164
“Al-Fa g‘oniy a lodla i” elek on ilmiy
ju nali
ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil
"Descendan s o Al-Fa ghani" elec onic scien i ic
jou nal.
ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 yea
Электронный научный журнал "Потомки Аль-
Фаргани"
ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год
h ps://al- a goniy.uz/
MATHEMATICAL FOUNDATIONS OF GPS AND RTK POSITIONING
Khon u ae Sa do bek,
PhD s uden o he Tashken S a e Technical Uni e si y
Email: sa do is oil[email p o ec ed]
Muhammadjono a Shuk onakhon,
S uden o he Fe gana S a e Technical Uni e si y
Abs ac : D ones equipped wi h Global Na iga ion Sa elli e Sys ems ha e ans o med mode n
geospa ial mapping, p ecision ag icul u e, and su eying. S anda d Global Posi ioning Sys em
echnology p o ides me e -le el accu acy, bu i s p ecision is o en insu icien o ca og aphic
applica ions. Real-Time Kinema ic echniques enhance GPS accu acy by applying ca ie -phase
co ec ions om e e ence s a ions, achie ing cen ime e -le el posi ioning. This pape examines he
in eg a ion o d ones wi h GPS and RTK, e iews ma hema ical o mulas unde lying loca ion
de e mina ion, and highligh s applica ions in geospa ial ca og aphy.
Keywo ds: GNSS, GPS, RTK, UAV, d one, eal- ime, posi ion, 3d su ace
In oduc ion.
Geospa ial posi ioning echnologies o m he
backbone o mode n mapping and su eying [1].
T adi ional me hods o g ound-based measu emen a e
labo -in ensi e and limi ed in spa ial co e age. The
ad en o d ones (UAVs) combined wi h sa elli e-
based posi ioning sys ems has e olu ionized da a
collec ion by o e ing apid, high- esolu ion, and
au oma ed geospa ial mapping [2].
While he Global Posi ioning Sys em (GPS) is
widely used, i s ypical e o ma gin o 5–10 me e s a e
insu icien o many scien i ic and enginee ing asks.
[3] To o e come his limi a ion, Real-Time Kinema ic
(RTK) posi ioning was de eloped, o e ing cen ime e -
le el accu acy by u ilizing co ec ion da a om a
e e ence s a ion [4]. D one pla o ms se e as ideal
ca ie s o GNSS/RTK senso s due o hei abili y o
co e di icul e ain and cap u e high- esolu ion da a
in eal ime [5].
Mo eo e , as Khalilo e al. [6] emphasize,
ad anced compu a ional echniques such as neu al
ne wo ks and sel -lea ning algo i hms may u he
enhance posi ional accu acy h ough adap i e e o
co ec ion and signal p ocessing.
This pape explo es how GPS and RTK
de e mine loca ion, which ma hema ical o mulas a e
used in coo dina e compu a ion, and how hese
me hods a e applied in d one-based geospa ial
ca og aphy.
Me hods.
The Global Posi ioning Sys em (GPS)
de e mines a ecei e ’s loca ion using ila e a ion
om sa elli e signals [7]. Each sa elli e ansmi s i s
posi ion (𝑥𝑖, 𝑦𝑖, 𝑧𝑖) and a p ecise imes amp. The
ecei e calcula es he a el ime o signals o es ima e
dis ances called pseudo anges.
The pseudo ange equa ion is [8]:
𝜌𝑖=√(𝑥−𝑥𝑖)2+(𝑦−𝑦𝑖)2+(𝑧−𝑧𝑖)2+𝑐∗∆𝑡
Whe e:
𝜌 = measu ed pseudo ange om sa elli e i
(x, y, z) = ecei e coo dina es (unknowns o be
sol ed)
(𝑥𝑖, 𝑦𝑖, 𝑧𝑖) = known sa elli e coo dina es
c = speed o ligh
Δ = ecei e clock bias
Wi h a leas ou sa elli es, he ecei e sol es
ou nonlinea equa ions o de e mine (x, y, z, Δ ) [3].
RTK enhances GPS by inco po a ing ca ie -
phase measu emen s [9]. Ins ead o only pseudo
anges, RTK measu es he phase o he ca ie wa e
[4]:
165
“Al-Fa g‘oniy a lodla i” elek on ilmiy
ju nali
ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil
"Descendan s o Al-Fa ghani" elec onic scien i ic
jou nal.
ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 yea
Электронный научный журнал "Потомки Аль-
Фаргани"
ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год
h ps://al- a goniy.uz/
𝜑𝑖=
1
𝜆(√(𝑥−𝑥𝑖)2+(𝑦−𝑦𝑖)2+(𝑧−𝑧𝑖)2+𝑐∗∆𝑡−
𝑁𝑖∗𝜆)
Whe e:
𝜑𝑖 = measu ed ca ie phase (in cycles)
𝜆 = ca ie wa eleng h
𝑁𝑖 = in ege ambigui y (unknown numbe o
whole wa eleng hs)
A base s a ion a a known loca ion p o ides
co ec ions by compa ing i s measu ed alues o ue
dis ances, elimina ing common-mode e o s such as
ionosphe ic delays [10]. The o e (d one) applies
hese co ec ions in eal ime o achie e cen ime e
accu acy.
D ones equipped wi h RTK-GNSS modules ac
as o e s, in eg a ing posi ioning da a wi h onboa d
senso s (e.g., came as, LiDAR, IMU) [11]. As
Khon u ae no es, his in eg a ion enhances geospa ial
ca og aphy by educing su ey ime and imp o ing
accu acy in inaccessible e ain [2].
Uljae , Ubaydullae , and Khon u ae [5] de ail
d one-based coo dina e de e mina ion echnologies,
showing ha UAVs combined wi h RTK can eplace
adi ional g ound su eying me hods while
main aining high p ecision.
Khalilo e al. [6] sugges ha ad anced neu al
ne wo k me hods can op imize weigh coe icien s o
e o educ ion, p o iding u u e pa hways o
au onomous calib a ion o d one na iga ion sys ems.
Resul s.
The accu acy o posi ioning sys ems a ies
signi ican ly depending on he me hod used.
S andalone GPS ypically p o ides an accu acy o
abou 5–10 me e s, which is su icien o e e yday
na iga ion in ca s, sma phones, and logis ics [7].
Howe e , his accu acy is s ongly in luenced by
ac o s such as ionosphe ic and oposphe ic delays,
mul ipa h in e e ence caused by signal e lec ions
om e ain o buildings, and he geome ic
con igu a ion o isible sa elli es (Dilu ion o
P ecision) [3]. While inexpensi e and globally
a ailable, s andalone GPS canno be used in asks
equi ing high p ecision such as cadas al mapping o
cons uc ion su eying.
To imp o e on his, Di e en ial GPS (DGPS)
was de eloped, which educes common-mode e o s
by using co ec ion da a om a s a iona y e e ence
ecei e loca ed a a p ecisely known posi ion. By
b oadcas ing he co ec ion in o ma ion o mobile
use s, DGPS can imp o e accu acy o abou 0.5–3
me e s. This makes DGPS sui able o applica ions
such as ag icul u al ield mapping, ma ine na iga ion,
and gene al geospa ial moni o ing. Howe e , e en his
me hod does no achie e he le el o p ecision needed
o geode ic su eying o enginee ing p ojec s.
The highes p ecision in eal- ime is ob ained
wi h Real-Time Kinema ic (RTK) GPS, which
achie es 1-3-cen ime e accu acy unde a o able
condi ions [4]. RTK uses no only pseud o ange
measu emen s bu also ca ie -phase obse a ions,
combined wi h co ec ion da a om a base s a ion [10].
This makes i possible o elimina e mos e o s,
including a mosphe ic delays and sa elli e clock biases.
The echnology is widely applied in geospa ial
ca og aphy [2], p ecision ag icul u e, cons uc ion,
and he na iga ion o d ones and au onomous ehicles
[5]. The main limi a ion is he equi emen o
con inuous, eliable communica ion be ween he o e
(e.g., d one) and he base s a ion o e e ence ne wo k.
Ano he ad anced me hod wo h men ioning is
P ecise Poin Posi ioning (PPP), which does no equi e
a base s a ion bu ins ead uses p ecise sa elli e o bi s
and clock co ec ions p o ided by global se ices [12].
PPP can achie e accu acies o abou 5–20 cen ime e s
a e a con e gence ime o 10–30 minu es. This
app oach is use ul o geode ic applica ions in emo e
a eas whe e local e e ence s a ions a e no a ailable.
Table 1. The compa ison o posi ioning
me hods.
Technology
Accu acy
Applica ions
S andalone
GPS
5–10 m
Sma phones, ca
na iga ion, logis ics [7]
DGPS
0.5–3 m
Ag icul u e, ma ine
na iga ion, mapping [8]
166
“Al-Fa g‘oniy a lodla i” elek on ilmiy
ju nali
ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil
"Descendan s o Al-Fa ghani" elec onic scien i ic
jou nal.
ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 yea
Электронный научный журнал "Потомки Аль-
Фаргани"
ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год
h ps://al- a goniy.uz/
Technology
Accu acy
Applica ions
RTK GPS
1–3 cm
Su eying, d ones,
p ecision ca og aphy [4]
PPP
5–20 cm
Geodesy, emo e
moni o ing, scien i ic use
[12]
Unmanned Ae ial Vehicles (UAVs) equipped
wi h Real-Time Kinema ic (RTK) sys ems ha e
ans o med ca og aphy by p o iding high-p ecision
spa ial da a o di e se applica ions. Thei abili y o
gene a e cen ime e -le el accu a e geospa ial da ase s
allows o de ailed analysis in opog aphy, ag icul u e,
disas e managemen , and u ban planning.
Topog aphic Mapping. UAV-RTK sys ems
p oduce high- esolu ion Digi al Ele a ion Models
(DEMs) by cap u ing o e lapping images and applying
pho og amme ic p ocessing. The ele a ion Z a a poin
(x, y) can be compu ed om s e eo image y using he
collinea i y equa ions:
𝑥−𝑥0
=−𝑓𝜏11(𝑋−𝑋𝑠)+𝜏12(𝑌−𝑌𝑠)+𝜏13(𝑍−𝑍𝑠)
𝜏31(𝑋−𝑋𝑠)+𝜏32(𝑌−𝑌𝑠)+𝜏33(𝑍−𝑍𝑠)
𝑦−𝑦0
=−𝑓𝜏21(𝑋−𝑋𝑠)+𝜏22(𝑌−𝑌𝑠)+𝜏23(𝑍−𝑍𝑠)
𝜏31(𝑋−𝑋𝑠)+𝜏32(𝑌−𝑌𝑠)+𝜏33(𝑍−𝑍𝑠)
whe e (𝑋𝑠, 𝑌𝑠, 𝑍𝑠) is he came a posi ion, 𝜏𝑖𝑗 a e
elemen s o he o a ion ma ix, and 𝑓 is he ocal
leng h. DEMs gene a ed can be isualized as 3D
su aces o con ou maps, p o iding essen ial
in o ma ion o slope analysis, wa e shed delinea ion,
and lood modeling. A ypical g aph is a con ou map
o ele a ion s. geog aphic coo dina es (x, y).
Ag icul u e. P ecision ag icul u e le e ages
UAVs o p oduce ege a ion indices such as he
No malized Di e ence Vege a ion Index (NDVI):
𝑁𝐷𝑉𝐼 =𝑁𝐼𝑅−𝑅𝑒𝑑
𝑁𝐼𝑅+𝑅𝑒𝑑
whe e NIR and Red a e he nea -in a ed and
ed e lec ance alues o c ops. NDVI maps enable
a me s o iden i y s essed a eas and gene a e
p esc ip ion maps o a iable- a e e iliza ion. G aphs
o NDVI alues o e ield coo dina es can highligh
spa ial he e ogenei y, allowing a ge ed in e en ions
o inc ease yield and educe chemical usage.
Disas e Response. UAVs p o ide apid
si ua ional awa eness by gene a ing o ho mosaics and
3D econs uc ions o a ec ed egions. The olume o
deb is o loodwa e can be es ima ed using:
𝑉 =∑(𝑍𝑖−𝑍𝑟𝑒𝑓)∗𝐴
𝑛
𝑖=1
whe e 𝑍𝑖 is he measu ed ele a ion, 𝑍𝑟𝑒𝑓 is a
e e ence su ace, and A is he a ea o each g id cell.
G aphs such as hea maps o ele a ion di e ences o
ime-se ies o a ec ed a ea co e age can assis
eme gency planne s in p io i izing in e en ions.
U ban Planning. In u ban en i onmen s, UAV-
de i ed poin clouds gene a e accu a e 3D ci y models.
Su ace models can be used o calcula e building
heigh s H and shadow leng hs L o sola exposu e
s udies: 𝐿 =𝐻∗ an(∅)
whe e ∅ is he sola ele a ion angle. G aphical
ou pu s can include 3D ende ings, shadow simula ion
plo s, and densi y maps o in as uc u e planning.
Addi ionally, in eg a ion wi h Geog aphic In o ma ion
Sys ems (GIS) allows o e laying mul iple da ase s o
mul i-c i e ia analysis.
In eg a ion o UAV Da a. A comp ehensi e
g aph can be a mul i-laye ed map showing ele a ion,
NDVI, and u ban in as uc u e simul aneously.
Ma hema ical in e pola ion me hods like K iging o
In e se Dis ance Weigh ing (IDW) can be applied o
gene a e con inuous su aces om disc e e UAV
poin s:
𝑍(𝑥0)=∑𝑍(𝑥𝑖)
𝑑(𝑥0,𝑥𝑖)𝑝
𝑛
𝑖=1
∑1
𝑑(𝑥0,𝑥𝑖)𝑝
𝑛
𝑖=1
whe e 𝑑(𝑥0,𝑥𝑖) is he dis ance be ween g id
poin 𝑥0 and measu emen 𝑥𝑖 and p is a weigh ing
exponen .
O e all, UAV-RTK sys ems p o ide
unpa alleled spa ial p ecision and e sa ili y, enabling
quan i a i e analysis and isualiza ion ha suppo
167
“Al-Fa g‘oniy a lodla i” elek on ilmiy
ju nali
ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil
"Descendan s o Al-Fa ghani" elec onic scien i ic
jou nal.
ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 yea
Электронный научный журнал "Потомки Аль-
Фаргани"
ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год
h ps://al- a goniy.uz/
opog aphy, ag icul u e, disas e managemen , and
u ban planning. The in eg a ion o ma hema ical
models, g aphs, and GIS ools maximizes he u ili y o
UAV-de i ed ca og aphic da a.
Discussion.
The in eg a ion o d ones wi h GPS and RTK
enables cen ime e -le el geospa ial mapping, which is
c i ical o mode n ca og aphy, cons uc ion, and
en i onmen al moni o ing [13]. While GPS p o ides
global co e age, i s accu acy is limi ed by a mosphe ic
delays, sa elli e geome y, and clock biases [3]. RTK
o e comes hese limi a ions bu equi es eliable
communica ion be ween base and o e [4].
Beyond accu acy, he combina ion o d ones
and RTK o e s signi ican e iciency ad an ages [11].
T adi ional su eying me hods o en equi e ex ensi e
manual labo and ime, whe eas d ones can quickly
cap u e high- esolu ion da a o e la ge and di icul - o-
access a eas. This allows o nea eal- ime moni o ing
o cons uc ion si es, ag icul u al ields, o
en i onmen al changes such as coas al e osion o
de o es a ion.
Howe e , he e ec i eness o RTK is
in luenced by se e al ope a ional ac o s. Signal
obs uc ion caused by dense ege a ion, u ban canyons,
o e ain can deg ade posi ioning accu acy [10].
Mul ipa h e o s, whe e GPS signals e lec o
su aces be o e eaching he ecei e , can also
in oduce disc epancies. To mi iga e hese issues, many
sys ems in eg a e addi ional senso s such as ine ial
measu emen uni s (IMUs), ba ome e s, o isual
odome y, p o iding edundancy and imp o ing
eliabili y in challenging en i onmen s [14].
The da a ob ained h ough RTK-enabled d ones
is no only accu a e bu also highly e sa ile [11]. I can
be used o gene a e digi al ele a ion models (DEMs),
o homosaic maps, and 3D models ha suppo
decision-making ac oss mul iple indus ies. Fo
ins ance, in p ecision ag icul u e, cen ime e -le el
mapping allows o op imized i iga ion planning, c op
moni o ing, and yield es ima ion. In cons uc ion, RTK
d ones can moni o si e p og ess, de ec de ia ions
om design plans, and enhance sa e y by educing he
need o manual inspec ions in haza dous a eas.
Finally, as d one echnology and RTK solu ions
con inue o e ol e, in eg a ion wi h cloud compu ing,
AI-d i en analy ics, and au onomous ligh sys ems
p omises u he enhancemen s in mapping speed, da a
accu acy, and ope a ional scalabili y [6]. These
ad ancemen s indica e a u u e whe e d ones wi h
RTK may become a s anda d ool o geospa ial
in elligence, en i onmen al moni o ing, and
in as uc u e managemen [12].
Conclusion.
The p ac ical implica ions o his enhanced
accu acy, as de ailed in he esul s and applica ions
sec ions, a e p o ound and span a di e se ange o
indus ies. In opog aphic mapping, he abili y o
gene a e high- esolu ion Digi al Ele a ion Models and
Digi al Te ain Models wi hou he in ensi e labo o
placing nume ous g ound con ol poin s has
e olu ionized su eying. The ma hema ical igo o
pho og amme y, go e ned by he collinea i y
equa ions, is now unde pinned by p ecise geo agging,
esul ing in eliable 3D models and con ou maps. In
p ecision ag icul u e, he combina ion o RTK
posi ioning wi h spec al analysis, such as he
calcula ion o he No malized Di e ence Vege a ion
Index, allows o he c ea ion o accu a e p esc ip ion
maps. This acili a es a ge ed in e en ions,
op imizing esou ce use and boos ing c op yields.
Fu he mo e, in cons uc ion and enginee ing, he
cen ime e -le el accu acy ensu es ha as-buil models
can be eliably compa ed agains design plans,
enabling p oac i e p ojec managemen and quali y
con ol. The capaci y o apid olume ic calcula ions
and change de ec ion in en i onmen al moni o ing and
disas e esponse u he unde sco es he
ans o ma i e u ili y o d one-RTK sys ems.
Howe e , he deploymen o hese sys ems is
no wi hou i s challenges. The ope a ional e icacy o
RTK is con ingen upon a s able and eliable
communica ion link be ween he base s a ion and he
o e , which can be comp omised in emo e a eas o
168
“Al-Fa g‘oniy a lodla i” elek on ilmiy
ju nali
ISSN 2181-4252. Tom: 1 | Son: 3 | 2025-yil
"Descendan s o Al-Fa ghani" elec onic scien i ic
jou nal.
ISSN 2181-4252. Vol: 1 | Iss: 3 | 2025 yea
Электронный научный журнал "Потомки Аль-
Фаргани"
ISSN 2181-4252. Том: 1 | Выпуск: 3 | 2025 год
h ps://al- a goniy.uz/
e ain wi h obs uc ed lines o sigh . Fac o s such as
signal mul ipa h, sa elli e geome y, and ini ializa ion
imes o in ege ambigui y esolu ion emain p ac ical
conside a ions. To mi iga e hese limi a ions, he
in eg a ion o RTK wi h supplemen a y senso s—
Ine ial Measu emen Uni s o a i ude and ajec o y
smoo hing, ba ome e s o al i ude alida ion, and e en
isual odome y sys ems—c ea es a obus , mul i-
senso posi ioning solu ion ha main ains accu acy
e en du ing empo a y signal ou ages.
Looking o wa d, he e olu ion o d one-based
posi ioning is poised o con inue i s apid ad ancemen .
As highligh ed by Khalilo e al. [6], he nex on ie
lies in he in eg a ion o a i icial in elligence and
neu al ne wo ks wi h adi ional GNSS/RTK sys ems.
These sel -lea ning algo i hms p omise o u he
enhance accu acy h ough adap i e e o modeling,
in elligen signal il e ing in noisy en i onmen s, and
au onomous sys em calib a ion. Fu he mo e,
echnologies like P ecise Poin Posi ioning and i s
hyb id a ian , PPP-RTK, o e a compelling u u e
di ec ion by po en ially deli e ing global, cen ime e -
le el accu acy wi hou he absolu e equi emen o a
local base s a ion, he eby inc easing ope a ional
lexibili y, especially in emo e o ma i ime
en i onmen s. In conclusion, d ones equipped wi h
RTK ha e i mly es ablished hemsel es as an
indispensable ool in he mode n geospa ial oolki . By
ansla ing complex ma hema ical p inciples in o
p ac ical, high-p ecision applica ions, hey ha e no
only democ a ized access o accu a e mapping bu ha e
also opened new ho izons o scien i ic inqui y,
indus ial e iciency, and in elligen en i onmen al
managemen . The con inued con e gence o his
echnology wi h AI and cloud-based analy ics p omises
a u u e whe e eal- ime, cen ime e -accu a e
geospa ial in elligence is seamlessly in eg a ed in o he
ab ic o sma in as uc u e, sus ainable ag icul u e,
and p oac i e en i onmen al s ewa dship.
Re e ences
1. Colomina, I. &. (2014). Unmanned ae ial
sys ems o pho og amme y and emo e sensing: A
e iew. ISPRS. Jou nal o Pho og amme y and
Remo e Sensing.
2. Ho mann-Wellenho , B. L. (2007). GNSS–
Global Na iga ion Sa elli e Sys ems: GPS,
GLONASS, Galileo, and mo e. Sp inge .
3. Kaplan, E. D. (2017). Unde s anding
GPS/GNSS: P inciples and applica ions. . A ech
House.
4. Khalilo , D., Bozo o a, S., Khon u ae , S.,
Khoi kulo , A., So oldie a, D., & Toshma o , S.
(2024). Sel -lea ning sys em and me hods o selec ion
o weigh coe icien s o neu al ne wo k. In E3S Web
o Con e ences (Vol. 508, p. 04011). EDP Sciences.
5. Leick, A. R. (2015). GPS sa elli e su eying.
John Wiley & Sons.
6. Mis a, P. &. (2011). Global Posi ioning
Sys em: Signals, measu emen s, and pe o mance.
Ganga-Jamuna P ess.
7. Siebe , S. &. (2014). Mobile 3D mapping o
su eying ea hwo k p ojec s using an Unmanned
Ae ial Vehicle (UAV) sys em. Au oma ion in
Cons uc ion.
8. Takasu, T. &. (2009). De elopmen o he
low-cos RTK-GPS ecei e wi h an open-sou ce
p og am package RTKLIB.
9. Teunissen, P. J. (2017). Sp inge handbook
o global na iga ion sa elli e sys ems. Sp inge .
10. Yigi , C. O. (2017). Expe imen al s udy on
he p ecise posi ioning wi h GPS, GLONASS and
GPS/GLONASS combined s a ic and kinema ic RTK.
Measu emen .
11. Za chan, P. (2019). Fundamen als o GPS.
P og ess in As onau ics and Ae onau ics.
12. Zhang, J. &. (2015). Visual-lida odome y
and mapping: Low-d i , obus , and as . IEEE
In e na ional Con e ence on Robo ics and Au oma ion.
13. Улжаев, Э., Убайдуллаев, У., &
Хонтураев, С. (2025). ТЕХНОЛОГИИ
ОПРЕДЕЛЕНИЯ КООРДИНАТ С ПОМОЩЬЮ
ДРОНОВ. Techscience. uz-Texnika anla ining
dolza b masalala i, 3(5), 25-29.
14. Хонтураев, С. (2025). ПРИМЕНЕНИЕ
ДРОНОВ В СОВРЕМЕННОЙ
ГЕОПРОСТРАНСТВЕННОЙ КАРТОГРАФИИ.
Techscience. uz-Texnika anla ining dolza b
masalala i, 3(4), 29-32.
