## Recent work in nonlinear control has drawn attention to drift-free systems with fewer controls than state variables. These arise in problems of motion planning for wheeled robots subject to nonholonomic constraints, models of kinematic drift (or geometric phase) effects in space systems subject to appendage vibrations or articulations, and models of self-propulsion of paramecia at low Reynolds numbers. There is a well-known Lie algebra rank condition from nonlinear control theory that allows one to determine whether or not such systems are controllable. However, unlike the linear setting where the controllability Grammian yields constructive controls, here the rank condition does not lead immediately to an explicit procedure for constructing controls. As a result, recent research has focused on constructing controls to achieve complete controllability. In our work, we address this problem of constructing controls using the abstract framework of motion control on Lie groups. Our goal is to develop a control strategy that combines small-amplitude, periodic, open-loop control with intermittent feedback corrections to achieve prescribed motion for the class of nonlinear mechanical systems described by drift-free, left-invariant systems on finite-dimensional Lie groups. Included in this class are the three-dimensional spacecraft attitude control problem with as few as two controls, the six-dimensional autonomous underwater vehicle motion control problem with as few as three controls and the motion planning problem for light-weight wheeled robots.

## The strategy for constructing controls for our systems can be summarized in four steps: (1) Choose intermediate target points between the initial position and desired final position of the system. (2) Specify open-loop, small-amplitude, periodic controls that will drive the state from the initial position to the first target point approximately. This can be done by specifying controls that will drive an average approximation of the state to the first target point exactly. (3) If desired, apply feedback, i.e., make appropriate modifications based on measurement of the new system state. (4) Repeat steps (2) and (3) for each successive target point until done. A critical ingredient for the realization of this strategy is the derivation of useful formulas for the average approximation that makes it possible to drive the average approximation exactly to a target point, while ensuring that the actual state stays close to the approximation for a sufficiently long time. We have derived such formulas and have shown that the formulas admit a geometric interpretation geometric interpretation has then been used to construct open-loop control algorithms.

## One major result of this project is a general algorithm for constructing open-loop controls to solve the complete constructive controllability problem for drift-free, left-invariant systems on Lie groups that satisfy the Lie algebra controllability rank condition with up to two iterations of Lie brackets. The controls are continuous, relatively low-frequency, small-amplitude, sinusoidal signals and are constructed using only a basic description of the system kinematics. Assuming that the application does not require a very rapid change in motion, the frequency of the sinusoidal control signals can be chosen relatively low. This can help avoid unintentionally exciting vibrational modes in the system. Additionally, the algorithm can be used so that the controller can adapt to the loss of a control input, e.g., the failure of an actuator, as long as the system remains controllable. For example, consider an autonomous underwater vehicle in which position and orientation in three dimensions are controlled with three rotational control inputs and one translational control. Suppose one of the rotational control actuators fails. In this scenario, the algorithm could automatically be adjusted so that the controller would continue to effect complete control over the vehicle's position and orientation. Simulations have been used to verify the theoretical results of this algorithm, and an experimental test of the algorithm is under development. An attitude control experiment will be run on the Supplemental Camera and Maneuvering Platform (SCAMP), an underwater robot in the neutral buoyancy tank of the Space Systems Laboratory at the University of Maryland.

## Our results are significant in showing how to compute the average solution to a drift-free, left-invariant system on a Lie group with periodic forcing. In particular, by using the framework of motion control on Lie groups, the average formulas for system response are coordinate independent and geometrically intuitive. This, as we show, is very useful in determining general open-loop algorithms for controlling the motion of a class of interesting nonlinear mechanical systems. This general formulation also makes for a more fault-tolerant control system as described above. Further, control designs that use small-amplitude periodic controls are motivated by the need in some contexts to explore new means of actuation and control, e.g., in the space program for control of micro-spacecraft. Recent technological advances in the area of vibratory actuation and sensing provide another source of motivation for this work in that our control algorithms could be used, for example, in the rectification of vibratory motion. Finally, in addition to showing how to construct controls, our average formulas also reveal how to compute drifts in system behavior caused by undesirable oscillations. Kinematic drift of a spacecraft caused by thermo-elastically induced vibrations in flexible attachments on the spacecraft is an example.

## Future work in this area could involve (1) extending this methodology to other nonlinear systems by using bilinearization and related Lie theory; (2) formalizing the process of combining open-loop and feedback controls by use of a higher level supervisory control strategy and (3) exploring adaptation schemes that involve a change of motion script.