Optimal Manoeuvres of Underactuated Linear Mechanical Systems: The Case of Controlling Gantry Crane Operations

A method of solving optimal manoeuvre control of linear underactuated mechanical systems is presented. The nonintegrable constraints present in such systems are handled by adding dummy actuators and then by applying Lagrange multipliers to reduce their action to zero. The open- and closed-loop control schemes can be analyzed. The method, referred to as the constrained modal space optimal control (CMSOC), is illustrated in the examples of gantry crane operations. 1. Introduction Underactuated mechanical systems have fewer independent actuators than degrees of freedom (DOFs) to be controlled [1]. Typical nonlinear examples of such systems, usually with only several DOFs, are rigid multilink robotic manipulators with passive joints or any manipulator with flexible links (described by at least one mode of vibration). Linear examples include vibrating structures with continuously distributed mass (i.e., with theoretically infinite number of DOFs to describe them) such as masts, antennas, buildings, brides, and car suspension, controlled by discrete actuators. This paper presents a method of analyzing and simulating optimal manoeuvres between two given configurations (often referred to as point-to-point manoeuvres) for linear underactuated systems. The method combines optimal control theory with computational mechanics and the finite element (FE) technique, in particular. The number of DOFs equal to the number of actuators will be referred to as actuated (after [1]), while all remaining DOFs will be referred to as underactuated (however, all DOFs are in fact controlled). The actuated and unactuated DOFs must satisfy a number of constraints equal to the number of unactuated DOFs and resulting from the equations governing the motion of such systems. For mechanical systems we assume that these constraints may be nonintegrable (nonholonomic), meaning unactuated DOFs cannot be explicitly eliminated. Many of the techniques presented in the literature deal with underactuated problems by applying the constraints to eliminate the unactuated DOFs and then by solving the reduced fully actuated problems [2–4]. These approaches are limited to particular problems where the constraints can be simplified to a form making such mathematical manipulations possible. The method presented here is capable of dealing with any linear system, as it does not require the elimination of unactuated DOFs. Instead, the underactuated system is formulated as if it were fully actuated by adding “dummy” (zero-valued) actuators to all unactuated DOFs. The modal space is used in modelling the