Purpose
How Spacecraft Point
Quaternion dynamics, reaction wheels, and keep-out constraints. This is Chapter 3 of Corbin's dissertation -- proving a spacecraft's boresight NEVER enters a dangerous cone during maneuvers.
The Physics
Euler's Equation
omega_dot = J^{-1} * (-omega x J*omega + tau)
Quaternion Kinematics
q_dot = 0.5 * H(q) * omega
Lyapunov Function (Corbin's Eq 3.11)
V = Kp*sigma^2 + 0.5*omega^T*J*omega + Kp*(1-q0)^2
Key Insight: If V(x) <= 18.26, the boresight is GUARANTEED safe. No simulation needed. This is formal verification applied to real hardware (ASTROS at Georgia Tech).
Historical
William Rowan Hamilton (1805-1865)
Invented quaternions in 1843. Carved i^2=j^2=k^2=ijk=-1 into a bridge in Dublin. Hubble: 0.007 arcsecond pointing -- like hitting a dime 200 miles away.
References
Wertz, Spacecraft Attitude Determination and Control | Corbin Klett dissertation, Chapter 3 | MIT OCW 16.346