Concurrent Contact Planning and Trajectory Optimization in One Step Walking Motion

Abstract

Bipeds’ trajectories and control during walking are closely coupled with the contact force distribution in time and space as an indeterminate problem. Therefore, generating the motion of redundant bipeds in presence of unilateral contact is usually formulated as a nonlinear constrained optimization problem. The optimal walking motion must be solved in terms of trajectories, control, contact status (i.e., when, where, and whether a foot is in contact), and contact response (i.e., ground reaction forces). The solution for this problem requires predictive methods within the general optimal motion planning framework. However, there is a lack of fully predictive methods that can concurrently solve for all the above mentioned unknowns. This represents an important challenge in the simulation, design, analysis, and control of general robotic systems. A novel approach for the optimal motion planning of multibody systems with contacts is developed, based on a Sequential Quadratic Programming (SQP) algorithm for Nonlinear Programming (NLP). The complete formulation is presented and demonstrated with numerical experiments on a simple planar biped with the assigned task of one complete step motion in forward progression.

A Computational Framework Towards the Tele-Rehabilitation of Balance Control Skills

“…Links’ length and mass distribution, joint strengths and range of motions, ground contact modeling, and stance-specific constraints are specified and the subject’s dynamics can then be described using common robotic modeling approaches for floating-base robotic systems with multiple degrees-of-freedom (DOF) ( Figure 4A ). The dynamics and stability of systems in multi-contact stances is usually more challenging to describe ( Del Prete et al, 2018 ; Orsolino et al, 2020 ), given the indeterminacy in the system’s foot reactions when the legs form a closed loop with the ground ( Mummolo et al, 2015a ). Different modeling choices (e.g., number of DOF, single vs. double stance, planar vs. three-dimensional) lead to different BoB and balanced regions.…”

A Computational Strategy for Trajectory Optimization of Underactuated Multibody Systems With Contacts

“…Once this preliminary optimization is solved, we proceed to the second stage, using the just computed solution to warm-start the final optimization problem, whose parameters are summarized in (9). The results we present show how this two-stage strategy, with a proper choice of the parameters in the preliminary optimization, can lead to a significant reduction in the computational time required to obtain the solution.…”

“…5.1. The obtained solution is then interpolated on a finer grid and used to initialize the solver of the second optimization, whose parameters are the reference ones listed in (9).…”

“…Namely, this solution was computed by solving the optimal control problem, characterized by the parameters detailed in (9), without exploiting the advantage of solving a preliminary optimization with a coarser time grid, nor a relaxed contact model. The slope of both the track and the ground was set to α = π/30.…”

“…In [7], a complementarity formulation of contact (resulting in a Linear Complementarity Problem, LCP) is employed, whereas in [8] two different contact models, respectively based on complementarity and on a continuous penalty-based formulation, are evaluated. In [9], where an optimization framework for trajectory planning is proposed and applied to a simplified biped system performing a one-step walking motion, smoothing functions are introduced to relax the complementarity conditions involving normal and tangential contact forces.…”