# 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.

CDS Technical
Report (PDF, 5011K, 128 pages)

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