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Many tasks in robot-assisted surgery, food handling, manufacturing, and other applications require planning and controlling the motions of manipulators or other devices that must interact with highly deformable objects. We present a unified approach for motion planning under uncertainty in deformable environments that maximizes probability of success by accounting for uncertainty in deformation models, noisy sensing, and unpredictable actuation. Unlike prior planners that assume deterministic deformations or treat deformations as a type of small perturbation, our method explicitly considers the uncertainty in large, time-dependent deformations. Our method requires a simulator of deformable objects but places no significant restrictions on the simulator used. We use a samplingbased motion planner in conjunction with the simulator to generate a set of candidate plans based on expected deformations. Our method then uses the simulator and optimal control to numerically estimate time-dependent state distributions based on uncertain parameters (e.g. deformable material properties or actuation errors). We then select the plan with the highest estimated probability of successfully avoiding obstacles and reaching the goal region. Using FEM-based simulation of deformable tissues, we demonstrate the ability of our method to generate high quality plans in two medical-inspired scenarios: (1) guiding bevel-tip steerable needles through slices of deformable tissue around obstacles for minimally invasive biopsies and drug-delivery, and (2) manipulating planar tissues to align interior points at desired coordinates for precision treatment.