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Research Papers

Discrete Mechanics and Optimal Control of Walking Gaits

[+] Author and Article Information
M. W. Koch

Chair of Applied Dynamics,
University of Erlangen-Nuremberg,
Erlangen 91058, Germany
e-mail: michael.koch@fau.de

M. Ringkamp

Chair of Applied Dynamics,
University of Erlangen-Nuremberg,
Erlangen 91058, Germany
e-mail: maik.ringkamp@fau.de

S. Leyendecker

Chair of Applied Dynamics,
University of Erlangen-Nuremberg,
Erlangen 91058, Germany
e-mail: sigrid.leyendecker@fau.de

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 20, 2015; final manuscript received November 8, 2016; published online December 2, 2016. Assoc. Editor: Zdravko Terze.

J. Comput. Nonlinear Dynam 12(2), 021006 (Dec 02, 2016) (12 pages) Paper No: CND-15-1444; doi: 10.1115/1.4035213 History: Received December 20, 2015; Revised November 08, 2016

In this work, we optimally control the upright gait of a three-dimensional symmetric bipedal walking model with flat feet. The whole walking cycle is assumed to occur during a fixed time span while the time span for each of the cycle phases is variable and part of the optimization. The implemented flat foot model allows to distinguish forefoot and heel contact such that a half walking cycle consists of five different phases. A fixed number of discrete time nodes in combination with a variable time interval length assure that the discretized problem is differentiable even though the particular time of establishing or releasing the contact between the foot and the ground is variable. Moreover, the perfectly plastic contact model prevents penetration of the ground. The optimal control problem is solved by our structure preserving discrete mechanics and optimal control for constrained systems (DMOCC) approach where the considered cost function is physiologically motivated and the obtained results are analyzed with regard to the gait of humans walking on a horizontal and an inclined plane.

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References

Safonova, A. , and Hodgins, J. , 2007, “ Construction and Optimal Search of Interpolated Motion Graphs,” ACM Trans. Graph. 26(3), p. 106.
Ren, C. , Zhao, L. , and Safonova, A. , 2010, “ Human Motion Synthesis With Optimisation-Based Graphs,” Computer Graphics Forum, 29(2), pp. 545–554.
Delp, S. L. , Anderson, F. C. , Arnold, A. S. , Loan, P. , Habib, A. , John, C. T. , Guendelmann, E. , and Thelen, D. G. , 2007, “ Opensim: Open-Source Software to Create and Analyze Dynamics Simulation of Movements,” IEEE Trans. Biomed. Eng., 54(11), pp. 1940–1950. [CrossRef] [PubMed]
Posa, M. , and Tedrake, R. , 2012, “ Direct Trajectory Optimization of Rigid Body Dynamical Systems Through Contact,” Workshop on the Algorithmic Foundations of Robotics, Springer, Berlin, Heidelberg, p. 16.
Posa, M. , Cantu, C. , and Tedrake, R. , 2014, “ A Direct Method for Trajectory Optimization of Rigid Bodies Through Contact,” Int. J. Rob. Res., 33(1), pp. 69–81. [CrossRef]
Yunt, K. , and Glocker, C. , 2005, “ Trajectory Optimization of Mechanical Hybrid Systems Using SUMT,” 2006 9th IEEE International Workshop on Advanced Motion Control, Mar. 27–29, pp. 665–671.
Nowak, I. , 2006, Relaxation and Decomposition Methods for Mixed Integer Nonlinear Programming, Springer Science & Business Media, Basel, Switzerland.
Sager, S. , 2005, “ Numerical Methods for Mixed-Integer Optimal Control Problems,” Ph.D. thesis, Der Andere Verl, Tönning, Germany.
Gerdts, M. , 2006, “ A Variable Time Transformation Method for Mixed-Integer Optimal Control Problems,” Optim. Control Appl. Methods, 27(3), pp. 169–182. [CrossRef]
Ringkamp, M. , Ober-Blöbaum, S. , and Leyendecker, S. , 2016, “ On the Time Transformation of Mixed Integer Optimal Control Problems Using a Consistent Fixed Integer Control Function,” Math. Program., pp. 1–31.
Ferris, M. C. , and Munson, T. S. , 2000, “ Complementarity Problems in GAMS and the PATH Solver1,” J. Econ. Dyn. Control, 24(2), pp. 165–188. [CrossRef]
Geyer, H. , Seyfarth, A. , and Blickhan, R. , 2006, “ Compliant Leg Behaviour Explains Basic Dynamics of Walking and Running,” Proc. R. Soc. B, 273, 2861–2867.
Mombaur, K. , 2009, “ Using Optimization to Create Self-Stable Human-Like Running,” Robotica, 27(03), pp. 321–330. [CrossRef]
Gross, D. , Hauger, W. , Schröder, J. , and Wall, W. A. , 2010, Technische Mechanik-Band 3: Kinetik, Springer-Verlag, Heidelberg, Germany.
Kraft, D. , 1985, “ On Converting Optimal Control Problems Into Nonlinear Programming Problems,” Computational Mathematics Programming (NATO ASI Series), Vol. 15, Springer, Berlin, Heidelberg, pp. 261–280.
Stryk, O. , and Bulirsch, R. , 1992, “ Direct and Indirect Methods for Trajectory Optimization,” Ann. Oper. Res., 37(1), pp. 357–373. [CrossRef]
Hairer, E. , Lubich, C. , and Wanner, G. , 2006, Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Science & Business Media, Berlin, Heidelberg.
Ober-Blöbaum, S. , Junge, O. , and Marsden, J. E. , 2011, “ Discrete Mechanics and Optimal Control: An Analysis,” ESAIM: Control, Optim. Calculus Var., 17(02), pp. 322–352. [CrossRef]
Campos, C. M. , Ober-Blöbaum, S. , and Trélat, E. , 2015, “ High Order Variational Integrators in the Optimal Control of Mechanical Systems,” Discrete Contin. Dyn. Syst., 35(9), pp. 4193–4223. [CrossRef]
Leyendecker, S. , Ober-Blöbaum, S. , Marsden, J. E. , and Ortiz, M. , 2009, “ Discrete Mechanics and Optimal Control for Constrained Systems,” Optim. Control Appl. Methods, 31(6), pp. 505–528. [CrossRef]
Leyendecker, S. , Pekarek, D. , and Marsden, J. E. , 2013, Structure Preserving Optimal Control of Three-Dimensional Compass Gait, Springer, Berlin, Heidelberg, Chap. 8.
Betsch, P. , and Steinmann, P. , 2001, “ Constrained Integration of Rigid Body Dynamics,” Comput. Methods Appl. Mech. Eng., 191(3–5), pp. 467–488. [CrossRef]
Betsch, P. , and 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(4), pp. 499–552. [CrossRef]
Terze, Z. , Müller, A. , and Zlatar, D. , 2014, “ Lie-Group Integration Method for Constrained Multibody Systems in State Space,” Multibody Syst. Dyn., 34, p. 275. [CrossRef]
Terze, Z. , Müller, A. , and Zlatar, D. , 2015, “ An Angular Momentum and Energy Conserving Lie-Group Integration Scheme for Rigid Body Rotational Dynamics Originating From Störmer–Verlet Algorithm,” ASME J. Comput. Nonlinear Dyn., 10(5), p. 051005. [CrossRef]
Koch, M. W. , and Leyendecker, S. , 2016, “ Structure Preserving Optimal Control of a Three-Dimensional Upright Gait,” Multibody Dynamics, Springer International Publishing, Switzerland, pp. 115–146.
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(50–52), pp. 5159–5190. [CrossRef]
Marsden, J. E. , and West, M. , 2001, “ Discrete Mechanics and Variational Integrators,” Acta Numer., 10, pp. 357–514. [CrossRef]
Lee, T. , Leok, M. , and McClamroch, H. , 2009, “ Discrete Control Systems,” Springer Encyclopedia of Complexity and System Science, Springer, New York, pp. 2002–2019.
Koch, M. W. , and Leyendecker, S. , 2013, “ Structure Preserving Simulation of Monopedal Jumping,” Arch. Mech. Eng., 60(1), pp. 127–146.
Anderson, F. C. , and Pandy, M. G. , 2001, “ Dynamic Optimization of Human Walking,” ASME J. Biomech. Eng., 123(5), pp. 381–390. [CrossRef]
Li, Y. , Wang, W. , Crompton, R. H. , and Gunther, M. M. , 2001, “ Free Vertical Moments and Transverse Forces in Human Walking and Their Role in Relation to Arm-Swing,” J. Exp. Biol., 204, pp. 47–58. [PubMed]
Park, J. , 2008, “ Synthesis of Natural Arm Swing Motion in Human Bipedal Walking,” J. Biomech., 41(7), pp. 1417–1426. [CrossRef] [PubMed]
Collins, S. H. , Adamczyk, P. G. , and Kuo, A. D. , 2009, “ Dynamic Arm Swinging in Human Walking,” Proc. R. Soc., 276(1673), pp. 3679–3688. [CrossRef]
Sasidharan, S. , Smitha, K. S. , and Thomas, M. , 2012, “ Human Gait Recognition Using Multisvm Classifier,” Int. J. Sci. Res., 11(3), pp. 1907–1913.
de Quervain, I. A. K. , Steussi, E. , and Stacoff, A. , 2008, “ Ganganalyse Beim Gehen und Laufen,” Schweiz. Z. Sportmed. Sporttraumatologie, 56(2), pp. 35–42.
Betts, J. T. , 2009, Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, 2nd ed., Cambridge University Press, New York.
Leyendecker, S. , 2011, “ On Optimal Control Simulations for Mechanical Systems,” Habilitationsschrift, TU Kaiserslautern, Kaiserslautern, Germany.
Uno, Y. , Kawato, M. , and Suzuki, R. , 1989, “ Formulation and Control of the Optimal Trajectory in Human Multijoint Arm Movement,” Biol. Cybern., 61(2), pp. 89–101. [CrossRef] [PubMed]
Simmons, G. , and Demiris, Y. , 2005, “ Optimal Robot Arm Control Using the Minimum Variance Model,” J. Rob. Syst., 22(11), pp. 677–690. [CrossRef]
Maas, R. , and Leyendecker, S. , 2013, “ Biomechanical Optimal Control of Human Arm Motion,” J. Multi-Body Dyn., 27(4), pp. 375–389.
Soechting, J. F. , Buneo, C. A. , Herrmann, U. , and Flanders, M. , 1995, “ Moving Effortlessly in Three Dimensions: Does Donders' Law Apply to Arm Movement?,” J. Neurosci., 15(9), pp. 6271–6280. [PubMed]
François, C. , and Samson, C. , 1996, “ Energy Efficient Control of Running Legged Robots—A Case Study: The Planar one-Legged Hopper,” Institut National de Recherché en Informatique et en Automatigue, Report No. 3027.
Roussel, L. , de Wit, C. C. , and Goswami, A. , 1998, “ Generation of Energy Optimal Complete Gait Cycles for Biped Robots,” IEEE International Conference on Robotics and Automation, pp. 2036–2041.
Fujimoto, Y. , 2004, “ Trajectory Generation of Biped Running Robot With Minimum Energy Consumption,” IEEE International Conference on Robotics and Automation, pp. 3803–3808.
Biess, A. , Liebermann, D. , and 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(48), pp. 13045–13064. [CrossRef] [PubMed]
Friedmann, T. , and Flash, T. , 2009, “ Trajectory of the Index Finger During Grasping,” Exp. Brain Res., 196(4), pp. 497–509. [CrossRef] [PubMed]
Felis, M. , and Mombaur, K. , 2013, “ Modeling and Optimization of Human Walking,” Modeling, Simulation and Optimization of Bipedal Walking, Springer, Berlin, Heidelberg.

Figures

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Fig. 2

Energy and angular momentum errors of the symplectic Störmer–Verlet (SV) and the Runge–Kutta integrator (R IIA) for the optimal control of a planar double pendulum upswing

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Fig. 1

Position errors of the symplectic Störmer–Verlet (SV) and the Runge–Kutta integrator (R IIA) for the forward dynamic simulation of a planar double pendulum

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Fig. 3

Illustration of the three-dimensional seven link bipedal walker model and its generalized coordinates (as in Ref. [26])

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Fig. 4

Illustration of the characteristic configurations at the transition between two contact phases for a half gait cycle (as in Ref. [26])

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Fig. 7

Combination of minimal specific kinetic energy and specific jerk: evolution of the contact Lagrange multipliers of the left and right leg during the half gait cycle

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Fig. 5

Combination of minimal specific kinetic energy and specific jerk: snapshots of the optimized bipedal motion

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Fig. 6

Combination of minimal specific kinetic energy and specific jerk: evolution of the feet center of mass' trajectory coordinates during a gait cycle

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Fig. 11

Minimal specific torque change: snapshots of the optimized bipedal motion on an inclined plane

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Fig. 12

Minimal specific torque change: evolution of the feet center of mass' trajectory coordinates during a gait cycle

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Fig. 18

Minimal specific jerk: evolution of the feet center of mass' trajectory coordinates during a gait cycle

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Fig. 19

Minimal specific jerk: evolution of the contact Lagrange multipliers of the left and right legs during the half gait cycle

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Fig. 8

Minimal specific control effort: snapshots of the optimized bipedal motion on an inclined plane

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Fig. 9

Minimal specific control effort: evolution of the feet center of mass' trajectory coordinates during a gait cycle

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Fig. 10

Minimal specific control effort: evolution of the contact Lagrange multipliers of the left and right legs during the half gait cycle

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Fig. 13

Minimal specific torque change: evolution of the contact Lagrange multipliers of the left and right legs during the half gait cycle

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Fig. 14

Minimal specific kinetic energy: snapshots of the optimized bipedal motion on an inclined plane

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Fig. 15

Minimal specific kinetic energy: evolution of the feet center of mass' trajectory coordinates during a gait cycle

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Fig. 16

Minimal specific kinetic energy: evolution of the contact Lagrange multipliers of the left and right legs during the half gait cycle

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Fig. 17

Minimal specific jerk: snapshots of the optimized bipedal motion on an inclined plane

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