Gait and Trajectory Optimization for Legged Systems Through Phase-Based End-Effector Parameterization

TLDR

The paper proposes a unified trajectory optimization framework that automatically determines gait sequence, step timings, footholds, swing‑leg motions, and body motion for legged robots on uneven terrain without extra modules. The method employs a phase‑based parameterization of foot motion and forces, a simplified centroidal dynamics model, and terrain‑dependent friction cone constraints to formulate a continuous decision‑variable optimization that is validated in simulation and on the real quadruped ANYmal. The resulting nonlinear programming solver produces highly dynamic motion plans with full flight phases for diverse legged morphologies efficiently, and the open‑source TOWR software is released for public use.

Abstract

We present a single trajectory optimization formulation for legged locomotion that automatically determines the gait sequence, step timings, footholds, swing-leg motions, and six-dimensional body motion over nonflat terrain, without any additional modules. Our phase-based parameterization of feet motion and forces allows to optimize over the discrete gait sequence using only continuous decision variables. The system is represented using a simplified centroidal dynamics model that is influenced by the feet's location and forces. We explicitly enforce friction cone constraints, depending on the shape of the terrain. The nonlinear programming problem solver generates highly dynamic motion plans with full flight phases for a variety of legged systems with arbitrary morphologies in an efficient manner. We validate the feasibility of the generated plans in simulation and on the real quadruped robot ANYmal. Additionally, the entire solver software TOWR, which used to generate these motions is made freely available.