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Motion planning of deformable objects is challenging due to the high degrees-of-freedom inherent in deformation as well as the computational cost of producing physically accurate deformations. This paper develops a method for fast, physically plausible deformations using a sampling-based planner in a reduced dimensionality configuration space containing position, orientation, and a reduced deformation space produced using principal component analysis. A key element of the presented approach is the use of energy constraints in the configuration space designed to remove implausible deformations that may be produced by the principal component analysis dimension reduction process. This approach is tested on several deforming robot planning tasks, such as a sphere fitting through slots, a bendable bar in a maze, and a deforming plate in tunnels, with collision free paths using plausible deformations found in a few minutes.