SENSOR NETWORK LOCALIZATION VIA ... - Semantic Scholar
SENSOR NETWORK LOCALIZATION VIA BOUNDARY PROJECTIONS Jing Wang and Phillip A. Regalia Dept. Electrical Engineering and Computer Science Catholic University of America Washington, DC 20064 {98wang,regalia}@cua.edu
ABSTRACT This work considers localization of nodes in a sensor network using distance measurements. Recent methods favor projection onto convex sets (POCS), since it overcomes the multimodality problem that plagues least-squares formulations. Previous efforts in this direction require either that the sensor be located in the convex hull of the anchor nodes, or that complicated hyperbolic geometric calculations be employed. Here we propose a new algorithm which projects onto the boundary of convex sets, and features a computationally simple update procedure. Both cyclic and random projection schedules are considered, and initial convergence proofs are offered. 1. INTRODUCTION Self localization of nodes in wireless sensor networks has numerous potential benefits in a wide array of application areas, and has thus met with significant research efforts in recent years. The basic problem refers to deducing the positions of specific sensors in a network where some of the nodes have known positions, refered to as anchor nodes. The remaining sensor nodes locate themselves based on the relative position information from the anchor nodes, which may be deduced from, e.g., signal strengths or ping times. Some prior works [1] have estimated the sensor locations with linear algorithms, which tend not to lead to high accuracy at convergence. Nonlinear estimation approaches have thus been pursued [2], [3], in terms of general estimatiom theory and the Cramer-Rao bound. An interesting development is identified in [4], giving a fast cyclic search method with an incremental subgradient. This work offers a trade-off between the accuracy required and the amount of communications, although susceptibility to local minima is not addressed. In [5], a new method is proposed that describes the geometric relations of distance measurements as the equality constraints and aims to formulate an objective function with respect to the error in the inaccurate measurements. This work was supported by the National Science Foundation under grant CCF 0728521.
A localization algorithm using projection onto convex sets (POCS) was recently proposed in [6], and several developments have been made based on this approach [7]–[10]. In [6], the authors formulate the problem as a convex feasibility problem, and show convergence provided the sensor location lies in the convex hull of the anchor nodes. A means to circumvent this limitation is developed in [7], requiring hyperbolic trigonometry in the calculations. In section 3, we propose a novel algorithm along with simulations, with some intial convergence results in section 4. Initialization aspects and performance with noisy measurements are also addressed. 2. BACKGROUND We consider a sensor network with N anchor nodes, whose positions are denoted as ci ∈ IR2 , i = 1, . . . , N . Let x ∈ IR2 be the unknown coordinate of a non-anchor node, and let di be the measured distance between the node and the i-th anchor node. We assume exact distance measurements in the initial formulation, and absorb the influence of noisy distance measurements subsequently. In [6], a cyclic projection algorithm onto convex sets is applied to formulate an appoximation to the nonlinear least squares objective function. The i-th convex set Ci is the disk of points within a distance di of the anchor node ci : Ci = y ∈ IR2 : ky − ci k ≤ di using the Euclidean norm k · k. It is shown that the cyclic projection algorithm converges to the correct location estimate provided the point x is in the convex hull of the anchor nodes, as given by ( ) N N X X 2 H = y ∈ IR : y = αi ci , αi ≥ 0, αi = 1 (1) i=1
i=1
For sensor nodes lying outside this convex hull, by contrast, the algorithm need not converge to a useful position estimate. To overcome this shortcoming, [7] proposed a hyperbolic POCS algorithm, which successfully converges (in the absence of noise) to the correct position estimate for sensors
lying outside the convex hull H. A performance penalty is observed, however, using the hyperbolic POCS algorithm on sensors inside the convex hull, thus motivating a hybrid algorithm [7] that aims to leverage the strengths of the hyperbolic algorithm and its circular predecessor [6]. We illustrate next section another variant by projecting instead onto the boundaries of specific convex sets.
5 4 3 2
d3 c3 +
1
d1
0
3. NEW ALGORITHM: PROJECTION ONTO BOUNDARY SETS With di denoting the distance between the true location x and the anchor node ci , the boundary set ∂Ci is given as ∂Ci = y ∈ IR2 : ky − ci k = di (2)
−2
using the Euclidean norm k · k. If the true distances di are used, then clearly the point x lies at the intersection of the sets {∂Ci }: N \ x∈ ∂Ci (3)
−5 −5
d2
+ c1
−1
+ c2
x
−3 −4 0
5
Fig. 1. Anchor nodes indicated by “+”, with true location as “◦”.
i=1
Moreover, save for degenerate situations (e.g., anchor nodes all on a line), the true position is the unique intersecting point of the boundary sets. Now, the projection of an arbitrary point y ∈ IR2 onto the boundary set ∂Ci , denoted Pi (y), is readily calculated as Pi (y) = ci + di
y − ci ky − ci k
and satisfies Pi (y) = arg min ky − zk . z ∈ ∂Ci
(4)
This motivates a cyclic projection algorithm that seeks the intersection of the boundaries, given as follows:
• Initialization: x0 is arbitrary, and set j1 = 1.
1. Initialization: x0 is arbitrary.
• Iterations: for k = 1, 2, . . . ,
2. Iterative step: for k = 1, 2, . . . , xk = Pk mod N (xk−1 )
iterates xk become trapped in a triangle. Such limit cycles have been observed for arbitrary numbers of anchor nodes for specific alignments (such as the equilateral triangle illustrated here, when using three anchor nodes). A slight repositioning of the anchor nodes may be observed to break the limit cycle, indicating that, in a probabilistic sense, the likelihood of encountering one may be small. The example of figure 2, however, indicates that this likelihood remains nonzero. We have observed in simulations that if we alter the sequence of projection steps, limit cycles can be broken. This suggests that a more effective way to eliminate such limit cycles is to use a random (rather than cyclic) projection algorithm, summarized as follows:
(5)
in which k mod N cycles through the set {1, 2, . . . , N }. Figure 1 shows three anchor nodes (indicated as “+”) along with the circles which collect points ∂Ci at the true distance di ; the true location (“◦”) lies at the intersection ∩i ∂Ci . Figure 2 shows a typical run, illustrating convergence of the position estimates to the true location, even though the true location lies outside the convex hull H, thus overcoming the shortcoming of [6] while remaining computationally simpler than the hyperbolic variant from [7]. An observed weakness of the proposed algorithm, however, is its susceptibility to limit cycles in some cases. Figure 2 likewise illustrates one such case, in which successive
xk jk+1
= Pjk (xk−1 ) =
rand {1, 2, . . . , N }\jk
in which rand(·) returns an element chosen at random from its argument list. Figure 3 shows the same situation as figure 2, using now a random projection sequence with many different initializations; each run is observed to convergence to the true location. 4. CONVERGENCE ASPECTS For coherence we consider first the case of two anchor nodes, and then extend the results to an arbitrary number of anchor nodes.
5
5
4
4
3
3
Limit cycle
2
2
1
1
0
0
−1
−1
−2
−2
−3
−3
−4 −5 −5
Correct convergence −4
−3
−2
−1
−4
0
1
2
3
4
5
−5 −5
0
5
Fig. 2. Illustrating convergence versus a limit cycle, depending on the initialization.
Fig. 3. Convergence for various initial points, using the random projection schedule. Each converges to the true location.
Figure 4 illustrates that, with two anchor nodes, the intersection ∂Ci ∩ ∂Cj gives two solutions: the true point x and its “conjugate” x∗ij . The half-plane of points closer to x than to + its conjugate x∗ij is denoted Rij , and is bounded by the line passing through ci and cj . We introduce a discrepancy function
2. The local convergence rate is linear: for Dij (ˆ xk−1 ) sufficiently small,
Dij (ˆ x) =
kˆ x − xk kˆ x − x∗ij k
ˆ = x. It is easy which is nonnegative and vanishes only at x + to check that Dij (ˆ x) is bounded in Rij : Dij (ˆ x) < 1
+ ˆ ∈ Rij x
⇔
Note that, with only two anchor nodes, the random projection algorithm effectively reduces to the periodic projection algorithm. Lemma 1 Projections between circles centered at ci and cj + ˆ k ∈ Rij preserve half-plane membership: x if and only if + k−1 ˆ x ∈ Rij . A proof is offered in the Appendix.
Dij (ˆ xk ) = Dij (ˆ xk−1 ) cos θij where θij is the subspace angle between x − ci and x − cj . ˆ 0 is in the same half-plane as x For the proof, suppose x + + 0 ˆ ∈ Rij ); by Lemma 1 all iterates x ˆ k remain in Rij (i.e., x as well. Consider the unit-norm vector u=
ˆ k−1 − cj x , kˆ xk−1 − cj k
which relates to successive iterates as ˆ k−1 x ˆk x
= cj + r u = cj + d j u
with r = kˆ xk−1 − cj k. The line cj + ρ u, parametrized by a ˆ k−1 and x ˆk nonnegative scalar ρ, then contains the iterates x k at ρ = r and ρ = dj , respectively. The inequality Dij (ˆ x )< Dij (ˆ xk−1 ) will follow upon showing that dj = arg min Dij (cj + ρ u). ρ
Theorem 1 Consider the update equations that alternately project onto circles centered at ci and cj . ˆ 0 for which Dij (ˆ 1. Given any initialization x x0 ) < 1, convergence is monotonic: Dij (ˆ xk ) < Dij (ˆ xk−1 ),
ˆ k 6= x ˆ k−1 ; for all x
To this end, a straightforward calculation shows that ∂ 2 hu, x − x∗ i [Dij (cj + ρ u)]2 = (ρ2 − d2j ) ∂ρ kcj + ρ u − x∗ k4 whose sole zero in ρ > 0 lies at ρ = dj . As D(ˆ x) ≤ 1 in + Rij , we have in particular that D(cj + ρ u) ≤ 1 for all ρ ≥ 0.
The “global” convergence (i.e., from points too far from x to allow the circles to be approximated by tangent lines) is more delicate when using more than two anchor nodes. One approach to studying convergence is to average, in some sense, the various pairwise discrepancyfunctions Dij (ˆ x). With N anchor nodes, we have M = N2 = N (N −1)/2 such discrepancy functions, and different average choices include:
!ij R− ij x∗ij
ci yx
F1 (ˆ x)
x cj
R+ ij
F2 (ˆ x)
1 X Dij (ˆ x) (arithmetic mean) M i,j Y 1/M = Dij (ˆ x) (geometric mean)
=
i,j
F3 (ˆ x) + − Fig. 4. Illustrating half-planes Rij and Rij .
M [1/D x)] ij (ˆ i,j
P
(harmonic mean)
It is straightforward to check that F1 (ˆ x) ≥ F2 (ˆ x) ≥ F3 (ˆ x),
xˆ k−1
x θ
Y F3 (ˆ x) = F2 (ˆ x)
Fig. 5. Zooming in near the intersection of two circles. Indeed, as the maximum Dij (cj + ρ u) = 1 is attained only at ρ = 0 and in the limit as ρ → ∞, the critical point ρ = dj must be a minimum. ˆ k+1 , lying Interchanging the role of i and j shows that x k+1 on the circle centered at ci , fulfills Dij (ˆ x ) < Dij (ˆ xk ) as well, to establish the first part of the theorem. For the second part, we may zoom in near the intersection x; locally the two circles may be replaced by their tangent lines, as in figure 5. As each projection gives a displacement normal to a tangent line, we see by inspection that kˆ xk − xk = kˆ xk−1 − xk cos θij .
ˆ. for all x
Indeed, F1 (ˆ x) ≥ F2 (ˆ x) follows from the arithmetic-mean– geometic-mean inequality, as does F2 (ˆ x) ≥ F3 (ˆ x) upon observing that
xˆ k+1 xˆ k
=
(6)
Moreover, points sufficiently close to x are nearly equidistant from x∗ij : kˆ xk − x∗ij k ≈ kˆ xk−1 − x∗ij k. This approximation becomes increasingly accurate as x is approached, so that dividing both sides of the equality (6) by kˆ xk − x∗ij k gives the second part of the theorem. When projecting between more than two circles, the local convergence properties follow similarly to theorem 1: if the ˆ k+1 goes to the circle centered at c` , then by next projection x the same geometric consideration of figure 5 we have kˆ xk+1 − xk = kˆ xk − xk cos θj` . Thus the convergence rate is asymptotically limited by the poorest angular separation maxi,j cos θij of anchor nodes, as seen from the true location.
1/M
kˆ x−
x∗ij k
i,j
1 X kˆ x − x∗ij k M i,j
We have observed in simulations that, when using more than two anchor nodes, monotonic convergence kicks in once Fi (ˆ x) < 1, although a proof in the general case is still lacking. 5. INITIALIZATION ˆ 0 will generally lead to A well-chosen initialization point x surer convergence to the correct position estimate. Two cases may be considered: • Case 1: The true location x lies in the convex hull of the anchor nodes (x ∈ H). In this case, convergence to the correct position estimate follows by arguments similar to those of [8]. • Case 2: The true location x lies outside the convex hull of the anchor node (x 6∈ H). For case 2, an effective initialization strategy consists in first sorting the available distance measurements to select the ˆ 0 in a smallest; let this index be denoted n. Then choose x vicinity of cn . Let i and j be the indices of two of the larger distances, and project back and forth between circles centered at ci and cj for a few iterations, and then switch to the random projection strategy of section 3. In essence, when the
true location lies outside H, this strategy gives a high proba+ bility of the initial position estimates lying in Rij , as desired in view of Theorem 1. The subsequent switch to the random projection scheme then relies on the local convergence of the algorithm. This same initialization strategy also appears effective for case 1, as the true position x is already in H anyway.
4 3
6. NOISY MEASUREMENTS 2
With random noise contaminating the distance measurements {di }, the position estimates become random variables. Let
1
di = di + δ i
0
in which di is the true distance to the i-th anchor, and δi is the measurement error. We assume for simplicity that δi is a zero-mean random variable, and we let σi2 = E[δi2 ], where the expectation is over the uncertainty mechanisms in the distance measurements. Now, the position estimate at iteration k depends on only one past position estimate, giving a Markov structure:
−1 −2 −3 −4
−3
−2
−1
0
1
2
3
4
Fig. 6. Position estimates using noisy distance measurements.
ˆ 0 ) = Pr(ˆ Pr(ˆ xk |ˆ xk−1 , . . . , x xk |ˆ xk−1 ). As such, if at step k we are projecting onto the circle centered at cj , only dj (and no other distance measurement) will affect ˆk. x Convergence (in probability) in this case takes the form of the position estimates approaching a neighborhood of x and then “rattling around” (to borrow terminology from [11]). Since the measurement errors are modeled as zero-mean random variables, the mean distance estimate may be shown unbiased; due to the Markov property, the variance is bounded by the worst-case variance in the distance measurements: E[(ˆ xk − x)2 ] ≤ max σi2 , i
for large k.
Figure 6 shows a simulation run in which the distance measurements are contaminated with errors, with σi2 = 0.04. The position estimates are seen to converge towards the true location, and then “rattle around”. By averaging the position estimates over many such runs, the mean position was observed to be unbiased, with a variance given by σi2 .
1 0.5 0 −0.5 −1 −1.5 −2 2 1
7. CONCLUDING REMARKS The modified projection algorithm proposed here, using projections to the boundaries of sets, is observed to perform well even when the true position lies outside the convex hull of the anchor nodes. The algorithm also features a simple update law and behaves benignly in the presence of measurement errors. Although we have focused on the two-dimensional case, the algorithm extends readily to three-dimensional data as well; figure 7 gives an example run.
2
0
1 0
−1
−1 −2
−2 −3
Fig. 7. Illustrating convergence for three-dimensional data.
8. APPENDIX: PROOF OF LEMMA 1
9. REFERENCES
To verify Lemma 1, consider the line `ij passing through anchor nodes ci and cj , parametrized by a scalar α: n o `ij = y : y = ci + α (cj − ci ) for some α .
[1] A. H. Sayed, A. Tarighat, and N. Khajehnouri, “Network based wireless location,” IEEE Signal Processing Magazine, pp. 24–40, July 2005.
The closest point to x on this line, denoted yx , is obtained from the optimization problem
[2] N. Patwari, J.N. Ash, S. Kyperountas, A.O. Hero, R.L. Moses, and N.S. Correal, “Locating the nodes: cooperative localization in wireless sensor networks,” IEEE Signal Processing Magazine, pp. 54–69, July 2005.
yx
=
arg min kx − yk2 y ∈ `ij
= ci +
(ci − cj ) (ci − cj )T (x − ci ) kci − cj k2
The difference x − yx gives a line segment normal to `ij (cf. Fig. 4): (ci − cj ) (ci − cj )T x − yx = I − (x − ci ) kci − cj k2 {z } | ∆
= Pij Now let z be any other point in the plane. It will lie on the + same side of `ij as x (i.e., z ∈ Rij ) iff the subspace angle φ between the vectors z − yx and x − yx is less than 90 degrees (or π/2 radians). From the subspace angle formula cos φ =
(z − yx )T (x − yx ) kz − yx k kx − yx k
clearly |φ| < π/2 if and only if (z − yx )T (x − yx ) > 0. By a straightforward calculation, (z − yx )T (x − yx ) T (ci − cj ) (ci − cj )T = z − ci − (x − ci ) Pij (x − ci ) kci − cj k2 = (z − ci )T Pij (x − ci ) + Rij .
This final form is thus positive if and only if z ∈ We now observe that Pij (ci − cj ) = 0 and therefore Pij (x − ci ) = Pij (x − ci ) + Pij (ci − cj ) = Pij (x − cj ) {z } | 0 Similarly, (z − ci )T Pij = (z − cj )T Pij . As such, from the update formula ˆ k − cj = d j x
xk−1 − cj kxk−1 − cj k
we have (ˆ xk − cj )T Pij (x − cj ) dj = (ˆ xk−1 − cj )T Pij (x − cj ) k−1 kˆ x − cj k dj = (ˆ xk−1 − ci )T Pij (x − ci ) kˆ xk−1 − cj k ˆ k−1 As the factor dj /kˆ xk−1 −cj k is positive, this shows that x k ˆ lie in the same half-plane with respect to `ij . and x
[3] N. Patwari, A. O. Hero, M. Perkins, N. S. Correal, and R. J. O’Dea, “Relative location estimation in wireless sensor networks,” IEEE Trans. Signal Processing, vol. 51, no. 8, pp. 2137–2148, Aug. 2003. [4] M.G. Rabbat and R.D. Nowak, “Decentralized source localization and tracking,” in Proc. Int. Conf. Acoustics Speech, Signal Processing, Montreal, Canada, May 2004, vol. III, pp. 921–924. [5] M. Cao, B. D. O. Anderson, and A. S. Morse, “Localization with imprecise distance information in sensor networks,” in Conf. Decision and Control, Dec. 2005, pp. 1829–1834. [6] A. O. Hero and D. Blatt, “Sensor network source localization via projection onto convex sets (POCS),” in Proc. Int. Conf. Acoustics Speech, Signal Processing, Philadelphia, Mar. 2005, vol. III, pp. 689–692. [7] M. Rydstrom, E. G. Strom, and A. Svensson, “Robust sensor network positioning based on projection onto circular and hyperbolic convex sets (POCS),” in Proc. SPAWC, Cannes, France, July 2006. [8] D. Blatt and A. O. Hero, “Energy-based sensor network source localization via projection onto convex sets,” IEEE Signal Processing Magazine, vol. 54, no. 9, pp. 3614–3619, 2006. [9] C. Meng, Z. Ding, and S. Dasgupta, “A semidefinite programming approach to source localization in wireless sensor networks,” IEEE Signal Processing Magazine, 2008, accepted. [10] B. Fidan, S. Dasgupta, and B. D. O. Anderson, “Guaranteeing practical convergence in algorithms for sensor and source localization,” IEEE Trans. Signal Processing, vol. 56, 2008, to appear. [11] W. A. Sethares and C. R. Johnson, Jr., “A comparison of two quantized state adaptive algorithms,” IEEE Trans. Acoustics, Speech and Signal Processing, vol. 37, no. 1, pp. 138–143, Jan. 1989.
ABSTRACT This work considers localization of nodes in a sensor network using distance measurements. Recent methods favor projection onto convex sets (POCS), since it overcomes the multimodality problem that plagues least-squares formulations. Previous efforts in this direction require either that the sensor be located in the convex hull of the anchor nodes, or that complicated hyperbolic geometric calculations be employed. Here we propose a new algorithm which projects onto the boundary of convex sets, and features a computationally simple update procedure. Both cyclic and random projection schedules are considered, and initial convergence proofs are offered. 1. INTRODUCTION Self localization of nodes in wireless sensor networks has numerous potential benefits in a wide array of application areas, and has thus met with significant research efforts in recent years. The basic problem refers to deducing the positions of specific sensors in a network where some of the nodes have known positions, refered to as anchor nodes. The remaining sensor nodes locate themselves based on the relative position information from the anchor nodes, which may be deduced from, e.g., signal strengths or ping times. Some prior works [1] have estimated the sensor locations with linear algorithms, which tend not to lead to high accuracy at convergence. Nonlinear estimation approaches have thus been pursued [2], [3], in terms of general estimatiom theory and the Cramer-Rao bound. An interesting development is identified in [4], giving a fast cyclic search method with an incremental subgradient. This work offers a trade-off between the accuracy required and the amount of communications, although susceptibility to local minima is not addressed. In [5], a new method is proposed that describes the geometric relations of distance measurements as the equality constraints and aims to formulate an objective function with respect to the error in the inaccurate measurements. This work was supported by the National Science Foundation under grant CCF 0728521.
A localization algorithm using projection onto convex sets (POCS) was recently proposed in [6], and several developments have been made based on this approach [7]–[10]. In [6], the authors formulate the problem as a convex feasibility problem, and show convergence provided the sensor location lies in the convex hull of the anchor nodes. A means to circumvent this limitation is developed in [7], requiring hyperbolic trigonometry in the calculations. In section 3, we propose a novel algorithm along with simulations, with some intial convergence results in section 4. Initialization aspects and performance with noisy measurements are also addressed. 2. BACKGROUND We consider a sensor network with N anchor nodes, whose positions are denoted as ci ∈ IR2 , i = 1, . . . , N . Let x ∈ IR2 be the unknown coordinate of a non-anchor node, and let di be the measured distance between the node and the i-th anchor node. We assume exact distance measurements in the initial formulation, and absorb the influence of noisy distance measurements subsequently. In [6], a cyclic projection algorithm onto convex sets is applied to formulate an appoximation to the nonlinear least squares objective function. The i-th convex set Ci is the disk of points within a distance di of the anchor node ci : Ci = y ∈ IR2 : ky − ci k ≤ di using the Euclidean norm k · k. It is shown that the cyclic projection algorithm converges to the correct location estimate provided the point x is in the convex hull of the anchor nodes, as given by ( ) N N X X 2 H = y ∈ IR : y = αi ci , αi ≥ 0, αi = 1 (1) i=1
i=1
For sensor nodes lying outside this convex hull, by contrast, the algorithm need not converge to a useful position estimate. To overcome this shortcoming, [7] proposed a hyperbolic POCS algorithm, which successfully converges (in the absence of noise) to the correct position estimate for sensors
lying outside the convex hull H. A performance penalty is observed, however, using the hyperbolic POCS algorithm on sensors inside the convex hull, thus motivating a hybrid algorithm [7] that aims to leverage the strengths of the hyperbolic algorithm and its circular predecessor [6]. We illustrate next section another variant by projecting instead onto the boundaries of specific convex sets.
5 4 3 2
d3 c3 +
1
d1
0
3. NEW ALGORITHM: PROJECTION ONTO BOUNDARY SETS With di denoting the distance between the true location x and the anchor node ci , the boundary set ∂Ci is given as ∂Ci = y ∈ IR2 : ky − ci k = di (2)
−2
using the Euclidean norm k · k. If the true distances di are used, then clearly the point x lies at the intersection of the sets {∂Ci }: N \ x∈ ∂Ci (3)
−5 −5
d2
+ c1
−1
+ c2
x
−3 −4 0
5
Fig. 1. Anchor nodes indicated by “+”, with true location as “◦”.
i=1
Moreover, save for degenerate situations (e.g., anchor nodes all on a line), the true position is the unique intersecting point of the boundary sets. Now, the projection of an arbitrary point y ∈ IR2 onto the boundary set ∂Ci , denoted Pi (y), is readily calculated as Pi (y) = ci + di
y − ci ky − ci k
and satisfies Pi (y) = arg min ky − zk . z ∈ ∂Ci
(4)
This motivates a cyclic projection algorithm that seeks the intersection of the boundaries, given as follows:
• Initialization: x0 is arbitrary, and set j1 = 1.
1. Initialization: x0 is arbitrary.
• Iterations: for k = 1, 2, . . . ,
2. Iterative step: for k = 1, 2, . . . , xk = Pk mod N (xk−1 )
iterates xk become trapped in a triangle. Such limit cycles have been observed for arbitrary numbers of anchor nodes for specific alignments (such as the equilateral triangle illustrated here, when using three anchor nodes). A slight repositioning of the anchor nodes may be observed to break the limit cycle, indicating that, in a probabilistic sense, the likelihood of encountering one may be small. The example of figure 2, however, indicates that this likelihood remains nonzero. We have observed in simulations that if we alter the sequence of projection steps, limit cycles can be broken. This suggests that a more effective way to eliminate such limit cycles is to use a random (rather than cyclic) projection algorithm, summarized as follows:
(5)
in which k mod N cycles through the set {1, 2, . . . , N }. Figure 1 shows three anchor nodes (indicated as “+”) along with the circles which collect points ∂Ci at the true distance di ; the true location (“◦”) lies at the intersection ∩i ∂Ci . Figure 2 shows a typical run, illustrating convergence of the position estimates to the true location, even though the true location lies outside the convex hull H, thus overcoming the shortcoming of [6] while remaining computationally simpler than the hyperbolic variant from [7]. An observed weakness of the proposed algorithm, however, is its susceptibility to limit cycles in some cases. Figure 2 likewise illustrates one such case, in which successive
xk jk+1
= Pjk (xk−1 ) =
rand {1, 2, . . . , N }\jk
in which rand(·) returns an element chosen at random from its argument list. Figure 3 shows the same situation as figure 2, using now a random projection sequence with many different initializations; each run is observed to convergence to the true location. 4. CONVERGENCE ASPECTS For coherence we consider first the case of two anchor nodes, and then extend the results to an arbitrary number of anchor nodes.
5
5
4
4
3
3
Limit cycle
2
2
1
1
0
0
−1
−1
−2
−2
−3
−3
−4 −5 −5
Correct convergence −4
−3
−2
−1
−4
0
1
2
3
4
5
−5 −5
0
5
Fig. 2. Illustrating convergence versus a limit cycle, depending on the initialization.
Fig. 3. Convergence for various initial points, using the random projection schedule. Each converges to the true location.
Figure 4 illustrates that, with two anchor nodes, the intersection ∂Ci ∩ ∂Cj gives two solutions: the true point x and its “conjugate” x∗ij . The half-plane of points closer to x than to + its conjugate x∗ij is denoted Rij , and is bounded by the line passing through ci and cj . We introduce a discrepancy function
2. The local convergence rate is linear: for Dij (ˆ xk−1 ) sufficiently small,
Dij (ˆ x) =
kˆ x − xk kˆ x − x∗ij k
ˆ = x. It is easy which is nonnegative and vanishes only at x + to check that Dij (ˆ x) is bounded in Rij : Dij (ˆ x) < 1
+ ˆ ∈ Rij x
⇔
Note that, with only two anchor nodes, the random projection algorithm effectively reduces to the periodic projection algorithm. Lemma 1 Projections between circles centered at ci and cj + ˆ k ∈ Rij preserve half-plane membership: x if and only if + k−1 ˆ x ∈ Rij . A proof is offered in the Appendix.
Dij (ˆ xk ) = Dij (ˆ xk−1 ) cos θij where θij is the subspace angle between x − ci and x − cj . ˆ 0 is in the same half-plane as x For the proof, suppose x + + 0 ˆ ∈ Rij ); by Lemma 1 all iterates x ˆ k remain in Rij (i.e., x as well. Consider the unit-norm vector u=
ˆ k−1 − cj x , kˆ xk−1 − cj k
which relates to successive iterates as ˆ k−1 x ˆk x
= cj + r u = cj + d j u
with r = kˆ xk−1 − cj k. The line cj + ρ u, parametrized by a ˆ k−1 and x ˆk nonnegative scalar ρ, then contains the iterates x k at ρ = r and ρ = dj , respectively. The inequality Dij (ˆ x )< Dij (ˆ xk−1 ) will follow upon showing that dj = arg min Dij (cj + ρ u). ρ
Theorem 1 Consider the update equations that alternately project onto circles centered at ci and cj . ˆ 0 for which Dij (ˆ 1. Given any initialization x x0 ) < 1, convergence is monotonic: Dij (ˆ xk ) < Dij (ˆ xk−1 ),
ˆ k 6= x ˆ k−1 ; for all x
To this end, a straightforward calculation shows that ∂ 2 hu, x − x∗ i [Dij (cj + ρ u)]2 = (ρ2 − d2j ) ∂ρ kcj + ρ u − x∗ k4 whose sole zero in ρ > 0 lies at ρ = dj . As D(ˆ x) ≤ 1 in + Rij , we have in particular that D(cj + ρ u) ≤ 1 for all ρ ≥ 0.
The “global” convergence (i.e., from points too far from x to allow the circles to be approximated by tangent lines) is more delicate when using more than two anchor nodes. One approach to studying convergence is to average, in some sense, the various pairwise discrepancyfunctions Dij (ˆ x). With N anchor nodes, we have M = N2 = N (N −1)/2 such discrepancy functions, and different average choices include:
!ij R− ij x∗ij
ci yx
F1 (ˆ x)
x cj
R+ ij
F2 (ˆ x)
1 X Dij (ˆ x) (arithmetic mean) M i,j Y 1/M = Dij (ˆ x) (geometric mean)
=
i,j
F3 (ˆ x) + − Fig. 4. Illustrating half-planes Rij and Rij .
M [1/D x)] ij (ˆ i,j
P
(harmonic mean)
It is straightforward to check that F1 (ˆ x) ≥ F2 (ˆ x) ≥ F3 (ˆ x),
xˆ k−1
x θ
Y F3 (ˆ x) = F2 (ˆ x)
Fig. 5. Zooming in near the intersection of two circles. Indeed, as the maximum Dij (cj + ρ u) = 1 is attained only at ρ = 0 and in the limit as ρ → ∞, the critical point ρ = dj must be a minimum. ˆ k+1 , lying Interchanging the role of i and j shows that x k+1 on the circle centered at ci , fulfills Dij (ˆ x ) < Dij (ˆ xk ) as well, to establish the first part of the theorem. For the second part, we may zoom in near the intersection x; locally the two circles may be replaced by their tangent lines, as in figure 5. As each projection gives a displacement normal to a tangent line, we see by inspection that kˆ xk − xk = kˆ xk−1 − xk cos θij .
ˆ. for all x
Indeed, F1 (ˆ x) ≥ F2 (ˆ x) follows from the arithmetic-mean– geometic-mean inequality, as does F2 (ˆ x) ≥ F3 (ˆ x) upon observing that
xˆ k+1 xˆ k
=
(6)
Moreover, points sufficiently close to x are nearly equidistant from x∗ij : kˆ xk − x∗ij k ≈ kˆ xk−1 − x∗ij k. This approximation becomes increasingly accurate as x is approached, so that dividing both sides of the equality (6) by kˆ xk − x∗ij k gives the second part of the theorem. When projecting between more than two circles, the local convergence properties follow similarly to theorem 1: if the ˆ k+1 goes to the circle centered at c` , then by next projection x the same geometric consideration of figure 5 we have kˆ xk+1 − xk = kˆ xk − xk cos θj` . Thus the convergence rate is asymptotically limited by the poorest angular separation maxi,j cos θij of anchor nodes, as seen from the true location.
1/M
kˆ x−
x∗ij k
i,j
1 X kˆ x − x∗ij k M i,j
We have observed in simulations that, when using more than two anchor nodes, monotonic convergence kicks in once Fi (ˆ x) < 1, although a proof in the general case is still lacking. 5. INITIALIZATION ˆ 0 will generally lead to A well-chosen initialization point x surer convergence to the correct position estimate. Two cases may be considered: • Case 1: The true location x lies in the convex hull of the anchor nodes (x ∈ H). In this case, convergence to the correct position estimate follows by arguments similar to those of [8]. • Case 2: The true location x lies outside the convex hull of the anchor node (x 6∈ H). For case 2, an effective initialization strategy consists in first sorting the available distance measurements to select the ˆ 0 in a smallest; let this index be denoted n. Then choose x vicinity of cn . Let i and j be the indices of two of the larger distances, and project back and forth between circles centered at ci and cj for a few iterations, and then switch to the random projection strategy of section 3. In essence, when the
true location lies outside H, this strategy gives a high proba+ bility of the initial position estimates lying in Rij , as desired in view of Theorem 1. The subsequent switch to the random projection scheme then relies on the local convergence of the algorithm. This same initialization strategy also appears effective for case 1, as the true position x is already in H anyway.
4 3
6. NOISY MEASUREMENTS 2
With random noise contaminating the distance measurements {di }, the position estimates become random variables. Let
1
di = di + δ i
0
in which di is the true distance to the i-th anchor, and δi is the measurement error. We assume for simplicity that δi is a zero-mean random variable, and we let σi2 = E[δi2 ], where the expectation is over the uncertainty mechanisms in the distance measurements. Now, the position estimate at iteration k depends on only one past position estimate, giving a Markov structure:
−1 −2 −3 −4
−3
−2
−1
0
1
2
3
4
Fig. 6. Position estimates using noisy distance measurements.
ˆ 0 ) = Pr(ˆ Pr(ˆ xk |ˆ xk−1 , . . . , x xk |ˆ xk−1 ). As such, if at step k we are projecting onto the circle centered at cj , only dj (and no other distance measurement) will affect ˆk. x Convergence (in probability) in this case takes the form of the position estimates approaching a neighborhood of x and then “rattling around” (to borrow terminology from [11]). Since the measurement errors are modeled as zero-mean random variables, the mean distance estimate may be shown unbiased; due to the Markov property, the variance is bounded by the worst-case variance in the distance measurements: E[(ˆ xk − x)2 ] ≤ max σi2 , i
for large k.
Figure 6 shows a simulation run in which the distance measurements are contaminated with errors, with σi2 = 0.04. The position estimates are seen to converge towards the true location, and then “rattle around”. By averaging the position estimates over many such runs, the mean position was observed to be unbiased, with a variance given by σi2 .
1 0.5 0 −0.5 −1 −1.5 −2 2 1
7. CONCLUDING REMARKS The modified projection algorithm proposed here, using projections to the boundaries of sets, is observed to perform well even when the true position lies outside the convex hull of the anchor nodes. The algorithm also features a simple update law and behaves benignly in the presence of measurement errors. Although we have focused on the two-dimensional case, the algorithm extends readily to three-dimensional data as well; figure 7 gives an example run.
2
0
1 0
−1
−1 −2
−2 −3
Fig. 7. Illustrating convergence for three-dimensional data.
8. APPENDIX: PROOF OF LEMMA 1
9. REFERENCES
To verify Lemma 1, consider the line `ij passing through anchor nodes ci and cj , parametrized by a scalar α: n o `ij = y : y = ci + α (cj − ci ) for some α .
[1] A. H. Sayed, A. Tarighat, and N. Khajehnouri, “Network based wireless location,” IEEE Signal Processing Magazine, pp. 24–40, July 2005.
The closest point to x on this line, denoted yx , is obtained from the optimization problem
[2] N. Patwari, J.N. Ash, S. Kyperountas, A.O. Hero, R.L. Moses, and N.S. Correal, “Locating the nodes: cooperative localization in wireless sensor networks,” IEEE Signal Processing Magazine, pp. 54–69, July 2005.
yx
=
arg min kx − yk2 y ∈ `ij
= ci +
(ci − cj ) (ci − cj )T (x − ci ) kci − cj k2
The difference x − yx gives a line segment normal to `ij (cf. Fig. 4): (ci − cj ) (ci − cj )T x − yx = I − (x − ci ) kci − cj k2 {z } | ∆
= Pij Now let z be any other point in the plane. It will lie on the + same side of `ij as x (i.e., z ∈ Rij ) iff the subspace angle φ between the vectors z − yx and x − yx is less than 90 degrees (or π/2 radians). From the subspace angle formula cos φ =
(z − yx )T (x − yx ) kz − yx k kx − yx k
clearly |φ| < π/2 if and only if (z − yx )T (x − yx ) > 0. By a straightforward calculation, (z − yx )T (x − yx ) T (ci − cj ) (ci − cj )T = z − ci − (x − ci ) Pij (x − ci ) kci − cj k2 = (z − ci )T Pij (x − ci ) + Rij .
This final form is thus positive if and only if z ∈ We now observe that Pij (ci − cj ) = 0 and therefore Pij (x − ci ) = Pij (x − ci ) + Pij (ci − cj ) = Pij (x − cj ) {z } | 0 Similarly, (z − ci )T Pij = (z − cj )T Pij . As such, from the update formula ˆ k − cj = d j x
xk−1 − cj kxk−1 − cj k
we have (ˆ xk − cj )T Pij (x − cj ) dj = (ˆ xk−1 − cj )T Pij (x − cj ) k−1 kˆ x − cj k dj = (ˆ xk−1 − ci )T Pij (x − ci ) kˆ xk−1 − cj k ˆ k−1 As the factor dj /kˆ xk−1 −cj k is positive, this shows that x k ˆ lie in the same half-plane with respect to `ij . and x
[3] N. Patwari, A. O. Hero, M. Perkins, N. S. Correal, and R. J. O’Dea, “Relative location estimation in wireless sensor networks,” IEEE Trans. Signal Processing, vol. 51, no. 8, pp. 2137–2148, Aug. 2003. [4] M.G. Rabbat and R.D. Nowak, “Decentralized source localization and tracking,” in Proc. Int. Conf. Acoustics Speech, Signal Processing, Montreal, Canada, May 2004, vol. III, pp. 921–924. [5] M. Cao, B. D. O. Anderson, and A. S. Morse, “Localization with imprecise distance information in sensor networks,” in Conf. Decision and Control, Dec. 2005, pp. 1829–1834. [6] A. O. Hero and D. Blatt, “Sensor network source localization via projection onto convex sets (POCS),” in Proc. Int. Conf. Acoustics Speech, Signal Processing, Philadelphia, Mar. 2005, vol. III, pp. 689–692. [7] M. Rydstrom, E. G. Strom, and A. Svensson, “Robust sensor network positioning based on projection onto circular and hyperbolic convex sets (POCS),” in Proc. SPAWC, Cannes, France, July 2006. [8] D. Blatt and A. O. Hero, “Energy-based sensor network source localization via projection onto convex sets,” IEEE Signal Processing Magazine, vol. 54, no. 9, pp. 3614–3619, 2006. [9] C. Meng, Z. Ding, and S. Dasgupta, “A semidefinite programming approach to source localization in wireless sensor networks,” IEEE Signal Processing Magazine, 2008, accepted. [10] B. Fidan, S. Dasgupta, and B. D. O. Anderson, “Guaranteeing practical convergence in algorithms for sensor and source localization,” IEEE Trans. Signal Processing, vol. 56, 2008, to appear. [11] W. A. Sethares and C. R. Johnson, Jr., “A comparison of two quantized state adaptive algorithms,” IEEE Trans. Acoustics, Speech and Signal Processing, vol. 37, no. 1, pp. 138–143, Jan. 1989.
