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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

Figures

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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. 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. 5

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

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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. 8

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

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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. 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. 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. 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. 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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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. 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. 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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