Linear-Quadratic Optimal Control in Maximal Coordinates

Abstract

The Linear-quadratic regulator (LQR) is an efficient control method for linear and linearized systems. Typically, LQR is implemented in minimal coordinates (also called generalized or "joint" coordinates). However, recent research suggests that there may be numerical and control-theoretic advantages when using higher-dimensional non-minimal state parameterizations for dynamical systems. One such parameterization is maximal coordinates, in which each link in a multi-body system is parameterized by its full SE(3) pose and joints between links are modeled with algebraic constraints. Such constraints can also account for closed kinematic loops or contact with the environment. This paper presents an extension to the standard LQR control method to handle dynamical systems with explicit physical constraints. Experiments comparing the basins of attraction and tracking performance of minimal-and maximal-coordinate LQR controllers suggest that maximal-coordinate LQR achieves greater robustness and improved tracking compared to minimalcoordinate LQR when applied to nonlinear systems.

Dojo: A Differentiable Simulator for Robotics

“…Summary: Dojo makes several advancements over previous robotics simulators: First, a maximal-coordinates state representation is coupled with a numerically improved variational integrator. This large, sparse representation can be efficiently optimized and provides more information from the simulator compared to minimal-coordinate representations [3]. As demonstrated in the examples, the variational integrator better conserves energy and momentum compared to higher-order explicit and lower-order implicit counterparts while being stable using relatively large time steps.…”

Linear-Time Contact and Friction Dynamics in Maximal Coordinates using Variational Integrators

“…Due to the intra-mechanism contact, a modification to the proposed method is required to achieve lineartime complexity. Additionally, an investigation whether the improved control performance demonstrated with maximal coordinates [24], [25] also extends to systems with contact interactions could be beneficial.…”

“…We solve (24) with an interior-point method (see [20] for details) to reduce the numerical difficulties of the contact constraints. However, the graph-based complexity argument holds for any matrix-based solver.…”

“…Solving (24) with any variant of Newton's method requires at each iteration to solve a linear system of equations…”