Optimizing CubeSatellite Orientation Using Magnetorquers

A Comparative Study of Attitude Control Systems for 1U Satellites

Arnav Venkatesh, Grade 11, NHVPS Bangalore PDF File:

Executive Summary

This white paper presents a comparative study of magnetorquer-based attitude control solutions for 1U CubeSatellites (CubeSats), with an emphasis on initial detumbling following deployment into Low Earth Orbit (LEO). Due to the spring-loaded ejection mechanisms in their launch vehicles, CubeSats experience very high initial angular velocities, often reaching 10–20°/s. To stabilize these small satellites efficiently under strict power, mass, and volume constraints, magnetic actuators known as magnetorquers are utilized. Their application, design trade-offs, and control strategies are examined in this paper.

The analysis begins by outlining three primary types of magnetorquers: air-core coils, embedded (PCB) coils, and ferromagnetic rod-core coils. These are compared across key performance parameters such as mass, torque generation, integration complexity, and power consumption. Subsequently, control laws including B-dot, bang-bang, and proportional-derivative (PD) are evaluated for their effectiveness in transitioning the spacecraft from an uncontrolled tumbling state to a stable orientation.

Using realistic assumptions and experimental values for Earth’s magnetic flux, satellite inertia, and actuator dipole strength, three configuration models are constructed and simulated. The results strongly suggest that rod-core magnetorquers paired with a bang-bang controller offer the fastest detumbling performance within feasible power budgets, capable of arresting the satellite's rotation in under 20 seconds.

Based on these findings, the recommended system for a resource-limited 1U CubeSat prioritizes a 3-axis rod-core magnetorquer array controlled by a hybrid bang-bang and B-dot scheme. This configuration achieves a practical and highly effective trade-off: it maximizes torque-per-watt, requires minimal onboard sensing, and avoids moving parts, all while enabling fast, autonomous detumbling with minimal software complexity.

1. Introduction

CubeSats have revolutionized access to space by standardizing satellite design parameters into compact 10 cm³, 1.33 kg units (1U satellites)¹. To place them in orbit, these satellites are ejected by PPOD (Poly-Picosatellite Orbital Deployer) springs from the launch vehicle, resulting in uncontrolled initial attitudes and angular rates often exceeding 10–20 °/s². Without reliable stabilization, essential functions like solar array alignment, antenna deployment, and stable imaging cannot be executed.

Conventional actuators, such as reaction wheels and control-moment gyros, offer precise pointing but incur prohibitive mass, power usage, and mechanical complexities for a 1U satellite. By contrast, magnetorquers generate a controllable magnetic dipole that interacts with Earth’s geomagnetic field to produce torque without any propellant or moving parts. This makes them uniquely suited to the critical tasks of detumbling and coarse pointing under the strict resource constraints of a CubeSat.

This white paper is not intended as a contribution to original scientific discovery but rather as a conceptual engineering study grounded in analytical evaluation and supported by established literature.

2. Foundational Concepts and Definitions

• Attitude (R): The orientation of the satellite body relative to an inertial reference frame, often represented using rotation matrices or quaternions.

• Magnetorquer: A device used in spacecraft for attitude control and detumbling. It consists of electromagnetic coils that, when energized, generate a magnetic dipole moment. This moment interacts with the ambient planetary magnetic field to produce the necessary control torque.

• Angular Velocity (ω): The rate at which a satellite’s orientation changes over time, measured in radians per second (rad/s).

• Detumbling: The process of reducing a satellite's initial, uncontrolled angular motion to a near-zero rotational velocity immediately after deployment.

• Magnetic Dipole Moment (m): A vector that quantifies the strength and direction of a magnetorquer’s field, given by the formula m=nIA, where n is the number of coil turns, I is the current, and A is the coil area.

• Torque (τ): The rotational force generated by the interaction of the magnetic dipole with Earth’s magnetic field, described by the vector cross product τ=m×B.

• Geomagnetic Field (B): The Earth’s magnetic field, which ranges from approximately 25 to 65 µT in Low Earth Orbit (LEO).

3. Literature Review

The field of CubeSat attitude control is built upon several foundational control laws and hardware designs.

One of the earliest and most widely cited methods is the B-dot controller. Mark L. Psiaki demonstrated that this approach—which uses the rate of change of the magnetic field (dtdB​) to determine the direction of the magnetic dipole—can effectively slow a spinning satellite without needing to estimate its exact orientation. His analysis showed that this method naturally stabilizes the spacecraft’s rotation over time by creating a damping effect³.

Leomanni, in his comparative analysis, exemplified a faster method known as bang-bang control. In this approach, the system switches the magnetic dipole directly to its maximum or minimum strength depending on the direction of the change in the magnetic field. This provides a much stronger and quicker torque, helping the satellite detumble faster than the standard B-dot method. However, at low spin rates, the abrupt on-off nature of this method can cause the satellite to oscillate. To prevent this, a small threshold known as a "deadband" is introduced, or the system can be designed to automatically switch to the gentler B-dot method once the satellite slows sufficiently⁴.

Another critical method, described by Wertz (2001), is Proportional-Derivative (PD) control. This method allows not only for detumbling but also for coarse pointing—for example, aligning the satellite’s solar panels toward the Sun. It calculates the required torque based on the current rotation rate and the difference between the current and desired orientations. Since magnetorquers can only create torque perpendicular to the Earth’s field, the calculated torque is projected onto the perpendicular plane. While PD control offers more precise pointing, it requires knowledge of the satellite’s attitude, necessitating additional sensors like sun sensors or star trackers and more complex estimation algorithms like Kalman filters⁵.

Regarding hardware, review papers by Bellini (2012) and a UPC Thesis by Salamat et al. (2024) compare various magnetorquer designs, highlighting the trade-offs in dipole strength per watt, mass density, and integration ease. They conclude that air-core and PCB coils offer minimal mass at the cost of lower torque, while rod-core coils deliver the highest dipole moment density but require demagnetization cycles to prevent core saturation.

4. The Attitude Control Problem

Upon deployment, 1U CubeSats typically enter an uncontrolled tumbling state with unpredictable spin axes and high angular velocities (10–20°/s)¹. The primary task of the Attitude Determination and Control System (ADCS) is to arrest this rotation—a process known as detumbling—and bring the spacecraft to a stabilized state from which it can perform mission-specific pointing maneuvers.

Magnetorquers are the most widely adopted actuators for CubeSat ADCS due to several practical advantages⁶:

• Low Mass and Volume: They are compact and lightweight.

• High Reliability: They have no moving parts, reducing the risk of mechanical failure.

• No Propellant: They rely solely on electrical power, allowing for a long operational life.
  

Magnetorquers generate a magnetic dipole moment, m, which interacts with Earth’s geomagnetic field, B, to produce torque via the cross product:

τ=m×B

This torque reorients the spacecraft. In LEO (altitudes ≤ 500 km), Earth’s magnetic field strength (30–50 µT) is sufficient for magnetorquers to generate meaningful torques. However, they are ineffective in deep space where no significant external magnetic field exists⁷.

For the initial detumbling phase, simple and robust algorithms like the B-dot controller are frequently employed. This controller passively aligns the magnetic dipole to oppose the satellite’s angular velocity, effectively damping the rotation without requiring precise attitude knowledge—only time derivatives of magnetometer measurements are needed, as explained by Psiaki (2001).

In summary, a CubeSat ADCS must quickly damp post-launch tumbling and then maintain or change orientation within strict power and mass limits. Magnetorquers offer a reliable, fuel-free method for detumbling and coarse pointing in LEO.

5. Magnetorquers: Principle and Action

A magnetorquer is an electromagnetic coil that produces a magnetic dipole m when current flows through it. The dipole interacts with Earth’s field B, generating a torque τ=m×B. The resulting torque attempts to align the coil’s dipole with the ambient field. By adjusting the currents on three orthogonal coils, one can produce torques about the two axes perpendicular to the local B-vector. No torque can be generated along the B-field axis (τ is zero if m∣∣B).

Magnetorquers are widely used in small satellites because they are light, reliable, and draw only electrical power⁵. However, they produce relatively weak torques, making attitude changes slow. The maximum torque is limited by the coil current and, for core-based designs, material saturation. Thus, magnetorquers are best suited for coarse control (detumbling) rather than fast maneuvers or precision pointing.

For a typical 1U CubeSat (inertia ~0.002 kg·m²) in LEO, a magnetorquer dipole of ~0.1 A·m² interacting with a ~50 µT field gives τ≈5×10−6 N·m, yielding an angular deceleration of ~2.5 ×10−3 rad/s². This can stop a 10°/s tumble in the order of 100 seconds (minutes). In practice, small CubeSats often carry three magnetorquers, one along each body axis, to allow for full 3-axis control⁸.

6. Common Magnetorquer Types

There are three main types of magnetorquers suitable for CubeSats, each with distinct trade-offs.

6.1. Air-Core Coils

These are simple wound wire loops around a non-magnetic frame. Air-core coils achieve moderate dipole moments with relatively low weight and are easy to fabricate⁹. However, achieving a high magnetic moment requires many turns or high current, which increases both mass and power demand¹⁰.

6.2. PCB (Embedded) Coils

Here, copper traces are printed onto the satellite’s circuit boards in a flat spiral, efficiently using space and adding almost no additional volume or mass¹¹. However, the thinness of PCB traces limits the number of turns, resulting in a much smaller magnetic dipole compared to a 3D coil. While multiple layers can be stacked, this increases manufacturing costs and requires high currents (and thus significant power) for relatively little torque¹¹,¹².

6.3. Rod-Core Coils (Torque Rods)

These are wire coils wound around a ferromagnetic core (e.g., iron alloy), which greatly amplifies the magnetic field, increasing the dipole moment for a given current. Rod-core torquers achieve the highest torque-per-watt of the three types¹⁰. The trade-off is higher mass due to the core and the need for a demagnetization cycle to counteract residual magnetism¹⁶.

6.4. Performance Summary

The choice of magnetorquer depends on the mission's priorities regarding torque, mass, power, and integration complexity.

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7. Control Algorithms for Magnetorquers

An effective ADCS requires not only hardware but also intelligent control algorithms to decide when and how to activate the magnetorquers.

7.1. B-dot Control

As previously mentioned, this algorithm is a simple and robust method for detumbling. It works by detecting the rate of change of the magnetic field (dtdB​) as the satellite spins. The controller generates a magnetic dipole in the opposite direction of this change to create a damping torque that slows the rotation. Its primary advantages are its simplicity and that it does not require attitude knowledge³. However, it cannot be used for active pointing and works best at higher spin rates.

7.2. Bang-Bang Control

This is a more aggressive, on-off version of B-dot control. It applies the maximum possible torque instantly when the magnetic field changes beyond a certain rate, maximizing deceleration and enabling the fastest detumble time⁴. Its main disadvantage is the risk of overshooting and inducing oscillations ("limit cycling") near zero velocity. In practice, a hybrid approach is often used, where bang-bang control is active at high spin rates before switching to a smoother B-dot or PD controller for final settling.

7.3. Proportional-Derivative (PD) Control

A PD controller is designed for pointing, not just detumbling. It calculates the necessary torque to move the satellite from its current orientation to a desired one. Because magnetorquers can only produce torques perpendicular to the local magnetic field, the algorithm must project the desired torque onto the plane orthogonal to B. This complexity, along with the need for full attitude estimation (requiring sun sensors, star trackers, and gyroscopes), makes PD control less suitable for the initial detumbling phase but essential for subsequent mission operations⁵.

Other advanced algorithms like LQR (Linear Quadratic Regulator) exist but are generally too complex for resource-constrained 1U satellites and are therefore omitted from this analysis¹³.

8. Comparative Configuration Analysis

This section analyzes three hypothetical 1U CubeSat ADCS configurations to compare their expected detumbling performance. Assumptions:

• Earth's magnetic field in LEO: ∣B∣≈4×10^−5 T

• 1U CubeSat moment of inertia: I≈2×10^−3 kg·m²

• Initial angular velocity: 10°/s (≈ 0.175 rad/s)

Configuration 1: Three-Axis Air-Coil with B-dot Control

With a practical dipole moment of m = 0.05 A·m², this configuration yields a maximum torque of τ≈2×10−6 N·m. This results in an angular deceleration of ≈1×10−3 rad/s², stopping the initial spin in approximately 175 seconds (about 3 minutes). This configuration offers moderate detumble time with low hardware complexity but may struggle with very high initial spin rates.

Configuration 2: Three-Axis Rod-Core with Bang-Bang Control

Rod-cores can easily achieve a dipole of m = 0.5 A·m² for the same current, generating a torque ten times higher: τ≈2×10−5 N·m. The resulting deceleration is ≈1×10−2 rad/s², stopping the spin in just ~17 seconds. Despite the added mass and complexity of demagnetization, this option provides vastly superior performance for rapid detumbling.

Configuration 3: Three-Axis PCB Coils with PD Control

PCB coils achieve a very low dipole of m ≈ 0.005 A·m², yielding minuscule torques (τ≈2×10−7 N·m) and extremely slow detumble times (hundreds of seconds). While a PD controller would enable precise pointing after stabilization, this configuration is highly inefficient for the initial detumbling phase.

Analytical Trade-offs: Configuration 2 (Rod-Core) produces by far the highest torque-to-power ratio, enabling detumbling in tens of seconds. Configuration 1 (Air-Core) is slower but simpler, while Configuration 3 (PCB) is the least effective for detumbling.

9. Recommended Pairing

For a 1U CubeSat requiring low complexity and reliable detumbling in LEO, the recommended configuration is a 3-axis array of rod-core magnetorquers controlled by a hybrid B-dot and bang-bang algorithm.

• Magnetorquer Choice: A ferromagnetic "torque rod" on each axis provides the largest magnetic moment per watt¹⁵. Despite a slight mass penalty, the high torque ensures that detumbling can be accomplished quickly and within realistic power limits¹⁰.

• Control Law: A hybrid controller leverages the strengths of both algorithms. A bang-bang variant is used at high spin rates for the fastest possible damping. Once the angular velocity drops below a predefined threshold, the system switches to a standard B-dot law for smooth final settling, avoiding the oscillations typical of pure bang-bang control.

This system maximizes detumble speed and reliability while minimizing hardware and software complexity, making it an ideal choice for resource-constrained CubeSat missions.

10. References

1. California Polytechnic State University. “CubeSat Design Specification, Rev. 13.” Cal Poly CubeSat Program, 2014. https://static1.squarespace.com/static/5418c831e4b0fa4ecac1bacd/t/56e9b62337013b6c063a655a/1458157095454/cds_rev13_final2.pdf.

2. Aerospace Corporation. “AeroCube 7‑OCSD‑A: Aerospace CubeSats report typical rotation rates of 10–20 °/s after deployment.” EOportal. https://www.eoportal.org/satellite-missions/aerocube-ocsd

3. Psiaki, Mark L. “Magnetic Torquer Attitude Control via Asymptotic Periodic Linear Quadratic Regulation.” Cornell University GPS Laboratory. Accessed August 3, 2025. https://gps.mae.cornell.edu/magtorquer_alqr.pdf.

4. Leomanni, Mirko. 2014. “Comparison of Control Laws for Magnetic Detumbling.” ResearchGate, June 2014. https://www.researchgate.net/publication/263008407_Comparison_of_Control_Laws_for_Magnetic_Detumbling

5. Wertz, James. “Spacecraft Attitude Determination and Control.” Astrophysics and Space Science Library, 1978. doi:10.1007/978-94-009-9907-7. https://www.academia.edu/108794040/Spacecraft_Attitude_Determination_and_Control

6. PW-Sat2 Team. Critical Design Review: Attitude Determination and Control System (CDR Report). PW‑SAT2 Project, 2014. https://pw-sat.pl/wp-content/uploads/2014/07/PW-Sat2-C-01.00-ADCS-CDR.pdf

7. Finlay, Christopher C., et al. 2010. “The CHAOS-4 Geomagnetic Field Model.” Geophysical Journal International 183 (3): 1216–30. https://doi.org/10.1111/j.1365-246X.2010.04804.x.

8. Praks, J. 2022. Design of Magnetorquer‑Based Attitude Control Subsystem for FORESAIL‑1 Satellite. Aalto University. https://acris.aalto.fi/ws/portalfiles/portal/88686047/Design_of_Magnetorquer_Based_Attitude_Control_Subsystem_for_FORESAIL_1_Satellite.pdf

9. Hauf, Paul Jannik, and Nadim Maraqten. "A Guide to Self‑Built Low‑Cost Magnetorquers as will be used in the 3U+ CubeSat SOURCE." 12th European CubeSat Symposium, Paris, 2021. https://www.researchgate.net/publication/361548818_A_Guide_to_Self-Built_Low-Cost_Magnetorquers_as_will_be_used_in_the_3U_CubeSat_SOURCE

10. Bellini, Niccolò. "Magnetic Actuators for Nanosatellite Attitude Control." Master’s thesis, University of Bologna, 2014. https://amslaurea.unibo.it/id/eprint/7506/1/Bellini_Niccol%C3%B2_Tesi.pdf

11. Suchantke, Ingolf, Sebastian Grau, and Klaus Briess. “A Comprehensive Study on Magnetic Actuator Design for CubeSat Missions.” ResearchGate, 2017. https://www.researchgate.net/publication/361463779_A_Comprehensive_Study_on_Magnetic_Actuator_Design_for_CubeSat_Missions

12. Sorensen, Nicholas J. “Efficiency‑Optimized Design of PCB‑Integrated Magnetorquers for CubeSats.” arXiv preprint, September 2020. https://arxiv.org/abs/2009.07981

13. S. Satellite for Optimal Control and Imaging (SOC‑i), “Satellite for Optimal Control and Imaging (SOC‑i): A CubeSat Demonstration of Optimization-Based Real-Time Constrained Attitude Control,” NASA SSSVI‑KB (2021), accessed August 3, 2025. https://s3vi.ndc.nasa.gov/ssri-kb/static/resources/SOC_i__A_CubeSat_Demonstration_of_Optimization_Based_Real_Time_Constrained_Attitude_Control.pdf

14. Sancho, Jordi. “Magnetorquers for Low Earth Orbit CubeSat’s Attitude Control.” Bachelor Thesis, Universitat Politècnica de Catalunya, April 2021. https://upcommons.upc.edu/bitstream/handle/2117/360868/TFG_report_JordiSancho.pdf

15. CubeSatShop. “NCTR‑M002 Magnetorquer Rod.” CubeSatShop.com (NewSpace Systems), accessed August 3, 2025. https://www.cubesatshop.com/product/nctr-m002-magnetorquer-rod/

16. Hao, Zhou. "Detumbling Control of ΔDsat Satellite: Experimental Results." Master’s Thesis. https://www.diva-portal.org/smash/get/diva2%3A1026679/FULLTEXT02.pdf