4. Computer Simulations
4.1 Comparison of RMSE with GN method
4.1.2 Comparison of Pre-Linear Methods and Joint-GN Method
In this Section, convergence and effect on number of mobiles are considered. In Figure 4.5, the convergence rate is considered.
(a)
(b)
Figure 4.5 RMSE vs. convergence rate for pre-linear methods and joint GN method in (a) 2-D case (b) 3-D case
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In Figure 4.5, six mobiles are considered. We can see that joint method and GN method need two iterations to converge both in 2-D and 3-D case, while sequential and parallel are about 5.
In the following figure, divided method for Jacobi and Gauss-Seidel methods
mentioned in Section 3.1.2 are compared to parallel and sequential methods in 3-D case. Note that parallel and sequential methods use the additional mapping information between target and auxiliary mobiles while Jacobi and Gauss-Seidel method regard the other mobiles as virtual BSs with known position. The number of mobiles is set to 3 and randomly located in
6 6 6 geometry.
(a)
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(b)
Figure 4.6 RMSE vs. convergence rate for (a) parallel method and Jacobi method (b) sequential and Gauss-Seidel method in 3-D case
Figure 4.6 compares proposed parallel and sequential methods with Jacobi and Gauss-Seidel method respectively. We can see that parallel method is better than Jacobi method in Fig 4.6 (a) and sequential method is better than Gauss-Seidel method in Fig 4.6 (b).
This simulation shows that the transformed mapping functions in parallel and sequential methods are useful to improve the accuracy of positions. However, the extra computation cost is needed for parallel and sequential methods.
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(a)
(b)
Figure 4.7 RMSE vs. the number of mobiles for pre-linear methods and GN method in (a) 2-D (b) 3-D case
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In (3.3), we expect that the cooperative terms can improve the localization accuracy because of the information exchange between mobiles. Number of mobile from 2 to 6 is set in our simulation. From Figure 4.7, the RMSE improves when the number of mobiles increases.
The worst RMSE occurs in number of mobiles equals to one, which means uncooperative localization. However, the quality of the cooperative information is important, or the localization accuracy may be degraded, which will be discussed in Section 4.2.
4.2 The Reliability of Cooperative Localization
We take two factors of cooperative localization into consideration in this Section. The discussion of noise variance of cooperative measurements is given in Section 4.2.1. The reliability of positions of mobiles is discussed in Section 4.2.2.
4.2.1 Measurement between Mobiles
We explore the cooperative localization when mobiles are in noisy channel. The measured distance is affected by measured noise, i.e., dij Aij nij.The noise variance of
measurement between mobile i and mobile j ij2 in (3.1) affects the accuracy of localization.
The simulation setup is as follows
The noise variances of uncooperative measurements are based on (4.2) with set to 0.3. we The cooperative measurements are set with the factor equals to 0.5 (reliable) and 0.1 (unreliable), respectively. We expect that the reliable cooperative measurements can improve the accuracy of position. The simulation results compare CDF (Cumulative Density Function) which are given in Figure 4.8 to Figure 4.10.
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(a)
(b)
Figure 4.8 Comparison of CDF of location error for joint pre-linear method in (a) 2-D (b) 3-D case
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(a)
(b)
Figure 4.9 Comparison of CDF of location error for parallel pre-linear method in (a) 2-D (b) 3-D case
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(a)
(b)
Figure 4.10 Comparison of CDF of location error for sequential pre-linear method in (a) 2-D (b) 3-D case
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We can see that the reliable measurements (red line) upgrade the performance no matter in joint pre-linear method in Figure 4.8, parallel pre-linear method in Figure 4.9 and
sequential pre-linear method in Figure 4.10. In addition, the positions of mobiles also affect the localization. The details will be given in Section 4.2.2.
4.2.2 Positions of Mobiles
In (3.22), (3.35) and (3.43), Fk, yk include the position of mobiles in k-th iteration and it also effect the estimation of mobiles. The position of mobiles plays a role that the measurements are unreliable if two mobiles are far away between each other. Futher, the poor geometry location cause the uncertain virtual BSs and lead to degradation of localization. In Figure 4.11(a), we discuss the influnce on parallel and sequential methods. Mobiles are placed at [1 1], [9 1], [9 9], [1 9] (distance between mobiles is far and in poor geometry) and [3 3], [7 3], [7 7], [3 7] respectively. In 3-D case, the mobiles are placed at [1 1 1], [5 1 5],[5 5 1], [1 5 5] and [2 2 2], [4 2 4], [4 4 2], [2 4 4]. The simulation results are shown in the following figures.
(a)
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(b)
Figure 4.11 Influence of positions of mobiles on different noise variance in (a) 2-D (b) 3-D case
In parallel and sequential pre-linear method, the positions of uncertain virtual BSs are used in (3.29) and (3.37) respectively, and it can affect the performance. The RMSE of parallel and sequential pre-linear methods are shown in Figure 4.11. We can see that the performance is better if the positions are set closer and near in the middle of geometry. The influence of uncertainty of mobile will be modified based on mobile selection and weighting compensation in Section 4.3 and Section 4.4.
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4.3 Effect on Target Mobile
In this Section, the target mobile selection schemes are considered. We want to modify the performance with the probable target mobile. In 2-D case, mobile 1 is selected at [2 2] and others are randomly selected in 2 2( ) m in the middle of geometry. Note that we select mobile1 as target mobile in general case, and pick a mobile from all candidates as target mobile in target mobile selection scheme. In 3-D case, the procedures are identical with 2-D case. We select mobile 1 at [2 2 2], and others are in 3.5 3.5 3.5( ) m in the middle of geometry.
(a)
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(b)
Figure 4.12 Influence of target mobile selection in (a)2-D (b) 3-D case
From Figure 4.12, we can see that the performance of mobile selection (solid line) is improved compared with the genaral case (dashed line), especially in parallel (blue) and sequential (green) pre-linear methods, while the joint method improves slightly. We infer that in joint method, the mapping function comes from the cooperation of all the auxiliary mobile, and the influence on target mobile is less than parallel and sequential methods..
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4.4 Weighting Compensation
In this Section, the effect on weighting are discussed. Section 4.4.1 shows the poor initial value degrades the performance. The improvement on weighting of noise variance is shown in Section 4.4.2. In Section 4.4.3, weighting compensation on virtual BSs and target mobile upgrades the RMSE of localization.
4.4.1 Initial Value
It is known that a good initial value is important when the Taylor-series expansion is used to linearize the nonlinear function, or the Taylor higher order truncation error can not be neglected. Our pre-linear methods are based on GN method which applies Taylor-series expansion to non-linear range function. The following figure shows the effect when the poor initial value is used.
Figure 4.13 RMSE vs. iteration for a poor initial value
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In Figure 4.13, the number of mobiles is equal to 2 and randomly located within 10 10 4( ) m in 12 12 6( ) m cube with four BSs at the corner like Figure 4.2. We give an initial value at [6 6 3] and [5 5 4]. Note that it is a special case of pre-linear methods in M=2 that there is only one way to generate the mapping function. We can see that in GN method (red line) and pre-linear method (blue line), the algorithm with variance of noise (solid line) can not improve the RMSE. The term nts ij, in (3.6) dominate the total error, i.e., nts ij, nij. In Section 4.4.2, a good initial value is used so that the noise dominates the total error.
4.4.2 Effect on Weighting of Noise Variance
Here, the statistics of noise variance ij is considered in the following figures
(a)
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(b)
Figure 4.14 The effect on weighting of noise variance in (a) 2-D (b) 3-D case
We can see that RMSE is improved with the weighting of noise variance obviously in three pre-linear methods. Different from 4.4.1, the uncooperative LLS estimator offers a not bad initial value in this section. From Figure 4.14, we further know that the noise variance dominates the localization rather than the Taylor truncation error, i.e., nij nts ij, .
4.4.3 Weighting Compensation
The compensation of uncertain position of virtual BSs and the target mobile in three pre-linear methods in (3.23), (3.36) and (3.44) are considered. The simulations are given in following figure.
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(a)
(b)
Figure 4.15 Weighting compensation on joint method in (a) 2-D (b) 3-D case
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(a)
(b)
Figure 4.16 Weighting compensation on parallel method in (a) 2-D (b) 3-D case
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(a)
(b)
Figure 4.17 Weighting compensation on sequential method in (a) 2-D (b) 3-D case
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From Figure 4.15 to 4.17, we can see that the compensation improves the RMSE. Note that there is an improvement for convergence in 3-D case. By iteration, the accurate positions of mobiles are obtained, the effect on location errors become smaller. The extra computation are 9(M1) for joint method, 9(M 1)2 for parallel and sequential methods. In parallel and sequential methods , it is still less than joint GN method, we conclude that the
compensation is useful in our proposed pre-linear methods.
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Chapter 5
Conclusions and Future Works
In cooperative localization, three pre-linear methods based on distance measurement have been proposed to reduce the dimensions of unknown parameters in this thesis. Using the concept of linear mapping from target mobile to auxiliary mobiles, we expect that the
complexity can be reduced. Compared with joint GN method, the total computation cost in each iteration saves roughly M3 multiplication in parallel and sequential methods when the number of mobiles is increased. Simulation results validate that the RMSE of proposed methods are still comparable with joint GN method, but the total cost is reduced greatly.
Simulations also show the influence on reliability of cooperative measurement; because the additional weighting compensation for uncertain position of mobiles not only improves the location accuracy, the convergence is also improved. Moreover, we can see that target mobile selection scheme enhances the RMSE. In a word, the contribution of this thesis is that we propose three low complexity pre-linear methods with good accuracy.
In fact, there exist lots of mapping relation that the mapping can be generated based on different localization requirement. Here, the linear mapping we proposed is based on the requirement for low complexity. Besides, the proposed methods can be also applied in NLOS environment. In the end, the theoretical analysis of proposed methods are another attractive issue to verify the performance.
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