12.5 Motion Planning

Motion planning for nonholonomic and underactuated systems has been the subject of a great deal of recent research, and the results could easily fill several books (see, e.g., the books edited by Li and Canny [288] and Laumond [266]). In this section we summarize a few useful approaches. The approaches can be classified by the type of robot to which they apply (e.g., the structure of the equations of motion, and with or without control constraints or drift) or the nature of the problem (with or without obstacles or cost function to be minimized). Motion-planning approaches with roots in control theory tend to apply to systems with particular structure and no obstacles, while approaches based on search algorithms are computationally intensive and are suited to finding collision-free trajectories among obstacles. Some approaches attempt to combine the benefits of control-theoretic and search-based methods.

The problem is to find a motion ( x( t), u( t)), t ? [0, t _f] satisfying the equations of motion (12.6) such that x(0) = x _start, x( t _f) = x _goal.In the presence of obstacles, where represents the free configuration space, we also require , t ? [0, t _f].