For industrial constrained manipulators, the interactive motion and force dynamics between the end effector and the constrained environment occur at both the normal space and the tangent space to the constraint manifold. In order to characterize such interactive behaviors, it is advantageous to develop a specific coordinate frame to facilitate the formulation of such constrained dynamics. Most previous approaches solved the problem from the algebraic viewpoint. This paper, on the other hand, develops a systematic method to construct a geometric constraint frame from the perspective of integral manifold theory. A constraint frame, distinct from previous orthogonal moving frames, behaves as a partially orthogonal coordinate system and is endowed with intrinsic geometric implications which are beneficial to the dynamic analysis and the controller design. The algebraic and geometric points of view concerning the constrained systems are compared and discussed. An illustrative example for constrained manipulators will be demonstrated.