attitude-control

name	attitude-control
description	Spacecraft attitude determination and control system (ADCS) — Euler equations of rigid body rotation, reaction wheels (momentum exchange), magnetorquers (magnetic dipole), thrusters (torque arm), attitude sensors (star tracker, sun sensor, magnetometer, gyroscope), attitude representations (quaternions, direction cosines, Euler angles), control law (PD, PID), detumbling algorithm (B-dot), pointing budget, and ECSS/NASA standards for ADCS design.
metadata	{"priority":7,"promptSignals":{"phrases":["attitude control","ADCS","reaction wheel","spacecraft attitude","attitude determination","spacecraft pointing"],"minScore":3}}

Spacecraft Attitude Determination and Control (ADCS) — Complete Skill

Euler's Equations of Motion

Rigid Body Rotational Dynamics

Euler's equations (body frame): I_x × ω̇_x − (I_y − I_z) × ω_y × ω_z = T_x [x-axis; T = external torque] I_y × ω̇_y − (I_z − I_x) × ω_z × ω_x = T_y I_z × ω̇_z − (I_x − I_y) × ω_x × ω_y = T_z

For small angular rates (linearized): I × ω̇ ≈ T [decoupled; each axis; valid for ω_body << ω_orbital]

Angular momentum: H = I × ω [H = angular momentum vector; I = inertia tensor; ω = angular velocity] For symmetric satellite: H = [I_xx 0 0; 0 I_yy 0; 0 0 I_zz] × ω

Euler instability: If I_x ≠ I_y ≠ I_z (all different): rotation about intermediate axis (I_y) is unstable Rotation about major (I_z max) or minor (I_x min) axis: stable Practical: use spin stabilization about major axis; or 3-axis control to avoid Euler instability

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Attitude Representations

Quaternions (Preferred for Flight Software)

Unit quaternion: q = [q₁ q₂ q₃ q₄]ᵀ [q₄ = scalar; q₁q₂q₃ = vector; q₁²+q₂²+q₃²+q₄² = 1] q₄ = cos(φ/2); [q₁ q₂ q₃] = sin(φ/2) × ê [φ = rotation angle; ê = unit axis vector] Advantages: no singularities (vs. Euler angles); compact; numerically efficient kinematics

Quaternion kinematics: q̇ = 0.5 × Ω(ω) × q [Ω = 4×4 skew-symmetric matrix from ω] ⎡q̇₁⎤ 0.5 × ⎡ 0 ω₃ -ω₂ ω₁⎤ ⎡q₁⎤ ⎣q̇₄⎦ ⎣-ω₁ -ω₂ -ω₃ 0 ⎦ ⎣q₄⎦

Euler angles (roll φ, pitch θ, yaw ψ) — intuitive but singularities: Gimbal lock at θ = ±90° for 3-2-1 sequence Attitude matrix (DCM) from Euler 3-2-1: C = C₃(ψ) × C₂(θ) × C₁(φ)

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Actuators

Reaction Wheels (Momentum Exchange)

Operation: electric motor spins flywheel; spacecraft counter-rotates (angular momentum conservation) H_wheel + H_spacecraft = constant [no external torques] Torque on spacecraft: T_sc = −T_wheel = −I_w × α_w [I_w = wheel inertia; α_w = wheel angular acceleration]

Wheel momentum capacity: H_max = I_w × ω_max [storage; spacecraft rotates until wheel hits speed limit → "saturation"] Desaturation: use magnetorquers or thrusters to dump momentum

Typical reaction wheel: Small CubeSat RW: H_max = 30 mN·m·s; torque = 1 mN·m; mass = 130 g; power = 0.3 W Medium satellite: H_max = 4 N·m·s; torque = 100 mN·m; mass = 1.8 kg; power = 5 W Large GEO platform: H_max = 40–100 N·m·s; for stationkeeping momentum storage

Redundancy: 4 wheels in skewed configuration (tetrahedral) → any 3 provide full 3-axis control

Magnetorquers (Magnetic Torque Rods)

Operation: current through coil generates magnetic dipole m; Earth's field B → torque T = m × B Limitation: torque is always perpendicular to B → cannot control about B direction (need ≥2 orbits for full attitude control) Used for: detumbling, momentum dumping

Torque magnitude: T = m × B [m = magnetic dipole moment [A·m²]; B = Earth field [Tesla]] At 400 km LEO: B ≈ 50 μT; m = 0.1 A·m² → T = 5 μN·m Moment of typical magnetorquer: 1–5 A·m² for 6U CubeSat to small satellite

B-dot detumbling algorithm: m_cmd = −k_B × Ḃ [k_B = gain; Ḃ = rate of change of measured B-field in body frame] Effective and robust: automatically reduces angular rate; requires no attitude knowledge; standard for initial detumbling

Thrusters (Torque from Moment Arms)

Torque from thruster: T = F_thrust × d [d = moment arm from center of mass to thruster; F = thrust force] Bipropellant thruster pair: 5 N each; moment arm 0.5 m → T = 2.5 N·m (large satellite maneuvers) Cold gas thrusters: 0.1–1 N; GN₂ or butane; simple; for fine attitude control

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Attitude Sensors

Sensor Summary

Sensor	Accuracy	Update Rate	Power
Star tracker	5–50 arcsec	1–10 Hz	3–10 W
Sun sensor (coarse)	0.1–2°	10 Hz	0.05 W
Sun sensor (fine)	0.01°	100 Hz	0.2 W
Magnetometer	1–2°	10 Hz	0.1 W
Gyroscope (MEMS)	0.1–1°/hr drift	100 Hz	0.5 W
Gyroscope (FOG)	0.001°/hr drift	1000 Hz	5–10 W
Earth horizon sensor	0.1–0.5°	5 Hz	1 W
GPS receiver	position ±10 m	1 Hz	2 W

Star tracker — most accurate: Measures: vector directions to multiple stars → compares to star catalog → full attitude solution Exclusion angle from Sun: 30–60° typically (bright source blinds detector) Accuracy: 5 arcsec (1σ) per axis → < 0.002° absolute pointing

Sensor fusion (Kalman filter): Extended Kalman Filter (EKF) for nonlinear attitude estimation: x̂_k+1 = f(x̂_k, u_k) + K × (z_k − h(x̂_k)) [K = Kalman gain; z = measurement; h = measurement model] State vector: quaternion (4) + gyro bias (3) = 7 states Typical: gyro propagation at 100 Hz; star tracker update at 1–10 Hz; magnetometer at 10 Hz

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Control Laws

PD Control (Standard 3-Axis)

Attitude error (quaternion error): q_err = q_cmd ⊗ q_meas⁻¹ [quaternion multiplication; small q_err → small angle error] θ_err ≈ 2 × [q_err_1; q_err_2; q_err_3] [for small angles; vector part ≈ half-angle-axis]

PD control law: T_cmd = −K_P × θ_err − K_D × ω_err [K_P = proportional gain; K_D = derivative gain; ω_err = angular rate error] Closed-loop natural frequency: ω_n = √(K_P/I); damping ratio: ζ = K_D/(2×√(K_P×I)) Target: ω_n = 0.1 × orbit rate (avoids orbital resonance); ζ = 0.7 (critically damped)

Example CubeSat: I_z = 0.01 kg·m² (10 cm × 10 cm × 20 cm); target ω_n = 0.05 rad/s K_P = ω_n² × I = (0.05)² × 0.01 = 2.5×10⁻⁵ N·m/rad K_D = 2×ζ×ω_n×I = 2×0.7×0.05×0.01 = 7×10⁻⁴ N·m·s/rad

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Pointing Budget

Error Sources

Pointing error budget (3σ, all sources RSS'd):

Error Source	Magnitude [arcsec]
Star tracker measurement noise	±15
Gyro noise during propagation	±10
Structural thermal flex	±20
Alignment calibration residual	±30
Control bandwidth limitation	±40
Total (RSS)	±55 arcsec ≈ 0.015°

APE (Absolute Pointing Error): total 3σ pointing error wrt inertial frame RPE (Relative Pointing Error): pointing stability over short time window (often tighter requirement than APE)

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Standards and References

Standard	Scope
ECSS-E-ST-60-30C	Attitude and orbit control (AOCS)
ECSS-E-ST-60-10C	Control engineering
NASA-RP-1288	Orbital mechanics for engineering students
MIL-STD-1540D	Environmental test methods for launch vehicles
ISO 26900	Space systems — attitude control
SMAD (Wertz)	Space Mission Analysis and Design, Chapter 11

Output

Provide: mission requirements (orbit: LEO/GEO/HEO; pointing requirement APE [arcsec]; stabilization mode: 3-axis/spin/gravity-gradient; mission: imaging/comms/science), inertia properties (I_xx/I_yy/I_zz [kg·m²]; products of inertia; stability check: rotating about max inertia axis?), actuators (reaction wheels: torque [mN·m]; momentum [N·m·s]; count; configuration; wheel saturation check with max disturbance torque × orbit period [N·m·s]; magnetorquers if: LEO, low-power, detumble; thrusters if: GEO, large maneuvers), sensors (star tracker accuracy [arcsec]; gyro type; drift rate [°/hr]; sensor fusion: EKF states), control law (PD or LQR; K_P = ω_n²×I; K_D = 2ζω_n×I; natural frequency ω_n [rad/s]; damping ζ; settling time [s]; control bandwidth vs. orbit rate), pointing budget (table: each error source [arcsec]; RSS total APE [arcsec]; meets requirement?), detumbling (B-dot algorithm; time to detumble from ω_0 = [rad/s] to ω_final < [rad/s]; requires [orbits]), and applicable standard (ECSS-E-ST-60-30C; ISO 26900; SMAD Chapter 11 for sizing).