On the Preliminary Design of Multiple Gravity-Assist Trajectories

Massimiliano Vasile; Paolo DePascale

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On the Preliminary Design of Multiple Gravity-Assist Trajectories

Massimiliano Vasile, Paolo DePascale

arXiv:1105.1822v1May 9, 2011math.OCcs.NEeess.SYphysics.space-ph

TL;DR

Designing multiple gravity-assist trajectories requires balancing model complexity against solution-space richness, because simplified models can omit interesting solutions. The paper develops a compromise trajectory model and a global search approach for automatic preliminary investigation of complex, multimodal domains.

Problem
Preliminary trajectory design must balance model complexity, computational cost, and solution-space richness without excluding potentially interesting solutions.
Method
The paper combines a trajectory model balancing computational cost and solution accuracy with a novel approach to automatically investigate multiple gravity-assist trajectories.
Results
The resulting search domains are typically non-convex and multimodal, while simplified modeling can exclude resonant swing-bys and free orbits before encounters.
Takeaways & Limitations
Effective preliminary investigation requires preserving sufficient search-space richness to identify multiple families of gravity-assist trajectory solutions.
Takeaways & Limitations
An oversimplified trajectory model can lead to the loss of interesting solutions.

Abstract

from arXiv · show

In this paper the preliminary design of multiple gravity-assist trajectories is formulated as a global optimization problem. An analysis of the structure of the solution space reveals a strong multimodality, which is strictly dependent on the complexity of the model. On the other hand it is shown how an oversimplification could prevent finding potentially interesting solutions. A trajectory model, which represents a compromise between model completeness and optimization problem complexity is then presented. The exploration of the resulting solution space is performed through a novel global search approach, which hybridizes an evolutionary based algorithm with a systematic branching strategy. This approach allows an efficient exploration of complex solution domains by automatically balancing local convergence and global search. A number of difficult multiple gravity-assist trajectory design cases demonstrates the effectiveness of the proposed methodology.

II. Trajectory Model Analysis · A. Two- and Three-Impulse Transfers

The trajectory-model analysis frames preliminary design as a trade-off between solution accuracy and computational cost. For two- and three-impulse transfers, adding a deep-space maneuver increases feasible-path diversity, while omitting it excludes potentially interesting solution classes.

II. Trajectory Model Analysis: Oversimplified trajectory models can lose interesting solutions, whereas accurate models require more efficient searches to identify relevant solutions.The analysis emphasizes balancing model completeness against search complexity.
II. Trajectory Model Analysis: The proposed modeling direction seeks a compromise between computational cost and solution accuracy while preserving the richness needed to identify potentially interesting solution families.The model includes full three-dimensional trajectories and gravity-assist maneuvers, deep-space maneuvers, and analytical celestial-body ephemerides.
II. Trajectory Model Analysis: Neglecting deep-space maneuvers prevents resonant or almost resonant swing-bys and free orbits before celestial-body encounters.These solution classes are excluded by the simpler model.
II. Trajectory Model Analysis: Patched-conic modeling decomposes a multiple gravity-assist transfer into finitely many subproblems whose search domains are typically non-convex and multimodal.This decomposition gives each subproblem a finite set of design variables while retaining complex search structure.
A. Two- and Three-Impulse Transfers: A two-impulse direct transfer is parameterized by departure date t0 and time of flight T, with departure and arrival impulses computed by solving Lambert’s problem.The transfer cost is the sum of the departure and arrival ∆v values.
A. Two- and Three-Impulse Transfers: Introducing a deep-space maneuver at M1 adds a velocity discontinuity ∆vs after analytically propagating the initial state for tds=εT, followed by a Lambert transfer to P2.The maneuver extends the direct-transfer model and contributes to total transfer cost.
A. Two- and Three-Impulse Transfers: A direct Earth–Mars transfer solution space contains all solutions with total ∆v lower than 15 km/s, and its structure depends on the bodies’ synodic period and orbital elements.The two-impulse solution-space structure is contrasted with the expanded three-impulse space.
A. Two- and Three-Impulse Transfers: Adding a deep-space maneuver increases the number of feasible paths from P1 to P2, as illustrated by the three-impulse solution-space plot.Figure 2 compares the two-impulse and three-impulse solution spaces.

B. Gravity-Assist Maneuvers

The paper models gravity assists with a linked-conic approximation and an auxiliary rotation angle, then characterizes how maneuver modeling and deep-space corrections shape a highly multimodal solution space. These findings motivate balancing model fidelity and complexity through global search.

B. Gravity-Assist Maneuvers: The linked-conic gravity-assist model treats the maneuver as instantaneous at a collapsed sphere-of-influence point, enforcing equal incoming and outgoing relative-speed magnitudes.The deflection angle depends on pericenter altitude h and incoming relative velocity.
B. Gravity-Assist Maneuvers: An auxiliary rotation angle η defines the otherwise undetermined hyperbolic-plane orientation and enables quaternion-based rotation from incoming to outgoing relative velocity.The orbit-plane normal and rotation angle parameterize the swing-by geometry without introducing nonlinear velocity constraints.
B. Gravity-Assist Maneuvers: The DSM-cost landscape is generally multimodal because η is periodic, and increasing ε significantly increases multimodality across different post-swing-by maneuver times.The landscape changes as the deep-space maneuver is delayed after the swing-by.
B. Gravity-Assist Maneuvers: Because simple models may omit required solutions while complex models make them difficult to find, the paper adopts a compromise between model complexity and fidelity with global search.The search algorithm uses a general-purpose strategy without problem-specific information.

III. Complete Trajectory Model and Problem Formulation

The paper assembles a complete multiple gravity-assist trajectory model from conic-arc phases joined by swing-bys and formulates its design as a box-constrained nonlinear program. It also provides a simplified Lambert alternative and extends the formulation to mixed integer-real optimization when encounter sequences are variable.

III. Complete Trajectory Model and Problem Formulation: The complete model divides a trajectory into phases between planetary encounters, propagates each phase analytically and with Lambert’s algorithm, and joins phases using a swing-by model.Each phase contains two conic arcs separated by a possible deep-space maneuver.
III. Complete Trajectory Model and Problem Formulation: A simplified model replaces the first leg with a Lambert arc without a deep-space maneuver, reducing optimization parameters for direct post-launch planetary transfers.The full model instead permits bounded escape velocity and asymptote direction, including a first-arc deep-space maneuver when required.
III. Complete Trajectory Model and Problem Formulation: The formulation represents multigravity-assist transfer design as a general nonlinear programming problem with simple box constraints.The solution vector includes departure time, phase durations, maneuver parameters, and gravity-assist characteristics.
III. Complete Trajectory Model and Problem Formulation: When planetary encounter sequences are free, the problem becomes mixed integer-real optimization, but encoding all sequence lengths together is discouraged when every sequence matters.The integer variables identify candidate planets, while the authors recommend decoupling combinatorial and real-valued components in that case.

IV. Global Search Through a Hybrid Evolutionary Algorithm

The section frames multiple gravity-assist trajectory optimization as too costly for exhaustive systematic search, especially in high-dimensional spaces. It motivates pruning unpromising regions by blending an evolutionary algorithm with systematic branching.

A. Search and Optimization Issues: The number of design parameters grows as n=4Np-2, typically reaching 18 to 22 for missions with two or three flybys.This dimensionality contributes to the difficulty of systematic exploration.
A. Search and Optimization Issues: 4,194,304 objective-function evaluations would be required for a four-flyby transfer when only two samples are taken per variable.The example concerns a Cassini trajectory and illustrates the cost of systematic discretization.
A. Search and Optimization Issues: A complete systematic search over a black-box real-valued problem would require infinitely many densely distributed samples.Completeness also requires proving that no better solutions exist, making exhaustive search practically impossible without additional landscape information.
A. Search and Optimization Issues: Blending an evolutionary-based algorithm with systematic branching prunes unpromising regions of the solution space.The hybrid strategy is introduced to guide sampling toward regions with higher probability of containing solutions.

B. The Evolutionary-Branching Principle

Evolutionary-Branching combines evolutionary exploration with deterministic domain partitioning to address mixed-integer, multimodal nonlinear programming problems. Its perception mechanism increases each individual’s local search capabilities while branching progressively narrows the search space.

The Evolutionary-Branching Principle: EB uses an evolutionary algorithm to explore the solution space, then branches it into subdomains for repeated searches until good minima, the global one, or infinitesimal partitioning is reached.The deterministic branching step is intended to reduce the search space at every new run because standard evolutionary algorithms require several runs for reliable results.
The Evolutionary-Branching Principle: EB represents individuals as mixed integer-real strings and associates each with a hypercube for local landscape exploration, while continuously regenerating part of the population for global search.The population contains integer values in the first m components and real values in the remaining n-m components.
The Evolutionary-Branching Principle: During evolution, individuals exchange findings and maintain a discoveries-resources balance by adjusting resources according to the number of good findings.Each individual’s resource level is reduced or increased depending on its good findings.
The Evolutionary-Branching Principle: The method introduces perception to increase the local search capabilities of each individual beyond standard evolutionary algorithms.This improvement is described as a novel mechanism called perception.

C. Environment Perception

The perception mechanism samples each migration region to improve individuals by moving them toward better expected resources. It adaptively adjusts sampling effort and migration-region radius based on whether improvements are found.

C. Environment Perception: Perception evaluates each migration region by generating samples and associating a new region with the best discovered value, guiding individuals toward better resources.Sampling begins with mutation, followed by linear extrapolation and second-order interpolation mating.
C. Environment Perception: Sampling repeats until an improvement is found or the resource limit is reached; the resource level increases after improvement and decreases otherwise, bounded from 1 to the number of coordinates.The mechanism therefore allocates more samples after successful searches and fewer after unsuccessful ones.
C. Environment Perception: The migration radius ρ controls contraction or expansion of each region and depends on the perception findings, using distances to region and domain boundaries.When no better sample is found, the radius is reduced according to a rule using ε=0.5 and the normalized distance of the best regional sample.
C. Environment Perception: When differential improvement increases between generations, the migration radius is recomputed by prediction with θ=2 in the implementation.The update uses the preceding radius and generation-level differential improvement.

D. Communication Mechanisms · E. Ranking · F. Branching Step

The method combines evolutionary communication and rank-dependent operators with systematic branching to address stochastic convergence limitations. Branching partitions subdomains, scores nodes using population density and fitness, and prioritizes further exploration.

D. Communication Mechanisms: Improved individuals exchange components or use extrapolation and interpolation operators with randomly selected population members after each full evolution step.
D. Communication Mechanisms: Interpolation also prevents crowding near the same solution by remating the worst individual with subdomain boundaries when reciprocal distance falls below a threshold.
E. Ranking: At each evolutionary step, the population is ranked; the best ne individuals perceive, while lower-ranked individuals are hibernated or mutated with mutation probability increasing downward.
F. Branching Step: Because stochastic evolution does not guarantee convergence, the method systematically decomposes the solution space using the evolutionary algorithm’s output.
F. Branching Step: Each subdomain is branched into 2^n nodes by cutting coordinates through its worst evolutionary individual, with safeguards against cuts too close to boundaries.
F. Branching Step: Nodes are scored from relative population density, volume, and fitness, with weighting factor σ controlling reliance on evolutionary results versus broader exploration.
F. Branching Step: σ=0 prioritizes low-fitness nodes, whereas σ=1 reflects unreliable evolutionary results caused by premature convergence or poor solution-space exploration.
F. Branching Step: The best node pair is selected, a second cut at the best converged individual adds two nodes, and the parent becomes three subdomains ranked for continued exploration.

G. Stopping Criteria · V. Case Studies

The method uses heuristic stopping criteria and optional SQP refinement, while acknowledging the absence of a rigorous global-convergence proof. IMAGO tests the resulting EPIC-based design tool on several challenging interplanetary mission cases.

G. Stopping Criteria: The algorithm stops based on function-evaluation limits, unproductive branching, branching level, or evolutionary-filter convergence.
G. Stopping Criteria: These stopping rules are heuristic rather than supported by a rigorous proof of global convergence.
G. Stopping Criteria: The branching scheme is designed to asymptotically partition the domain into infinitesimal subdomains, supporting global convergence in principle.
G. Stopping Criteria: Solution accuracy is enhanced by launching SQP local searches from each best solution identified by the evolutionary-branching algorithm.
G. Stopping Criteria: For direct transfers or single flybys, SQP typically cannot improve the evolutionary-branching result because local convergence has already been reached.
V. Case Studies: IMAGO is a Matlab preliminary-design tool combining the trajectory model with the evolutionary-branching algorithm EPIC.
V. Case Studies: The tool is evaluated on Jupiter transfer options, Cassini, NEO interception, and Rosetta mission design cases using a 2GHz Intel Centrino processor with Windows XP.

A. Missions to Jupiter

For Jupiter transfers, IMAGO identified both established and unconventional gravity-assist trajectories. In fixed-sequence EVEEJ tests, it found solutions that improved on literature references in total ∆v and transfer time.

A. Missions to Jupiter: The broader free-sequence run generated about 1000 solutions with different planet combinations and launch dates after eight hours of computation.The run used a population of 40 individuals, a filter dimension of 20, and four branching levels.
A. Missions to Jupiter: The free-sequence search identified typical Jupiter mission sequences such as EVVEJ and EVEEJ, with promising escape and arrival velocity solutions.The algorithm was run three times and produced alternative valid trajectories to those reported in prior work.
A. Missions to Jupiter: The search also discovered an EMMJ Mars-resonant transfer, despite lacking prescriptions or a dedicated model for generating resonant swing-by solutions.The trajectory requires a large deep space correction but could be implemented with low-thrust propulsion.

B. Cassini Trajectory Design

The Cassini case tests IMAGO on a highly complex outer-planet multiple-gravity-assist trajectory under limited preliminary information and stringent mission constraints. IMAGO identifies diverse low-∆v solutions and outperforms the simple multistart approach.

B. Cassini Trajectory Design: The design allowed broad transfer-time bounds because preliminary information on good options was lacking, while arrival velocity and launch escape velocity remained constrained.The launch escape velocity was limited below 4 km/s, and a deep-space maneuver could occur up to 90% of each planet’s transfer time.
B. Cassini Trajectory Design: The preliminary search minimized Eq. (13) using 40 individuals, a filter dimension of 25, and two branching levels.The reported search generated around 200 solutions after a little bit more than 800,000 function evaluations.
B. Cassini Trajectory Design: Most local minima clustered around -800 MJD, the optimal launch window, with total ∆v approximately 10 km/s.The approximately 10 km/s total ∆v matches Cassini’s total ∆v.
B. Cassini Trajectory Design: The first IMAGO solution resembled the reference in launch dates and total ∆v, while the second reduced arrival velocity at the cost of higher transfer time.These comparisons are reported for two of the best IMAGO solutions against the reference Cassini mission solution.
B. Cassini Trajectory Design: IMAGO used a little bit more than 800,000 evaluations and produced around 200 solutions, while multistart used about 1,600,000 evaluations and found fewer good solutions; its best solutions were worse.The results show that a simple random approach with simple heuristics cannot efficiently solve this problem.

C. Missions to Asteroid (10302) -1989ML · D. Rosetta Trajectory Design

The asteroid mission analysis identifies recurrent 180-day Earth-to-Earth flyby trajectories in two transfer families. The Rosetta study uses broad preliminary searches and refined exploration to characterize solution groups and obtain varied solutions comparable with the reference mission.

C. Missions to Asteroid (10302) -1989ML: The asteroid reference sequence was EEV-asteroid, with a first Earth swing-by after 180 days to separate spacecraft launched together.Earth-to-Earth transfer times ranged from 80 to 400 days, allowing investigation of alternative options including a one-to-one resonant Earth flyby.
C. Missions to Asteroid (10302) -1989ML: The asteroid study retained trajectories with arrival velocity greater than 10 km/s while searching a 2011 launch window.The optimization used an evolutionary population of 30 individuals and a maximum of 100,000 function evaluations.
D. Rosetta Trajectory Design: Rosetta was redesigned after the failed 2003 launch, requiring new characterization of trajectory opportunities and a new target with limited prior knowledge.The adopted sequence was EEMEE-comet, matching the solution used for the new Rosetta mission.
D. Rosetta Trajectory Design: Using two branching levels and up to about 800,000 total evaluations, the first Rosetta search extensively characterized possible solutions and local optima.The search used a population of 40 individuals and up to 200,000 evaluations per evolutionary step.
D. Rosetta Trajectory Design: Rosetta’s extensive search found two groups of good solutions near launch dates around 1455 and 1730 MJD, with total ∆v of about 2–2.5 km/s.Less-crowded alternatives also appeared around 1530 MJD, motivating reduced search bounds for further investigation.

VI. Conclusions

The paper presents a general global optimization methodology for preliminary multiple gravity-assist trajectory design, combining an evolutionary algorithm with deterministic branching. Tests show it handles highly multimodal problems effectively, reconstructs reference trajectories accurately, and supports future search-space reduction.

VI. Conclusions: A simple trajectory model balances preliminary-design accuracy with sufficient completeness to preserve important classes of possible solutions.The underlying search spaces are generally multimodal and non-convex as a consequence of model complexity.
VI. Conclusions: The proposed approach blends an evolutionary algorithm with deterministic branching to partition and explore complex solution spaces.It is presented as a general, problem-independent global optimization approach.
VI. Conclusions: The method correctly reconstructed the nominal Cassini trajectory from very large design-parameter bounds and found cheaper options in total ∆v.Rosetta and asteroid-impact cases produced diverse trajectories with total ∆v comparable to reference solutions.
VI. Conclusions: Free-sequence searches produced diverse solution families, including unexpected cases, while Cassini and Rosetta comparisons showed high accuracy in the main trajectory features.The reference solutions represent the optimal trajectories actually flown.
VI. Conclusions: The methodology effectively tackles highly multimodal multigravity-assist transfers without becoming trapped in local minima, while maintaining solution quality and low computational time.These properties make the approach appealing for preliminary design tools.
VI. Conclusions: Future work will improve search by incrementally solving sub-problems and pruning the search domain from early trajectory-development stages.The proposed reduction is intended for long gravity-assist sequences involving up to 15 or more celestial bodies.