Smooth Parameterization of Rigid-Body Inertia

Abstract

In this letter, we propose a parameterization of the rigid-body inertia tensor that is singularity free, guarantees full physical consistency, and has a straightforward physical interpretation. Based on a version of log-Cholesky decomposition of the pseudo-inertia matrix, we construct a smooth isomorphic mapping from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathbb {R}^{10}$</tex-math></inline-formula> to the set of fully physically consistent inertia tensors. This facilitates inertial estimation via unconstrained optimization on a vector space and avoids the non-uniqueness and singularities which we show are inherent to parameterizations of the inertia tensor based on eigenvalue decomposition and principal moments. The elements of our parameterization have straightforward physical meanings in terms of geometric transformations applied to a reference body. While adopting this new parameterization breaks the linear least squares structure of the system identification, theoretical results guarantee that all local optima of the resulting problems remain global optima. We compare the performance of three different parameterizations of inertia by performing inertial estimation on test data from a set of 1000 simulations of randomly sampled rigid bodies subject to an external time-varying wrench and measurement noise. We also investigate the performance of the log-Cholesky parameterization on a dataset from MIT Cheetah 3 to empirically demonstrate the theoretical results. Overall, the results indicate that the log-Cholesky parameterization achieves fast convergence while providing a simple and intuitive parametric description of inertia.