Foundations — Mechanization equations — integrating IMU to get position, velocity, attitude

Before you can read the parent note Mechanization equations, you must own every symbol it throws at you. This page builds each one from nothing, in the order they depend on each other.

0. The most basic idea: a frame (a set of directions)

Everything in navigation is a set of three arrows: "which way is forward, which way is right, which way is down". A frame is just a named choice of those three arrows.

We use four frames. Picture each as a little tripod of arrows:

• Inertial frame ($i$) — a tripod that does not spin with the Earth. It stays fixed relative to the distant stars. This is the "honest" frame where Newton's law $\text{force}=\text{mass}\times \text{acceleration}$ works without corrections.
• Earth frame ($e$) — a tripod glued to the planet; it spins once per day with the Earth.
• Navigation frame ($n$) — a tripod at your current spot pointing North, East, Down (NED), a right-handed set (N → E → D). As you travel, it rides along and slowly tips because the ground curves.
• Body frame ($b$) — a tripod bolted to the vehicle: nose (forward), right wing, belly (down), also right-handed. The IMU lives here.

1. Reading the sub/superscript code

The parent note is full of symbols like ${\mathbf{\omega }}_{ib}^b$ and $C_b^n$. This looks scary; it is actually a tidy filing system.

2. Vectors and their components

Bold letters like $\mathbf{v}$, $\mathbf{f}$, $\mathbf{g}$ mean vectors. In navigation we most often write them as a stack of three: ${\mathbf{v}}^n=\left[\begin{array}{c}{v}_N\\ v_E\\ v_D\end{array}\right]\text{(velocity North, East, Down).}$

The topic needs vectors because velocity, force, and gravity all have both size and direction — a single number could never say "20 m/s toward the North-East and slightly down".

3. Angular rate $\mathbf{\omega }$ — the gyro's language

Why a vector for spinning? A spin needs both an axis (a direction) and a speed. Packing both into one arrow lets us do arithmetic on rotations. A gyroscope is a sensor that outputs exactly this: ${\mathbf{\omega }}_{ib}^b$, the body's spin relative to inertial space, written in body axes.

A radian is the natural angle unit: sweep an arc equal to the radius, that's 1 radian ($\approx 57.3^{\circ }$). We use radians because arc-length $=$ radius $\times$ angle only works cleanly in radians — and that fact is the whole of Layer 3 (position).

4. Specific force $\mathbf{f}$ — the accelerometer's language

This single fact is why the parent note must "add gravity back in software". The sensor gives $\mathbf{f}$; the true motion needs $\mathbf{f}+\mathbf{g}$.

5. Gravity vector $\mathbf{g}$

6. The cross product $\times$ — "how a spin moves a point"

The parent note's core rule is $\dot{\mathbf{u}}=\mathbf{\omega }\times \mathbf{u}$. To read it, you need the cross product.

Why the topic needs it: a rotating rigid body carries every attached arrow around a circle. The cross product is the exact formula for how fast each point on that arrow moves. That is why attitude propagation (Layer 1) is built from it.

7. The skew-symmetric matrix $[\mathbf{\omega }\times ]$

We want to apply the cross product to three arrows at once (the three columns of a rotation matrix). A neat trick turns "cross with $\mathbf{\omega }$" into an ordinary matrix multiply.

Why bother? Because $\dot{C}=C[\mathbf{\omega }\times ]$ then handles all three axes of the attitude matrix in one clean line.

8. The direction-cosine matrix $C_b^n$ (attitude)

The picture: each column of $C_b^n$ is one body axis (nose, wing, belly) drawn in NED numbers. Read the columns and you know exactly how the vehicle is tilted.

9. The derivative and the integral — "rate now" vs "add it all up"

Two calculus ideas glue the whole recipe together. Both are simple pictures.

Why this is the heart of mechanization: the IMU only reports rates (spin rate, and — after fixing gravity — acceleration). To get orientation, velocity, and position we must integrate: rate → total, three times over. Each integration is one "layer" (Layer 1, 2, 3) of the parent note.

10. Earth's spin and transport rate

Two small rotation vectors correct for living on a spinning, curved planet.

• ${\mathbf{\omega }}_{ie}$ — Earth-spin rate, ${\Omega }_e\approx 7.292\times 10^{-5}\text{rad/s}$ (one turn per sidereal day). This is why the honest inertial frame and the Earth frame disagree.
• ${\mathbf{\omega }}_{en}$ — transport rate: as you travel over the round Earth, your local "Down" slowly tips to keep pointing at Earth's centre. The nav tripod itself rotates. Its size depends on speed and the Earth's radii of curvature.

Together these produce the Coriolis / transport corrections you see in the velocity equation (Layer 2). Deeper story: Coriolis and Centrifugal Effects. Whether the IMU box is bolted down or on a gimbal changes how these enter — see Strapdown vs Gimbaled INS.

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer aloud.