Motion planning for spatially constrained robots is difficult due to additional constraints placed on the robot, such as closure constraints for closed chains or requirements on end effector placement for articulated linkages. It is usually computationally too expensive to apply sampling-based planners to these problems since it is difficult to generate valid configurations. We overcome this challenge by redefining the robot's degrees of freedom and constraints into a new set of parameters, called reachable distance space (RD-space), in which all configurations lie in the set of constraint-satisfying subspaces. This enables us to directly sample the constrained subspaces with complexity linear in the robot's number of degrees of freedom. In addition to supporting efficient sampling, we show that the RD-space formulation naturally supports planning, and in particular, we design a local planner suitable for use by sampling-based planners. We demonstrate the effectiveness and efficiency of our approach for several systems including closed chain planning with multiple loops, restricted end effector sampling, and on-line planning for drawing/sculpting. We can sample single-loop closed chain systems with 1000 links in time comparable to open chain sampling, and we can generate samples for 1000-link multi-loop systems of varying topology in less than a second.