On the Emergence of Pendular Structure in Multi-Contact Locomotion

TL;DR

This paper explores how pendular force patterns naturally emerge in multi-contact legged locomotion control, linking classical models with optimal control solutions and validating findings on simulated and real robots.

Contribution

It explicitly connects LIPM modeling choices with optimal control solutions, revealing conditions under which pendular structures emerge in locomotion.

Findings

Optimal solutions tend toward pendular force patterns with full-rank stance.

Friction cone constraints impose a lower bound on angular momentum rate.

Adding a task term shifts the optimal solution away from pendular patterns.

Abstract

LIPM is everywhere in legged-locomotion control, but almost always as a modeling choice rather than as something the controller's cost actually prefers. This note tries to make that link more explicit. Working from a small centroidal OCP that penalizes the rate of angular momentum, we look at what its optimum tends to look like. Three things come out. With full-rank stance, the optimum drifts toward a pendular force pattern at a rate determined by the SVD of the moment Jacobian; the constant is set by foot-span geometry and matches the experiments to within 16%. With N=2 stance, as in trot, the friction cone introduces a lower bound on $\|\dot{H}_G\|$ that no amount of weight tuning fixes; we also see a non-smooth feasibility kink at a critical horizontal acceleration that we can write in closed form. Adding a task term that asks for a nonzero $\dot{H}_G$ moves the optimum off the pendular set in a predictable way. None of this is far from the classical ZMP/DCM picture. We test these claims on a point-mass quadruped and on the Unitree Go1 in MuJoCo (open-loop QP and a torque-level closed-loop controller), and we note where the asymptotic story stops being a good description of what the closed loop actually does.