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Motion Control of Drift-Free, Left-Invariant Systems on Lie Groups

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N.E. Leonard and P.S. Krishnaprasad

*IEEE Transactions on Automatic Control*, Vol. 40, No. 9,
September, 1995, p. 1539-1554.
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Abstract

In this paper we address the
constructive controllability problem for drift-free, left-invariant
systems on finite-dimensional
Lie groups with fewer controls than state dimension.
We consider small ($\epsilon$) amplitude, low-frequency, periodically
time-varying controls and derive average solutions for system behavior.
We show how the $p$th-order average formula can be used to construct
open-loop controls for point-to-point maneuvering of systems that
require up to $(p-1)$ iterations of
Lie brackets to satisfy the Lie algebra controllability rank
condition. In the cases $p = 2,3$, we give algorithms for
constructing these controls as a function of structure constants that
define the control authority, i.e., the
actuator capability, of the system. The algorithms
are based on a geometric interpretation of the average formulas and
produce sinusoidal controls that solve the constructive controllability
problem with $O(\epsilon^p)$ accuracy in general
(exactly if the Lie algebra is
nilpotent).
The methodology is applicable to a variety of control problems and
is illustrated for the motion control
problem of an autonomous underwater vehicle with as few as three
control inputs.

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