Overview

In Chapter 10 we described dynamic models for robot systems. Equipped with such a dynamic model, the trajectory planning problem is to find control (force) inputs u( t) yielding a trajectory q( t) that avoids obstacles, takes the system to the desired goal state, and perhaps optimizes some objective function while doing so. This can be considered a complete "motion-planning" problem, as opposed to a "path-planning" problem that only asks for a feasible curve q( s)in the configuration space, without reference to the speed of execution.

In this chapter we study two approaches to trajectory planning for a dynamic system: the decoupled approach, which involves first searching for a path in the configuration space and then finding a time-optimal time scaling for the path subject to the actuator limits; and the direct approach, where the search takes place in the system's state space. Examples of the latter approach include optimal control and numerical optimization, grid-based searches, and randomized probabilistic methods. In this chapter we focus on fully actuated systems-systems where there is an actuator for each degree of freedom.

In section 11.1 we provide some definitions used throughout the chapter. In section 11.2 we describe an algorithm for finding the time-optimal execution of a path subject to actuator limits, and describe how this can be used in a decoupled trajectory planner. In section 11.3 we outline several approaches to trajectory planning directly in the system state space, including optimal control, gradient-based numerical methods,...