A PSO-Based Layout Method for GNSS Pseudolite System Jian Wang*
Hongxin Li
Jinzhi Lu
University of Electronic Science and Technology of China, Chengdu, China, 611731
Shanghai aerospace electronic technology institute Shanghai, China, 201109
KTH-Royal of Institute and Technology Stockholm, Sweden, 10044
[email protected]. cn
[email protected]
[email protected]
Kun Li, Huan Li, Lian Yang, Yubai Li University of Electronic Science and Technology of China, Chengdu, China, 611731
kilometers away from the earth ground. On the other hand, the user positioning accuracy heavily depends on the number of observable satellites and their layout. It implies that when a user is in a special scenario, e.g., indoors, tunnels and city canyons, etc., it may not be able to receive enough satellites signals at the same time, leading to inaccuracy positioning result or even worse case – it cannot be successful positioned. Fig. 1 give an example to illustrate this situation. Apparently, when a user is indoor, the radio signals from satellites are blocked and the user cannot be positioned by using the GNSS system. To overcome these weaknesses, indoor pseudolites, which can work as a complementary of satellites or even replace the navigation satellite to get the user position independently [1], [2], are introduced to enforce the GNSS system.
ABSTRACT In order to improve the user positioning accuracy in GNSS (Global Navigation Satellite System) pseudolite system, we propose a PSO (Particle Swarm Optimization)-based method to optimize the pseudolite layout in this paper. In detail, given the pseudolite layout information, we calculate the system GDOP (Geometric Dilution of Precision) and then minimize it by using a PSO-based algorithm with N particles. Here the first particle indicates the classical layout under the given scenario and the other particles separately represent N-1 randomly generated layouts. In each iteration of our PSO-based algorithm, these particles move to a direction to reduce the GDOP value. After several iterations, the GDOP value can be minimized and the optimal pseudolite layout is found out as well. To evaluate the merits of our method, we perform some experiments. The experimental results show that compared to the classical pseudolite layout, our method can reduce the GDOP by 13.4%. This, with no doubt, improves the user positioning accuracy. For example, when the pseudo-range error is about 1%, the user positioning accuracy in our optimized layout can be improved by 12.4% against the classical layout.
Beidou satellite
Beidou satellite
Outdoor Indoor
CCS Concepts Information systems➝Information systems applications
User
Keywords
Pseudolite
Figure 1. Pseudolites enforcing GNSS system
Pseudolite, Layout, Positioning, GDOP, PSO algorithm
On the other hand, the research shows that the accuracy of pseudolite positioning is related to the following two aspects [3], one is the measurement error and the other is the pseudolites geometric distribution, namely pseudolites layout. As it known to us all, the larger the measurement error is, the worse the positioning result will be. Besides, from the perspective of pseudolites layout, a reasonable layout supply promising merits for high accuracy user positioning. Moreover, in GNSS system, GDOP is often used to evaluate the reasonability of pseudolite layout [4].
1. INTRODUCTION Nowadays, GNSS is widely used in various areas, such as navigation, positioning and measurement, etc. However, in the GNSS system, the vertical positioning accuracy is a challenge problem since the satellites in the space are about 20000 Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from
[email protected]. ICIT 2017, December 27–29, 2017, Singapore, Singapore © 2017 Association for Computing Machinery. ACM ISBN 978-1-4503-6351-8/17/12…$15.00
In this paper, we propose a PSO-based method to minimize the GDOP value for a given pseudolite system, which effectively improves the accuracy of user positioning as well. The rest parts of the paper are organized as follows. In Section II, we briefly present the related work on multi-point positioning system, including GNSS, wireless sensor network and Wi-Fi network, etc. In Section III, we detail our PSO-based method. In Section IV, we perform some experiments to evaluate the merits of our proposed
DOI: https://doi.org/10.1145/3176653.3176668
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algorithm, and then we compare it with the classical layout technique. Finally, we conclude our work in Section V.
1) We assume that the pseudolite system is in a rectangular space, which could be a classroom or a lab room. Besides, in the space, there are no obstacles hindering the transmission of pseudolite signals. Moreover, any users in the space can receive pseudolite signals with random pseudo-range errors, all of which may deviate the real range less than 1.5 meters, about 1% code-loop error.
2. RELATED WORK In recent years, multi-point positioning system is almost everywhere in our daily life, and the layout of base stations, which significantly impacts the system positioning accuracy, attracts a lot of attentions. Many existing work have been done to address how to find out the optimal layout for a multi-point positioning system.
2) Some factors affecting the accuracy of positioning, such as pseudolites synchronization, multi-paths and near-far effect, etc, can be resolved by existing techniques. For example, in [17], the authors propose a method called bidirectional measuring. By using this method, the time error between any two pseudolites can be eliminated, such that all pseudolites in a system is time synchronies with each other. In [18], Yedukondalu et al. design a linear recursive least squares adaptive filter, which can be used to address the multi-paths issue in pseudolite system. In [19], a novel technique, namely subspace projection method, successfully solves the near-far effect.
In [5], Sonia et al. propose an sensor placement and motion coordination strategy for mobile sensor networks. They optimize the layout of sensors by using a new decentralized control rate. In [6], Adrian et al. analyze the geometric relationship between sensors and users in the two-dimensional plane and then propose a layout algorithm to minimize the CramerRao boundary. In [7], Zheng et al. illustrate the influence of beacon placement strategy on the accuracy of positioning and then propose an enhanced TOA algorithm based on it. By using this method, the positioning error can be decreased by 60%. In [8], Monteiro et al. propose a layout strategy based on genetic algorithm to find out the proper positions for all transmitters in ADS-B system, such that the system coverage areas is extended. In [9], Boci et al. proposed a method based on frequency spectrum analyzing, which help them to construct an optimal layout for radio stations. The simulation results show that the proposed method provides promising merits for station layout optimization.
3.2 Layout Algorithm In the layout algorithm, we use the PSO algorithm to find out the optimal layout, in which the three dimensional position of N pseudolites are combined to a Q-dimensional particle Si=(si1,si2, ,siQ), Note that Q=N×3. For each particle, there is a velocity vi (vi1,vi2, ,viQ) representing the particle’s speed and direction, and we also need the user position which can be expressed as U=(u1,u2,u3). Some notations, which are frequently used in our PSO-based method, are listed in Table 1.
In [10], Yongcai et al. mathematically analyze the position matrix and conclude that increasing the number of pseudolites can optimize system GDOP and improve the layout of pseudolite system. In [11], Chien-Sheng et al. propose genetic algorithm based on the minimum error estimation, which could be used to optimize the GDOP. In [12], Mosavi et al. use an adaptive filtering based evolutionary algorithm to calculate the GDOP, and then discuss how to use machine learning methods, e.g., genetic algorithm and simulated annealing algorithm, to optimize the pseudolite layout. In [13], Kelly proposes a ridge regression algorithm. While reducing the overall mean square error of the positioning results, the method also reduces the positioning error variance. In [14], Xu et al. use the singular value decomposition to solve the anomaly caused by the shortage of navigation satellites, and increased the number of pseudolites to optimize the geometric distribution of the navigation satellites. Finally, they obtain an optimal geometric distribution and high precision system clock. In [15] and [16], the authors propose a method to construct pseudolites network by solving the observation matrix combined with tetrahedral volume. Then, they give out a layout method for indoor positioning system with four or six pseudolites.
Table 1. Notations in our method Notations
Description
ω
Inertia weight factor remaining particle velocity
vi
The velocity of the particle i
m
Iteration time
M
Maximum times for iteration
c1
Cognitive acceleration coefficients
c2
Social acceleration coefficients
N
Particle swarm size
Si
Layout corresponding to particle i
S0
Pseudolites classical layout
User position The optimal layout found by a particle i in Lmi iteration m m Lg The optimal layout in all Lmi Our layout algorithm is defined as follows. U
3. PSUEDOLITE LAYOUT METHOD Given
The pseudolite layout greatly affects the accuracy of system positioning. In this section, we first give an overview about the pseudolite system in our paper, and then design a PSO-based method to automatically find out the optimal positions for all pseudolites under given scenarios.
1) the user position U; 2) the classical pseudolites layout S0; 3) Particle swarm size N; Find out
3.1 Overview of Pseudolite System
The optimized GDOP value and the corresponding layout LMg .
Before designing the layout optimization method, we give out an overview about our pseudolite system.
Here GDOP is defined as follow,
314
GDOP tr ((GT G )1 )
to each particle based on the layout of the particle (line 14); Finally, we update Lim and Lgm (line 15). After M iterations, the optimal layout and GDOP is obtained.
(1)
where tr() indicates the matrix trace, and G is the geometric matrix, as mentioned in [6].
m vidm1 ωvidm c1ξ (lidm sidm ) c2 η(lgd sidm )
Table 2. Pseudo-code of our PSO algorithm PSO Algorithm
sidm1 sidm vidm1
1
Initialize U , S , ω , c1 , c2 and m = 0;
2
L0i S0 ; L0g S0 ; while(m M) do
3
(7)
0 i
4. SIMULATION RESULTS In order to verify the advantages of our method, we assign four synchronized pseudolites to a 10m×8m×4m room space and then evaluate the system GDOP as well as the user positioning accuracy. Two layouts, one is generated by using our method and the other is generated by using the classical method in [16], are realized for comparison.
for swarm i = 1 to N do
4
randomly initialize ξ and η ;
5
calculate each particle’s speed vi by using formula (6);
6
if (vid > vmax)do
7
vid = vmax;
8
z
9
end
10
if (vid < vmin)do
4
Pseudolite User
vid = vmin;
11 12
end
13 14
update the Sim by using fomula (7); calculate GDOP by using formula (1), (2) and (3);
15
update Lmi and Lmg by using formula (4) and (5);
16
end
17
m=m+1;
18
(6)
8 y (4,2,1) 10 x (a) Classical layout
z
end
4
19
Get the optimal layout L ;
20
Calculate the optimal GDOP for LMg ;
M g
Pseudolite User
8 y (4,2,1)
The pseudo-code of PSO algorithm is shown in Table 2. First, we initialize the relevant parameters (line 1), including the user position U , pseudolite classical layout parameters S 0 , the layout
10 x
Si0 corresponding to each particle and PSO parameters (c1, c2, ω , m g
m i
m i
m, etc.). Then, we initialize L and L (line 2), where L
(b) Our optimal layout Figure 2. Pseudolite layout
can
Fig. 2 shows the detailed layout information for a system with four synchronized pseudolites. The Cartesian coordinate system is established along with the length (x), width (y) and height (z) of the room.
be expressed as L (l , l , , l ) and Lg can be expressed m i
m i1
m i2
m iQ
m m m m as Lg (l g1 , l g 2 , , l gQ ) , i indicates a particle and m represents
the number of iteration, their definitions are as follows. L Lmi m Si
m1 i
Lmg Lmk
m1 i
if GDOP( S ) GDOP( L ) m i
otherwise
i 1, 2, , N
s.t. GDOP( Lmk ) GDOP( Lmi ), i 1, 2,..., N
Fig. 2(a) is the classical layout, in which the four pentagons separately represent four pseudolites, namely s1, s2, s3 and s4, and the solid circle indicates a user. From Fig. 2(a), we can find that by using classical method, one pseudolite is assigned to the room ceiling, s4 for (5, 4, 4), and the other three pseudolites are in the floor. In details, s1, s2 and s3 are located at (0,0,0), (10,4,0) and (0,8,0), respectively. Fig. 2(b) is our optimal layout, in which one pseudolite is at room ceiling and the others are in floor, too. However, the detailed position of s1, s2, s3 and s4 is not the same as the classical layout. Specifically, s1, s2, s3 and s4 are located at (3.4502, 0.8124, 0), (8.9993, 3.1811, 0), (0, 8.07, 0.02) and (3.6040, 1.1812, 3.9972). For the user at (4, 2, 1), we can calculate the system GDOP with the two layouts, which is 1.871 in classical layout and 1.6204 in our layout. Therefore, our method reduce the GDOP value by up to ((1.871-1.6204)/1.871) = 13.4% .
(4)
(5)
In the third step (line 3-18), we enter the iterative process of PSO algorithm to find out the optimal GDOP. In this step, we first calculate the velocity vi of each particle i (line 6-12). If the velocity vid exceeds the threshold (vmax and vmin), the velocity is set to be vmax and vmin (line 7-12). Then we update the layout Sim (line 13). The particle’s velocity and layout update formula are shown in the following equations (6) and (7) (Where ξ and η are
the random numbers between 0 and 1 ， c1 and c2 are commonly set to 2.05). We calculate the GDOP corresponding
315
According to the optimized layout as shown in Fig2, we evaluate the improvement on user positioning accuracy. To emulate the real world environment, we add random pseudo-range errors to all pseudolites. Note that the added error is less than 1.5 meters, which is a traditional pseudo-range error caused by code loop of pseudolites. For the two layouts, we solve out the user position and repeat the simulation for 300 times.
6. ACKNOWLEDGEMENT The authors would like to thank the support of the National Foundation of China [61201005] and [61671110].
7. REFERENCES [1] J. Wang. “Pseudolite Applications in Positioning and Navigation: Progress and Problems”. Journal of Global Positioning Systems, 2002, 1(1): 48-56. [2] L. Dai, J. Wang and C. Rizos, et al. “Pseudo-Satellite Applications in Deformation Monitoring”. GPS Solutions, 2002, 5(3): 80-87. [3] G. Xie. “Principles of GPS and receiver design”. The press of Electronics Industry, 2009. [4] D. Yun, H. Jun and B. Parkinson, et al. “Centimeteraccuracy indoor navigation using gps-like pseudolites”. GPS World, 2001. [5] S. MartNez, F. Bullo. “Optimal sensor placement and motion coordination for target tracking”. Automatica, 2006, 42(4): 661-668.
(a) Classical Layout
[6] A. N. Bishop, B. Fidan and. D. O. Anderson, et al. “Optimality Analysis of Sensor-Target Geometries in Passive Localization:Part2-Time-of-Arrival Based Localization”. Automatica, 2007, 46(3): 479-492. [7] Y. Zheng, H. Wang and L. Wan, et al. “A Placement Strategy for Accurate TOA Localization Algorithm”. 7th IEEE Communication Networks and Services Research Conference, 2009: 166-170. [8] M. Monteiro, A. Barreto and Research Division, et al. “Detecting malicious ADS-B broadcasts using wide area multilateration”. IEEE Digital Avionics Systems Conference, 2015: 4A3-1-4A3-12.
(b) Our Layout
Figure 3. Comparison of positioning error results
[9] E. Boci, S. Sarkani and T. A. Mazzuchi. “Optimizing ADSB RF coverage”. IEEE Integrated Communications, Navigation and Surveillance Conference, 2009: 1-10.
The positioning results of classical and optimal layout are shown in Fig. 3(a) and (b), respectively. In two figures, the x-axis is the number of position times, and the y-axis represents positioning error in unit meter. From Fig. 3, we can get that the user positioning error in our optimal layout is smaller than that of the classical layout. In details, the average user positioning error under our layout is 0.0825m, which gains 12.4% improvement against to 0.0942m under classical layout. Moreover, the variance of position error, which is decreased from 0.00140 to 0.00076, is improved by up to 46%, too.
[10] Y. Teng, J. Wang and Q. Huang. “Minimum of Geometric Dilution of Precision (GDOP) for five satellites with dualGNSS constellations”. Advances in Space Research, 2015, 56(2): 229-236. [11] C. S. Chen, K. S. Chen and J. F. Huang, et al. “Using Genetic Algorithms to Approximate Weighted Geometric Dilution of Precision”. International Symposium on Computer, Consumer and Control. 2016:895-898.
5. CONCLUSION
[12] M. R. Mosavi, M. Divband. “Calculation of Geometric Dilution of Precision Using Adaptive Filtering Technique Based on Evolutionary Algorithms”. IEEE International Conference on Electrical and Control Engineering, 2010:4842-4845.
In this paper, we propose a layout strategy for pseudolite system based on PSO algorithm, which improves the user positioning accuracy. For a given scenario, our method can find out the optimal location for all pseudolites automatically. The simulation results show that our optimized layout can reduce system GDOP by up to 13.4% compared to the classical layout. Meanwhile, with the same pseudo-range error, a user under our optimal layout can get more accuracy position than the classical layout. In details, the average positioning error and the variance of positioning error are improved by 12.4% and 46%, respectively. Therefore, our method can be used to improve the performance of pseudolite system.
[13] R. J. Kelly. “Additional results on reducing geometric dilution of precision using ridge regression”. IEEE Transactions on Aerospace & Electronic Systems, 1990, 26(4):677-681. [14] C. H. Xu, J. Wang and J. X. Gao, et al. “Dilution of Precision and Clock of Simulated Pseudolites Augmented GPS Signals”. IEEE International Congress on Image and Signal Processing, 2009:1-4. [15] D. Y. Hsu. “Relations between dilutions of precision and volume of the tetrahedron formed by four satellites”. IEEE
316
[18] K. Yedukondalu, A. D. Sarma and A. Kumar. “Mitigation of GPS multipath error using recursive least squares adaptive filtering Circuits and Systems”, 2010 IEEE Asia Pacific Conference, 2010:104-107.
Position Location and Navigation Symposium, 1994:669676. [16] Q. Song, B. Zhang and S. Li. “Study of Configuration Technology of Ground Pseudolite”. Computer Measurement & Control, 2013.
[19] F. M. G. Sousa, F. D. Nunes. “Near-far effect mitigation for GNSS software receivers using subspace projection”. 2010: 1-6.
[17] Y. Han, Z. Huang and Y. Zhao. “Research on High-precision Time Synchronization of Pseudolites Based on Bidirectional Ranging”. Journal of Telemetry Tracking & Command, 2010.
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