In this dissertation we examine a class of systems where nonholonomic kinematic constraints are combined with periodic shape variations, giving rise to a snake-like undulating motion of the system. Within this class, we distinguish two subclasses, one where the system possesses enough kinematic constraints to allow the control of its motion to be based entirely on kinematics and another which does not; in the latter case, the dynamics plays a crucial role in complementing the kinematics and in making motion control possible. An instance of these systems are the Nonholonomic Variable Geometry Truss (NVGT) assemblies, where shape changes are implemented by parallel manipulator modules, while the nonholonomic constraints are imposed by idler wheels attached to the assembly. We assume that the wheels roll without slipping on the ground, thus constraining the instantaneous motion of the assembly. These assemblies can be considered as land locomotion alternatives to systems based on legs or actuated wheels. Their propulsion combines features of both biological systems like skating humans and snakes, and of man-made systems like orbiting satellites with manipulator arms. The NVGT assemblies can be modeled in terms of the Special Euclidean group of rigid motions on the plane. Generalization to nonholonomic kinematic chains on other Lie groups (G) gives rise to the notion of G-Snakes.
Moreover, we examine systems with parallel manipulator subsystems which can be used as sensor-carrying platforms, with potential applications in exploratory and active visual or haptic robotic tasks. We concentrate on specifying a class of configuration space path segments that are optimal in the sense of a curvature-squared cost functional, which can be specified analytically in terms of elliptic functions and can be used to synthesize a trajectory of the system.
In both cases, a setup of the problem which involves tools from differential geometry and the theory of Lie groups appears to be natural. In the case of G-Snakes, when the number of nonholonomic constraints equals the dimension of the group G, the constraints determine a principal fiber bundle connection. The geometric phase associated to this connection allows us to derive (kinematic) motion control strategies based on periodic shape variations of the system. When the G-Snake assembly has one constraint less than the dimension of the group G, we are still able to synthesize a principal fiber bundle connection by taking into account the Lagrangian dynamics of the system through the so-called nonholonomic momentum. The symmetries of the system are captured by actions of non-abelian Lie groups that leave invariant both the constraints and the Lagrangian and play a significant role in the definition of the momentum and the specification of its evolution. The (dynamic) motion control is now based on periodic shape variations that build up momentum and allow propulsion and steering, as described by the geometric and dynamic phases of the system.