Trajectory Optimization with Sparse Gauss-Hermite Quadrature

Abstract

This paper develops a new Differential Dynamic Programming (DDP) algorithm using a quadrature-based numerical integration scheme for improved accuracy and efficiency. The key innovation is replacing numerical differentiation with Sparse Gauss-Hermite Quadrature (SGHQ), which utilizes Smolyak’s rule, to approximate gradients and Hessians using sparsely selected sampling points in the Bellman equation. This eliminates the need for first-order derivatives of system dynamics, enhancing performance in complex systems. To extend applicability, the algorithm integrates a penalty function approach based on the Augmented Lagrangian Method (ALM) for handling constrained optimal control problems. Additionally, a Flexible Final-Time (FFT) adjustment algorithm is introduced to dynamically modify the time horizon based on optimality conditions. The final framework combines SGHQ-based integration, an augmented cost function, and adaptive time adjustment. Numerical simulations confirm the proposed method’s effectiveness and computational advantages.

Keywords: Trajectory optimization, Differential Dynamic Programming, Sparse Gauss-Hermite Quadrature