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A key challenge to ensuring the rapid transition of robotic systems from the industrial sector to more ubiquitous applications is the development of algorithms that can guarantee safe operation while in close proximity to humans. Motion planning and control methods, for instance, must be able to certify safety while operating in real-time in arbitrary environments and in the presence of model uncertainty. This paper proposes Wrench Analysis for Inertial Transport using Reachability (WAITR), a certifiably safe motion planning and control framework for serial link manipulators that manipulate unsecured objects in arbitrary environments. WAITR uses reachability analysis to construct over-approximations of the contact wrench applied to unsecured objects, which captures uncertainty in the manipulator dynamics, the object dynamics, and contact parameters such as the coefficient of friction. An optimization problem formulation is presented that can be solved in real-time to generate provably-safe motions for manipulating the unsecured objects. This paper illustrates that WAITR outperforms state of the art methods in a variety of simulation experiments and demonstrates its performance in the real-world.