3 Control of under-actuated mechanical systems

Summary form only. The research on under-actuated systems is widely motivated by the study of mobile robots and particularly of walking robots. Numerous robotic tasks are associated with underactuation. In many situations, the mechanical models have a cyclic variable, the cyclic variable is unactuated, and all shape variables are independently actuated. In this talk we introduce and solve two control problems for this class of models. If the generalized momentum conjugate to the cyclic variable is not conserved, conditions are found for the existence of a set of outputs that yields a system with a one-dimensional exponentially stable zero dynamicsx i.e., an exponentially minimum-phase systemx along with a dynamic extension that renders the system locally input-output decouplable. If the generalized momentum conjugate to the cyclic variable is conserved, a reduced system is constructed and conditions are found for the existence of a set of outputs that yields an empty zero dynamics, along with a dynamic extension that renders the system feedback linearizable. A common element in these two feedback problems is the construction of a scalar function of the configuration variables that has relative degree three with respect to one of the input components. The function arises by partially integrating the conjugate momentum. The results are illustrated on two balancing tasks and on a ballistic flip motion. From a mathematical point of view, it is shown that Pfaff-Darboux Theorem plays a key role to compute the minimal number of actuators necessary to control the structure of a general number of rigid links.