Simple mechanical control systems with constraints and symmetry

We develop tools for studying the control of underactuated mechanical systems that evolve on a configuration space with a principal fiber bundle structure. Taking the viewpoint of affine connection control systems, we derive reduced formulations of the Levi-Civita and the nonholonomic affine connections, along with the symmetric product, in the presence of symmetries and nonholonomic constraints. We note that there are naturally two kinds of connections to be considered here, affine and principal connections, leading to what we term a "connection within a connection." These results are then used to describe controllability tests that are specialized to simple, underactuated mechanical systems on principal fiber bundles, including the notion of fiber configuration controllability. We present examples of the use of these tools in studying the planar rigid body with a variable direction (vectored) thruster and the snakeboard robot.