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One of the fundamental problems in the field of robotic motion planning is to safely and efficiently drive the end effector of a robotic manipulator to a specified goal position. Here, safety refers to the requirement that the robotic manipulator must have no collision with surrounding obstacles, and efficiency requires that some predefined cost function is minimized. In addition, kinematic and dynamic constraints have to be satisfied. These requirements lead to non-convex optimization problems, which may be approximated by mixed-integer linear programs (MILPs). The solution of the latter, however, is often intolerably complex due to a huge number of binary decision variables. In the present paper, we consider motion planning scenarios with polyhedral obstacles and velocity constraints for the joint positions of the robotic manipulator. We provide a geometric result whose application leads to MILPs with drastically reduced numbers of binary decision variables. Computational efficiency is demonstrated for two- and three-link manipulators interacting with obstacles, where the number of simplex steps during the MILP solution is reduced by a factor of roughly 200 over previous methods. We also demonstrate the application of the proposed method to an industrial robot.