The state of a vehicle relative to a chosen reference frame. The state typically contains position, velocity, and attitude.
\(\hat{\mathbf{x}}\) is the state estimate. \(\mathbf{P}\) is the state error covariance, which quantifies how much we trust each component. Reporting an estimate without an honest uncertainty hides the most operationally relevant information and breaks every downstream integrity calculation.
Truth (\(\mathbf{x}_{\text{truth}}\)) is the actual state of the vehicle (approximated in test by a higher-grade reference). Measurement (\(\mathbf{z}\)) is the noisy sensor output, often a nonlinear function of the truth. Estimate (\(\hat{\mathbf{x}}\)) is what the navigation system reports.
\(\mathbf{e} = \hat{\mathbf{x}} - \mathbf{x}_{\text{truth}}\). Flight test evaluates a navigation system by collecting many samples of \(\mathbf{e}\) and asking whether the resulting empirical distribution meets requirements.
Earth-Centered, Earth-Fixed. A global Cartesian frame attached to the rotating Earth with origin at the Earth's center of mass, \(X\) through (lat 0°, lon 0°), \(Y\) through the equator at lon 90°E, and \(Z\) along the spin axis through the North Pole. Natural for GNSS geometry; less intuitive for local error metrics.
North-East-Down. A local tangent-plane frame defined at a chosen reference latitude/longitude/height. \(N\) along the local meridian, \(E\) along the local parallel, \(D\) normal to the local-level plane. Standard frame for aviation navigation and error reporting.
Radial-Transverse-Normal: a local-orbit reference frame centered on a spacecraft. \(R\) radial outward from Earth's center, \(T\) along-track in the direction of motion, \(N\) cross-track normal to the orbital plane (right-hand rule). RTN rotates with the orbit; for spacecraft it plays the same role NED plays for aircraft.
Orbital relative motion: rendezvous and proximity operations (RPO), formation flying, debris collision-risk reporting. The Clohessy-Wiltshire (Hill's) equations of relative motion are written in RTN, and any "radial / along-track / cross-track" decomposition of an orbital error implicitly uses the RTN frame.
A frame rigidly attached to the airframe. SY6301 uses forward-right-down (FRD): \(X\) forward along the nose, \(Y\) out the right wing, \(Z\) down through the belly. Aircraft sensors physically measure in this frame.
A frame rigidly attached to the spacecraft structure (origin at center of mass or principal-inertia reference point). Conceptually identical to the aircraft body frame, but with mission-dependent axis assignments instead of FRD: \(X\) typically aligned with primary instrument boresight or nadir, \(Y\) along a solar array or side panel, \(Z\) completes the right-hand rule. It is the working frame for star trackers, gyros, sun sensors, and thrusters, and the input to attitude determination, control, and pointing analysis.
It rotates the components of a vector from frame \(a\) to frame \(b\): \(\mathbf{v}^b = \mathbf{C}_a^b \mathbf{v}^a\). The same physical arrow is expressed in two different sets of axes. The vector itself does not move; only its numerical representation changes.
A DCM is orthonormal: (1) \(\mathbf{C}^\top \mathbf{C} = \mathbf{I}\); (2) \(\left(\mathbf{C}_a^b\right)^{-1} = \left(\mathbf{C}_a^b\right)^{\top} = \mathbf{C}_b^a\). The inverse equals the transpose. To go from \(b\) back to \(a\), transpose; do not invert.
\(\displaystyle \mathbf{C}_a^b(\theta) = \begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}\). Apply it to a vector and the components change while the magnitude is preserved (within floating-point round-off).
A point requires both rotation and translation. If \(\mathbf{r}_{a\rightarrow b}\) is the position of frame \(b\)'s origin expressed in frame \(a\), then \(\mathbf{p}^b = \mathbf{C}_a^b \left(\mathbf{p}^a - \mathbf{r}_{a\rightarrow b}\right)\). The translation step is what is missing if you treat a point like a free vector.
A sequence of three elementary rotations about the \(x\), \(y\), and \(z\) axes. The Euler 321 convention used in this course is: \(\mathbf{C}_a^b(\phi, \theta, \psi) = \mathbf{R}_z(\psi) \mathbf{R}_y(\theta) \mathbf{R}_x(\phi)\). Order matters. RPY suffers from gimbal lock at \(\theta = \pm 90^\circ\), which is one motivation for using quaternions in flight software.
\(\displaystyle \mathbf{e}_{\text{NED}} = \hat{\mathbf{p}}_{\text{NED}} - \mathbf{p}_{\text{NED, truth}} = \begin{bmatrix} e_N \\ e_E \\ e_D \end{bmatrix}\). Horizontal error: \(e_H = \sqrt{e_N^2 + e_E^2}\). Vertical error: \(e_V = \lvert e_D \rvert\). Most accuracy requirements are written as upper 95% confidence bounds on these two scalars.
The components change because they are expressed in different axes. The magnitude \(\lVert \mathbf{e} \rVert\) does not change, because rotation preserves length. The split of error into horizontal and vertical only makes sense in NED, because horizontal-versus-vertical is defined by the local-level plane.
Statistics on inconsistent frames mix apples and oranges: a 1 m bias in NED north is not comparable to a 1 m bias in body forward, and a covariance ellipse in ECEF cannot be interpreted as a horizontal protection level. Every integrity test (Mahalanobis distance, protection level, HMI exposure) implicitly assumes its inputs share a common frame, common units, and a common reference point.
TLE is the difference between an estimated target position and the true target position, expressed in a local-level frame so it is operationally interpretable. The standard chain is: (1) convert truth and estimate from LLH to ECEF, (2) convert both ECEF positions to NED about a chosen reference (typically the truth point), (3) subtract: \(\mathbf{e}_{\text{NED}} = \hat{\mathbf{p}}_{\text{NED}} - \mathbf{p}_{\text{NED, truth}}\), (4) reduce to \(e_H\) and \(e_V\). The MATLAB demo TargetLocationError.m walks through this end to end.