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

Abstract

We present a single Trajectory Optimization formulation for legged locomotion that automatically determines the gait-sequence, step-timings, footholds, swing-leg motions and 6D body motion over non-flat 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 NLP 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 used to generate these motions is made freely available.

Trajectory Optimization With Implicit Hard Contacts

“…While this eliminates the hybrid nature of the problem, the sequence of contact configurations (i.e., the gait) must be known a priori. Recently, a per-leg phase switching parametrization was proposed [13], also allowing the gait sequence to be optimized in a continuous optimization framework without the use of complementarity constraints. This appears to be a promising approach, although it remains unclear how the formulation scales to a larger number of contacts points.…”

“…A method [17] using strongly simplified humanoid dynamics also achieve similar speed while optimizing over the gait sequence. Most competing approaches for synthesizing dynamic contact invariant motions [2], [13], [14], [21] report times that are orders of magnitude larger, albeit using models with more DOFs in some cases which makes comparisons difficult.…”

Rolling in the Deep – Hybrid Locomotion for Wheeled-Legged Robots Using Online Trajectory Optimization

“…Fortunately, research in traditional legged locomotion offers solutions to bridge this gap. The quadrupedal robot ANYmal (without wheels) performs highly dynamic motions using MPC [16], [17] and TO [18], [19] approaches. Impressive results are shown by MIT Cheetah, which performs blind locomotion over stairs [20] and jumps onto a desk with the height of 0.76 m [21].…”

Guided Constrained Policy Optimization for Dynamic Quadrupedal Robot Locomotion

“…One of the common approaches leverages mathematical optimization techniques [7] to generate reference trajectories by solving an optimal control (OC) [8] problem with objectives such as minimization of energy consumption, and constraints that consider the dynamics of the robotic systems. Authors of [9] presented a TO formulation for legged locomotion that automatically generates reference motions without requiring any prior footstep planning.…”