INS error propagation — error state equations

3.5.17 · Physics › Guidance, Navigation & Control (GNC)

1. "Error state" kya hota hai?

$\mathbf{\psi }$ KYA HAI? INS ek computed body-to-nav rotation rakhta hai. Agar yeh thoda galat ho, toh computed nav frame, true nav frame se ek small rotation vector $\mathbf{\psi }$ ke barabar tilt hota hai. Yahi crucial coupling variable hai.

2. First principles se error equations derive karna

Hum local navigation frame (North-East-Down) mein kaam karte hain. Exact INS mechanization hai:

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

jahan ${\mathbf{f}}^b$ accelerometers se measured specific force hai, $C_b^n$ body-to-nav rotation hai, aur $\mathbf{g}$ gravity hai.

2.1 Position error

KAISE: Position rate bilkul velocity hai. $\mathbf{r}\to \mathbf{r}+\delta \mathbf{r}$, $\mathbf{v}\to \mathbf{v}+\delta \mathbf{v}$ perturb karo:

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

Yeh step kyun? $\dot{\mathbf{r}}=\mathbf{v}$ pehle se hi linear hai, isliye perturbation exact hai — koi approximation ki zaroorat nahi.

2.2 Velocity error

KAISE: Computed acceleration computed attitude ${\hat{C}}_b^n$ use karta hai. Kyunki frame $\mathbf{\psi }$ se tilt hai:

${\hat{C}}_b^n=(I-[\mathbf{\psi }\times ])C_b^n$

Kyun? Ek small rotation $\mathbf{\psi }$ ek matrix ko $I-[\mathbf{\psi }\times ]$ (skew-symmetric matrix of $\mathbf{\psi }$) ke zariye map karta hai. $\dot{\mathbf{v}}$ mein substitute karo aur true equation subtract karo. Accelerometer mein bhi error $\delta {\mathbf{f}}^b$ hai (bias + noise). First order rakhte hue:

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

${\mathbf{f}}^n=C_b^n{\mathbf{f}}^b$ likhte hue (nav frame mein specific force):

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

$-[\mathbf{\psi }\times ]{\mathbf{f}}^n$ term kyun? Ek tilt INS ko (bade) specific force ko galat resolve karwata hai — yahi dominant coupling hai: attitude error, velocity error mein jaata hai.

2.3 Attitude error

KAISE: Computed frame rotation rate mein gyro error $\delta \mathbf{\omega }$ hota hai. Tilt definition differentiate karo aur first order rakhte huo:

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

jahan ${\mathbf{\omega }}_{in}={\mathbf{\omega }}_{ie}+{\mathbf{\omega }}_{en}$ nav-frame rotation rate hai aur $\delta {\mathbf{\omega }}_{ib}^b$ gyro error hai (drift + noise).

Yeh form kyun? Tilt vector transport rate ${\mathbf{\omega }}_{in}$ ke saath rotate karta hai (pehla term) aur continuously gyro drift se corrupt hota hai (doosra term). Gyro drift hi long-term navigation error ki jad hai.

3. Errors KYUN BADHTE HAIN — key insight

4. Worked examples

5. Common mistakes

6. Flashcards

Connections

• Kalman Filter for INS-GPS Integration — is $F$ ko state-transition model ke roop mein use karta hai
• Strapdown INS Mechanization Equations — woh nonlinear system jise humne perturb kiya
• Schuler Tuning and Oscillation — horizontal errors kyun bound hote hain
• Gyro and Accelerometer Error Models — $w$ ke sources
• Direction Cosine Matrix and Small-Angle Rotations — $I-[\psi \times ]$ ki origin

Concept Map