Rigid-Body Noncertainty Equivalence Adaptive Control on the Tangent Bundle of SE(3)

Abstract

This paper addresses an adaptive rigid-body pose-tracking control problem with unknown system parameters. The proposed control is based on a noncertainty equivalence (NCE) adaptive principle applied to the configuration manifold of rigid-body motion, i.e., special Euclidean group SE(3), and its tangent bundle TSE(3). The controller drives the system to follow an arbitrary reference trajectory with bounded time derivatives. The system states are composed of the elements of SE(3), consisting of rotations and translations, and the six-dimensional vector of angular and translational velocities in Euclidean space. Almost-global-asymptotic stability of the system is demonstrated using a Morse–Lyapunov stability analysis. Then, the performance of the proposed controller is verified and is compared to the results obtained by a certainty-equivalence (CE) adaptive control applied to exponential coordinates associated with the linear space of the Lie algebra . It is shown that the TSE(3)-based NCE adaptive control design can eliminate the performance degradation due to the cancellation of uncertain parameter effects that would otherwise appear when the CE adaptive controller is used. As a result, the proposed NCE-based adaptive pose-tracking controller recovers the closed-loop pose-tracking controller performance with the system parameters fully known to the controller.