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An Algebraic Solution to the Multilateration Problem See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/275027725 An Algebraic Solution to the Multilateration Problem Research · April 2015 DOI: 10.13140/RG.2.1.1681.3602 READS 570 1 author: Abdelmoumen Norrdine RWTH Aachen University 16 PUBLICATIONS 47 CITATIONS SEE PROFILE All in-text references underlined in blue are linked to publications on ResearchGate, Available from: Abdelmoumen Norrdine letting you access and read them immediately. Retrieved on: 18 April 2016 2012 International Conference on Indoor Positioning and Indoor Navigation, 13-15th November 2012 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 By using the constraint x ∈ E it follows: With additional distances s4, s5 …. sn to the reference points 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) b ± b2 4 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) x1= x p +t1 x h The projection of p on the column space of A is: (9) 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.
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