ECI, ECEF, local navigation, body frames. Euler angles, DCMs, quaternions, Earth models, and gravity.
You're standing on the Earth. Relative to the ground beneath you, you're stationary. Relative to the Sun, you're hurtling through space at 30 km/s. Relative to a satellite overhead, you're rotating at 460 m/s. Same person, three different answers for "velocity." The difference? The coordinate frame.
Navigation is fundamentally a multi-frame problem. Inertial sensors measure motion relative to inertial space. GPS measures position relative to a satellite constellation. But you want to know where you are relative to the Earth's surface. Getting from one to another requires careful mathematics.
Four frames dominate navigation:
| Frame | Symbol | Origin | Axes |
|---|---|---|---|
| Earth-Centered Inertial | i | Earth's center | Fixed relative to stars |
| Earth-Centered Earth-Fixed | e | Earth's center | Rotate with Earth |
| Local Navigation | n | User position | North, East, Down |
| Body | b | User position | Forward, Right, Down |
The Earth-Centered Inertial (ECI) frame has its origin at the Earth's center and its axes are fixed relative to distant stars. Newton's laws hold directly in this frame — no fictitious forces.
The z-axis points along the Earth's spin axis (toward the North Pole). The x-axis points toward the vernal equinox (where the Sun's apparent path crosses the equator at the March equinox). The y-axis completes the right-handed system.
Gyroscopes measure angular rate relative to this frame. Accelerometers measure specific force, which is the acceleration due to all forces except gravity — equivalent to the inertial-frame acceleration minus gravitational acceleration.
The Earth-Centered Earth-Fixed (ECEF) frame also has its origin at Earth's center, but its axes rotate with the Earth. The z-axis is the same as ECI (spin axis). The x-axis points through the intersection of the prime meridian (0° longitude) and the equator. The y-axis completes the right-hand system.
ECEF is the natural frame for GPS. Satellite positions are computed in ECEF coordinates, and the standard GPS position solution is in ECEF. A point fixed on Earth's surface has constant ECEF coordinates (ignoring tectonic motion).
The relationship between ECI and ECEF is a simple rotation about the z-axis at the Earth's rotation rate, ωie ≈ 7.292 × 10−5 rad/s:
Where Cie is the coordinate transformation matrix from ECI to ECEF, and Rz is a rotation about the z-axis.
You probably think about position as latitude, longitude, and height — not as Cartesian x,y,z from the Earth's center. The local navigation frame (n-frame) formalizes this intuition.
Its origin is at the user's current position. Its axes point North (x), East (y), and Down (z). This is called the NED convention.
The n-frame is the most natural frame for expressing a navigation solution: "I'm moving 5 m/s north and 2 m/s east" is more useful than Cartesian ECEF velocities.
Converting between ECEF and the local navigation frame requires the user's latitude L and longitude λ. The transformation matrix involves sines and cosines of these angles.
The body frame (b-frame) is attached to the vehicle (or person). Axes: x forward, y right, z down. In a strapdown INS, the accelerometers and gyroscopes are fixed to the body, so they naturally measure in body-frame axes.
The relationship between the body frame and the local navigation frame is the vehicle's attitude: roll, pitch, and heading. This is what the INS must track to transform accelerometer measurements from body axes to navigation axes.
How do we describe the orientation of one frame relative to another? Three main representations, each with pros and cons.
Euler angles: roll (φ), pitch (θ), and heading/yaw (ψ). Intuitive and compact (3 numbers), but suffer from gimbal lock at ±90° pitch, and the rotation order matters.
Coordinate Transformation Matrix (DCM, also called Direction Cosine Matrix): a 3×3 orthogonal matrix Cbn that transforms a vector from body axes to navigation axes. No singularities, but 9 numbers with 6 constraints (orthogonality).
Key DCM properties:
| Property | Expression |
|---|---|
| Inverse = transpose | Cnb = (Cbn)T |
| Determinant | det(C) = +1 |
| Chain rule | Cbn = Cen Cbe |
Quaternions represent rotation using four numbers: a scalar part and a 3D vector part. They avoid gimbal lock, are more compact than DCMs (4 vs. 9 numbers), and are efficient for computation.
A quaternion q = (q0, q1, q2, q3) satisfies the constraint q02 + q12 + q22 + q32 = 1 (unit quaternion). It represents a rotation by angle μ about axis e as:
Quaternion multiplication composes rotations. The DCM can be directly computed from the quaternion (and vice versa). Most modern INS implementations use quaternions for attitude propagation.
The rotation vector is another representation: a 3-element vector whose direction is the rotation axis and magnitude is the rotation angle. It is used in some attitude update algorithms but suffers from a singularity at ±2π.
| Representation | Elements | Singularity? | Composition |
|---|---|---|---|
| Euler angles | 3 | Yes (±90° pitch) | Complex |
| DCM | 9 (6 constraints) | No | Matrix multiply |
| Quaternion | 4 (1 constraint) | No | Quaternion multiply |
| Rotation vector | 3 | Yes (±2π) | Complex |
Navigation requires a mathematical model of the Earth's surface. The Earth isn't a perfect sphere — it bulges at the equator. The standard model is an oblate spheroid (ellipsoid of revolution).
Two radii define it: the equatorial radius R0 ≈ 6,378,137 m and the polar radius RP ≈ 6,356,752 m. The flattening f = (R0 − RP)/R0 ≈ 1/298.257.
Geodetic latitude L is the angle between the ellipsoid normal and the equatorial plane — this is what GPS gives you. Geocentric latitude is the angle from the center — different from geodetic due to flattening. Longitude λ is the angle east of the prime meridian. Height h is above the ellipsoid.
The geoid is a surface of constant gravitational potential that best fits mean sea level. Height above the geoid (orthometric height) is what we mean by "elevation" in everyday life. It differs from ellipsoid height by up to ±100 m.
This distinction trips up many people. Gravitation (γ) is the attractive force due to Earth's mass — what Newton's law gives you. Gravity (g) is what you actually feel when standing still: gravitation minus the centrifugal acceleration due to Earth's rotation.
At the surface, total gravity is about 9.8 m/s2. The centrifugal component contributes up to 0.034 m/s2 (at the equator). Small but significant for precision navigation.
Accelerometers measure specific force f, which is the acceleration due to all forces except gravity. For a stationary object on Earth's surface, the specific force is the reaction to gravity (pointing up), not zero. The relationship:
where a is the inertial acceleration. This means the INS must know gravity to get acceleration from the accelerometer output.
The WGS 84 Somigliana model gives surface gravity as a function of latitude. Height variation follows an inverse-square law. For high-precision applications, spherical harmonic gravity models or gravity anomaly databases are used.
Click a frame to highlight its properties.