Gauss-Newton Runge-Kutta Integration for Efficient Discretization of Optimal Control Problems with Long Horizons and Least-Squares Costs

This work proposes an efficient treatment of continuous-time optimal control problem (OCP) with long horizons and nonlinear least-squares costs. The Gauss-Newton Runge-Kutta (GNRK) integrator is presented which provides a high-order cost integration. Crucially, the Hessian of the cost terms required within an SQP-type algorithm is approximated with a Gauss-Newton Hessian. Moreover, L2 penalty formulations for constraints are shown to be particularly effective for optimization with GNRK. An efficient implementation of GNRK is provided in the open-source software framework acados. We demonstrate the effectiveness of the proposed approach and its implementation on an illustrative example showing a reduction of relative suboptimality by a factor greater than 10 while increasing the runtime by only 10 %.