University study material for Spacecraft Attitude Dynamics and Control in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Department of Aerospace Engineering Spacecraft Attitude Dynamics Domande Esame Author: Mattia Bonanomi Academic Year: 2025-2026 Contents Acronyms I 1 Attitude dynamics and kinematics, disturbances 1 1.1 Attitude Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
University study material for Spacecraft Attitude Dynamics and Control in the Aerospace Engineering degree programme at Politecnico di Milano. The document covers: Department of Aerospace Engineering Spacecraft Attitude Dynamics Domande Esame Author: Mattia Bonanomi Academic Year: 2025-2026 Contents Acronyms I 1 Attitude dynamics and kinematics, disturbances 1 1.1 Attitude Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Department of Aerospace Engineering Spacecraft Attitude Dynamics Domande Esame Author: Mattia Bonanomi Academic Year: 2025-2026
1 - Attitude dynamics and kinematics, disturbances D1.2.7 Euler angles for kinematics, explain also for small rotations Euler angles (𝜙, 𝜃, 𝜓) parametrize attitude through three consecutive independent rotations. The exact non-linear kinematic differential equation mapping the body angular rates 𝝎 to the Euler angle rates is given by: 𝜙¤ cos 𝜃 sin 𝜙 sin 𝜃 cos 𝜙 sin 𝜃 𝜔 𝑥 1 𝜃¤ = (1.7) cos 𝜃 0 cos 𝜙 cos 𝜃 − sin 𝜙 cos 𝜃 𝜔 𝑦 𝜓¤ 0 sin 𝜙 cos 𝜙 𝜔 𝑧 This formulation possesses a geometric singularity (gimbal lock) when 𝜃 = ±90◦ . For small rotations (𝜙, 𝜃, 𝜓 1), we approximate cos(·) ≈ 1 and sin(·) ≈ 0. The kinematic relation simplifies to a linear, decoupled system: 𝜙¤ 𝜔 𝑥 𝜃¤ ≈ 𝜔 𝑦 (1.8) 𝜓¤ 𝜔 𝑧 Under these conditions, consecutive rotations commute. In our project control design, this small-rotation approximation allows us to write the linearized attitude error matrix 𝑨𝑒 as: 1 𝛼𝑧 −𝛼 𝑦 (1.9) 𝑨𝑒 = 𝑨 𝐵/𝑁sens 𝑨𝑇target = −𝛼𝑧 1 𝛼 𝑥 𝛼𝑦 −𝛼 𝑥 1 where 𝛼 𝑥 , 𝛼 𝑦 , 𝛼𝑧 denote the small tracking error angles. Traduzione: Gli angoli di Eulero parametrizzano l’assetto attraverso tre rotazioni indipendenti consecutive. L’equazione dif- ferenziale cinematica esatta e non lineare che mappa le velocità angolari del corpo sulle velocità degli angoli di Eulero presenta una singolarità geometrica (gimbal lock) a ±90◦ . Per piccole rotazioni, approssimiamo i coseni a 1 e i seni a 0. La relazione cinematica si semplifica in un sistema lineare disaccoppiato (le velocità degli angoli eguagliano le velocità del corpo). In queste condizioni, le rotazioni consecutive commutano. Nel design del nostro progetto, questa…
2 - Attitude Sensors, Actuators, and Determination D2.1.7 How was the sun sensor modelled? (project related) The sun sensor was modeled in Simulink by first calculating the true Sun vector in the inertial frame using an ephemeris block. This vector was then rotated into the spacecraft body frame using the true DCM 𝑨 𝐵/𝑁 . Finally, to simulate the sensor output, we added white Gaussian noise to the vector components and re-normalized the vector to ensure it remained a unit vector. Traduzione: Il sensore solare è stato modellato partendo dal vettore solare vero (calcolato con efemeridi), ruotandolo nel sistema del corpo tramite la DCM reale, aggiungendo poi rumore gaussiano e infine ri-normalizzando il vettore risultante per mantenerlo unitario. D2.1.8 Does the sun sensor output a normalized vector? (project related) Yes, typical digital 2-axis sun sensors process the internal array data to output a 3D unit vector 𝒔ˆ𝑏 pointing toward the Sun in the body frame. Because it relies purely on the angle of incidence of the incoming light, it cannot measure the distance to the Sun; therefore, the output is strictly a normalized directional vector. Traduzione: Sì, i sensori digitali a 2 assi elaborano i dati per fornire un versore 3D unitario che punta al Sole. Basandosi solo sull’angolo di incidenza, non possono calcolarne la distanza, per cui l’output è rigorosamente un vettore direzionale normalizzato. D2.1.9 Does this sun sensor output a 3D vector? Yes. While a 1-axis sun sensor only provides a 2D angle (azimuth or elevation), a standard 2-axis digital sun sensor provides both. These two angles are mathematically converted inside the sensor’s processing unit to output a 3D normalized unit vector in the sensor’s reference frame. Traduzione: Sì. Mentre quelli a 1 asse…
2 - Attitude Sensors, Actuators, and Determination D2.3.13 Magnetotorquers: derivation and Reaction Wheel direction Magnetotorquers are electromagnetic coils that generate a controllable magnetic dipole moment 𝒎. When interacting with the Earth’s magnetic field 𝑩, they produce a mechanical torque: 𝑻=𝒎×𝑩 (2.4) Because the cross product means they can never generate torque parallel to the 𝑩 field, they cannot control all 3 axes simultaneously. Thus, Reaction Wheels (RWs) are sized and oriented (often in a pyramid or tetrahedral configuration) to guarantee 3-axis control, while the MTQs are used purely to dump the secular momentum accumulated by the RWs over the orbit. Traduzione: Creano dipoli magnetici tramite bobine. Interagendo col campo terrestre generano una coppia perpendicolare ad esso. A causa del prodotto incrociato, perdono sempre il controllo sull’asse parallelo al campo magnetico. Le ruote di reazione servono al controllo completo 3D, mentre questi magneti si usano solo per ”scaricare” il momento immagazzinato dalle ruote. 2.3.4 Thrusters D2.3.14 Thrusters configuration and null space vector While 6 thrusters can theoretically provide minimum 3-axis translation and rotation, 12 thrusters are standard to provide pure uncoupled torques (using thruster pairs to create force-free couples) and full redundancy. A null space vector 𝒘 is a mathematical combination of thruster firings that results in exactly zero net force and zero net torque on the spacecraft (𝑨𝑡 ℎ𝑟𝑢𝑠𝑡𝑒𝑟 𝑠 𝒘 = 0). This is highly useful for managing propellant evenly across tanks or satisfying minimum impulse bit constraints without disrupting the attitude. Traduzione: Sebbene bastino 6 propulsori, se ne usano 12 per fornire coppie pure disaccoppiate e ridondanza. Il vettore spazio…
3 - State space control and attitude control D3.2.12 Schmidt Trigger for thruster control (what is the aim and why parabolic profile) The aim of the Schmidt trigger is to maintain pointing within a deadband without exhausting fuel via valve chattering. In the phase plane (𝜃 vs 𝜃), ¤ the trajectory is a parabola because a constant thruster torque applies a constant angular acceleration (𝜃¥ = const). Integrating constant acceleration yields velocity proportional to time, and position proportional to time squared, resulting in 𝜃¤2 ∝ 𝜃. Traduzione: Crea un range morto di risparmio carburante per thruster. Sul piano delle fasi forma archi di parabole matematiche in quanto l’accelerazione lineare data dall’accensione a potenza immutata è proporzionale a parabola se sbarrata alla coordinata posizione/spazio nel tempo integrato. D3.2.13 Control logic with Schmidt trigger and what is the main assumption The logic turns the thruster ON when error exceeds a set upper bound and turns it OFF when it drops below a smaller lower bound. The main assumption is that the spacecraft acts as a pure rigid body, and that the thruster instantly delivers a perfect, constant square-wave torque impulse without rise-time delays or transient dynamics. Traduzione: Spara ”ON” solo quando gli sforamenti passano i limiti netti superiori. Spegne ”OFF” molto più all’interno e lontano rispetto alla barra in rientro. Assume logicamente però, che la scocca spaziale regga un’onda quadra a picco pulita senza vibrare e il liquido iniettato sia sincrono con zero lag e senza rampe dinamiche intermedie. D3.2.14 Control with thruster and describe the logic of the Schmidt Trigger control The logic relies on a relay with hysteresis and a deadband. If the attitude error exceeds a certain high threshold, the…
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