Optimization of human generated trajectories for safety controller synthesis

The aim of the optimal safety controller synthesis problem is to synthesize a feedback controller that results in closed-loop trajectories that meet certain criteria, namely, the state or output trajectories terminate in a goal set without entering an unsafe set while optimizing some function. Our previous work presented a method for using finitely many human generated trajectories to synthesize a non-optimal safety controller. We propose a formal method for optimizing the human generated trajectories used to synthesize the controller. Our method is based on the calculus of variations, but is different from other similar algorithms in that it uses a gradient descent based approach to directly solve the optimization problem without formulating the optimality conditions given by the Pontryagin Minimum Principle. This method provides a tool for improving the performance of a controller synthesized using the methods outlined in our previous work. We present an example of optimizing a human generated trajectory for a nonlinear system, specifically a quadrotor, and quantify the improvements it is able to generate.