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Spacecraft Attitude Determination & Precision Pointing

Instrument pointing is an EM measurement problemOptical and RF boresight orientation determines which field/radiance direction is sampled. Pointing error, attitude jitter and misalignment can dominate scientific uncertainty even with perfect detectors.

1. Attitude dynamics and measurements

For body angular rate $\boldsymbol\omega$ and scalar-first unit quaternion $q=(q_0,\mathbf q)$, one consistent convention is

\[\dot q=\frac12\,q\otimes(0,\boldsymbol\omega),\]

provided $q$ maps body to inertial coordinates with the associated multiplication convention. Quaternion handedness and transformation conventions vary across software: document them.

Euler rigid-body dynamics are

\[\mathbf J\dot{\boldsymbol\omega}+\boldsymbol\omega\times(\mathbf J\boldsymbol\omega)=\boldsymbol\tau_\mathrm{ext}+\boldsymbol\tau_\mathrm{ctrl}.\]

Disturbance torques include gravity gradient, solar pressure, atmospheric drag, residual magnetization and mechanism motion.

2. Sensors and actuators

Component Output or function Calibration concern
Star tracker Absolute inertial attitude from star patterns Lost-in-space acquisition, boresight misalignment
Sun sensor Sun direction Field of view, albedo, calibration
IMU/gyro Angular-rate estimate Bias, scale, random walk
Magnetometer Local magnetic-field vector Spacecraft magnetic contamination
Reaction wheel Smooth torque via momentum exchange Jitter, momentum saturation
Magnetic torquer Dipole interacting with Earth’s magnetic field No torque parallel to local magnetic field
Thruster Torque or momentum dumping Impulse variability and disturbance

3. Kalman filtering and pointing budget

An error-state EKF combines star-tracker attitude updates with gyro propagation; bias estimates and measurement covariance matter. Report knowledge (estimated attitude error) separately from control (actual jitter). Optical point-spread smearing depends on angular jitter relative to pixel angular scale and exposure time.

Worked check: for focal length 1 m and 10 µm pixel pitch, geometric pixel angular scale is $\approx10$ µrad ≈2.06 arcsec/pixel under small-angle approximation. A 1 arcsec line-of-sight drift is ~0.485 pixel.

Exercise: integrate quaternion attitude from a biased gyro, periodically correct with noisy star-tracker observations, and compare error with/without bias estimation.

4. References