Control of underactuated mechanical systems

Abstract

Underactuated mechanical systems, or system having more degrees of freedom than actuators, are abundant in real-life. Examples of such systems include, but are not limited to, road vehicles such as cars and trucks, mobile robots, underactuated robot manipulators, surface vessels, underwater vehicles, helicopters and spacecraft. In certain cases, these underactuated mechanical systems are subject to second-order nonholonomic constraints. A second-order nonholonomic constraint is known as an acceleration constraint which is non-integrable, which means that the constraint can not be written as the time-derivative of some function of the generalized coordinates and velocities. Therefore, the second-order nonholonomic constraint can not be eliminated by integration and this constraint forms an essential part of the dynamics. The interest for underactuated mechanical systems with second-order nonholonomic constraints can be motivated by the fact that, in general, the stabilization problem can not be solved by smooth (or even continuous) time-invariant state feedbacks. Typically, a first indication for this obstruction follows form the fact that the linearization around equilibrium points is not controllable. The control of this class of underactuated mechanical system is thus a challenging problem for which many open problems exist. To date, many researches have only considered the stabilization problem and the tracking control problem has received less attention. However, in practice, the tracking control problem is more important than the stabilization problem because one does not only want the system to move from one point to another, but the system should also move along a specified path. This specified path may be necessary in order to avoid obstacles or to satisfy requirements which are imposed on the motion of the system. The tracking control problem can be solved by imposing additional requirements on the trajectory to be tracked. In general, the reference trajectory has to satisfy a so-called persistence of excitation condition, meaning that the reference trajectory is not allowed to converge to a point. This means that the tracking and stabilization problems require different approaches and have to be treated separately. In this thesis, the tracking and stabilization problem are considered for a class of underactuated mechanical systems. This class consists of second-order nonholonomic mechanical systems that can be transformed into a canonical form, called the second-order chained form, by a suitable coordinateand feedback transformation. The second-order chained form facilitates controller design for secondorder nonholonomic systems because the dynamics of the system are considerably simplified and provides the possibility to design controllers for a whole class of second-order nonholonomic systems instead of a specific mechanical system. The tracking control problem for the second-order chained form, in which the controlled system should move along a specified reference trajectory, can be solved by application of a combined cascade and backstepping approach, provided that the trajectory to be tracked does not converge to a point. This approach results in a linear time-varying controller that stabilizes the second-order chained form system to the desired trajectory with exponential convergence. In addition to the tracking control problem, also some methods for generating state-to-state trajectories are presented which additionally give an explicit way of showing controllability for such underactuated mechanical systems. These methods allow the generation of feasible trajectories that connect an initial state and a desired final state and which are optimal in some sense, i.e., by formulating the trajectory generation as an optimal control problem the resulting trajectory is a local minimum of a certain cost-criterion. The stabilization problem for the second-order chained form, in which the system should be stabilized to a desired equilibrium point, can also be solved by application of a combined averaging and backstepping approach for homogeneous systems. It is well-known that the stability analysis of nonlinear time-varying systems can be quite involved and, in general, is very hard to solve. If the nonlinear time-varying system is homogeneous, the theory of homogeneous systems can be used, under additional requirements, to investigate its stability properties. A homogeneous system is associated with a corresponding homogeneous norm. In addition, a homogeneous system, under certain conditions, shares the same properties as a linear system in the sense that asymptotic stability implies exponential stability and local stability implies global stability. The combined averaging and backstepping approach results in a continuous homogeneous controller that stabilizes the system to a desired equilibrium point. To date and to our knowledge, this homogeneous controller is the only one capable of ensuring Lyapunov stability as well as exponential convergence of the second-order chained form system with respect to the corresponding homogeneous norm. It is well-known that homogeneous controllers are not robust with respect to parameter uncertainties. Therefore a periodically updated version of the homogeneous stabilizing controller has been given in which the states of the system are periodically updated at discrete time instants. This controller is robust with respect to a class of additive perturbations that includes perturbations resulting from certain parameter uncertainties, but excludes non-smooth effects, such as friction, or measurement noise. In order to successfully apply the controllers, they should first be tested in experiments with reallife second-order nonholonomic systems. The developed tracking and stabilizing controllers have been validated on an experimental set-up that consists of an underactuated H-Drive manipulator. This experimental set-up has the same dynamics as a planar horizontal underactuated PPR manipulator, or in other words a manipulator with two prismatic and one unactuated rotational joint. This experimental setup can be used as a benchmark set-up for controllers of second-order nonholonomic systems. In the experiments the goal is to use the two control inputs to control the two planar positions as well as the orientation of the link. The experimental results correspond to the simulation results and show the validity of the control design approaches in the sense that the system can be controlled to a region around the desired trajectory or equilibrium. Due to disturbances, mainly resulting from friction in the rotational link, measurement noise and gravitational disturbances, the closed-loop system is not asymptotically stable, but instead, oscillates around the desired trajectory or equilibrium. The size of the region around the desired trajectory or equilibrium, to which the system is controlled, depends on the magnitude of the disturbances. This shows the need for controllers that are robust with respect to perturbations, including non-smooth effects such as friction, or controllers which include disturbance adaptation or compensation. In most research dealing with the control of underactuated mechanical systems with second-order nonholonomic constraints the influence of perturbations on the closed-loop dynamics has generally not been taken into account. Nevertheless, the experimental results show that underactuated mechanical systems are more susceptible to perturbations than fully actuated mechanical systems. This is caused by the fact that no actuator is available to directly compensate (part of) the perturbations acting on the un-actuated degree of freedom. Therefore, the development of robust controllers for underactuated mechanical systems is an important issue that should be a subject of further research.