Motion planning for closed-chain systems is particularly difficult due to additional closure constraints placed on the system. In fact, the probability of randomly selecting a set of joint angles that satisfy the closure constraints is zero. We propose Planning with Reachable Distance (PRD) to overcome this challenge by first precomputing the subspace satisfying the closure constraints, then directly sampling in it. To do so, we represent the chain as a hierarchy of sub-chains. Then we calculate the ``closure'' sub-space as appropriate reachable distance ranges of sub-chains satisfying the closure constraints. This provides two distinct advantages over traditional approaches: (1) configurations are quickly sampled and converted to joint angles using basic trigonometry functions instead of more expensive inverse kinematics solvers, and (2) configurations are guaranteed to be closed.
In this paper, we describe this hierarchical chain representation and give a sampling algorithm with complexity linear in the number of links. We provide the necessary motion planning primitives for most sampling-based motion planners. Our experimental results show our method is fast, making sampling closed configurations comparable to sampling open chain configurations that ignore closure constraints. Our method is general, easy to implement, and also extends to other distance-related constraints besides the ones demonstrated here.