Linear-Quadratic Optimal Control in Maximal Coordinates

TL;DR

This paper explores the use of maximal coordinates in linear-quadratic regulator (LQR) control for multi-body systems, demonstrating potential advantages in robustness and tracking over traditional minimal-coordinate approaches.

Contribution

It introduces and analyzes maximal-coordinate LQR control laws, highlighting their benefits over minimal-coordinate methods in nonlinear system control.

Findings

Maximal-coordinate LQR shows greater robustness.

Improved tracking performance with maximal coordinates.

Case study and simulations support advantages over minimal 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, other coordinates are possible and 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 six degrees of freedom and joints between links are modeled with algebraic constraints. Such constraints can also represent closed kinematic loops or contact with the environment. This paper investigates the difference between minimal- and maximal-coordinate LQR control laws. A case study of applying LQR to a simple pendulum and simulations 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 performance compared to minimal-coordinate LQR when applied to nonlinear systems.