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An Algebraic Solution to the Multilateration Problem
Research · April 2015 DOI: 10.13140/RG.2.1.1681.3602
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Abdelmoumen Norrdine RWTH Aachen University
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An Algebraic Solution to the Multilateration Problem
Abdelmoumen Norrdine Institute of Geodesy RWTH Aachen University, Germany [email protected]
Abstract—Across the spectrum of known algorithm for position N estimation there is no favorite method. Some algorithms require intensive computation capabilities, while other algorithms could s be implemented in devices with limited resources such as sensor 3 nodes. In this paper an approach for solving nonlinear problems s 1 P on the example of multilateration is presented in both cases with s 3 and without over determination. Thereby neither approximation, 2 nor iterative solutions are used. In the proposed method, the P nonlinear elements of the equations system are treated as 1 additional unknowns, which represent simultaneously a constraint. Thus a new equations system is created, which is solved by mean of linear algebra methods with low computational complexity. The algorithm was implemented and P 2 tested in conjunction with a developed UWB indoor positioning Figure 1: Trilateration problem system.
Index Terms—Localization, Trilateration, Multilateration, non (x, y, z) of the point N is equivalent to finding the solutions to linear least square, Ultra Wide Band (UWB), sensor networks the following system of quadratic equations. (1) x x2 + y y 2 + z z 2 = s2 I. INTRODUCTION 1 1 1 1 2 2 2 2 The calculation of the spatial coordinates of unknown x x2 + y y 2 + z z 2 = s 2 points from its distances to other known points is a common 2 2 2 2 operation, known as multilateration. In recent years numerous x x3 + y y 3 + z z 3 = s 3 studies on the solution of the multilateration range equation (1) can be arranged as: have been published. As an example, we can refer to [1] and (2) [2] which listed a number of procedures and presented an 2 2 2 2 2 2 2 algebraic approach. This paper shows an alternative method for x+y+z 2 xx1 2 yy 1 2 zz=s 1 1 x 1 y 1 z 1 solving the multilateration range equation with low x+y+z2 2 22 xx 2 yy 2 zz=s 2 x 2 y 2 z 2 computational complexity. The algorithm was first published 2 2 2 2 2 2 2 by the author in German language [3]. In this contribution the 2 2 2 2 2 2 2 x+y+z2 xx 2 yy 2 zz=s x y z algorithm is applied to real measurement data of an UWB 3 3 3 3 3 3 3 indoor positioning system. Or in matrix representation: The suggested method is described in section II in the case (3) 2 2 2 of the trilateration problem. In section III the algorithm is x + y + z 2 2 2 2 1 2x 2 y 2 z s x y z extended to the multilateration problem based on range 1 1 1 1 1 1 1 x 2 2 2 2 measurements to more than three reference points. 1 2x2 2 y 2 2 z 2 s 2 x 2 y 2 z 2 y Subsequently, as shown in section III, by using a recursive 2 2 2 2 1 2x3 2 y 3 2 z 3 s 3 x 3 y 3 z 3 least square approach additional range measurements can be z added gradually to lead to a balanced solution. In section IV Thus, (3) is represented in the known form two examples illustrate the application of the algorithm in A= xb (4) conjunction with real measurement data. 0 0 With the constraint: x ∈ E T OLUTION BASED ON THREE REFERENCE POINTS Where E= x,x,x,x 4/ x=x+x+x 2 2 2 II. S 0 1 2 3 0 1 2 3 Given are the three reference points P1(x1, y1, z1), P2(x2, y2, A. Solution of the equation system (4) z2), P3(x3, y3, z3) and the range measurements s1, s2 and s3 to the point N (cf. Fig.1). The determination of the coordinates Case 1: P1, P2 and P3 do not lie on a straight line. Then Rang(A )=3 and dim(Kern(A ))=1. 0 0 2012 International Conference on Indoor Positioning and Indoor Navigation, 13-15th November 2012
The general solution of (4) is then 0000 x= x +t x (5) ph 0 1 0 0 With the real parameter t. Where x is a particular solution of I p 0 0 1 0 (4) and xh is a solution of the homogeneous system A0.x=0 i.e. xh is a Basis of Kern(A0). 0 0 0 1 The vectors xp and xh can be computed using the Gaussian elimination method. The particular solution xp can also be Case 2: P1, P2 and P3 lie on a straight line. determined using the pseudo inverse of the matrix A0. The Then Rang(A0)=2 and dim(Kern(A0))=2. The general solution pseudo-inverse gives the solution with the minimum norm [4]. of (4) is then Determination of the parameter t: (11) x= xp +t x h1 +k x h2 Let T , T and x p = xp0 ,x p1 ,x p2 ,x p3 xh = xh0 ,x h1 ,x h2 ,x h3 With real parameters t and k T x = x ,x ,x ,x xp is a particular solution of (4) and xh1 and xh2 are two 0 1 2 3 solutions of the homogeneous system A .x=0 . They are When inserted into (5) then it becomes: 0 linearly independent solutions and form therefore a basis of (6) Kern(A0). Since there is only one constraint equation, the x0 = xp0 +t x h0 multilateration problem has infinitely many solutions. x1 = xp1 +t x h1 x = x +t x 2 p2 h2 III. SOLUTION BASED ON MORE THAN THREE REFERENCE POINTS x3 = xp3 +t x h3 With additional distances s , s …. s to the reference points By using the constraint x ∈ E it follows: 4 5 n P4, P5 … Pn , (3) can be extend as follows:
2 2 2 (12) 2 2 2 2 x+tx=x+txp0 h0 p1 h1 +x+tx p2 h2 +x+tx p3 h3 1 2x1 2 y 1 2 z 1 2 2 2 s1 x 1 y 1 z 1 x + y + z 2 2 2 2 1 2x 2 y 2 z s x y z 2 2 2 2 2 2 2 and thus x 2 2 2 2 1 2x3 2 y 3 2 z 3 s x y z (7) y 3 3 3 3 2 2 2 2 tx+x+x +t2 xx+ 2 xx+ 2 xx x z h1 h2 h3 p1 h1 p2 h p3 h3 h0 2 2 2 2 1 2xn 2 y n 2 z n sn x n y n z n +x2 + x 2 + x 2 x = 0 p1 p2 p3 p0
This is a quadratic equation in the form at2 +bt+c= 0 with That means in the known form the solutions A= xb (13) 2 b ± b4 ac (8) t=1/2 With the constraint x ∈ E. The solution xˆ of (13) in the sense 2a of least squares method is: The solutions of the equation system (4) are: 1 xbˆ = ATT A A (14) x= x +t x 1 p1 h (9) The projection of p on the column space of A is: 1 x= x +t x TT 2 p2 h pb= A A A A (15) The coordinates of p on the column space Col(A) represent the If the multilateration problem cannot be solved (too short distances), so there are no real solutions. In this case, the real solution xˆ T -1 T part is used as an approximation for the solution. With this To note is that all elements in matrix L = (A A) A are approximation, the constraint x1/2 ∈ E is not met. Thus the derived from reference points coordinates only. Moreover, difference: vector b consists of distances between the unknown point N 222 (10) and all the reference points. Especially in static sensor d = x0 x 1 + x 2 + x 3 networks the computation of the entire localization can be accomplished in the nodes themselves, since the computation is a measure of the solvability of the multilateration problem, is restricted to a matrix vector multiplication of matrix L and where x , x , x and x the coordinates of the solution x of (4). 0 1 2 3 vector b. Solutions of the multilateration problem are the points: If, however, the measurements are uncorrelated but have N =x I and N =x I , where 1 1 2 2 different uncertainties, the Weighted Least Square (WLS) is used [5]. The solution is given by the following equation: 2012 International Conference on Indoor Positioning and Indoor Navigation, 13-15th November 2012
1 xbˆ = ATT V11 A A V (16) V is the covariance matrix of random errors. Since the solution should also belong to E, and because E does not form a vector space, a closed formula for the projection of p on Col(A) ∩ E is not given. Thus only candidates for approximate solutions are selected. The solution candidate N, which minimizes the error square sum (17) is then chosen
2222 (17) min N P11 s N Pnn s N
Figure 2: Mobile station with attached WLAN bridge First candidate: Equation (13) is solved using (14) or (16).
Further candidates by using the recursive least square: 15,0 P First the non over determined system of equations (4) have 43 to be solved. Since (4) has two solutions, the solution which is Reference station 20,0 Mobile station P closer to the first candidate is selected. This solution is used as 31 P101 P208 a starting point for the Recursive Least Square (RLS) [5]. Let P102 X [m] X P36 x0 be the initial solution. By every incoming distance, x0 is 25,0 P331 updated in x1 by using RLS. P38
The approach enables a simultaneous execution of distance P37 measurement and positioning calculation. Hence a position 30.0 assignment can be started, although not all distances are -10,0 -5,0 0,0 5,0 Y [m] available. Thereby unnecessary waiting time is avoided and Figure 3: Test field at the Geodetic Institute (2nd floor) the positioning calculation is speed up.
IV. NUMERICAL EXAMPLES TABLE I To test the numerical method, distance measurements in a DISTANCE MEASUREMENTS TO THREE REFERENCE POINTS specially created test field at the University Institute were Reference X[m] Y [m] Z [m] dmeasured dtrue [m] dtrue - points [m] d performed using a UWB Indoor Local Positioning System measured [cm] (UWB-ILPS). The main hardware components of UWB-ILPS are Time Domain PulsON 210 UWB Radios [6]. Fig. 2 shows P37 27,297 -4,953 1,470 3,851 3,857 0,6 the mobile station. Distance measurement between the stations takes place via UWB, communication and control is via P331 25,475 -6,124 2,360 3,875 3,988 -11,3 WLAN. For more details on UWB-ILPS, please refer to [7] P102 22,590 0,524 1,200 3,514 3,497 -1,7 and [8]. Fig. 3 demonstrates the location of the reference stations as well as the mobile station located on points P36 and P38 , where The general solution of (4) is x = xp + t xh , where t is the the x-axis points to north direction, y-axis to east direction and solution of the quadratic equation (7): z-axis points down to the center of earth. (0,00457...)t2 + 5,5087... t+ 1658,74... = 0 A. Solution based on three reference points: In this case t1 = -603,2647 and t2 = -601,1773. (4) has then the following solutions: The coordinates of three reference points and the measured x = (601,8863 ; 24,3506 ; −2,4811 ; 1,6667)T and distances to the point P are listed in Table I. The true 1 36 x = (599,8010 ; 24,3123 ; −2,5205 ; 1,5365)T coordinate of the unknown point are 2 The corresponding differences according to (10) are P = (24,335 ; -2,506 ; 1,130). 36 d =d =0. The small size of the differences confirmed the The particular solution of (4) and the solution of the 1 2 solvability of the problem. homogeneous system are respectively: The solutions of the multilateration problem are in meter: xp = (0,0000 ; 13,2890 ; -13,4807 ; -35,9246) and N1 = (24,3506 ; −2,4811 ; 1,6667) and xh = (0,9977 ; 0.0183 ; 0,0188 ; 0,0623) N2 = (24,3123 ; −2,5205 ; 1,5365) 2012 International Conference on Indoor Positioning and Indoor Navigation, 13-15th November 2012
B. Solution based on more than three reference points: TABLE II DISTANCE MEASUREMENTS TO SIX REFERENCE POINTS The coordinates of six reference points and the measured distances to the point P are listed in Table II. The true Reference X[m] Y [m] Z [m] dmeasured dtrue dtrue - 38 points [m] [m] d coordinate of the mobile station are measured [cm] P38 = (26,7590 ; -1,342 ; 1,130)
The first three distances provide the following solutions: P37 27,297 -4,953 1,470 3,652 3,666 1,47 x01 = (720,6931 ; 26,7726 ; −1,3389 ; 1,4584) and P31 20,693 -4.849 1,93 7,036 7,052 1,63 x02 = (716,7319 ; 26,7234 ; −1,4152 ; 0,7685) The starting value for the RLS is taken from the x and x 01 02 P102 22,590 0,524 1,200 4,586 4,568 1,79 solution that is compatible to the fourth distance (distance to P43), in this case x0 = x01 P43 17,113 -3,003 2,17 9,960 9,843 11,69 Furthermore it can be shown from the location of the P208 22,554 4,727 1,77 7,542 7,411 13,09 solutions that the lines to the points P43 and P208 run through walls and therefore the distances to these points have to be P101 22,45 -7,880 1,6 7,883 7,846 3,7 corrected according to UWB wave excess delay estimation equation proposed in [9]. The calculated delays are in this case 12 cm and 14 cm respectively. TABLE III CALCULATED DIFFERENCES AND SUM OF SQUARES By using the RLS we get the following solutions: Solutions x01 x02 x1 x2 x3 x0 = (720,6931 ; 26,7726 ; −1,3389 ; 1,4584) Difference d [m2] 0 0 0,37 0,29 0,44 x1 = (720,6933 ; 26,7659 ; −1,3377 ; 1,4576) 2 x2 = (720,6934 ; 26,7680 ; −1,3267 ; 1,4564) Error square sum [m ] 0,0421 0,1246 0,0485 0,0432 0,0537 x3 = (720,6877 ; 26,7629 ; −1,3189 ; 1,5005) is proposed. Thereby neither approximation, nor iterative Table III shows the difference d and the sum of squares by solutions are used. The algorithm is based on linear algebraic using (10) and (17) respectively method, has low computational complexity and can be The solution which minimizes the error square sum is x01. applicable to real-time applications and in wireless sensor Therefore The solution of the multilateration problem is the nodes. The measurement results performed with our UWB point N = (26,7726 ; -1,3389 ; 1,4584) positioning system show that the algorithm is highly effective C. Computational efficiency and has low estimation error. However the raging update rate of the used UWB transceivers is relatively low, and doesn’t Generally for indoor positioning, the limited number of permit to test the algorithm by moving objects. Therefore participating reference stations doesn’t cause a big challenge further works consist of the expansion of the positioning to the computational resource for multilateration. However, for example, in the case of a sensor network node, every algorithm, regarding kinematic scenarios. reduction in the computation time can lead to an increase in battery lifetime. REFERENCES The computational efficiency of the algorithm is compared to [1] E. Grafarend, P. Lohse and B. Schaffrin, Dreidimensionaler the Gauss–Newton algorithm (GNA) applied on (1). The Rückwärtsschnitt. Z.f. Verm.wesen 114,61ff.; 1989 [2] J. Awange and E. Grafarend, Algebraic Solution of GPS Pseudo-Range results indicate that the average execution of the proposed Equations. Journal of GPS Solutions 5 (4) pp. 20 – 32, 2002. algorithm is significantly faster than GNA. However the [3] A. Norrdine, Direkte Loesung des raeumlichen Bogenschnitts mit convergence of GNA depends on the start solution. Assuming Methoden der Linearen Algebra, AVN, Heft 1/2008, pp.7-9 that the point N is within the reference point’s volume, the test [4] G. Strang and K. Borre, Linear Algebra, Geodesy, and GPS , Wellesley- Cambridge Press, 1997 field center is selected as start solution for GNA. For [5] G. Strang, Introduction to applied mathematics,Wellesley-. Cambridge comparison, the execution time of the algorithm based on the Press, 1986 last numerical example average to 1.9 µs while GNA needs 18 [6] Time Domain PulsON 210, http://www.timedomain.com/product_p210, µs. To note the evaluation is based on the execution time last accessed August 2012. [7] J. Blankenbach, Z. Kasmi, A. Norrdine, H. Schlemmer, Indoor- performed with MATLAB on a 2.53 GHz dual-core PC. Positionierung auf Basis von UWB – Ein Lokalisierungsprototyp zur Baufortschrittsdokumentation, AVN, Vol 8-9/2008, pp.292-300 [8] J. Blankenbach, A. Norrdine: Mobile Building Information Systems based on precise Indoor Positioning, Journal of Location Based Services, V. CONCLUSION Volume 5 Issue 1, Taylor & Francis, 2011 In this contribution, an alternative solution for the [9] I. Oppermann, H. Matti and J. Inatti, UWB Theory and Applications, pp. multilateration problem, with and without over determination, 189-190, John Wiley & Sons Ltd, 2004