A Hybrid Differential Dynamic Programming Algorithm for Robust Low-Thrust Optimization

Low-thrust propulsion is becoming increasingly considered for future space missions, but optimization of the resulting trajectories is very challenging. To solve such complex prob-lems, differential dynamic programming is a proven technique based on Bellman’s Principle of Optimality and successive minimization of quadratic approximations. In this paper, we build upon previous and existing optimization strategies to present an alternative hybrid variant of differential dynamic programming for robust low-thrust optimization. It uses first- and second-order state transition matrices to take advantage of an efficient discretiza-tion scheme and obtain the partial derivatives needed to perform the minimization. Unlike the traditional formulation, the state transition approach provides valuable constraint sen-sitivities and furthermore is naturally amenable to parallel computation. The method includes also a smoothing strategy to improve robustness of convergence when starting far from the optimum, as well as the capability to handle efficiently both soft and hard constraints. Procedures to drastically reduce the computation cost are mentioned. Pre-liminary numerical results are presented and compared to existing algorithms to illustrate the performance and the accuracy of our approach. Nomenclature ∆ Trust Region radius ∆max Maximum trust region radius ∆min Minimum trust region radius ǫ Step reduction parameter η Energy penalty paramter λ Lagrange multiplier vector λmin Minimum eigenvalue L Lagrangian x Nominal state vector: x ∈ ℜnx Φ1 First-order state transition matrix Φ2 Second-order state transition matrix π Control law ρ Cost reduction ratio ϕ Terminal loss function F ̃ Augmented transition function g ̃ Hard equality and active inequality constraint vector: g ̃ : ℜnx ×ℜnu → ℜm g̃c Vector of current values of active constraints: g̃c: ℜ