On octopus arm and multigrid techniques for nonlinear constrained minimization

Our objective is to calculate a curve (configuration) in two dimensions that solves an optimization problem of minimizing a generalized squared curvature functional subject to global constrained. We allow the given rigidity coefficient to vary along the curve. The global constraints include the length of the curve and its initial and ending positions and directions. We use a multigrid approach to efficiently solve the resulting discretized system of non-linear equations. This approach is in particular convenient for introducing the desired global constraints and solving the entire problem in linear time complexity. Since our work is motivated by the idea of building a soft octopuslike robotic arm, we then produce a dynamic sequence of such configurations depending on time to resemble a complete movement of the octopus arm, for instance in a reaching movement. The obtained sequence involves bend propagation as well as elongation of the arm as is typical to the octopus. We suggest the use of our results as input to the robotic arm in order to enable its basic movements. The numerical approach itself can be generalized to much more complex problems in higher dimensions.