Optimal Path Following for Differentially Flat Robotic Systems Through a Geometric Problem Formulation

Path following deals with the problem of following a geometric path with no predefined timing information and constitutes an important step in solving the motion-planning problem. For differentially flat systems, it has been shown that the projection of the dynamics along the geometric path onto a linear single-input system leads to a small dimensional optimal control problem. Although the projection simplifies the problem to great extent, the resulting problem remains difficult to solve, in particular in the case of nonlinear system dynamics and time-optimal problems. This paper proposes a nonlinear change of variables, using a time transformation, to arrive at a fixed end-time optimal control problem. Numerical simulations on a robotic manipulator and a quadrotor reveal that the proposed problem formulation is solved efficiently without requiring an accurate initial guess.