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In many manipulator configurations, where the end effector of the manipulator is in contact with a fixed object, a complete mathematical model for the manipulator dynamics should include the effects of the resulting contact force between the end effector and the fixed object. Equations for such a closed chain manipulator are developed, where the end effector constraint is defined by a smooth manifold. These equations are shown to be complete in the sense that the direct dynamics problem and the inverse dynamics problem are well-posed. This formulation suggests a new approach to planning and tracking control for manipulators in a constrained environment.