  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.
