Time of Arrival (TOA)-Based Direct Location Method Mohamed Khalaf-Allah Umm Al-Qura University P.O. Box 715, Makkah, 21955, SAUDI ARABIA
[email protected] Abstract: The presented direct location method computes the 3D position of an object using exactly four time-of-arrival (TOA) measurements, i.e. range measurements, to four reference points or stations. This operation is also commonly known as multilateration. Traditionally, this problem can also be solved using three TOA measurements to three reference stations, i.e. trilateration. However, a twofold ambiguity, usually of the vertical component, arises, which cannot be resolved without some prior knowledge about the general location of the object or an extra TOA measurement. The developed algorithm is simpler and faster than conventional ones, thus, complexity in terms of time, power consumption, and chip area is reduced. Therefore, the algorithm is suitable for applications with limited computational and power resources.
1. Introduction Trilateration is the determination of an object’s position based on simultaneous time of arrival (TOA) or range measurements from three stations at known locations. It is simply the problem of finding the intersection of three circles in the 2D case or the intersection of three spheres in the 3D case, where a system of quadratic, i.e. nonlinear, equations is involved. An exact solution is not easy to obtain. However, many algebraic and numerical solutions are available in the open literature for the 2D [4-16] and 3D [1, 2, 17-21] cases respectively. These approaches involve complex geometric computations [1, 3], which usually require a relatively long execution time and, thus, are not suitable for low-cost applications with constrained power and computational resources. Statistical approaches, e.g. the least-squares (LS) method, that are usually implemented by an iterative algorithm [22] can miss the global minima and instead converge to undesired local minima. Algorithms that use three TOA measurements with three stations to obtain a 3D position solution face a twofold ambiguity, usually of the vertical component, and can only be resolved if some information about the general location of the object is available, i.e. above or below the stations’ planes. Using four TOA measurements with four stations yields 3D position solutions without ambiguities. In [19], various positioning algorithms for range-based TOA localization have been analyzed. The positioning algorithms include the analytical method, the least-squares method, Taylor series method, the approximate maximum likelihood method, two-stage maximum likelihood method and the genetic algorithm. The described analytical (direct) method has three possibilities for the solution of x, i.e. one solution, two solutions and no solution, unlike our developed method, which yields a direct, exact and unique solution. This is due to using exactly four TOA measurements rather than three TOA measurements as described in [19], and thus any ambiguities can be resolved. A comprehensive review of different TOA-based localization algorithms, based on maximum likelihood and least-squares techniques, is given in [23].
The presented method has not been found, currently pending international patent [24], and it exactly and uniquely solves the 3D components of an object’s position. Moreover, the solution does not require any matrix computations. The algorithm is advantageous in terms of implementation simplicity and computational cost. Therefore, it can be easily implemented in many low-power and low-cost wireless applications, e.g. 3D sensor networks. It uses four TOA measurements with four stations to avoid object location ambiguities, which is a problem associated with using three TOA measurements with three stations. Another advantage of this direct method is that it needs exactly four TOA measurements to compute 3D position solutions, where in many situations more than four measurements are not available or the availability of more than four measurements is not important for the accuracy requirements of the application at hand. Thus, the algorithm suits very well applications with constrained computational and power resources. The 3D position algorithm delivers position solutions that are dependent only on the given measurements and information, in case the interest is only in a single coordinate, e.g. the vertical component.
2. Derivation of the Direct Location Method In a TOA positioning system, the clocks of the four stations and the object are assumed synchronized, where the stations’ positions are well known. The TOA measurements between the object and the stations are multiplied by the known signal propagation speed in the media to yield range measurements. Thus, ( x i , y i , z i ), i = 1,..., 4 is the known position of station i, ri , i = 1,..., 4 is the range measurement between the object and station i, and ( x , y , z ) is the unknown position of the object. The TOA measurement equation is written as ( x − xi ) 2 + ( y − y i ) 2 + ( z − z i ) 2 = ri 2 , i = 1,..., 4 .
(1)
Substituting i = 2,3, 4 in Eq. (1) and subtracting each of the three equations from Eq. (1) when i = 1 , we get three equations of the form 1 A j , i = 2,3,4, . 2 2 2 2 2 2 2 2 2 A j = ( r1 − ri ) + ( xi − x1 ) + ( y i − y1 ) + ( z i − z1 ), j = 1, 2,3
( xi − x1 ) x + ( y i − y1 ) y + ( z i − z1 ) z =
(2)
From the set of three equations expressed in (2) z and y can be cancelled out using straight forward algebra to get an explicit expression for x independent of y and z as 1 I4I5 − I3I6 . (3) x= 2 I 2 I 5 − I1 I 6
Similarly, we can obtain an explicit expression for y independent of x and z as 1 I1 I 4 − I 2 I 3 . y= 2 I1 I 6 − I 2 I 5
(4)
And an explicit expression for z independent of x and y as z=
Where
1 I 7 I 10 − I 8 I 9 . 2 I 7 I 12 − I 8 I 11
(5)
I 1 = ( z 3 − z 1 )( x 2 − x1 ) − ( z 2 − z1 )( x 3 − x1 ) .
I 2 = ( z 4 − z1 )( x 2 − x1 ) − ( z 2 − z1 )( x 4 − x1 ) . I 3 = ( z 3 − z1 ) A1 − ( z 2 − z 1 ) A2 . I 4 = ( z 4 − z 1 ) A1 − ( z 2 − z1 ) A3 . I 5 = ( z 3 − z1 )( y 2 − y1 ) − ( z 2 − z1 )( y 3 − y1 ) . I 6 = ( z 4 − z 1 )( y 2 − y1 ) − ( z 2 − z1 )( y 4 − y1 ) . I 7 = ( y 3 − y 1 )( x 2 − x1 ) − ( y 2 − y1 )( x 3 − x1 ) . I 8 = ( y 4 − y 1 )( x 2 − x1 ) − ( y 2 − y1 )( x 4 − x1 ) . I 9 = ( y 3 − y1 ) A1 − ( y 2 − y1 ) A2 . I 10 = ( y 4 − y1 ) A1 − ( y 2 − y1 ) A3 . I 11 = ( y 3 − y 1 )( z 2 − z1 ) − ( y 2 − y 1 )( z 3 − z 1 ) .
I 12 = ( y 4 − y1 )( z 2 − z1 ) − ( y 2 − y1 )( z 4 − z1 ) .
(6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17)
Eq’s (3), (4) and (5) are the mathematical formulas of a computer algorithm to calculate the 3D position of an object using TOA measurements. The computation of each position component is independent of the other components. Thus, allowing to only calculating the position component of interest, e.g. the vertical position (altimeter mode). The developed algorithm can also be used to initiate more computationally intensive iterative positioning algorithms to help overcome divergence problems.
3. Numerical Example Four fixed transmitters were located at (0,0,0), (10,0,5), (10,10,0) and (0,10,5). The error-free range measurements were respectively 75 , 50 , 75 and 50 , where all coordinates and range measurements are in meters. Substituting the given transmitters’ positions and the errorfree range measurements into Equations (3-5) yields the exact true receiver’s position at (5,5,5).
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