Dynamics and control of underactuated mechanical systems: analysis and simple experimental verification

Underactuated mechanical systems are systems with fewer control inputs than the degrees of freedom, m < n, the relevant technical examples being e.g. cranes, aircrafts and flexible manipulators. The determination of an input control strategy that forces an underactuated system to complete a set of m specified motion tasks (servo-constraints) is a demanding problem. The solution is conditioned to differential flatness of the problem, denoted that all 2n state variables and m control inputs can algebraically be expressed, at least theoretically, in terms of the desired m outputs and their time derivatives up to a certain order. A more practical formulation, motivated hereafter, is to pose the problem as a set of differential-algebraic equations, and then obtain the solution numerically. The theoretical considerations are illustrated by a simple two-degree-of-freedom underactuated system composed of two rotating discs connected by a flexible rod (torsional spring), in which the pre-specified motion of the first disc is actuated by the torque applied to the second disc, n = 2 and m = 1. The determined control strategy is then verified experimentally on a laboratory stand representing the two-disc system. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)