Variational Attitude and Pose Estimation Using the Lagrange-d’ Alembert Principle

Abstract

Stable estimation of rigid body motion states from noisy measurements, without any knowledge of the dynamics model, is treated using the Lagrange-d’ Alembert principle from variational mechanics. From body-fixed sensor measurements, a Lagrangian is obtained as the difference between a kinetic energy-like term that is quadratic in velocity estimation errors and an artificial potential function of pose (attitude and position) estimation errors. An additional dissipation term that is linear in the velocity estimation errors is introduced, and the Lagrange-d’ Alembert principle is applied to the Lagrangian with this dissipation. This estimation framework is shown to be almost globally asymptotically stable in the state space of rigid body motions. It is discretized for computer implementation using the discrete Lagrange-d’ Alembert principle, as a first order Lie group variational integrator. In the presence of bounded measurement noise from sensors, numerical simulations show that the estimated states converge to a bounded neighborhood of the actual states. Ongoing and future work will explore finite-time stable extensions of this framework for nonlinear observer design, with applications to rigid body and multi-body systems.