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Numerical bifurcation analysis of the bipedal spring-mass model. (English) Zbl 1335.37056
Summary: The spring-mass model and its numerous extensions are currently one of the best candidates for templates of human and animal locomotion. However, with increasing complexity, their applications can become very time-consuming. In this paper, we present an approach that is based on the calculation of bifurcations in the bipedal spring-mass model for walking. Since the bifurcations limit the region of stable walking, locomotion can be studied by computing the corresponding boundaries. Originally, the model was implemented as a hybrid dynamical system. Our new approach consists of the transformation of the series of initial value problems on different intervals into a single boundary value problem. Using this technique, discontinuities can be avoided and sophisticated numerical methods for studying parametrized nonlinear boundary value problems can be applied. Thus, appropriate extended systems are used to compute transcritical and period-doubling bifurcation points as well as turning points. We show that the resulting boundary value problems can be solved by the simple shooting method with sufficient accuracy, making the application of the more extensive multiple shooting superfluous. The proposed approach is fast, robust to numerical perturbations and allows determining complete manifolds of periodic solutions of the original problem.

MSC:
37N05 Dynamical systems in classical and celestial mechanics
92C15 Developmental biology, pattern formation
34C25 Periodic solutions to ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
37M05 Simulation of dynamical systems
Software:
RWPKV; RWPM; SOCS
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References:
[1] Seyfarth, A.; Grimmer, S.; Häufle, D.; Maus, H.-M.; Peuker, F.; Kalveram, K.-T., Biomechanical and neuromechanical concepts for legged locomotion: computer models and robot validation, (Gollhofer, A.; Taube, W.; Nielsen, J. B., Routledge Handbook of Motor Control and Motor Learning, Routledge International Handbooks, (2012)), 90-110, (Chapter 5)
[2] Seyfarth, A.; Lipfert, S. W.; Rummel, J.; Maus, H.-M.; Maykranz, D., Walking and running: how leg compliance shapes the way we move, (Mombaur, K.; Berns, K., Modeling, Simulation and Optimization of Bipedal Walking, Cognitive Systems Monographs, vol. 18, (2013), Springer), 211-222
[3] Blickhan, R., The spring-mass model for running and hopping, J. Biomech., 22, 11-12, 1217-1227, (1989)
[4] McMahon, T. A.; Cheng, G. C., The mechanics of running: how does stiffness couple with speed?, J. Biomech., 23, Suppl. 1, 65-78, (1990)
[5] Geyer, H.; Seyfarth, A.; Blickhan, R., Compliant leg behaviour explains basic dynamics of walking and running, Proc. R. Soc. Biol. Sci. Ser. B, 273, 1603, 2861-2867, (2006)
[6] Rummel, J.; Seyfarth, A., Stable running with segmented legs, Int. J. Robot. Res., 27, 8, 919-935, (2008)
[7] Maykranz, D.; Grimmer, S.; Lipfert, S. W.; Seyfarth, A., Foot function in spring mass running, (Dillmann, R.; Beyerer, J.; Stiller, C.; Zöllner, J. M.; Gindele, T., Autonome Mobile Systeme, Informatik Aktuell, (2009), Springer Karlsruhe, Germany), 81-88
[8] Maus, H.-M.; Lipfert, S. W.; Gross, M.; Rummel, J.; Seyfarth, A., Upright human gait did not provide a major mechanical challenge for our ancestors, Nature Commun., 1, 6, 70, (2010)
[9] Blum, Y.; Lipfert, S. W.; Rummel, J.; Seyfarth, A., Swing leg control in human running, Bioinspiration Biomimetics, 5, 2, 026006, (2010), (11pp)
[10] Riese, S.; Seyfarth, A., Stance leg control: variation of leg parameters supports stable hopping, Bioinspiration Biomimetics, 7, 1, 016006, (2012)
[11] Rummel, J.; Blum, Y.; Seyfarth, A., Robust and efficient walking with spring-like legs, Bioinspiration Biomimetics, 5, 4, 046004, (2010), (13pp)
[12] Merker, A.; Rummel, J.; Seyfarth, A., Stable walking with asymmetric legs, Bioinspiration Biomimetics, 6, 4, 045004, (2011)
[13] Strogatz, S. H., Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering, (1994), Westview Press
[14] Marx, B.; Vogt, W., Dynamische systeme—theorie und numerik, (2011), Spektrum Akademischer Verlag · Zbl 1213.37002
[15] Moore, G.; Spence, A., The calculation of turning points of nonlinear equations, SIAM J. Numer. Anal., 17, 4, 567-576, (1980) · Zbl 0454.65042
[16] Keener, J. P.; Keller, H. B., Perturbed bifurcation theory, Arch. Ration. Mech. Anal., 50, 159-175, (1973) · Zbl 0254.47080
[17] Moore, G., The numerical treatment of non-trivial bifurcation points, Numer. Funct. Anal. Optim., 2, 6, 441-472, (1980) · Zbl 0459.65040
[18] Weber, H., On the numerical approximation of secondary bifurcation problems, (Allgower, E. L.; Glashoff, K.; Peitgen, H.-O., Numerical Solution of Nonlinear Equations, Lecture Notes in Mathematics, vol. 878, (1981), Springer), 407-425
[19] Weber, H., Numerische behandlung von verzweigungsproblemen bei gewöhnlichen differentialgleichungen, Numer. Math., 32, 17-29, (1979) · Zbl 0413.65064
[20] Wallisch, W.; Hermann, M., (Numerische Behandlung von Fortsetzungs- und Bifurkationsproblemen bei Randwertaufgaben, Teubner-Texte zur Mathematik, vol. 102, (1987), Teubner)
[21] Lipfert, S. W., Kinematic and dynamic similarities between walking and running, (2010), Verlag Dr. Kovac
[22] Alexander, R. M.; Jayes, A. S., Vertical movements in walking and running, J. Zool., 185, 1, 27-40, (1978)
[23] Alur, R.; Courcoubetis, C.; Halbwachs, N.; Henzinger, T. A.; Ho, P.-H.; Nicollin, X.; Olivero, A.; Sifakis, J.; Yovine, S., The algorithmic analysis of hybrid systems, Theoret. Comput. Sci., 138, 1, 3-34, (1995) · Zbl 0874.68206
[24] Holmes, P.; Full, R. J.; Koditschek, D.; Guckenheimer, J., The dynamics of legged locomotion: models, analyses, and challenges, Dynamics, 48, 2, 207-304, (2006) · Zbl 1100.34002
[25] Ghigliazza, R. M.; Holmes, P., Towards a neuromechanical model for insect locomotion: hybrid dynamical systems, Regul. Chaotic Dyn., 10, 2, 193-225, (2005) · Zbl 1084.34050
[26] Hermann, M.; Kaiser, D., RWPM: a software package of shooting methods for nonlinear two-point boundary value problems, Appl. Numer. Math., 13, 1-3, 103-108, (1993) · Zbl 0785.65089
[27] Hermann, M.; Kaiser, D.; Schröder, M., Theoretical and numerical studies of nonlinear shell equations, Physica D, 132, 1-2, 19-39, (1999) · Zbl 0951.74021
[28] Hermann, M.; Timokha, A., Modal modelling of the nonlinear resonant fluid sloshing in a rectangular tank I: a single-dominant model, Math. Models Methods Appl. Sci., 15, 09, 1431-1458, (2005) · Zbl 1098.76012
[29] Rummel, J.; Blum, Y.; Maus, H.-M.; Rode, C.; Seyfarth, A., Stable and robust walking with compliant legs, (Proceedings of International Conference on Robotics and Automation (ICRA), (2010), IEEE Anchorage, USA), 5250-5255
[30] Poincaré, H., LES méthodes nouvelles de la mécanique céleste, Bull. Amer. Math. Soc., 1, 9, 206-214, (1892)
[31] Guckenheimer, J.; Holmes, P., (Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, vol. 42, (1983), Springer) · Zbl 0515.34001
[32] Rummel, J.; Blum, Y.; Seyfarth, A., From walking to running, (Dillmann, R.; Beyerer, J.; Stiller, C.; Zöllner, J. M.; Gindele, T., Autonome Mobile Systeme, Informatik Aktuell, (2009), Springer Karlsruhe, Germany), 89-96
[33] Martinez, H. R.; Carbajal, J. P., From walking to running: a natural transition in the SLIP model using the hopping gait, (Proceedings of International Conference on Robotics and Biomimetics (ROBIO), (2011), IEEE Phuket, Thailand), 2163-2168
[34] Doedel, E. J.; Govaerts, W.; Kuznetsov, Y. A.; Dhooge, A., Numerical continuation of branch points of equilibria and periodic orbits, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15, 3, 841-860, (2005) · Zbl 1081.37054
[35] Hermann, M., Numerik gewöhnlicher differentialgleichungen: anfangs- und randwertprobleme, (2004), Oldenbourg · Zbl 1097.65076
[36] Zeidler, E., Nonlinear functional analysis and its applications I: fixed-point theorems, (1998), Springer
[37] Spence, A.; Werner, B., Non-simple turning points and cusps, IMA J. Numer. Anal., 2, 4, 413-427, (1982) · Zbl 0539.65043
[38] Jepson, A.; Spence, A., Folds in solutions of two parameter systems and their calculation. part I, SIAM J. Numer. Anal., 22, 2, 347-368, (1985) · Zbl 0576.65052
[39] Hermann, M., Numerische Mathematik, (2011), Oldenbourg · Zbl 1254.65001
[40] Stoer, J.; Bulirsch, R., Introduction to numerical analysis, (1993), Springer · Zbl 0771.65002
[41] Hopf, E., Abzweigung einer periodischen Lösung von einer stationären Lösung eines differentialsystems, Ber. Math.-Phys. Klasse Sächs. Akad. Wiss. Lepizig, 94, 1-22, (1942)
[42] Marsden, J. E.; McCracken, M., (The Hopf Bifurcation and its Applications, Applied Mathematical Sciences, vol. 19, (1976), Springer) · Zbl 0346.58007
[43] Hermann, M.; Saravi, M., A first course in ordinary differential equations. analytical and numerical methods, (2014), Springer · Zbl 1296.34001
[44] Seydel, R., A continuation algorithm with step control, (Küpper, T.; Middelmann, H. D.; Weber, H., Numerical Methods for Bifurcation Problems, International Series of Numerical Mathematics, vol. 70, (1984), Birkhäuser), 480-494 · Zbl 0544.65055
[45] Hermann, M.; Ullrich, K., RWPKV: a software package for continuation and bifurcation problems in two-point boundary value problems, Appl. Math. Lett., 5, 2, 57-61, (1992) · Zbl 0743.65073
[46] Andrada, E.; Nyakatura, J.; Müller, R.; Rode, C.; Blickhan, R., Grounded running: an overlooked strategy for robots, (Levi, P.; Zweigle, O.; Häußermann, K.; Eckstein, B., Autonomous Mobile Systems, Informatik Aktuell, (2012), Springer Stuttgart, Germany), 79-87
[47] Segers, V.; Aerts, P.; Lenoir, M.; De Clerq, D., Kinematics of the transition between walking and running when gradually changing speed, Gait Posture, 26, 3, 349-361, (2007)
[48] Sadeghi, H.; Allard, P.; Prince, F.; Labelle, H., Symmetry and limb dominance in able-bodied gait: a review, Gait Posture, 12, 1, 34-45, (2000)
[49] Gurney, B., Leg length discrepancy, Gait Posture, 15, 2, 195-206, (2002)
[50] Gregg, R.; Dhaher, Y.; Degani, A.; Lynch, K., On the mechanics of functional asymmetry in bipedal walking, IEEE Trans. Biomed. Eng., 59, 5, 1310-1318, (2012)
[51] Allgower, E. L.; Schwetlick, H., A general view of minimally extended systems for simple bifurcation points, ZAMM Z. Angew. Math. Mech., 77, 2, 83-97, (1997) · Zbl 0915.65042
[52] Pönisch, G.; Schwetlick, H., Computing turning points of curves implicitly defined by nonlinear equations depending on a parameter, Computing, 26, 107-121, (1981) · Zbl 0463.65036
[53] Renjewski, D.; Manoonpong, P.; Seyfarth, A.; Wörgötter, F., From biomechanical concepts towards fast and robust robots, (Proceedings of the Eleventh International Conference on Climbing and Walking Robots (CLAWAR), (2008), World Scientific), 630-637
[54] Renjewski, D.; Seyfarth, A., Robots in human biomechanics—a study on ankle push-off in walking, Bioinspiration Biomimetics, 7, 3, 036005, (2012)
[55] Lipfert, S. W.; Günther, M.; Renjewski, D.; Grimmer, S.; Seyfarth, A., A model-experiment comparison of system dynamics for human walking and running, J. Theoret. Biol., 292, 0, 11-17, (2012)
[56] Diehl, M.; Bock, H. G.; Diedam, H.; Wieber, P.-B., Fast direct multiple shooting algorithms for optimal robot control, (Diehl, M.; Mombaur, K., Fast Motions in Biomechanics and Robotics, Lecture Notes in Control and Information Sciences, vol. 340, (2006), Springer), 65-93 · Zbl 1310.70002
[57] Bock, H. G.; Plitt, K. J., A multiple shooting algorithm for direct solution of optimal control problems, (Proceedings of 9th IFAC World Congress, (1984), Pergamon Press Budapest), 243-247
[58] Betts, J. T., (Practical Methods for Optimal Control and Estimation using Nonlinear Programming, Advances in Design and Control, (2010), SIAM)
[59] Minetti, A. E.; Ardigo, L. P.; Saibene, F., The transition between walking and running in humans: metabolic and mechanical aspects at different gradients, Acta Physiol. Scand., 150, 3, 315-323, (1994)
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