Study On Motion Control Of Underactuated Mechanical Systems

An underactuated mechanical system (UMS) is a mechanical system that has fewer actuators than degrees of freedom (DOF). Because there are fewer actuators, these kinds of systems are lighter, less energy-consuming, and more flexible than fully actuated ones. However, this characteristic also restricts the states of a UMS to an uncertain manifold of the motion space. As a result, the control design of such a system is very difficult; and the problem of controlling a UMS is a challenging task in the field of engineering control.This dissertation concerns the motion control of UMSs, which includes system stabilization control, motion planning, and tracking control. The main results and innovations of this study are as follows:(1) An equivalent-input-disturbance-(EID-) based stabilization control method was developed for2-DOF UMSs.A mechanical system that has two DOFs and only one actuator is the simplest type of UMS. First,2-DOF UMSs were divided into three categories based on the inertia matrix of the system. And then, a homeomorphic coordinate transformation was devised for each category. It transforms the original system into a new one with a relatively simple structure. After the controllability, observability, arrangement of zero points, and other properties of the transformed system were thoroughly analyzed, the EID approach was used to globally stabilize the transformed system at the zero equilibrium point. The stabilizing control objective of the original system at the zero point is guaranteed by the homeomorphism of the coordinate transformation. The EID-based approach uses the state variables of position, but not velocity, to ensure global stabilization of2-DOF UMSs. This not only reduces the cost of the control system, but also avoids the influence of speed noise on control performance. Simulation results show that this method yields satisfactory results for stabilization time, motion process, and other measures of control performance for2-DOF UMSs.(2) The EID-based control method was extended to the global stabilization of UMSs with multiple DOFs.The results of the study on the EID-based stabilization control of2-DOF UMSs were used as a basis for examining the application of the EID-based approach to a multi-DOF UMS. First, a coordinate transformation for a3-DOF UMS was devised, and the properties of the transformed system were analyzed. Next, an EID-based control system was developed that stabilizes the3-DOF UMS at the zero equilibrium point. Then, the EID-based approach was applied to an n-DOF UMS (n>3), thereby further extending the applicability of this control method.(3) A torque-coupling-based stabilization control strategy was devised for a class of multi-DOF UMSs.An analysis of the structural characteristics of a class of multi-DOF UMSs was used to work out the coupling relationship between control torques. This relationship decouples some state variables of the system from others and changes the original system into a cascade-connected system consisting of a driving subsystem and a2-DOF driven subsystem. Next, the stability of the driving subsystem and the passivity of the driven subsystem were analyzed, and the results were used to convert the problem of asymptotically stabilizing the multi-DOF UMS at the zero equilibrium point into the problem of stabilizing the2-DOF driven subsystem at the zero equilibrium point. Finally, a passivity-based method was used to stabilize the driven subsystem at the zero point. The proposed method is employed on the global stabilization of a horizontal three-link manipulator. Simulation results show that the manipulator is quickly and smoothly stabilized at the target point. This demonstrates the validity and practicability of the method. In addition, this torque-coupled-based control strategy transforms the stabilization of an n-DOF UMS (n≥3) into that of a2-DOF UMS, and it might enable a control method for stabilizing a2-DOF UMS to be generalized to the stabilization of a multi-DOF one.(4) For an underactuated Acrobot system, a method of constructing a time-optimal desired trajectory was devised; and a tracking controller was designed that asymptotically stabilizes the Acrobot at the target position along the desired trajectory. A two-link Acrobot is a typical UMS. A common control objective is to swing it up from the straight-down position and stabilize it at the straight-up position. First, an artificial friction and rewinding approach was used to construct a time-optimal trajectory from the start position to the target position. Then, the stabilization control problem was converted into a tracking control problem; and an error dynamic equation was obtained based on the desired motion trajectory. Finally, a tracking controller was designed that asymptotically stabilized the Acrobot at the target position along the desired trajectory. In this study, a time-optimal trajectory for a nonlinear Acrobot system with a second-order nonholonomic constraint was constructed. It provides a guideline for the motion planning of other underactuated nonholonomic mechanical systems. Moreover, the trajectory-tracking-based stabilization control strategy is simple and efficient. It does not require division of the motion space of the Acrobot, and it uses just a single controller for motion control. Most importantly, the stabilizing movement and the settling time of the Acrobot can be accurately predicted from the desired trajectory. These can be used as guidelines for selecting actuators as well as for evaluating an experimental Acrobot system design.