⚡ electrical engineering

Polynomial-Based Solutions to Targeting Problems for Onboard Applications

This paper proposes a polynomial-based targeting framework leveraging differential algebra and moment-sum-of-squares optimization to achieve globally optimal, accurate, and reliable solutions for both impulsive and continuous low-thrust spacecraft maneuvers, making it particularly suitable for autonomous onboard applications where traditional methods struggle with convergence and large nonlinearities.

Original authors: Adam Evans, Alberto Fossa, Roberto Armellin, Didier Henrion, Renato Zanetti

Published 2026-07-20

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Adam Evans, Alberto Fossa, Roberto Armellin, Didier Henrion, Renato Zanetti

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to steer a spaceship through a cosmic obstacle course. In the real world, space isn't empty; it's a chaotic dance floor where gravity from planets, moons, and even the Sun pulls and pushes your ship in complex, twisting ways. To get from point A to point B, or to stay in a specific orbit without crashing, you need to calculate the perfect push (a "maneuver") at the exact right moment. This is the heart of astrodynamics, the science of moving spacecraft.

The tricky part is that the math describing these movements is incredibly messy. It's like trying to predict the path of a leaf swirling in a storm while also trying to catch it with a net. Traditionally, computers solve this by making a "best guess" and then tweaking it over and over again. This is called Nonlinear Programming (NLP). Think of it like trying to find the lowest point in a foggy mountain valley by feeling your way down. You might get stuck in a small dip (a local minimum) and think you've reached the bottom, when a much deeper valley is just over the next hill. For a spaceship, getting stuck in a "local dip" could mean missing your target or running out of fuel.

This paper tackles that problem by asking: "Can we find the absolute best path, every single time, without getting lost in the fog?" The authors propose a new way to do the math that turns the messy, swirling equations of space travel into a structured puzzle that guarantees you find the deepest valley, not just a shallow one. They tested this on two types of space travel: "impulsive" (giving the ship a quick, sharp kick) and "low-thrust" (giving it a gentle, long push like a slow-motion engine). Their goal is to make these calculations fast and reliable enough to run directly on a spaceship's computer, allowing the ship to fix its own path without waiting for instructions from Earth.

The Paper's Story: Turning Space Math into a Puzzle

The authors, a team of researchers from universities in New Zealand, the US, and France, are essentially trying to upgrade the navigation software for future autonomous spacecraft. They argue that while current methods are good, they can fail when the spaceship has to make a huge turn or travel for a long time, because the old math relies on "linear approximations." Imagine trying to draw a perfect circle using only straight lines; it works okay for a tiny arc, but if you try to draw a whole circle, it looks like a jagged polygon. Similarly, standard math treats space curves as straight lines, which breaks down when the curves get too twisty.

The Big Idea: The Polynomial Transformation
The paper's main trick is to stop treating the spaceship's movement as a messy, continuous flow and instead turn it into a Polynomial Optimization Problem (POP).

Think of a polynomial like a recipe with ingredients raised to different powers (like $x^2$ , $x^3$ ). The authors use a clever mathematical tool called Differential Algebra (DA). You can think of DA as a "super-magnifying glass" that zooms in on the spaceship's path and writes down a high-order recipe (a Taylor expansion) that describes exactly how the ship will move based on its current speed and direction. Instead of simulating the flight second-by-second (which is slow), they create a giant algebraic equation that predicts the future position instantly.

Once they have this equation, the problem becomes: "Find the specific values for our control knobs (the thrusters) that make this equation equal the target, while using the least amount of fuel."

The Solution: The Moment-SOS Hierarchy
Here is where the magic happens. The authors use a method called Moment-Sum-of-Squares (SOS) optimization.

The Analogy: Imagine you are looking for the lowest point in a landscape, but the landscape is made of invisible, shifting fog. Standard methods (NLP) send a hiker to feel around; they might get stuck in a small hole. The SOS method is like sending a drone that scans the entire landscape at once, building a mathematical "safety net" that proves where the absolute lowest point is. It doesn't just guess; it mathematically guarantees that no deeper valley exists within the rules they set.
The Result: The paper shows that this method finds the exact same answer as the traditional, slow "hiker" method (NLP) but with a superpower: it promises to find the global best solution, not just a local one. It also handles "non-convex" problems (those twisty, multi-valley landscapes) much better than older methods.

Testing the Theory: Two Scenarios
The team tested their idea in two different "cosmic playgrounds":

The Quick Kick (Impulsive Targeting):
They simulated a spaceship in a standard two-body system (like Earth and a satellite) and a more chaotic three-body system (Earth, Moon, and a satellite).
- The Finding: When the spaceship needed to make a small adjustment, all methods worked. But when the ship needed to make a large maneuver or travel for a long time, the old linear methods (the "straight line" guesses) failed miserably, missing the target by a wide margin. The new polynomial method, however, stayed accurate even when the ship had to twist and turn significantly.
- The Comparison: They compared their method against "Map Inversion" (another advanced technique). The results showed that their new method was just as accurate as the map inversion but offered a stronger mathematical guarantee that the solution was the best possible one.
The Gentle Push (Low-Thrust Station Keeping):
They then moved to a more complex scenario: a spaceship using a weak, continuous engine to stay in a specific orbit around the Moon (in the Earth-Moon system). This is harder because the engine is always on, and the ship is constantly being pulled by both Earth and Moon.
- The Simulation: They simulated a scenario where the spaceship was hit with random "errors" (like a sudden gust of wind or a sensor glitch) that pushed it off course.
- The Outcome: The new algorithm successfully corrected the ship's path for 62 revolutions (over a year of simulated time), keeping it close to its intended orbit. The ship only used 0.56 kg of fuel to do this. In contrast, a ship without this correction crashed into the Moon (or rather, descended below its surface) after just 11 revolutions.

What They Rule Out
The paper explicitly argues against relying solely on linear approximations (the "straight line" math) for complex space missions. They show that while linear math is fast, it is too fragile for large maneuvers or long durations. They also note that while standard solvers (like IPOPT) are fast, they cannot guarantee finding the best solution if the problem is too complex; they might get stuck in a "local minimum." The authors don't rule out these solvers entirely but suggest that for critical onboard autonomy, the guarantee of the Moment-SOS method is worth the extra computational setup.

How Sure Are They?
The authors are very confident in their results, but they are careful to frame them as simulations. They didn't fly a real rocket; they ran thousands of computer simulations.

They proved mathematically that their method can find the global optimum.
They showed through simulation that their method is three orders of magnitude more accurate than linear methods for large errors.
They demonstrated that the method works even when the ship is thrown off course by significant random errors.

Why This Matters
The ultimate goal of this research is autonomy. Currently, if a spaceship gets lost, it often has to wait for a human on Earth to do the math and send a new command. This takes time and bandwidth. The authors' method is designed to be robust and reliable enough to run on the spaceship's own computer. If a ship can calculate its own "perfect path" instantly and know for a fact that it's the best one, it can fix itself in real-time, making deep-space exploration safer and more independent.

In short, this paper offers a new mathematical toolkit that turns the chaotic, foggy problem of steering a spaceship into a solvable puzzle, promising that future spacecraft won't just guess their way through the stars—they'll know the exact best path to take.

Technical Summary: Polynomial-Based Solutions to Targeting Problems for Onboard Applications

Problem Statement

The paper addresses the critical challenge of spacecraft trajectory targeting and station-keeping (SK) in the context of increasing onboard autonomy. Traditional approaches rely on Nonlinear Programming (NLP) solvers to handle highly nonlinear orbital dynamics. However, these methods face significant limitations:

Computational Constraints: NLPs require numerical integration at every iteration, which is computationally expensive for onboard systems with limited resources.
Convergence Issues: The highly nonlinear nature of orbital dynamics (especially in multi-body environments like the Circular Restricted Three-Body Problem, CR3BP) creates cost landscapes with multiple local minima. Standard gradient-based solvers often converge to local optima rather than the global optimum.
Model Fidelity vs. Efficiency: Linear approximations are computationally efficient but fail to capture dynamics accurately during large maneuvers or long propagation times. Higher-order map inversion techniques improve accuracy but struggle with complex path constraints and non-convex landscapes.

The objective is to develop an optimization framework that balances the flexibility of NLPs with the efficiency and convergence guarantees of convex methods, specifically tailored for autonomous onboard applications.

Methodology

The proposed methodology recasts trajectory optimization problems into Polynomial Optimization Problems (POPs) and solves them using Moment-Sum-of-Squares (SOS) optimization.

1. Problem Recasting via Differential Algebra (DA)

The core of the approach involves leveraging Differential Algebra to compute high-order Taylor expansions of the nonlinear dynamics and constraints with respect to the decision variables (e.g., impulsive maneuvers $\Delta v$ or control profiles).

Impulsive Targeting: The final state $x(t_f)$ is expanded as a high-order Taylor polynomial of the maneuver $\Delta v$ . The original equality constraints (trajectory propagation) and inequality constraints (targeting accuracy) are replaced by polynomial inequalities derived from these expansions.
Continuous Low-Thrust: For low-thrust SK, the control vector and thrust duration are treated as DA variables. A Picard iteration scheme is employed within the DA framework to generate high-order polynomial expansions of the state flow with respect to both control inputs and time-of-flight variations.

2. Moment-SOS Optimization

Once formulated as a POP, the problem is solved using the Moment-SOS hierarchy.

Relaxation: The non-convex POP is relaxed into a sequence of convex Semidefinite Programming (SDP) problems. This hierarchy constructs increasingly tight lower bounds on the global optimum.
Global Optimality: Under mild assumptions (compactness of the feasible set), the method guarantees convergence to the global optimum. The solution is recovered from the underlying relaxation, and global optimality is verified via rank conditions on the moment matrix (e.g., if the moment matrix has rank one, the solution is globally optimal).
Convex Relaxation: For scenarios where a second-order expansion of the dynamics provides sufficient accuracy, the paper proposes an efficient convex relaxation based on a quadratic approximation of the constraints, further reducing computational complexity.

3. Numerical Implementation

The algorithms are implemented in Julia using:

GTPSA.jl for computing polynomial expansions.
MomentOpt.jl and SumOfSquares.jl for modeling the Moment-SOS problems.
Mosek as the SDP solver.
IPOPT is used as a benchmark for traditional NLP solutions.

Key Contributions

Systematic Recasting: The paper demonstrates a systematic method to transform complex targeting and SK problems (including those in the CR3BP) into POPs using high-order DA Taylor expansions.
Convex Relaxation: An efficient convex relaxation based on second-order dynamics is proposed for scenarios where high-order expansions are not strictly necessary, offering a balance between accuracy and speed.
Global Convergence Guarantee: The application of Moment-SOS optimization provides a mathematically rigorous path to finding the global optimum, a feature often lacking in standard NLP solvers for non-convex astrodynamics problems.
Versatility: The methodology is successfully extended from impulsive targeting to continuous low-thrust station-keeping, handling variable time segments and non-convex control constraints.

Results

The paper evaluates the proposed methods against traditional NLP solvers (IPOPT), linear approximations, and DA map inversion techniques in two primary scenarios:

1. Impulsive Targeting in Keplerian Dynamics (R2BP)

Accuracy: The Moment-SOS (MOP) and convex relaxation approaches yielded solutions with magnitudes and components virtually identical to the NLP solution (differences in the order of $10^{-6}$ to $10^{-9}$ ).
Constraint Satisfaction: All methods satisfied terminal constraints within solver tolerances.
Conclusion: The polynomial-based methods are viable alternatives that combine NLP-level accuracy with the structural benefits of convex optimization.

2. Station-Keeping in the Earth-Moon CR3BP

Accuracy vs. Linearization: In a Target Point Approach (TPA) scenario, the linearized solution showed relative errors in maneuver magnitude up to three orders of magnitude worse than the polynomial-based solutions (POP and DA map inversion) when compared to the NLP benchmark.
Handling Nonlinearity: The polynomial approach captured complex, non-concentric structures in the optimal maneuver space that linear methods failed to resolve, particularly for large initial deviations.
Computational Trade-offs:
- Single Solution: Linear methods are fastest. NLP is ~20x slower. Polynomial methods incur a high upfront cost to compute the Taylor map.
- Multiple Solutions (e.g., Monte Carlo): Once the high-order map is computed, DA map inversion becomes the most efficient method for evaluating multiple perturbation realizations.
- Global Optimality: While solving the POP as a generic NLP (using IPOPT) reduces runtime, it sacrifices the guarantee of global convergence. The Moment-SOS approach maintains this guarantee.

3. Low-Thrust Station-Keeping

Robustness: The method was tested on a 368-day simulation (62 revolutions) of an Earth-Moon Near Rectilinear Halo Orbit (NRHO) with significant unmodeled state errors (position $\sigma=10$ km, velocity $\sigma=0.3$ m/s).
Performance: The Moment-SOS based SK successfully maintained the spacecraft near the nominal trajectory, consuming only 0.56 kg of fuel. In contrast, the uncorrected trajectory diverged rapidly, resulting in a collision with the Moon by the 11th revolution.
Handling Constraints: The approach successfully managed the non-convex control magnitude constraints and the coupling between thrust duration and coast time without linearization.

Significance and Claims

The paper claims that the Moment-SOS approach bridges the gap between high-fidelity physical modeling and reliable numerical convergence. Its primary significance lies in its suitability for autonomous onboard applications where:

Reliability is paramount: The guarantee of global convergence (under mild assumptions) mitigates the risk of the spacecraft failing to find a valid trajectory due to local minima.
Non-convexity is inherent: The method can directly handle non-convex constraints and complex nonlinear dynamics without relying on linear approximations that degrade during large maneuvers or long propagation times.
Versatility: The framework is applicable to both impulsive and continuous-thrust regimes, including complex multi-body environments like the CR3BP.

The authors conclude that while the upfront computational cost of generating high-order expansions is non-trivial, the resulting ability to solve complex, non-convex targeting problems with guaranteed convergence makes this approach a strong candidate for future autonomous spacecraft operations. Future work is identified as extending these methods to multi-impulsive scenarios and more complex objective functions.