Indoor localization based on trilateration method uses at least three receivers for an accurate localization in 2-D. We performed indoor localization in 2-D using only two receivers, combining algebraic equations for signal strengths into one quadratic equation with transmitter position as unknown and using a specific receiver placement at the two adjacent corners of the room. This receiver arrangement assures unique coordinates of the transmitter position inside a room, rejecting automatically the other solution which appears outside the room volume. The accuracy of the method is numerically tested in a room with dimensions of 9.7m×4.7m×3m and shows a mean reconstruction error of 3.4cm.

Introduction

In this paper, we introduce a new source localization scheme based on quadratic equation approach and using only two receivers placed at the corners of the ceiling of the room. Such a recommendation for sensor arrangement could easily be followed in every room very quickly. Furthermore, this arrangement assures that one of the roots of quadratic equation is outside the room volume, provided that the required transmitter position is unique. We briefly describe the method in Sect. 2, show the simulation results in Sect. 3, and derive the conclusions in Sect. 4.

Methods

Considering two receivers, their placement, and a forward model, a quadratic equation will be formulated and solved. The localization task will be exemplified using transceivers commonly used in wireless sensor networks (WSNs). The target of localization or tracking scenario can be a person or a mobile robot equipped with a transceiver.

Analytical solution

The received power Pr at the position of the receiver Ri is related to the transmitted power Pt at the position rj according to the forward model (Eq. ). It has been assumed in this equation that there is a direct path between the receiver and the transmitter and no signal interference occurrence :

Pr=Pt(λ4πr)2GrGt, where r=|Ri-rj| represents the distance between the transmitter and the receiver, λ is the wavelength of propagation, and Gr and Gt are the gains of the receiver and the transmitter antenna, respectively.

Considering that all parameters in Eq. () are constants except the distance, we can write this equation in the following form: Pr=c1r2, where c1=PtGrGtλ2/(4π)2. Using Eq. (), the received power Pr1 of the first receiver at the position R1 and the received power Pr2 of the second receiver at the position R2 can be written as follows:

Two receivers are placed at 3m in height and presented by blue points. The path of the moving transmitter (18 positions) is presented by red points, joined by a red solid line.

Pr1=c1r12=c1(xs1-xt)2+(ys1-yt)2+(zs1-zt)2,Pr2=c1r22=c1(xs2-xt)2+(ys2-yt)2+(zs2-zt)2, where xs1, ys1, and zs1 are the Cartesian coordinates of receiver S1; xs2, ys2, and zs2 are the coordinates of receiver S2; and xt, yt, and zt are unknown coordinates of the transmitter. By subtracting and reordering (see Appendix ), we get axt2+bxt+c=0. By solving Eq. (), we obtain two solutions: xt1 and xt2. Then, by substituting them in Eq. (), yt1 and yt2 are concluded. So, (xt1, yt1) is the first solution of transmitter localization and (xt2, yt2) is the second one. For a specific receiver placement, there will be only one correct solution for all possible positions of the transmitter.

Simulation I: reconstruction of the path when receivers S1 and S2 are located along the y axis at the corners of the ceiling of the room. Estimated paths denoted by × and represent the accepted and rejected path of the transmitter, respectively.

Simulation study Description of the use case

The indoor localization problem will be investigated by observing a room with dimensions of 9.7m×4.7m×3m. Two receivers will be placed at the ceiling of the room for logical and practical reasons (this has the purpose of not having a problem with obstacles like furniture or an existence of other objects). Two simulations will be investigated: simulation I is noise-free. Simulation II will be studied under the effect of additive white Gaussian noise (AWGN). For simulation I, we present an irregular source path consisting of 18 discrete positions at 1m in height and with a step of approximately 0.5m (see Fig. ). For simulation II, the plane where the transmitter can move (at 1m in height) will be discretized in a grid of 40×40 points. That means the resolution in x direction is about 0.24m and in y direction about 0.12m. For simulation I, the receivers are placed along y axis at the corners of the ceiling with coordinates S1 (0,0,3m) and S2 (0,4.7,3m). In simulation II, the same receiver placement will be used as simulation I but with AWGN.

The radio transceiver AT86RF230 from  is considered. This transceiver works with a frequency of 2.4GHz (ZigBee/IEEE802.15.4 applications). The transmitter antenna is simulated by the maximum output power of the transmitter of Pt=+3dBm with a gain of Gt=-0.5dBi. The gain of the receiving antenna is set also to Gr=-0.5dBi.

Simulation II: the plane where the transmitter with a height of 1m could move is discretized in 40×40 grid points resulting in a resolution of about 0.24m in x direction and about 0.12m in y direction. Receivers are placed along the width of the ceiling and at the corners as in simulation I.

Result and discussion Noise-free simulation

Simulation I with noise-free data has the purpose of representing the effect of receiver placement on the localization using analytical solution. Solving the quadratic Eq. () gives two possible roots for x coordinate of the transmitter position, one inside the boundaries of the specified room and the other one outside the room. The root inside the room (xt1) is chosen automatically as the right one, whereas the other (xt2) has to be rejected. Then, the y coordinate of the transmitter position yt1 is calculated according to Eq. () given in Appendix . So, such a placement of receivers in combination with analytical solution method guarantees that the solution (xt1, yt1) will always be the right one for all possible transmitter positions in the room, and the second solution (xt2, yt2) will always be outside of the room.

Simulation II: distribution of reconstruction error in the whole room at 1m in height. The unit of the color bar is meters. Receivers are placed along the width of the ceiling and at the corners (left side).

Simulation II: distribution of reconstruction error in the whole room at 1m in height using the second solution of Eq. (). Two receivers are placed at the right ceiling's corners and along the width in Fig. . The unit of the color bar is meters.

Simulation with AWGN

In simulation II, we discretize the plane where the transmitter can move in a grid of 40×40 points, resulting in a resolution of about 0.24m in x direction and about 0.12m in y direction (Fig. ), respectively. Receivers are placed as in simulation I at the corners of the ceiling of the room (Fig. ). This has the purpose of assessing the robustness of our method by considering the reconstruction error in the whole room. So, the transmitter will be placed each time at one point of the grid and the reconstruction error will be calculated under the AWGN for signal-to-noise ratio (SNR=130dB). Figure  shows the reconstruction error at all the points of the grid. With such an SNR, the mean of reconstruction error for all the points is 0.0340m. The unit of the color bar is meters. The points far away from receiver positions have higher reconstruction error. If an accuracy like the one in the half of the room near the receivers S1 and S2 is desired in the other half, two more receivers can be used at the ceiling's corner (on the right side of Fig. ). So, the reconstruction error of the whole room will be flipped as seen in Fig. . That means, in this case, four receivers will be implemented, and each time just two of them having larger signal strength will be used to reconstruct the source position (two on the left or two on the right). However, in the case of four receivers, a recently introduced scanning method  can also be applied.

Conclusions

In this paper, we presented a new procedure of indoor localization based on the RSSI approach using only two receivers. Our specific arrangement of sensors at the corners of the ceiling of the room solves the ambiguity problem of the roots of the quadratic equation, putting one of the roots outside the boundaries of the room volume. This allows the other root to be a unique solution of the transmitter position. Realistic conditions of the proposed procedure are simulated by adding white Gaussian noise. In that case the reconstruction error averaged over 1600 possible transmitter positions in the room was 3.4cm, showing the high accuracy of the proposed method.

Data used for transceiver AT86RF230 from Amtel (2009) are available at http://www.atmel.com/images/doc5131.pdf.