Interplanetary Trajectories Research Articles

Lambert-Free Solution of Multiple-Gravity-Assist Optimization Problem

A new method to find optimized multiple-gravity-assist (MGA) interplanetary trajectories that does not require the solution of Lambert’s problem is presented. The method, applicable to MGA sequences with ballistic transfer arcs and flybys, exploits Godal’s hyperbolic locus of incoming and outgoing velocities connecting the different trajectory legs combined with the v∞ sphere flyby constraint. The intersection between the two loci, when it exists, can be used to find feasible candidate solutions for each interplanetary trajectory segment and can be obtained analytically after solving a quartic equation. A time-of-flight compatibility condition can then be introduced that replaces the classical C3-matching constraint for locating properly phased transfer arcs, whereby opening the door to a very significant improvement in computational efficiency. The implementation of the method and its applicability to interplanetary trajectory optimization and moon tours are discussed in detail using representative test cases.

An approach for end-to-end optimization of low-thrust interplanetary trajectories using collinear libration points

This article presents an approach to solve the problem of end-to-end optimization of the low-thrust interplanetary trajectory. At first, the problem of optimizing low-thrust interplanetary trajectory, which passes through the collinear libration points near the planets of departure and arrival is considered. The obtained solutions from this problem can be used as an initial guess to solve the end-to-end optimization problem, i.e., to calculate interplanetary low-thrust trajectories that satisfy the necessary optimality conditions at the match points of the heliocentric and planetocentric segments of trajectory. The minimum-fuel problem is considered. An indirect method based on the maximum principle, and the continuation method is applied to optimize interplanetary spacecraft trajectories. Numerical examples of trajectories from the Earth orbit to the orbit around the destination planet (Mars and Jupiter) are given. The possibility of a significant reduction in the required characteristic velocity is shown in comparison with the estimates obtained by using the zero-radius sphere of influence model.

Modelization of Low-Cost Maneuvers for an Areostationary Preliminary Mission Design

The aim of this paper is to analyze the determination of interplanetary trajectories from Earth to Mars to evaluate the cost of the required impulse magnitudes for an areostationary orbiter mission design. Such analysis is first conducted by solving the Lambert orbital boundary value problem and studying the launch and arrival conditions for various date combinations. Then, genetic algorithms are applied to investigate the minimum-energy transfer orbit. Afterwards, an iterative procedure is used to determine the heliocentric elliptic transfer orbit that matches at the entry point of Mars’s sphere of influence with an areocentric hyperbolic orbit imposing specific conditions on inclination and periapsis radius. Finally, the maneuvers needed to obtain an areostationary orbit are numerically computed for different objective condition values at the Mars entry point to evaluate an areostationary preliminary mission cost for further study and characterization. Results show that, for the dates of the minimum-energy Earth–Mars transfer trajectory, a low value for the maneuvers to achieve an areostationary orbit is obtained for an arrival hyperbola with the minimum possible inclination and a capture into an elliptical trajectory with a low periapsis radius and an apoapsis at the stationary orbit. For a 2026 mission with a TOF of 304 for the minimum-energy Earth–Mars transfer trajectory, for a capture with a periapsis of 300 km above the Mars surface the value achieved will be 2.083 km/s.

Robust interplanetary trajectory design under multiple uncertainties via meta-reinforcement learning

This paper focuses on the application of meta-reinforcement learning to the robust design of low-thrust interplanetary trajectories in the presence of multiple uncertainties. A closed-loop control policy is used to optimally steer the spacecraft to a final target state despite the considered perturbations. The control policy is approximated by a deep recurrent neural network, trained by policy-gradient reinforcement learning on a collection of environments featuring mixed sources of uncertainty, namely dynamic uncertainty and control execution errors. The recurrent network is able to build an internal representation of the distribution of environments, thus better adapting the control to the different stochastic scenarios. The results in terms of optimality, constraint handling, and robustness on a fuel-optimal low-thrust transfer between Earth and Mars are compared with those obtained via a traditional reinforcement learning approach based on a feed-forward neural network.

Designing Low-Energy Low-Thrust Flight to the Moon on a Temporary Capture Trajectory

The study considers the problem of calculating the low-energy trajectories of a low-thrust spacecraft to the Moon during the ballistic capture. The transfer is carried out using a transit trajectory in the vicinity of one of the collinear libration points L1 or L2 of the Earth-Moon system. Using a transit trajectory allows us to reduce fuel consumptions for the transfer by applying spacecraft dynamic in the Earth-Moon system. After exit from the orbit of ballistic capture, depending on the goal of mission the required lunar orbit can be formed, or the maneuver can be completed for inserting into the required interplanetary trajectory. A method for solving the problem is proposed, which consists in selecting the suitable transit trajectory to ensure sufficiently long duration of staying a spacecraft in the sphere of influence of the Moon, and in calculating the optimal low-thrust trajectories from initial lunar orbit to the transit trajectory to the Moon. To solve the problem of optimal control and calculate the optimal exit point to the transit trajectory, the Pontryagin’s maximum principle is used in combination with the continuation method by parameter. Numerical examples are given for calculating low-energy trajectories to the Moon during the ballistic capture with the optimization of exit point to the transit trajectory.

Optimization of a Low-Thrust Heliocentric Trajectory between the Collinear Libration Points of Different Planets

The aim of this study is to optimize a low-thrust interplanetary trajectory using collinear libration points L1 and L2 as the junction points of the geocentric or planetocentric segments of the trajectory with the heliocentric segment. The problem of optimizing the heliocentric segment of perturbed low-thrust interplanetary transfer is considered in the four-body ephemeris model, which includes the Sun, Earth, target planet, and spacecraft. To optimize the trajectories, an indirect approach is used based on Pontryagin’s maximum principles and the continuation method. The possibility of reducing the characteristic velocity in comparison with the estimates obtained through the method of zero sphere of influence is shown.

Reinforcement learning-based hybrid differential evolution for global optimization of interplanetary trajectory design

The global optimization and design of interplanetary trajectories is one of the most important tasks in deep space exploration. Its search space is often characterized by multi-constraints, extreme non-linearity and sensitivity to initial conditions. To cope with these difficulties, an improved reinforcement learning-based hybrid differential evolution (DE) named RL_HDE is proposed in this paper. In RL_HDE, a novel multi-mutation strategy LSHADE based on an adaptive Q-Learning framework is proposed for global exploration. To further balance the global exploration and local exploitation of RL_HDE, a new parameter adaptive strategy based on Q-Learning is designed to control two trigger parameters. The performance of RL_HDE is verified by the well-known Global Trajectory Optimization Problems (GTOP), which are developed by Advanced Concepts Team of European Space Agency (ESA-ACT). A comparison of RL_HDE with three sets of algorithms, including ten state-of-the-art hybrid evolutionary algorithms, four interplanetary trajectory design algorithms, and seven reinforcement learning-based hybridizations. Experimental statistical results demonstrate that RL_HDE outperforms other competitors in terms of convergence efficiency and accuracy. RL_HDE has better performance for solving complex interplanetary trajectory design problems such as Cassini2 and Messenger-full.

Indirect Optimization of Optimal Continuous Trajectory for Electric Sails With Refined Thrust Model in the Multi-Target Exploration Scenario

Electric sail-based propulsion is an innovative propellant-less propulsion technology that generates continuous thrust through the interaction between an artificial electric field and the solar wind. In an electric sail, the propulsive acceleration is adjusted by controlling the attitude of its normal plane and the coefficient determining the maximum thrust. However, the attitude adjustment speed of the electric sail is relatively small and the direction of the normal vector is constrained. Consequently, the electric sail is required to maintain a continuous propulsive acceleration vector when flying by the intermediate targets in multi-target interplanetary exploration. Therefore, an indirect optimization of three-dimensional optimal continuous interplanetary trajectory for electric sails with refined thrust model is investigated in this study. First, the optimal propulsive angles and thrust adjustment coefficient of electric sails with a refined thrust model are derived using Pontryagin's minimum principle. Second, a homotopy function is introduced in the process of trajectory optimization with an indirect method to approximate the step of the thrust adjustment coefficient to improve the accuracy of numerical integration. Additionally, the initial costates of the electric sail are transformed from the numerical simulation results of the Bezier shaping approach (BSA) and integrative Bezier shaping approach (IBSA) using the Karush-Kuhn-Tucker (KKT) condition. The numerical simulation results for the Earth-Mars rendezvous mission and the multi-target flyby mission reveal that the initial values of the costates are effective to implement an indirect optimization process of the optimal continuous trajectory for the rapid convergence of the electric sail. According to the numerical simulation results for multi-target mission, the indirect method is on average 2.5% better than the Gauss pseudospectral method (GPM) in overall index, and the calculation time is only 1.06% of GPM.

Optimization of Interplanetary Trajectory for Direct Fusion Drive Spacecraft

The Direct Fusion Drive (DFD) technology, which is being developed at present, will allow fast and affordable interplanetary travel. This is a result of the very high specific impulse and the low specific mass of DFD thrusters which outperform more conventional Nuclear Electric Propulsion (NEP), with which it shares the ability of providing a low (albeit higher than in the case of NEP) continuous thrust. It is well known that, to optimize the payload fraction, the thruster should operate in Variable Exhaust Velocity (VEV) mode and that the lower is the specific mass, the higher should be the maximum specific impulse the thruster can produce. A low thrust interplanetary travel, from the orbit around the starting planet to the orbit around the destination planet, can be considered as made of three parts: a first planetocentric phase, a second heliocentric phase and finally a third planetocentric phase; in all of them the trajectory is a sort of a spiral, but while in the first and third the spacecraft makes several (or even a large number) turns about the two planets, the second consists of a fraction of a turn about the Sun. In the first and last one the optimal specific impulse is not much variable and should remain quite low, while in the second one it must go through large variations, reaching a very high value at roughly midway between the planets. To show the potentialities of DFD, three typical fast missions are studied: to the Moon, to Mars and to Titan, showing that this propulsion device will allow humans to reach practically the whole solar system in a reasonable time. Keywords: Interplanetary travel, Human Mars Exploration, Direct Fusion Drive, Trajectory Optimization, Specific Impulse Optimization

Optimization of Interplanetary Trajectory for Direct Fusion Drive Spacecraft

The Direct Fusion Drive (DFD) technology, which is being developed at present, will allow fast and affordable interplanetary travel. This is a result of the very high specific impulse and the low specific mass of DFD thrusters which outperform more conventional Nuclear Electric Propulsion (NEP), with which it shares the ability of providing a low (albeit higher than in the case of NEP) continuous thrust. It is well known that, to optimize the payload fraction, the thruster should operate in Variable Exhaust Velocity (VEV) mode and that the lower is the specific mass, the higher should be the maximum specific impulse the thruster can produce. A low thrust interplanetary travel, from the orbit around the starting planet to the orbit around the destination planet, can be considered as made of three parts: a first planetocentric phase, a second heliocentric phase and finally a third planetocentric phase; in all of them the trajectory is a sort of a spiral, but while in the first and third the spacecraft makes several (or even a large number) turns about the two planets, the second consists of a fraction of a turn about the Sun. In the first and last one the optimal specific impulse is not much variable and should remain quite low, while in the second one it must go through large variations, reaching a very high value at roughly midway between the planets. To show the potentialities of DFD, three typical fast missions are studied: to the Moon, to Mars and to Titan, showing that this propulsion device will allow humans to reach practically the whole solar system in a reasonable time. Keywords: Interplanetary travel, Human Mars Exploration, Direct Fusion Drive, Trajectory Optimization, Specific Impulse Optimization

Optimal interplanetary trajectories for Sun-facing ideal diffractive sails

A diffractive sail is a solar sail whose exposed surface is covered by an advanced diffractive metamaterial film with engineered optical properties. This study examines the optimal performance of a diffractive solar sail with a Sun-facing attitude in a typical orbit-to-orbit heliocentric transfer. A Sun-facing attitude, which can be passively maintained through the suitable design of the sail shape, is obtained when the sail nominal plane is perpendicular to the Sun–spacecraft line. Unlike an ideal reflective sail, a Sun-facing diffractive sail generates a large transverse thrust component that can be effectively exploited to change the orbital angular momentum. Using a recent thrust model, this study determines the optimal control law of a Sun-facing ideal diffractive sail and simulates the minimum transfer times for a set of interplanetary mission scenarios. It also quantifies the performance difference between Sun-facing diffractive sail and reflective sail. A case study presents the results of a potential mission to the asteroid 16 Psyche.

Generation of Multiple-Revolution Many-Impulse Optimal Spacecraft Maneuvers

Impulsive trajectories provide time- and -reachability insights. Two novel homotopy-based methods are proposed for generating optimal many-impulse, multirevolution maneuvers. The first method is based on the continuation over the specific impulse value, which is shown to enhance convergence performance of the resulting two-point boundary-value problems. The second method is based on the formulation of impulsive trajectories using a linear acceleration term. The two methods are used in a hybrid framework. The utility of the proposed methods is demonstrated on four problems: Two interplanetary trajectories 1) from Earth to Mars, 2) from Earth to asteroid Dionysus, 3) a planet-centric transfer maneuver from a geostationary transfer orbit (GTO) to geostationary orbit (GEO) consisting of 21 revolutions, and 4) a 50-revolution transfer from the considered GTO to a circular orbit at the first Lagrange point (L1) of the Earth–moon system. The last two problems leading to 18 and 50 impulses are tackled to give an optimal near-impulsive solution. The impulsive solution with 18 impulses is shown to satisfy Lawden’s conditions.

OPTIMAL TWO-IMPULSE INTERPLANETARY TRAJECTORIES: EARTH-VENUS AND EARTH-MARS MISSIONS

This work describes several models to design optimal interplanetary trajectories. The transfer problem consists in transferring a space vehicle from a circular low Earth orbit (LEO) to a circular low orbit around a destiny planet (Venus or Mars). Models based on the two-body, four-body, and five-body problems are considered. Also, several versions of the patched-conic approximation are utilized including a detailed version that designs a lunar swing-by maneuver. The results show that the optimal trajectories for Earth-Mars and Earth-Venus missions collide with the Moon if a lunar swing-by maneuver with an unspecified altitude of the closest approach is included in the trajectory design; however, sub-optimal trajectories that do not collide with the Moon exist, presenting a smaller fuel consumption than the trajectories without lunar swing-by and with no greater changes in the time of flight.

A method for constructing an interplanetary trajectory of a spacecraft to Venus using resonant orbits to ensure landing in the desired region

A problem of constructing the trajectory of a spacecraft flight to Venus within the framework of a mission including landing of a lander in a given region of the planet's surface is being considered. A new celestial mechanics related method based on the use of gravity assist maneuver near Venus is proposed to transfer the spacecraft to a heliocentric orbit resonant with the orbit of Venus, so that, at the next approach the planet, the given region of the surface becomes attainable for landing. It is shown that the best resonant orbit in terms of the cost of the characteristic velocity is an orbit with a 1:1 ratio of the period to the orbital period of Venus. A procedure for choosing one of possible resonant orbits depending on coordinates of the desired landing point on the surface and the launch date of the mission is described. An example of calculating the flight trajectory that ensures landing in the Vellamo-South region of the Venus surface at launch from the Earth in 2031 is considered.

A dynamical definition of the sphere of influence of the Earth

The concept of sphere of influence of a planet is useful in both the context of impact monitoring of asteroids with the Earth and of the design of interplanetary trajectories for spacecraft. After reviewing the classical results, we propose a new definition of the sphere which is suitable for the classical patched-conic method: the new definition depends on the position and velocity of the small body for given values C of the Jacobi constant. Here, we compare the orbit of the small body, obtained in the framework of the circular restricted three-body problem, with orbits computed by patching two-body solutions. Our definition is based on an optimisation process, minimising a suitable target function with respect to the assumed radius of the sphere of influence. For different values C we represent the results in the planar case: we show the values of the selected radius as a function of two angles characterising the orbit. In this case, we also produce a database of radii of the sphere of influence for several initial conditions, allowing an interpolation.

Line Sampling Procedure for Extensive Planetary Protection Analysis

Exploration missions to other planets have to satisfy planetary protection requirements to limit the probability of impacts between mission-related objects and celestial bodies, with the goal of reducing the risk of contaminating them with biological material coming from Earth. The verification of these requirements can become a lengthy and computationally expensive task when addressed with common methods such as Monte Carlo simulations, as they involve analysing the interplanetary trajectories and the uncertainties associated with them for time spans up to 100 years, and estimating small probabilities with strict confidence levels. This paper presents novel improvements of the line sampling method, already introduced for the verification of planetary protection requirements as a way to estimate the impact probabilities more efficiently and with greater accuracy than achieved with standard Monte Carlo. These newly developed techniques are presented, with the aim of making the analysis with Line Sampling more effective, and providing more information about the distribution of impacts in the initial uncertainty distribution: an algorithm to identify the time intervals where most close approaches are clustered, and an algorithm to improve the determination of the main sampling direction and increase the accuracy of the probability estimation.

A Flagship-class Uranus Orbiter and Probe mission concept using aerocapture

The conceptual design for a Flagship-class Uranus Orbiter and Probe (UOP) mission using aerocapture is presented. Uranus is historically the least studied destination for aerocapture, primarily attributed to the lack of an engineering-level atmosphere model until UranusGRAM was released by NASA in 2021. The present study is the first detailed end-to-end study of a Uranus aerocapture mission concept taking into account constraints arising from launch vehicle performance, interplanetary trajectory, aerocapture vehicle design, thermal protection system, and probe delivery. The mission concept uses a Falcon Heavy Expendable launcher and a high-energy, fast-arrival V∞ Earth–Earth–Jupiter–Uranus (EEJU) gravity assist trajectory to deliver a 1400 kg orbiter and a 300 kg entry probe to Uranus. The aerocapture vehicle is a derivative of the Mars Science Laboratory entry system with extensive flight heritage, and uses the state-of-the-art HEEET thermal protection system. Compared to the current baseline UOP mission using conventional propulsive orbit insertion with a 13-year flight time and 5-year orbital mission at Uranus, the proposed 8-year aerocapture mission concept enables a 10-year orbital mission at Uranus within the budgetary and schedule constraints of a Flagship-class mission.

Double Tisserand graphs for low-energy lunar transfer design

Tisserand graphs are a widespread tool for interplanetary trajectory and Moon tour design. They are based on the Jacobi constant being an integral of motion in the Circular Restricted Three Body Problem (CR3BP); as such, the classical Tisserand graph does not include the perturbation of bodies other than the flyby bodies. Low-energy transfers in the Earth-Moon system make use of the combined Earth, Moon, and Sun gravities, exploiting the third body perturbations to reach the Moon with a reduced transfer Δv. The paper, therefore, proposes a novel double Tisserand graph, where the level lines of both the Earth-Moon and the Sun-Earth CR3BPs are superimposed, portraying the complex 4-body dynamics into a single plot. Paths along such level lines correspond to trajectories utilizing the dynamical effect of the Moon or the Sun or a combination of both. We show how such a graph can be efficiently used for preliminary design of Weak Stability Boundary transfers, lunar resonance transfers, lunar flybys, weak lunar captures or any combination of them.

Expanding Interplanetary Transfer Opportunities from Geostationary Transfer Orbits via Earth Synchronous Orbits

Geostationary transfer orbits (GTOs) are geocentric orbits widely considered for kick-stage operations of satellites for deep space missions. A piggyback spacecraft departing from GTOs requires a low transfer cost. The type of GTO and launch timings of piggyback missions, on the other hand, are typically determined by the primary mission that carried the piggyback spacecraft. This research introduces a new transfer strategy that coordinates piggyback spacecraft’s departure timing from Earth to deep space via an Earth synchronous orbit (ESO) after departure from a GTO. It also enables the spacecraft to change its velocity direction by introducing Earth gravity assists. Connecting several GTOs and interplanetary trajectories via ESOs reduces the required for the transfers. It leads to using transfer opportunities previously considered unsuitable for missions or miniaturizing the kick motor. As a result, ESOs allow for a broader launch opportunity for deep space piggyback probes. In addition, the feasibility of ESOs is shown numerically by comparing direct and ESO-assisted transfers from GTOs to Mars with the required .

Rapid optimization of continuous trajectory for multi-target exploration propelled by electric sails

As an innovative spacecraft propellantless propulsion technology, the electric sail can generate electric fields that interact with the solar wind, generating continuous propulsive thrust. However, unlike conventional electric thrusters, the thrust vector of the electric sail is constrained, and the attitude cannot be adjusted quickly. Consequently, the position and velocity of each flight trajectory at the connecting node and the direction of propulsion acceleration must be continuous. In this study, the continuous trajectory optimization of the electric sail–based probe in multi-target exploration is investigated. An integrative Bezier shaping approach (IBSA) is proposed to realize the rapid optimization of a multi-target exploration trajectory with continuous acceleration. The relationship between boundary constraints, intermediate constraints, and Bezier basis function coefficients is derived, transforming the original continuous optimal control problem of multiple stages into a problem of nonlinear programming that naturally satisfies the boundary and intermediate constraints. Numerical simulation results demonstrate that the IBSA can obtain a better interplanetary trajectory with a shorter calculation time than that obtained with the traditional Bezier shaping approach using piecewise optimization. Furthermore, the simulation results are successfully applied to the initial value estimation of the Gauss pseudospectral method (GPM). Compared with the GPM, the proposed shaping approach differs from the objective function by approximately 2%–6%, but requires approximately 1.5% of the computational time. During the preliminary mission design stage, the IBSA can be used for rapid feasibility assessments of different electric sail–based probe mission profiles.