Optimal control of robotic manipulators

This dissertation addresses the problem of finding the desired optimal trajectories and the optimal control inputs for a rigid multi-degrees-of-freedom robotic manipulator in a positioning mode. For a general, highly coupled and nonlinear dynamic model an analytical control law is derived and structures of controllers are suggested. The equations of motion of a multilink mechanism and its actuators are unified into a common dynamic model. A nonlinear feedback transformation which globally linearizes, decouples and places the poles of the closed loop system is derived. Each of the decoupled subsystems represents a single degree of freedom of the manipulator motion. This transformation is realized by on-line computations based on the measurements of the manipulator's state variables and reference to a look-up table, and requires no matrix inversion. The set of uncoupled subsystems in a form of triple integrators allows a solution of various optimal control problems for each degree of freedom independently. The time optimal positioning problem of a multi-degrees-of-freedom robotic manipulator with bounded velocity, acceleration and jerk is posed in a form which guarantees uniqueness of the control and state trajectories. The solution is demonstrated via single and double link examples. The look-up table implementing the feedback transformation introduces non-analytic (quantization) errors. The time optimal trajectories of a double link manipulator are shown to be rather insensitive to this type of error. For a class of errors an existence theorem of the linear feedback controller is proven which guarantees closed loop asymptotic stability at the state space origin. Finally, outlines of various controller structures are discussed. This dissertation is the first application of the following principles: (1) Instead of optimizing the real system, a fictitous system should be introduced which is more convenient for optimization and can be obtained from the original system, for example, by a state space transformation. (2) The state space feedback transformation can be realized for decoupling, linearization and pole-placement simultaneously. (3) The programmed open loop control for a deterministic system can be devised with the assumption that deviations are compensated by a separated subsystem for "control in-the-small."