INS error propagation — error state equations: Foundations

The parent note (parent topic) throws a lot of notation at you at once: $δ r$ , $C_{b}^{n}$ , $[ψ \times]$ , $ω_{i e}$ , $f^{n}$ . Here we earn each one, in order, from a smart-12-year-old starting point.

1. A vector and its arrow

Picture it: an arrow from where you think the origin is to where the object is.

Why the topic needs it: an INS tracks three things at once (which way is North, how fast, how high), and each is naturally an arrow in 3D space. A single letter that carries all three numbers keeps the equations short.

Recall What does a bold

$v$ mean vs plain $v$ ? Bold $v$ ::: the whole arrow (3 numbers, direction + length). Plain $v$ would be just one number (a length or one component).

2. The dot on top — a rate of change

Why a derivative and not something simpler? Because the INS is given rates (how fast you turn, how hard you accelerate) and must recover the totals (angle, speed, place). The dot is the language of "rate," and integration (Section 8) is how you climb back from rate to total.

Recall Read

$\dot{v}$ aloud in plain words. $\dot{v}$ ::: "how much the velocity changes in one second" — i.e. acceleration.

Picture it: two sets of arrows sharing one origin — the ground set stays put, the vehicle set rolls with the aircraft.

Why two frames? The sensors live in the vehicle (body frame), but you want the answer on the map (nav frame). Every INS equation is really a translation job between these two.

The little superscript tells you the frame: $f^{b}$ is force felt in the body, $f^{n}$ is that same force rewritten on the map.

Recall What does the superscript in

$f^{b}$ tell you? superscript $b$ ::: the vector is expressed in the body frame (the vehicle's own axes).

4. The rotation matrix $C_{b}^{n}$

Picture it: feed in an arrow described by the vehicle's tilted axes; out comes the same physical arrow described by North-East-Down.

Why a matrix? A rotation mixes all three components (turning changes North and East together). Multiplication by a $3 \times 3$ matrix is exactly the tool that does this mixing linearly. See Direction Cosine Matrix and Small-Angle Rotations for how its 9 numbers are built.

5. Specific force $f$ (not acceleration!)

Why this subtlety matters: the INS must add gravity back in by hand ( $\dot{v} = C_{b}^{n} f^{b} + g + \dots$ ). If your stored value of $g$ or your tilt is slightly off, the force is mis-added — and that mistake is the seed of the whole error story.

Recall A free-falling accelerometer reads what?

free fall ::: zero — the only push it can't feel is gravity, and in free fall that's the only force.

6. Angular rates $ω$ — the turning symbols

Why bother with Earth's tiny spin? Over hours it matters (it sets the Schuler behaviour, see Schuler Tuning and Oscillation); over a 2-minute missile flight it's negligible. Knowing which $ω$ to keep is half of practical INS work.

Recall What is

$ω_{in}$ made of? $ω_{in}$ ::: $ω_{i e} + ω_{e n}$ — Earth's spin plus the transport rate from moving over the globe.

7. The cross product and the skew matrix $[\cdot \times]$

Why rewrite a cross product as a matrix? The whole error model is $\dot{δ x} = F δ x$ , one big matrix multiply. To fit cross-products (like $ψ \times f^{n}$ ) into that matrix $F$ , we turn each cross product into its skew matrix. That's the only reason the bracket notation exists.

Recall Why does

$[a \times]$ have zeros on its diagonal? diagonal zeros ::: an arrow crossed with itself is zero, so the "same-axis" entries vanish.

8. Integration — turning rates into totals

Why this is the villain of the whole topic: integrating a steady error makes it grow. A constant velocity error $c$ integrated once gives $c t$ (a straight ramp). A constant acceleration error integrated twice gives $\frac{1}{2} c t^{2}$ (a parabola that runs away). Every "quadratic" and "cubic" growth in the parent note is just counting how many integrations sit between a sensor flaw and the position.

9. Small errors and the $δ$ symbol

Why "small" is the magic word: for tiny angles, curves look straight — so the messy nonlinear rotation equations flatten into linear ones ( $F δ x$ ). That linearity is precisely what makes the Kalman filter able to estimate the errors. Details of the sensor errors that feed $δ f^{b}$ and gyro drift live in Gyro and Accelerometer Error Models.

Recall What kind of error is

$ψ$ , and why is it special? $ψ$ ::: a small rotation error (tilt of the computed frame); it's special because it mis-projects the large specific force into velocity.

How the foundations feed the topic

Each box is one section above; every arrow means "you need the left idea before the right one makes sense." The two roots — geometry (vectors → frames → rotations) and rates (omega, integration) — merge at the small-error idea, which is where the parent note begins.

Equipment checklist

Test yourself — you're ready for the parent note when every line reads easily.

Read

\dot{r} = v

in plain words

"the rate position changes is the velocity."

What does the superscript in

f^{n}

mean

the vector is written in the nav (NED) frame.

Which way does

C_{b}^{n}

translate

from body frame to nav frame.

Specific force sitting on a table reads

about

9.8 m/s^{2}

upward (it can't feel gravity, only the table's push).

Value and meaning of

ω_{i e}

Earth's spin,

7.3 \times 1 0^{- 5}

rad/s (one turn/day).

Purpose of the skew matrix

[a \times]

to write a cross product as a matrix so it fits inside

F

Why does gyro drift give cubic position error

three integrations separate drift from position (drift→tilt→velocity→position).

Meaning of the prefix

δ

the small error = computed minus true value.

What is

ψ

the small tilt (attitude) error between computed and true frames.

Why can we linearize at all

the errors are tiny, so first-order (straight-line) terms suffice.

Foundations — INS error propagation — error state equations