Lie Group Observer Design for Robotic Systems: Extensions, Synthesis, and Higher-Order Filtering

Abstract

The kinematics and dynamics of many robotic systems evolve on differential manifolds, rather than strictly in Euclidean space. Lie groups, a class of differential manifold with a group structure, arise naturally in the study of rigid-body kinematics. This dissertation studies the design of state observers for systems whose state evolves on a Lie group. State observers, or state estimators, are a crucial part of the guidance, navigation, and control algorithms necessary for autonomous operation of many ground, air, and marine vehicles. The design of state observers on Lie groups is therefore a highly practical exercise. One such nonlinear observer, the gradient-based observer, has generated significant interest in the literature due to its computational simplicity and stability guarantees. The first part of this dissertation explores several applications of the gradient-based observer, including both the attitude estimation problem and the simultaneous localization and mapping (SLAM) problem. By modifying the cost function associated with the observer, several novel attitude estimators are introduced that provide faster convergence when the initial attitude error is large. Further, a SLAM algorithm with guaranteed convergence is introduced and tested in both simulation and experiment.

In the second part of this dissertation, the state of the art in Lie group observer design is extended by the development of a higher-order filter on a Lie group. By analogy to the classical linear complementary filter, the proposed method can be interpreted as a nonlinear complementary filter on a Lie group. A disturbance observer that accounts for constant and harmonic disturbances in the group velocity measurements is also considered. Local asymptotic stability about the desired equilibrium point is demonstrated. In addition, an H2-optimal filter synthesis method is derived and disturbance rejection via the internal model principle is considered. A numerical example that demonstrates the desirable properties of the higher-order nonlinear complementary filter, as well as the synthesis techniques, is presented in the context of rigid-body attitude estimation.