Visual walkthrough — INS error propagation — error state equations

3.5.17 · D2 · Physics › Guidance, Navigation & Control (GNC) › INS error propagation — error state equations

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Step 0 — Do frames, aur unke beech ka rotation

Ek inertial navigation system (INS) kabhi bhi window se bahar nahi dekhta. Yeh andheron mein baithta hai aur do cheezein jo yeh feel karta hai, unpar arithmetic karta hai:

• kitna hard push ho raha hai (the accelerometers),
• kitni tezi se yeh turn kar raha hai (the gyros).

Kisi bhi equation se pehle, hume woh do coordinate frames confirm karne honge jisme yeh poora subject rehta hai.

Sensors se, INS compute karta hai ki woh kahan sochta hai woh hai. Lekin body-frame push ko North-East-Down acceleration mein convert karne ke liye, use $C_b^n$ apply karna hoga — aur agar uska stored $C_b^n$ thoda galat hai, toh sab kuch downstream galat hai. Machine ke stored, computed nav frame ko computed frame kahte hain, aur honest real wale ko true frame.

Neeche sab kuch $\mathbf{\psi }$ ke janam ki kahani hai, yeh velocity mein kaise leakta hai, aur velocity position mein kaise leakti hai.

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Step 1 — Teen errors jo hum track karte hain

Jo naam nahi diya usse fix nahi kar sakte. Teen disagreements jo matter karte hain:

Teen kyun, aur is order mein kyun? Kyunki yeh ek chain banate hain: tilt velocity ko corrupt karta hai, velocity position ko corrupt karti hai. Unhe top-to-bottom position, velocity, tilt likhne se coupling ko stack mein upar ki taraf flow ke roop mein padha ja sakta hai.

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Step 2 — Position error easy wala hai (yeh exact hai)

Us ek relation se shuru karo jise koi approximation nahi chahiye. True motion follow karta hai "position, velocity ki rate se badlta hai": $\dot{\mathbf{r}}=\mathbf{v}$ Upar ka dot, $\dot{}$, ka matlab hai "rate of change per second" — the derivative. Hum derivative yahan use karte hain aur abhi kahin nahi kyunki akela sawaal yeh hai "position arrow kitni tezi se move ho raha hai?", aur yahi bilkul wahi hai jo ek derivative answer karta hai.

HUM KYA KARTE HAIN: dono sides ko thoda nudge karo. True position ko ek chhoti si error $\delta \mathbf{r}$ gain karne do aur true velocity ko $\delta \mathbf{v}$ gain karne do. Nudged one mein se true equation subtract karo: $\boxed{\dot{\delta \mathbf{r}}=\delta \mathbf{v}}$

• $\dot{\delta \mathbf{r}}$ — woh rate jis par position error grow karta hai.
• $\delta \mathbf{v}$ — current velocity error, use directly feed karta hai.

YEH EXACT KYUN HAI: $\dot{\mathbf{r}}=\mathbf{v}$ ek straight-line (linear) relation hai — koi products nahi, koi sines nahi. Jab ek relation already linear hota hai, error usi relation ko follow karta hai bina koi leftover terms ke. Kuch bhi throw away nahi kiya gaya.

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Step 3 — Pehle rotation rates aur sensor errors ko naam dena

Velocity equation se pehle, hum har woh symbol milte hain jo woh use karega — koi bhi symbol bina earn kiye nahi aa sakta.

ABHI NAAM KYUN DEIN? Velocity equation neeche tilted frame se true force multiply karta hai aur accelerometer ki apni lie add karta hai. Agar humne $\delta {\mathbf{f}}^b$ aur $\mathbf{\omega }$'s ko pehle naam nahi diya hota, woh mystery symbols ke roop mein appear hote. Ab nahi karenge.

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Step 4 — Tilt velocity ko kaise poison karta hai (iska dil)

Ab important coupling. True velocity mechanization equation follow karti hai (Strapdown INS Mechanization Equations se): $\dot{\mathbf{v}}=C_b^n{\mathbf{f}}^b+\mathbf{g}-(2{\mathbf{\omega }}_{ie}+{\mathbf{\omega }}_{en})\times \mathbf{v}$ jahan $\mathbf{g}$ nav frame mein nominal gravity vector hai — gravity ka ek known model jo (mostly) Down ki taraf point karta hai, jiski value INS apni position se look up karta hai; rotation-rate vectors Step 3 mein define kiye gaye. ${\mathbf{f}}^n=C_b^n{\mathbf{f}}^b$ likho: felt push, sahi se North-East-Down mein rotate ki gayi.

INS, however, apna computed orientation ${\hat{C}}_b^n$ use karta hai, jo $\mathbf{\psi }$ se true $C_b^n$ se tilted hai.

Toh computed nav-frame force hai $(I-[\mathbf{\psi }\times ]){\mathbf{f}}^n={\mathbf{f}}^n-\mathbf{\psi }\times {\mathbf{f}}^n$.

VELOCITY-ERROR EQUATION KAISE MILTI HAI. INS ka computed rate likho computed orientation use karke, computed velocity $\mathbf{v}+\delta \mathbf{v}$, aur accelerometer ki apni error $\delta {\mathbf{f}}^b$ (Step 3 mein define ki gayi). True $\dot{\mathbf{v}}$ subtract karo aur sirf first-order terms rakho:

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

Term by term:

• $\dot{\delta \mathbf{v}}$ — woh rate jis par velocity error badh raha hai.
• $-[\mathbf{\psi }\times ]{\mathbf{f}}^n=-\mathbf{\psi }\times {\mathbf{f}}^n$ — show ka star: tilt × (bada specific force) velocity mein leak ho jaata hai.
• $C_b^n\delta {\mathbf{f}}^b$ — accelerometer error $\delta {\mathbf{f}}^b$ Step 3 se, $C_b^n$ ke through body→nav rotate ki gayi.
• $\delta \mathbf{g}$ — gravity-model error: (INS ka looked-up gravity) − (true gravity), Step 4b mein develop kiya gaya.
• $-(2{\mathbf{\omega }}_{ie}+{\mathbf{\omega }}_{en})\times \delta \mathbf{v}$ — Coriolis/transport term ka perturbation; iska origin Step 4c hai.

YEH TERM DOMINANT KYUN HAI: ${\mathbf{f}}^n$ bada hai — gravity ke paas, lagbhag $9.8{\text{m/s}}^2$. Even ek milliradian tilt ko isse multiply karo aur tumhe real acceleration error milti hai. Tilt ek lever hai, aur ${\mathbf{f}}^n$ uska lamba arm hai.

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Step 4b — $\delta \mathbf{g}$ kya hai, aur yeh position se kaise couple back karta hai

INS gravity $\mathbf{g}$ us position par evaluate karta hai jahan woh sochta hai woh hai. Agar woh guess $\delta \mathbf{r}$ se off hai, toh looked-up gravity bhi off hai.

KAISE: gravity position ka ek smooth function hai, toh ek chhoti si position error ek chhoti si gravity error deti hai — first-order Taylor rule "output error = (slope) × (input error)": $\delta \mathbf{g}\approx \frac{\partial \mathbf{g}}{\partial \mathbf{r}}\delta \mathbf{r}\equiv G_g\delta \mathbf{r}$

• $\frac{\partial \mathbf{g}}{\partial \mathbf{r}}$ — gravity gradient $G_g$: ek $3\times 3$ table jo batata hai ki gravity har metre move karne par kitni change hoti hai. Hum yahan derivative use karte hain usi wajah se jaise hamesha — yeh exact tool hai "ek metre input shift hone par is output mein kitna shift hoga?" ke liye.

YEH USUALLY TINY KYUN HAI: Earth ke paas, $G_g$ lagbhag $1.5\times 10^{-6}{\text{s}}^{-2}$ hai (gravity metres ke upar muskil se change hoti hai), toh chhoti flights mein yeh term negligible hai. Lekin yeh $\delta \mathbf{r}$ ko $\delta \mathbf{v}$ mein couple back karta hai, toh full matrix mein yeh velocity-row, position-column slot mein $+G_g$ ke roop mein baithta hai — ek chhota feedback jo hum Step 5 mein explicitly place karenge. Yeh gravity feedback Schuler Tuning and Oscillation ke Schuler oscillation ka ek ingredient hai.

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Step 4c — Coriolis term kahan se aata hai (aur uska sign)

True velocity equation carry karta hai $-(2{\mathbf{\omega }}_{ie}+{\mathbf{\omega }}_{en})\times \mathbf{v}$; rate vectors Step 3 mein define kiye gaye the.

ERROR TERM KAISE ARISE KARTA HAI. Yeh term true equation mein appear karta hai kyunki nav frame khud rotate ho raha hai, toh isme likhi velocity ko Coriolis-type corrections milti hain. Jab hum $\mathbf{v}\to \mathbf{v}+\delta \mathbf{v}$ perturb karte hain aur true equation subtract karte hain, yeh exactly apna perturbation contribute karta hai: $-(2{\mathbf{\omega }}_{ie}+{\mathbf{\omega }}_{en})\times (\mathbf{v}+\delta \mathbf{v})-[-(2{\mathbf{\omega }}_{ie}+{\mathbf{\omega }}_{en})\times \mathbf{v}]=-(2{\mathbf{\omega }}_{ie}+{\mathbf{\omega }}_{en})\times \delta \mathbf{v}.$ True-$\mathbf{v}$ part cancel ho jaata hai; sirf $\delta \mathbf{v}$ part bachta hai. Toh sign aur form directly true equation se inherited hain — kuch assume nahi kiya gaya. (${\mathbf{\omega }}_{ie},{\mathbf{\omega }}_{en}$ ko fixed treat karna unke khud ke tiny perturbations drop karta hai, jo ek legitimate first-order move hai.)

YEH USUALLY IGNORABLE KYUN HAI: ${\mathbf{\omega }}_{ie}$ bahut chhoti hai; ek chhoti flight par ${\mathbf{\omega }}_{ie}t\ll 1$ (Step 7). Lekin hours mein yeh feedback exactly wahi hai jo growing horizontal error ko Schuler oscillation mein bend karta hai.

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Step 5 — Tilt khud kahan se aata hai (gyro drift)

Humne $\mathbf{\psi }$ use kiya; ab hum uski apni equation earn karte hain. Computed nav frame gyros se spin hota hai. Agar ek gyro ka turn-rate report $\delta {\mathbf{\omega }}_{ib}^b$ (uska drift + noise, Step 3 mein define kiya gaya) se galat hai, toh woh error continuously tilt mein dump hoti rehti hai.

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

• $\dot{\mathbf{\psi }}$ — woh rate jis par tilt badh raha hai.
• $-{\mathbf{\omega }}_{in}\times \mathbf{\psi }$ — existing tilt carried around hoti hai jab nav frame khud ${\mathbf{\omega }}_{in}={\mathbf{\omega }}_{ie}+{\mathbf{\omega }}_{en}$ rate par rotate karta hai (Step 3).
• $-C_b^n\delta {\mathbf{\omega }}_{ib}^b$ — long-term error ka engine: gyro drift, $C_b^n$ se body→nav rotate kiya gaya, steadily $\mathbf{\psi }$ mein pour karta rehta hai.

GYRO QUALITY LONG RUN KYUN RULE KARTA HAI: drift $\mathbf{\psi }$ ko feed karta hai, $\mathbf{\psi }$ $\delta \mathbf{v}$ ko feed karta hai (Step 4), $\delta \mathbf{v}$ $\delta \mathbf{r}$ ko feed karta hai (Step 2). Ek row mein teen tanks — Step 8 dekho.

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Step 6 — Matrix $F$ assemble karna

Teen boxed equations, har stacked error ke liye ek. Unhe line up karo aur coefficients padho — yeh hi $\dot{\delta x}=F\delta x+Gw$ mein system matrix $F$ hai.

Ek repackaging: velocity equation mein $-[\mathbf{\psi }\times ]{\mathbf{f}}^n$ hai, lekin $F$ mein tilt $\mathbf{\psi }$ woh state hai jise multiply kiya ja raha hai, toh hume woh term (matrix)$\times \mathbf{\psi }$ ke roop mein likhna hoga. Flip rule $\mathbf{a}\times \mathbf{b}=-\mathbf{b}\times \mathbf{a}$ use karke: $-[\mathbf{\psi }\times ]{\mathbf{f}}^n=-\mathbf{\psi }\times {\mathbf{f}}^n=+{\mathbf{f}}^n\times \mathbf{\psi }=+[{\mathbf{f}}^n\times ]\mathbf{\psi }.$ Sign flip hota hai kyunki humne cross product ka order swap kiya $\mathbf{\psi }$ ko right ki taraf kheenchne ke liye, aur order swap karne se sign flip hota hai. Toh $F$ ka velocity-vs-tilt block $+[{\mathbf{f}}^n\times ]$ hai.

Har block collect karte hue (Step 4b se gravity feedback $G_g$ include karke): $F=\left[\begin{array}{ccc}0& I& 0\\ G_g& -[(2{\mathbf{\omega }}_{ie}+{\mathbf{\omega }}_{en})\times ]& +[{\mathbf{f}}^n\times ]\\ 0& 0& -[{\mathbf{\omega }}_{in}\times ]\end{array}\right]$

Har row ko "is error ko kya grow karta hai" ke roop mein padhte hue:

• Row 1 (position): akela $I$ kehta hai $\dot{\delta \mathbf{r}}=\delta \mathbf{v}$ — Step 2.
• Row 2 (velocity): tilt column mein $+[{\mathbf{f}}^n\times ]$ Step 4 ki coupling hai (repackaged); position column mein chhota $G_g$ Step 4b ka gravity feedback hai; Step 4c ka Coriolis/transport perturbation diagonal par baitha hai.
• Row 3 (attitude): sirf Step 5 ka transport-swirl $-[{\mathbf{\omega }}_{in}\times ]$. Drift noise channel ke through enter karta hai: $w=\left[\begin{array}{c}\delta {\mathbf{f}}^b\\ \delta {\mathbf{\omega }}_{ib}^b\end{array}\right],G=\mathrm{diag}(0,C_b^n,-C_b^n)$ ($w$ ki dono entries wahi same sensor errors hain jo Step 3 mein define ki gayi theen — poore mein ek consistent naam.)

POORI CHEEZ LINEAR KYUN HAI (parent ne yeh stress kiya tha): upar har step ne sirf first-order pieces rakhe — ek term mein chhoti error ka sirf ek factor, do small errors ka product kabhi nahi. Small × small drop kiya gaya. Yeh "linearize" karne ka poora matlab hai, aur yahi woh hai jo ek Kalman filter ko chahiye. $[\mathbf{\psi }\times ]$ ki small-angle machinery seedha Direction Cosine Matrix and Small-Angle Rotations se aati hai.

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Step 7 — Degenerate aur edge cases (kabhi surprise mat ho)

Ek accha map har corner cover karta hai. Special inputs walk karo:

• Zero tilt, zero errors ($\delta x=0$): har rate zero hai → INS perfect rehta hai. Correct: bina error aur noise ke ek system khud se koi invent nahi karta.
• Zero specific force (${\mathbf{f}}^n=0$, true free-fall): $[{\mathbf{f}}^n\times ]$ block vanish ho jaata hai — tilt ab velocity ko poison nahi kar sakta. Lever arm chala gaya. (Practice mein gravity ${\mathbf{f}}^n\ne 0$ rakhti hai, toh yeh rarely hota hai.)
• Short flight (missile, ~2 min): Earth rate ${\omega }_{ie}=7.3\times 10^{-5}\text{rad/s}$ deta hai ${\omega }_{ie}t\approx 0.0088$ rad — ek radian ke sauven hisse se kam, aur $G_g$ bhi negligible. ${\mathbf{\omega }}_{ie},{\mathbf{\omega }}_{en},G_g$ drop karo; sirf $+[{\mathbf{f}}^n\times ]$ aur sensor errors rakho. Woh akela block short-flight error ka ~80% carry karta hai.
• Long flight (hours): dropped Earth-coupling aur gravity-gradient terms wapas life mein aate hain aur growing horizontal error ko ek bounded oscillation mein bend karte hain — Schuler period ≈ 84 min of Schuler Tuning and Oscillation. Errors explode karne ki jagah sway karte hain.
• $\mathbf{\psi }$ small nahi (crash-level misalignment): hamara first-order picture toot jaata hai — dropped small×small terms matter karte hain — aur tumhe re-align karna hoga, linearize nahi.

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Step 8 — Integrations count karna (quadratic kyun, cubic kyun)

Growth rate punchline hai. Ek constant source ko Steps 2, 4, 5 ki tank chain mein follow karo.

Accelerometer bias $b_a$ velocity par enter karta hai (Step 4, $C_b^n\delta {\mathbf{f}}^b$ channel). Position tak do tanks: $\delta v(t)=b_{a}t,\delta r(t)=\frac{1}{2}{b}_a{t}^2(\text{quadratic})$ Ek $100\mu g$ bias $\approx 10^{-3}{\text{m/s}}^2$ deta hai $\delta r(60\text{s})=\frac{1}{2}(10^{-3})(60{)}^2=1.8\text{m}$.

Gyro drift $ϵ$ ek tank pehle enter karta hai — tilt par (Step 5). Tilt $\psi =ϵt$, phir mis-resolved gravity $\dot{\delta v}=-g\psi =-gϵt$. Position tak teen tanks: $\delta v=-\frac{1}{2}gϵt^2,\delta r=-\frac{1}{6}gϵt^3(\text{cubic})$

DIFFERENCE KYUN: source se position ko alag karne wale integrations count karo. Bias → 2 → $t^2$. Drift → 3 → $t^3$. Long run par, $t^3$ $t^2$ ko crush karta hai: gyros dominate karte hain.

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Ek-picture summary

Ek nazar, poori chain: gyro drift → tilt $\mathbf{\psi }$ → (× bada ${\mathbf{f}}^n$) → velocity error $\delta \mathbf{v}$ → position error $\delta \mathbf{r}$, accelerometer bias velocity stage par jump in karta hai, gravity-gradient feedback $G_g$ position se velocity ko nudge karta hai, aur Earth-rate feedback (dashed) sab kuch hours mein ek Schuler sway mein bend karta hai.

Recall Feynman: poora walkthrough simple shabdon mein

Ek robot ko imagine karo jo aankhein bandh karke chal raha hai, "kaun si taraf up aur North hai" ka ek mental picture rakhte hue. Woh mental picture asliyat mein ek chhota sa conversion table hai — $C_b^n$ — jo body ko jo feel hota hai usse North-East-Down mein turn karta hai. Uski inner compass har second thodi si drift karti hai (yahi gyro drift hai). Toh uski mental picture धीरे धीरे tilt hoti hai — woh tilt $\mathbf{\psi }$ hai. Ab, robot ek strong push feel karta hai (gravity aur thrust, bada arrow ${\mathbf{f}}^n$). Kyunki uska mental up tilted hai, woh us push ko thodi galat direction mein file karta hai — aur kyunki push strong hai, even ek tiny tilt uska ek real chunk sideways smear kar deta hai. Woh sideways smear ek fake acceleration hai, toh robot ka guessed speed off drift karta hai. Aur galat speed, time ke saath add karte hue, uske guessed position ko true path se off walk kara deta hai. Position ek bias ke baad doosra bucket hai, ek drift ke baad teesra — isliye drift ki error faster grow hoti hai (cubic) bias se (quadratic). Ek gentle feedback bhi hai: galat position matlab robot galat jagah par gravity look up karta hai, uski speed error ko thoda nudge karta hai — tiny, lekin hours mein yeh runaway ko ek slow 84-minute sway mein turn karne mein help karta hai. Aur is sab mein humne sirf chhote first-order pieces rakhe — yahi exactly woh wajah hai jab messy nonlinear reality ek tidy linear machine mein collapse ho jaati hai jise ek Kalman filter steer kar sakta hai.

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Recall Quick self-check

$\dot{\delta \mathbf{r}}=\delta \mathbf{v}$ exact kyun hai? ::: Kyunki $\dot{\mathbf{r}}=\mathbf{v}$ already linear hai, toh error usi relation ko follow karta hai bina kuch drop kiye. $F$ ka woh single block kaun sa hai jo tilt ko velocity mein couple karta hai, aur uska sign kya hai? ::: $+[{\mathbf{f}}^n\times ]$ — positive kyunki $\mathbf{\psi }$ ko right pull karne par $-\mathbf{\psi }\times {\mathbf{f}}^n$ flip ho kar $+{\mathbf{f}}^n\times \mathbf{\psi }$ ban jaata hai; ${\mathbf{f}}^n$ bada hai (~$g$) toh coupling strong hai. $\delta \mathbf{g}$ kya hai aur $F$ mein kahan baithta hai? ::: Gravity-model error $\delta \mathbf{g}\approx G_g\delta \mathbf{r}$ galat position par gravity evaluate karne se; yeh velocity-row, position-column block mein $+G_g$ ke roop mein baithta hai. Convection term $-{\mathbf{\omega }}_{in}\times \mathbf{\psi }$ negative kyun hai? ::: Kyunki $\mathbf{\psi }$ turning nav frame mein bookkeep ki jaati hai; ek still tilt frame ke rotation ke opposite direction mein swing hoti dikhti hai, isliye minus aata hai. Bias vs drift: kaun position error faster grow karta hai aur kyun? ::: Drift (cubic $t^3$) bias (quadratic $t^2$) ko beat karta hai kyunki drift chain mein ek integration pehle baitha hai.