Efficient Constrained Dynamics Algorithms based on an Equivalent LQR Formulation using Gauss' Principle of Least Constraint

TL;DR

This paper introduces efficient constrained dynamics algorithms derived from an LQR formulation using Gauss' principle, enabling faster simulations of complex robots by leveraging novel mathematical connections and optimized solvers.

Contribution

It extends the PV solver to floating-base kinematic trees, introduces new O(n + m) complexity solvers, and connects LQR dual Hessian with OSIM for improved computational efficiency.

Findings

Significant speed-up in high-dimensional robot simulations.

Extension of PV solver to floating-base kinematic trees.

Development of two new O(n + m) complexity solvers.

Abstract

We derive a family of efficient constrained dynamics algorithms by formulating an equivalent linear quadratic regulator (LQR) problem using Gauss principle of least constraint and solving it using dynamic programming. Our approach builds upon the pioneering (but largely unknown) O(n + m^2d + m^3) solver by Popov and Vereshchagin (PV), where n, m and d are the number of joints, number of constraints and the kinematic tree depth respectively. We provide an expository derivation for the original PV solver and extend it to floating-base kinematic trees with constraints allowed on any link. We make new connections between the LQR's dual Hessian and the inverse operational space inertia matrix (OSIM), permitting efficient OSIM computation, which we further accelerate using matrix inversion lemma. By generalizing the elimination ordering and accounting for MUJOCO-type soft constraints, we derive two original O(n + m) complexity solvers. Our numerical results indicate that significant simulation speed-up can be achieved for high dimensional robots like quadrupeds and humanoids using our algorithms as they scale better than the widely used O(nd^2 + m^2d + d^2m) LTL algorithm of Featherstone. The derivation through the LQR-constrained dynamics connection can make our algorithm accessible to a wider audience and enable cross-fertilization of software and research results between the fields