This paper presents a trajectory generation method for contact-constrained robotic systems such as manipulators and legged robots. Contact-constrained systems are affected by the interaction forces between the robot and the environment. In turn, these forces determine and constrain state reachability of the robot parts or end effectors. Our study subdivides the trajectory generation problem and the supporting reachability analysis into tractable subproblems consisting of a sampling problem, a convex optimization problem, and a nonlinear programming problem. Our method leads to significant reduction of computational cost. The proposed approach is validated using a realistic simulated contact-constrained robotic system.