Discrete Mechanics and Optimal Control for Passive Walking with Foot Slippage

TL;DR

This work develops a geometric, discrete-mechanics framework for passive walking with foot slip by introducing simple hybrid holonomically constrained forced Lagrangian systems and corresponding forced variational integrators. It constructs a reduced foot-slip model on a constraint submanifold $N$ (avoiding Lagrange multipliers), derives both continuous and discrete Euler–Lagrange equations with non-conservative forces, and provides explicit forms for the discrete Lagrangian and forces under a midpoint rule. The authors then formulate a discrete-time optimal-control problem to generate sub-optimal trajectories that track a reference, including an explicit treatment of impacts through a discretized impact map, enabling trajectory generation within a structure-preserving integrator. The approach yields a tractable pathway to trajectory optimization and control for hybrid mechanical systems like passive walkers, with insights into energy changes across impacts and the role of momentum balance in fall avoidance. The methodology has potential implications for design and control of bipedal locomotion with slip and impact dynamics in robotics and biomechanics.

Abstract

Forced variational integrators are given by the discretization of the Lagrange-d'Alembert principle for systems subject to external forces, and have proved useful for numerical simulation studies of complex dynamical systems. In this paper we model a passive walker with foot slip by using techniques of geometric mechanics, and we construct forced variational integrators for the system. Moreover, we present a methodology for generating (locally) optimal control policies for simple hybrid holonomically constrained forced Lagrangian systems, based on discrete mechanics, applied to a controlled walker with foot slip in a trajectory tracking problem.

Figures (6)

• Figure 1: Leg and foot: The coordinates of the foot are given by $(\overline{x},\overline{y})$, the center of mass are $(x,y)$ and $\theta$ is the angle between the leg of length $\ell$ and the vertical axis.
• Figure 2: Depiction of the impact. The position of the foot is continuous at the impact. Resetting the angle forces a reset on the position of the center of mass.
• Figure 3: Horizontal components of the position of the center of mass and the reference trajectory as functions of time.
• Figure 4: Angular components of the position of the center of mass and the reference trajectory as functions of time.
• Figure 5: Trajectories of the center of mass and the foot compared with the reference trajectory.

Theorems & Definitions (4)

• Remark 1
• Definition 2
• Remark 3
• Remark 4