INS error propagation — error state equations: Foundations

3.5.17 · D1 · Physics › Guidance, Navigation & Control (GNC) › INS error propagation — error state equations

Parent note (parent topic) ek saath bahut saara notation throw karta hai: $δ r$ , $C_{b}^{n}$ , $[ψ \times]$ , $ω_{i e}$ , $f^{n}$ . Yahan hum inhe ek-ek karke earn karte hain, order mein, ek smart-12-year-old ke starting point se.

1. Ek vector aur uska arrow

Imagine karo: ek arrow jahan se aap sochte ho origin hai wahan se object tak.

Topic ko iske kyun zaroorat hai: ek INS ek saath teen cheezein track karta hai (North kaunsi direction hai, kitni speed hai, kitni height hai), aur har ek naturally 3D space mein ek arrow hai. Ek single letter jo teeno numbers carry kare, equations ko chhota rakhta hai.

Recall Bold

$v$ aur plain $v$ mein kya fark hai? Bold $v$ ::: poora arrow (3 numbers, direction + length). Plain $v$ sirf ek number hoga (ek length ya ek component).

2. Upar ki dot — change ki rate

Derivative kyun aur kuch simple kyun nahi? Kyunki INS ko rates diye jaate hain (aap kitni tezi se mude, kitni tezi se accelerate kiya) aur use totals recover karne hote hain (angle, speed, jagah). Dot "rate" ki language hai, aur integration (Section 8) woh tarika hai jisse aap rate se total tak wapas jaate ho.

Recall

$\dot{v}$ ko plain words mein padho. $\dot{v}$ ::: "ek second mein velocity kitni badal jaati hai" — matlab acceleration.

Imagine karo: do arrows ke sets ek origin share karte hain — zameen wala set tika rehta hai, vehicle wala set aircraft ke saath roll karta hai.

Do frames kyun? Sensors vehicle mein rehte hain (body frame), lekin aapko jawab map par chahiye (nav frame). Har INS equation asal mein inhi dono ke beech ek translation ka kaam hai.

Chhota superscript aapko frame batata hai: $f^{b}$ woh force hai jo body mein feel hoti hai, $f^{n}$ wahi force hai map par rewrite ki gayi.

Recall

$f^{b}$ mein superscript aapko kya batata hai? superscript $b$ ::: vector body frame mein express kiya gaya hai (vehicle ke apne axes).

4. Rotation matrix $C_{b}^{n}$

Imagine karo: vehicle ke tilted axes se described ek arrow feed karo; bahar nikalta hai wahi physical arrow jo North-East-Down ke terms mein describe kiya gaya hai.

Matrix kyun? Ek rotation teeno components ko mix karta hai (turning North aur East dono ko saath badalta hai). Ek $3 \times 3$ matrix se multiply karna exactly woh tool hai jo yeh mixing linearly karta hai. Iske 9 numbers kaise bante hain, yeh Direction Cosine Matrix and Small-Angle Rotations mein dekho.

5. Specific force $f$ (acceleration nahi!)

Yeh subtlety kyun matter karti hai: INS ko gravity haath se wapas jodhni padti hai ( $\dot{v} = C_{b}^{n} f^{b} + g + \dots$ ). Agar $g$ ki aapki stored value ya aapka tilt thoda sa galat hai, toh force galat tarike se add hoti hai — aur wahi galti poori error story ka beej hai.

Recall Free-fall mein ek accelerometer kya read karta hai?

free fall ::: zero — woh sirf wahi push feel nahi kar sakta jo gravity hai, aur free fall mein wahi ek force hai.

6. Angular rates $ω$ — ghoomne ke symbols

Earth ki choti si spin se pareshan kyun hon? Ghanton mein yeh matter karta hai (yeh Schuler behaviour set karta hai, Schuler Tuning and Oscillation dekho); ek 2-minute missile flight mein yeh negligible hai. Kaunsa $ω$ rakhna hai yeh jaanna, practical INS work ka aadha hissa hai.

Recall

$ω_{in}$ kis cheez se bana hai? $ω_{in}$ ::: $ω_{i e} + ω_{e n}$ — Earth ki spin plus globe par move karne se transport rate.

7. Cross product aur skew matrix $[\cdot \times]$

Cross product ko matrix ke roop mein kyun rewrite karein? Poora error model $\dot{δ x} = F δ x$ hai, ek badi matrix multiply. Cross-products (jaise $ψ \times f^{n}$ ) ko us matrix $F$ mein fit karne ke liye, hum har cross product ko uski skew matrix mein badal dete hain. Bracket notation isi ek kaam ke liye exist karta hai.

Recall

$[a \times]$ ke diagonal par zeros kyun hote hain? diagonal zeros ::: ek arrow ko khud se cross karne par zero aata hai, isliye "same-axis" entries vanish ho jaati hain.

8. Integration — rates se totals banana

Yeh poore topic ka villain kyun hai: ek steady error ko integrate karne se woh barhta hai. Ek constant velocity error $c$ ek baar integrate karne par $c t$ deta hai (ek seedha ramp). Ek constant acceleration error do baar integrate karne par $\frac{1}{2} c t^{2}$ deta hai (ek parabola jo bhaag jaata hai). Parent note mein har "quadratic" aur "cubic" growth bas yeh count karna hai ki ek sensor flaw aur position ke beech kitne integrations hain.

9. Choti errors aur $δ$ symbol

"Chota" magic word kyun hai: chote angles ke liye, curves seedhe dikhte hain — isliye messy nonlinear rotation equations linear ho jaate hain ( $F δ x$ ). Wahi linearity exactly woh cheez hai jo Kalman filter ko errors estimate karne deti hai. $δ f^{b}$ aur gyro drift ko feed karne wale sensor errors ki details Gyro and Accelerometer Error Models mein hain.

Recall

$ψ$ kis tarah ki error hai, aur yeh special kyun hai? $ψ$ ::: ek chota rotation error (computed frame ka tilt); yeh special hai kyunki yeh badi specific force ko velocity mein galat project karta hai.

Foundations topic ko kaise feed karte hain

Har box upar ka ek section hai; har arrow ka matlab hai "daayaan waali cheez samajhne se pehle baayin waali cheez chahiye." Do roots — geometry (vectors → frames → rotations) aur rates (omega, integration) — choti-error idea par milte hain, jahan se parent note shuru hota hai.

Equipment checklist

Khud test karo — aap parent note ke liye ready ho jab har line aasaani se padhi jaaye.

\dot{r} = v

ko plain words mein padho

"position ke change ki rate velocity hai."

f^{n}

mein superscript ka kya matlab hai

vector nav (NED) frame mein likha hua hai.

C_{b}^{n}

kis direction mein translate karta hai

body frame se nav frame mein.

Table par baithe hue specific force kya read karta hai

lagbhag

9.8 m/s^{2}

upward (woh gravity feel nahi kar sakta, sirf table ki push).

ω_{i e}

ki value aur matlab

Earth ki spin,

7.3 \times 1 0^{- 5}

rad/s (ek turn/day).

Skew matrix

[a \times]

ka purpose

cross product ko matrix ke roop mein likhna taaki woh

F

ke andar fit ho sake.

Gyro drift cubic position error kyun deta hai

teen integrations drift aur position ke beech hain (drift→tilt→velocity→position).

Prefix

δ

ka matlab

choti error = computed minus true value.

ψ

kya hai

computed aur true frames ke beech chota tilt (attitude) error.

Hum linearize kar paate hain kyun

errors tiny hain, isliye first-order (straight-line) terms kaafi hain.

Foundations — INS error propagation — error state equations

1. Ek vector aur uska arrow

2. Upar ki dot — change ki rate

3. Frames: body vs navigation

4. Rotation matrix $C_{b}^{n}$

5. Specific force $f$ (acceleration nahi!)

6. Angular rates $ω$ — ghoomne ke symbols

7. Cross product aur skew matrix $[\cdot \times]$

8. Integration — rates se totals banana

9. Choti errors aur $δ$ symbol

Foundations topic ko kaise feed karte hain

Equipment checklist