GNSS — GPS, GLONASS, Galileo, BeiDou: Foundations

3.5.19 · D1 · Physics › Guidance, Navigation & Control (GNC) › GNSS — GPS, GLONASS, Galileo, BeiDou

Yeh page kuch bhi assume nahi karta. Parent note GNSS index padhne se pehle, aapko har woh symbol pehle se obvious lagna chahiye jo woh use karta hai. Isliye hum har ek ko ek picture se build karte hain. Upar se neeche padho — har block mein sirf wahi symbols hain jo usse pehle define ho chuke hain.

0. Woh alphabet jo hum earn karne wale hain

Yahan parent note ke saare characters ka poora cast hai. Hum har ek ko order mein define karte hain taaki koi bhi symbol apne build hone se pehle na aaye.

Symbol	Saada naam	Kahan bana
$c$	speed of light (hamara ruler)	§1
$t_{t x}, t_{r x}$	transmit time, receive time	§2
$(x_{i}, y_{i}, z_{i})$	known satellite position	§3
$(x, y, z)$	unknown receiver position	§3
$r_{i}$	sachi geometric distance	§4
$b$	receiver clock bias	§5
$ρ_{i}$	pseudorange (measured)	§5
$ε_{i}$	bacha hua error	§5
$\hat{u}_{i}$	line-of-sight unit vector	§6
$H$	geometry matrix	§6
$Δ x$	guess mein chhota correction	§6

1. $c$ — speed of light, hamara measuring stick

Woh picture. Ek lamp se light nikalte socho. Ek second mein yeh $300, 000$ km sweep kar leti hai — Moon ki distance ke lagbhag. Ek nanosecond ( $1 0^{- 9}$ s) mein yeh sirf $30$ cm chalti hai — ek ruler ki length.

Figure — GNSS — GPS, GLONASS, Galileo, BeiDou

Topic ko kyun chahiye yeh. GNSS seedha tape se distance measure nahi kar sakta. Iski jagah woh time measure karta hai aur $c$ se multiply karta hai. Toh $c$ "signal ko kitna time laga?" aur "satellite kitna dur hai?" ke beech ka conversion factor hai.

2. $t_{t x}$ aur $t_{r x}$ — do clocks

Woh picture. Ek runner ka note paas karte socho. Satellite signal ke andar likhta hai "Maine yeh $t_{t x}$ par bheja." Aapka receiver apni ghadi dekhta hai jab note milta hai aur use $t_{r x}$ kehta hai.

Travel time woh gap hai: $Δ t = t_{r x} - t_{t x}$

Do alag clocks kyun matter karti hain. Satellite ek atomic clock le jaata hai (ek billionth of a second tak accurate). Aapke phone mein ek sasta crystal hai jo drift karta hai. Yahi mismatch ek poori wajah hai ki hume baad mein ek symbol $b$ chahiye hoga (§5). Abhi yeh thought rokke raho.

3. Positions: $(x_{i}, y_{i}, z_{i})$ known, $(x, y, z)$ unknown

Woh picture. Ek street corner par khade ho. Kisi bhi point ko pin karne ke liye teen numbers chahiye: kitna right, kitna forward, kitna upar. Woh triple ek coordinate hai.

$(x_{i}, y_{i}, z_{i})$ — known. Satellite apni khud ki position broadcast karta hai (ek data se jo ephemeris kehlaata hai). Ek badi clue: iska subscript $i$ hai, toh ek per satellite hai.
$(x, y, z)$ — unknown. Yeh aap HO. Koi subscript nahi, kyunki ek hi receiver hai. Yahi hum solve karne ki koshish kar rahe hain.

4. $r_{i}$ — sachi seedhi-line distance (3D mein Pythagoras)

Square root kyun? Kyunki 3D mein distance sirf Pythagorean theorem do baar stack karna hai. 2D mein do points ke beech distance $(Δ x)^{2} + (Δ y)^{2}$ hai — ek right triangle ka hypotenuse. Teesra axis add karo aur root ke neeche ek aur squared term aa jaata hai.

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

Ise zor se padho: East-gap lo, square karo; North-gap, square karo; Up-gap, square karo; jodhdo; square root lo. Yeh aapse satellite tak arrow ki length hai.

5. $b$ , $ρ_{i}$ , $ε_{i}$ — clock ka jhooth aur "pseudo" range

Ab woh twist jo GNSS ko clever banata hai.

Woh picture. Aapki ghadi 12:00:00.000001 dikhati hai jab sahi time 12:00:00.000000 hai. Aap $b = 1$ microsecond aage hain. Aap jo bhi travel time compute karte hain woh $b$ se zyada bada hai.

Us time error ko $c$ se multiply karo ise distance ke roop mein dekhne ke liye:

Kyunki clock $b$ seconds ki fake delay add karta hai, aur $c \cdot b$ metres ki fake distance:

$b$ har equation mein same kyun hai. Sirf EK receiver clock hai, toh wahi same bias $b$ har satellite ki measurement ko equally kharaab karta hai. Yahi trick hai: kyunki $b$ shared hai, ek extra satellite kaafi hai ise pin karne ke liye.

Unknowns count karo. Ab hamare paas chaar unknowns hain: $x, y, z$ aur $b$ . Chaar unknowns ko chaar equations chahiye. Ek equation per satellite ⇒ aapko kam se kam 4 satellites chahiye. (Dekho Least Squares Estimation jab 4 se zyada hain.)

6. $\hat{u}_{i}$ aur $H$ — geometry ko matrix mein badalna

$ρ_{i} = r_{i} + c b$ equation mein ek square root hai, jo solve karna annoying hai. Parent note "linearize" karta hai — curve ko ek guess ke paas seedhi-line approximation se replace karta hai. Iske liye use ek unit vector aur ek matrix chahiye.

Woh picture. Apni ungli seedhi ek satellite ki taraf point karo. Aapki ungli jis direction mein point karti hai woh $\hat{u}_{i}$ hai. Iske teen components batate hain us direction ka kitna East, North, aur Up hai.

Unit vector kyun aata hai. Jab aap apni guessed position ko thoda sa hilate ho, toh distance $r_{i}$ kitni change hoti hai? Sirf aapke nudge ka woh part jo satellite ki line ke along point karta hai matter karta hai. Woh "kitni motion satellite ki taraf hai" yahi exactly hai jo ek unit-vector direction measure karta hai. (Iska formal naam derivative $\partial r_{i} / \partial x$ hai — parent note dekho; answer $\hat{u}_{i}$ ka ek component hai.)

Prerequisite map

Left par har foundation apne right wale box ko feed karta hai; sab kuch final GNSS position fix mein funnel hota hai. Yahan se aap parent note ke liye ready hain, phir time ke saath fixes fuse karne ke liye Kalman Filter aur gaps fill karne ke liye Inertial Navigation System (INS).

Equipment checklist

Khud test karo — agar aap har ek answer kar sakte hain, toh parent note ke liye taiyaar hain.

c

physically kya convert karne deta hai?

Time (measured) aur distance (wanted):

distance = c \times time

t_{t x}

aur

t_{r x}

alag clocks par kyun padhte hain?

t_{t x}

satellite ki atomic clock hai (signal mein);

t_{r x}

aapki sasti receiver clock hai — mismatch se bias

b

banta hai.

Kaunsi position known hai aur kaunsi unknown, aur aap kaise pata lagate hain?

(x_{i}, y_{i}, z_{i})

subscript

i

ke saath known satellite position hai;

(x, y, z)

bina subscript ke unknown receiver hai.

r_{i}

mein square root kyun hai?

Yeh 3D Pythagorean theorem hai — teen perpendicular gaps ke paas hypotenuse ki length.

Ek akela distance

r_{i}

aapko kaunsi surface par rakhta hai?

Satellite

i

par centred

r_{i}

radius ki ek sphere.

ρ_{i}

ko pseudorange kyun kehte hain?

Kyunki yeh ek biased clock se compute kiya jaata hai, toh

ρ_{i} = r_{i} + c b

hai, sachi distance

r_{i}

nahi.

b = 1 μ s

kitne metres ka error inject karta hai?

c b = 300

Har satellite ki equation mein same

b

kyun aata hai?

Sirf ek receiver clock hai, toh iska bias saari measurements ko equally kharaab karta hai — isliye ek extra satellite ise pin karne ke liye kaafi hai.

4 satellites kyun chahiye, 3 nahi?

Chaar unknowns

x, y, z, b

ko chaar equations chahiye; ek per satellite.

\hat{u}_{i}

mein hat ka matlab kya hai, aur vector kya store karta hai?

Length exactly 1; yeh sirf direction (satellite

i

ki taraf) store karta hai, distance nahi.

Matrix

H

kya encode karta hai?

Pure geometry — satellites ki sky directions, plus shared clock bias ke liye

1

s ka ek column.