⚡ electrical engineering

Robust Sampling-Based Covariance Steering for Aerocapture Guidance

This paper presents a robust sampling-based covariance steering algorithm for aerocapture guidance that leverages sampled nonlinear trajectories to effectively address atmospheric and entry uncertainties, achieving a 5–15% reduction in high-percentile and worst-case delta-V requirements for Mars and Uranus missions compared to state-of-the-art methods.

Original authors: Alex Rose, Christopher Jewison, Jonathan P. How

Published 2026-06-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alex Rose, Christopher Jewison, Jonathan P. How

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 park a massive, fragile spaceship in orbit around a distant planet like Mars or Uranus. The problem? You can't just slow down with your engines alone; you don't have enough fuel. Instead, you have to dive into the planet's atmosphere and use the air itself like a brake pad to slow you down. This maneuver is called Aerocapture.

Think of it like a skier trying to stop at the bottom of a steep hill. If they just dig their skis in (using engines), they might not stop in time. Instead, they might drag their hands in the snow (using atmospheric drag) to scrub off speed. But here's the catch: the snow isn't uniform. Sometimes it's deep powder, sometimes it's hard-packed ice. If the skier doesn't know exactly what the snow will be like, they might stop too early (and crash) or not stop at all (and fly off into space).

The Problem: The "Unknown Snow"

The paper explains that space missions face a similar problem. We don't know the atmosphere of a planet perfectly. Is it denser than we thought? Is it thinner? These uncertainties are like "unknown snow conditions."

If a spacecraft's guidance system assumes the air is a certain way, but it's actually different, the ship might miss its target orbit. To fix this, the ship has to fire its engines later to correct the mistake. This uses up precious fuel (called Delta-V or $\Delta V$ ). The more fuel you use, the less cargo you can carry, or the heavier your rocket needs to be to launch.

The Old Way vs. The New Way

The authors compare two ways of guiding the ship:

The Old Way (The "Best Guess" Approach): Imagine a driver who looks at a map, assumes the road is perfectly smooth, and drives a straight line. If they hit a pothole (an unexpected atmospheric change), they swerve hard to correct it. This works okay for smooth roads, but if the road is bumpy, the driver might over-correct, wasting energy and risking a crash. In technical terms, this method assumes the errors are small and linear, which isn't true for the chaotic, bumpy ride of entering a planet's atmosphere.
The New Way (The "Robust Sampling" Approach): The authors propose a new algorithm that acts like a driver who doesn't just look at one map, but simulates hundreds of different possible road conditions before even starting the car.
- They imagine: "What if the air is 10% thicker? What if it's 10% thinner? What if I enter at a slightly different angle?"
- They run these "what-if" scenarios (called sigma points) through a computer model of the physics.
- Instead of planning for the average road, they plan for the worst-case scenarios among those simulations. They steer the ship in a way that ensures it survives even the bumpiest, most unpredictable atmospheric conditions without needing a massive fuel correction at the end.

How It Works (The Metaphor)

Think of the spacecraft's path as a tightrope walk.

The Old Method tries to walk the tightrope by balancing perfectly in the center, assuming the wind won't blow hard. If a gust hits, the walker stumbles and has to make a huge, desperate leap to stay on the rope (using lots of fuel).
The New Method practices walking the tightrope while wearing a blindfold, simulating strong winds from every direction. They learn to shift their weight proactively so that even if a strong gust hits, they stay balanced. They don't need that desperate leap at the end.

The Results: Saving Fuel

The researchers tested this new method on two planets: Mars (where the atmosphere is better understood) and Uranus (where the atmosphere is very uncertain and the mission is much harder).

They ran thousands of computer simulations (like running the tightrope walk 5,000 times in a row) to see how much fuel was needed to land safely.

The Finding: The new "Robust Sampling" method saved 5% to 15% of the fuel needed for the final correction burn compared to the best existing method.
Why it matters: In the world of space travel, saving 15% of fuel is huge. It means you can carry heavier scientific instruments, or you can launch a mission that was previously too heavy for our current rockets.
The "Worst-Case" Win: The biggest improvement was seen in the "worst-case" scenarios (the hardest 1% or 0.3% of missions). For these difficult cases, the new method was significantly better at ensuring the ship didn't run out of fuel or miss its orbit.

Summary

This paper introduces a smarter way to guide spacecraft through planetary atmospheres. Instead of guessing the weather and hoping for the best, the new algorithm simulates thousands of different weather patterns and plans a route that is safe for the worst of them. This "safety-first" planning actually turns out to be more efficient, saving valuable fuel and making ambitious space missions (like bringing Mars rocks back to Earth or visiting Uranus) much more feasible.

Technical Summary: Robust Sampling-Based Covariance Steering for Aerocapture Guidance

Problem Statement
Aerocapture is a critical maneuver for enabling long-duration missions with heavy payloads, such as Mars Sample Return and the proposed Uranus Orbiter and Probe. By utilizing atmospheric drag to reduce velocity, aerocapture offers significant fuel savings and reduced transit times compared to fully propulsive insertion or aerobraking. However, the maneuver is highly sensitive to uncertainties in the initial entry state and atmospheric density.

Existing deterministic guidance algorithms (predictor-corrector architectures) are sensitive to atmospheric profile mismatches, often failing when actual density deviates from predictions. While chance-constrained covariance steering methods have been developed to model atmospheric uncertainty as a Gaussian random field, prior approaches suffer from two main limitations:

They often linearize dynamics, failing to account for nonlinearities caused by the state-dependence of atmospheric uncertainty and the inherent nonlinear aerocapture dynamics.
They typically assume the objective function (required $\Delta V$ ) is well-approximated by a Gaussian distribution, which may not hold true for highly nonlinear systems.

The authors address the stochastic optimization problem of minimizing the $X$ -th percentile of the required $\Delta V$ (typically 99th or 99.7th) subject to probabilistic bank angle constraints, Gaussian initial state dispersion, and state-dependent atmospheric disturbances modeled as a Gaussian random field.

Methodology
The paper proposes a Robust Sampling-Based Covariance Steering algorithm that integrates successive convexification with a sampling-based objective evaluation. The core approach involves:

Iterative Linearization and Convexification:
The algorithm adopts a successive convexification approach. It generates a nominal control trajectory and linearizes the nonlinear aerocapture dynamics around this trajectory and the mean atmospheric disturbance. This transforms the stochastic problem into a convex covariance steering problem solvable by commercial solvers. The state covariance evolution is modeled using a feedback control law parameterized by a decision variable $L$ , allowing the optimization of the covariance propagation.
Robust Sampling-Based Objective:
To address the nonlinearity of the $\Delta V$ objective function and the state-dependence of disturbances, the authors introduce a sampling-based cost function:
- Sigma Point Propagation: The algorithm samples $2n$ sigma points symmetrically on the covariance contour of the initial state distribution.
- Nonlinear Trajectory Rollout: These sigma points are propagated through the true nonlinear system dynamics (including the state-dependent atmospheric disturbance) to generate a collection of trajectories.
- Worst-Case Approximation: The objective function is evaluated at the exit states of these sampled trajectories. The cost function is formulated to minimize the worst-case $X$ -th percentile $\Delta V$ over this collection of Gaussian trajectories. This effectively penalizes sigma point trajectories associated with high $\Delta V$ and accounts for variations in the objective function that linearization alone would miss.
Trust Region Constraints:
To ensure the validity of the linearized convex approximation, the algorithm enforces trust region constraints on the control inputs and state variables (specifically dynamic pressure and exit apoapsis). The process iterates (Algorithm 1), re-linearizing and re-solving the convex problem until convergence.

Key Contributions

Algorithm Development: The development of a new robust sampling-based covariance steering algorithm specifically designed for aerocapture guidance under state-dependent environmental disturbances.
Nonlinearity Handling: The method explicitly addresses nonlinearities arising from both the aerocapture dynamics and the state-dependence of atmospheric uncertainty by propagating sigma points through the nonlinear system rather than relying solely on linearized approximations for the objective evaluation.
Objective Function Reformulation: The formulation of a cost function that approximates the worst-case percentile $\Delta V$ over a collection of sampled trajectories, improving robustness to initial state errors and dynamic nonlinearities.

Results
The authors evaluated the proposed algorithm against a state-of-the-art iterative nonlinear covariance steering baseline (Ridderhof & Tsiotras) using Monte Carlo simulations across three scenarios:

Mars (Small Dispersion): A scenario with small initial state dispersion.
Mars (Large Dispersion): A more challenging scenario with large initial velocity dispersion.
Uranus: A realistic scenario with dispersed initial states, highly elliptical target orbits, and high atmospheric uncertainty (using UranusGRAM).

Performance Metrics:

Mars (Small Dispersion): The proposed method reduced the 99.7th-percentile and worst-case $\Delta V$ by 15% and the 99th-percentile $\Delta V$ by 5% compared to the baseline.
Mars (Large Dispersion): The method reduced the worst-case $\Delta V$ by 6.5%, the 99.7th-percentile $\Delta V$ by 14%, and the 99th-percentile $\Delta V$ by 15%.
Uranus: The method reduced the worst-case $\Delta V$ and 99.7th-percentile $\Delta V$ by approximately 12%, and the 99th-percentile $\Delta V$ by 13.5%.

In all scenarios, the algorithm demonstrated significant reductions in the tail-end of the $\Delta V$ distribution (the most critical cases) without increasing the mean $\Delta V$ . The improvements were most pronounced in the hardest 5% of Monte Carlo cases.

Significance and Claims
The paper claims that the proposed robust sampling-based covariance steering algorithm offers a 5-15% reduction in the 99th-percentile, 99.7th-percentile, and worst-case $\Delta V$ required for aerocapture compared to existing state-of-the-art methods. The authors emphasize that this improvement is achieved by better modeling the nonlinearities inherent in the system and the state-dependence of atmospheric disturbances, thereby increasing the robustness of the guidance solution to initial state errors and atmospheric uncertainties. The work suggests that this approach enables safer and more efficient aerocapture for high-priority missions like Mars Sample Return and Uranus exploration.

Future Work
The authors note that future work could involve updating the estimated atmosphere and uncertainty online to re-run the algorithm, potentially improving performance in highly uncertain initial conditions. Additionally, the method could be extended to handle non-Gaussian initial state dispersions.