GNSS — GPS, GLONASS, Galileo, BeiDou — Hinglish notes

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

GNSS kyun exist karta hai / kaunsa problem solve karta hai?

KYA: GNSS (Global Navigation Satellite System) satellites ka ek constellation hai jo timed signals broadcast karta hai taaki Earth par koi bhi receiver apni position, velocity aur time (PVT) compute kar sake.

KYUN char systems? Ye sab independent national/regional systems hain jo sovereignty ke liye banaye gaye hain (tum nahi chahoge ki wartime mein tumhara navigation kisi doosre desh par depend kare):

GPS — USA, ~24+ satellites, 6 orbital planes, altitude ~20,200 km.
GLONASS — Russia, ~24 satellites, 3 planes, ~19,100 km.
Galileo — EU (civilian), ~24+ satellites, 3 planes, ~23,222 km.
BeiDou (BDS) — China, MEO + GEO + IGSO ka mix, ~21,500 km MEO.

HOW — ek receiver apni position kaise dhundta hai — scratch se derive karo

Step 1 — Receiver actually kya measure karta hai

Satellite $i$ jaani-mani position $(x_{i}, y_{i}, z_{i})$ par baitha hai (apne broadcast ephemeris se). Wo signal mein apna transmit time $t_{t x}$ stamp karta hai. Receiver arrival time $t_{r x}$ apni clock par note karta hai.

$ρ_{i} = c (t_{r x} - t_{t x})$

Ye step kyun? Distance = speed × time, aur signal $c$ (speed of light) par travel karta hai. $ρ_{i}$ ko pseudorange kehte hain — "pseudo" isliye kyunki $t_{r x}$ ek biased clock par measure hota hai.

Step 2 — True geometry ko model karo

Receiver $(x, y, z)$ se satellite $i$ tak ki true geometric distance:

$r_{i} = (x - x_{i})^{2} + (y - y_{i})^{2} + (z - z_{i})^{2}$

Ye step kyun? Ye sirf 3D mein Euclidean distance hai — satellite par centered $r_{i}$ radius ki ek sphere.

Step 3 — Clock bias daalo

Maano receiver clock true time se $b$ (seconds) age hai. To measured travel time $b$ se zyada badi hai, isliye:

$ρ_{i} = r_{i} + c b + ε_{i}$

Step 4 — System solve karo (linearize)

Equations nonlinear hain (square roots). Hum ek position $x_{0}$ guess karte hain aur Taylor expansion se linearize karte hain. $r_{i}$ ka $x$ ke saath partial derivative:

$\frac{\partial r _{i}}{\partial x} = \frac{x - x _{i}}{r _{i}} = - \overset{u}{^}_{i x}$

Ye step kyun? $\frac{( x _{i} - x )}{r _{i}}$ unit vector hai jo receiver se satellite ki taraf point karta hai. To geometry matrix line-of-sight directions se bani hai.

Corrections $Δ x = (Δ x, Δ y, Δ z, c Δ b)$ stack karo:

\mathbf{H}=\begin{bmatrix} -\hat u_{1x} & -\hat u_{1y} & -\hat u_{1z} & 1\\ \vdots & & & \vdots\\ -\hat u_{nx}&-\hat u_{ny}&-\hat u_{nz}&1\end{bmatrix}$$ Least squares se solve karo aur iterate karo: $$\Delta\mathbf{x} = (\mathbf{H}^{\mathsf T}\mathbf{H})^{-1}\mathbf{H}^{\mathsf T}\,\Delta\boldsymbol\rho$$ *Least squares kyun?* 5+ satellites ke saath system overdetermined ho jaata hai; least squares woh position dhundta hai jo total squared residual minimize kare. ![[3.5.19-GNSS-—-GPS,-GLONASS,-Galileo,-BeiDou.png]] --- ## Geometry matter karti hai: DOP > [!definition] Dilution of Precision (DOP) > Ek number jo batata hai ki satellite **geometry** measurement error ko position error mein kitna amplify karti hai. Satellites ek jagah bunch ho jaayein = bura (high DOP); sky mein faile hue = accha (low DOP). $$\sigma_{\text{position}} = \text{DOP}\times \sigma_{\text{measurement}},\qquad \text{GDOP}=\sqrt{\operatorname{tr}\!\big((\mathbf H^{\mathsf T}\mathbf H)^{-1}\big)}$$ > [!intuition] Spread-out satellites kyun help karte hain > Wahi reason jis wajah se ek carpenter table ko *faile hue* pairo se support deta hai: paas-paas wale supports wobble karte hain. Widely separated lines-of-sight $\mathbf H^{\mathsf T}\mathbf H$ ko well-conditioned banate hain, isliye ek chhoti si range error ek badi position error mein nahi badlti. **Zyada constellations (GPS+GLONASS+Galileo+BeiDou saath mein) = zyada visible satellites = lower DOP**, khaaskar urban canyons mein. --- ## Relativity — woh correction jo log bhool jaate hain > [!formula] GPS satellites par clock rate correction > **Special relativity** (satellite fast move kar raha hai) uski clock slow kar deta hai: > $$\frac{\Delta t_{SR}}{t}\approx -\frac{v^2}{2c^2}\;(\text{clock slow chalti hai})$$ > **General relativity** (upar weak gravity) use tez kar deta hai: > $$\frac{\Delta t_{GR}}{t}\approx +\frac{\Delta\Phi}{c^2}=+\frac{GM}{c^2}\Big(\frac1{r_E}-\frac1{r_{sat}}\Big)$$ > Net effect: satellite clocks **~+38 μs/day tezi se** tick karti hain. Ignore karo → position error ~11 km/day badhta hai! *Kyun matter karta hai:* light 1 nanosecond mein 30 cm travel karti hai; 38 μs bahut bada hai. Satellite oscillator ko compensate karne ke liye ground par hi *pahle se slower* tune kar diya jaata hai. --- ## Signals aur char systems, comparison | System | Owner | Access method | Nominal freq | |---|---|---|---| | GPS | USA | **CDMA** (har sat, same freq, unique code) | L1 1575.42 MHz | | Galileo | EU | CDMA | E1 1575.42 MHz | | BeiDou | China | CDMA | B1 ~1561 MHz | | GLONASS | Russia | **FDMA** (same code, alag freq per sat) | ~1602 MHz + k·0.5625 | > [!definition] CDMA vs FDMA > **CDMA** = sab satellites ek frequency share karte hain lekin har ek unique orthogonal *pseudo-random code* use karta hai; receiver unhe correlation se alag karta hai. **FDMA** = har satellite thodi alag *frequency* use karta hai (purana GLONASS design). --- ## Worked examples > [!example] Example 1 — 4 satellites kyun (dimension count) > **Q:** Satellite aur receiver *dono* mein perfectly synchronized atomic clocks hain. Ab kitne satellites? > **Reasoning:** Clock bias nahi ⇒ $b=0$ jaana hua hai ⇒ sirf 3 unknowns $(x,y,z)$ ⇒ **3 satellites**. > *Ye step kyun?* Har unknown ek equation khaata hai. $b$ hatao to ek zaroorat equation bhi hat jaati hai. > [!example] Example 2 — Pseudorange numbers > **Q:** Signal aane mein 67 ms lagte hain. Distance? > $$\rho = c\cdot t = (3\times10^8)(67\times10^{-3}) = 2.01\times10^7 \text{ m} \approx 20{,}100\text{ km}.$$ > *Ye step kyun?* MEO altitude se consistent hai — sanity check pass ho gayi. > [!example] Example 3 — Clock bias in metres > **Q:** Receiver clock 1 μs off hai. Kitna range error inject hota hai? > $$c\,b = (3\times10^8)(1\times10^{-6}) = 300\text{ m}.$$ > *Ye step kyun?* Dikhata hai kyun timing GNSS accuracy dominate karta hai — ek microsecond ek football-stadium-sized error hai. Isliye hi $b$ ko solve karna zaroori hai, assume nahi karna. > [!example] Example 4 — DOP intuition > **Q:** Char satellites sab ek direction mein horizon ke paas hain. Accha ya bura? > **A:** Bura — near-parallel line-of-sight vectors ⇒ $\mathbf H^{\mathsf T}\mathbf H$ almost singular ⇒ huge GDOP ⇒ poor vertical accuracy. > *Ye step kyun?* Vertical error khaaskar bahut badhti hai kyunki height constrain karne ke liye koi satellite "neeche" ya sidha upar nahi hai. --- > [!mistake] Common errors ko steel-man karo > **"3D position ke liye 3 satellites kaafi hain."** > *Kyun sahi lagta hai:* 3 unknowns $(x,y,z)$, 3 equations — classic algebra. **Fix:** receiver clock ek *hidden 4th unknown* $b$ add karta hai; ise skip karo to hundreds of metres ka error aata hai (Example 3). 4 chahiye. > > **"Pseudorange true distance hai."** > *Kyun sahi lagta hai:* ye "distance = c × time" hai aur time measure hota hai. **Fix:** time ek *biased* clock par measure hota hai, isliye $\rho = r + cb$. "Pseudo" yahi toh baat hai. > > **"Relativity engineering ke liye negligible hai."** > *Kyun sahi lagta hai:* daily life mein relativistic effects tiny hote hain. **Fix:** 30 cm/ns precision par, 38 μs/day drift ⇒ ~11 km/day error. GNSS arguably GR ka sabse bada roz-marra ka proof hai. > > **"Agar main already 4 GPS sats dekh raha hoon to zyada constellations help nahi karte."** > *Kyun sahi lagta hai:* 4 minimum hai. **Fix:** extra satellites **DOP** kam karte hain aur faulty measurements detect karne ke liye redundancy dete hain (RAIM). Zyada genuinely better hota hai. --- > [!recall]- Feynman: ek 12-saal ke bachche ko explain karo (click to open) > Socho tum ek giant field mein raat ko kho gaye ho, aur char dost uunche towers par se har ek chillata hai "Main tower A par hoon!" aur bilkul usi second clap karta hai. Awaaz dhire chalti hai, isliye door wale dost ki clap tumtak baad mein pahunchti hai. Har clap *kitni der se* aayi, ye compare karke tum exactly jaan sakte ho tum kahan khade ho. GPS yehi satellites se radio "claps" ke saath karta hai, awaaz ki jagah light use karke. Tricky part ye hai: *tumhari* ghadi thodi galat hai, isliye tumhe apni ghadi bhi fix karne ke liye ek extra dost ki clap chahiye — isliye **char** chahiye, teen nahi. > [!mnemonic] Sab yaad karo > **"Please Give Great Big Clocks"** → > **P**seudorange, **G**eometry (DOP), **G**PS/**G**LONASS, **B**eiDou, **C**lock-bias-4th-unknown & relativity **Clocks**. > Aur 4 systems ke liye, owner region ke roughly alphabetical order mein: **G**PS(US)–**G**LONASS(Russia)–**G**alileo(EU)–**B**eiDou(China) = "**GGG-B**". --- ## #flashcards/physics GNSS ka full form kya hai aur ye kya compute karta hai? ::: Global Navigation Satellite System; ye Position, Velocity aur Time (PVT) compute karta hai. Receiver ko minimum 4 satellites kyun chahiye? ::: Teen $x,y,z$ ke liye plus ek unknown receiver-clock bias $b$ solve karne ke liye (4 unknowns). Pseudorange define karo aur uski equation do. ::: Biased clock par measured range: $\rho_i = r_i + cb + \varepsilon_i$, jahan $r_i$ true geometric distance hai. Ise "pseudo" range kyun kehte hain? ::: Kyunki ye receiver ki biased clock use karta hai, isliye ye true distance nahi hai jab tak $cb$ nahi hataya jaata. DOP kya hai aur uska formula? ::: Dilution of Precision — satellite geometry measurement error ko kitna amplify karti hai; $\text{GDOP}=\sqrt{\operatorname{tr}((H^TH)^{-1})}$. Spread-out ya clustered satellites better accuracy dete hain? ::: Spread-out (low DOP); clustered geometry $H^TH$ ko ill-conditioned banati hai (high DOP). GPS satellites par per day net relativistic clock drift aur uska consequence? ::: About +38 μs/day tezi se; uncorrected to ~11 km/day position error aata hai. Kaun se do relativistic effects kaam karte hain, aur unke signs? ::: SR (motion) sat clock slow karta hai ($-v^2/2c^2$); GR (weaker gravity) use fast karta hai ($+\Delta\Phi/c^2$); GR dominate karta hai. CDMA vs FDMA — kaun sa system FDMA use karta hai? ::: GLONASS (classic) FDMA use karta hai (har satellite ke liye alag frequency); GPS/Galileo/BeiDou CDMA use karte hain (shared freq, unique codes). DOP ke through measurement error ka position-error relation? ::: $\sigma_{\text{pos}} = \text{DOP} \times \sigma_{\text{meas}}$. 1 μs receiver clock offset kitna range error cause karta hai? ::: $cb = 3\times10^8 \times 10^{-6} = 300$ m. Geometry matrix $H$ mein kya hota hai? ::: Receiver se har satellite tak ke line-of-sight unit vectors, plus clock term ke liye 1's ka ek column. --- ## Connections - [[Trilateration and Multilateration]] - [[Kalman Filter]] — smooth PVT ke liye GNSS ko IMU ke saath fuse kiya jaata hai - [[Inertial Navigation System (INS)]] — GNSS/INS integration - [[Special Relativity — Time Dilation]] - [[General Relativity — Gravitational Time Dilation]] - [[Least Squares Estimation]] - [[Orbital Mechanics — MEO/GEO/IGSO]] - [[Signal Modulation — CDMA & FDMA]] - [[Dilution of Precision (DOP)]] ## 🖼️ Concept Map ```mermaid flowchart TD GNSS[GNSS constellation] -->|broadcasts| SIG[Timed signals] GNSS -->|four systems| SYS[GPS GLONASS Galileo BeiDou] SYS -->|orbit in| MEO[MEO ~19000-24000 km] SIG -->|receiver measures| PR[Pseudorange rho_i] PR -->|equals c times tx to rx| DIST[Distance = c x time] PR -->|modeled as| OBS[rho = r_i + c x b + err] OBS -->|contains 4 unknowns| UNK[x, y, z, clock bias b] UNK -->|needs one eqn each| FOUR[>= 4 satellites] OBS -->|nonlinear, solved by| LIN[Linearize + Taylor] LIN -->|yields| PVT[Position Velocity Time] UNK -->|resolves| PVT ```