Abstract-We present a proof for the probabilistic completeness of RRT-based algorithms when planning with constraints on end-effector pose. Pose constraints can induce lowerdimensional constraint manifolds in the configuration space of the robot, making rejection sampling techniques infeasible. RRT-based algorithms can overcome this problem by using the sample-project method: sampling coupled with a projection operator to move configuration space samples onto the constraint manifold. Until now it was not known whether the sampleproject method produces adequate coverage of the constraint manifold to guarantee probabilistic completeness. The proof presented in this paper guarantees probabilistic completeness for a class of RRT-based algorithms given an appropriate projection operator. This proof is valid for constraint manifolds of any fixed dimensionality.