Hyperbolic Positioning Using RIPS Measurements for Wireless Sensor Networks Xuezhi Wang∗ , Bill Moran∗ and Marcus Brazil‡ ∗ Melbourne
Systems Laboratory Dept of Electrical and Electronic Engineering, The University of Melbourne, Melbourne, Australia Email:
[email protected],
[email protected] ‡
ARC Special Research Centre for Ultra-Broadband Information Networks (CUBIN) an affiliated program of National ICT Australia, Department of Electrical and Electronic Engineering, The University of Melbourne, Victoria 3010, Australia, Email:
[email protected]
Abstract—Measurements based on a Radio Interferometric Positioning System (RIPS) can determine distance differences between pairs of motes in a wireless sensor network. Such measurements allow one to localise motes in the network when the locations of at least three motes are known. In the 2-dimensional case, a RIPS trilateration approach, which we describe in this paper, appears to be the most efficient way of localising all motes with respect to a set of known anchors. An important issue associated with this method is the sensitivity of the trilateration to RIPS measurement noise. In this paper we present an analytic study of the impact of RIPS measurement noise on the localisation error, the results of which can be used as a guide for choosing RIPS measurements that minimise the localisation error. A simulated example demonstrates the underlying idea, and its effectiveness.
I. I NTRODUCTION Precise localisation of wireless sensor networks, composed of small sensor motes with limited information processing ability, has become an increasingly important research issue with applications in areas such as defence, homeland security and remote area monitoring and control [1]. To date, the most widely discussed localisation techniques in the literature are based on directly estimating the distances between pairs of motes by taking measurements such as time difference of arrival (TDOA) or time of arrival (TOA), angle of arrival (AOA) or received signal strength (RSS) [2], [3]. Attempts have been made to optimise the efficiency of this methodology by using the principles of graph rigidity [4] to minimise the number of measurements required [2], [5], [6], [7]. For small sensors, however, such an approach presents enormous practical challenges in terms of computational and constructional cost, sensing ability, and accuracy [8]. The Radio Interferometric Positioning System (RIPS) measurement technique, initial reported in [9] and further discussed in [10], utilizes radio frequency interference to obtain a sum of distance differences between the locations of a quartet of motes. The sum of distance differences is referred as a RIPS measurement. As reported in [10], the system requires no extra resource for wireless mote networks and the RIPS measurements yield smaller errors and covers longer distances than those resulting from RSS based small sensor measurement systems.
1-4244-1230-7/07/$25.00 © 2007 IEEE
In [11], the problem of localising mote networks using RIPS measurements is considered in an optimisation framework. RIPS measurements together with their RSS measurements are used to estimate pairwise distance measurements, which are then fed into the iterations of a traditional least square optimisation algorithm to find all locations of free motes based on a few landmark motes known as anchors. In [9], a genetic algorithm is described for estimating locations of free motes using RIPS measurements only. The convergence of these approaches requires large numbers of RIPS measurements being used, which is computationally demanding for a mote network. Experiments and simulations have shown that satisfactory localisation error can be achieved, but the number of measurements required is high. Using a method similar to trilateration with pairwise distance measurements, a closed form mote localisation algorithm was developed in [12] and [13]. In this paper, we refer to this algorithm as RIPS trilateration. The approaches using similar principle have appeared in the target tracking and related literature for applications of TDOA or TOA measurements [5], [6], [7], [14]. The RIPS trilateration algorithm is promising as it exhibits a closed form and is computationally efficient. Moreover, it attains the minimum number of RIPS measurements required to localise a free mote. For localising N free motes, it requires at most 3N RIPS measurements. A distributed algorithm is available in [13], which can efficiently divide the transmission network into cliques so that the required RIPS measurements can be obtained with a minimal number of transmissions. However, in practical applications there are a number of serious issues that need to be considered: 1) attempting to localise a single free mote with respect to three given anchor motes can result in ambiguous solutions, even if exact measurement are possible; 2) the localisation is inevitably subject to error, which can be large in some regions, due to the presence of RIPS measurement noise. In this paper, these two problems are addressed. We show that the solution ambiguity can be fully resolved by taking a single extra RIPS measurement, and we present an analysis of the impact of RIPS measurement noise on the localisation error, which can be used as a guide for choosing RIPS
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measurements that will lead to a minimal degree of localisation error. The remainder of paper is arranged as follows. In Section II, RIPS trilateration for mote localisation is reviewed and some issues are discussed. In Section III, an efficient method to resolve solution ambiguity for the RIPS trilateration algorithm is presented. The localisation error of the RIPS trilateration algorithm due to RIPS measurement noise is analysed in Section IV. The application of the analytical results are demonstrated in a simulated example in Section V which is followed by the conclusion in Section VI. II. RIPS T RILATERATION Let S denote a set of N motes in a sensor network whose associated communication graph is a clique; in other words, such that each mote in S is within transmission range of every other mote in S. Throughout this paper we assume that each element of S lies in R2 . Given A ∈ S we denote by xA ∈ R2 the location state of A, which we represent by its Cartesian coordinates: xA = [xA , yA ] . We consider a collection of four motes A, B, C, D ∈ S, such that A, B and C are anchor motes, that is, their location states are already known, and D is a free mote. These motes are illustrated in Figure 1. A
Unknown
XA
B
which uses motes A and C as a transmitter pair and B and D as a receiver pair, is a second independent measure. Every other RIPS measurements on this quartet is a linear combination of (2) and (3). Suppose S contains m anchor motes. We define the mote localisation problem as the problem of localising all (N − m) free motes in S using RIPS measurements. Efficient algorithms for solving this problem in the absence of noise have appeared in [12] and [13], the latter paper generalising the problem to the case where S is no longer a clique. In [12], a closed form solution for solving xD from (2) and (3) for zero noise case is derived; we refer to this as the RIPS trilateration algorithm in this paper, and summarize it below. After a coordinate transformation, which sets xA to the origin, (2) and (3) can be written as dAD − dBD (4) β = dAD − dCD 2 2 2 2 xD + yD + (xB − xD ) + (yB − yD ) = 2 + x2D + yD (xC − xD )2 + (yC − yD )2 where
β=
β1 β2
=
kA,B,C,D − dBC + dAC kA,C,B,D − dBC + dAB
The RIPS trilateration algorithm solves (4) for xD xD = yD
XB
Anchors
in the form xD = a + bdAD
Fig. 1.
where
D
XC
a =
RIPS measurements
A single RIPS measurement, as described in [9], involves four motes. Two motes act as transmitters, sending a pure sine wave at slightly different frequencies. This results in an interference signal at low beat frequency which is received by the other two motes (acting as receivers). A sum of range differences between the four motes can be obtained from the phase difference of the received interference signals at the two receiver locations. If motes A and B serve as transmitters and motes C and D form the receiver pair, then the corresponding RIPS measurement, denoted kA,B,C,D , measures the distance differences kA,B,C,D
(5)
XD
C
= ||xD − xA || − ||xD − xB || + ||xB − xC || − ||xA − xC ||
(1)
which may be simply written as kA,B,C,D = dAD − dBD + dBC − dAC .
(2)
It was observed in [9], [10], [12], that for the clique S at most 12 N (N − 3) independent RIPS measurements can be made. In particular, for the quartet of motes A, B, C, D, there are at most two independent RIPS measurement. For example, kA,C,B,D = dAD − dCD + dBC − dAB
(3)
S=
S −1 z
(6) S −1 β 2 2 1 xB + yB − β12 , and z = 2 2 2 xC + yC − β22 b =
xB xC
yB yC
As indicated in [12], the quadratic equation (5) will provide a unique solution for xD when bT b ≤ 1. Otherwise, there will be two solutions. Remarks: 1) The RIPS Trilateration algorithm requires two independent RIPS measurements. In general these measurements may be taken from two different quartets of motes, each containing three anchors, providing the two quartets share a common free mote and at least one common anchor mote, which will be used as a transmitter for each of the two measurements. 2) A single RIPS measurement kA,B,C,D defines a branch of a hyperbola in R2 which contains all possible values of the free mote location xD . The transmitter pair A and B are the foci of this hyperbola. The RIPS trilateration algorithm uniquely determines xD from the RIPS measurement pair kA,B,C,D and kA,C,B,D if the two single branch hyperbolas intersect a single point, or, equivalently, bT b ≤ 1 (see Figure 2).
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the RIPS measurement pair kA,B,C,D and kA,C,B,D in (5). We address this problem in the next section by showing that solution ambiguity can be resolved by taking an extra RIPS measurement.
120
100 A
B
80
m
kA,B,C,D = 40.5168
III. R ESOLVING RIPS T RILATERATION S OLUTION A MBIGUITY
kA,C,B,D = 42.7479
60
40
C
D
20
0
0
20
40
60 m
80
100
120
Fig. 2. The philosophy of the RIPS trilateration method. A, B, C are anchor motes. D is a free mote.
We show below that a RIPS measurement is bounded by a constant b which can be calculated from knowledge of the locations of the anchor motes. Theorem 1: For a RIPS system of anchor motes A, B, C and free mote D, the RIPS measurement kA,B,C,D is bounded by (7) −bABC ≤ kA,B,C,D ≤ bABC where bABC = |dBC − dAC | + dAB .
For a pair of RIPS measurements on a quartet of motes A, B, C, D (where A, B and C are anchor motes), it is possible to identify regions in R2 for the location of the free mote D where the RIPS trilateration algorithm will generate ambiguous solutions. An example of such regions (which depend on the relative positions of A, B and C) is shown in Figure 3. As discussed in the previous section, a pair of independent RIPS measurements results in at most two solutions for xD . We will show that these two solutions can be resolved by taking a single extra RIPS measurement via a fifth mote. This expands on a brief discussion given in [13]. Suppose that we obtain two solutions xD1 and xD2 for free mote D from the RIPS measurement pair (2) and (3) via the RIPS trilateration algorithm. Suppose, in addition, that there exists a fifth mote E ∈ S and that the five motes are in general position (ie, no three of them are collinear). Using motes A and E as transmitter pair we may take an extra RIPS measurement kA,E,B,D , which results in the following equation. kA,E,B,D = dAD − dED + dEB − dAB
Proof. In view of Equation (2) and Figure 2, we may write |kA,B,C,D | = |dBC − dAC + dAD − dBD | ≤ |dBC − dAC | + |dAD − dBD | ≤ |dBC − dAC | + dAB . As mentioned earlier, one of the crucial issues to be considered in RIPS trilateration is the existence of solution ambiguity when the condition bT b < 1 for (5) is not met. Figure 3 illustrates the extent of this problem for a clique of three
(8)
If xE , the location of E, is known, then we can immediately find the correct solution xDi , (i = 1 or 2) using (8), since Equation (8) will only hold for the correct location of D. If, on the other hand, E is a free mote, and an attempt to compute its location (from A, B and C via RIPS trilateration) results in a pair of possible solutions xE1 and xE2 , we can still verify the correct solution pair xDi and xEj using (8). It can be shown that the four solution pairs (xDi , xEj ) i = 1, 2, j = 1, 2 for motes D and E must be different, and that only one of these solution pairs satisfies (8). This is illustrated in Figure 4. When taking the extra RIPS measurement, either
120 110
180
100 160
90
E A
140
single solution area 70
.
100
A
.
D2
80
C
50
A,B,C,D
D1
120
B
60
k
true location of D
80
C
B 60
40 30
kA,C,B,D
40
two solution area
20
k
A,E,B,D
20
10
0
20
40
60
80
100
0
20
40
60
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100
120
140
160
180
120
Fig. 3. Solution ambiguity areas for fixed locations of anchor motes A, B and C when using RIPS measurement pair kA,B,C,D and kA,C,B,D in the RIPS trilateration algorithm.
anchors A, B and C, where two solutions for the free mote D are found if D is in the yellow colored area when using
Fig. 4. The ambiguity solutions D1 and D2 can be resolved by taking an extra RIPS measurement kA,E,C,D even without knowing the location of mote E.
mote D or mote E should be chosen as a transmitter, ensuring that the new RIPS measurement is not be a linear combination of the existing two RIPS measurement pairs. In the case shown
427
From the inverse function theorem, we have
or
100
δac
= =
dBC − dAC (xb − xc )2 + (yb − yc )2 − (xa − xc )2 + (ya − yc )2
38
6 41
4.2
50 40
5.5362
20
3 34
10
6.8 07
3 9.4
25 1.64
16
43 0.3 B 0.3 43 47 1.6 41 6 2.939
C
5.5362 5.59.4307 36 2
4.238
38
4.2 20
38
8
3
30
98
.93 2A 47
40
60
5.5362
1.6
70 5.5 8.1 362 60 325
4.2
8
43 83 6.
=
39
5.5362
38
98
=
2.9
80
4.2
93
δab
38
4.2
90
343
6.8
2.
where xD = [x, y] , and
2
5.536
110
5.5362
(17)
25
=
|∇k xD | = |(∇x k)−1 |
8. 13
120
kA,B,C,D (9) k(xD ) = kA,C,B,D 2 2 2 2 δab + (xa − x) + (ya − y) − (xb − x) + (yb − y) δac + (xa − x)2 + (ya − y)2 − (xc − x)2 + (yc − y)2
(16)
The value of the absolute determinant (17) reflects the uncertainty volume of the localised xD with respect to a unit uncertainty volume of the RIPS measurement pair. Figure 5 shows the error contours for localising mote D over the field of a clique of size 120 × 120 m2 . These are calculated on a logarithmic scale using (17), based on the knowledge of the locations of anchor motes A, B and C. We can easily demonstrate that error contours will keep a same patten but with smaller values for most area in Figure 5 when the triangle ABC is enlarged.
A. Error prediction via the Inverse Jacobian We may write the two independent RIPS measurements as
∇k xD = (∇x k)−1
8.13
In the presence of RIPS measurement noise, the localisation error of RIPS trilateration on a quartet of motes is strongly dependent on the relative locations of the anchor motes and the free mote. Therefore, it is useful to understand the noise behavior of RIPS trilateration in order to control the localisation error whenever possible. In this section, we discuss how to determine the uncertainty of localisation using the RIPS trilateration method when Gaussian zero-mean RIPS measurement noise is present. The analysis is based on the RIPS measurement pair for four motes A, B, C, D, where D is a free mote. Ideally, we would like to derive an expression for the Jacobian of the RIPS trilateration solution (5) with respect to kA,B,C,D . the input RIPS measurement pair, i.e., xD ∼ kA,C,B,D Unfortunately, this is not straightforward. Instead, we find the inverse Jacobian of the desired vector valued function and obtain an analytic result based on the inverse function theorem.
ya − y ∂kA,C,B,D yc − y − (15) = ∂y (xc − x)2 + (yc − y)2 (xa − x)2 + (ya − y)2
34
IV. RIPS M EASUREMENT N OISE E FFECT
xa − x xc − x ∂kA,C,B,D − (14) = ∂x (xc − x)2 + (yc − y)2 (xa − x)2 + (ya − y)2
6.8
in the figure, the new RIPS measurement is kA,E,B,D , and the corresponding hyperbola passes through only one of the two possible locations for D. The above method is summarized by following theorem. Theorem 2: For the set S suppose N ≥ 5 and the motes in S are in general position. Suppose further that there are m ≥ 3 anchor motes in S. Then all N motes can be localised via RIPS trilateration using at most 3 RIPS measurements. The proof of this result is straightforward and is not given here.
80
100
120
Fig. 5. Localisation error contours — the inverse of absolute Jacobian determinant of (11) on a logarithmic scale, where a larger value indicates a (potentially) larger localisation error when using a RIPS measurement pair with Gaussian noise of fixed standard deviation.
dBC − dAB (xb − xc )2 + (yb − yc )2 − (xa − xb )2 + (ya − yb )2
120 KA,B,C,D 100 KA,C,B,D
are both known constants (since the locations of the anchor motes are known). The Jacobian of (9) is defined as ∂k ∂k A,B,C,D
∇xD k =
∂x
∂kA,C,B,D ∂x
80
B
meter
A C
60
A,B,C,D
∂y ∂kA,C,B,D ∂y
approximated variance contour
40
(10)
D
20
and its absolute determinant is given by ∂k ∂kA,B,C,D ∂kA,C,B,D A,B,C,D ∂kA,C,B,D |∇xD k| = − ∂x ∂y ∂y ∂x
0
(11)
where ∂kA,B,C,D xa − x xb − x − (12) = ∂x (xb − x)2 + (yb − y)2 (xa − x)2 + (ya − y)2 ∂kA,B,C,D ya − y yb − y − (13) = ∂y (xb − x)2 + (yb − y)2 (xa − x)2 + (ya − y)2
0
20
40
60 meter
80
100
120
Fig. 6. An example of a point D on the extended line BC outside the triangle ∆ABC, where the determinant of (10) approaches to zero.
We can specify precisely when the Jacobian matrix is singular.
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Theorem 3: Let LABC be the union of the three lines representing the extensions of the sides of the triangle ABC, but not including the interiors of the edges of ABC. Then, the Jacobian ∇xD k is singular if and only if xD ∈ LABC . Proof. (Sketch) Let θ1 , θ2 and θ3 denote the polar angles of the vectors from A, B and C, respectively, to D. The absolute determinant of the Jacobian, (10), can be expressed in terms of these angles as follows: |∇xD k| = |sin(θ1 − θ2 ) + sin(θ2 − θ3 ) + sin(θ3 − θ1 )|. Clearly, this expression equals 0 if and only if θi = θj for some i, j ∈ {1, 2, 3} (i = j). This in turn means that D is collinear with two of the anchors A, B, C, but not on the interior of an edge of ABC, since then the polar angles would differ by π. Geometrically, as a free mote approaches LABC the two hyperbolas resulting from the RIPS measurements become almost parallel at the point of intersection (as shown in Figure 6), and exactly parallel once it reaches LABC . This results in the singularity of the Jacobian on LABC , and a large localisation error at points close to LABC . More generally, we observe, from Figure 5, that the localisation error is small if the free mote is located inside the triangle ABC and generally increases at a steady exponential rate as the free mote moves away from the triangle, unless it is close to LABC . The consequence of this is that if there is some choice in the locations of anchors they should be chosen to be as far apart as possible, close to the boundary of the clique. This property is further investigated in the simulation study in the next section, and is exploited in the algorithm for choosing pseudo-anchors in [13].
V. S IMULATION E XAMPLE In this section, we demonstrate the significance of the noise issue within the RIPS trilateration algorithm by analyzing a simulation scenario used in [12] and illustrated in Figure 7. Here we attempt to localise 50 randomly deployed motes via 25 uniformly located anchor motes whose locations are known (for example, by GPS). In this simulation the network is not a clique; the maximum communication range between two motes is assumed to be 80 m. Hence the localisation is done progressively. Each free mote can detect which motes lie within its communication range. If there are three anchor motes in this set of neighboring motes the free mote is localised via the RIPS trilateration algorithm, using at most three measurements. The noise of a RIPS measurement is assumed to be zero-mean Gaussian with a constant standard deviation σ = 0.2 m. After this procedure has been repeated for every free mote, each newly localised free mote is added to the list of anchors using its estimated location as its location state. The algorithm then repeats the above method for localising the free motes using the updated anchor list.
75
50
21
64 56 41
70
4722
23
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24
39 38 69 62
16
71 2917
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74 18
19 46
42 11
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59 31
4
65 52
15
45 637 58
60 66 1
72
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20 54
26
30 34 40
5 67
61 43
B. Remarks and Discussion 1) As indicated in (12)–(15), the localisation error caused by the effects of noise on a single RIPS measurement is a function of the locations of the two transmitters. However, the localisation error caused by a RIPS measurement pair depends on the locations of both transmitters and receivers. 2) We can obtain an estimate of the Cramer-Rao Lower Bound (CRLB) via the RIPS trilateration measurement model. If we use the RIPS trilateration algorithm to produce a “measurement” for estimating the location of a free mote, we may write the measurement model as y = k(x) + w
(18)
where the RIPS measurement noise is assumed to be a zero-mean Gaussian, w ∼ N (0, σ 2 I2×2 ), and k(x) is given by (9). When using an unbiased estimator to estimate a free mote location x based on (18), the minimum achievable error variance (the CRLB) is the inverse of Fisher information matrix (JF IM )−1 , where JF IM = −E ∇x ∇Tx ln p(y|x) 1 (∇x k) (∇x k)T . (19) = σ2
Fig. 7. A realisation of the simulated scenario, where square markers represent GPS enabled motes and triangles indicate free motes.
For each free mote the localisation error will depend on the choice of three anchors for the RIPS trilateration. We examine three strategies for making this choice of anchors: Case 1: Select three anchor motes randomly from the available neighbor anchor list. Case 2: Select three anchor motes from the available neighbor anchor list such that the area of the triangle formed by those anchors is maximum. Case 3: Select three anchor motes from the available neighbor anchor list such that the sum of symmetric differences for the neighbor sets of each pair of anchors is maximum. The last two cases are designed to reduce localisation error based on the principle that RIPS measurements are most accurate when the free mote lies within or close to the triangle formed by the three anchors, as discussed in the previous section. The maximum symmetric difference method (Case 3) is included because of its importance in [13], where RIPS measurements are taken with respect to pseudo-anchors whose positions are unknown, as well as anchors. It will tend to be most effective under the assumption that the free motes are uniformly distributed.
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The performance of the localisation algorithm for each run is measured in terms of number of rounds required, number of unlocalised motes left, and average root-meansquare (RMS) location error. Results are averaged over 100 runs. The statistical result comparison is shown in Table V. After each run, the free motes are assigned indices from 1 to 50 in order of the size of their RMS location error. The average RMS location error (over 500 runs) for free motes in order of increasing index is given in Figure 8. Case 1 2 3
No. of RIPS meas 127.81 111.38 105.91
Table V: Statistical Result Comparison No. of No. of No. of localised motes rounds failure Round 1 Round 2 Round 3 2.78 0.26 34.91 13.39 1.07 2.26 0.29 38.53 10.56 0.35 2.25 0.28 38.60 10.49 0.35
VII. ACKNOWLEDGMENTS The authors thank Drs Barbara La Scala, Mark Moreland and Jia Weng for taking part in many discussions on this topic. This work was supported in part by the Defense Advanced Research Projects Agency of the US Department of Defense and was monitored by the Office of Naval Research under Contract No. N00014-04-C-0437.
2
10
localisation error can be significant. In this work, we extend the RIPS trilateration algorithm by resolving solution ambiguity. Furthermore, the localisation error of the RIPS trilateration algorithm is characterized via the inverse Jacobian determinant. These analytic results can help us to better understand the localisation error performance of the RIPS trilateration algorithm when noise is present in RIPS measurements and provides a guide for choosing anchor motes in such a way as to minimis localisation error. A simulated example for localising 50 free motes demonstrates the effectiveness of our approach.
Case 1 Case 2 Case 3
1
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metre
R EFERENCES 0
10
−1
10
5
Fig. 8.
10
15
20 25 30 35 Number of Free Nodes
40
45
50
Average RMS localisation error comparison.
Remarks: 1) The number of RIPS measurements indicates the overall extent of solution ambiguity; it would be 100 if all free motes could be unambiguously localised from two RIPS measurements. From Table V, it appears to be correlated with the number of rounds required. The selection of anchors based on the criterion of maximum symmetric difference (Case 3) appears to be the most effective way of minimising the number of measurements required. 2) Figure 8 shows comparative average RMS location errors for the three strategies. The horizontal axis lists the indices of the free nodes (ordered by increasing location error), and the vertical axis the extent of the error (on an exponential scale). Here Case 2 has best the RMS location error performance, marginally better than that of Case 3, while randomly selecting anchors as in Case 1 resulted in substantially larger RMS location errors. 3) Our result indicates that the average propagation error grows approximately exponentially with the number of propagation rounds. VI. C ONCLUSION In this paper, the hyperbolic positioning using RIPS trilateration method for 2 dimensional mote networks is studied. Given three anchor motes, the RIPS trilateration algorithm presented in [12] may localise a free mote within a clique using a pair of RIPS measurements. However, there are regions where solution ambiguity exists and there are positions where
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