Exercises — GNSS — GPS, GLONASS, Galileo, BeiDou

• ${\rho }_i$ = satellite $i$ tak ka pseudorange: woh distance jo tumne socha ki tune measure ki, apni imperfect clock use karke.
• $r_i$ = true geometric distance (seedhi line) tumse satellite $i$ tak.
• $cb$ = clock-bias distance: tumhari clock error $b$ (seconds) ko $c$ se multiply karke metres mein convert kiya.
• ${\epsilon }_i$ = leftover errors (atmosphere bending, receiver noise).

Chaar. Unknowns hain tumhare teen coordinates $x,y,z$ aur tumhara clock bias $b$ — yani 4 unknowns. Har satellite ek equation deta hai, toh tumhe kam se kam 4 equations chahiye ⇒ 4 satellites.

CDMA (Code Division Multiple Access). Har satellite ek hi frequency share karta hai lekin ek unique pseudo-random code carry karta hai; receiver incoming signal ko har known code ke against correlate karta hai taaki ek ek satellite ko pick out kar sake.

Distance = speed × time: $\rho =ct=(3\times 10^8)(70\times 10^{-3})=2.1\times 10^7\text{m}=21,000\text{km}.$ Kyun: $\rho =ct$ bas "light us time mein kitni door jaati hai" hai. $21,000$ km MEO band ke andar aata hai (~19,000–24,000 km, dekho Orbital Mechanics — MEO/GEO/IGSO), toh sanity check pass ho jaata hai.

$cb=(3\times 10^8)(2\times 10^{-6})=600\text{m}.$ Kyun: observation equation mein bias term literally $cb$ hai. Ek microsecond $300$ m hota hai; do microseconds, $600$ m. Isliye timing — geometry nahi — raw GNSS error ko dominate karta hai, aur isliye $b$ ko solve karna padta hai, kabhi bhi zero assume nahi karna.

$b=0$ ke saath pseudoranges sach mein true distances hain, toh tum dono circles pe ho: $x^2+(y-20000{)}^2=25000^2,(x-30000{)}^2+(y-20000{)}^2=25000^2.$ Dono equations subtract karo — $(y-20000{)}^2$ terms cancel ho jaate hain: $x^2-(x-30000{)}^2=0\Rightarrow 60000x-30000^2=0\Rightarrow x=15000\text{km}.$ $x=15000$ ko circle 1 mein wapas daalo: $(15000{)}^2+(y-20000{)}^2=25000^2\Rightarrow (y-20000{)}^2=25000^2-15000^2=20000^2.$ Toh $y-20000=\pm 20000\Rightarrow y=40000$ km ya $y=0$ km. Figure dekho: do circles do points pe milti hain. Earth pe tum physically impossible wala discard karte ho (jaise satellites ke upar wala point, ya planet ke bahar). Yeh 2D shadow hai uss reason ka ki ambiguity khatam karne ke liye ek extra satellite kyun chahiye.

Ek uncorrected clock error $b$ ek range error $cb$ ban jaata hai: $cb=(3\times 10^8)(38\times 10^{-6})=1.14\times 10^4\text{m}\approx 11.4\text{km per day.}$ Kyun: wahi $\rho =r+cb$ machinery — $38\mu \text{s}$ ki timing error $\approx 11$ km ki distance error hai. Isliye satellite oscillator ko ground pe pre-detune kiya jaata hai: fix launch se pehle hi bake in ho jaata hai. Yeh Special Relativity — Time Dilation (clock slow hoti hai) aur General Relativity — Gravitational Time Dilation (clock fast hoti hai) ko blend karta hai, jisme GR term jeet jaata hai.

Special relativity moving clock ko $\frac{v^2}{2c^2}$ fraction se slow karta hai per unit time. Ek din mein ($86400$ s $=8.64\times 10^{10}\mu \text{s}$): $\frac{v^2}{2c^2}=\frac{3874^2}{2(3\times 10^8{)}^2}=8.34\times 10^{-11}.$ $\Delta t_{SR}=-(8.34\times 10^{-11})(8.64\times 10^{10}\mu \text{s})\approx -7.2\mu \text{s/day.}$ Net: $+45+(-7.2)\approx +37.8\approx +38\mu \text{s/day}$. ✓ Kyun: SR (motion) aur GR (upar kamzor gravity) opposite directions mein push karte hain; gravitational speed-up jeet jaata hai, aur famous $+38$ milta hai.

Geometry matrix $\mathbf{H}$ ki har row tumse ek satellite ki taraf unit line-of-sight vector hoti hai (plus clock column ke liye ek $1$). Agar chaar satellites almost ek hi direction mein hain, toh woh unit vectors almost parallel hain. Parallel rows ${\mathbf{H}}^{\mathsf{T}}\mathbf{H}$ ko almost singular bana deti hain (uska inverse blow up ho jaata hai). Kyunki $\text{GDOP}=\sqrt{\mathrm{tr}(({\mathbf{H}}^{\mathsf{T}}\mathbf{H}{)}^{-1})},$ near-singular matrix GDOP ko bada kar deta hai, aur ${\sigma }_{\text{position}}=\text{DOP}\times {\sigma }_{\text{measurement}}$ phir ek chhoti si range error ko ek badi position error mein amplify kar deta hai. Vertical sabse zyada suffer karta hai kyunki koi satellite overhead ya neeche nahi hai jo tumhari height ko "pin" kare — figure dekho: jab sab sight-lines horizon skim kar rahi hain, tumhe upar ya neeche move karne se koi bhi range barely change hoti hai. Dekho Dilution of Precision (DOP) aur Least Squares Estimation.

Widely spread supports wobble resist karte hain kyunki ek chhota push structure ko ab collapse nahi kar sakta — mathematically, spread-out directions geometry matrix ko well-conditioned banate hain taaki errors amplify na hon. Phone yeh achieve karta hai sab chaar constellations ek saath use karke (GPS + GLONASS + Galileo + BeiDou), toh zyada satellites aur zyada directions mein visible hote hain, DOP kam hota hai — urban canyons mein bahut zaroori.

(a) Unknown correction vector $\Delta \mathbf{x}=(\Delta x,\Delta y,\Delta z,c\Delta b)$ — teen position nudges aur clock-bias nudge (metres mein).

(b) Satellite $i$ ke liye, $\mathbf{H}$ ki uski row hai $(-{\hat{u}}_{ix},-{\hat{u}}_{iy},-{\hat{u}}_{iz},1)$, jahan ${\hat{u}}_i$ unit vector hai jo tumhare guessed position se satellite ki taraf point karta hai. Last column mein woh $1$ kehta hai "ek clock error saari pseudoranges ko equally shift karta hai."

(c) Measured-minus-predicted residuals $\Delta \mathbf{\rho }$ ke saath: $\Delta \mathbf{x}=({\mathbf{H}}^{\mathsf{T}}\mathbf{H}{)}^{-1}{\mathbf{H}}^{\mathsf{T}}\Delta \mathbf{\rho }.$

(d) Hum iterate karte hain kyunki true equations nonlinear hain (square roots). Taylor linearization sirf guess ke paas accurate hai. Toh hum: guess karo → linearize karo → $\Delta \mathbf{x}$ solve karo → guess update karo → repeat karo jab tak $\Delta \mathbf{x}$ ek tolerance se neeche na aa jaaye (usually 2–3 loops). Yeh exactly Least Squares Estimation hai ek loop mein wrapped; ek moving receiver ke liye tum ise ek Kalman Filter mein feed karte ho ek Inertial Navigation System (INS) ke saath.

Haan. Altitude constraint ek free extra equation hai ("tumhari height 0 hai"). Count karo: unknowns abhi bhi $x,y,z,b$ = 4 hain. Equations = 3 pseudoranges + 1 altitude = 4. Chaar equations, chaar unknowns ⇒ solvable. Yeh "altitude aiding" hai, ek real trick jo ships, barometers wale aircraft, aur known-terrain database wale receivers use karte hain.

Standard GNSS fix mein unknowns ki sankhya ::: Chaar — $x,y,z$ aur clock bias $b$. $1\mu \text{s}$ clock error se range error ::: $300$ m. Kaun sa relativity effect GPS clock ko fast chalata hai ::: General relativity (upar kamzor gravity) — yeh special-relativity slowdown ko outweigh karta hai. High GDOP physically kya matlab hai ::: Satellites ek jagah bunched hain, toh chhoti range errors badi position errors mein blow up ho jaati hain. 3 satellites phir bhi fix kaise de sakte hain ::: Ek external constraint add karo (jaise known altitude) 4th equation ke roop mein.