INS error propagation — error state equations

1. What is an "error state"?

WHAT is $\boldsymbol{\psi}$? The INS keeps a computed body-to-nav rotation. If it's slightly wrong, the computed nav frame is tilted from the true nav frame by a small rotation vector $\boldsymbol{\psi}$. This is the crucial coupling variable.

2. Deriving the error equations from first principles

We work in the local navigation frame (North-East-Down). The exact INS mechanization is:

$\dot{\mathbf{r}} = \mathbf{v}, \qquad \dot{\mathbf{v}} = C_b^n \mathbf{f}^b + \mathbf{g} - (2\boldsymbol{\omega}_{ie}+\boldsymbol{\omega}_{en})\times\mathbf{v}$

where $\mathbf{f}^b$ is the specific force measured by accelerometers, $C_b^n$ the body-to-nav rotation, and $\mathbf{g}$ gravity.

2.1 Position error

HOW: Position rate is exactly velocity. Perturb $\mathbf{r}\to\mathbf{r}+\delta\mathbf{r}$, $\mathbf{v}\to\mathbf{v}+\delta\mathbf{v}$:

$\dot{\delta\mathbf{r}} = \delta\mathbf{v}$

Why this step? $\dot{\mathbf r}=\mathbf v$ is linear already, so the perturbation is exact — no approximation needed.

2.2 Velocity error

HOW: The computed acceleration uses the computed attitude $\hat C_b^n$. Because the frame is tilted by $\boldsymbol{\psi}$:

$\hat C_b^n = (I - [\boldsymbol{\psi}\times])\,C_b^n$

Why? A small rotation $\boldsymbol\psi$ maps a matrix via $I-[\boldsymbol\psi\times]$ (skew-symmetric matrix of $\boldsymbol\psi$). Substitute into $\dot{\mathbf v}$ and subtract the true equation. The accelerometer also has error $\delta\mathbf{f}^b$ (bias + noise). Keeping first order:

$\dot{\delta\mathbf{v}} = -[\boldsymbol{\psi}\times]\,C_b^n\mathbf{f}^b + C_b^n\,\delta\mathbf{f}^b + \delta\mathbf{g} - (2\boldsymbol{\omega}_{ie}+\boldsymbol{\omega}_{en})\times\delta\mathbf{v}$

Writing $\mathbf{f}^n = C_b^n\mathbf{f}^b$ (specific force in nav frame):

$\boxed{\dot{\delta\mathbf{v}} = -[\boldsymbol{\psi}\times]\,\mathbf{f}^n \;+\; C_b^n\,\delta\mathbf{f}^b \;+\;\delta\mathbf{g}\;-\;(2\boldsymbol{\omega}_{ie}+\boldsymbol{\omega}_{en})\times\delta\mathbf{v}}$

Why the $-[\boldsymbol\psi\times]\mathbf f^n$ term? A tilt makes the INS mis-resolve the (large) specific force — this is the dominant coupling: attitude error feeds velocity error.

2.3 Attitude error

HOW: The computed frame rotation rate has gyro error $\delta\boldsymbol{\omega}$. Differentiating the tilt definition and keeping first order:

$\boxed{\dot{\boldsymbol{\psi}} = -\boldsymbol{\omega}_{in}\times\boldsymbol{\psi} - C_b^n\,\delta\boldsymbol{\omega}_{ib}^b}$

where $\boldsymbol{\omega}_{in}=\boldsymbol{\omega}_{ie}+\boldsymbol{\omega}_{en}$ is the nav-frame rotation rate and $\delta\boldsymbol{\omega}_{ib}^b$ is the gyro error (drift + noise).

Why this form? The tilt vector rotates with the transport rate $\boldsymbol\omega_{in}$ (first term) and is continuously corrupted by gyro drift (second term). Gyro drift is the root source of long-term navigation error.

3. Why errors GROW — the key insight

4. Worked examples

5. Common mistakes

6. Flashcards

Connections

• Kalman Filter for INS-GPS Integration — uses this $F$ as the state-transition model
• Strapdown INS Mechanization Equations — the nonlinear system we perturbed
• Schuler Tuning and Oscillation — why horizontal errors bound
• Gyro and Accelerometer Error Models — sources of $w$
• Direction Cosine Matrix and Small-Angle Rotations — origin of $I-[\psi\times]$

Concept Map

Hinglish (regional understanding)