Estimating Symmetric, Positive Definite Matrices in Robotic Control

In a number of contexts relevant to control problems, including estimation of robot dynamics, covariance, and smart structure mass and stiffness matrices, we need to solve an over-determined set of linear equations AX ≈ B with the constraint that the matrix X be symmetric and positive definite. In the classical least squares method, the measurements of A are assumed to be free of error. Hence, all errors are confined to B. Thus, the “optimal” solution is given by minimizing ‖AX − B‖F . However, this assumption is often impractical. Sampling errors, modeling errors, and, sometimes, human errors bring inaccuracies to A as well. In this paper, we introduce a different optimization criterion, based on area, which takes the errors in both A and B into consideration. The analytic expression of the global optimizer is derived. The algorithm is applied to identify the joint space mass-inertia matrix of a Gough-Stewart platform. Experimental results indicate that the new approach is practical, and improves performance. Keywords— Mass-inertia estimation, covariance estimation, stiffness estimation, symmetric positive definite matrix.