Robust Range-Free Localization in Wireless Sensor Networks

Abstract

In wireless sensor networks, sensors should have some mechanisms to learn their locations since sensed data without associated location information may be meaningless. While many sensor localization algorithms have been proposed, security issues in sensor localization are usually not addressed in their original design. Secure sensor localization is very challenging due to limited computation and energy resources in sensors. It is highly desirable that a localization scheme is robust and is able to detect malicious attacks without using complex cryptographic operations. In this paper, we present and analyze detection methods purely based on geometric constraints in sensor networks. Our detection methods can protect the localization algorithm from malicious attacks by detecting and eliminating the negative impact of fake information.

Appendix

1.1 A. Proof of Theorem 1

Based on Remarks 1, 2, and 3, the probability that an anchor node, A, is detected as a cheater is lowered bounded by a function of l, the distance between the real location and the fake location of A. The function is a monotonically increasing function of l due to the fact that the size of the shadowed area in Fig. 2a, b, c and d monotonically increases with the increase of l.

Case 1—Fig.  2 a: The size of the shadowed area can be calculated with Eq. 2, which is clearly a monotonically increasing function of l because

$$\frac{\partial f_1}{\partial l} = \sqrt{4R^2-l^2}> 0$$

when l < 2R. When l ≥ 2R, the shadowed area reached its maximum, πR ².

Case 2—Fig.  2 b: The size of the shadowed area can be calculated with Eq. 4, which is,

$$\begin{array}{rll}f_2\! &=&\! 2*\left(\int_{-R}^{d-2R} \sqrt{R^2\!-\!x^2} dx\right. \\ &&{\kern18pt} \left.+\int_{d-2R}^{\frac{d^2\!-\!3R^2}{2d}} \sqrt{R^2\!-\!x^2}-\sqrt{4R^2\!-\!(x-d)^2} \right) dx \\ &=&\!2{\kern-1pt}*{\kern-2.3pt}\left(\int_{-R}^{\frac{d^2\!-\!3R^2}{2d}}\!\! \sqrt{R^2\!-\!x^2} dx \!-\!\!\int_{d-2R}^{\frac{d^2\!-\!3R^2}{2d}}\!\! \sqrt{4R^2\!-\!(x\!-\!d)^2} dx \right) \\ &=&\!2{\kern-1.5pt} *{\kern-1.5pt} \left(\int_{\!-\!R}^{\frac{d^2\!-\!3R^2}{2d}}\!\! \sqrt{R^2\!-\!x^2} dx \!-\!\int_{-2R}^{\frac{-d^2\!-\!3R^2}{2d}}\!\! \sqrt{4R^2\!-\!x^2} dx \right)\end{array}$$

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where d is the distance between S and A′ and R < d ≤ 2R. For given ρ and θ, f ₂ is a monotonically increasing function of l because (1) d is a monotonically increasing function of l and (2) f ₂ is a monotonically increasing function of d. The first reason is obvious since $d=\sqrt{(l-\sqrt{\rho}cos\theta)^2+\rho sin^2\theta}$. The second reason is true since

$$\begin{array}{rll}\frac{\partial f_2}{\partial d} &=& 2\sqrt{-d^4+10d^2R^2\!-\!9R^4} \\ &&\times\left(\left(\frac{d^2\!-\!3R^2}{2d}\right)'\!+\!\left(\frac{d^2\!+\!3R^2}{2d}\right)'\right) \\ &=&2\sqrt{-d^4\!+\!10d^2R^2\!-\!9R^4} \end{array}$$

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It is easy to see that $\frac{\partial f_2}{\partial d}>0$ when R < d ≤ 2R, because

$$-d^4+10d^2R^2-9R^4 = (9R^2-d^2)(d^2-R^2)>0.$$

Case 3—Fig.  2 c (l ≤ R): The shadowed area is simply the union of the shadowed areas of Case 1 and Case 2, and as such the size of the shadowed area in Case 3 monotonically increases with the increase of l.

Case 4—Fig.  2 d (R < l ≤ 2R): The shadowed area is the union of the shadowed areas of Case 1 and Case 2 minus the stroked area. Although the stroked area can be calculated numerically in Matlab, its exact mathematical expression is hard to get. Nevertheless, that the size of the shadowed area in Fig. 2d monotonically increases with the increase of l is equivalent to that the size of the shadowed area in Fig. 14 monotonically decreases with the increase of l.

Figure 14

Analysis on cooperative detection

The shadowed area in Fig. 14 is the union of the starred area and the stroked area. It is easy to see that the size of the starred area monotonically decreases with the increase of l because its size equals $\pi R^2-f_1$, where f ₁ is the monotonically increasing function of l in Case 1.

Next we prove that the stroked area in Fig. 14 also monotonically decreases with the increase of l. As shown in Fig. 14, we use line SA′ as the x-axis and S as the origin. The y-axis is selected upward if S sits above line AA′ and downward otherwise. We denote points B, D, E, and A as (B _x ,B _y ), (D _x ,D _y ), (E _x ,E _y ), and (A _x , A _y ) respectively. Note that B _x ,B _y ,D _x , and D _y all are a function of l, but E _x , E _y , A _x , and A _y do not change with l. If we use d to denote the distance between S and A′ as before, d is also a function of l. The size of the stroked area, denoted as f ₄, can be calculated as:

$$\begin{array}{rll}f_4 &=& \int_{B_x}^{D_x}\sqrt{4R^2-(x\!-\!d)^2}dx \\ &&- \int_{B_x}^{D_x}\left(A_y+\sqrt{R^2-(x\!-\!A_x)^2}\right) dx \\ && + \int_{D_x}^{E_x}\!\!\sqrt{R^2\!-\!x^2}dx \!-\! \int_{D_x}^{E_x}\!\left(A_y\!+\!\sqrt{R^2\!-\!(x\!-\!A_x)^2}dx \right. \\ &=& \int_{B_x-d}^{D_x-d}\sqrt{4R^2\!-\!x^2}dx \!+\! \int_{D_x}^{E_x}\sqrt{R^2-x^2}dx \\ &&- \int_{B_x}^{E_x}\left(A_y\!+\!\sqrt{R^2-(x-A_x)^2}\right) dx \end{array}$$

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Therefore,

$$\begin{array}{rll} \frac{\partial f_4}{\partial l} &=& D_y(D_x-d)' - B_y(B_x-d)' - D_yD_x'+ B_yB_x' \\ &=& B_yd'-D_yd' \\ &=& (B_y-D_y)\frac{l-\sqrt{\rho}\cos\theta}{\sqrt{l^2+\rho-2l\sqrt{\rho}\cos\theta}} \end{array}$$

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Note that in Case 4, R < l ≤ 2R and $\sqrt{\rho}\cos\theta \leq R$ (S must be able to hear A). Thus $\frac{l-\sqrt{\rho}\cos\theta}{\sqrt{l^2+\rho-2l\sqrt{\rho}\cos\theta}}>0$. Due to the way that we set the direction of y-axis, B _y  < D _y . Therefore $\frac{\partial f_4}{\partial l} <0$, which means, the size of the stroked area in Fig. 14 monotonically decreases with the increase of l. □