Research / URSlab
Autonomous navigation
INS/GNSS algorithms, orthodromic flight and coordinated turns on the WGS 84 ellipsoid, real-time route planning.
A unique model
Turns along a closed route — from take-off at WP2 to landing at WP9 — via Kummer's equation
Classical turn algorithms treat the Earth as flat or spherical and fly the aircraft along a constant-radius arc. Here the turn is a clothoid on the actual WGS 84 ellipsoid, and the heading law is expressed analytically through the Kummer function. To our knowledge, this is the first closed-form solution of a coordinated turn between geodesic orthodromes that accounts for both the shape and the rotation of the Earth.
ψT(t) = ψpart + (α1 − ψpart) · M(a, ½, −q t²/2), 0 ≤ t ≤ T*
Scientific contributions
- The condition for a coordinated, slip-free turn (centrifugal force balanced by weight along the lateral axis) is introduced into the differential equations of the geodesic line on WGS 84 through the geodesic curvature kg = g·tan γ / W² (14). This yields system (15) for flight along a clothoid as a function of time, accounting simultaneously for bank, the principal radii of curvature G and Q, the Earth rotation Ω and the convergence of meridians.
- The vertical channel is extended with a differential equation for pitch (in (15)) that matches pitch to the ellipsoid curvature along the current track angle. The turn is thus flown at constant height above WGS 84, with no "dip" below the horizon on long trajectories.
- The heading equation is shown to reduce to a special case of Kummer's confluent hypergeometric equation (b = ½). A closed-form heading law ψ(t) is obtained through the Kummer function M(a, ½, z) — an analytical solution where numerical integration or spherical, constant-radius approximations have been used so far.
- Boundary conditions at the detachment point b₁ and the interception point b₂ yield a transcendental equation (31) for the turn duration T; its root uniquely determines the roll-rate parameter k (32) and the whole trajectory (33).
Scientific-applied contributions
- An algorithm for the Flight Management System (FMS): T, k and the distance to the detachment point are computed in advance, before the WP is reached — with no numerical integration in flight.
- Smooth turn entry: lateral acceleration builds up from zero, without the step inherent in the classical constant-radius turn. This reduces dynamic loads, eases autopilot tracking and stabilises on-board sensors.
- Applicable to autonomous unmanned aircraft systems and to piloted aircraft; demonstrated on a 7-waypoint route over Northern Bulgaria (W = 150 m/s, γmax = 30°).
Source: A. Madzharov, „Application of Kummer's Equation for Flight Between Two Geodesic Orthodromes in a Coordinated Turn“, AMiTaNS 2026 — AIP Conference Proceedings (submitted).
Computational model
All values below are computed live in the browser from the equations of the paper. Change the speed or maximum bank — turn duration, roll rate and detachment point are recomputed for every WP. The comparison is with the classical constant-radius turn.
Select a turn on the map or in the table.
Clothoid from Kummer equation (31), (33)Classical constant-radius turn (1), (2)
| WP | B, L [°] | Δα [°] | TK [s] | kK [1/s] | dK [km] | Tcl [s] | TADcl [km] |
|---|
Tₖ and kₖ are the root of the transcendental equation (31) with the Kummer function M(a, ½, z), computed in the browser; dₖ is the distance from the detachment point b₁ to the WP. Radii of curvature G₀, Q₀ are WGS 84 values at the turn point. The map is for display only (local projection); the computation does not depend on it.
Publications in this area
Application of Kummer's Equation for Flight Between Two Geodesic Orthodromes in a Coordinated Turn
A. Madzharov
AMiTaNS 2026 — IOP Publishing, Journal of Physics: Conference Series
A coordinated turn between two adjacent orthodromes is described by a clothoid on the WGS 84 ellipsoid. The heading equations reduce to a special case of Kummer's confluent hypergeometric equation, giving a closed-form heading law that lets the FMS compute turn duration and roll rate in advance.
Precise Calculations of Gravity Anomalies from the Geometric Height for Autonomous Air Navigation
A. Madzharov
AMiTaNS 2026 — IOP Publishing, Journal of Physics: Conference Series
An analytical gravity model on WGS84 constants with an exact geometric relation for the vertical line. It matches WGS84/Somigliana within ±1.5 mGal at sea level and removes a latitude-dependent bias of up to 260 mGal at altitudes up to 10 km.
A Highly Accurate Calculation of the Difference Between Geocentric and Geodetic Latitude in Flight
A. Madzharov, S. Hristozov
EnviroRISKs 2026 (N. Dobrinkova, S. Fidanova, Eds.) — Springer, Lecture Notes in Networks and Systems 2240 · DOI 10.1007/978-3-032-39070-7_6 (not yet active)
Geocentric distance from the prime-vertical radius with a coefficient proportional to e⁴, including flight altitude — yielding high-accuracy analytical geocentric–geodetic latitude differences and a differential equation of the meridian ellipse for geodesic-line computation.
Compensations for Horizontal Inertial Components of INS/GNSS with Flight Altitude
Madzharov, A., Hristozov, S., Gaidarski, I.
Engineering Proceedings 150(1), 70, MDPI · DOI 10.3390/engproc2026150070
URSlab Website and Information System for Reporting the Publication Activity of a Scientific Department of BAS: Source Code (v1.1.0)
A. Madzharov
Zenodo — Software (files with restricted access) · DOI 10.5281/zenodo.23111566
Version 1.1.0 extends the URSlab website into an information system for recording, validating and reporting the publication activity of a scientific department of BAS: yearly publication pages categorised as in the BAS reporting system, combinable filters with a live reporting summary and CSV export, per-member summary reports, and entry of new publications by team members with APA validation, GitHub-based identity and automatic re-validation.
URSlab Website and Interactive Coordinated-Turn Model: Source Code (v1.0.0)
A. Madzharov
Zenodo — Software (files with restricted access) · DOI 10.5281/zenodo.23086274
Source code of the URSlab website, including the interactive browser model of a clothoid coordinated turn between two adjacent orthodromes, with comparison to a constant-radius arc and the curvature profile along the route.
Assessment of GNSS Performance and Error Bounding for SAIL III UAS Operations
González-deSantos, L. M., Bruzual, J., Socías, D., Lacarra Arcos, E., Santos, M., González, R., Gil, E., Moreno López, G., Hristozov, S., Karas, J., Vyshnevskyy, M., Gebhardt, J., Haro, P., Bellingham, S. R.
Drones 9(11), 776, MDPI · DOI 10.3390/drones9110776
The Applicability Of Quantum Gyroscopes For Navigation
Madzharov, A.
Engineering Sciences LXII(1), Bulgarian Academy of Sciences, pp. 37–44 · DOI 10.7546/engsci.lxii.25.01.06
Design of an Unmanned Helicopter System for Collecting and Processing of Geophysical Information
Madzharov, A., Hristozov, S., Chehlarova, N., Georgiev, R., Gaidarski, I., Alexandrov, A.
Complex Control Systems 7, IR-BAS
Design of a testing model for evaluation the levels of automation and autonomy of a helicopter autopilot
Hristozov, S., Madzharov, A., Chehlarova, N.
Complex Control Systems 7, IR-BAS
Структура и приложение на спътниковите навигационни системи [Structure and Application of Satellite Navigation Systems]
Madzharov, A.
В. Търново: НВУ „Васил Левски“, 177 с.
The influence of the free air anomaly over the stability of the vertical channel in strapdown inertial navigation system (SINS)
Madzharov, A. N., & Biliderov, S. S.
Proceedings of the First International Conference on Cartography & GIS, 25–28 January 2006, Borovets, Bulgaria, pp. 1–8
Applying corrections from the theoretical model of the geoid when differential mode of GPS is used in landing
Madzharov, A. N.
Proceedings of the First International Conference on Cartography & GIS, 25–28 January 2006, Borovets, Bulgaria, pp. 1–8
Моделиране траекторното движение на летателен апарат по зададен маршрут и профил на полета [Modelling the trajectory motion of an aircraft along a given route and flight profile]
Madzharov, A. N., & Biliderov, S. S.
XII Международна научно-техническа конференция Trans & MOTAUTO’05+, 23–25 ноември 2005 г., В. Търново, Proceedings „Technics and Technologies“, т. 2, с. 138–141
Моделиране движението на ортогонален репер върху геодезична повърхност [Modelling the motion of an orthogonal frame on a geodesic surface]
Madzharov, A. N., & Biliderov, S. S.
Курсантско-студентска научна конференция, 24–25 март 2005 г., В. Търново: НВУ „В. Левски“, с. 91–98
Theoretical research of possibilities for gravitation measurements in motion using the inertial navigation system
Madzharov, A. N., Panova, P. V., & Getsov, P. S.
55th International Astronautical Congress of the IAF, the IAA and the IISL (IAC 2004), Vancouver, Canada, 4–8 October 2004, paper IAC-04-J.P.08, AIAA, Reston, VA (published online 29 Nov 2012) · DOI 10.2514/6.IAC-04-J.P.08
Алгоритъм за съвместна обработка на сигналите от GPS приемник и аерометричен датчик за височина в интегрално изпълнение [An algorithm for joint processing of the signals from a GPS receiver and an air-data altitude sensor in an integrated design]
Madzharov, A. N., & Biliderov, S. S.
Втора международна научна конференция ХЕМУС 2004, Пловдив. Сборник доклади „Изследвания и технологии за нуждите на отбраната и въоръжените сили“, София: Военна академия – ИПО, с. 81–92
Жироскопи и инерциални навигационни системи [Gyroscopes and Inertial Navigation Systems]
Madzharov, A.
Д. Митрополия: ВВВУ „Георги Бенковски“, 422 с.
Структура и приложение на глобалните навигационни системи NAVSTAR и ГЛОНАСС [Structure and Applications for Global Navigation Systems NAVSTAR and GLONASS]
Madzharov, A., & Todorov, P.
Д. Митрополия: ВВВУ „Георги Бенковски“ (печат: София, Военно издателство), 191 с.