Optimization of Spacecraft Trajectories Using Nonlinear Programming

Low-thrust interplanetary spacecraft trajectory optimization poses a uniquely difficult problem to solve because of the inherent nonlinearities of the dynamics and constraints as well as the large size of the search space of possible solutions. Tools currently exist that optimize low-thrust interplanetary trajectories, but these tools are rarely openly available to the public, and when they are available they require multiple interfaces between multiple different packages. The goal of this work is to present a new piece of low-thrust interplanetary spacecraft trajectory optimization software that is open-source and entirely self-contained so that more people can have access to the ability to design interplanetary trajectories. To achieve this goal, a gradient-descent based nonlinear programming method, called the interior point method, was used. The nonlinear programming method was chosen so that results from this work could be compared and contrasted with results from Spacecraft Trajectory Optimization Suite (STOpS), which uses heuristics to iterate towards a solution. Interior point methods are popular because of their ability to handle large amounts of equality and inequality constraints, which is a characteristic that is valuable for low-thrust interplanetary spacecraft trajectories. The software developed, Interior Point Optimizer (IP Optimizer), was then validated against test cases with known solutions to ensure that the software delivered the intended results. Lastly, a constraint satisfaction, a minimum-time, and a maximum-final-mass optimization problem were solved and compared with literature to illustrate the advantages of IP Optimizer and the methods it employs. For the constraint satisfaction problem, IP Optimizer was able to find a solution that exactly satisfied the desired terminal constraints whereas STOpS had an error of 2.29 percent. In this case, IP Optimizer had a reduced runtime of 15 percent compared to STOpS as well. When minimizing time for a spacecraft transfer, IP Optimizer improved upon the solution found by STOpS by 5.3 percent. The speed of convergence for IP Optimizer was almost twice as fast as STOpS for this case. These results show that IP Optimizer is faster than STOpS at converging on a solution and the solution it converges to has a better objective value and more accurately satisfies the terminal constraints than STOpS. Lastly, the maximum-final-mass problem resulted in an objective value that was only 0.5 percent lower than the value found in literature.

As an increasing number of asteroids are being discovered, detecting them using limited propulsion resources and time has become an urgent problem in the aerospace field. However, there is no universal fast asteroid sequence selection method that finds the trajectories for multiple low-thrust spacecraft for detecting a large number of asteroids. Furthermore, the calculation efficiency of the traditional trajectory optimization method is low, and it requires a large number of iterations. Therefore, this study combines Monte Carlo tree search (MCTS) with spacecraft trajectory optimization. A fast MCTS pruning algorithm is proposed, which can quickly complete asteroid sequence selection and trajectory generation for multispacecraft exploration of multiple asteroids. By combining the Bezier shape-based (SB) method and MCTS, this study realizes the fast search of the exploration sequence and the efficient optimization of the continuous transfer trajectories. In the simulation example, compared with the traversal algorithm, the MCTS pruning algorithm obtained the global optimal detection sequence of the search tree in a very short time. Under the same conditions, the Bezier SB method obtained the transfer trajectory with a better performance index faster than the finite Fourier series SB method. Performances of the proposed method are illustrated through a complex asteroid multiflyby mission design.

A System for Collaborative Design, Analysis, and Optimization of Spacecraft Trajectories

In this chapter, the problem of improving convergence and finding suitable tentative solutions for the indirect optimization of spacecraft trajectories is discussed. The application of theory of optimal control to spacecraft trajectories transforms the optimal control problem into a multi-point boundary value problem, which is usually solved by means of an iterative procedure. The convergence radius of the problem may be small and convergence to the optimal solution is only obtained if the tentative solution, which is used to start the procedure, is sufficiently close to the optimum. The definition of a suitable solution is often the hardest part of the solution procedure for the optimization problem. Several cases and examples are presented in this chapter to illustrate the measures that could be adopted for the most common difficulties, which may be found during the optimization of space trajectories.

This paper presents an approach for multi-spacecraft trajectory planning, optimization and control. Maneuver planning as a global optimization problem is solved using genetic algorithms (GA). Methods were devised to reduce the dimensionality of the decision space, yet retain adequate generality of maneuver possibilities. A compact formulation based on thruster switching-times was used for generic point-to-point spacecraft maneuvers. Optimal control is implicitly satisfied by "bang-coast-bang" actuation schemes. Maneuver profiles, including line-of-sight and orthogonal collision avoidance, were developed. A GA optimizer selects the optimal parameter set for each scenario. Simulation case studies were performed for 2, 3 and 5-spacecraft formation initialization tasks. Objective criteria used in the evaluation function included: endpoint errors; collision avoidance; path lengths; maneuvering times; fuel usage and equalization. In all cases, a nominal GA computed feasible trajectories. Objective criteria trade-offs were demonstrated by selective weighting. Ongoing work includes multi-objective optimization of multiple spacecraft trajectories using niched-Pareto genetic algorithms.

In this paper I describe the utility of, and present an accessible derivation of, the classic Leitmann equation. This is an analytic,approximate equation which relates propulsion system requirements to space mission requirements for a power-limited orvariable-specific-impulse rocket. It is valid in the limiting case that the rocket thrust acceleration is much larger than that ofgravity, which is true for certain categories of interplanetary transfers. This equation has been known since at least 1959, but its utility to early-stage mission design and scoping studies may no longer be universally appreciated. The accessible derivation is an elementary application of the Calculus of Variations and does not require detailed knowledge of that field, making it suitable for application by students and for building understanding and physical insight. I show that this equationcan be used to produce approximate designs and scoping studies for crewed missions to Mars and the outer planets, and torobotic flybys of other star systems. Finally I summarize a few other scenarios approximately susceptible to this accessibleapproach: Ascending to orbit from a circular, airless world; starting and ending an interplanetary transfer in a low orbit; and accomplishing higher-energy missions by jettisoning (staging) propulsion units. Keywords: Spacecraft Trajectory Optimization, Variable Specific Impulse, Mars

In the present research, optimal control problem for low thrust spacecraft orbital transfer is investigated by combination of analytical method and Artificial Bee Colony (ABC) algorithm, which is a population-based method. Spacecraft trajectory optimization is defined for orbital transfer from Low Earth Orbit (LEO) to Geosynchronous Earth Orbit (GEO). In the proposed analytical method co-state variables are simplified by employing concepts of optimal control and specific conditions which are considered for the problem, consequently, an adjoint differential equation is added to set of differential equations and these equations are solved simultaneously that result in optimum transfer trajectory. Dynamic modeling of the problem is represented based on Edelbaum equations and trajectory design is carried out considering circular orbits. Optimization problem is defined for different kinds of objective functions such as minimum time, minimum control effort, minimum time with minimum control effort as well as both minimum time and minimum control effort considering different optimal weighted coefficients. These coefficients are searched by means of artificial bee colony algorithm in weighted minimum time minimum effort scheme. Initial values of control variable and terminal time for all defined objective functions are foraged through artificial bee colony algorithm. The rate of changes of the states and control variables are obtained and depicted during orbit transfer. The obtained results demonstrate considerable convergence as well as sufficient accuracy.

According to the disadvantages of indirect method for trajectory optimization, that initial value the for conjugate variables are highly sensitive and can not obtain the global optimal solution, this paper introduced hybrid genetic algorithm to search initial value of conjugate variables, solved equations to obtain the track of Near Spacecraft at unpowered gliding phase. In the optimization process, the author takes the constraints of heating rate, dynamic pressure and overload into consideration, and optimizes the trajectory of under different initial factors. Numerical simulations confirm the applicability of the presented algorithm. Considering practical applications, some typical spacecraft gliding trajectory are optimized. The discussion of the numerical simulations gives some valuable opinions in practical applications.

The direct transcription or collocation method has demonstrated notable success in the solution of trajectory optimization and optimal control problems. This approach combines a sparse nonlinear programming algorithm with a discretization of the trajectory dynamics. A challenging class of optimization problems occurs when the spacecraft trajectories are characterized by thrust levels that are very low relative to the vehicle weight. Low thrust trajectories are demanding because realistic forces due to oblateness, aerodynamic drag, and third-body perturbions often dominate the thrust. Furthermore because the thrust is so low, significant changes to the orbits require very long duration trajectories. When a collocation method is applied to a problem of this type, the resulting nonlinear program is very large because the trajectories are long, and very nonlinear because of the perturbing forces.KeywordsOptimal Control ProblemMission OrbitOrbit TransferDirect TranscriptionPark OrbitThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A EROSPACE engineering applications greatly stimulated the development of optimal control theory during the 1950s and 1960s, where the objective was to drive the system states in such a way that some defined cost was minimized. This turned out to have very useful applications in the design of regulators (where some steady state is to be maintained) and in tracking control strategies (where some predetermined state trajectory is to be followed). Among such applications was the problem of optimal flight trajectories for aircraft and space vehicles. Linear optimal control theory in particular has been very well documented and widely applied, where the plant that is controlled is assumed linear and the feedback controller is constrained to be linear with respect to its input. However, the availability of powerful low-cost microprocessors has spurred great advantages in the theory and applications of nonlinear control. The competitive era of rapid technological change, particularly in aerospace exploration, now demands stringent accuracy and cost requirements in nonlinear control systems. This has motivated the rapid development of nonlinear control theory for application to challenging, complex, dynamical real-world problems, particularly those that bear major practical significance in aerospace, marine, and defense industries. Infinite-time horizon nonlinear optimal control (ITHNOC) presents a viable option for synthesizing stabilizing controllers for nonlinear systems by making a state-input tradeoff, where the objective is to minimize the cost given by a performance index. The original theory of nonlinear optimal control dates from the 1960s. Various theoretical and practical aspects of the problem have been addressed in the literature over the decades since. In particular, the continuous-time nonlinear deterministic optimal control problem associated with autonomous (time-invariant) nonlinear regulator systems that are affine (linear) in the controls has been studied by many authors. The long-established theory of optimal control offers quite mature and well-documented techniques for solving this control-affine nonlinear optimization problem, based on dynamic programming or calculus of variations, but their application is generally a very tedious task. Bellman’s dynamic programming approach reduces to solving a nonlinear first-order partial differential equation (PDE), expressed by the Hamilton–Jacobi–Bellman (HJB) equation. The solution to the HJB equation gives the optimal performance/cost value (or storage) function and determines an optimal control in feedback form under some smoothness assumptions. Alternatively, in the classical calculus of variations, optimal control problems can be characterized locally in terms of the Hamiltonian dynamics arising from Pontryagin’s minimum principle. These are the characteristic equations of the HJB PDE, which result in a nonlinear, constrained two-point boundary value problem (TPBVP) that, in general, can only be solved by successive approximation of the optimal control input using iterative numerical techniques for each set of initial conditions. Numerically, even though the nonlinear TPBVP is somewhat easier to solve than the HJBPDE, control signals can only be determined offline and are thus best suited for feedforward control of plants for which the state trajectories are known a priori. Therefore, contrary to the dynamic programming approach, the resultant control law is not generally in feedback form. Open-loop control, however, is sensitive to random disturbances and requires that the initial state be on the optimal trajectory. In contrast, nonlinear optimal feedback has inherent robustness properties (inherent in the sense that it is obtained by ignoring uncertainty and disturbances). The potential difficulty with the HJB approach is that no efficient algorithm is available to solve the PDE when it is nonlinear and the problem dimension is high, making it impossible to derive exact expressions for optimal controls for most nontrivial problems of interest. The optimal can only be computed in special cases, such as linear dynamics and quadratic cost, or very low-dimensional systems. In particular, if the plant is linear time invariant (LTI) and the (infinite-time) performance index is quadratic, then the corresponding HJB equation for this infamous linear-quadratic regulator (LQR) problem reduces to an algebraic Riccati equation (ARE). Contrary to the well-developed and widely applied theory and computational tools for theRiccati equation (for example, see [1]), theHJB equation is difficult, if not impossible, to solve for most practical applications. The exact solution for the optimal control policies is very complex

Low-thrust spacecraft trajectory optimization for the many-revolution orbital transfer problem is especially challenging due to the high problem dimension and perturbing accelerations that prevent or complicate accurate analytical solutions. This analysis seeks to simplify the calculation of the nonaveraged spiral trajectory through the use of the well-known Q-Law guidance algorithm. Here, Q-Law is used to seed phases of the direct trajectory optimization problem. We explore the trade space between the fast calculation speed of Q-Law and the time or mass optimality of the full optimization problem using the common example of a geostationary transfer orbit to geostationary transfer. We find that a computationally efficient, near-optimal solution can be achieved by using Q-Law for an initial fraction of a spiral trajectory and direct optimization for the remainder of the trajectory. This reduces the direct optimization problem to a lower dimension and enables the spacecraft to reach a specific final state. Q-Law is particularly effective for portions of the trajectory where there are many eclipses, which can challenge direct optimization methods.

A technique of searching for optimal trajectories with gravity assisted maneuvers (GAMs) for interplanetary transfers of spacecraft (SC) with an electric propulsion system (EPS) is proposed. In this case, the indirect optimization method is used. A distinctive feature of this technique is the combination of optimality conditions at the point of GAMs within a single boundary value problem for two cases, when the height of the flyby hyperbola with the GAM is less than or equal to the maximum one. This approach makes it possible to considerably reduce the volume of necessary calculations in optimizing SC interplanetary trajectories that include GAMs. It considers end-to-end trajectory optimization with an analysis of the full set of optimality conditions at the point of the GAM. The efficiency of the proposed approach is demonstrated by the example of optimization of interplanetary trajectories from Earth to Mercury with a GAM in the vicinity of Venus and from Earth to Jupiter with a GAM near Earth.

This paper deals with the trajectory optimization, through a direct method, for the SIMONE mission study. The mission consists of six identical microsatellites, each targeting a rendezvous with a different type of NEO using low-thrust propulsion both for deep space navigation and for Earth escape phase, departing from an Earth orbit. The direct method used consists of a transcription using polynomial base functions in which trajectory and control laws are reduced to a finite set of unknown states and controls, then a parametric optimization is carried out on these unknowns with an NLP technique. Finally, the optimal trajectory is recovered from these unknown states and controls by a polynomial interpolation. The mission has been conceptually divided in two segments inside the Earth sphere of influence, dealing with the perigee rising and with the escape from Earth gravity field through a Moon gravity assist, and an interplanetary segment, dealing with the transfer to each of the six selected targets. Two different cases were analyzed for each trajectory, one which considers the engine characteristics fixed for the entire mission, and the second which instead considers maximum thrust level and specific impulse dependent on the distance from Sun.

A high-precision method of calculating gravitational interactions is applied in order to determine optimal trajectories. A number of problems, necessary for determination of optimal parameters at a launch of a spacecraft and during its flyby near celestial bodies, are considered. The spacecraft trajectory was determined by numerical integration of the equations of passive motion of the spacecraft and of the equations of motion for planets, the Sun, and the Moon. The optimal trajectory of the spacecraft approaching the Sun is determined by fitting its initial conditions.

The Unified State Model is a method for expressing orbits using a set of seven elements. The elements consist of a quaternion and three parameters based on the velocity hodograph. The equations of this model and the background theory necessary to understand them have been shown here. Numerical simulations comparing the Unified State Model with the traditional Cartesian coordinates have been carried out for perturbed orbits, orbits with low-thrust propulsion, and a solar sailing trajectory. The Unified State Model outperforms Cartesian coordinates for all cases in terms of accuracy and computational speed, except for highly eccentric perturbed orbits. The performance of the Unified State Model is exceptionally better for the case of orbits with continuous low-thrust propulsion with CPU simulation time being an order of magnitude lower than for the simulation using Cartesian coordinates. This makes the Unified State Model especially suited for use in trajectory simulators and optimizers.