Mauro Pontani

Particle Swarm Optimization Applied to Space Trajectories

The particle swarm optimization technique is a population-based stochastic method developed in recent years and successfully applied in several fields of research. It represents a very intuitive (and easy to program) methodology for global optimization, inspired by the behavior of bird flocks while searching for food. The particle swarm optimization technique attempts to take advantage of the mechanism of information sharing that affects the overall behavior of a swarm, with the intent of determining the optimal values of the unknown parameters of the problem under consideration. In this research the method is applied to a variety of space trajectory optimization problems, i.e., the determination of periodic orbits in the context of the circular restricted three-body problem, and the optimization of (impulsive and finite thrust) orbital transfers. Despite its simplicity and intuitiveness, the particle swarm algorithm proves to be quite effective in finding the optimal solution to all of the applications considered in the paper, with great numerical accuracy.

Numerical solution of the three-dimensional orbital Pursuit-evasion game

The problem of interception of an evasive spacecraft by a pursuing spacecraft is formulated as a differential game. Each spacecraft is given a modest capability to maneuver in the three-dimensional space. Interception concludes the game and occurs if the pursuing spacecraft reaches the instantaneous position of the evading spacecraft. The objective of the pursuer is to minimize the time for interception, whereas the evader tries to delay it indefinitely. Saddle-point equilibrium solutions are found using a recently developed direct numerical method that uses the analytical necessary conditions (unlike ordinary direct methods) to find the optimal control for one of the players. The method requires an initial guess of the solution, and this is provided by generating an approximate solution using genetic algorithms. The evolutionary algorithm is employed as a preprocessing technique and is very useful in this context because the trial-and-error selection of first-attempt values for the variables involved is very challenging for the problem at hand. The numerical method is tested on a variety of different starting conditions related to the distinct initial orbits of the two spacecraft. It successfully finds the saddle-point trajectories of the two spacecraft, thus proving its effectiveness and robustness in solving a quite complicated problem such as the three-dimensional orbital game at hand.

Particle Swarm Optimization of Multiple-Burn Rendezvous Trajectories

The particle swarm algorithm is a population-based heuristic method successfully applied in several fields of research, only recently to aerospace trajectories. It represents a very intuitivemethodology for optimization, inspired by the behavior of bird flocks while searching for food. In this work, the method is applied to (impulsive and finitethrust) multiple-burn rendezvous trajectories. First, the technique is employed to determine the globally optimal four-impulse rendezvous trajectories for two challenging test cases. Second, the same problems are solved under the assumption of using finite thrust. In this context, the control function is assumed to be a linear combination of B-splines. Themethod at hand is relatively straightforward to implement anddoes not require an initial guess, unlike gradient-based solvers. Despite its simplicity and intuitiveness, the particle swarm methodology proves to be quite effective in finding the optimal solution to orbital rendezvous optimization problems with considerable numerical accuracy.

Optimal Finite-Thrust Rendezvous Trajectories Found via Particle Swarm Algorithm

The particle swarm optimization technique is a population-based stochastic method developed in recent years and successfully applied in several fields of research. The particle swarm optimization methodology aims at taking advantage of the mechanism of information sharing that affects the overall behavior of a swarm, with the intent of determining the optimal values of the unknown parameters of the problem under consideration. This research applies the technique to determining optimal continuous-thrust rendezvous trajectories in a rotating Euler–Hill frame. Five distinct applications, both in two dimensions and in three dimensions, are considered. Hamiltonian methods are employed to translate the related optimal control problems into parameter optimization problems. The transversality condition, which is an analytical condition that arises from the calculus of variations, is proven to be ignorable for these problems, and this property greatly simplifies the solution process. For each of the five applicati...

Optimal Interception of Evasive Missile Warheads: Numerical Solution of the Differential Game

The problem of interception of a ballistic missile warhead by a defending missile is formulated as a differential game. Each missile is given a modest postlaunch capability to maneuver. Interception concludes the game and occurs if the interceptor reduces the distance between it and the warhead to a specified value. The objective of the warhead is to minimize the final distance to the target, which lies on the Earths surface but is not necessarily collocated with the interceptor launch site. The objective of the interceptor is to maximize this same distance at the capture time. Saddle-point equilibrium solutions are found using a recently developed, direct numerical method that nevertheless uses the analytical necessary conditions to find the optimal control for one of the players. The method requires an initial guess of the solution, and this is provided by generating an approximate solution using genetic algorithms. For initial conditions yielding eventual capture, we derive the necessary condition that the saddle-point trajectories terminate on the usable part of the terminal hypersurface; this condition is shown to have an interesting physical interpretation. The numerical method successfully finds the saddle-point trajectories and is used to determine the sensitivity of the value of the game to the capability of the interceptor, for design purposes, and also to gauge the robustness of the numerical method for the solution of this dynamic game.

Optimal Low-Thrust Orbital Maneuvers via Indirect Swarming Method

In the last decades, heuristic techniques have become established as suitable approaches for solving optimal control problems. Unlike deterministic methods, they do not suffer from locality of the results and do not require any starting guess to yield an optimal solution. The main disadvantages of heuristic algorithms are the lack of any convergence proof and the capability of yielding only a near optimal solution, if a particular representation for control variables is adopted. This paper describes the indirect swarming method, based on the joint use of the analytical necessary conditions for optimality, together with a simple heuristic technique, namely the particle swarm algorithm. This methodology circumvents the previously mentioned disadvantages of using heuristic approaches, while retaining their advantageous feature of not requiring any starting guess to generate an optimal solution. The particle swarm algorithm is chosen among the different available heuristic techniques, due to its apparent simplicity and the recent promising results reported in the scientific literature. Two different orbital maneuvering problems are considered and solved with great numerical accuracy, and this testifies to the effectiveness of the indirect swarming algorithm in solving low-thrust trajectory optimization problems.

Simple Method to Determine Globally Optimal Orbital Transfers

This research is intended to describe a simple analytical approach to determine the globally optimal impulsive transfer between two arbitrary Keplerian trajectories, including the case of initial and final elliptic orbits. First, Hohmann and bielliptic transfers are proved to be two possible global optimal transfers between two elliptic orbits without any restriction on the number of impulses. This result is achieved by using ordinary calculus in conjunction with a simple graphical construction. The final choice between these two transfers depends on the apoapse and periapseradiioftheinitialand finalellipses.Inthispaper,asimpleanalyticalprocedureisproposedthatiscapableof determining the optimal choice between a Hohmann and bielliptic transfer for arbitrary initial and final orbits. An approach similar to that used for elliptic orbits leads to results of a global nature for transfers that involve both elliptic orbits and escape trajectories. Nomenclature a = semimajor axis E = trajectory (specific) energy EB = specific energy associated with the generic point B E0 = specified value for the specific energy E e = eccentricity f = true anomaly h = specific angular momentum h = magnitude of the specific angular momentum r = position vector r = radius rA = apoapse radius (for elliptic orbits only) rAB = apoapse radius associated with the generic point B

Variable-Time-Domain Neighboring Optimal Guidance, Part 2: Application to Lunar Descent and Soft Landing

In recent years, several countries have shown an increasing interest toward both manned and automatic lunar missions. The development of a safe and reliable guidance algorithm for lunar landing and soft touchdown represents a very relevant issue for establishing a real connection between the Earth and the Moon surface. This paper applies a new, general-purpose neighboring optimal guidance algorithm, proposed in a companion paper and capable of driving a dynamical system along a specified nominal, optimal path, to lunar descent and soft landing. This new closed-loop guidance, termed variable-time-domain neighboring optimal guidance, avoids the usual numerical difficulties related to the occurrence of singularities for the gain matrices, and is exempt from the main drawbacks of similar algorithms proposed in the past. For lunar descent, the nominal trajectory is represented by the minimum-time path departing from the periselenium of a given elliptic orbit and arriving at the Moon with no residual velocity. Perturbations arising from the imperfect knowledge of the propulsive parameters and from errors in the initial conditions are considered. At specified, equally spaced times the state displacements from the nominal flight conditions are evaluated, and the guidance algorithm yields the necessary control corrections. Extensive robustness and Monte Carlo tests are performed, and definitely prove the effectiveness, robustness, and accuracy of the new guidance scheme at hand, also in comparison with the well-established linear tangent steering law.

Minimum-Fuel Finite-Thrust Relative Orbit Maneuvers via Indirect Heuristic Method

Fuel-optimal space trajectories in the Euler–Hill frame represent a subject of great relevance in astrodynamics, in consideration of the related applications to formation flying and proximity maneuvers involving two or more spacecraft. This research is based upon employing a Hamiltonian approach to determining minimum-fuel trajectories of specified duration. The necessary conditions for optimality (that is, the Pontryagin minimum principle and the Euler–Lagrange equations) are derived for the problem at hand. A switching function is also defined, and it determines the optimal sequence and durations of thrust and coast arcs. The analytical nature of the adjoint variables conjugate to the dynamics equations leads to establishing useful properties of these trajectories, such as the maximum number of thrust arcs in a single orbital period and a remarkable symmetry property, which holds in the presence of certain boundary conditions. Furthermore, the necessary conditions allow translating the optimal control p...

Variable---Time---Domain Neighboring Optimal Guidance, Part 1: Algorithm Structure

This paper presents a general purpose neighboring optimal guidance algorithm that is capable of driving a dynamical system along a specified nominal, optimal path. This goal is achieved by minimizing the second differential of the objective function along the perturbed trajectory. This minimization principle leads to deriving all the corrective maneuvers, in the context of a closed-loop guidance scheme. Several time-varying gain matrices, referring to the nominal trajectory, are defined, computed offline, and stored in the onboard computer. Original analytical developments, based on optimal control theory, in conjunction with the use of a normalized time scale, constitute the theoretical foundation for three relevant features: (i) a new, efficient law for the real-time update of the time of flight (the so called time-to-go), (ii) a new termination criterion, and (iii) a new analytical formulation of the sweep method. This new guidance, termed variable–time–domain neighboring optimal guidance, is rather general, avoids the usual numerical difficulties related to the occurrence of singularities for the gain matrices, and is exempt from the main disadvantages of similar algorithms proposed in the past. For these reasons, the variable–time–domain neighboring optimal guidance has all the ingredients for being successfully applied to problems of practical interest.