We study the problem of computing an exact motion plan for the snakeboard, an underactuated system subject to nonholonomic constraints, by exploiting its kinematic controllability properties and its decoupling vector fields. Decoupling vector fields allow us to plan motions for the underactuated dynamic system as if it were kinematic, and rest-to-rest paths are the concatenation of integral curves of the decoupling vector fields. These paths can then be time-scaled according to actuator limits to yield fast trajectories. Switches between decoupling vector fields must occur at zero velocity, so, to find fast trajectories, we wish to find paths minimizing the number of switches. In this paper, we solve the minimum-switch path-planning problem for the snakeboard. We consider two problems: 1) finding motion plans achieving a desired position and orientation of the body of the snakeboard and 2) the full problem of motion planning for all five configuration variables of the snakeboard. The first problem is solvable in closed form by geometric considerations, while the second problem is solved by a numerical approach with guaranteed convergence. We present a complete characterization of the snakeboard's minimum-switch paths.

We study the problem of computing an exact motion plan for the snakeboard by exploiting its kinematic controllability properties and its decoupling vector fields. Decoupling vector fields allow us to treat the underactuated dynamic system as a kinematic one, and rest-to-rest paths are the concatenation of integral curves of the decoupling vector fields. These paths can then be time-scaled according to actuator limits to yield fast trajectories. Switches between decoupling vector fields must occur at zero velocity, so to find fast trajectories, we wish to find paths minimizing the number of switches. In this paper we solve the minimum switch path planning problem for the snakeboard. We consider two problems: (1) finding motion plans achieving a desired position and orientation of the body of the snakeboard, and (2) the full problem of motion planning for all five configuration variables of the snakeboard. The first problem is solved in closed form by geometric considerations, while the second problem is solved by a numerical approach with guaranteed convergence. We present a complete characterization of the snakeboard's optimal paths in terms of the number of switches.

We consider the controllability problem, i.e., the existence of a suitable control input that achieves a desired reconfiguration, for underactuated mechanical systems. Since there is no general analytic tool for investigating this natural controllability property in nonlinear systems, one possibility is to study small-time local controllability (STLC), a property which is sufficient for stating controllability. The available STLC conditions require the computation of Lie brackets on the classical state-space form of the dynamic model equations. In this paper, we provide a simple sufficient condition for testing STLC in underactuated mechanical systems with n degrees of freedom and n - 1 control inputs, directly based on the terms of the system inertia matrix. As an application, we analyze the STLC of planar robots with n rotational joints, one of which is passive.

We consider the trajectory planning problem for the class of XYnR~ planar underactuated robots, having the first two (rotational or prismatic) joints actuated and the last n rotational joints passive. Under the assumption that each passive link is attached at the center of percussion of the previous passive link, the dynamic model assumes a simplified form and we show how to recursively design a dynamic feedback that completely linearizes the system equations. This result allows to plan smooth rest-to-rest motions using polynomial interpolation. As an example, we report the numerical results obtained for trajectory planning of an RR2R~ robot.

We consider the stabilization problem for an underactuated prismatic-rotational (PR) robot with the second joint passive and moving on the horizontal plane. After a controllability analysis, a nilpotent approximation of the system is derived and used for designing an open-loop polynomial command that reduces the state error in finite time. Under suitable hypotheses, the iterative application of this command, computed as a function of the state at the end of each iteration, leads to exponential convergence to the desired equilibrium configuration. Simulation results are reported, also in the presence of unmodeled viscous friction.

We consider the trajectory tracking control problem for a 4-wheel differentially driven mobile robot moving on an outdoor terrain. A dynamic model is presented accounting for the effects of wheel skidding. A model-based nonlinear controller is designed, following the dynamic feedback linearization paradigm. An operational nonholonomic constraint is added at this stage, so as to obtain a predictable behavior for the instantaneous center of rotation thus preventing excessive skidding. The controller is then robustified, using conventional linear techniques, against uncertainty in the soil parameters at the ground-wheel contact. Simulation results show the good performance in tracking spline-type trajectories on a virtual terrain with varying characteristics.