Nonlinear Control of Mechanical Systems: A Reimannian Geometry Approach

Francesco Bullo
PhD Dissertation, Caltech, August 1998

Nonlinear control of mechanical systems is a challenging discipline that lies at the intersection between control theory and geometric mechanics. This thesis sheds new light on this interplay while investigating motion control problems for Lagrangian systems. Both stability and motion planning aspects are treated within a unified framework that accounts for a large class of devices such as robotic manipulators, autonomous vehicles and locomotion systems.

One distinguishing feature of mechanical systems is the number of control forces. For systems with as many input forces as degrees of freedom, many control problems are tractable. One contribution of this thesis is a set of trajectory tracking controllers designed via the notions of configuration and velocity error. The proposed approach includes as special cases a variety of results on joint and workspace control of manipulators as well as on attitude and position control of vehicles.

Whenever fewer input forces are available than degrees of freedom, various control questions arise. The main contribution of this thesis is the design of motion algorithms for vehicles, i.e., rigid bodies moving in Euclidean space. First, an algebraic controllability analysis characterizes the set of reachable configurations and velocities for a system starting at rest. Then, provided a certain controllability condition is satisfied, various motion algorithms are proposed to perform tasks such as short range reconfiguration and hovering.

Finally, stabilization techniques for underactuated systems are investigated. The emphasis is on relative equilibria, i.e., steady motions for systems that have a conserved momentum. Local exponential stabilization is achieved via an appropriate splitting of the control authority.

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