This article surveys the usual techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems. In practice the knowledge resulting from the maximum principle is often insufficient for solving the problem, in particular because of the well-known problem of initializing adequately the shooting method. In this survey article it is explained how the usual tools of optimal control can be combined with other mathematical techniques to improve significantly their performances and widen their domain of application. The focus is put onto three important issues. The first is geometric optimal control, which is a theory that has emerged in the 1980s and is combining optimal control with various concepts of differential geometry, the ultimate objective being to derive optimal synthesis results for general classes of control systems. Its applicability and relevance is demonstrated on the problem of atmospheric reentry of a space shuttle. The second is the powerful continuation or homotopy method, consisting of deforming continuously a problem toward a simpler one and then of solving a series of parameterized problems to end up with the solution of the initial problem. After having recalled its mathematical foundations, it is shown how to combine successfully this method with the shooting method on several aerospace problems such as the orbit transfer problem. The third one consists of concepts of dynamical system theory, providing evidence of nice properties of the celestial dynamics that are of great interest for future mission design such as low-cost interplanetary space missions. The article ends with open problems and perspectives.

/pdf/optimal-control-and-applications-to-aerospace-some-results-88kz8kwgy9.pdf

Optimal Control and Applications to Aerospace: Some Results and Challenges

The optimization of a near-circular low-Earth-orbit multispacecraft refueling problem is studied. The refueling sequence, service time, and orbital transfer time are used as design variables, whereas the mean mission completion time and mean propellant consumed by orbital maneuvers are used as design objectives. The J2 term of the Earth’s nonspherical gravity perturbation and the constraints of rendezvous time windows are taken into account. A hybrid-encoding genetic algorithm, which uses normal fitness assignment to find the minimum mean propellant-cost solution and fitness assignment based on the concept of Pareto-optimality to find multi-objective optimal solutions, is presented. The proposed approach is demonstrated for a typical multispacecraft refueling problem. The results show that the proposed approach is effective, and that the J2 perturbation and the time-window constraints have considerable influences on the optimization results. For the problems in which the J2 perturbation is not accounted f...

Multispacecraft Refueling Optimization Considering the J2 Perturbation and Window Constraints

Successive missions must be planned to clean the near-Earth space from the heaviest debris. The problem mixes combinatorial optimization to select and order the debris, and continuous optimization to define the orbital maneuvers. In order to reduce the costs all missions have to be achieved by identical expendable vehicles with a minimum fuel requirement. The solution method proposed consists in three stages. Firstly the orbital transfer problem is solved for all pairs of debris and for discretized dates, considering a generic transfer strategy suited either to a high-thrust or to a low-thrust vehicle. The results are stored in cost matrices defining a response surface. Secondly a simulated annealing algorithm is applied to find the optimal mission planning. The cost is assessed by interpolation on the response surface. The convergence is quite fast, yielding an optimal mission planning. Thirdly the successive missions are re-optimized in terms of transfer maneuvers without changing the debris order. These continuous problems yield a refined performance requirement for designing the removal vehicle. The solution method is illustrated on a representative application case.

Multiple Space Debris Collecting Mission: Optimal Mission Planning

The space close to our planet is getting more and more polluted. Orbiting debris are posing an increasing threat to operational orbits and the cascading effect, known as Kessler syndrome, may result in a future where the risk of orbiting our planet at some altitudes will be unacceptable. Many argue that the debris density at the Low Earth Orbit (LEO) has already reached a level sufficient to trigger such a cascading effect. An obvious consequence is that we may soon have to actively clean space from debris. Such a space mission will involve a complex combinatorial decision as to choose which debris to remove and in what order. In this paper, we find that this part of the design of an active debris removal mission (ADR) can be mapped into increasingly complex variants to the classic Travelling Salesman Problem (TSP) and that they can be solved by the Inver-over algorithm improving the current state-of-the-art in ADR mission design. We define static and dynamic cases, according to whether we consider the debris orbits as fixed in time or subject to orbital perturbations. We are able, for the first time, to select optimally objects from debris clouds of considerable size: hundreds debris pieces considered while previous works stopped at tens.

/pdf/evolving-solutions-to-tsp-variants-for-active-space-debris-295v13aui3.pdf

Evolving Solutions to TSP Variants for Active Space Debris Removal

Mission design for multiple debris removal is performed by selecting the most favorable sequences of objects to be removed. Debris items among a population with similar inclination values are consi...

Optimization of active debris removal missions with multiple targets

This paper investigates the cost requirement for a space debris collecting mission aimed at removing heavy debris from low Earth orbits. The problem mixes combinatorial optimization to select the debris among a list of candidates and functional optimization to define the orbital manoeuvres. The solving methodology proceeds in two steps: Firstly, a specific transfer strategy with impulsive manoeuvres is defined so that the problem becomes of finite dimension; secondly the problem is linearized around an initial reference solution. A Branch and Bound algorithm is then applied iteratively to optimize simultaneously the debris selection and the orbital manoeuvres, yielding a new reference solution. The optimal solutions found are close to the initial guess despite a very complicated design space. The method is exemplified on a representative application case.

Multiple Space Debris Collecting Mission—Debris Selection and Trajectory Optimization

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Multiple Space Debris Collecting Mission—Debris Selection and Trajectory Optimization