Multi-impulse Transfers Research Articles

Deep networks as approximators of optimal low-thrust and multi-impulse cost in multitarget missions

In the design of multitarget interplanetary missions, there are always many options available, making it often impractical to optimize in detail each transfer trajectory in a preliminary search phase. Fast and accurate estimation methods for optimal transfers are thus of great value. In this paper, deep feed-forward neural networks are employed to estimate optimal transfer costs to three types of optimization problems: the transfer time of time-optimal low-thrust transfers, the fuel consumption of fuel-optimal low-thrust transfers, and the total Δv of minimum-Δv J2-perturbed multi-impulse transfers. To generate the training data, both considered categories of low-thrust trajectories are optimized using the indirect method, and the J2-perturbed multi-impulse trajectories are optimized using J2 homotopy and particle swarm optimization. The hyper-parameters of deep networks are determined by grid search, random search, and the tree-structured Parzen estimators approach. Results show that deep networks are capable of estimating the final mass or time of optimal transfers with a mean relative error of less than 0.5% for low-thrust transfers and less than 4% for multi-impulse transfers. Our results are also compared with other off-the-shelf machine-learning algorithms and the generalization capabilities of the developed deep networks for predicting cases well outside the training data are investigated. Applications in multitarget mission design are also investigated.

An evolutionary discretized Lambert approach for optimal long-range rendezvous considering impulse limit

Abstract In this paper, an approach is presented for finding the optimal long-range space rendezvous in terms of fuel and time, considering limited impulse. In this approach, the Lambert problem is expanded towards a discretized multi-impulse transfer. Taking advantage of an analytical form of multi-impulse transfer, a feasible solution that satisfies the impulse limit is calculated. Next, the obtained feasible solution is utilized as a seed for generating individuals for a hybrid self-adaptive evolutionary algorithm to minimize the total time, without violating the impulse limit while keeping the overall fuel mass the same as or less than the one associated with the analytical solution. The algorithm eliminates similar individuals and regenerates them based on a combination of Gaussian and uniform distribution of solutions from the fuel-optimal region during the optimization process. Other enhancements are also applied to the algorithm to make it auto-tuned and robust to the initial and final orbits as well as the impulse limit. Several types of the proposed algorithm are tested considering varieties of rendezvous missions. Results reveal that the approach can successfully reduce the overall transfer time in the multi-impulse transfers while minimizing the fuel mass without violating the impulse limit. Furthermore, the proposed algorithm has superior performance over standard evolutionary algorithms in terms of convergence and optimality.

A new shape-based multiple-impulse strategy for coplanar orbital maneuvers

A new shape-based geometric method (SBGM) is proposed for generation of multi-impulse transfer trajectories between arbitrary coplanar oblique orbits via a heuristic algorithm. The key advantage of the proposed SBGM includes a significant reduction in the number of design variables for an N-impulse orbital maneuver leading to a lower computational effort and energy requirement. The SBGM generates a smooth transfer trajectory by joining a number of confocal elliptic arcs such that the intersections share common tangent directions. It is proven that the well-known classic Hohmann transfer and its bi-elliptic counterpart between circular orbits are special cases of the proposed SBGM. The performance and efficiency of the proposed approach is evaluated via computer simulations whose results are compared with those of optimal Lambert maneuver and traditional methods. The results demonstrate a good compatibility and superiority of the proposed SBGM in terms of required energy effort and computational efficiency.

Multi-Impulse Transfer from Mars Approach to Martian Moon's Orbit

Multi-Impulse Transfer from Mars Approach to Martian Moon's Orbit

Optimal circle-to-rectilinear orbit transfer with circumferential thrust

This paper investigates the optimal transfer trajectories from a circular parking orbit towards the apocenter of a rectilinear ellipse, where the spacecraft reaches a quasi-stationary condition relative to an inertial reference frame. The spacecraft is equipped with a propulsion system that provides a circumferential continuous propulsive acceleration, that is, an acceleration whose direction is perpendicular to the primary body-spacecraft line. The performance index to minimize is the total flight time, and an indirect method is used to analyze the transfer trajectories. In this context, the optimal transfer performance is obtained as a function of the spacecraft propulsive acceleration magnitude through an interpolation procedure of numerical simulations. The results obtained with a continuous thrust propulsion system are also compared with those derived from a multi-impulse transfer. Finally, the paper investigates a heliocentric mission scenario in which the spacecraft minimizes the flight time required to reach a rectilinear ellipse with a given value of the aphelion radius.

Asteroid Rendezvous Mission Design Using Multiobjective Particle Swarm Optimization

A new preliminary trajectory design method for asteroid rendezvous mission using multiobjective optimization techniques is proposed. This method can overcome the disadvantages of the widely employed Pork-Chop method. The multiobjective integrated launch window and multi-impulse transfer trajectory design model is formulated, which employes minimum-fuel cost and minimum-time transfer as two objective functions. The multiobjective particle swarm optimization (MOPSO) is employed to locate the Pareto solution. The optimization results of two different asteroid mission designs show that the proposed approach can effectively and efficiently demonstrate the relations among the mission characteristic parameters such as launch time, transfer time, propellant cost, and number of maneuvers, which will provide very useful reference for practical asteroid mission design. Compared with the PCP method, the proposed approach is demonstrated to be able to provide much more easily used results, obtain better propellant-optimal solutions, and have much better efficiency. The MOPSO shows a very competitive performance with respect to the NSGA-II and the SPEA-II; besides a proposed boundary constraint optimization strategy is testified to be able to improve its performance.

Optimal ballistically captured Earth–Moon transfers

The optimality of a low-energy Earth–Moon transfer terminating in ballistic capture is examined for the first time using primer vector theory. An optimal control problem is formed with the following free variables: the location, time, and magnitude of the transfer insertion burn, and the transfer time. A constraint is placed on the initial state of the spacecraft to bind it to a given initial orbit around a first body, and on the final state of the spacecraft to limit its Keplerian energy with respect to a second body. Optimal transfers in the system are shown to meet certain conditions placed on the primer vector and its time derivative. A two point boundary value problem containing these necessary conditions is created for use in targeting optimal transfers. The two point boundary value problem is then applied to the ballistic lunar capture problem, and an optimal trajectory is shown. Additionally, the problem is then modified to fix the time of transfer, allowing for optimal multi-impulse transfers. The tradeoff between transfer time and fuel cost is shown for Earth–Moon ballistic lunar capture transfers.

Qualitative Properties of Orbits for Minimum-Fuel Impulsive Transfers between Circular Coplanar Orbits with a Given Launch Time

The coplanar problem of minimizing propellant consumption in impulsive transfer between circular boundary orbits is investigated. The launch time and the initial configuration of objects on the boundary orbits are specified arbitrarily. The qualitative properties of optimal two-impulse trajectories and their optimality in the class of multi-impulse transfers are studied.

Definition and analysis of properties of optimal three-impulse point-to-orbit transfers under time constraint

This paper deals with energetically optimal multi-impulse transfers of a spacecraft in the central Newtonian gravitational field near a planet. The transfer from a point on initial orbit to the final orbit with the given angular momentum and energy constants is considered. The transfer time is bounded above. With the distance from spacecraft to planet limited and the time free, such parameters of given orbits are chosen that the 3-impulse apsidal transfer T r is optimal with an intermediate impulse at the maximum distance. On the basis of necessary optimality conditions an algorithm is developed to numerically determine the desired optimal transfer trajectory T t under time constraint, the apsidal trajectory T r being taken as initial approach. From the geometry and energy viewpoints, both trajectories T t and T r are close to each other. The trajectory T t is also 3-impulsive, all impulses on it are nonapsidal. The distance from the planet is larger and the sum of impulses is less for this trajectory than for the initial trajectory T r with the same transfer time. The simplified solution of the problem is constructed producing good approximation to the exact numerical optimization results. The solution asymptotics is found when the transfer time tends to infinity.

Optimal impulse point-to-elliptical orbit transfers of a spacecraft

The paper deals with energetically optimal multi-impulse transfer of a spacecraft in the central Newtonian gravity field near a planet. At the initial state of the transfer the distance from the spacecraft to the center of attraction, its radial and transversal velocity projections are known. At the end of the transfer the spacecraft must be located in the elliptical orbit with the given area and energy constants. The distance from the spacecraft to the center of attraction is bounded above and below, the transfer time being unspecified. The initial orbit intersects the inner boundary of the given ring. All the optimal solutions have been obtained by analytical way. A number of new solutions has been found for the given problem in comparison with the case of the transfer from the orbit at the free initial point. Up to five impulses can be applied on the optimal trajectories. The numerical simulation of the problem is carried out. It shows that all obtained solutions give not only local but global optimal energetic input on the corresponding conditions.

Globally Optimal Coplanar Orbit Transfer

This paper considers the problem of determining which time-free of a rocket between fixed elliptical orbits in a central force field requires absolutely minimum AK The paper describes the operation of a program designed to determine all globally optimal (multi-impulse) solutions by joining extremal fields in state space, and it illustrates the program results by completing the solution to the coplanar version of the minimum A V problem. The coplanar extremals are presented both in parameter space and in state space. The advantages of parameter space are illustrated by showing the difficult problem of how many successive impulses can be joined in optimum transfer can be handled there. The results presented in state space illustrate all the regions of different types of optimal transfers and map the manner in which the regions join one another. The value of this extremalgeneration method is demonstrated by 'finding that, even in this extensively analyzed coplanar problem, the complete solution still yields some new regions and some previously unguessed results; e.g., that some one-impulse transfers which continuously maximize Lawden's primer vector can be improved by finite multiimpulse transfers.

Analytical guidance in the neighborhood of optimal multi-impulse trajectories

A first-order minimum propellant guidance law is developed for multi-impulse trajectories in an inverse-square gravitational field. A second-order variational analysis is used to formulate the guidance problem as an accessory minimum problem, i.e. minimize a quadratic form (second-variation in propellant consumption) subject to linear constraints (variational equations of motion and deterministic boundary conditions). Solution of the accessory minimum problem provides the optimal guidance law in feedback form. It is emphasized that this analysis takes into account the nominal impulse programme when calculating the optimal guidance corrections. It is shown that for multi-impulse transfers it is in general, non-optimal to add impulses. All corrections to the trajectory should be made by a combination of small changes in timing, magnitude and direction of the nominal impulses.