This dissertation presents novel vibrational control strategies for mechanical control-affine systems with high-frequency, high-amplitude inputs. Since these control systems use high-frequency, zero-mean, periodic inputs, averaging techniques are widely used in the analysis of their dynamics. By studying their time-averaged approximations, new properties of the averaged dynamics of this class of systems are revealed. Using these properties, the problem of input optimization of vibrational control systems was formulated and solved by transforming the problem to a constrained optimization one. Geometric control theory provides powerful tools for studying the control properties of control-affine systems. Using the concepts of vibrational and geometric controls and averaging tools, a closed-loop control strategy for trajectory tracking of a class of underactuated mechanical control-affine systems is developed. In the developed control law, the fact that for underactuated systems, the actuated coordinates together with the corresponding generalized velocities can be considered as generalized inputs for the unactuated dynamics plays the main role. Using the developed control method, both actuated and unactuated coordinates of the system are able to follow slowly time-varying prescribed trajectories on average. The developed control method is applied for altitude control of flapping wing micro-air vehicles by considering the sweeping (flapping) angle of the wings as the inputs. Using the feathering (pitch) angles of the wings as additional inputs, and using non-symmetric flapping, the control method is then extended for three-dimensional flight control of flapping wing micro-air vehicles. / Ph. D.

In this thesis, we develop a feedback-invariant theory of local controllability for affine distributions. We begin by developing an unexplored notion in control theory that we call proper small-time local controllability (PSTLC). The notion of PSTLC is developed for an abstraction of the well-known notion of a control-affine system, which we call an affine system. Associated to every affine system is an affine distribution, an adaptation of the notion of a distribution. Roughly speaking, an affine distribution is PSTLC if the local behaviour of every affine system that locally approximates the affine distribution is locally controllable in the standard sense. We prove that, under a regularity condition, the PSTLC property can be characterized by studying control-affine systems. The main object that we use to study PSTLC is a cone of high-order tangent vectors, or variations, and these are defined using the vector fields of the affine system. To better understand these variations, we study how they depend on the jets of the vector fields by studying the Taylor expansion of a composition of flows. Some connections are made between labeled rooted trees and the coefficients appearing in the Taylor expansion of a composition of flows. Also, a relation between variations and the formal Campbell-Baker-Hausdorff formula is established. After deriving some algebraic properties of variations, we define a variational cone for an affine system and relate it to the local controllability problem. We then study the notion of neutralizable variations and give a method for constructing subspaces of variations. Finally, using the tools developed to study variations, we consider two important classes of systems: driftless and homogeneous systems. For both classes, we are able to characterize the PSTLC property. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2010-01-11 20:11:45.466