News Physiol. Sci., 14 (1999), pp. 168-174.
220. F. Lacquaniti, C. Terzuolo, and P. Viviani, The law relating the kinematic and figural aspects of drawing movements, Acta Psychol., 54 (1983), pp. 115-130.
221. A. Lamperski, O, Loh, B. Kutscher, and N. Cowan, Dynamical wall-following for a wheeled robot using a passive tactile sensor, in Proceedings of the IEEE International Conference on Robotics and Automation, 2005.
222. G. Larsen, S. Frazier, and S. Zill, The tarso-pretarsal chordotonal organ as an element in cockroach walking, J. Comp. Physiol., 180 (1997), pp. 683-700.
223. G. Lauder, Evolutionary innovations and constraints: Functional analyses and phylogenetic test, J. Morphology, 248 (2001), p. 201.
224. A. Lewis, J. Ostrowski, R. Murray, and J. Burdick, Nonholonomic mechanics and locomotion: The snakeboard example, in International Conference on Robotics and Automation, IEEE Press, Piscataway, NJ, 1994, pp. 2391-2400.
225. T. Lewis and J. Rinzel, Dynamics of spiking neurons connected by both inhibitory and electrical coupling, J. Comput. Neurosci., 14 (2003), pp. 283-309.
226. C.-S. Li, C. Padoa-Schioppa, and E. Bizzi, Neuronal correlates of motor performance and motor learning in the primary motor cortex of monkeys adapting to an external force field, Neuron, 30 (2001), pp. 593-607.
227. G. Loeb, I. Brown, and E. Cheng, A hierarchical foundation for models of sensorimotor control, Exp. Brain Res., 126 (1999), pp. 1-18.
228. I. Malkin, Methods of Poincaré and Linstedt in the Theory of Nonlinear Oscillations, Gostexisdat, Moscow, 1949 (in Russian).
229. I. Malkin, Some Problems in Nonlinear Oscillation Theory, Gostexisdat, Moscow, 1956 (in Russian).
230. E. Mallett, G. Yamaguci, J. Birch, and K. Nishikawa, Feeding motor patterns in Anurans: Insight from biomechanical modeling, Amer. Zool., 41 (2001), pp. 1364-1374.
231. C. Mangum, and P. Hochachka, New directions in comparative physiology and biochemistry: Mechanisms, adaptations, and evolution, Physiol. Zool., 71 (1998), pp. 471-484.
232. E. Marder, Motor pattern generation, Current Opinion in Neurobiol., 10 (2000), pp. 691-698.
233. R. Margaria, Biomechanics and Energetics of Muscular Exercise, Oxford University Press, Oxford, UK, 1976.
234. J. Marsden, Lectures on Mechanics, London Math. Soc. Lecture Note Ser. 174, Cambridge University Press, Cambridge, UK, 1992.
235. J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed., Springer-Verlag, New York, 1999.
236. T. McGeer, Passive bipedal running, Proc. Roy. Soc. London Ser. B, 240 (1990), pp. 107-134.
237. T. McGeer, Passive dynamic walking, Int. J. Robotics Res., 9 (1990), pp. 62-82.
238. T. McGeer, Passive walking with knees, in Proceedings of the IEEE International Conference on Robotics and Automation, Cincinnati, OH, 1990, IEEE Press, Piscataway, NJ, pp. 1640-1645.
239. T. McGeer, Principles of walking and running, in Advances in Comparative and Environmental Physiology, Vol. 11: Mechanics of Animal Locomotion, R. Alexander, ed., Springer-Verlag, Berlin, 1992, pp. 113-139.
240. T. McGeer, Dynamics and control of bipedal locomotion, J. Theoret. Biol., 16 (1993), pp. 277-314.
241. R. McGehee, Triple collisions in Newtonian gravitational systems, in Dynamical Systems, Theory and Applications, J. Moser, ed., Lecture Notes in Phys. 38, Springer-Verlag, Berlin, 1975, pp. 550-572.
242. T. McMahon, Muscles, Reflexes and Locomotion, Princeton University Press, Princeton, NJ, 1984.
243. T. McMahon and J. Bonner, On Size and Life, Scientific American Books/ W.H. Freeman, New York, 1983.
244. T. McMahon and G. Cheng, The mechanics of running: how does stiffness couple with speed?, J. Biomechanics, 23 (suppl 1) (1990), pp. 65-78.
245. K. Meijer. Personal communication, 2001.
246. K. Meijer, H. Grootenboer, H. Koopman, B. van der Linden, and P. Huijing, A Hill type model of rat medial gastrocnemius muscle that accounts for shortening history effects, J. Biomech., 31 (1998), pp. 555-564.
247. S. Mochon and T. McMahon, Ballistic walking, J. Biomech., 13 (1980), pp. 49-57.
248. S. Mochon and T. McMahon, Ballistic walking: An improved model, Math. Biosci., 52 (1981), pp. 241-260.
249. K. Mombaur, R. Longman, H. Bock, and J. Schlöder, Stable one-legged hopping without feedback and with a point foot, in Proceedings of the IEEE International Conference on Robotics and Automation, Vol. 4, 2002, pp. 3978-3983.
250. C. Morris and H. Lécar, Voltage oscillations in the barnacle giant muscle, Biophys. J., 35 (1981), pp. 193-213.
251. J. Murray, Mathematical Biology, 3rd ed., Springer-Verlag, New York, 2001.
252. F. Mussa-Ivaldi, Modular features of motor control and learning, Current Opinion in Neurobiol., 9 (1999), pp. 713-717.
253. Y. Nakamura, Advanced Robotics: Redundancy and Optimization, Addison-Wesley, Reading, MA, 1991.
254. J. Nakanishi, T. Fukuda, and D. Koditschek, A brachiating robot controller, IEEE Trans. Robotics and Automat., 16 (2000), pp. 109-123.
255. J. Neǐmark and N. Fufaev, Dynamics of Nonholonomic Systems, AMS, Providence, RI, 1972.
256. J. Ostrowski and J. Burdick, Gait kinematics for a serpentine robot, in International Conference on Robotics and Automation, IEEE Press, 1996, Piscataway, NJ, pp. 1294-1299.
257. K. Pearson, Central programming and reflex control of walking in the cockroach, J. Exp. Biol., 56 (1972), pp. 173-193.
258. K. Pearson, Common principles of motor control in vertebrates and invertebrates, Ann. Rev. Neurosci., 16 (1993), pp. 265-297.
259. K. Pearson, Proprioceptive regulation of locomotion, Current Opinion in Neurobiol., 5 (1995), pp. 786-791.
260. K. Pearson, Motor systems, Current Opinion in Neurobiol., 10 (2000), pp. 649-654.
261. K. Pearson and J. Iles, Discharge patterns of coxal levator and depressor motoneurones in the cockroach periplaneta americana, J. Exp. Biol., 52 (1970), pp. 139-165.
262. K. Pearson and J. Iles, Innervation of the coxal depressor muscles in the cockroach periplaneta americana, J. Exp. Biol., 54 (1971), pp. 215-232.
263. K. Pearson and J. Iles, Nervous mechanisms underlying intersegmental co-ordination of leg movements during walking in the cockroach, J. Exp. Biol., 58 (1972), pp. 725-744.
264. K. Pearson, R. Wong, and C. Fourtner, Connections between hairplate afferents and motoneurons in the cockroach leg, J. Exp. Biol., 64 (1976), pp. 251-266.
265. E. Phipps, Taylor Series Integration of Differential-Algebraic Equations: Automatic Differentiation as a Tool for Simulating Rigid Body Mechanical Systems, Ph.D. thesis, Cornell University, Ithaca, NY, 2003.
266. I. Poulakakis, J. Smith, and M. Buehler, Experimentally validated bounding models for the Scout II quadrupedal robot, in International Conference on Robotics and Automation, Vol. 3, IEEE Press, Piscataway, NJ, 2004, pp. 2595-2600.
267. I. Poulakakis, J. Smith, and M. Buehler, Modeling and experiments of untethered quadrupedal running with a bounding gait: The Scout II robot, Int. J. Robotics Res., 24 (2005), pp. 239-256.
268. M. Raibert, Legged Robots that Balance, MIT Press, Cambridge, MA, 1986.
269. M. Raibert, M. Chepponis, and H. Brown Jr, Running on four legs as though they were one, IEEE J. Robotics and Automat., 2 (1986), pp. 70-82.
270. M. Reed, Why is mathematical biology so hard?, AMS Notices, 51 (2004), pp. 338-342.
271. M. Richardson and T. Flash, Comparing smooth arm movements with the two-thirds power law and the related segmented-control hypothesis, J. Neurosci., 22 (2002), pp. 8201-8211.
272. A. Ridgel, S. Frazier, R. A. DiCaprio, and S. Zill, Encoding of forces by cockroach tibial campaniform sensilla: Implications in dynamic control of posture and locomotion, J. Comp. Physiol. A, 186 (1998), pp. 359-374.
273. R. Riener and J. Quintern, A physiologically based model of muscle activation verified by electrical stimulation, Bioelectrochemistry and Bioenergetics, 43 (1997), pp. 257-264.
274. J. Rinzel, A formal classification of bursting mechanisms in excitable systems, in Mathematical Topics in Population Biology, Morphogenesis, and Neurosciences, E. Teramoto and M. Yamaguti, eds., Lecture Notes in Biomath., Springer-Verlag, New York, 1987, pp. 267-281.
275. J. Rinzel and G. Ermentrout, Analysis of excitability and oscillations, in Methods in Neuronal Modeling: From Ions to Networks, C. Koch and I. Segev, eds., MIT Press, Cambridge, MA, 1999, Chap. 7, pp. 251-291.
276. J. Rinzel and Y. Lee, On different mechanisms for membrane potential bursting, in Nonlinear Oscillations in Biology and Chemistry, H. Othmer, ed., Lecture Notes in Biomath. 66, Springer-Verlag, Berlin, 1986, pp. 19-33.
277. J. Rinzel and Y. Lee, Dissection of a model for neuronal parabolic bursting, J. Math. Biol., 25 (1987), pp. 653-675.
278. R. Ritzmann, S. Gorb, and R. D. Quinn, eds., Arthropod locomotion systems: From biological materials and systems to robotics, Arthropod Structure and Development, 33 (2004), special issue.
279. A. Rizzi and D. Koditschek, Further progress in robot juggling: Solvable mirror laws, in Proceedings of the International Conference on Robotics and Automation, IEEE Press, Piscataway, NJ, 1994, pp. 2935-2940.
280. H. Robinson and N. Kawai, Injection of digitally-synthesized synaptic conductance transients to measure the integrative properties of neurons, J. Neurosci. Methods, 49 (1993), pp. 157-165.
281. R. Rose and J. Hindmarsh, The assembly of ionic currents in a thalamic neuron I: The three-dimensional model, Proc. Roy. Soc. London Ser. B, 237 (1989), pp. 267-288.
282. A. Ruina, Non-holonomic stability aspects of piecewise holonomic systems, Rep. Math. Phys., 42 (1998), pp. 91-100.
283. P. Saltiel, K. Wyler-Duda, A. D'Avella, M. Tresch, and E. Bizzi, Muscle synergies encoded within the spinal cord: Evidence from focal intraspinal NMDA iontophoresis in the frog, J. Neurophys., 85 (2001), pp. 605-619.
284. U. Saranli, M. Buehler, and D. Koditschek, RHex: A simple and highly mobile hexapod robot, Int. J. Robotics Res., 20 (2001), pp. 616-631.
285. U. Saranli and D. Koditschek, Template based control of hexapedal running, in Proceedings of the IEEE International Conference on Robotics and Automation, Vol. 1, Taipei, Taiwan, IEEE Press, Piscataway, NJ, 2003, pp. 1374-1379.
286. U. Saranli, W. Schwind, and D. Koditschek, Toward the control of a multi-jointed, monoped runner, in Proceedings of the IEEE International Conference on Robotics and Automation, Leuven, Belgium, IEEE Press, Piscataway, NJ, 1998, pp. 2676-2682.
287. S. Schaal and D. Sternad, Origins and violations of the 2/3 power law in rhythmic three-dimensional arm movements, Exp. Brain Res., 136 (2001), pp. 60-72.
288. S. Schaal, D. Sternad, and C. Atkeson, One-handed juggling: A dynamical approach to a rhythmic movement task, J. Motor Behavior, 28 (1996), pp. 165-183.
289. K. Schmidt-Nielsen, Scaling: Why Is Animal Size So Important?, Cambridge University Press, New York, 1984.
290. J. Schmitt, M. Garcia, R. Razo, P. Holmes, and R. Full, Dynamics and stability of legged locomotion in the horizontal plane: A test case using insects, Biol. Cybern., 86 (2002), pp. 343-353.
291. J. Schmitt and P. Holmes, Mechanical models for insect locomotion: Dynamics and stability in the horizontal plane - Theory, Biol. Cybern., 83 (2000), pp. 501-515.
292. J. Schmitt and P. Holmes, Mechanical models for insect locomotion: Dynamics and stability in the horizontal plane - Application, Biol. Cybern., 83 (2000), pp. 517-527.
293. J. Schmitt and P. Holmes, Mechanical models for insect locomotion: Stability and parameter studies, Phys. D, 156 (2001), pp. 130-168.
294. J. Schmitt and P. Holmes, Mechanical models for insect locomotion: Active muscles and energy losses, Biol. Cybern., 89 (2003), pp. 43-55.
295. A. Schwartz and D. Moran, Motor cortical activity during drawing movements: Population representation during lemniscate tracing, J. Neurophys., 82 (1999), pp. 2705-2718.
296. W. Schwind, Spring Loaded Inverted Pendulum Running: A Plant Model, Ph.D. thesis, University of Michigan, Ann Arbor, MI, 1998.
297. W. Schwind and D. Koditschek, Characterization of monoped equilibrium gaits, in Proceedings of the IEEE International Conference on Robotics and Automation, Albuquerque, NM, IEEE Press, Piscataway, NJ, 1997, pp. 1986-1992.
298. W. Schwind and D. Koditschek, Approximating the stance map of a 2 DOF monoped runner, J. Nonlinear Sci., 10 (2000), pp. 533-568.
299. J. Seipel and P. Holmes, Running in three dimensions: Analysis of a point-mass springy-leg model, Int. J. Robotics Res., 24 (2005), pp. 657-674.
300. J. Seipel and P. Holmes, Three-dimensional dynamics and stability of multi-legged runners, Int. J. Robotics Res., submitted.
301. J. Seipel and P. Holmes, Three-dimensional running is unstable but easily stabilized, in Climbing and Walking Robots: Proceedings of the 7th International Conference CLAWAR 2004, M. A. Armada and P. González de Santos, eds., Springer-Verlag, Berlin, 2005, pp. 585-592.
302. J. Seipel, P. Holmes, and R. Full, Dynamics and stability of insect locomotion: A hexapedal model for horizontal plane motions, Biol. Cybern., 91 (2004), pp. 76-90.
303. A. Selverston, R. Elson, M. Rabinovich, R. Huerta, and H. Abarbanel, Basic principles for generating motor output in the stomatogastric ganglion, in Neuronal Mechanisms for Generating Locomotor Activity, O. Kiehn, R. M. Harris-Warrick, L. M. Jordan, H. Hultborn, and N. Kudo, eds., Ann. New York Acad. Sci. 860, New York Academy of Sciences, New York, 1998, pp. 35-50.
304. A. Seyfarth and H. Geyer, Natural control of spring-like running - optimized self-stabilization, in Proceedings of the Fifth International Conference on Climbing and Walking Robots (CLAWAR 2002), Professional Engineering Publishing Limited, 2002, pp. 81-85.
305. A. Seyfarth, H. Geyer, R. Blickhan, and H. Herr, Does leg retraction simplify control in running?, in the Fourth World Congress of Biomechanics, Calgary, Canada, Omnipress, 2002 (CD-ROM).
306. A. Seyfarth, H. Geyer, M. Günther, and R. Blickhan, A movement criterion for running, J. Biomech., 35 (2002), pp. 649-655.
307. A. Seyfarth, H. Geyer, and H. Herr, Swing leg retraction: A simple control model for stable running, J. Exp. Biol., 206 (2003), pp. 2547-2555.
308. A. Sharp, M. O'Neil, L. Abbott, and E. Marder, Dynamic clamp: Computer-generated conductances in real neurons, J. Neurophys., 69 (1993), pp. 992-995.
309. A. Sharp, F. Skinner, and E. Marder, Mechanisms of oscillation in dynamic clamp constructed two-cell half-center circuits, J. Neurophys., 76 (1996), pp. 867-883.
310. Y. Sinai, Hyperbolic billiards, in Proceedings of the International Congress of Mathematicians, Kyoto, 1990, Vol. I, Springer-Verlag, Berlin, 1991, pp. 249-260.
311. D. Singer, Stable orbits and bifurcations of maps of the interval, SIAM J. Appl. Math., 35 (1978), pp. 260-267.
312. P. Swanson, R. Burridge, and D. Koditschek, Global asymptotic stability of a passive juggler. A parts feeding strategy, Math. Probab. Engrg., 1 (1995), pp. 193-224.
313. S. P. Swinnen, Intermanual coordination: From behavioural principles to neural-network interactions, Nature Neurosci., 3 (2002), pp. 348-359.
314. F. Theunissen and J. Miller, Temporal encoding in nervous systems - a rigorous definition, J. Comput. Neurosci., 2 (1995), pp. 149-162.
315. F. Theunissen, J. Roddey, S. Stufflebeam, H. Clague, and J. Miller, Information theoretic analysis of dynamical encoding by four primary sensory interneurons in the cricket cercal system, J. Neurophys., 75 (1996), pp. 1345-1359.
316. J. Thompson, Instabilities and Catastrophes in Science and Engineering, Wiley, New York, 1982.
317. L. Ting, R. Blickhan, and R. Full, Dynamic and static stability in hexapedal runners, J. Exp. Biol., 197 (1994), pp. 251-269.
318. E. Todorov, Direct cortical control of muscle activation in voluntary arm movements: A model, Nature Neurosci., 3 (2000), pp. 391-398.
319. E. Todorov and M. Jordan, Smoothness maximization along a predefined path accurately predicts the speed profIles of complex arm movements, J. Neurophys., 80 (1998), pp. 696-714.
320. E. Todorov and M. Jordan, Optimal feedback control as a theory of motor coordination, Nature Neurosci., 5 (2002), pp. 1226-1235.
321. A. Tryba and R. Ritzmann, Multi-joint coordination during walking and foothold searching in the Blaberus cockroach. I. Kinematics and electromyograms, J. Neurophys., 83 (2000), pp. 3323-3336.
322. A. Tryba and R. Ritzmann, Multi-joint coordination during walking and foothold searching in the Blaberus cockroach. II. Extensor motor patterns, J. Neurophys., 83 (2000), pp. 3337-3350.
323. V. Utkin, Variable structure systems with sliding modes, IEEE Trans. Automat. Control, AC-22 (1977), pp. 212-231.
324. F. Valero-Cuevas, Large index-fingertip forces are produced by subject-independent patterns of muscle excitation, J. Biomech., 31 (1998), pp. 693-703.
325. F. Valero-Cuevas, Predictive modulation of muscle coordination pattern magnitude scales fingertip force magnitude over the voluntary range, J. Neurophys., 83 (2000), pp. 1469-1479.
326. F. Valero-Cuevas, An integrative approach to the biomechanical function and neuromuscular control of the fingers, J. Biomech., 38 (2005), pp. 673-684.
327. F. Valero-Cuevas, N. Smaby, M. Venkadesan, M. Peterson, and T. Wright, The strength-dexterity test as a measure of dynamic pinch performance, J. Biomech., 36 (2003), pp. 265-270.
328. K. Waldron, Force and motion management in legged locomotion, IEEE J. Robotics and Automat., RA-2 (1986), pp. 214-220.
329. K. J. Waldron and V. J. Vohnout, Configuration design of the adaptive suspension vehicle, Int. J. Robotics Res., 3 (1984), pp. 37-48.
330. J. Watson and R. Ritzmann, Leg kinematics and muscle activity during treadmill running in the cockroach, blaberus discoidalis. I. Slow running, J. Comp. Physiol. A, 182 (1998), pp. 11-22.
331. J. Watson and R. Ritzmann, Leg kinematics and muscle activity during treadmill running in the cockroach, blaberus discoidalis. II. Transition to fast running, J. Comp. Physiol. A, 182 (1998), pp. 23-33.
332. B. Webb, R. Harrison, and M. Willis, Sensorimotor control of navigation in arthropod and artificial systems, Arthropod Structure and Development, 33 (2004), pp. 301-329.
333. J. Weingarten, R. Groff, and D. Koditschek, A framework for the coordination of legged robot gaits, in Proceedings of the IEEE Conference on Robotics, Automation and Mechatronics, IEEE Press, Singapore, 2004, pp. 679-686.
334. E. Westervelt, J. Grizzle, and D. Koditschek, Hybrid zero dynamics of planar biped walkers, IEEE Trans. Automat. Control, 48 (2003), pp. 42-56.
335. S. Wickl, D. Hoyt, E. Cogger, and G. Myers, The energetics of the trot-gallop transition, J. Exp. Biol., 206 (2003), pp. 1557-1564.
336. D. Wilson, Insect walking, Ann. Rev. Entomol., 7 (1966), pp. 103-122.
337. D. Wilson, An approach to the problem of control of rhythmic behavior, in Invertebrate Nervous System, C. Wiersma, ed., The University of Chicago Press, Chicago, IL, 1967, Chap. 17, pp. 219-229.
338. A. Winfree, The Geometry of Biological Time, Springer-Verlag, New York, 2001.
339. R. Wong and K. Pearson, Properties of the trochanteral hair plate and its function in the control of walking in the cockroach, J. Exp. Biol., 64 (1976), pp. 233-249.
340. C. Woolsey and N. Leonard, Stabilizing underwater vehicle motion using internal rotors, Automatica, 38 (2002), pp. 2053-2062.
341. P. Wriggers, Computational Contact Mechanics, Wiley, New York, 2002.
342. X. Xu, W. Cheng, D. Dudek, M. Cutkosky, M. Hatanaka, and R. Full, Material modeling for shape deposition manufacturing of biomimetic components, in ASME Proceedings, DETC/DFM 2000, Baltimore, MD, 2000.
343. F. Zajac, Muscle and tendon: Properties, models, scaling, and application to biomechanics and motor control, Critical Rev. Biomed. Eng., 17 (1989), pp. 359-411.
344. E. Zehr and R. Stein, What functions do reflexes serve during human locomotion?, Prog. Neurobiol., 58 (1999), pp. 185-205.
345. S. Zill, Proprioceptive feedback and the control of cockroach walking, in Feedback and Motor Control in Invertebrates and Vertebrates, W. Barnes and M. Galden, eds., Croom Helm, London, 1985, pp. 187-208.
346. S. Zill, Mechanoreceptors: Exteroceptors and proprioceptors, in Cockroaches as Models for Neurobiology: Applications in Biomedical Research, Vol. 2, I. Huber, E. P. Masler, and B. R. Rao, eds., CRC Press, Boca Raton, FL, 1990, Chap. 25, pp. 247-267.
347. S. Zill, J. Schmitz, and A. Büschges, Load sensing and control of posture and locomotion, Arthropod Structure and Development, 33 (2004), pp. 273-286.