Inertial navigation — accelerometer measures non-gravitational specific force

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

KYA measure ho raha hai?

KAISE: Newton ke 2nd law se specific force derive karo

KYO derive karo? Taaki "gravity subtract karo" rule hamare upar forced ho, na ki yaad kiya hua ho.

Proof mass $m$ consider karo. Iske upar do categories ki force act karti hain:

Gravity $m g$ (ek body force jo har atom par act karti hai).
Contact / applied force $F_{c}$ (spring / electrostatic pickoff jo isse case mein hold karta hai).

Newton ka second law ek inertial frame mein, jahan $a$ mass ki true acceleration hai:

$m a = F_{c} + m g$

Contact force per unit mass ke liye solve karo — yahi exactly spring read karta hai:

$f \equiv \frac{F _{c}}{m} = a - g$

Yeh step kyun? ($\mathbf{f}$ ke liye solve karo) Spring sirf $F_{c}$ supply karta hai; gravity spring ko kabhi nahi chhurti. Isliye readable quantity hai $F_{c} / m = a - g$ .

Do "surprising" special cases (Pehle Forecast karo, phir Verify karo)

Forecast: Accelerometer kya read karega (a) table par chup pada ho, (b) free fall mein?

Case A — table par resting. True acceleration $a = 0$ . Gravity $g = (0, 0, - g)$ (neeche). $f = a - g = 0 - (0, 0, - g) = (0, 0, + g)$ Yeh $+ g$ upward read karta hai — spring mass ko support karne ke liye upar push kar raha hai. Verified: ek still accelerometer $9.81 m/s^{2}$ read karta hai, zero nahi.

Case B — free fall. Sirf gravity act karti hai, isliye $a = g$ . $f = a - g = g - g = 0$ Zero read karta hai — yahi "weightlessness" hai. Astronauts float karte hain kyunki unke body ka accelerometer $f = 0$ feel karta hai, bhale hi woh Earth ki taraf accelerate kar rahe hon. Yeh sensor form mein Einstein ka equivalence principle hai.

Inertial navigation mein position paane ke liye sensor output ko do baar integrate karte hain:

\;\xrightarrow{\int dt}\; \mathbf{v} \;\xrightarrow{\int dt}\; \mathbf{r}$$ - Agar aap $\mathbf{g}$ add karna bhool gaye, toh galat acceleration integrate karoge → position error ==$\tfrac12 g\,t^2$== ki tarah badhta hai (drift seconds mein explode ho jaata hai). - Kyunki $\mathbf{g}=\mathbf{g}(\mathbf{r})$ position par depend karta hai, aur $\mathbf{g}$ jaanne ke liye aapko position chahiye... yeh equations ko couple karta hai — yeh ek real navigation loop hai. ![[3.5.13-Inertial-navigation-—-accelerometer-measures-non-gravitational-specific-force.png]] > [!example] Worked: upar accelerate karta elevator > Ek elevator $2\,\text{m/s}^2$ upar accelerate karta hai. Floor-mounted accelerometer (z-axis up) kya read karega? > > - **Step 1:** true acceleration $\mathbf{a}=(0,0,+2)$. *Kyun?* Elevator physically upar accelerate kar raha hai. > - **Step 2:** gravity $\mathbf{g}=(0,0,-9.81)$. *Kyun?* Motion chahe kuch bhi ho, hamesha neeche point karti hai. > - **Step 3:** $f_z = a_z - g_z = 2 - (-9.81) = 11.81\,\text{m/s}^2$. *Kyun?* Spring ko weight support karna bhi padta hai ($9.81$) aur mass accelerate bhi karna hota hai ($2$). > - **Interpretation:** aap "zyada bhaari" feel karte ho — reading mein exactly woh extra $2\,\text{m/s}^2$ dikhta hai. > [!example] Worked: aircraft level cruise mein, banked turn > Ek plane ek level, coordinated turn (constant altitude) fly karta hai. Vertical accel $a_z=0$ hai, lekin centripetal accel horizontal $a_x = v^2/R$ hai. > - $f_z = 0 - (-g) = +g$ (weight abhi bhi support ho raha hai). > - $f_x = a_x - 0 = v^2/R$. > - Magnitude $|\mathbf{f}| = \sqrt{g^2 + (v^2/R)^2} > g$: yahi **load factor / g-force** hai jo pilots feel karte hain. *Kyun?* Seat (contact force) ko pilot ko upar bhi rakhna hai aur unka path curve bhi karna hai. --- ## Common mistakes (Steel-man + fix) > [!mistake] "Accelerometer acceleration measure karta hai, isliye still device 0 read karta hai." > **Kyun sahi lagta hai:** naam kehta hai *accelero*meter, aur intuitively "nahi hil raha = koi acceleration nahi = zero." **Galti:** yeh specific force $\mathbf{f}=\mathbf{a}-\mathbf{g}$ measure karta hai, $\mathbf{a}$ nahi. Rest par $\mathbf{a}=0$ lekin $\mathbf{g}\neq0$, isliye $+g$ upward read karta hai. > **Fix:** hamesha poochho "spring kaunsi force supply kar raha hai?" Wahi reading hai. > [!mistake] "Free fall mein accelerometer $g$ downward read karta hai." > **Kyun sahi lagta hai:** object $g$ par neeche accelerate kar raha hai, toh surely sensor $g$ dikhayega. **Galti:** free fall mein *sirf* gravity force hai, aur gravity spring ke liye invisible hai — koi contact force nahi → **0 read karta hai**. > **Fix:** free fall ⇒ $\mathbf{a}=\mathbf{g}$ ⇒ $\mathbf{f}=0$. > [!mistake] "Bas raw sensor output ko do baar integrate karo toh position milega." > **Kyun sahi lagta hai:** position acceleration ka double integral hai. **Galti:** raw output $\mathbf{f}$ hai, $\mathbf{a}$ nahi. Pehle $\mathbf{g}(\mathbf{r})$ add karna hoga. > **Fix:** $\mathbf{a}=\mathbf{f}+\mathbf{g}$, *tab* integrate karo. --- > [!recall]- Feynman: ek 12-saal ke bacche ko samjhao > Socho tum ek shoebox ke andar ek spring par ball pakde ho. Jab tum still baithe ho, spring ko thoda upar dabana padta hai ball ko gravity ke against hold karne ke liye — spring "kaam kar raha hai," isliye box kehta hai "mujhe kuch feel ho raha hai!" Lekin agar tum poora box drop kar do, toh ball, spring, aur box sab saath mein girte hain, isliye spring ko bilkul push nahi karna padta — box kehta hai "mujhe *kuch bhi* feel nahi ho raha," bhale hi yeh zameen ki taraf zoom kar raha ho. Isliye astronauts float karte hain: sab kuch saath girta hai, isliye unki andar ki "spring" zero feel karti hai. Box kabhi directly gravity feel nahi kar sakta — kisi ko (computer ko) *yaad rakhna hota hai* ki gravity wahan hai aur use add karna hota hai. > [!mnemonic] > **"A = F + G"** → *"**A**ll **F**light needs **G**ravity added back."* > Aur sign ke liye: **"At rest, points to the sky"** (still accelerometer $+g$ *upward* read karta hai, gravity ke opposite direction mein). --- ## Active recall #flashcards/physics Accelerometer asal mein kaunsi physical quantity measure karta hai? ::: Specific force $\mathbf{f}$ — non-gravitational (contact) force per unit mass, acceleration nahi. Fundamental accelerometer equation likho. ::: $\mathbf{f} = \mathbf{a} - \mathbf{g}$, isliye $\mathbf{a} = \mathbf{f} + \mathbf{g}$. Gravity accelerometer ke liye invisible kyun hai? ::: Gravity proof mass aur case dono par barabar act karti hai, isliye spring (contact force) ko kabhi iske against nahi ladhna padta. Sirf contact forces read hote hain. Stationary accelerometer (z up) kya read karta hai? ::: $+g \approx 9.81\,\text{m/s}^2$ upward (spring weight support karta hai; $\mathbf{a}=0$, $\mathbf{f}=-\mathbf{g}$). Free fall mein accelerometer kya read karta hai? ::: Zero — sirf gravity force hai, jo spring feel nahi kar sakta. Yahi weightlessness / equivalence principle hai. Inertial navigation mein, integrate karne se pehle sensor output ke saath kya karna hota hai? ::: Modeled gravity vector $\mathbf{g}(\mathbf{r})$ add karo: $\mathbf{a}=\mathbf{f}+\mathbf{g}$. Elevator $2\,\text{m/s}^2$ upar accelerate karta hai; uska z-accelerometer kya read karega? ::: $f_z = 2-(-9.81)=11.81\,\text{m/s}^2$. Agar gravity add karna bhool gaye toh position error kya hoga? ::: Yeh $\tfrac12 g t^2$ ki tarah badhta hai — drift seconds mein explode ho jaata hai. --- ## Connections - [[Equivalence Principle]] — free-fall reading zero hona gravity ≡ acceleration hai. - [[Newton's Second Law]] — derivation ka starting point. - [[Strapdown Inertial Navigation System]] — jahan $\mathbf{a}=\mathbf{f}+\mathbf{g}$ integrate hota hai. - [[Gyroscope and Attitude Determination]] — $\mathbf{g}$ add karne se pehle $\mathbf{f}$ ko nav frame mein rotate karne ke liye zaruri. - [[Gravity Model (WGS-84 / J2)]] — $\mathbf{g}(\mathbf{r})$ provide karta hai. - [[Dead Reckoning and Error Drift]] — integration ka consequence. ## 🖼️ Concept Map ```mermaid flowchart TD ACC[Accelerometer: mass on spring] PM[Proof mass] SF[Specific force f] CONTACT[Contact force Fc: spring] GRAV[Gravity mg] N2[Newton 2nd law: ma = Fc + mg] EQ[f = a - g] NAV[Nav computer adds g] A[True acceleration a] CaseA[Case A: at rest reads +g] CaseB[Case B: free fall reads 0] ACC -->|contains| PM PM -->|displacement gives| SF CONTACT -->|per unit mass equals| SF GRAV -->|invisible to| SF N2 -->|solve for f| EQ EQ -->|so a = f + g| NAV NAV -->|recovers| A A -->|integrated for| NAV EQ -->|predicts| CaseA EQ -->|predicts| CaseB ```

KYA measure ho raha hai?

KAISE: Newton ke 2nd law se specific force derive karo

Do "surprising" special cases (Pehle Forecast karo, phir Verify karo)

Full navigation chain (YEH note kyun matter karta hai)