Real-world environments are inherently uncertain, and to operate safely in\nthese environments robots must be able to plan around this uncertainty. In the\ncontext of motion planning, we desire systems that can maintain an acceptable\nlevel of safety as the robot moves, even when the exact locations of nearby\nobstacles are not known. In this paper, we solve this chance-constrained motion\nplanning problem using a sequential convex optimization framework. To constrain\nthe risk of collision incurred by planned movements, we employ geometric\nobjects called $\\epsilon$-shadows to compute upper bounds on the risk of\ncollision between the robot and uncertain obstacles. We use these\n$\\epsilon$-shadow-based estimates as constraints in a nonlinear trajectory\noptimization problem, which we then solve by iteratively linearizing the\nnon-convex risk constraints. This sequential optimization approach quickly\nfinds trajectories that accomplish the desired motion while maintaining a\nuser-specified limit on collision risk. Our method can be applied to robots and\nenvironments with arbitrary convex geometry; even in complex environments, it\nruns in less than a second and provides provable guarantees on the safety of\nplanned trajectories, enabling fast, reactive, and safe robot motion in\nrealistic environments.\n