Deformable robots are notoriously difficult to model or control due to its high-dimensional configuration spaces. Direct trajectory optimization suffers from the curse-of-dimensionality and incurs a high computational cost, while learning-based controller optimization methods are sensitive to hyper-parameter tuning. To overcome these limitations, we hypothesize that high fidelity soft robots can be both simulated and controlled by restricting to low-dimensional spaces. Under such assumption, we propose a two-stage algorithm to identify such simulation- and control-spaces. Our method first identifies the so-called simulation-space that captures the salient deformation modes, to which the robot's governing equation is restricted. We then identify the control-space, to which control signals are restricted. We propose a multi-fidelity Riemannian Bayesian bilevel optimization to identify task-specific control spaces. We show that the dimension of control-space can be less than for a high-DOF soft robot to accomplish walking and swimming tasks, allowing low-dimensional MPC controllers to be applied to soft robots with tractable computational complexity.
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