Motion Planning for Highly Constrained Spaces

In many motion planing problems, the feasible subspace becomes thin in some directions. This is often due to kinematic closure constraints, which restrict the feasible configurations to a lower-dimensional manifold or variety. This may also be simply due to the way the obstacles are arranged. For example, is planning for a closed chain different from planning a sliding motion for a washer against a rod? An illustration for the two instances of such problems is shown on Fig. 27.1. Traditionally, the two problems are solved with different methods in motion planning. However, the two seemingly different problems have similar algebraic and geometric structure of the sets of feasible configurations. In either case, there may exist functions of the form |f i (q)| ≤ e i , e i ≥ 0, that contain most or all of the feasible set. Typically, e i is very small, and in the case of closed chains, e i = 0. When this occurs, a region of the feasible space has small intrinsic dimensionality, in comparison to the ambient configuration space. Note that in the problem on Fig. 27.1(b) and in similar motion planning problems, the f i are not necessarily given.