Geometric analysis and control of underactuated mechanical systems is a multidisciplinary field of study that overlaps diverse research areas in engineering and applied mathematics. These areas include differential geometry, geometric mechanics and nonlinear control theory. Many challenging applications exist such as robotics, autonomous aerospace and marine vehicles, multi-body systems, constrained systems and legged locomotion. These systems are characterized by the fact that one or more degrees of freedom are unactuated. The unactuated nature gives rise to many interesting control problems which require fundamental nonlinear approaches. This thesis contains contributions to modeling, analysis and algorithm design for underactuated mechanical systems.

We provide two novel differential geometric formulations of the nonlinear control models for underactuated mechanical systems. The key feature of each formulation is the partitioning of the equations of motion into those associated with the actuated and unactuated dynamics. Both formulations are constructed using control forces and the kinetic energy metric inherent in the classic problem formulation. Interestingly, each formulation gives rise to an intrinsic vector-valued symmetric bilinear form that can be associated with an underactuated mechanical control system.

The first formulation models an underactuated mechanical system evolving on an affine foliation of the tangent bundle. The affine foliation decomposes the velocity curve of the underactuated system into affine and linear components. We show that the affine component represents the unactuated velocity states and the linear component represents the actuated velocity states. In this framework, the ability to move from leaf to leaf in the affine foliation is characterized by the definiteness of the intrinsic symmetric bilinear form.

The second formulation utilizes two linear connections. Specifically, we introduce the actuated and unactuated connections which provide a coordinate-invariant representation of the actuated and unactuated dynamics. We show that feedback linearization of the actuated dynamics gives rise to a control-affine system whose drift vector field is the geodesic spray of the unactuated connection. We call this control-affine system the geometric normal form for underactuated mechanical systems. The geometric normal form is the starting point for our reachability analysis and motion algorithms for mechanical systems underactuated by one.

Our main analytical contribution is a unique characterization of the set of reachable velocities from an arbitrary initial configuration and velocity (possibly nonzero velocity) for mechanical systems underactuated by one control. The characterization is computable and dependent upon the definiteness of the intrinsic symmetric bilinear form. The proof of the existence of a control law that will drive a mechanical system underactuated by one control from velocity to velocity is constructive. Therefore, our main result gives rise to a velocity to velocity motion planning algorithm. The algorithm is applied to various examples of nonlinear mechanical systems underactuated by one control.