UC Berkeley

In this dissertation, we study two important subjects in robotics: (i) time-optimal trajectory planning, and (ii) optimal control synthesis methodologies for trajectory tracking. In the first subject, we concentrate on a rather specific sub-class of problems, the time-optimal trajectory planning along predetermined geometric paths. In this kind of problem, a purely geometric path is already known, and the task is to find out how to move along this path in the shortest time physically possible. In order to generate the true fastest solutions achievable by the actual robot manipulator, the complete nonlinear dynamic model should be incorporated into the problem formulation as a constraint that must be satisfied by the generated trajectories and feedforward torques. This important problem was studied in the 1980s, with many related methods for addressing it based on the so-called velocity limit curve and variational methods. Modern formulations directly discretize the problem and obtain a large-scale mathematical optimization problem, which is a prominent approach to tackle optimal control problems that has gained popularity over variational methods, mainly because it allows to obtain numerical solutions for harder problems.

We contribute to the referred problem of time-optimal trajectory planning, by extending and improving the existing mathematical optimization formulations. We successfully incorporate the complete nonlinear dynamic model, including viscous friction because for the fastest motions it becomes even more significant than Coulomb friction; of course, Coulomb friction is likewise accommodated for in our formulation. We develop a framework that guarantees exact dynamic feasibility of the generated time-optimal trajectories and feedforward torques. Our initial formulation is carefully crafted in a rather specific manner, so that it allows to naturally propose a convex relaxation that solves exactly the original problem formulation, which is non-convex and therefore hard to solve. In order to numerically solve the proposed formulation, a discretization scheme is also developed. Unlike traditional and modern formulations, we motivate the incorporation of additional criteria to our original formulation, with simulation and experimental studies of three crucial variables for a 6-axis industrial manipulator. Namely, the resulting applied torques, the readings of a 3-axis accelerometer mounted at the manipulator end-effector, and the detrimental effects on the tracking errors induced by pure time-optimal solutions. We therefore emphasize the significance of penalizing a measure of total jerk and of imposing acceleration constraints. These two criteria are incorporated without destroying convexity. The final formulation generates near time-optimal trajectories and feedforward torques with traveling times that are slightly larger than those of pure time-optimal solutions. Nevertheless, the detrimental effects induced by pure time-optimality are eliminated. Experimental results on a 6-axis industrial manipulator confirm that our formulation generates the fastest solutions that can actually be implemented in the real robot manipulator.

Following the work done on near time-optimal trajectories, we explore two controller synthesis methodologies for trajectory tracking, which are more suitable to achieve trajectory-tracking under such fast trajectories. In the first approach, we approximate the discrete-time nonlinear dynamics of robot manipulators, moving along the state-reference trajectory, as an affine time-varying (ATV) dynamical system in discrete-time. Therefore, the problem of trajectory tracking for robot manipulators is posed as a linear quadratic (LQ) optimal control problem for a class of discrete-time ATV dynamical systems. Then, an ATV control law to achieve trajectory tracking on the ATV system is developed, which uses LQ methods for linear time-varying (LTV) systems. Since the ATV dynamical system approximates the nonlinear robot dynamics along the state-reference trajectory, the resulting time-varying control law is suitable to achieve trajectory tracking on the robot

manipulator. The ATV control law is implemented in experiments for the 6-axis industrial manipulator, tracking the near time-optimal trajectory. Experimental results verify the better performance achieved with the ATV control law, but also expose its shortcomings.

The second approach to address trajectory tracking is related in spirit, but different in crucial aspects, which ultimately endow this approach with its superior features. In this novel approach, the highly nonlinear dynamic model of robot manipulators, moving along a state-reference trajectory, is approximated as a class of piecewise affine (PWA) dynamical systems. We propose a framework to construct the referred PWA system, which consists in: (i) choosing strategic operating points on the state-reference trajectory with their respective (local) linearized system dynamics, (ii) constructing ellipsoidal regions centered at the operating points, whose purpose is to facilitate the scheduling strategy of controller gains designed for each local dynamics. Likewise, in order to switch controller gains as the robot state traverses in the direction of the state-reference trajectory, a simple scheduling strategy is proposed. The controller synthesis near each operating point is an LQR-type that takes into account the local coupled dynamics. The referred PWA control law is implemented in experiments for the 6-axis manipulator tracking the near time-optimal trajectory. The experimental results show the feasibility and superiority of the PWA control law over the typical PID controller and the ATV control law.