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TDOA MEASUREMENT PROCESSING FOR POSITIONING USING UNMANNED AERIAL VEHICLES
DOI 10.24411/2072-8735-2018-10121
Grigoriy A. Fokin,
The Bonch-Bruevich St. Petersburg State University of Telecommunications, St. Petersburg, Russia grihafokin@gmail.com
Abdulwahab H. Al-Odhari,
The Bonch-Bruevich St. Petersburg State University of Telecommunications, St. Petersburg, Russia, abdwru2011@yandex.ru
Keywords: TDOA (Time Difference of Arrival), UAV (Unmanned Aerial Vehicles), RMSE (Root Mean Square Error), Positioning, Measurement Processing, Radio Source.
In this paper we refine Time Difference of Arrival (TDOA) measurement processing model for positioning of radio source using Unmanned Aerial Vehicles (UAVs). Developed mathematical model was realized in simulation model and allowed to estimate the accuracy of the fixed and moving radio source based on TDOA using UAVs in three-dimensional space. Coordinates of radio source were estimated using Nonlinear Least Squares (NLS) technique.
The aim of this work is to formalize the analytical relationships of the TDOA for the scenario of positioning using UAV, as well as to obtain quantitative estimates by means of simulation for the following cases: a) using fixed receiving points for positioning the stationary radio source; b) using UAVs as receiving points for positioning a stationary radio source; c) using UAV as receiving point for positioning moving radio source. Obtained estimates for the case with UAVs and stationary receiving points are compared for different versions of the UAV trajectory. Based on Root Mean Square Error (RMSE) results of the simulation it is shown that the use of UAVs improves positioning accuracy from 60 meters for scenario with five stationary receiving points to 10 meters for a scenario with three stationary points and two UAVs. The material in the paper is organized in the following order. In Section 2 we present TDOA measurement processing model using UAV. In Section 3 we pre-sent UAV and radio source movement trajectory models. Simulation results are described in Section 4. Finally, we draw the conclusions in Section V.
Для цитирования:
Фокин Г.А., Аль-Одхари А.Х. Обработка РДМ измерений для позиционирования с использованием беспилотных летательных аппаратов // T-Comm: Телекоммуникации и транспорт. 2018. Том 12. №7. С. 52-58.
For citation:
Fokin GA., Al-Odhari A.H. (2018). TDOA measurement processing for positioning using unmanned aerial vehicles. T-Comm, vol. 12, no.7, pр. 52-58.
Introduction
Time Difference of Arrival (TDOA) positioning method in the context under consideration is termed passive geolocation and is based on measuring the propagation time of a signal from radio source to receiving point []]. Actual trends in current networks development includes joint cooperation of flying segment based on Unmanned Aerial Vehicles (UAVs) with terrestrial segment [2], if positioning is carried out in such network in three-dimensional space, at least four receiving points are required. Estimation of the positioning accuracy by TDOA in cellular networks on the plane was carried out in [3]. UAV-based Localization in three-dimensional space was considered in [4], however scenarios of radio source movement were not evaluated.
Existing accuracy results achieves the order of tens and hundreds of meters [5]. To improve the accuracy of the radio source positioning the use of UAVs was evaluated in [6J. This work is a development of investigation [7] with the aim to 1) develop mathematical and simulation mode! in [8] for TDOA measurement processing for positioning using UAVs in three-dimensional space and 2) refine known estimates by means of simulation for the following cases: a) using fixed receiving points for positioning the stationary radio source; b) using UAVs as receiving points for positioning a stationary radio source; c) using UAV as receiving point for positioning moving radio source,
TDOA Measurement Processing Model using UAV
Suppose that the radio source emits a signal at an unknown time to, then the time of arrival at the receiving point i is ti = lo+cJ/c, where dj is the distance between the radio source and the receiving point i; c is the speed of light; the time of arrival at the receiving point j is lj=to+d/c, where i,j = 1,2,,,.M is the total number receiving points, M>3. TDOA measurement processing model using unmanned aerial vehicles in three-dimensional space is depicted in Fig. I.
Fig. 1. TDOA measurement processing model using UAVs in three-dimensional space
There are M (M -l)/2 distinct TDOAs from all possible receiving point - pairs, denoted by /¡,- ={t—i^)—{tj—ti^~t—tj, where i>j. However, there are only (M~\) non-redundant real time
differences which are calculated with respect to one receiving point, called the reference point, for example, the first receiving point, then we can write the following expressions for range differences of the signals [0]
In practice, the primary measurements of the range difference are subject to errors nj, taking into account the measurement errors, then range difference measurements deduced from the TDOAs are modeled as
rj=dJ*+nJ (2)
Denote unknown coordinates of moving radio source in three-dimensional space as Ei/„—[xi/„7y\/a,Zi/n]T, n-\,2,...,N, N is the number of measurement points in time (space); Sy„= [Xj/fl, ypn Zj/i:]T are coordinates of (moving) receiving points, j - 1,2,...,A/, M is the total number receiving points. The actual distance between the radio source and the j receiving point denoted by d\j/n is given by
4>=Vc
fy -xjt*)" + Ov„ - y„nY + (-,/„ - zJ/n) , j = \,2,..„M,n = 1,2,...,N
In matrix form observed range difference measurement mod el is [5]
r = f(x) + n
)
where
nT
(3)
(4)
r -
'2,1/1
^2,1/2
rn
2mn
'3.1/1
'3,1/2
'3*1 IN
'M, l/l
' M.I/2
M.I/A'
(5)
where r is the measured ranae difference between the radio
j.Uti
source and the fu receiving point and between the radio source and the 1sl receiving point at the n[h time.
Actual range difference between the radio source and the/h receiving point and between the radio source and the 1st receiving point at the «lh time are defined by
f(x) =
Ac/,
A4,,
¿4,
/W„
Ad.
iW.IW
A^-.i/n = yj(x,,„ - XjJ2 + (yUll - yj/nf + (zVn - ZPS
- >/(*]/„ (yVn -}\Y + (Zyn --if,
j = 2,3,..„M, n = \,2,...,N.
(6)
(7)
Range difference measurement errors are determined by the expression
t
n =
n
2.3/1 ^2,1/2
»3,1/
n,
'3.1/,V
n
A/,l/l
>h
'M. I/A'
(8)
where n. |/)( is the range difference error between Hie radio
source and the /h receiving point and between the radio source and the lsl receiving point at the nlh lime. To estimate radio source coordinates it is necessary to solve a system of nonlinear equations (4),
The Nonlinear Least Squares (NLS) technique is one of the methods of minimizing the cost functions (9) obtained from equation (4). The cost function for TDOA positioning under consideration is [7]
V^(x)=(r-f(i)')(r-f(i)) ^
Coordinates estimate of the radio source obtained by NLS denoted by x are
nls (x )
x = argiiiin^
(10)
In general case, there is no complete analytical solution for (10), therefore, iterative Levetiberg-Marquardt (LM) [9] algorithm is used, which is a modification of the classical Gauss-Newton method [5J and is determined by
x"'+l - x'" + jG' (f (x"r))G (f (x'")) + //n,l) ' x (1])
xG' (f(x"'))(r-f(x"'))
where I is an identity matrix; is the corrective multiplier (the Marqunrdt factor), which is recalculated at each iteration m.
In the next section let's present UAV and radio source movement trajectory models to develop mathematical and simulation model for TDOA measurement processing for positioning using UAVs in three-dimensional space.
UAV and Radio Source Trajectory Model
Suppose radio source moves in a three-dimensional space along a smooth trajectory, which is a vector of coordinates in time, and contains information about the position, parameters, and nature ofthe movement of the radio source. The trajectory of the moving radio source can be approximated by a A?1' degree polynomial [6, 8]:
rh
= xvo + YjtaVr >1/» =
(12)
~y>/0 + fy/p Z\in ZM0 + S'ii Cltp
/>=I f=I
where xuo, yyQ, z]/V are coordinates of the radio source at the initial time t = 0; clip. bt,p. c\/p are the parameters (velocity, acceleration, etc.) ofthe moving radio source along the x, y, z-axes, respectively.
Let's analyze following variants of moving radio source, determined by the degree K ofthe polynomial (12) [6, 8|.
For stationary radio source K. = 0 and equation of a stationary radio source becomes
X\hi ~ *l/0 » y\in ~ -T'l/01 Z|/„ = Zl/Q _ (]3)
For uniform movement of the radio source K — I and equation of mov ing radio source becomes:
XUn ~ *li<i ^ av\ '«» Xi/n ~ _ 4)
=>'li0 + Kte zv» = *m+ cm**
UAV dynamics as a moving receiving point is given by the approximation a polynomial ofthe Qxh degree:
-A
xj„,= Xjm+ ZAaj«r
q= I
v , ' (l5)
yjh," }'j/0 + Zjl^jlq*
O
ZJln= ZJI0 +
a
cj>9
where Xj/g, yyo, zjm are the coordinates of the moving receiving points (UAV) at the initial time f=0; a,-.^, Cj/q are the parameters (velocity, acceleration, etc.) ofthe moving receiving points along the J', y, z-axes, respectively.
Approximation ofthe trajectory of moving of receiving point by a circle is:
where R is the radius ofthe circle, co = 2?z//,\. In the next section let's describe simulation model scenarios for positioning using UAVs in three-dimensional space.
Simulation Results
The simulation described further is based on the assumption of ideal synchronization between receiving points (stations) of the terrestrial and flying segments aboard UAV, which gather primary time of arrival (TOA) measurements. Primary TOA measurements are instantly sent to the central processing unit with respective time stamps for further TDOA estimation by means of pair-wise signals cross correlation |7j.
Simulation model was realized in MatLab and included arrangement, estimation and visualization subsystems. The positioning of the radio source was performed for scenarios when it is located in three-dimensional space in an area with a size of (10 km * 10 km x 5 km). Circle and straight line trajectories ofthe UAV movement were chosen for simulation. This choice is due to the fact that these are the most frequently encountered for analyzing the trajectory ofthe UAV when positioning using the air segment [1, 4, 10]. Further, the simulation is carried out according to the following scenarios: a) the fixed receiving points and the stationary source; b) using of UAVs as receiving points for the positioning of a stationary radio source; e) using of UAVs as receiving points for positioning the moving radio source.
The root-mean-square error (RMSE) of coordinates estimate ofthe radio source is
RMSE = ^E^ix-xf +{y-yf +{z-if
(17)
where x — is the actual current radio source position
and v=
c,y,z] is
the current radio source position NLS LM
estimate.
Initial value for the iterative LM positioning algorithm is calculated as the mean value of the coordinates of the receiving points
I M
Mjt J
(1)
350
I Error in V
50
100
150
200
250
300
350
100
S 50 a:
ill. I
Error in 2
' УН
50
100
150 200 Time (s)
250
300
350
Fig. 12. RMSE of current moving radio source estimates according to UAV flight time
Results for moving source in Fig, 12 showed that when it moves from the center to boundaries of the positioning area, error, especially along the z-axis, increases [1! |.
Conclusion
In this investigation we developed mathematical and simulation model for TDOA positioning using UAVs in three-dimensional space and refined known estimates [1,4] for several scenarios of terrestrial and flying segments arrangement and several UAVs movement trajectories. Obtained results showed that the use of UAVs allows to substantially increase the accuracy of the positioning of the radio source especially along the z-axis by 6 times from 60 meters for scenario with five stationary receiving points to 10 meters for a scenario with three stationary points and two UAVs.
References
1. Du, H. J., Lee J. P. Y. (2004). Passive geolocation using TDOA method from UAVs and ship/land-based platforms for maritime and littoral area surveillance. Defence R&D Canada-Ottawa.
2. Koucheryavy, A„ Vladyko, A., Kirichek, R. (2015). State of the art and research challenges for public flying ubiquitous sensor networks. In: 13 a I and in, S., Andreev, $,, Koucheryavy, Y. (eds.) NEW2AN/ ruSMART 2015. LNCS, vol. 9247, pp. 299-308. Springer, Heidelberg.
3. Mensing, C„ PI ass, S. (2006). Positioning Algorithms for Cellular Networks Using TDOA, In: Proceedings of the 2006 IEEE International Conference on Acoustics, Speech and Signal Processing, 1CASSP 2006, vol. 4.
4. Kim, D. H„ Lee, K„ Park, M. Y„ Lim, J. (2013). UAV-Based Localization Scheme for Battlefield Environments. In: MILCOM2013 -2013 IEEE Military Communications Conference, San Diego, CA, pp. 562-567.
5. Zekavat, R„ Buehrer, R. M.: Handbook of position location. (2011). Theory, practice and advances. John Wiley & Sons.
6. Fokin, G, A., Alodhari, A. H. (2017). Positioning of the moving radiation source using Time Difference of Arrival method. T-Contm, vol, 11, № 4, pp. 41-46. Media Publisher,
7. Sivers M., Fokin G. (2015). LTE Positioning Accuracy Performance Evaluation. In: Balandin S„ Andreev S„ Koucheryavy Y, (eds.) NEW2AN/ruSMART 2015. LNCS, vol 9247, pp. 393-406. Springer, Heidelberg.
8. Al-odhari, A. H. A., Fokin, G„ Kireev, A. (2018). Positioning of the radio source based on time difference of arrival method using unmanned aerial vehicles. In: 2018 Systems of Signals Generating and Processing in the Field of on Board Communications, Moscow, pp. I -5.
9. Levenberg, K. (1944). A Method for the Solution of Certain Problems in Least-Squares. Quarterly Applied Math. 2, pp. 164-168.
10. Kirichek R., Paramonov A., Vareldzhyan K. (2015). Optimization of the UAV-P's Motion Trajectory in Public Flying Ubiquitous Sensor Networks (FUSN-P), In: Balandin S., Andreev S., Koucheryavy Y. (eds) NEW2AN/ruSMART 2015. LNCS, vol. 9247, pp. 352-366. Springer, Heidelberg.
11. Mashkov, G., Borisov, E., Fokin, G. (2016). Experimental validation of multipoint joint processing of range measurements via software-defined radio testbed. In: 18th international Conference on Advanced Communication Technology (ICACT), Pyeongchang, pp. 268-273.
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ОБРАБОТКА РДМ ИЗМЕРЕНИЙ ДЛЯ ПОЗИЦИОНИРОВАНИЯ С ИСПОЛЬЗОВАНИЕМ БЕСПИЛОТНЫХ ЛЕТАТЕЛЬНЫХ АППАРАТОВ
Фокин Григорий Алексеевич, Санкт-Петербургский государственный университет телекоммуникаций им. проф. М.А.Бонч-Бруевича, Санкт-Петербург, Россия, grihafokin@gmail.com
Аль-Одхари Абдулвахаб Хуссейн, Санкт-Петербургский государственный университет телекоммуникаций им. проф. М.А.Бонч-Бруевича, Санкт-Петербург, Россия, abdwru2011@yandex.ru
Аннотация
Разработана математическая модель на основе измерения разности времен прихода сигналов (ТЭОД) для позиционирования источника радиоизлучения с использованием беспилотных летательных аппаратов (БПЛА). Разработанная математическая модель была реализована в имитационной модели и позволила оценить точность стационарного и движущегося источника радиоизлучения на основе ТЭОД с использованием беспилотных летательных аппаратов в трехмерном пространстве. Координаты источника радиоизлучения оценивались с использованием метода нелинейных наименьших квадратов.
Цель данной работы состояла в формализации аналитических отношения ТЭОД для сценария позиционирования с использованием БПЛА, а также получения количественных оценок с помощью моделирования для следующих случаев: а) использование стационарных пунктов приема для позиционирования неподвижного источника радиоизлучения; б) использование БПЛА в качестве пункта приема для позиционирования неподвижного источника радиоизлучения; в) использование БПЛА в качестве пункта приема для позиционирования подвижного источника радиоизлучения. Полученные оценки для случая с БПЛА и стационарных пунктов приема сравнивались для разных вариантов траектории БПЛА. Результаты моделирования по критерию среднеквадратичной ошибкой показали, что использование БПЛА позволяет повысить точность позиционирования с 60 метров для сценария из пяти стационарных пунктов приема до 10 метров для сценария из трех стационарных и двух двумя БПЛА. Материал настоящей работы организован следующим образом: в разделе 2 представлена математическая модель обработки ТЭОД измерений; в разделе 3 представлены модели траектории движения БПЛА и источника радиоизлучения; Результаты моделирования описаны в разделе 4.
Ключевые слова: метод разности времен прихода сигналов (TDOA), беспилотный летательный аппарат (БПЛА), среднеквадратическая ошибка, обработка измерений, источник радиоизлучения.
Литература
1. Du, H. J., Lee J. P. Y.: Passive geolocation using TDOA method from UAVs and ship/land-based platforms for maritime and littoral area surveillance. Defence R & D Canada-Ottawa (2004).
2. Koucheryavy, A., Vladyko, A., Kirichek, R.: State of the art and research challenges for public flying ubiquitous sensor networks. In: Balandin, S., Andreev, S., Koucheryavy, Y. (eds.) NEW2AN/ruSMART 2015. LNCS, vol. 9247, pp. 299-308. Springer, Heidelberg (2015).
3. Mensing, C., Plass, S.: Positioning Algorithms for Cellular Networks Using TDOA. In: Proceedings of the 2006 IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2006, vol. 4 (2006).
4. Kim, D. H., Lee, K., Park, M. Y., Lim, J.: UAV-Based Localization Scheme for Battlefield Environments. In: MILCOM 2013 - 2013 IEEE Military Communications Conference, San Diego, CA, pp. 562-567 (2013).
5. Zekavat, R., Buehrer, R.M.: Handbook of position location: Theory, practice and advances. John Wiley & Sons (2011).
6. Fokin, G. A., Alodhari, A. H.: Positioning of the moving radiation source using Time Difference of Arrival method. T-Comm, vol. 11, № 4, pp. 41-46. Media Publisher (2017).
7. Sivers M., Fokin G.: LTE Positioning Accuracy Performance Evaluation. In: Balandin S., Andreev S., Koucheryavy Y. (eds.) NEW2AN/ruSMART 2015. LNCS, vol. 9247, pp. 393-406. Springer, Heidelberg (2015).
8. Al-odhari, A. H. A., Fokin, G., Kireev, A.: Positioning of the radio source based on time difference of arrival method using unmanned aerial vehicles. In: 2018 Systems of Signals Generating and Processing in the Field of on Board Communications, Moscow, pp. 1-5 (2018).
9. Levenberg, K.: A Method for the Solution of Certain Problems in Least-Squares. Quarterly Applied Math. 2, pp. 164-168 (1944).
10. Kirichek R., Paramonov A., Vareldzhyan K.: Optimization of the UAV-P's Motion Trajectory in Public Flying Ubiquitous Sensor Networks (FUSN-P). In: Balandin S., Andreev S., Koucheryavy Y. (eds) NEW2AN/ruSMART 2015. LNCS, vol. 9247, pp.352-366. Springer, Heidelberg (2015).
11. Mashkov, G., Borisov, E., Fokin, G.: Experimental validation of multipoint joint processing of range measurements via software-defined radio testbed. In: 18th International Conference on Advanced Communication Technology (ICACT), Pyeongchang, pp. 268-273 (2016).
