Optimal path planning for continuum systems

When we think about robots, especially in the context of industry, we tend to consider bulky systems made from rigid materials brought to life with electromagnetic motors. This kind of system is pervasive in the field owing to its relative simplicity in most aspects as compared to the alternatives. The cost of their simplicity is the inability to penetrate some workspaces, leaving room for improvement and the introduction of the field of soft robotics. A soft robot is one made from compliant materials (e.g. silicone) and often they take strong inspiration from biological systems like octopus arms. By removing the constraints of rigid construction, we gain the potential for accomplishing new tasks in dense environments executed with inherent safety owing to the ‘soft’ build of the robot. The cost of this promise comes in the complexity of questions about how the system is moved, manufactured, and modeled. In this thesis, we tackle a component of this challenge, focusing on planning trajectories that a soft robot can take to accomplish a given task while adhering to the constraints of it constructions and environment. This is achieved through the formulation and solution to an optimal control problem using the Bernstein polynomial basis. While this has been accomplished for problems governed by ordinary differential equations, this work extends to partial differential equations which capture system changes over spatial and temporal derivatives.