Abstract
The optimal control of human locomotion requires simulation techniques, which handle the contact’s establishing and releasing between foot and ground. In this work, our aim is to optimally control the human upright gait using a structure preserving variational integrator, whereby different physiologically motivated cost functions are chosen and the obtained results are analysed with regard to the gait of humans. Thereby, the implemented three-dimensional rigid multibody system enables us to model forefoot as well as heel contact. The contacts between feet and ground are modelled as perfectly plastic impact and the orientation of the contact forces prevent penetration of the ground. To guarantee the structure preservation and the geometrical correctness, the non-smooth problem is solved including the contact configuration, time and force, in contrast to relying on a smooth approximation of the contact problem via a penalty potential. The applied mechanical integrator is based on a discrete constrained version of Lagrange-d’Alembert principle, which yields a symplectic momentum preserving method (see Leyendecker et al., Optim Control Appl Methods 31:505–528, 2009, [31] for details).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ackermann M, Schiehlen W (2006) Dynamic analysis of human gait disorder and metabolical cost estimation. Arch Appl Mech 75:569–594
Aerospace Medical Research Laboratory (1975) Investigation of inertial properties of the human body. National Highway Traffic Safety Administration
Anderson FC, Pandy MG (2001) Dynamic optimization of human walking. J Biomech Eng 123:381–390
Arnold VI (1989) Mathematical methods of classical mechanics. Springer, New York
Betsch P (2005) The discrete null space method for the energy consistent integration of constrained mechanical systems. Part I: holonomic constraints. Comput Methods Appl Mech Eng 194:5159–5190
Betsch P, Leyendecker S (2006) The discrete null space method for the energy consistent integration of constrained mechanical systems. Part II: multibody dynamics. Int J Numer Methods Eng 67:499–552
Betsch P, Steinmann P (2001) Constrained integration of rigid body dynamics. Comput Methods Appl Mech Eng 191:467–488
Betts JT (2009) Practical methods for optimal control and estimation using nonlinear programming, 2nd edn. Cambridge University Press, Cambridge
Biess A, Liebermann D, Flash T (2007) A computational model for redundant human three-dimensional pointing movements: integration of independent spatial and temporal motor plans simplifies movement dynamics. J Neurosci 27:13045–13064
BostonDynamics (2013) PETMAN. http://www.bostondynamics.com/robot_petman.html
Chevallereau C, Aoustin Y (2001) Optimal reference trajectories for walking and running of a biped robot. Robotica 19:557–569
Collins SH, Adamczyk PG, Kuo AD (2009) Dynamic arm swinging in human walking. Proc R Soc Biol Sci 276:3679–3688
Delp SL, Anderson FC, Arnold AS, Loan P, Habib A, John CT, Guendelmann E, Thelen DG (2007) Opensim: open-source software to create and analyze dynamics simulation of movements. IEEE Trans Biomed Eng 54:1940–1950
Djoudi D, Chevallereau C, Aoustin Y (2005) Optimal reference motions for walking of a bipedal robot. In: Proceedings of the 2005 IEEE international conference, vol 1, pp 2002–2007
Espiau B, Goswami A (1994) Compass gait revisited. In: IFAC symposium on robot control
Felis M, Mombaur K (2013) Modeling and optimization of human walking. In: Mombaur K, Berns K (eds) Modeling simulation and optimization of bipedal walking. Springer, New York
Felis M, Mombaur K, Kadone H, Berthoz A (2013) Modeling and identification of emotional aspects of locomotion. J Comput Sci 4:255–261
François C, Samson C (1996) Energy efficient control of running legged robots. A case study: the planar one-legged hopper. Institut National de Recherch en Informatique et en Automatigue
Friedmann T, Flash T (2009) Trajectory of the index finger during grasping. Exp Brain Res 196:497–509
Fujimoto Y (2004) Trajectory generation of biped running robot with minimum energy consumption. In: IEEE international conference on robotics and automation
Geyer H, Herr H (2010) A muscle-reflex model that encodes principles of legged mechanics produces human walking dynamics and muscle activities. IEEE Trans Neural Syst Rehabil Eng 18:263–273
Geyer H, Seyfarth A, Blickhan R (2003) Positive force feedback in bouncing gaits. Proc R Soc B: Biol Sci 270:2173–2183
Geyer H, Seyfarth A, Blickhan R (2006) Compliant leg behaviour explains basic dynamics of walking and running. Proc R Soc B: Biol Sci 273:2861–2867
Gross D, Hauger W, Schrder J, Wall WA (2010) Modeling, Technische Mechanik - Band 3: Kinetik. Springer, Berlin
Günther M (1997) Computersimulation zur Synthetisierung des muskulär erzeugten menschlichen Gehens unter Verwendung eines biomechanischen Mehrkörpermodells. Eberhard-Karls-Universität zu Tübingen
Hicheur H, Kadone H, Grzes J, Berthoz A (2013) The combined role of motion-related cues and upper body posture for the expression of emotions during human walking. In: Mombaur K, Berns K (eds) Modeling, simulation and optimization of bipedal walking. Springer, New York
Koch MW, Leyendecker S (2013) Structure preserving simulation of monopedal jumping. Arch Mech Eng LX:127–146
Kraft D (1985) On converting optimal control problems into nonlinear programming problems. Comput Math Program F15:261–280
Leyendecker S (2011) On optimal control simulations for mechanical systems. Technische Universität Kaiserslautern
Leyendecker S, Marsden JE, Ortiz M (2008) Variational integrators for constrained dynamical systems. Zeitschrift für Angewandte Mathematik und Mechanik 88:677–708
Leyendecker S, Ober-Blöbaum S, Marsden JE, Ortiz M (2009) Discrete mechanics and optimal control for constrained systems. Optim Control Appl Methods 31:505–528
Leyendecker S, Pekarek D, Marsden JE (2013) Structure preserving optimal control of three-dimensional compass gait. In: Mombaur K, Berns K (eds) Modeling, simulation and optimization of bipedal walking. Springer, New York
Li Yu, Wang W, Crompton RH, Gunther MM (2001) Free vertical moments and transverse forces in human walking and their role in relation to arm-swing. J Exp Biol 204:47–58
Lim CL, Jones NB, Spurgeon SK, Scott JJA (2003) Modelling of knee joint muscles during the swing phase of gait - a forward dynamics approach using MATLAB/Simulink. Simul Model Pract Theory 11:91–107
Luksch T, Berns K (2013) Modeling, simulation and optimization of bipedal walking - modeling and control of dynamically walking bipedal robots. Springer, New York
Maas R, Leyendecker S (2013) Biomechanical optimal control of human arm motion. J Multi-body Dyn. doi:10.1177/1464419313488363
Marsden JE, Ratiu TS (1999) Introduction to mechanics and symmetry. Springer, New York
Marsden JE, West M (2001) Discrete mechanics and variational integrators. Acta Numerica 10:357–514
Mianoski K, Schmitz N, Berns K (2007) Mechatronics of the humanoid robot roman. In: Kozłowski K (ed) Robot motion and control. Springer, London
Olver P (1986) Appliance of lie groups to differential equations. Springer, New York
Pandy MG, Zajac FE, Sim E, Levine WS (1990) An optimal control model for maximum-height human jumping. J Biomech 23:1185–1198
Pandy MG, Garner BA, Anderson FC (1995) Optimal control of non-ballistic muscular movements: a constraint-based performance criterion for rising from a chair. J Biomech Eng 117:15–26
Park J (2008) Synthesis of natural arm swing motion in human bipedal walking. J Biomech 41:1417–1426
Ren C, Zhao L, Safonova A (2010) Human motion synthesises with optimisation-based graphs. Comput Graph Forum 29:1321–1328
Rose J, Gamble JG (1994) Human walking. Williams & Wilkins, Baltimore
Roussel L, Candas-de Wit C, Goswami A (1998) Generation of energy optimal complete gait cycles for biped robots. In: Proceedings IEEE conference on robotics and automation
Safanova A, Hodgins JK (1992) Construction and optimal search of interpolated motion graphs. Ann Oper Res 57–373
Safanova A, Hodgins JK (2012) Human gait recognition using multisvm classifier. Int J Sci Res (IJSR)
Simmons G, Demiris Y (2015) Optimal robot arm control using the minimum variance model. J Robot Syst 22:677–690
Soechting JF, Buneo CA, Herrmann U, Flanders M (1995) Moving effortlessly in three dimensions: does Donders’ law apply to arm movement? J Neurosci 15:6271–6280
Spektrum Akademischer Verlag (1999) Aufrechter Gang. http://www.spektrum.de/lexikon/biologie/aufrechter-gang/6056
Uno Y, Kawato M, Suzuki R (1989) Formulation and control of the optimal trajectory in human multijoint arm movement. Biol Cybern 61:89–101
von Stryk O, Bulirsch R (1992) Direct and indirect methods for trajectory optimization. Ann Oper Res 37:357–373
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Koch, M.W., Leyendecker, S. (2016). Structure Preserving Optimal Control of a Three-Dimensional Upright Gait. In: Font-Llagunes, J. (eds) Multibody Dynamics. Computational Methods in Applied Sciences, vol 42. Springer, Cham. https://doi.org/10.1007/978-3-319-30614-8_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-30614-8_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-30612-4
Online ISBN: 978-3-319-30614-8
eBook Packages: EngineeringEngineering (R0)