3.5.18 · D1GPS — pseudorange, trilateration, dilution of precision
Ek GPS receiver aasman mein maujood satellites se radio "time stamps" sun-ta hai aur measure karta hai ki signal aane mein kitna time laga — isi se wo pata lagata hai ki har satellite kitni door hai . Kyunki uski apni clock sasti hoti hai aur ek unknown amount se galat hoti hai, use apni position aur apni clock kitni galat hai — dono ek saath solve karne padte hain — aur yahi ek extra unknown hai jiske wajah se poora subject exist karta hai.
Yeh page ek toolbox hai. Isse pehle ki tum parent topic padho, usme use hone wala har letter aur symbol tumhe pehle se jaana-pehchaana lagna chahiye. Hum har ek cheez ko zero se banate hain, use ek picture se anchor karte hain, aur clearly batate hain ki GPS ko yeh kyun chahiye.
Definition 3D space mein ek point( x , y , z ) ek exact jagah ko pin karte hain. Socho ek giant invisible box Earth ke around kheenchi gayi hai jisme teen rulers ek corner par milte hain: kitna East (x y z
Intuition GPS ko yeh kyun chahiyeGPS ka poora kaam ek aisa triple output karna hai: tumhara ( x , y , z ) Reference Frames — ECEF and WGS84 — lekin abhi ke liye bas "teen perpendicular rulers" imagine karo.
Distance-from-a-corner Ek point ( x , y , z )
Hum satellite ki jagah ( x i , y i , z i ) ( x , y , z )
Intuition Square root of squares kyun — yeh bas do baar Pythagoras haiFlat paper par, do dots ke beech straight-line distance ( across ) 2 + ( up ) 2 Square root woh tool hai jo "side lengths" ko wapas "straight-line length" mein badalta hai. Pehle square karne se har gap positive ho jaata hai (− 3 + 3
Definition Yahan ke symbols
r i i true geometric distance ("range"). Chhota i label hai: satellite 1, 2, 3, 4...x i , y i , z i x , y , z
Subscript i Yeh ek index hai jo batata hai kaunsa satellite; r 3
Dekho Time of Flight and Ranging — distance physically kaise measure hoti hai, sirf compute nahi.
c c ≈ 3 × 1 0 8 fixed constant hai.
Intuition Picture: light ek time ki tape measure ke jaisiEk radio pulse satellite se nikalti hai aur c t c × t c time maapti hai aur c
Worked example Ek chhoti si time error kitni badi hoti hai?Ek microsecond 1 0 − 6 c ( 3 × 1 0 8 ) ( 1 0 − 6 ) = 300 ek millisecond galat ho, tumhare position ko teen football fields ki length jitna galat bata deti hai. Yeh ek line hi wajah hai ki clock bias itna zyada matter karta hai.
Definition Transmit aur receive times
t t x bheja , uske atomic clock se stamped (bahut accurate — dekho Clock Bias and Atomic Clocks ).t r x receive kiya, hamare saste receiver clock se padha gaya.Subscripts t x r x t ransm it aur r ec eive ke liye.
Socho ek runner start line se t t x t r x t r x − t t x c ( t r x − t t x )
b Clock bias b off hai, seconds mein measure ki gayi. Agar b aage chal rahi hai; negative hai toh peeche .
Intuition Ek number sab kuch kyun bigaad deta hai — aur yeh gift kyun haiTumhare paas ek receiver clock hai. Wo jitna bhi fast ya slow ho, har t r x b c b same unknown har equation mein appear karta hai, ek extra satellite tumhein usse pin down karne ke liye kaafi information deta hai. Ek shared error ek solvable error hai.
Saare satellites ke liye clock error same kyun hoti hai Sirf ek hi receiver clock hoti hai, isliye uski offset b
ρ i = r i + c b Pseudorange true distance r i plus clock se hone wali error c b ρ "range jaisi lagti hai lekin abhi true nahi hai."
Socho har satellite ke around ek sphere draw karo jiska radius pseudorange ke barabar hai. c b b
Common mistake "Pseudorange = satellite tak distance"Yeh sahi lagta hai kyunki yeh c × c b r i b
Intuition Ek range = ek sphereAgar tum sirf jaante ho ki tum satellite 1 se 20,000 km door ho, tum us sphere par kahin bhi ho sakte ho jiska radius itna hai. Satellite 2 add karo: tum dono spheres par hona chahiye → unka overlap ek circle hai. Satellite 3 add karo: teen spheres (generally) do points par milte hain, aur Earth ek ko throw away kar deti hai. Yeh classic trilateration hai.
fourth sphere kyun chahiye
Teen spheres ek point pin karte hain agar radii exact hon. Lekin hamare radii pseudoranges hain — sab c b b Geometry kehti hai 3; clocks ki physics kehti hai 4. Zyada detail Trilateration and Multilateration mein.
Do spheres mein intersect hote hain... circle mein. Teen (exact radii se) do points mein.
Definition Unknowns vs equationsUnknowns: x , y , z , b Equations: har satellite se ek pseudorange. 4 unknowns solve karne ke liye tumhe kam se kam 4 equations chahiye, isliye kam se kam 4 satellites.
Mnemonic Question marks count karoHar "?" ko apni equation chahiye. Teen space coordinates plus ek clock = char question marks = minimum char satellites.
Definition Vector aur unit vectorVector ek arrow hai: iska ek direction aur ek length hoti hai. Unit vector ek aisa arrow hai jise exactly 1 direction rakhta hai, length throw away kar deta hai.
Intuition Line-of-sight unit vector( e i x , e i y , e i z ) i e i x , e i y , e i z "satellite mere liye kis direction mein hai?" Parent topic dikhata hai ki yeh exactly r i x − x i rate of change mein bas "us nudge ka kitna hissa satellite ki line ki taraf point karta hai" hi matter karta hai. Direction, distance nahi, geometry control karta hai.
Line-of-sight vector ki length 1 kyun hoti hai Hume sirf direction ki parwah hai; r i
Definition Matrix ek grid ke roop meinMatrix numbers ki ek rectangular grid hai, bade bracket mein likhi. Yahan har row ek satellite ki line-of-sight direction hai plus clock ke liye ek 1 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 1 1 ⋮ 1
Intuition Matrix kyun use karte hain?Char satellites ke saath tumhare paas char intertwined equations hain. Matrix bas woh bookkeeping hai jo charon ek saath solve karti hai. Jab zyada char satellites hon, system over-determined ho jaata hai aur tum best compromise fit karte ho Least Squares Estimation use karke; ( G ⊤ G ) − 1 G ⊤
G Clock-bias direction — har satellite b
Definition Measurements ki randomness
Noise = har real measurement mein chhota random wobble.σ standard deviation , us wobble ki typical size.σ UERE Covariance = ek table jo describe karta hai ki kai numbers ki errors kaise spread aur lean karte hain. Ise solution mein propagate karna Covariance Propagation se hota hai.
Intuition Picture: ek fuzzy dot, sharp point nahiNoise ki wajah se tumhari computed position ek crisp dot nahi balki ek chhota fuzzy cloud hai. σ DOP (Dilution of Precision) woh number hai jo batata hai ki satellite geometry us fuzz ko kitna stretch karti hai — achchi geometry use tight round blob rakhti hai, buri geometry use ek lambi streak mein smear kar deti hai.
DOP ya σ UERE DOP — yeh sirf satellite directions par depend karta hai, measurement quality par nahi.
Transmit and receive times
Four unknowns need four satellites
Spheres and intersections
Unit line of sight vectors
Noise sigma and covariance
Sab kuch jo aage hai — linearised solve, DOP formula, Kalman Filter in GNC jo baad mein in fixes ko time ke saath fuse karta hai — yeh sab upar ke baarah pieces par tikaa hai.
Kya tum memory se bata sakte ho ki har symbol ka kya matlab hai aur GPS ko yeh kyun chahiye? Check karne ke liye reveal karo.
( x , y , z ) Receiver ki unknown position — teen perpendicular ruler-distances; woh final answer jo GPS output karta hai.
r i Satellite i
c Speed of light, 3 × 1 0 8 time ko distance mein convert karta hai.
t t x , t r x Transmit time (satellite ka atomic clock) aur receive time (hamaari sasti clock); unka difference times c
b Receiver clock bias seconds mein — ek shared unknown jo har measurement ko same c b
ρ i Pseudorange = true range r i c b
4 satellites kyun, 3 nahi Char unknowns x , y , z , b
Sphere picture Har range ek satellite ke around ek sphere hai; unka intersection tumhari location hai.
( e i x , e i y , e i z ) Unit line-of-sight vector — satellite se receiver ki taraf pure direction , length 1.
G G ⊤ G ( G ⊤ G ) − 1 Directions ki geometry matrix (+clock column); uska transpose-product aur inverse least-squares solver banate hain.
Δ "Mein ek chhota change" — nudges Δ x Δ ρ
σ UERE Single-measurement error size, aur geometry-only amplification factor; inhe multiply karo toh position error milti hai.