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A practical guide to implementing zero-order-hold interplanetary trajectory legs

This paper presents a practical guide for implementing robust zero-order-hold interplanetary trajectory optimizations by introducing four key innovations: a forward-backward shooting construction, a singularity-free throttle parameterization, a differentiable softmax time-grid encoding, and the comprehensive TOPS benchmark suite.

Original authors: Dario Izzo, Harry Holt, Giacomo Acciarini, Laurent Beauregard, Yuri Shimane

Published 2026-05-13

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CC BY 4.0

Original authors: Dario Izzo, Harry Holt, Giacomo Acciarini, Laurent Beauregard, Yuri Shimane

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 guide a spaceship from Earth to Mars. You have a map (the laws of physics) and a destination, but the spaceship's engine is finicky. It doesn't run smoothly like a car; instead, it works best when you turn it on full blast for a while, then turn it off completely, then turn it on again. This is called a "bang-bang" control strategy.

The paper you're asking about is a practical guidebook for computer programs that try to figure out the best path for these spaceships. The authors are saying, "We found a way to make these computer calculations much more reliable, so they don't need a human expert to tweak them for every single new mission."

Here is a breakdown of their four main "secrets" using simple analogies:

1. The "Two-Way Street" Strategy (Forward-Backward Shooting)

The Problem: Usually, to find a path, computers guess a starting point and try to drive forward to the finish. If they miss the target, they have to guess again. This is like trying to throw a ball into a basket from across the gym by just guessing the angle. If you miss, you try again. It's slow and frustrating.

The Solution (ZOHα): The authors suggest a "two-way street" approach. Imagine you have two people: one starts at the launch pad and drives forward, while another starts at the destination and drives backward in time. They meet in the middle.

Why it works: By meeting in the middle, the computer gets clues from both ends of the trip. It's like having two people trying to solve a maze from opposite sides; they are much more likely to find the exit quickly than someone starting from just one side. The paper found that meeting exactly in the middle (50/50 split) is usually the sweet spot.

2. The "Magic Knob" vs. The "Broken Switch" (Throttle Parameterization)

The Problem: The spaceship engine has a "throttle" (how hard it pushes). When the engine is off (throttle = 0), the math used by the computer breaks down. It's like trying to divide by zero. The computer gets confused and stops working, especially on long stretches where the ship coasts without engines.

The Solution: The authors invented a new way to describe the throttle. Instead of just one number for "how hard," they use four numbers (a redundant 4D parameterization).

The Analogy: Imagine trying to describe a direction using a compass. If you are standing still, "North" doesn't make sense. But if you describe your movement as "a speed knob" and "a direction knob" separately, you can set the speed to zero without the direction breaking the math. This trick keeps the computer's math smooth and working even when the engine is completely off.

3. The "Smart Timer" (Softmax Time-Grid)

The Problem: To solve the path, the computer breaks the trip into time chunks (segments). Usually, these chunks are all the same size, like a ruler with equal marks. But what if the spaceship needs to make a sharp turn in 1 second, but then coast for 100 seconds? A ruler with equal marks is too rigid; it either misses the sharp turn or wastes time on the coast.

The Solution: They use a "Smart Timer" (Softmax encoding).

The Analogy: Instead of a rigid ruler, imagine a stretchy rubber band. The computer can stretch some parts of the timeline to be very short (for sharp turns) and other parts very long (for coasting), but it does this in a way that the math stays smooth. It's like a camera that automatically zooms in on the action and zooms out when nothing is happening, ensuring the computer never gets confused by the changing speed of the timeline.

4. The "Universal Test Lab" (TOPS Benchmark)

The Problem: Many researchers test their new methods on just one or two specific space missions. It's like a car manufacturer testing a new engine only on a race track in California. It might work there, but will it work in the snow or the mud?

The Solution: The authors created TOPS (Trajectory Optimisation Problems in Space).

The Analogy: This is a massive, standardized "obstacle course" for spaceship software. It includes 28 different challenges: flying around Earth, navigating the gravity of three bodies (like the Sun, Earth, and Moon), and using solar sails (engines powered by sunlight). By testing their method on all 28, they proved it works reliably across the board, not just in one lucky scenario.

The Bottom Line

The paper claims that by combining these four tricks—meeting in the middle, using a "magic knob" for the engine, using a "stretchy timer," and testing on a huge variety of problems—they have created a method (ZOHα) that is robust.

"Robust" here means: You can hand this tool to a computer, give it a random starting guess, and it will likely find a good solution without needing a human expert to whisper instructions or fix errors. It turns a delicate, art-like process into a reliable, systematic engineering tool.

They also compared two famous computer solvers (IPOPT and SNOPT) and found that while SNOPT is the industry standard, the new method actually helps the "underdog" solver (IPOPT) perform just as well, or even better, in many cases.

Technical Summary: A Practical Guide to Implementing Zero-Order-Hold Interplanetary Trajectory Legs

Problem Statement

The paper addresses the practical implementation of Zero-Order-Hold (ZOH) transcriptions for spacecraft trajectory optimization. While ZOH methods—where the control input is parameterized as a piecewise-constant function and the state is computed via numerical integration (shooting)—are conceptually simple, previous studies have often relied on problem-specific tuning and homogeneous dynamical settings. This reliance limits scalability, automation, and reproducibility. The authors aim to identify a set of design principles that render ZOH transcriptions robust across a broad class of dynamical settings (including two-body, modified equinoctial elements, circular restricted three-body, and solar sailing) without requiring extensive manual intervention or expert-crafted initial guesses.

Methodology

The authors propose a specific forward-backward ZOH scheme, denoted as ZOHα, and integrate it with four key technical ingredients to enhance robustness and differentiability:

Forward-Backward Shooting Construction (ZOHα):
Instead of purely forward or backward shooting, the time horizon $[t_0, t_f]$ is partitioned into $N$ segments. The first $N_{fwd}$ segments are integrated forward from the initial state $x_0$ , and the remaining $N_{bck}$ segments are integrated backward from the final state $x_f$ . A mismatch constraint is enforced at the junction point where the two arcs meet. The parameter $\alpha \in [0, 1]$ controls the ratio of forward to backward segments. This approach reduces gradient sensitivity amplification compared to single shooting while avoiding the need for intermediate state guesses required by classical multiple shooting.
Redundant Four-Dimensional Throttle Parameterization:
To address the singularity of the control influence matrix along ballistic arcs (where thrust is zero), the authors introduce a redundant parameterization. The throttle vector is split into a scalar magnitude $\tau \in [0, 1]$ and a unit-norm direction vector $d \in \mathbb{R}^3$ . The mass-flow equation is modeled using only $\tau$ , intentionally decoupling it from the norm of $d$ . This ensures the control influence matrix remains smooth even when $\tau = 0$ , while the optimality conditions (via Pontryagin's Maximum Principle) naturally enforce $\|d\|=1$ at the solution, preserving physical consistency.
Softmax Time-Grid Encoding:
To avoid ordering constraints on segment durations ( $t_i \le t_{i+1}$ ) while maintaining full differentiability, the authors employ a softmax transformation. Segment durations are derived from an unconstrained vector $w$ via $\Delta t_i = e^{w_i} / \sum e^{w_j}$ . This guarantees strictly positive durations and a full-rank Jacobian, facilitating gradient-based optimization without affine inequality constraints on time.
The TOPS Benchmark:
To validate these principles, the authors introduce the TOPS (Trajectory Optimisation Problems in Space) benchmark. This suite comprises 28 distinct problems across four dynamical models: two-body Cartesian, modified equinoctial elements (MEE), circular restricted three-body problem (CR3BP), and solar sailing. The problems vary in difficulty, covering multiple revolutions, inclination changes, and high eccentricities.

Experimental Setup

The study evaluates the ZOHα transcription using 128 random initial guesses per problem instance, solved with two standard NLP solvers: IPOPT and SNOPT. Crucially, no solver-specific tuning is performed; both are run with default configurations. Performance is measured using:

Expected Run Time (ERT): A hardware-normalized cost accounting for both successful and failed runs, penalizing setups with low success rates.
Expected Objective ( $E[J]$ ): The average final spacecraft mass ratio among successful runs.
Performance is visualized via non-dominated fronts in the $(ERT, E[J])$ plane, allowing for a trade-off analysis between speed and solution quality.

Key Results

The extensive experimental campaign (over 300,000 solver runs) yielded several significant findings:

Forward-Backward Superiority: Transcriptions with $\alpha = 0.5$ (equal forward and backward segments) consistently dominated purely forward ( $\alpha=1$ ) or purely backward ( $\alpha=0$ ) schemes. This is attributed to reduced gradient sensitivity and the simultaneous exploitation of both boundary conditions during the optimization of random initial guesses.
Solver Performance: While SNOPT is the de facto standard in aerospace, IPOPT demonstrated competitive, and in some cases superior, performance across the benchmark suite, particularly when the transcription was fully differentiable and problems were properly scaled. SNOPT retained a consistent advantage specifically for solar sailing problems.
Time Grids: Variable time grids generally provided higher accuracy at the cost of computation, making them suitable as refinement steps ("finishers") for solutions initialized with uniform grids. However, for solar sailing problems, the softmax time-grid encoding outperformed the uniform grid, likely due to its ability to contract segments with detrimental sail orientations.
Throttle Constraints: Inequality constraints on the throttle magnitude generally outperformed equality constraints. Equality constraints were only comparable or superior in cases where the optimal trajectory was predominantly ballistic with very short powered arcs.

Significance and Claims

The paper does not claim to introduce a new family of transcriptions but rather identifies a set of implementation principles that make existing ZOH methods robust and scalable. The authors argue that by adopting the ZOHα scheme with the specific parameterizations and benchmarking strategies outlined, trajectory optimization can shift from a "problem-dependent art" requiring extensive manual tuning to a "systematic computational methodology."

The work positions ZOH not merely as a simplistic baseline but as a flexible and competitive strategy, particularly when implemented with the numerical care described (e.g., handling singularities via redundant parameterization and ensuring end-to-end differentiability). The introduction of the TOPS benchmark is intended to facilitate future extensions and reproducible research in the field, moving beyond the narrow, homogeneous test cases often found in previous literature.