GPS — pseudorange, trilateration, dilution of precision

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

1. GPS ko "pseudo"-ranges ki ZAROORAT kyun hai?

$c b$ term har satellite ke liye same kyun hai: receiver ke paas ek clock hoti hai. Woh jitni bhi fast/slow ho, woh sabhi measurements ko identically corrupt karti hai. Yeh ek shared unknown hi poora trick hai — isse $b$ solvable banta hai.

2. Trilateration — position scratch se derive karna

Unknowns

Hamare paas 4 unknowns hain: $x, y, z$ (position) aur $b$ (clock bias). Toh humein 4 equations → 4 satellites chahiye.

ρ_{i} = (x - x_{i})^{2} + (y - y_{i})^{2} + (z - z_{i})^{2} + c b, i = 1, 2, 3, 4

KAISE solve karein: ek guess ke around linearize karo (yahi real receivers karte hain)

Formula Linearized GPS equation derive karna

KYA: khatarnak square-root equations ko ek linear system mein badlo by Taylor-expanding around a guess $x_{0} = (x_{0}, y_{0}, z_{0}, b_{0})$ .

Step 1 — true range as a function. Likho $r_{i} (x) = (x - x_{i})^{2} + (y - y_{i})^{2} + (z - z_{i})^{2}$ . Kyun? Hume linearize karne ke liye iske derivatives chahiye.

Step 2 — the gradient. Differentiate karo: $\frac{\partial r _{i}}{\partial x} = \frac{x - x _{i}}{r _{i}} \equiv e_{i x}$ aur similarly $e_{i y}, e_{i z}$ . Kyun? $(e_{i x}, e_{i y}, e_{i z})$ exactly woh unit vector hai satellite $i$ se receiver ki taraf — "line-of-sight direction."

Step 3 — first-order Taylor. Guess ke paas, $ρ_{i} \approx ρ_{i}^{(0)} + e_{i x} Δ x + e_{i y} Δ y + e_{i z} Δ z + c Δ b$ jahan $ρ_{i}^{(0)}$ woh pseudorange hai jo guess se predict hoti hai. $c Δ b$ kyun? Kyunki $\partial ρ_{i} / \partial b = c$ har satellite ke liye.

Step 4 — the residual. $Δ ρ_{i} = ρ_{i}^{measured} - ρ_{i}^{(0)}$ lo. Sabhi satellites stack karo: $Δ ρ = G Δ x$ geometry matrix ke saath $G = e_{1 x} e_{2 x} ⋮ e_{n x} e_{1 y} e_{2 y} e_{n y} e_{1 z} e_{2 z} e_{n z} 11 ⋮ 1 .$ 1's ka column kyun? Woh $\partial ρ / \partial (c b)$ hai — clock direction.

Step 5 — solve & iterate. $n \geq 4$ satellites ke saath, least squares deta hai $Δ x = (G^{⊤} G)^{- 1} G^{⊤} Δ ρ,$ update karo $x_{0} \leftarrow x_{0} + Δ x$ , aur repeat karo jab tak $Δ x \to 0$ . Iterate kyun? Problem nonlinear hai; har iteration ek better point pe re-linearize karta hai.

Figure — GPS — pseudorange, trilateration, dilution of precision

3. Dilution of Precision (DOP) — geometry errors ko kaise amplify karti hai

4. Worked examples

5. Common mistakes (steel-manned)

6. Flashcards

Pseudorange kya hota hai?

Measured satellite-to-receiver distance

c (t_{r x} - t_{t x})

, jo true range plus

c b

ke barabar hai jahan

b

unknown receiver clock bias hai.

GPS ko minimum 4 satellites kyun chahiye (3 nahi)?

4 unknowns hain — 3 position coordinates aur 1 clock bias — isliye 4 equations (satellites) chahiye.

Receiver clock bias har pseudorange mein same kyun hota hai?

Ek single receiver clock hoti hai, isliye uska offset

b

sabhi measurements ko identically contaminate karta hai — isse yeh ek shared unknown ke roop mein solvable banta hai.

Geometry matrix

G

kis cheez se bana hota hai?

Rows har satellite se receiver tak ke line-of-sight unit vectors hain, clock (

c b

) term ke liye ek final column of 1's ke saath.

Position solution ka covariance batao.

Cov (Δ x) = σ_{UERE}^{2} (G^{⊤} G)^{- 1}

PDOP define karo aur position-error relation batao.

PDOP

= q_{xx} + q_{y y} + q_{z z}

from

Q = (G^{⊤} G)^{- 1}

; position error

σ_{p os} = PDOP \times σ_{UERE}

Kya DOP measurements pe depend karta hai?

Nahi — sirf satellite geometry pe (

G

). Lower DOP = better geometry.

Kaun si geometry bad (high) DOP deti hai?

Satellites jo clustered hain / nearly collinear line-of-sight vectors, jisse

G^{⊤} G

near-singular ho jaata hai (small determinant).

1 μ s

clock bias ko range error mein convert karo.

c b = 3 \times 1 0^{8} \times 1 0^{- 6} = 300

Recall Feynman: ek 12-saal ke bachhe ko samjhao

Imagine karo tum blindfolded ho aur door ke doston ne ek saath "NOW!" cheekha. Tum har cheekh thodi der baad sunte ho kyunki awaaz mein time lagta hai — dost jitna door, utni zyada delay. Agar tumhe pata hota ki har cheekh exactly kitna time lagi, tum figure out kar sakte the ki tum kahan khade ho. GPS yahi radio "NOW!"s ke saath satellites se karta hai. Ek problem: tumhari stopwatch toot gayi hai aur hamesha ek si thodi galat rahti hai. Khushqismati se, agar tum chaar doston ko instead of teen suno, to math dono figure out kar sakta hai — kahan ho tum aur tumhari stopwatch kitni tooti hui hai, simultaneously. Aur agar chaaon yaar sky ke ek hi kone mein ikthe hain, tumhara guess fuzzy hoga; agar woh tumhare charo taraf spread hain, tumhara guess sharp hoga. Uss spreading-out quality ko DOP kehte hain.

Connections

Trilateration and Multilateration — method ki geometric root.
Least Squares Estimation — over-determined $Δ ρ = G Δ x$ solve karta hai.
Kalman Filter in GNC — recursive successor jo time ke saath pseudoranges fuse karta hai.
Clock Bias and Atomic Clocks — 4th unknown kyun exist karta hai.
Covariance Propagation — DOP derivation ke peeche ki machinery.
Reference Frames — ECEF and WGS84 — $(x_{i}, y_{i}, z_{i})$ ke liye coordinate system.
Time of Flight and Ranging — $c Δ t$ physics.