⚡ electrical engineering

Low-Thrust Orbital Trajectory Guidance and Control Using Mean Orbital Elements

This paper investigates the application of mean orbital elements in formulating a directional adaptive guidance law for low-thrust orbital transfers, demonstrating their robustness and effectiveness compared to osculating elements in handling high-dimensional optimization problems with natural perturbations across various maneuver types.

Original authors: Houman Hakima, Michael C.F. Bazzocchi

Published 2026-09-08

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CC BY 4.0

Original authors: Houman Hakima, Michael C.F. Bazzocchi

Original paper licensed under CC BY 4.0 (https://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

Space travel has long relied on powerful, short bursts of fuel to move spacecraft from one orbit to another, much like a sprinter exploding off the starting blocks. However, a quieter, more efficient method has gained prominence in recent decades: electric propulsion. These systems, often called low-thrust engines, do not fire with a sudden bang but instead provide a gentle, continuous push for months or even years. While they consume far less fuel than traditional chemical rockets, they require much longer times to complete a journey. The challenge for engineers is that this slow, steady push must be perfectly timed and directed over thousands of orbits. If the guidance system is too sensitive to the tiny, rapid wobbles of the spacecraft's path caused by Earth's uneven shape or atmospheric drag, the spacecraft might waste fuel constantly correcting its course. The question becomes how to steer a vessel that is being pushed by a whisper rather than a shout, ensuring it reaches its destination without exhausting its limited fuel supply.

In a recent study, researchers investigated a specific way to solve this steering problem by changing how they view the spacecraft's position. Instead of tracking the spacecraft's exact, moment-to-moment location, which is constantly jittering due to gravitational tugs and air resistance, they proposed using a smoothed-out average of the orbit. They tested a guidance method known as directional adaptive guidance, which calculates the best angle to point the thruster by weighing the importance of different orbital goals, such as changing the orbit's size or its tilt. The team compared two approaches: one that used the raw, jittery data of the spacecraft's actual path, known as osculating elements, and another that used the calculated average path, known as mean orbital elements. By running detailed computer simulations of three different types of space journeys, they aimed to see which method allowed the spacecraft to reach its target more efficiently and with less strain on its steering mechanisms.

The researchers simulated three distinct missions to test their theory. The first involved raising a satellite from a low altitude of 400 kilometers to a higher altitude of 1,500 kilometers. The second simulated a complex trip from a transfer orbit to a final geostationary orbit, requiring changes to the orbit's size, shape, and tilt. The third case study was a transfer to a highly elliptical orbit used for communications, where the spacecraft had to adjust its speed and shape carefully to avoid dipping too low into the atmosphere. In each scenario, the spacecraft was modeled with a mass of 500 kilograms and equipped with a high-efficiency electric thruster capable of producing a tiny but steady force. The team tracked how much fuel was consumed, how long the maneuvers took, and how much the spacecraft had to rotate its body to follow the guidance commands.

The results showed that both methods could successfully guide the spacecraft to its destination, but they offered different trade-offs. When the guidance system used the raw, moment-to-moment data, the spacecraft reached its target slightly faster and used a tiny fraction less fuel. This is expected, as the system was reacting to the true, immediate state of the orbit. However, this approach required the spacecraft's attitude control system to make much more frequent and rapid adjustments to the thruster's angle. In contrast, the guidance system based on the smoothed, average orbit required the spacecraft to change its pointing direction much more slowly and steadily. The maximum rate at which the spacecraft had to turn its thruster was significantly lower when using the average orbit data. For example, in the simulation of the trip to the geostationary orbit, the average rate of change for the steering angle was about 0.2 degrees per second with the smoothed method, compared to slightly higher values with the raw data method.

The study concluded that while the raw data method is technically more precise in terms of fuel and time, the smoothed average method is often more practical for real-world missions. Spacecraft have limited mechanical ability to rotate their bodies quickly, and constantly fighting against rapid, jittery guidance commands can wear out components or exceed the vehicle's physical limits. By using the average orbit, the guidance system provides a gentler, more predictable path for the spacecraft to follow, reducing the demand on its steering hardware. The researchers found that for orbits near the equator, where the Earth's shape causes the most significant wobbles, this difference in control effort was particularly noticeable. The study suggests that for many future missions, especially those involving small satellites with limited maneuvering power, designing the guidance system around the average behavior of the orbit rather than its instantaneous fluctuations offers a robust and efficient solution. This approach allows engineers to plan long-duration electric propulsion missions with greater confidence that the spacecraft will not be overwhelmed by the need to constantly correct its course.

Technical Summary: Low-Thrust Orbital Trajectory Guidance and Control Using Mean Orbital Elements

Problem Statement
Low-thrust propulsion systems, characterized by high specific impulse, are increasingly vital for missions requiring significant delta-v, such as orbit transfers, station-keeping, and interplanetary travel. However, optimizing low-thrust trajectories is a high-dimensional challenge complicated by perturbative accelerations (e.g., Earth's oblateness, atmospheric drag) that prevent accurate analytical solutions. Traditional guidance laws often rely on osculating orbital elements, which represent the instantaneous "true" orbit. The authors argue that osculating elements are suboptimal for long-term control because they fluctuate rapidly due to short-periodic perturbations (such as $J_2$ and drag). These rapid fluctuations necessitate frequent, high-rate adjustments in spacecraft attitude to maintain the desired thrust direction, potentially exceeding the capabilities of the Attitude Determination and Control Subsystem (ADCS) and increasing control effort.

Methodology
The paper investigates the application of mean orbital elements—parameters that filter out short-periodic variations to represent the slow-varying average orbit—in formulating a Directional Adaptive Guidance (DAG) law.

Dynamics and Perturbations: The study utilizes Gauss' Variational Equations (GVEs) to model orbital motion. It explicitly accounts for $J_2$ perturbations (Earth's oblateness) in the derivation of mean elements and includes both $J_2$ and atmospheric drag in numerical simulations to ensure fidelity.
Mean Element Derivation: The authors define mean elements ( $\bar{\epsilon}$ ) by subtracting short-periodic terms ( $\Delta \epsilon_{sp}$ ) derived from $J_2$ perturbations from the osculating elements ( $\epsilon_{osc}$ ). Analytical expressions for these short-periodic variations are provided for all six classical orbital elements (semimajor axis, eccentricity, inclination, RAAN, argument of perigee, and true anomaly).
Guidance Law Formulation: The DAG law is employed, which synthesizes a net thrust direction as a weighted sum of optimal thrust directions for specific orbital elements. The thrust direction is defined by in-plane ( $\alpha$ $α$ ) and out-of-plane ( $\beta$ $β$ ) steering angles.
- Optimum Steering Angles: Analytical solutions for the angles that maximize the rate of change for each element are derived. The paper addresses practical implementation issues, such as correcting for the range limitations of the single-argument inverse-tangent function and selecting the shortest path for angular elements (e.g., RAAN, argument of perigee) to minimize propellant usage.
- Adaptation: The law includes adaptation parameters ( $R_{\epsilon}$ ) that dynamically adjust the weighting of thrust directions based on the error between current and desired element values.
Numerical Simulation: Three case studies were simulated using a 500 kg spacecraft with a 235 mN gridded ion thruster ( $I_{sp} = 4,220$ $I_{s p} = 4, 220$ s):
- Case 1: Low-altitude LEO to high-altitude LEO (semimajor axis increase).
- Case 2: Geostationary Transfer Orbit (GTO) to Geostationary Orbit (GEO) (simultaneous semimajor axis, eccentricity, and inclination reduction).
- Case 3: Parking orbit to Molniya orbit (simultaneous changes in semimajor axis, eccentricity, and argument of perigee).

Key Contributions

Mean vs. Osculating Comparison: The paper provides a comparative assessment of guidance laws based on mean elements versus those based on osculating elements in the presence of natural perturbations.
Refined Steering Logic: It details the derivation of optimum steering angles for mean elements and introduces specific logic to handle discontinuities in steering angles (e.g., wrapping issues in inverse-tangent functions and minimizing angular error for periodic elements).
Weighting Strategy: The study demonstrates a dynamic weighting scheme where the priority of correcting certain elements (like eccentricity) is delayed until the orbit size increases, thereby avoiding low-perigee passages that would incur excessive atmospheric drag.

Results
The numerical results across the three case studies indicate:

Propellant and Time: Guidance laws based on osculating elements yielded slightly better performance in terms of total propellant consumption and maneuver duration (e.g., Case 2: 43.70 kg vs. 43.80 kg; 65.30 days vs. 65.46 days). This is attributed to osculating elements representing the true instantaneous state, allowing for theoretically optimal thrusting at every moment.
Control Effort: Guidance laws based on mean elements demonstrated significantly smoother steering angle profiles. The maximum and average rates of change for the steering angles were lower when using mean elements.
- In Case 2 (GTO to GEO), the mean-element approach reduced the maximum rate of change for the in-plane eccentricity steering angle ( $\dot{\alpha}_e$ ) from 4.4525 deg/s to 4.4471 deg/s and the average rate from 0.6466 deg/s to 0.6438 deg/s.
- In Case 3 (Molniya transfer), the mean-element approach reduced the maximum rate of change for the argument of perigee in-plane angle ( $\dot{\alpha}_\omega$ ) from 13.0630 deg/s to 12.9823 deg/s.
Convergence: The difference between osculating and mean elements was most pronounced at lower altitudes where $J_2$ perturbations are stronger, but the trends generally converged as the orbit size increased, except in high-eccentricity transfers where perigee drag effects caused divergence.

Significance and Claims
The authors conclude that while osculating-element-based guidance is marginally more efficient in terms of fuel and time, mean-element-based guidance is more applicable for practical space missions, particularly for spacecraft with limited attitude control authority. The smoother evolution of steering angles required by mean-element guidance reduces the demand on the spacecraft's attitude control system, avoiding the need for rapid, high-frequency attitude adjustments that osculating-element guidance might demand due to short-periodic oscillations.

The paper modestly claims that for low-inclination orbits, the difference in control effort is not significant, but for equatorial and near-equatorial orbits, the difference is significant. It suggests that future work should incorporate additional perturbation sources (such as drag, higher-order zonal harmonics $J_3, J_4$ , and third-body effects) to further refine the short-periodic terms used in the mean element calculations.

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