## 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 (∊) 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.

Original language | English (US) |
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Pages (from-to) | 1539-1554 |

Number of pages | 16 |

Journal | IEEE Transactions on Automatic Control |

Volume | 40 |

Issue number | 9 |

DOIs | |

State | Published - Sep 1995 |

## All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Computer Science Applications
- Electrical and Electronic Engineering