A partial state collocation method for covariance optimal control

Abstract

An overview is presented for two methods of incorporating the covariance in an optimal control problem. Including the covariance in the optimal control problem can be beneficial in the field of navigation where it is desirable to design trajectories which either minimize navigation error or maximize observability for instrument calibration. The full state collocation method uses Legendre Gauss Radau collocation to discretize the deterministic states and controls as well as the unique elements of the covariance matrix. The problem is then transcribed to a nonlinear progamming problem (NLP) and is solved with an NLP solver. This method, however, results in problems with many constraints and variables, which is computationally expensive. The partial state collocation method, the main focus of this thesis, collocates the deterministic states and controls but uses a shooting method to incorporate the covariance matrix. The problem is then transcribed to a nonlinear programming problem, which has fewer constraints and variables than the full state collocation method. Both of these methods are demonstrated by solving for the trajectory that minimizes the final position uncertainty for a spacecraft reentering Earth's atmosphere. The problem is tested with different sized covariance matrices, which shows how the time it takes to solve the problem increases as the covariance matrix increases in size. The partial state collocation method is generally faster and converges in fewer NLP iterations than the full state collocation method. As the covariance matrix increases in size, the time it takes to solve the problem increases at a smaller rate for the partial state collocation method.