Trajectory optimization

Draft

Mission design past the two-impulse case is an optimization problem: find the control history that reaches the target while minimising propellant, subject to constraints on launch date, arrival conditions, thrust and time of flight. This page covers the standard formulations and how they are actually solved.

Lambert’s problem

Given two position vectors and the time of flight between them, find the connecting conic. This is the workhorse of impulsive mission design, and it has been studied since Lambert stated it in 1761.

Lambert’s theorem says the transfer time depends only on the semi-major axis, the chord length between the two points, and the sum of the two radii, and not otherwise on the geometry. That reduces the problem to a single scalar root-find.

Solution count: For a given transfer angle and time, there are two solutions, short-way and long-way, plus multiple-revolution families for longer times.
Singularities: A 180-degree transfer angle is degenerate: the plane is undefined and any inclination is a solution.
Standard algorithms: Gauss for short transfer angles, universal variables for general robustness, Izzo for speed and reliability across all cases.
Use: Wrap it in a two-dimensional sweep over departure and arrival dates and you have a porkchop plot.

Porkchop plots

Sweep departure date on one axis and arrival date on the other, solve Lambert at every grid point, and contour the resulting departure energy $C_3$ . The contours form the closed lobes that give the plot its name.

The plot answers the questions a mission needs answered: when does a window open, how long does it stay open, what launch energy is required, and what does each day of slip cost. Arrival $v_\infty$ is usually contoured on the same axes, because the launch and arrival costs trade against each other.

Windows recur at the synodic period, the interval between successive identical alignments:

S = \frac{1}{\left|1/T_1 - 1/T_2\right|}

Target	Synodic period	Window cadence
Mars	2.135 years	Every 780 days
Venus	1.599 years	Every 584 days
Jupiter	1.092 years	Every 399 days

Mars is the constraining case in practice. A slipped Mars window costs 26 months, which is longer than many programmes can absorb, and it is why Mars launch campaigns are unusually schedule-driven.

Not all windows are equal. Because Mars has an eccentric orbit, the required $C_3$ varies substantially between windows on roughly a 15-year cycle, so some opportunities carry appreciably more payload than others.

Gravity assists

A close pass by a body rotates the spacecraft’s velocity relative to that body without changing its magnitude. In the heliocentric frame, that rotation changes the speed, because the body’s own orbital velocity is added back in.

The turn angle follows from hyperbolic geometry:

Gravity assist turn angle

\sin\frac{\delta}{2} = \frac{1}{1 + r_p v_\infty^2/\mu}

The velocity change achievable in the heliocentric frame is

\Delta v = 2v_\infty\sin\frac{\delta}{2}

Two consequences follow. A slow approach turns more sharply, so a low $v_\infty$ gives a larger turn but a smaller lever arm. And a close periapsis turns more sharply, so the achievable assist is bounded by the planet’s radius and atmosphere.

Energy is conserved overall: the spacecraft gains what the planet loses, in a ratio so extreme that the planet’s orbit is unmeasurably affected.

Gravity assists are what make the outer solar system reachable at all. Voyager 2 used Jupiter, Saturn and Uranus in sequence. Cassini flew Venus-Venus-Earth-Jupiter over nearly seven years. Parker Solar Probe uses seven Venus flybys to shed the angular momentum needed to approach the Sun, which is the harder direction: getting close to the Sun costs far more than escaping it.

Patched conics

The standard decomposition treats the trajectory as a sequence of two-body problems: heliocentric between planets, planetocentric inside each sphere of influence, with the state rotated and re-referenced at the boundary. See Gravitation and the two-body problem.

The method is approximate, since the boundary is a modelling construct rather than a physical surface, and it produces a small discontinuity. That is acceptable because patched conics are used for preliminary design, to find the structure of a solution, which is then handed to a numerical optimiser working with the full force model.

Direct and indirect methods

Once the problem is genuinely an optimal control problem, there are two families, and the split is fundamental.

Indirect methods apply Pontryagin’s maximum principle, introducing costate variables and deriving necessary conditions for optimality. The result is a two-point boundary value problem.

Direct methods discretise the trajectory and the control, converting the problem into a large but finite nonlinear program.

	Indirect	Direct
Formulation	Costates, maximum principle, TPBVP	Discretise, then solve an NLP
Optimality	Satisfies necessary conditions exactly	Optimal only to the discretisation
Convergence basin	Very small; costates have no physical meaning to guess	Large and forgiving
Constraints	Awkward to add; each one changes the derivation	Natural; just more NLP constraints
Problem size	Small	Large, thousands to millions of variables
Used for	Low-thrust problems where optimality must be provable	Nearly all operational work

The convergence problem with indirect methods is the practical dealbreaker. Initial costate guesses are unguessable, since they are Lagrange multipliers with no physical interpretation, and the sensitivity is extreme. Direct methods dominate operational use for this reason alone.

Within direct methods:

Single shooting: Integrate from the initial state, vary the controls to hit the target. Simple, but sensitive over long arcs.
Multiple shooting: Break the trajectory into segments with continuity constraints. Far better conditioned.
Collocation: Represent the state as polynomials and enforce the dynamics at collocation points. Hermite-Simpson is the common choice.
Pseudospectral: Global orthogonal polynomials with nodes at Gauss quadrature points. Spectral convergence for smooth problems.
Sims-Flanagan: Model low thrust as a sequence of small impulses. Fast enough for broad searches over launch dates and flyby sequences.

Low-thrust trajectories

Electric propulsion turns mission design from a small number of impulses into a continuous control problem over months, and the character changes completely.

The classical closed-form result is Edelbaum’s, for circular-to-circular transfer with a simultaneous plane change:

\Delta v = \sqrt{v_1^2 + v_2^2 - 2v_1v_2\cos\left(\frac{\pi}{2}\Delta i\right)}

For anything more complex, two approaches are used in practice. Q-law and similar Lyapunov feedback controllers steer the osculating elements toward the target with a control law rather than an optimisation, which is fast and robust but suboptimal, and is often used to seed a proper optimiser. Full optimal-control solutions use direct collocation or pseudospectral methods on the modified equinoctial elements, which stay well scaled and non-singular throughout.

The additional constraints that make low-thrust problems hard are physical rather than mathematical: available power falls as the inverse square of solar distance, thrusters have finite throughput and duty cycles, eclipses interrupt thrust, and the spacecraft may need to point its arrays and its thruster in incompatible directions.

Toolchains

Tool	Origin	Character
GMAT	NASA, open source	General mission analysis, scripting, optimisation
Copernicus	NASA JSC	Trajectory design and optimisation, human exploration heritage
STK Astrogator	Ansys, commercial	Interactive mission design within a broader analysis suite
Orekit	Open source, Java	Flight-dynamics library, well validated
pykep and PyGMO	ESA, open source	Global optimisation over flyby sequences and launch dates
poliastro	Open source, Python	Teaching and preliminary analysis
Nyx	Open source, Rust	Modern flight dynamics and orbit determination

References

Battin, R. H. An Introduction to the Mathematics and Methods of Astrodynamics, revised ed., AIAA, 1999, on Lambert’s problem.
Izzo, D. “Revisiting Lambert’s problem”, Celestial Mechanics and Dynamical Astronomy, 121:1, 2015.
Betts, J. T. Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, 2nd ed., SIAM, 2010.
Conway, B. A. (ed.) Spacecraft Trajectory Optimization, Cambridge University Press, 2010.
Petropoulos, A. E. “Low-Thrust Orbit Transfers Using Candidate Lyapunov Functions with a Mechanism for Coasting”, AIAA 2004-5089. The Q-law.
Sims, J. A. and Flanagan, S. N. “Preliminary Design of Low-Thrust Interplanetary Missions”, AAS 99-338.