Averaging and Motion Control of Systems on Lie Groups
Abstract
In this dissertation, the author studies motion control problems in the framework of systems on finite-dimensional Lie groups. Nonholonomic motion control problems are challenging because nonlinear controllability theory does not provide an explicit procedure for constructing controls and linearization techniques. This approach, distinguished from previous motion control research, is to exploit the Lie group framework since it provides a natural and mathematically rich setting for studying nonholonomic systems. In particular, the author uses the framework to develop explicit, structured formulas that describe system behavior, and from these formulas she derives a systematic way of synthesizing controls to achieve desired motion. As her main tool she derives averaging theory for left-invariant systems on finite-dimensional Lie groups. This theory provides basis-independent formulas that approximate system behavior on the Lie group to an arbitrarily high order in small-amplitude, periodically time-varying control inputs. She interprets the average formulas geometrically and exploits this interpretation to prove a constructive controllability theorem for the average system. The proof of this theorem provides a constructive control synthesis methodology for drift-free systems that she uses to derive algorithms that synthesize sinusoidal open-loop controls. She applies the algorithms to several under-actuated mechanical control problems, including problems in spacecraft attitude control, unicycle motion control, and autonomous underwater vehicle control. She illustrates the effectiveness of the synthesized controls through simulation and experimentation. She shows further that as a consequence of the geometry inherited from the average formulas, her algorithms can be used to produce motion controls that adapt to changes in control authority, such as loss of an actuator.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1994
- Accession Number
- ADA452002
Entities
People
- Naomi Ehrich Leonard
Organizations
- University of Maryland