Constrained Path Planning for Soft Continuum Robots with Bernstein Surfaces

This manuscript presents a framework for trajectory generation for soft continuum robots using principles from optimal control. The problem is constrained over the partial differential kinematic equations of the Cosserat rod model, capturing all modes of deformation which soft continuum systems can achieve. The derived optimal control problem is transformed to a nonlinear programming problem which can be solved using the Bernstein polynomial basis. Non-unit quaternions are used to discretely apply rotational transformations to values approximated over Bernstein polynomials, allowing individual components of strain to be constrained separately. Included within this manuscript are numerical results as well as validation through experimental results.