This paper proposes a new motion planning algorithm for robot manipulator systems with path constraints. The constraint function of a manipulator determines the subspace of its joint space, and a proposed sampling-based algorithm can find a path that connects valid samples in the subspace. These valid samples can be obtained by projecting the samples onto the subspace defined by the constraint function. However, these iteratively generated samples easily fall into local optima, which degrades the search performance. The proposed algorithm uses the local geometric information and expands the search tree adaptively to avoid the local convergence problem. It increases the greediness of the search tree when it expands toward an unexplored area, which produces the benefit of reducing computational time. In order to demonstrate the performance of the algorithm, it is applied to two example problems: a maze problem using PUMA 560 under predefined constraints and a closed-chain problem using two Selective Compliance Assembly Robot Arms. The results are compared with those obtained with an existing algorithm to show the improvement in performance.