Motion Control of Drift - Free , Left - Invariant Systems on Lie Groups , Part II : A General Constructive Control Algorithm

In this paper we address the constructive controllability problem for drift-free, leftinvariant systems on nite-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 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 de ne 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( ) 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. This research was supported in part by the National Science Foundation's Engineering Research Centers Program: NSFD CDR 8803012, by the AFOSR University Research Initiative Program under grant AFOSR90-0105, and by the Army Research O ce under Smart Structures URI Contract No. DAAL03-92-G-0121. Supported in part by the Zonta International Foundation.