# Linear Matrix Inequalities for Physically-Consistent Inertial Parameter Identification: A Statistical Perspective on the Mass Distribution

## TL;DR

This paper introduces Linear Matrix Inequalities (LMIs) based on covariance constraints for physically consistent inertial parameter identification in robotic links, improving robustness and convergence in model-based control.

## Contribution

It formulates novel LMIs for inertial parameters based on covariance, connecting to the classical moments problem, and demonstrates their effectiveness in robot link identification.

## Key findings

- LMIs improve convergence speed in inertial parameter estimation.
- The approach enhances robustness to noisy data.
- Global optimality achieved via semidefinite programming.

## Abstract

With the increased application of model-based whole-body control in legged robots, there has been a resurgence of research interest into methods for accurate system identification. An important class of methods focuses on the inertial parameters of rigid-body systems. These parameters consist of the mass, first mass moment (related to center of mass location), and rotational inertia matrix of each link. The main contribution of this paper is to formulate physical-consistency constraints on these parameters as Linear Matrix Inequalities (LMIs). The use of these constraints in identification can accelerate convergence and increase robustness to noisy data. It is critically observed that the proposed LMIs are expressed in terms of the covariance of the mass distribution, rather than its rotational moments of inertia. With this perspective, connections to the classical problem of moments in mathematics are shown to yield new bounding-volume constraints on the mass distribution of each link. While previous work ensured physical plausibility or used convex optimization in identification, the LMIs here uniquely enable both advantages. Constraints are applied to identification of a leg for the MIT Cheetah 3 robot. Detailed properties of transmission components are identified alongside link inertias, with parameter optimization carried out to global optimality through semidefinite programming.