TY - JOUR

T1 - Motion Control of Drift-Free, Left-Invariant Systems on Lie Groups

AU - Leonard, Naomi Ehrich

AU - Krishnaprasad, P. S.

N1 - Funding Information:
Manuscript received February 11, 1994; revised January 11, 199.5. Recommended by Associate Editor, A. M. Bloch. This work was supported in part by National Science Foundation's Engineering Research Centers Program: NSFD CDR 8803012, the AFOSR University Research Initiative Program Grant AFOSR-90-0105, the Amy Research Office under Smart Structures URI Contract DAAL03-92-G-0121, and the Zonta Intemational Foundation.

PY - 1995/9

Y1 - 1995/9

N2 - 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 (∊) amplitude, low-frequency, periodically time-varying controls and derive average solutions for system behavior. We show how the pth-order average formula can be used to construct open-loop controls for point-to-point maneuvering of systems which 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(∊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.

AB - 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 (∊) amplitude, low-frequency, periodically time-varying controls and derive average solutions for system behavior. We show how the pth-order average formula can be used to construct open-loop controls for point-to-point maneuvering of systems which 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(∊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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U2 - 10.1109/9.412625

DO - 10.1109/9.412625

M3 - Article

AN - SCOPUS:0029375604

SN - 0018-9286

VL - 40

SP - 1539

EP - 1554

JO - IEEE Transactions on Automatic Control

JF - IEEE Transactions on Automatic Control

IS - 9

ER -