Mechanization equations — integrating IMU to get position, velocity, attitude

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

IMU actually KYA measure karta hai?

Sabse bada trap yahan chupta hai: ek accelerometer gravity sense nahi kar sakta. Free fall mein yeh zero read karta hai. Isliye gravity ko software mein wapas add karna padta hai.

Teen-layer ka integration chain

gyro ω_{ib}^{b} \to ATTITUDE C_{b}^{n} accel f^{b} C_{b}^{n} f^{n} \to VELOCITY v^{n} \to POSITION

Pehle attitude lena zaroori hai, kyunki gravity subtract karne se pehle accelerometer ki body-frame specific force ko navigation frame mein rotate karne ke liye uski zaroorat padti hai.

Layer 1 — Attitude: gyros ko integrate karna

Matrix/quaternion kyun, sirf angles kyun nahi? Kyunki rotation rates scalars ki tarah simply add nahi hote — orientation ek curved space $S O (3)$ par rehti hai. Hum direction-cosine matrix (DCM) $C_{b}^{n}$ (body→nav) ko propagate karte hain.

Yeh step kyun? $[ω \times] u = ω \times u$ exactly hota hai, isliye teen columns stack karne par poora matrix derivative milta hai.

Lekin gyro $ω_{ib}^{b}$ (vs inertial space) measure karta hai, aur hum chahte hain $ω_{nb}^{b}$ (vs nav frame, jo khud move karta hai jab Earth ghoomti hai aur jab hum curved surface par travel karte hain):

ω_{nb}^{b} = ω_{ib}^{b} - C_{n}^{b} (Earth spin ω_{i e}^{n} + transport rate ω_{e n}^{n})

$ω_{i e}^{n}$ : Earth $Ω_{e} \approx 7.292 \times 1 0^{- 5} rad/s$ par rotate karti hai.
$ω_{e n}^{n}$ (transport rate): jab aap north/east fly karte ho, local "down" tilt ho jaata hai kyunki Earth round hai.

Layer 2 — Velocity: rotate karo, gravity hatao, Coriolis correct karo

Term-by-term KYUN:

$C_{b}^{n} f^{b}$ — felt force ko nav axes mein rotate karo (Layer 1 chahiye).
$- (2 ω_{i e} + ω_{e n}) \times v$ — Coriolis + transport correction, rotating/curving frame mein motion observe karne ke liye.
$+ g^{n}$ — gravity wapas add karo (woh hissa jo accelerometer sense nahi kar saka). Yahan $g^{n}$ plumb-bob gravity hai (gravitation minus centrifugal), "down" ki taraf point karti hai.

Layer 3 — Position: curved Earth par velocity integrate karna

Ek sphere/ellipsoid par, latitude/longitude un rates par change hote hain jo Earth ki curvature radii par depend karti hain (meridian $R_{M}$ , transverse $R_{N}$ ):

Flat-Earth / short-baseline approximation ke liye aap transport aur curvature drop kar sakte ho aur bas use kar sakte ho: $\dot{r}^{n} = v^{n}, \dot{v}^{n} = C_{b}^{n} f^{b} + g^{n}, \dot{C}_{b}^{n} = C_{b}^{n} [ω_{ib}^{b} \times] .$ Yeh 80/20 core hai — is trio ko memorize karo; Earth terms sirf long-range navigation ke liye add karo.

Worked Example 1 — Static IMU sanity check

Ek IMU flat aur still rakha hua hai. Accels $f^{b} = (0, 0, - 9.81)$ read karte hain NED body axes mein (Down = $+ z$ ), gyros $0$ read karte hain.

Attitude: $\dot{C} = C [0 \times] = 0$ → orientation constant. Kyun? koi rotation feel nahi hui.
Velocity (flat-Earth): $\dot{v} = C_{b}^{n} f^{b} + g^{n} = (0, 0, - 9.81) + (0, 0, + 9.81) = 0$ . Kyun? upar ki taraf table ka push (NED "Down" mein $- 9.81$ ) gravity ( $+ 9.81$ Down) ko cancel karta hai. Vehicle wahin rehti hai ✔.

Recall Ek

level accelerometer Down axis par $- 9.81$ kyun read karta hai, $0$ kyun nahi? Kyunki table use upar push karti hai; specific force Down convention mein us support ke opposite point karti hai. Free-fall mein $0$ milta.

Worked Example 2 — Constant forward acceleration, 1-D

Ek track par rocket, koi rotation nahi, nav $x$ ke saath aligned. Accel $f_{x} = 5 m/s^{2}$ read karta hai (gravity doosre axes par handle ho rahi hai).

$\overset{v}{˙}_{x} = f_{x} = 5$ . Kyun? koi Coriolis nahi (short run), $x$ par koi gravity nahi.
Integrate karo: $v_{x} (t) = 5 t$ . Kyun? constant accel.
$\overset{x}{˙} = v_{x} \Rightarrow x (t) = \frac{1}{2} (5) t^{2} = 2.5 t^{2}$ . $t = 4 s$ par: $x = 40 m$ , $v = 20 m/s$ .

Yeh kyun important hai: clean double-integration dikhata hai — lekin koi bhi tiny accel bias $b$ position error $\frac{1}{2} b t^{2}$ produce karta hai, jo $t^{2}$ ki tarah badhta hai. Yahi INS drift hai.

Worked Example 3 — Gyro bias se attitude drift

Constant gyro bias $ε = 0.01 rad/s$ ek axis ke baare mein, static vehicle. Attitude error $δ θ = εt$ hai. $10 s$ baad, $δ θ = 0.1 rad \approx 5. 7^{\circ}$ .

Ab projected gravity horizontal velocity mein leak ho jaati hai: $δ a \approx g sin (δ θ) \approx 9.81 (0.1) = 0.98 m/s^{2}$ . Yeh step kyun? tilted attitude gravity ko misrotate karti hai, isliye $g$ horizontal channel ko contaminate karta hai — yeh dominant INS error path hai.

Flashcards

Accelerometer physically kya measure karta hai?

Specific force (non-gravitational contact acceleration), gravity nahi; static reading

+ g

opposite the support hoti hai.

Mechanization mein attitude, velocity se pehle kyun compute karni padti hai?

Accelerometer force body axes mein hai; gravity subtract/add karne se pehle use nav frame mein rotate karne ke liye

C_{b}^{n}

chahiye.

DCM propagation equation

\dot{C}_{b}^{n} = C_{b}^{n} [ω_{nb}^{b} \times]

, body rate ka skew-symmetric cross-product matrix.

Full nav-frame velocity mechanization equation

\dot{v}^{n} = C_{b}^{n} f^{b} - (2 ω_{i e}^{n} + ω_{e n}^{n}) \times v^{n} + g^{n}

Transport rate

ω_{e n}^{n}

kya hai?

Local nav frame ka rotation jab vehicle curved Earth surface par move karta hai.

2 ω_{i e} \times v

term kyun aata hai?

Rotating Earth/nav frame mein motion express karne ka Coriolis correction.

Geodetic latitude rate

\dot{ϕ} = v_{N} / (R_{M} + h)

— meridian radius of curvature par arc-rate.

Static hone par accel ka raw double-integration kyun fail hota hai?

Yeh

+ g

specific-force bias integrate karta hai, fake motion produce karta hai; pehle nav frame mein gravity wapas add karni padti hai.

Gyro bias

ε

position ko kaise corrupt karta hai?

Attitude error

εt

gravity vector ko tilt karta hai,

\approx g εt

horizontal accel inject karta hai → error

t^{3}

ki tarah badhta hai.

Flat-Earth core mechanization trio

\dot{C}_{b}^{n} = C_{b}^{n} [ω_{ib}^{b} \times]

\dot{v}^{n} = C_{b}^{n} f^{b} + g^{n}

\dot{r}^{n} = v^{n}

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho tum ek car mein aankhon par patti baandhe ho. Jab yeh speed up, brake, ya turn karti hai toh tum pushes feel kar sakte ho — lekin road ko pass hote nahi feel kar sakte. Yeh andaza lagane ke liye ki tum kahan ho, pehle tum kis taraf face kar rahe ho figure out karo har turn track karke (yeh gyro/attitude part hai). Phir jo pushes feel kiye unse pata lagao ki tum kitni tezi se ja rahe ho, yaad rakhte hue ki still baithte waqt push sirf gravity hai jo tumhe seat mein rok rahi hai — isliye use subtract karo. Apni speed ko time ke upar add karo aur pata chalega kitna move kiya. "Tum kis taraf face kar rahe ho" mein tiny mistakes gravity ko tumhari speed guess mein sneeak in karne deti hain, isliye thodi der baad drift hone lagte ho — isliye cars GPS bhi use karti hain guess ko honest rakhne ke liye.

Connections

Strapdown vs Gimbaled INS — strapdown sensors ko body par rakhta hai, isliye mechanization software mein "virtual gimbal" ka kaam karti hai.
Direction Cosine Matrix and Quaternions — woh attitude representation jo propagate ho rahi hai.
Coriolis and Centrifugal Effects — $2 ω_{i e} \times v$ term ka origin.
Gravity Model and Geoid — $g^{n}$ supply karta hai.
Kalman Filter for INS-GNSS Integration — yahan derive kiya gaya $t^{2} / t^{3}$ drift correct karta hai.
Radii of Curvature of the Earth Ellipsoid — position rates mein $R_{M}, R_{N}$ .