Rapid Low-Thrust Trajectory Optimization in Deep Space Based on Convex Programming

Lowering the costs of interplanetary space missions through the use of small satellites (e.g., CubeSats) will be of key importance for solar system science and human space flight. As CubeSats can provide only very low thrust, new challenges arise. In particular, the guidance design is a complex optimization problem and usually takes several days or even months to complete. It is always performed on ground, and every deficiency of the algorithm has to be resolved by the operator. It would be desirable, however, to develop CubeSats that are able to autonomously approach minor or major bodies in the solar system without human intervention. Shifting the guidance task on board poses a great challenge as the algorithm must repeatedly recompute the reference trajectory and guarantee a (near-optimal) solution in real-time. Because of the nonlinear dynamics and lowthrust propulsion characteristics, this results in complex and computationally burdensome optimization tasks. The criteria of feasibility (convergence to a solution), optimality (cost function minimization), and sustainability (compatibility with available resources) must be traded off when the guidance is to be computed on-the-fly. Current techniques focus on optimality due to the large power capabilities of working stations that allow to compute optimal solutions offline, without the need of having real-time capabilities. Only little attention has been paid to designing a computationally simple, nonfailing algorithm that can potentially be implemented on board. As a consequence, standard techniques have mainly been used to calculate low-thrust trajectories: metaheuristics, indirect, direct, or feedbackdriven methods. Metaheuristics (optimization algorithms that employ heuristic rules to find an optimal solution, e.g., evolutionary algorithms [1]) and indirect methods (based on the calculus of variations, see [2,3]) are generally capable of finding locally optimal solutions. Yet, they are not suitable for on-board applications as they suffer from poor robustness and computational difficulties. Although improvements have been proposed during the past years (e.g., through smoothing techniques [4]), the convergence issues remain unsolved. Feedback-driven methods are computationally simple and hence a popular choice for preliminary trajectory design [5]. Still, they are in general not optimal and do not guarantee convergence. More recent approaches make use of artificial intelligence techniques to design robust, but less optimal and less flexible on-board guidance schemes [6]. Direct methods transcribe the infinite-dimensional optimal control problem (OCP) into a (often large-scale) finite-dimensional constrained optimization problem [7]. On-board computers, however, lack in general of the computational capability to solve the full nonlinear program. Still, because of the higher robustness compared with indirect methods, evolutionary algorithms, and similar techniques [8], some studies refined the existing methods to further lower the computational effort and increase accuracy and reliability [9]. Nevertheless, those solvers cannot guarantee convergence to a local minimum and still do not operate in real-time. As a consequence, novel sequential convex programming (SCP) methods [10] have been developed to overcome these issues. The original nonlinear problem is transformed into an equivalent convex program that is iteratively solved using sophisticated interior point methods [11]. Because of their rapid calculation speed and the fact that convex programs are shown to converge to the global minimum under certain conditions [12], such techniques are a popular choice for real-time applications, especially within the path planning of robots and quad-rotors [13,14]. Because of the high demand for more and more autonomy in aerospace vehicles, tremendous effort was made to exploit the advantages of convex optimization in aerospace applications. Therefore, power descent landing guidance [15,16] and entry trajectory optimization problems [17,18] have recently been solved using SCP. In this context, an improved Radau pseudospectral discretization scheme has been applied in [19] to increase the sparsity of the powered descent and landing problem, and thus lower the computational effort. In contrast to the majority of researchers who use a modeling language for convex programming [20] to facilitate the SCP implementation, the work in [21] aims to improve the computational performance by tailoring the algorithm to the actual flight code requirements. With regard to low-thrust trajectory design, Wang and Grant [22,23] applied SCP to solve time- and fuel-optimal transfers for the first time. Their simple numerical examples show that the computational time can be reduced considerably compared with standard nonlinear programming (NLP) solvers while still getting nearoptimal solutions with rather poor initial guesses. Yet, no conclusion can be drawn on how SCP performs when more complex interplanetary transfers are addressed. Moreover, extensive testing would be necessary to assess the robustness against poor initial guesses. Even simple examples show that bang-off-bang control structures cannot be captured accurately. This Note presents an improved method based on convex programming to generate complex interplanetary trajectories for lowthrust spacecraft in deep space. The goal is to have a computationally simple and robust algorithm that produces near-optimal solutions in little time. Building on the work of [19,24], we employ an adaptive flipped Radau pseudospectral method (FRPM) to lower the computational time and add a mesh refinement strategy for bang-off-bang control structures. A shape-based method generates initial guesses of various quality and several numerical simulations assess the overall robustness of the algorithm. The Note is structured as follows. Section II states the OCP for space flight and its transcription into a convex program. In Sec. III, the adaptive FRPM is explained and Sec. IV addresses the mesh refinement method. The results of numerical simulations are presented and discussed in Sec. V to assess the performance of the proposed method when compared with state-of-the-art solvers. Final remarks are given in Sec. VI.