## 5 Numerical Examples
### Simulation Verification  
In the simulation experiment, the bias estimation is based on the Monte Carlo method, five positioning system configurations are tested, which differ in the anchor-tag positions and redundancy. The coordinates of the anchor and tag are shown in Figure 2. The distance \(d\) between the tag and the anchor in Configurations 1, 2 and 3 are all the same, which is calculated to be \(240\sqrt{6}\) cm. The simulated measurement error obeys the normal distribution which is an independent Gaussian variable with zero mean and equal variance. The standard deviation of the first four sets and the sixth set is 5 cm. Both the fourth and fifth set of tests are based on Configuration 4. The fifth set of test is carried out by simulating large noise with the standard deviation of 18 cm, representing the variety of the observation variance. The observation model equation is simulated 10,000 times for detecting the bias in nonlinear least-squares estimator. The frequency histograms of parameter estimator bias are given in Figure 3. The average PDOP, HDOP and VDOP are shown in Figure 4.  
In Test 1 based on Configuration 1, there are adequate redundant observations, and the anchors are uniformly distributed in the three-dimensional (3D) space. The three types of DOP values in this group are the smallest. The average bias of coordinate components x, y and z are close to zero. This indicates that the least-squares estimator is approximately unbiased, with the reason that the amount of nonlinearity is orthogonal to the column space of the design matrix in this configuration. It makes us know that Configuration 1 is suitable for the traditional indoor positioning, and it also can achieve satisfactory accuracy. Figure 3: _Cont._  
Figure 2: The tests and positioning configurations. In the next five sets of tests, there is only one redundant observation with anchors distributed on a planar ground. This makes the number of parameters to be estimated relatively more for the particular observation model, and the nonlinear strength increases with the decreases of DOF. As similar to Test 1, the coordinate components x and y are also unbiased in Tests 2, 3, 4 and 5, but the coordinate component z is biased. It indicates that the corresponding component does not satisfy the orthogonal condition with the zero bias because of the ill-conditioned positioning configuration. Moreover, when the distance between the tag and the anchor and the ranging measurement noise are fixed in the Test 2 and 3 based on Configuration 2 and 3, it makes known that their amount of nonlinearity is equal. The average VDOP in Test 3 is smaller than that in the Test 2, and it has smaller bias in the coordinate component z. In this case, it indicates that the better the system geometry, the smaller the bias in the nonlinear least-squares estimator. In addition, the Tests 3, 4 and 5 have the identical geometry. The difference is that Test 4 based on Configuration 4 is scaled down compared with Configuration 3 and Test 5 has larger ranging measurement noise than that of the Test 4. The bias in Test 4 is greater than that of the Test 3, and the bias in Test 5 is largest among these three sets of tests. The results verify that the bias is inversely proportional to the ranging distance and directly proportional to the ranging error. Figure 4: The average position dilution of precision (PDOP), horizontal dilution of precision (HDOP) and vertical dilution of precision (VDOP) for each set of tests. Figure 3: The frequency histograms of parameter estimator bias. In Test 6, the tag has the same z coordinate as Test 5, but it is located outside the square enclosed by the anchors. The coordinate components x is unbiased, but both the coordinate component y and z are biased. The three types of DOP values in this group are larger than those in the previous five sets of tests. As we can see, Test 6 based on Configuration 5 has the worst system geometry.