We consider the nonprehensile object transportation task known as the waiter's problem - in which a robot must move an object on a tray from one location to another - when the transported object has uncertain inertial parameters. In contrast to existing approaches that completely ignore uncertainty in the inertia matrix or which only consider small parameter errors, we are interested in pushing the limits of the amount of inertial parameter uncertainty that can be handled. We first show how constraints that are robust to inertial parameter uncertainty can be incorporated into an optimization-based motion planning framework to transport objects while moving quickly. Next, we develop necessary conditions for the inertial parameters to be realizable on a bounding shape based on moment relaxations, allowing us to verify whether a trajectory will violate the constraints for any realizable inertial parameters. Finally, we demonstrate our approach on a mobile manipulator in simulations and real hardware experiments: our proposed robust constraints consistently successfully transport a 56 cm tall object with substantial inertial parameter uncertainty in the real world, while the baseline approaches drop the object while transporting it.

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