Improved Extreme Learning Machine Based UWB Positioning for Mobile Robots with Signal Interference

: For the purpose of tackling ultra-wideband (UWB) indoor positioning with signal interference, a binary classiﬁer for signal interference discrimination and positioning errors compensation model combining genetic algorithm (GA) and extreme learning machine (ELM) are put forward. Based on the distances between four anchors and the target which are calculated with time of ﬂight (TOF) ranging technique, GA-ELM-based binary classiﬁer for judging the existence of signal interference, and GA-ELM-based positioning errors compensation model are built up to compensate for the result of the preliminary evaluated positioning model. Finally, the datasets collected in the actual scenario are used for veriﬁcation and analysis. The experimental results indicate that the root-mean-square error (RMSE) of positioning without signal interference is 14.5068 cm, which is reduced by 71.32% and 59.72% compared with those results free of compensation and optimization, respectively. Moreover, the RMSE of positioning with signal interference is 28.0861 cm, which is decreased by 64.38% and 70.16%, in comparison to their counterparts without compensation and optimization, respectively. Consequently, these calculated results of numerical examples lead to the conclusion that the proposed method displays its wide application, high precision and rapid convergence in improving the positioning accuracy for mobile robots.


Introduction
The indoor positioning for mobile robots can hardly be realized by global positioning system (GPS) technology used mainly for outdoor positioning because GPS accuracy can degrade significantly in indoor scenarios [1][2][3]. Currently, several common optional technologies for indoor positioning of mobile robots have been proposed, including visible light communication (VLC) [4], Wi-Fi [5], magnetic patterns [6,7], radio frequency identification technology (RFID) [8], which can be used as potential alternatives to GPS. More importantly, localization using Wi-Fi and magnetic field does not necessitate the configuration of extra hardware. Although VLC and RFID are highly accurate, they are susceptible to interference from ambient light and objects containing metal, respectively. Zigbee, as another positioning technology, has been used in the indoor positioning of mobile robots with a precision of up to 0.636 m (in the range of 4 m × 4 m) [9]. Wireless local area network (WLAN) [10,11], also, has been utilized as a candidate solution for indoor localization, yet the location accuracy it offers needs to be improved. Weighing and considering the relationship between cost and accuracy, the original intention of this paper is to achieve higher positioning accuracy at a lower cost, even in signals being in strong interference. Therefore, UWB positioning, a technology more suitable for indoor positioning, because of its advantages of high precision [12], low loss [13], collaborative localization [14,15], this paper, a UWB indoor high-precision positioning model based on GA-optimized ELM is proposed, as a component of which, a binary classifier based on GA-ELM is established to judge the existence of signal interference, and the positioning errors compensation is then performed by using the GA-ELM model to realize the precise positioning of the mobile robot target in complex indoor environments.

Preliminary Positioning and Assessment Model
The UWB positioning anchor receives data packets with less interference, so hundreds to thousands of positioning tags can work in the same area at the same time. Concerning the influence of the number of anchors, it is necessary to discuss first. Theoretically, the more the number of positioning base stations, the higher the positioning accuracy of the target to be measured, but there is a lower limit to the number of base stations. Since the equation expressing the distance between the target and the anchors contains quadratic terms, at least N p + 1 sets of equations are required to solve the N p -dimensional quadratic equations, i.e., the number of anchors N B and unknown quantities in the set of equations, and also the spatial dimension must satisfy the following equation: Equation (1) shows that indoor three-dimensional spatial positioning requires four or more base stations, so four anchors are chosen to be adopted in this paper.
In the actual scene, the TOF ranging principle, a technique that uses the time of flight of radio waves between two objects to calculate distance, is first used to acquire the distances between the four anchors that transmit the signals and the target that receives the signals. The schematic diagram of the experimental scene, as well as the operational principle of data collection, are shown in Figure 1. In particular, the position coordinates of the four anchors are fixed at different heights indoors, while the mobile robot with the target (i.e., Tag in Figure 1) moves in the indoor environment, and the four anchors and the target sense each other through UWB pulses. Each movement of the target can be recorded with the information occurring in these four-communication links consisting of four anchors and the target. Through the mutual communication between the four anchors and the target, the spatial distance between each pair is accordingly measured with TOF technology.
i.e., abnormal data fluctuations usually caused by time delay will cause indoor positioning to be inaccurate. Therefore, the precise positioning of UWB in signal interference (e.g., obscuration between anchors and target) is regarded as an urgent problem to be solved. In this paper, a UWB indoor high-precision positioning model based on GA-optimized ELM is proposed, as a component of which, a binary classifier based on GA-ELM is established to judge the existence of signal interference, and the positioning errors compensation is then performed by using the GA-ELM model to realize the precise positioning of the mobile robot target in complex indoor environments.

Preliminary Positioning and Assessment Model
The UWB positioning anchor receives data packets with less interference, so hundreds to thousands of positioning tags can work in the same area at the same time. Concerning the influence of the number of anchors, it is necessary to discuss first. Theoretically, the more the number of positioning base stations, the higher the positioning accuracy of the target to be measured, but there is a lower limit to the number of base stations. Since the equation expressing the distance between the target and the anchors contains quadratic terms, at least Np + 1 sets of equations are required to solve the Np-dimensional quadratic equations, i.e., the number of anchors NB and unknown quantities in the set of equations, and also the spatial dimension must satisfy the following equation: Equation (1) shows that indoor three-dimensional spatial positioning requires four or more base stations, so four anchors are chosen to be adopted in this paper.
In the actual scene, the TOF ranging principle, a technique that uses the time of flight of radio waves between two objects to calculate distance, is first used to acquire the distances between the four anchors that transmit the signals and the target that receives the signals. The schematic diagram of the experimental scene, as well as the operational principle of data collection, are shown in Figure 1. In particular, the position coordinates of the four anchors are fixed at different heights indoors, while the mobile robot with the target (i.e., Tag in Figure 1) moves in the indoor environment, and the four anchors and the target sense each other through UWB pulses. Each movement of the target can be recorded with the information occurring in these four-communication links consisting of four anchors and the target. Through the mutual communication between the four anchors and the target, the spatial distance between each pair is accordingly measured with TOF technology. X Anchor 1 (x 1 , y 1 , z 1 ) Next, according to the collected ranging dataset of the target, a preliminary positioning mathematical model is established to initially calculate the three-dimensional coordinates of the target by following these detailed steps: Step 1: The distance D A t (obtained by TOF ranging) between the target and anchor t is denoted as Step 2: Squaring both sides of Equation (2) lead to by the definition simultaneously: Step 3: The two adjacent terms of Equation (3) are subtracted to eliminate the quadratic term, hence, the target position Tag (x, y, z) in matrix form is: Step 4: The RMSE, which is used for evaluating the positioning model accuracy of the mobile robot, is calculated as: where RMSE j represents the root-mean-square error of the coordinates y i andŷ i are the true value and estimated value of the i-th target coordinate, respectively.
Step 5: The degree of the fitting test is used to verify the fitting degree of the prediction model to the sample observations value. The sum of squares of the mean difference between the original and predicted data is given by: As a satisfactory prediction model, the sample observation values should be well fitted, i.e., S c in S t should be as small as possible, so the degree of fitting can be defined as: The closer R 2 is to 1, the stronger the explanatory ability of the independent variable of the analytical formula to the dependent variable, which means that there is a better fitting effect of the equation and a higher fitting degree of the data.

Precise Positioning Model
In this section, the details of the UWB-based precise positioning model for mobile robots proposed in this paper are illustrated as follows. First, the general framework of the method proposed in this contribution is presented. Afterward, the ELM model is established, followed by the GA-optimized ELM model is constructed for improving the positioning accuracy of mobile robots.

Construction of Overall Model
No matter whether the signal interference exists, accurate positioning (three-dimensional coordinates) of the target can be obtained by the GA-optimized ELM model for positioning errors compensation and its specific process is shown in Figure 2.

Data collection
Data preprocessing  It is worthwhile mentioning, in the workflow of the overall method proposed in this paper, that the GA-optimized ELM model is adopted three times, in which signal discrimination, Z-axis errors compensation and ranging errors compensation, respectively, embodies the reusability of the same method regarding the GA-ELM model.

Construction of the ELM Model
Compared with traditional training methods (e.g., BP neural network), ELM has the advantages of higher learning speed and better generalization performance [49]. In addition, ELM can not only predict the positioning errors of the target but also judge whether signal interference exists during the positioning process of the mobile robot using UWB technology.
In this research, the ELM model is established to train the nonlinear relationship between ranging data (between targets and anchors) and category or error, as shown in Figure 3. The input variable x is actual ranging data, and the output variable t is error or category. The input layer, hidden layer and output layer has n, l and m neurons, respectively.  Suppose there are N groups of training samples for ranging, betw (A0, A1, A2, A3) and the target, is (xi, ti) ∈ R n × R m . Subsequently, the E expressed as: ( ) , , , , Where g(x) is the excitation function, wi = [wi1, wi2, …, win] T is the of the connection between the i-th node of the hidden layer and the in vi2, …, vim] T is the weight coefficient of the connection between the i-th n layer and the output layer, and bi is the threshold of hidden layer node In the case of l = N, Equation (9) is defined as: If the number of datasets is large, l < N is taken to reduce the com network model of ELM is calculated as: Suppose there are N groups of training samples for ranging, between four anchors (A 0 , A 1 , A 2 , A 3 ) and the target, is (x i , t i ) ∈ R n × R m . Subsequently, the ELM model can be expressed as: where g(x) is the excitation function, w i = [w i1 , w i2 , . . . , w in ] T is the weight coefficient of the connection between the i-th node of the hidden layer and the input layer, v i = [v i1 , v i2 , . . . , v im ] T is the weight coefficient of the connection between the i-th node of the hidden layer and the output layer, and b i is the threshold of hidden layer nodes.
In the case of l = N, Equation (9) is defined as: where N×m . If the number of datasets is large, l < N is taken to reduce the computation, and the network model of ELM is calculated as: where g(x) is infinitely differentiable, the weight coefficient w i and the threshold b i are all random numbers limited in the interval [−0.5, 0.5], and the training process of the ELM can be regarded as solving the least square solution of the linear system Hv = T, namely: where H + is the Moore-Penrose generalized inverse matrix of H. Eventually, substituting the results of w, v and b into Equation (9) yield the result of the output layer.

Construction of GA-Optimized ELM Model
Since the weight coefficients and thresholds of ELM are generated by random numbers, which leads to the prediction results of the model not being accurate enough, an optimization algorithm should be used to obtain the optimal value of weight coefficients and thresholds. Similarly, GA simulates the biological evolution process in nature rather than the traditional search method and employs the evolution method to perform a random optimization search on the target. In view of its global search optimal ability [50], this paper adopts GA to optimize the weight coefficients and thresholds of ELM, which effectively improves the positioning accuracy of the target and the correct rate of judging the presence and absence of signal interference.
According to the input and output parameters of ELM, the number of GA optimization parameters is determined, so as to set the code length of population individuals. In the following, the fitness function adopts a linearly ordered fitness distribution function, which is defined as: where ρ is the individual serial number, χ expresses the population size of the objective function value, η max indicates the selective pressure.
The selection operator uses random traversal sampling, the crossover operator uses single-point crossover, and the mutation operator generates the number of variant genes with a certain probability (i.e., change code 1 to 0 or change code 0 to 1).
The objective function of signal interference identification is to minimize the error rate, specifically: min arg where n t and N t represent the correct number and the total number of the test sets respectively, respectively. The objective function of the target coordinate errors compensation is to minimize RMSE, specifically: min arg where ε c and ε r mean the error prediction value and the true value of the test set, respectively. Combining the advantages of GA and ELM, GA-optimized ELM based signal interference discrimination and positioning errors compensation algorithms are established, whose workflow chart is shown in Figure 4.
In this model, the weight coefficients and thresholds of ELM are optimized by GA to derive the optimal objective function value, which is used for the discrimination of signal interference and the positioning errors compensation with and without signal interference in mobile robots' positioning.

Start
Determine the network structure of ELM

Experimental Results and Analysis
In this section, measured experiments and analyses are carried out to obtain the evaluation results of the model proposed in Section 3. Data preprocessing is firstly performed to gain reliable and available datasets about the position of the target on the mobile robot, and then 324 datasets with and without signal interference, i.e., corresponding to two scenarios of NLOS and LOS, respectively, are tested and analyzed on the signal discrimination and the positioning errors compensation model proposed in this work.
In this experiment, a high-precision positioning base station (four anchors) and target with model DWM1000 UWB transceivers from Decawave are adopted to verify GA-ELM experimentally, which uses coherent receiving technology to enable the positioning system to realize object positioning, as well as has the advantages of low power consumption and high positioning accuracy. The basic parameters of Decawave-DWM1000 are shown in Table 1.

Data Preprocessing
In the experimental scenario mentioned above, the UWB ranging data of the target at 324 different positions with and without signal interference were collected, respectively, that is, two datasets of data were collected for each position with and without signal interference, respectively, and the ranging of each group of the target at the identical position was collected multiple times to obtain the arithmetic mean for avoiding random errors of the measurement system.
Typically, in the process of data measurement, regardless of the presence and absence of signal interference, the measured value may be affected by the environment, resulting in deviations in the measurement data, i.e., abnormal values, which will interfere with subsequent model predictions. Correspondingly, according to the number of measurements, 3σ criterion is selected for performing data processing to obtain usable data, whose calculation process is as follows.
Step 1: The mean value of the ranging samples is calculated to mitigate systematic random errors as follows: where n represents the number of samples, d i is the distance measurement between the target and anchors in the i-th sample.
Step 2: Sample standard deviation S is obtained by: If the residual error of the measured value d m satisfies d m − d > 3σ, it is considered as an abnormal value that must be eliminated.
As mentioned above, the standard deviation comparison of each location ranging datasets without and with signal interference before and after processing is shown in Figures 5 and 6, respectively.
From the comparison of Figures 5 and 6, it can be directly seen that the standard deviation after data preprocessing has been reduced, especially for anchors A 0 and A 3 , indicating that the selected approach is correct and effective. The most remarkable result is that the standard deviation of the datasets is large in the case of signal interference, illustrating that there are numerous abnormal values in the measured datasets in the condition of signal interference, which can effectively verify the effectiveness and accuracy of the aforementioned positioning algorithm.  From the comparison of Figures 5 and 6, it can be directly seen that the standard deviation after data preprocessing has been reduced, especially for anchors A0 and A3, indicating that the selected approach is correct and effective. The most remarkable result is that the standard deviation of the datasets is large in the case of signal interference, illustrating that there are numerous abnormal values in the measured datasets in the condition of signal interference, which can effectively verify the effectiveness and accuracy of the aforementioned positioning algorithm.

Identification of Signal Interference
Firstly, these 518 samples of the training set for model training and 130 samples of the test set for model testing, were generated by the sequential shuffling method from 648 (i.e., 324 × 2) sets of measurement datasets. The GA-ELM model is then created and trained, and finally, the error rate of the training model is calculated and analyzed by comparison.  From the comparison of Figures 5 and 6, it can be directly seen that the standard deviation after data preprocessing has been reduced, especially for anchors A0 and A3, indicating that the selected approach is correct and effective. The most remarkable result is that the standard deviation of the datasets is large in the case of signal interference, illustrating that there are numerous abnormal values in the measured datasets in the condition of signal interference, which can effectively verify the effectiveness and accuracy of the aforementioned positioning algorithm.

Identification of Signal Interference
Firstly, these 518 samples of the training set for model training and 130 samples of the test set for model testing, were generated by the sequential shuffling method from 648 (i.e., 324 × 2) sets of measurement datasets. The GA-ELM model is then created and trained, and finally, the error rate of the training model is calculated and analyzed by comparison. Before processing After processing Figure 6. The standard deviation comparison of the samples before and after processing with signal interference.

Identification of Signal Interference
Firstly, these 518 samples of the training set for model training and 130 samples of the test set for model testing, were generated by the sequential shuffling method from 648 (i.e., 324 × 2) sets of measurement datasets. The GA-ELM model is then created and trained, and finally, the error rate of the training model is calculated and analyzed by comparison.
In the following operation, the minimum error rate of the binary classifier for judging the existence of UWB signal interference is calculated, based on GA to optimize ELM, in the activation function of Sigmoid is 3.2819%. The evolution curve is shown in Figure 7, and the comparison of the discriminant results of the test set is shown in Figure 8. In the following operation, the minimum error rate of the binary classifier for judging the existence of UWB signal interference is calculated, based on GA to optimize ELM, in the activation function of Sigmoid is 3.2819%. The evolution curve is shown in Figure 7 and the comparison of the discriminant results of the test set is shown in Figure 8.  In Figure 8, the numerical value on the ordinate represents the judgment of the presence and absence of signal interference. Specifically, 1 and 2 represent no signa interference and signal interference, respectively. It can be concluded that the correct rate of GA-ELM is 96.72%, which is 16.78% higher than that of ELM, manifesting that the optimization effect of GA-ELM is better than that of ELM.
More classically, support vector machine (SVM) and logistic regression (LR) can also be used as binary classifiers for signal interference discrimination. The LR model is a generalized linear regression analysis model, whose independent variables can be either continuous or categorical [51], while the SVM model is a class of generalized linear classifiers that perform binary classification of data according to supervised learning [52] In order to further explore the superiority of the GA-optimized ELM binary classifier, the LR classifier and SVM classifier are established to compare with the GA-ELM classifier proposed in this paper, whose results of the comparison are listed in Table 2. In the following operation, the minimum error rate of the binary classifier for judging the existence of UWB signal interference is calculated, based on GA to optimize ELM, in the activation function of Sigmoid is 3.2819%. The evolution curve is shown in Figure 7, and the comparison of the discriminant results of the test set is shown in Figure 8.  In Figure 8, the numerical value on the ordinate represents the judgment of the presence and absence of signal interference. Specifically, 1 and 2 represent no signal interference and signal interference, respectively. It can be concluded that the correct rate of GA-ELM is 96.72%, which is 16.78% higher than that of ELM, manifesting that the optimization effect of GA-ELM is better than that of ELM.
More classically, support vector machine (SVM) and logistic regression (LR) can also be used as binary classifiers for signal interference discrimination. The LR model is a generalized linear regression analysis model, whose independent variables can be either continuous or categorical [51], while the SVM model is a class of generalized linear classifiers that perform binary classification of data according to supervised learning [52]. In order to further explore the superiority of the GA-optimized ELM binary classifier, the LR classifier and SVM classifier are established to compare with the GA-ELM classifier proposed in this paper, whose results of the comparison are listed in Table 2. In Figure 8, the numerical value on the ordinate represents the judgment of the presence and absence of signal interference. Specifically, 1 and 2 represent no signal interference and signal interference, respectively. It can be concluded that the correct rate of GA-ELM is 96.72%, which is 16.78% higher than that of ELM, manifesting that the optimization effect of GA-ELM is better than that of ELM.
More classically, support vector machine (SVM) and logistic regression (LR) can also be used as binary classifiers for signal interference discrimination. The LR model is a generalized linear regression analysis model, whose independent variables can be either continuous or categorical [51], while the SVM model is a class of generalized linear classifiers that perform binary classification of data according to supervised learning [52]. In order to further explore the superiority of the GA-optimized ELM binary classifier, the LR classifier and SVM classifier are established to compare with the GA-ELM classifier proposed in this paper, whose results of the comparison are listed in Table 2. By comparing these classifiers' accuracy rate of the training set and testing set, it is found that the accuracy rate of the GA-ELM classifier is the highest among these classifiers, accordingly, the GA-ELM classifier is preferred to discriminate whether the UWB signals are interfered with.

Preliminary Positioning Results and Analysis
Based on the preliminary positioning model established in Section 2, a rigorous numerical comparison chart concerning the true value and the estimated value of the target coordinates with and without signal interference are shown in Figures 9 and 10, respectively. Additionally, the RMSE of the target coordinates with and without signal interference are tabulated in Table 3.  By comparing these classifiers' accuracy rate of the training set and testing set, it is found that the accuracy rate of the GA-ELM classifier is the highest among these classifiers, accordingly, the GA-ELM classifier is preferred to discriminate whether the UWB signals are interfered with.

Preliminary Positioning Results and Analysis
Based on the preliminary positioning model established in Section 2, a rigorous numerical comparison chart concerning the true value and the estimated value of the target coordinates with and without signal interference are shown in Figure 9 and Figure  10, respectively. Additionally, the RMSE of the target coordinates with and without signal interference are tabulated in Table 3.  In the comparison of Figures 9 and 10 and Table 3, it is vividly depicted that errors in X and Y directions are small, only reaching centimeter-level accuracy, while errors in Z-direction are large, which reached an accuracy of the decimeter level. The reason is that the quadratic term is eliminated in Equation (5), so the target coordinates are no longer on the spherical surface with the target as the center and the measurement distance as the radius, which can potentially cause larger ranging errors. Secondly, the condition number of the coefficient matrix is much greater than one, consequently, a slight disturbance will produce a considerable error. In this case, the coordinate difference z i -z i−1 is one order of magnitude smaller than x i -x i−1 and y i -y i−1 , which makes the errors in the Z-direction more sensitive. Therefore, it is indispensable that the Z-axis coordinates errors compensation is carried out to accomplish accurate positioning of the target.  In the comparison of Figure 9 and Figure 10 and Table 3, it is vividly depicted that errors in X and Y directions are small, only reaching centimeter-level accuracy, while errors in Z-direction are large, which reached an accuracy of the decimeter level. The reason is that the quadratic term is eliminated in Equation (5), so the target coordinates are no longer on the spherical surface with the target as the center and the measurement distance as the radius, which can potentially cause larger ranging errors. Secondly, the condition number of the coefficient matrix is much greater than one, consequently, a slight disturbance will produce a considerable error. In this case, the coordinate difference zi-zi−1 is one order of magnitude smaller than xi-xi−1 and yi-yi−1, which makes the errors in the Zdirection more sensitive. Therefore, it is indispensable that the Z-axis coordinates errors compensation is carried out to accomplish accurate positioning of the target.

Positioning Errors Compensation Model
The 324 groups of samples are divided into 250 training set samples and 74 test set samples. For the absence of signal interference, only the errors compensation towards the Z-axis is enough, whose training sample has four input parameters (i.e., ranging from four anchors) and one output parameter (i.e., Z-axis errors). Successively, in the case of the presence of different interfering signals, the ranging errors caused by signal interference

Positioning Errors Compensation Model
The 324 groups of samples are divided into 250 training set samples and 74 test set samples. For the absence of signal interference, only the errors compensation towards the Z-axis is enough, whose training sample has four input parameters (i.e., ranging from four anchors) and one output parameter (i.e., Z-axis errors). Successively, in the case of the presence of different interfering signals, the ranging errors caused by signal interference are firstly compensated and the Z-direction errors are then compensated. The input layer and an output layer of the ranging errors compensation model each have four nodes, which are the ranging of four anchors and the ranging errors of four anchors, respectively. The following procedure is to calculate the position of the target with and without signal interference, respectively.

Without Signal Interference
In terms of no signal interference, the input of function expression f : R n×m → R n×k of the GA-optimized ELM based positioning errors compensation model is 324 groups of target position distances, each of which has distances measurement of four anchors (i.e., n = 324, m = 4), and the output is 324 groups Z-axis errors of the target (i.e., k = 1).
The GA-optimized ELM based positioning errors compensation model is applied to model training, and the minimum RMSE of the test dataset is 14.5068 cm. Then the evolution curve is shown in Figure A1, and a comparison chart of the prediction results of the Z-axis coordinate errors of the target in the test set is drawn, as shown in Figure 11. Additionally, due to the enormous number of standard and extended machine learning models used in positioning, the BP neural network is added to justify and compare the usage of the GA-ELM model for experimentation.
are firstly compensated and the Z-direction errors are then compensated. The input layer and an output layer of the ranging errors compensation model each have four nodes, which are the ranging of four anchors and the ranging errors of four anchors, respectively. The following procedure is to calculate the position of the target with and without signal interference, respectively.

Without Signal Interference
In terms of no signal interference, the input of function expression : n m n k f  → of the GA-optimized ELM based positioning errors compensation model is 324 groups of target position distances, each of which has distances measurement of four anchors (i.e., n = 324, m = 4), and the output is 324 groups Z-axis errors of the target (i.e., k = 1).
The GA-optimized ELM based positioning errors compensation model is applied to model training, and the minimum RMSE of the test dataset is 14.5068 cm. Then the evolution curve is shown in Figure A1, and a comparison chart of the prediction results of the Z-axis coordinate errors of the target in the test set is drawn, as shown in Figure 11. Additionally, due to the enormous number of standard and extended machine learning models used in positioning, the BP neural network is added to justify and compare the usage of the GA-ELM model for experimentation. From Figure 11, it is observed that the overall effect of GA-ELM is relatively ideal, and its training set degree of fitting is 0.9147. Moreover, the minimum RMSE of GA-ELM is 14.5068 cm in the case of LOS, which is 71.32% and 59.72% less than the original estimate and the unoptimized one, respectively. Consequently, the Z-direction coordinate can be compensated according to the training prediction errors of the measurement data.

With Signal Interference
As for signal interference, the input of function expression : n m n k f  → of the GA-optimized ELM, based on the ranging errors compensation model, is 324 groups of target position distances, each of which has the measured distances of four anchors (i.e., n = 324, m = 4), and the output is 324 ranging errors, each of which has four errors of anchors (i.e., k = 4). Then, the input and output of the errors compensation model for the Z-direction are the same as those without signal interference.
As detailed in Section 3, errors compensation is performed on the distances from the four anchors to the target, and the minimum of RMSE optimized by the GA-ELM model is 134.2459 cm. Then the evolution curve is shown in Figure A2 From Figure 11, it is observed that the overall effect of GA-ELM is relatively ideal, and its training set degree of fitting is 0.9147. Moreover, the minimum RMSE of GA-ELM is 14.5068 cm in the case of LOS, which is 71.32% and 59.72% less than the original estimate and the unoptimized one, respectively. Consequently, the Z-direction coordinate can be compensated according to the training prediction errors of the measurement data.

With Signal Interference
As for signal interference, the input of function expression f : R n×m → R n×k of the GA-optimized ELM, based on the ranging errors compensation model, is 324 groups of target position distances, each of which has the measured distances of four anchors (i.e., n = 324, m = 4), and the output is 324 ranging errors, each of which has four errors of anchors (i.e., k = 4). Then, the input and output of the errors compensation model for the Z-direction are the same as those without signal interference.
As detailed in Section 3, errors compensation is performed on the distances from the four anchors to the target, and the minimum of RMSE optimized by the GA-ELM model is 134.2459 cm. Then the evolution curve is shown in Figure A2, and a comparison diagram of the predicted value and the true value of the ranging errors of each anchor in the training set are shown in Figure 12.  The estimated ranging errors are then substituted into the positioning model for Zaxis errors compensation, and the comparison figure before and after the compensation of range errors with signal interference is shown in Figure 13. Successively, the RMSE after errors compensation is calculated and compared with the one before compensation in Table 4.  The estimated ranging errors are then substituted into the positioning model for Z-axis errors compensation, and the comparison figure before and after the compensation of range errors with signal interference is shown in Figure 13. Successively, the RMSE after errors compensation is calculated and compared with the one before compensation in Table 4.  The estimated ranging errors are then substituted into the positioning model for Zaxis errors compensation, and the comparison figure before and after the compensation of range errors with signal interference is shown in Figure 13. Successively, the RMSE after errors compensation is calculated and compared with the one before compensation in Table 4.   From the rigorous numerical comparison of Figure 13 and Table 4, it can be concluded that after the ranging errors compensation with signal interference, the RMSE of the Z-direction is reduced by nearly 50%, which shows the effectiveness of the errors compensation model. However, there is still room for improvement in the errors of the Z-axis compared with those precision free of signal interference. Therefore, compensation for the Z-axis errors is equivalent to dual errors compensation for signal interference, which can effectively improve the positioning accuracy of the three-dimensional space coordinates of the target on the mobile robot.
The Z-axis errors are compensated by using the data after ranging errors compensation, and finally, the minimum RMSE is 37.9811 cm, of which the evolution curve is shown in Figure A3. Then, as shown in Figure 14, the Z-axis errors of the target in the test with signal interference are compared with the predicted results by forming a point plot. Meanwhile, the RMSE comparison before and after errors compensation with signal interference is shown in Table 5.  From the rigorous numerical comparison of Figure 13 and Table 4, it can be concluded that after the ranging errors compensation with signal interference, the RMSE of the Z-direction is reduced by nearly 50%, which shows the effectiveness of the errors compensation model. However, there is still room for improvement in the errors of the Zaxis compared with those precision free of signal interference. Therefore, compensation for the Z-axis errors is equivalent to dual errors compensation for signal interference, which can effectively improve the positioning accuracy of the three-dimensional space coordinates of the target on the mobile robot.
The Z-axis errors are compensated by using the data after ranging errors compensation, and finally, the minimum RMSE is 37.9811 cm, of which the evolution curve is shown in Figure A3. Then, as shown in Figure 14, the Z-axis errors of the target in the test with signal interference are compared with the predicted results by forming a point plot. Meanwhile, the RMSE comparison before and after errors compensation with signal interference is shown in Table 5.   Figure 14. The comparison of Z-axis test set errors compensation with signal interference. From Table 5, it can be clearly seen that the RMSE of the spatial coordinates after compensation of the GA-ELM model is 28.0861 cm in the case of NLOS, which is 64.38% and 70.16% significantly lower than those without compensation and optimization, respectively.
In order to better compare the performance of different algorithms, it is necessary to draw a cumulative distribution function (CDF) graph of the algorithm positioning errors. Figure 15 shows the CDF comparison of different algorithms in the two cases of LOS and NLOS, where the localization errors of LOS and NLOS are uniformly distributed numbers between 0~180 cm and 0~350 cm, respectively. As can be seen from Figure 15, compared to other algorithms, the GA-ELM model performs the best with an absolute error of less than 20 cm and 50 cm at over 80% quantile, in LOS and NLOS, respectively. From Table 5, it can be clearly seen that the RMSE of the spatial coordinates after compensation of the GA-ELM model is 28.0861 cm in the case of NLOS, which is 64.38% and 70.16% significantly lower than those without compensation and optimization, respectively.
In order to better compare the performance of different algorithms, it is necessary to draw a cumulative distribution function (CDF) graph of the algorithm positioning errors. Figure 15 shows the CDF comparison of different algorithms in the two cases of LOS and NLOS, where the localization errors of LOS and NLOS are uniformly distributed numbers between 0~180 cm and 0~350 cm, respectively. As can be seen from Figure 15, compared to other algorithms, the GA-ELM model performs the best with an absolute error of less than 20 cm and 50 cm at over 80% quantile, in LOS and NLOS, respectively. In the following, the numerical results of state-of-the-art approaches provided by other researchers in comparison with the GA-ELM model proposed in this paper are summarized in Table 6. The numerical results in Table 6 demonstrate that the method proposed in this paper has high accuracy in comparison with other methods. Finally, as depicted in Figure 16, the target positioning trajectory is connected into a smooth transition three-dimensional curve by denoising the trajectory of the mobile robot, and views of XOZ and XOY are drawn to make the trajectory of the target in the Z-axis more clearly expressed. In the following, the numerical results of state-of-the-art approaches provided by other researchers in comparison with the GA-ELM model proposed in this paper are summarized in Table 6. The numerical results in Table 6 demonstrate that the method proposed in this paper has high accuracy in comparison with other methods. Finally, as depicted in Figure 16, the target positioning trajectory is connected into a smooth transition three-dimensional curve by denoising the trajectory of the mobile robot, and views of XOZ and XOY are drawn to make the trajectory of the target in the Z-axis more clearly expressed.   Appendix B Figure A2. The evolution curve of ranging errors with signal interference.
Appendix C Figure A3. The evolution curve of Z-axis errors compensation with signal interference. Figure A3. The evolution curve of Z-axis errors compensation with signal interference.