Question bank — INS error propagation — error state equations

False. Bias ek baar integrate hoke ek linear velocity error deta hai, phir doosri baar integrate hoke position, isliye position $t^2$ ki tarah badhta hai — quadratic (upar wala $t^2$ curve dekho). Integrations count karo, source nahi.

True growth order mein. Gyro drift ko teen integrations chahiye (drift → tilt → mis-resolved gravity → velocity → position), jo $t^3$ deta hai, accel bias ke $t^2$ ke versus — isliye growth figure ke crossover ke baad, gyro quality dominate karti hai.

False. Errors chhote hote hain, isliye true trajectory ke baare mein ek first-order Taylor expansion linear $\dot{\delta x}=F\delta x+Gw$ deta hai jo §0 mein print hai — exactly woh form jo ek linear Kalman filter ko chahiye.

False. $F$ mein off-diagonal block $-[{\mathbf{f}}^n\times ]$ unhe couple karta hai: ek tilt specific force ko ek spurious acceleration of size $\approx g\psi$ mein mis-resolve karta hai, isliye $\mathbf{\psi }$ directly $\dot{\delta \mathbf{v}}$ ko drive karta hai.

False (bounded, oscillatory). Gravity horizontal errors par feedback karta hai, upar draw kiya gaya ek bounded Schuler oscillation (≈84 min) produce karta hai; sirf unmodeled biases upar ek slow ramp add karte hain. Dekho Schuler Tuning and Oscillation.

False. $\dot{\mathbf{r}}=\mathbf{v}$ already linear hai, isliye ise perturb karna exact hai — $F$ ki top row (ek clean identity block $I$) mein koi truncation error nahi hai.

False. $w=0$ ke saath bhi $\dot{\delta x}=F\delta x$ rehta hai: koi bhi initial error (alignment tilt, initial position uncertainty) phir bhi propagate karta hai aur, Schuler coupling ke zariye, oscillate karta hai. Sirf $\delta x(0)=0$ aur $w=0$ zero error deta hai.

Aam taur par galat, short flights ke liye aksar sahi. Ek 2-minute flight ke liye ${\omega }_{ie}t\approx 0.009$, isliye yeh negligible hai; lekin ghanton ki run par yeh Earth-rate (${\mathbf{\omega }}_{ie}$) aur transport-rate (${\mathbf{\omega }}_{en}$) terms Schuler behaviour ko shape karte hain aur inhe drop nahi kiya ja sakta.

Signs galat hain. Sahi equation hai $\dot{\mathbf{\psi }}=-{\mathbf{\omega }}_{in}\times \mathbf{\psi }-C_b^n\delta {\mathbf{\omega }}_{ib}^b$ ($F$ ki bottom row plus $G$ ka $-C_b^n$ block); dono terms negative hain kyunki tilt ${\mathbf{\omega }}_{in}$ se transported hoti hai aur gyro error se corrupted hoti hai jo isme drain hoti hai.

Galat operand. Yeh $-[{\mathbf{f}}^n\times ]\mathbf{\psi }$ hai, full specific force ${\mathbf{f}}^n$ (size $\sim g$) use karke, tiny error $\delta {\mathbf{f}}^b$ nahi. Do chhoti quantities ko multiply karna ($\psi \cdot \delta f$) second order hai aur drop ho jaata hai — tilt bade specific force par act karta hai, exactly jaise geometry figure dikhata hai.

Yeh dominant coupling delete kar deta hai. Poora point yeh hai ki ${\hat{C}}_b^n=(I-[\mathbf{\psi }\times ])C_b^n$; $-[\mathbf{\psi }\times ]$ piece ko rakhna hi $F$ ka $-[{\mathbf{f}}^n\times ]\mathbf{\psi }$ block produce karta hai. Unhe equal set karna attitude→velocity feedback erase kar deta hai.

Gyro block ka sign galat hai. Yeh $G=\mathrm{diag}(0,C_b^n,-C_b^n)$ hai (dekho §0); attitude equation $-C_b^n\delta {\mathbf{\omega }}_{ib}^b$ carry karta hai, isliye uska noise gain negative hai.

Galat row. $G$ mein, accelerometer error velocity row ke through enter karta hai ($C_b^n\delta {\mathbf{f}}^b$ ke zariye); attitude row sirf gyro error se driven hoti hai ($-C_b^n\delta {\mathbf{\omega }}_{ib}^b$). Unhe mix karna confuse karta hai ki kaunsi sensor spec kaunsi state ko limit karta hai.

Ek off. Top row $\dot{\delta \mathbf{r}}=\delta \mathbf{v}$ velocity ko pehla integral banata hai; position bias ka doosra integral hai. Yeh single miscount ek $t^2$ law ko ek galat $t^1$ law mein badal deta hai.

Ek tilted frame large specific-force vector (size $\sim g$) ko mis-project karta hai isliye ek horizontal slice $\approx g\psi$ fake acceleration ki tarah appear hoti hai — woh $-[{\mathbf{f}}^n\times ]\mathbf{\psi }$ term jo figure s02 mein geometrically dikhaya gaya hai. Perfect accelerometers help nahi karte agar frame jisme woh report karte hain woh tilted hai.

Gyro drift position se teen integrations door hai ($t^3$ growth) aur gravity-mis-resolution coupling ko feed karta hai, isliye figure s03 ke crossover ke baad uska cubic ramp accelerometer ke quadratic ko overtake kar leta hai.

Nonlinearities (rotations, gravity variation) depend karti hain ki vehicle actually kahan hai; true state ke baare mein linearize karna neglected terms ko genuinely second-order-small rakhta hai, yahi constant-per-instant $F$ ko justify karta hai.

Gravity feedback loop ki ek restoring "stiffness" $g/R$ (Earth radius $R$) se set hoti hai, aur $2\pi \sqrt{R/g}\approx 84$ min — ek natural pendulum-of-planet-size frequency jo $F$ coupling mein baked in hai. Dekho Schuler Tuning and Oscillation.

Linear model $\dot{\delta x}=F\delta x+Gw$ predict karta hai ki errors time ke saath kaise pattern karte hain; ek external fix (jaise GPS) kuch combination observe karta hai, aur jaana-maana $F$ filter ko unobserved states back out karne deta hai. Dekho Kalman Filter for INS-GPS Integration.

Yeh woh ek state hai jo gyro error accumulate bhi karta hai aur $-[{\mathbf{f}}^n\times ]$ ke zariye velocity error mein inject bhi karta hai, isliye yeh woh bridge hai ($F$ ka middle-right block) jo sensor imperfection ko position drift se link karta hai.

Nahi. Stationary hone par bhi, accelerometers gravity ko oppose karne wali specific force read karte hain, isliye ${\mathbf{f}}^n\approx [0,0,-g{]}^{\top }$ nonzero hai; ek tilt phir bhi ise horizontal velocity error mein mis-resolve karta hai — isi tarah gyro drift standstill par bhi leak karta hai.

Yeh vanish ho jaata hai: $-[{\mathbf{f}}^n\times ]\mathbf{\psi }=0$. Koi specific force mis-resolve karne ke liye nahi hai, ek tilt ab velocity corrupt nahi karta — ek genuine degenerate case jahan dominant coupling switch off ho jaati hai.

Haan. $\dot{\delta x}=F\cdot 0+G\cdot 0=0$, isliye yeh ek fixed point hai. Koi bhi real system yeh sirf isliye fail karta hai kyunki $\delta x(0)\ne 0$ (imperfect alignment) ya $w\ne 0$ (real sensors) hote hain.

Haan — ~2 min par unka contribution ($\omega t\approx 0.009$) negligible hai; lekin tumhe $-[{\mathbf{f}}^n\times ]$ attitude→velocity block aur sensor-error inputs zaroor rakhne chahiye, jo short-flight error ka ~80% carry karte hain.

Constant bias se ek extra power: ek ramp $\propto t$ → velocity $\propto t^2$ → position $\propto t^3$, isliye cubic — jo constant gyro drift ke order se match karta hai.

Nahi. Reorient karna sirf yeh change karta hai ki ${\mathbf{f}}^n$ ke kaunse components bade hain; cross-product $[{\mathbf{f}}^n\times ]$ phir bhi perpendicular tilt components ko velocity mein couple karta hai — orientation ek nonzero specific force ko zero nahi kar sakti.