Performance Comparison of Localization Algorithms for UWB Measurements with Closely Spaced Anchors

Transcription

1 Performance Comparison of Localization Algorithms for UWB Measurements with Closely Spaced Anchors Max Nilsson Space Engineering, master's level 2018 Luleå University of Technology Department of Computer Science, Electrical and Space Engineering

2 Abstract Tracking objects or people in an indoor environment has a wide variety of uses in many different areas, similarly to positioning systems outdoors. Indoor positioning systems operate in a very different environment however, having to deal with obstructions while also having high accuracy. A common solution for indoor positioning systems is to have three or more stationary anchor antennas spread out around the perimeter of the area that is to be monitored. The position of a tag antenna moving in range of the anchors can then be found using trilateration. One downside of such a setup is that the anchors must be setup in advance, meaning that rapid deployment to new areas of such a system may be impractical. This thesis aims to investigate the possibility of using a different setup, where three anchors are placed close together, so as to fit in a small hand-held device. This would allow the system to be used without any prior setup of anchors, making rapid deployment into new areas more feasible. The measurements done by the antennas for use in trilateration will always contain noise, and as such algorithms have had to be developed in order to obtain an approximation of the position of a tag in the presence of noise. These algorithms have been developed with the setup of three spaced out anchors in mind, and may not be sufficiently accurate when the anchors are spaced very closely together. To investigate the feasibility of such a setup, this thesis tested four different algorithms with the proposed setup, to see its impact on the performance of the algorithms. The algorithms tested are the Weighted Block Newton, Weighted Clipped Block Newton, Linear Least Squares and Non-Linear Least Squares algorithms. The Linear Least Squares algorithm was also run with measurements that were first run through a simple Kalman filter. Previous studies have used the algorithms to find an estimated position of the tag and compared their efficiency using the positional error of the estimate. This thesis will also use the positional estimates to determine the angular position of the estimate in relation to the anchors, and use that to compare the algorithms. Measurements were done using DWM1001 Ultra Wideband (UWB) antennas, and four different cases were tested. In case 1 the anchors and tag were 10 meters apart in line-of-sight, case two were the same as case 1 but with a person standing between the tag and the anchors. In case 3 the tag was moved behind a wall with an adjacent open door, and in case 4 the tag was in the same place as in case 3 but the door was closed. The Linear Least Squares algorithm using the filtered measurements was found to be the most effective in all cases, with a maximum angular error of less than 5 in the worst case. The worst case here was case 2, showing that the influence of a human body has a strong effect on the UWB signal, causing large errors in the estimates of the other algorithms. The presence of a wall in between the anchors and tag was found to have a minimal impact on the angular error, while having a larger effect on the spatial error. Further studies regarding the effects of the human body on UWB signals may be necessary to determine the feasibility of handheld applications, as well as the effect of the tag and/or the anchors moving on the efficiency of the algorithms.

3 Project relevance to the aerospace specialization at the Master Programme in Space Engineering at LTU A large part of the work done during this thesis is to program a microcontroller to use the antennas to gather data as well as performing some calculations before outputting the result. This is similar to work within Space Engineering where microcontrollers are commonly used, and has been a part of the education during the Master Programme. The technology investigated in the thesis can be applied in aerospace on drones flying in swarm-configurations, and though the idea arose from a need to locate emergency personnel both areas require high reliability and failsafe products. Localization algorithms using trilateration can be used in aerospace for applications such as radar and similar technologies.

4 Contents 1 Introduction Problem Formulation Structure Theory Trilateration UWB Algorithms Linear Least Squares (LLS) Non-Linear Least Squares (NLLS) Weighted Block Newton (WBN) Weighted Clipped Block Newton (WCBN) Kalman Filter Performance Evaluation Metrics Accuracy Precision Mean Absolute Error and Root Mean Square Error Method and Materials DWM Experimental Setup Results Bias Correction Optimizing WBN and WCBN Case 1: Line-of-Sight Case 2: Non-Line-of-sight - Body block Case 3: Non-Line-of-sight - With Multipath Case 4: Non-Line-of-sight - No Multipath Numerical Summary Discussion Algorithm Performance LLS and NLLS Wall Interference Human Body Interference Future work Conclusion 30 References 31

5 List of Figures Geometric visualization of three anchor nodes with perfect measurements to a tag Visualization of three anchor nodes with noisy measurements to a tag Visualization of three anchor nodes with noisy measurements to a tag. The tag is very far outside the perimeter of the tags Spectral Power Density of a UWB signal and a Narrowband (NB) signal IEEE Setup of the anchors The experimental setup of the different cases Measurement data before bias correction Measurement data after bias correction The setup of the anchors and the tag for cases 1 and 2. The tag is placed on the chair at the far wall The Cumulative Distribution Function of the angular accuracy of the algorithms for case The Cumulative Distribution Function of the angular precision of the algorithms for case The Cumulative Distribution Function of the spatial accuracy of the algorithms for case The Cumulative Distribution Function of the spatial precision of the algorithms for case The Cumulative Distribution Function of the angular accuracy of the algorithms for case The Cumulative Distribution Function of the angular precision of the algorithms for case The Cumulative Distribution Function of the spatial accuracy of the algorithms for case The Cumulative Distribution Function of the spatial precision of the algorithms for case The setup of the tag for case 3. The anchors are through the wall and at the other end of the next room The Cumulative Distribution Function of the angular accuracy of the algorithms for case The Cumulative Distribution Function of the angular precision of the algorithms for case The Cumulative Distribution Function of the spatial accuracy of the algorithms for case The Cumulative Distribution Function of the spatial precision of the algorithms for case The Cumulative Distribution Function of the angular accuracy of the algorithms for case The Cumulative Distribution Function of the angular precision of the algorithms for case The Cumulative Distribution Function of the spatial accuracy of the algorithms for case

6 4.6.4 The Cumulative Distribution Function of the spatial precision of the algorithms for case List of Tables The Root Mean Square Error of each algorithm for case The Root Mean Square Error of each algorithm for case The Root Mean Square Error of each algorithm for case The Root Mean Square Error of each algorithm for case Numerical values of the accuracy of each algorithm for each case Numerical values of the precision of each algorithm for each case The Root Mean Square Error of the angular estimate from each algorithm The Mean Absolute Error of the estimates from each algorithm

7 Abbreviations UWB WBN WCBN LLS NLLS GPS RMSE MAE CDF RF LOS NLOS ToF NB Ultra Wide Bandwidth Weighted Block Newton Weighted Block Newton Linear Least Squares Non-Linear Least Squares Global Positioning System Root Mean Square Error Mean Absolute Error Cumulative Distribution Function Radio Frequency Line-Of-Sight Non-Line-Of-Sight Time-of-Flight Narrow Band

8 1 Introduction Positioning and tracking technologies are useful in a large number of different areas. Outdoors the most commonly known positioning system is GPS, used by many every day. Indoors, a positioning system can help blind or visually impaired with navigation and track expensive equipment [1]. Especially of interest for this thesis is the ability to track emergency personnel in e.g. a burning building or other harsh environments. Performing positioning indoors is very different from outdoors. In an outdoor environment, there a usually few obstructions to line-of-sight, there are few objects that can cause multipath interference, and a positioning system can use satellites. When using positioning inside a building, walls will deny line-of-sight and can, together with other objects indoors reflect signals and cause multipath interference. Satellites are of no use due to the inability of such signals to penetrate buildings. As such, an indoor positioning system needs to be able to handle more local sources of noise than an outdoor system. Furthermore, positioning indoors will in general require a higher accuracy. Indoor positioning systems today generally rely on a system of stationary anchor antennas with known and fixed positions that are set up in advance. A mobile tag can then measure the distance between itself and the anchors and, assuming there are enough anchors in range, determine its relative position. To find the position of the tag, it first obtains the distance to each anchor, which is the used in a trilateration algorithm that calculates where the tag is located in relation to the anchors. The distances are found by measuring the Time-of-Flight (ToF) of the signal by comparing the time when the transmission is sent to the time when the transmission is received, usually by using several transmissions in order to minimize errors caused by drift in clock frequency of the antenna modules. Since the signal travels at the known speed of light, the distance between the antennas can be easily calculated by d = c t; d is the distance to be determined, c is the speed of light and t is the measured ToF. The anchors are set up in a perimeter and the tag will then move within or very close to this perimeter. Studies have been made to compare different technologies [2] as well as different algorithms for calculating the position of the tag [3, 4, 5]. Ref. [4] suggests a solution for tracking equipment in three dimensions inside a mine using several anchors placed around the mine, but found that the accuracy degraded when locating points outside the perimeter of the anchors. Ferreira et al. [5] compared algorithms for locating both stationary and moving tags in a room, with anchors placed in three corners of the room, but does not move outside the perimeter of the anchors. Cotera et al. [6] used trilateration algorithms and a grid of anchors to position a robot on a surface, but moved only within on close to the perimeter of the anchors. Systems like this, with preexisting stationary anchors in a fixed area, are less useful in the cases such as emergency personnel in burning buildings. If there are time-constraints or the exact area in which positioning is required is unknown, a more mobile system would be of more use. Such a system may also be useful on drones to allow several drones to their position relative to each other. The problem is that this would put the tag very far outside the perimeter of the anchors, which may introduce inaccuracies or make the system more sensitive to noise. Many algorithms used in indoor positioning systems are developed to solve the problem of trilateration or multilateration, the method of using measurements of distance from three or more stationary anchors with known positions to a tag of unknown position. These algorithms are developed for the case where the anchors are spread out far 1

9 and the tag is within the perimeter of the anchors, and have little testing for the case used in this thesis where the tags are placed closely together with the tag far away. 1.1 Problem Formulation This thesis will examine how well existing algorithms perform in determining the position of a tag in two dimensions when the anchors are placed very closely together, and the tag is located far outside the perimeter of the anchors. The idea is that the anchors can be placed in a hand-held or carried device, so that no previous setup of anchors is necessary. This would allow for rapid and flexible localization in emergency environments, at the cost of only being able to obtain the position of the tag relative to the mobile anchors, instead of obtaining its position in a rigid coordinate system spanned by stationary anchors. Four algorithms will be compared based on four performance metrics; Accuracy, Precision, Root Mean Square Error and Mean Absolute Error. Previous studies comparing algorithms have used the distance between the estimated position and the true position to determine the performance of an algorithm. This thesis will also use the angular error of the estimate, that is the angle between the true tag position and the estimated tag position as seen from the position of the anchors. In a localization system, this would give a user holding the anchors a direction in which the tag is located. The aim is to find an algorithm that can reliably estimate the direction to the tag with less than 15 error. 1.2 Structure Section 2 contains the theory behind the trilateration problem, some information about UWB as well as a description of the algorithms that will be used in this thesis. Below that section 3 describes the UWB antennae used and the setup of the experiment. Section 4 contains the results of the experiment and section 5 has a discussion of the results as well as a brief conclusion of the thesis. 2 Theory 2.1 Trilateration The 2D coordinates of a tag node at an unknown position can be determined by finding the intersection point of three circles. Three anchor nodes at known positions each report a distance measurement to the unknown node, which can be seen geometrically as a circle of possible locations around the anchor. With perfect measurements, the point at which all three of these circles intersect will be the location of the tag, as can be seen in figure

10 Figure 2.1.1: Geometric visualization of three anchor nodes with perfect measurements to a tag. Mathematically the coordinates of the tag can be found by solving the system of equations created by (x i x) 2 + (y i y) 2 = r 2 i (i = 1, 2, 3), (1) where x and y are the coordinates of the tag, x i and y i are the coordinates of the ith anchor node, and r i is the distance from the ith anchor to the tag. This becomes problematic when there is noise present in the measurements, as the circles will, in general, not all intersect in the same point, or even at all, as visualized in figure This will cause Eq. (1) to have no solution, and instead an approximation must be made. 3

11 Figure 2.1.2: Visualization of three anchor nodes with noisy measurements to a tag. The problems shown above are further exaggerated when the tag is far outside of the perimeter of the anchors, as will be the case in this thesis. Even with perfect measurements the circles are very close to overlapping at every point, making approximations with imperfect measurements more difficult. Another issue is that it only takes small additive noise, where one anchor overestimates the distance to the tag and another underestimates it, to make one circle be completely inside another circle, causing large errors in any position estimates. This problem is illustrated in figure 2.1.3, where one anchor has negative noise and another positive noise. 4

12 Figure 2.1.3: Visualization of three anchor nodes with noisy measurements to a tag. The tag is very far outside the perimeter of the tags. 2.2 UWB Ultra Wideband is defined as an RF signal that occupies a part of the frequency spectrum that is greater than 20% of the center carrier frequency, or a signal that has a bandwidth greater than 500 MHz. Because it uses such a wide portion of the frequency spectrum, a UWB signal can transmit large amounts of data while keeping the transmit energy very low. As seen in figure the UWB signal has a much lower Spectral Power Density than a narrow band signal, and will appear as noise. As such it should not interfere with most other RF signals [7, 8]. 5

13 Figure 2.2.1: Spectral Power Density of a UWB signal and a Narrowband (NB) signal IEEE UWB is especially suitable for indoor positioning due to several factors. The high bandwidth and very short pulse length of a UWB signal significantly reduces the effect of interference caused by multipathing and the low-frequency pulses in the broad range of the UWB frequency spectrum allows the signal to effectively pass through walls and objects [2]. This combines to make UWB potentially accurate to sub-centimeters, and in recent years has become a popular solution for indoor positioning systems due to its accuracy, low cost and low power consumption. 2.3 Algorithms Several algorithms have been developed in order to find the best approximate solution. The algorithms used in this thesis will be described in short in this section. The algorithms have been taken from other works where they are proposed and described in greater detail than they will be here Linear Least Squares (LLS) The linear least squares solution is a common method of solving mathematical and statistical problems. A version of it will be used here, though a more efficient version that also works in three dimensions can be found in Ref. [9]. Starting from Eq. (1), the equations can be linearized and then solved with A x = b, (2) 6

14 where [ ] x1 x A = 3 y 1 y 3, (3) x 2 x 3 y 2 y 3 [ ] r 2 b = 1 x 2 1 y1 2 r3 2 + x y3 2 r2 2 x 2 2 y2 2 r3 2 + x y3 2, (4) [ ] x x =. (5) y The position of the tag is then obtained through the inverse of A x = A 1 b. (6) This algorithm is relatively simple, and unlike the other algorithms in this thesis does not use iterations Non-Linear Least Squares (NLLS) The Non-Linear Least Square algorithm is more complex than the LLS algorithm, using derivation and Newton iteration to achieve greater accuracy and precision. The NLLS algorithm used here is taken from Ref. [4], and it should be noted that the authors of that paper states that there is a significant loss of accuracy as the tag moves outside the perimeter of the anchors when using this algorithm. The algorithm has been modified for use in this thesis by using only the x and y coordinates. The NLLS algorithm aims to find a good approximation by using iteration to minimize the sum of the squares of the errors, that is to minimize the equation f i (x, y) = n (r i,actual r i,est ) 2 (i = 1, 2, 3) (7) i=1 where r i,actual is the true distance between the tag and the ith anchor, r i,meas is the distance measured by the ith anchor, and x and y are the coordinates of the tag. The equation to iterate will be where R = [x, y], and J is the Jacobian matrix R (k+1) = R ( 1 (k) J T (k) (k)) J J T (k) f(k), (8) J = f 1 f = f 2, (9) f 3 f 1 x f 2 x f 3 x f 1 y f 2 y f 3 y. (10) The number of iterations for the NLLS algorithm was chosen to be five. 7

15 2.3.3 Weighted Block Newton (WBN) The Weighted Block Newton method (WBN), proposed by Park and Chang [3], is an iterative method to produce an accurate position estimate. A summary of the derivation follows below. Modify Eq. (1) to add noise. It then becomes (x i x) 2 + (y i y) 2 + n i = d i + n i = r i (i = 1, 2, 3), (11) where n i is the measurement noise of the ith anchor. Squaring (11) and rearranging yields x i x + y i y 0.5R + m i = 0.5 ( x 2 i + yi 2 ri 2 ), (i = 1, 2, 3), (12) where R = x 2 + y 2 and m i = d i n i 1 2 n2 i. This equation can be simplified and represented in matrix form as Mx + q = V where q = [m 1, m 2, m 3 ] T and x = [x y R] T, M = x 1 y 1 x 2 y 2 0.5, (13) x 3 y and V = 1 x y1 2 r1 2 x y2 2 r2 2. (14) 2 x y3 2 r3 2 Park and Chang [3] proposes a cost function to minimize as ( V M x) T Q 1 (V M x ), (15) which is then solved by Here ( 1 x (k+1) = x (k) µ M T Q M) (k) M T Q (k) e (k). (16) e (k) = V M x (k) (17) and Q (k) is the weighting matrix given by [ Q (k) 1 1 = diag (b 1,k a T 1 x(k) ) 2 (b 2,k a T 2 x(k) ) 2 ] 1 (b 3,k a T 3 x(k) ) 2, (18) in which b 1,k, b 2,k, b 3,k are the respective rows of V for the kth measurement. Similarly, a T 1...a T 3 are the rows of M. µ is a step-size, and will be 0 < µ < 1. This value and the number of iterations used for each estimate can be varied to get different results Weighted Clipped Block Newton (WCBN) In addition to this method a clipped version will also be used, called the Weighted Clipped Block Newton (WCBN) method. It is made less computationally complex than the WBN method by using only the sign of the values of A. This means that Eq. (16) becomes ( 1 x (k+1) = x (k) µ sign(m) T Q sign(m)) (k) sign(m) T Q (k) e (k). (19) 8

16 2.3.5 Kalman Filter Kalman Filters are widely used in many applications to improve noisy or incomplete measurements. It cannot by itself compute and approximation of a position, but since the main source of error in the approximations of other algorithms is noise, filtering the measurements before running them through a localization algorithm should improve the position approximation. In this thesis measurements filtered with a Kalman filter will be used with the Least Linear Solution algorithm. This thesis will use a very simple Kalman filter to reduce the noise present in the range measurements. It will be implemented in MATLAB following the process described in Ref. [10]. The range measurements of each anchor is filtered independently and the signal is assumed to be constant and consisting of only the state value and Gaussian noise. This creates the following Time Update functions and Measurement Update functions ˆx k = ˆx k 1 (20) P k = P k 1 (21) P k K k = P k + R (22) ˆx k = ˆx k + K ( k zk ˆx ) k (23) P k = (1 K k ) P k. (24) The values used that will be used in the filter are R = 1, P 0 = ˆx and ˆx 0 will be the first measurement in a set. Each set of 2000 measurements is run through this filter to obtain 2000 new filtered measurements. 2.4 Performance Evaluation Metrics To properly compare the different algorithms, four performance metrics will be used, accuracy, precision, mean absolute error and root mean square error. This is based on a similar approach done by Ferreira et al. [5] Accuracy Accuracy is a measure of how far off an estimation is from the true value. In this thesis the angular error will be examined and Cumulative Distribution Function will be used to find the maximum error of the best 95% of estimates, which will be used for comparison. The spatial accuracy of an estimate is measured as the Euclidean distance between the actual position of the tag and the estimated position, calculated by A sp = (x est x actual ) 2 + (y est y actual ) 2, (25) where x est, y est are the coordinates of the estimate and x actual, y actual are the true coordinates of the tag. The angular accuracy will be measured by the angular distance 9

17 between the estimate and the actual position, seen from the center of the anchors, which is calculated with A ang = θ actual θ est, (26) where θ actual is the true angular position of the tag, and theta est is the estimated angular position. The CDF will be obtained by first using the MATLAB functions fitdist to fit the errors to a normal distribution, and the function cdf to get the actual CDF values, which are then plotted using MATLABs plot command Precision The precision metric shows how far from the median value any estimate is. Like accuracy, precision will be examined using a CDF to determine the max deviation from the median of the closest 95%. The spatial precision is given by the Euclidean distance between an estimate and the median coordinates of all estimates in a set, P sp = (x est x median ) 2 + (y est y median ) 2, (27) x median and x median are the median value of the coordinates of all estimates in a set. The angular precision of an estimate is given by P ang = θ median θ est, (28) where θ median is the median value of a set of angular estimates Mean Absolute Error and Root Mean Square Error The Mean Absolute Error is the average error without taking direction into account, making it so that errors of opposite signs do not cancel each other out. The MAE will be calculated using the equation n m=1 MAE ang = θ actual θ m,est, (29) n where n is the number of estimates and θ m,est is the mth angular estimate. The spatial MAE is calculated similarly, using the spatial error instead of the angular error. The Root Mean Square Error is a measure of the average error that gives higher weight to large errors. This means that an algorithm that has a higher tendency to give large errors will have a correspondingly larger RMSE. The angular RMSE is calculated with the equation n m=1 RMSE = (θ actual θ m,est ) 2. (30) n For the spatial error, the RMSE can be calculated separately for the X and Y coordinates, n m=1 RMSE i = (Est i Actual i ) 2, (31) n where i is the coordinate axis. A net RMSE can be calculated from RMSE X and RMSE Y, using RMSE Net RMSE 2 X + RMSE2 Y. (32) 10

18 3 Method and Materials Ranging measurements were performed using four DecaWave DWM1001 modules. The modules were each soldered onto PCBs for easy access to the pins used. Two MSP432P401R microcontrollers were used for communicating with, and programming the DWM1001 modules through UART and for communicating the data gathered to a laptop computer via USB. 3.1 DWM1001 The DWM1001 module uses the DecaWave DW1000 Ultra-Wideband (UWB) transceiver to perform accurate ranging measurements. Also included in the module, though not used in this thesis, is a Bluetooth antenna and a 3-axis accelerometer. The module can achieve a reported accuracy of 10cm and using the UWB channel 5 at 6.8 Mbps data rate has a typical range of 60m in line-of-sight (LOS) conditions [11]. The ranging method measures the Time of Flight using a Two-Way Ranging scheme and then calculates the range between two modules. The module also has a built-in location engine using it s own algorithm to calculate its position. This algorithm is not given out by DecaWave however, and thus will not be used in this thesis. The DWM1001 module ranging works by setting up one anchor as initiator. This anchor looks for other modules in range and adds them to its network. Any anchor in the network will have its coordinates pre-programmed and is assumed to be fixed. Any tag in the network will, at a rate set by the user up to 10 Hz, send a ranging request to any anchors in its range. The range to the anchors and the calculated position of the tag can then be obtained by the user through UART, SPI or Bluetooth. 3.2 Experimental Setup Three DWM1001 modules were connected to an MSP432 and set up around a set point, meant as the origin of a coordinate system, which will from here on be used as a reference for the placement of the modules. Two modules were put along the x-axis, each 15 cm from the origin in opposite directions. The third was placed on the y-axis 15 cm from the origin in the positive direction. The setup is shown in figure These three modules were used as the anchors in the ranging measurements. 11

19 Figure 3.2.1: Setup of the anchors The last DWM1001 module was connected to a separate MSP432 and placed along the y-axis at a set distances from the origin. This MSP432 was then connected to a laptop. Measurements were done in four different cases illustrated in figure 3.2.2: in case one the tag was placed at 10 meters distance and in line-of-sight of the anchors; in case two the tag was also placed at 10 meters distance, but with a person was standing between the tag and the anchors. Cases three and four were placed at 12 meters distance, with a wall 15 cm thick in between the tag and the anchors. In case three a door slightly to the side was open to allow for multipath signals (see figure 3.2.2), whereas in case four this door was closed. These four cases were used to see the performance of the algorithms in different noise environments that may arise in real world applications. One set of measurements were made for each of the four cases, each set consisting of 2000 measurements from the tag to each anchor. The measurements are taken simultaneously, initiated by the tag broadcasting a ranging request to all anchors in range. 12

20 Figure 3.2.2: The experimental setup of the different cases. 4 Results 4.1 Bias Correction When looking at the ranging measurements before they are put into algorithms, shown in figure 4.1.1, it is clear that there is some bias in one or more of the DWM1001 modules. 13

21 Figure 4.1.1: Measurement data before bias correction. It is clear since the experiment was setup so that two anchors were placed symmetrically about the y-axis, and the tag placed on the y-axis. As such, the measurements from those two anchors should be very close to equal, excluding noise. As seen in figure however, no two sets of measurements are close to overlapping. This bias will cause a large error in the results, as the algorithms cannot account for the bias of the modules. Since the aim of this thesis is to investigate the impact of noise on the performance of the algorithms, this bias will be corrected before proceeding. To correct the bias without affecting the noise, the mean of all measurements were used as a basis for the correction. Two modules were known to lie along the x-axis, call them module A and module B. With no bias present, the means of the measurements from A and B should be roughly equal. Assuming that the measurements of A are without bias, the difference between the means of A and B are subtracted from all measurements of B. Further, the measurements from the third module, call it module C, shows it to lie further from modules A and B than the 15 centimeters used in the experiment setup. This was corrected first corrected in the same way as module B, and then 15 centimeters were subtracted from each range measurement. The data shown in figure is the same data shown in figure but corrected for bias using the process described above. This should ensure that the noise is unaffected, and that any error seen after using the algorithms should depend solely on the noise. 14

22 Figure 4.1.2: Measurement data after bias correction. 4.2 Optimizing WBN and WCBN Varying the step-size µ in Eq. (16) and/or the number of iterations yields different results for both the clipped and regular Weighted Block Newton algorithms. An initial guess, x (0) must also be chosen. A simple initial guess, x (0) = [0, 0] was found to work well while requiring no extra calculations. While a higher iteration count will generally give better results, it would also result in a slower system. Large values of 1/µ will tend to move the estimate toward the initial guess. For the WBN algorithm, seven iterations will be used, with a stepsize µ W BN = 1/7. The WCBN algorithm is slightly different; seven iterations were again found to be effective, but the step-size will instead be µ W CBN = This was found to be the values for which the increase in accuracy and precision started to diminish, and was considered to be a good compromise to maintain efficiency. 4.3 Case 1: Line-of-Sight The anchors were placed at one end of a room, and the tag placed at the other end of the room, 10 meters distant. A picture of this setup can be seen in figure

23 Figure 4.3.1: The setup of the anchors and the tag for cases 1 and 2. The tag is placed on the chair at the far wall. Measurements were taken continuously until 2000 sets of measurements had been completed. The measurements were put into MATLAB and the Cumulative Distribution Functions obtained using the method described in section Figure shows the accuracy of the algorithms, and figure shows the precision. The Root Mean Square Errors were obtained as described in section 2.4 and are shown in table Figure 4.3.2: The Cumulative Distribution Function of the angular accuracy of the algorithms for case 1. 16

24 Figure 4.3.3: The Cumulative Distribution Function of the angular precision of the algorithms for case 1. Figure 4.3.4: The Cumulative Distribution Function of the spatial accuracy of the algorithms for case 1. 17

25 Figure 4.3.5: The Cumulative Distribution Function of the spatial precision of the algorithms for case 1. Table 4.3.1: The Root Mean Square Error of each algorithm for case 1. WBN WCBN LLS LLS Kalman NLLS Angular Net Spatial 2.16m 2.39m 2.14m 0.077m 1.28m Spatial X 0.40m 0.38m 1.27m 0.046m 1.27m Spatial Y 2.12m 2.35m 1.73m 0.061m 0.13m 4.4 Case 2: Non-Line-of-sight - Body block Case 2 used the same setup as case 1, but for the duration of the test a person was standing close to the tag, directly in between the tag and the anchors, effectively blocking the LOS path of the signals. Again 2000 sets of measurements were taken and processed in MATLAB. The CDF of the accuracy of each algorithm is shown in figure and precision in figure The RMSE of the algorithms can be found in table

26 Figure 4.4.1: The Cumulative Distribution Function of the angular accuracy of the algorithms for case 2. Figure 4.4.2: The Cumulative Distribution Function of the angular precision of the algorithms for case 2. 19

27 Figure 4.4.3: The Cumulative Distribution Function of the spatial accuracy of the algorithms for case 2. Figure 4.4.4: The Cumulative Distribution Function of the spatial precision of the algorithms for case 2. 20

28 Table 4.4.1: The Root Mean Square Error of each algorithm for case 2. Angular Net Spatial Spatial X Spatial Y 4.5 WBN m 2.25m 4.13m WCBN m 2.17m 4.11m LLS m 3.86m 9.02m LLSKalman m 0.26m 1.14m NLLS m 30.89m 20.27m Case 3: Non-Line-of-sight - With Multipath For case 3 and 4 the tag was moved beyond the far wall seen in figure and placed in the adjoining room about 2 meters beyond the wall. The door between the rooms was left open, figure shows a picture of the new placement. Figure 4.5.1: The setup of the tag for case 3. The anchors are through the wall and at the other end of the next room. As in the other cases, 2000 sets of measurements were taken and put into MATLAB. Results are shown in figures and table

29 Figure 4.5.2: The Cumulative Distribution Function of the angular accuracy of the algorithms for case 3. Figure 4.5.3: The Cumulative Distribution Function of the angular precision of the algorithms for case 3. 22

30 Figure 4.5.4: The Cumulative Distribution Function of the spatial accuracy of the algorithms for case 3. Figure 4.5.5: The Cumulative Distribution Function of the spatial precision of the algorithms for case 3. 23

31 Table 4.5.1: The Root Mean Square Error of each algorithm for case 3. WBN WCBN LLS LLS Kalman NLLS Angular Net Spatial 2.31m 2.57m 2.91m 0.57m 1.57m Spatial X 0.52m 0.50m 1.54m 0.064m 1.45m Spatial Y 2.25m 2.52m 2.47m 0.57m 0.59m 4.6 Case 4: Non-Line-of-sight - No Multipath Case 4 used the same setup as case 3, but the door seen in figure was closed so as to prevent multipathing between the tag and anchors sets of measurements were taken and MATLAB was used to obtain the results below. Figures and shows the angular accuracy and precision, figures and show the spatial accuracy and precision, and table shows the RMSE of each algorithm. Figure 4.6.1: The Cumulative Distribution Function of the angular accuracy of the algorithms for case 4. 24

32 Figure 4.6.2: The Cumulative Distribution Function of the angular precision of the algorithms for case 4. Figure 4.6.3: The Cumulative Distribution Function of the spatial accuracy of the algorithms for case 4. 25

33 Figure 4.6.4: The Cumulative Distribution Function of the spatial precision of the algorithms for case 4. Table 4.6.1: The Root Mean Square Error of each algorithm for case 4. WBN WCBN LLS LLS Kalman NLLS Angular Net Spatial 2.34m 2.59m 3.09m 0.57m 1.60m Spatial X 0.52m 0.50m 1.60m 0.089m 1.48m Spatial Y 2.28m 2.54m 2.65m 0.56m 0.61m 4.7 Numerical Summary Here the numerical values for the accuracy and precision have been gathered and put into tables for easy comparison. Table shows the numerical value of each algorithm representing the highest angular and spatial error of the 95% closest estimates. Table shows the same values but for precision, and table is the angular and net spatial RMSE also found in tables Table contains the Mean Average Error of the estimates from each algorithm. 26

34 Table 4.7.1: Numerical values of the accuracy of each algorithm for each case. Case WBN WCBN LLS LLS Kalman NLLS Angular Accuracy [ ] Spatial Accuracy [m] > Table 4.7.2: Numerical values of the precision of each algorithm for each case. Case WBN WCBN LLS LLS Kalman NLLS Angular Precision [ ] Spatial Precision [m] > Table 4.7.3: The Root Mean Square Error of the angular estimate from each algorithm. Case WBN WCBN LLS LLS Kalman NLLS Root Mean Square Error [ ] Net Root Mean Square Error [m]

35 Table 4.7.4: The Mean Absolute Error of the estimates from each algorithm. Mean Average Error [ ] Case WBN WCBN LLS LLS Kalman NLLS Mean Average Error [m] Discussion 5.1 Algorithm Performance In all cases except case 2, 95% or more of the estimates from all algorithms were able to estimate the direction to the tag within 15. In both LOS and NLOS conditions, the closer spacing of the anchors in relation to the tag is handled well by the algorithms. A comparison between the results of the LLS algorithm in this thesis with the results of an LLS algorithm used in Ref. [4] show that there is a clear loss of accuracy and precision when using the configuration in this thesis. The results from Ref. [4] show an accuracy at 95% of 0.56 meters versus 2.27 meters found in this thesis, and if the 0.56 meters is assumed to be in a direction that maximizes the angular error, it would result in an angular offset of 3.2, much smaller than the accuracy of found here. The loss of accuracy was expected, and using the other more robust algorithms the accuracy loss is mitigated greatly. The simple Kalman filter implemented was very effective in reducing noise, making the LLS algorithm using the filtered data the best algorithm in every metric for every case, achieving a high accuracy and precision even with human body interference. One drawback of this algorithm may arise if using it with a mobile tag, as it may be slow follow the change in distance. In such a case a more advanced filter may be needed, such as the Extended Kalman Filter used by Ref. [5]. The WBN and WCBN should have less trouble adapting to a moving tag, depending on the sampling rate of the system. Since they both need 5-10 measurements to get a good estimate, fast changes or a slow system may result in poor estimates. Both the Linear and Non-linear Least Squares methods use single measurements for their estimates, and should behave similarly in stationary and mobile cases, with the note that NLLS is the slower of the two. NLLS runs the LLS algorithm in its entirety before iterating to find a good estimate, resulting in a longer runtime per estimate. The same can be said for the Kalman filter, which has to run every measurement through the filtering algorithm before running the LLS algorithm. 28

36 5.1.1 LLS and NLLS An interesting result is that of the similarity in results between the Linear and Non-Linear Least Squares algorithm. Ref. [4] found the NLLS method to be much more accurate than the LLS. With the closely spaced anchors however, the non-linearity and iterations of the NLLS gave only a small improvement over the LLS. One reason for this is that the NLLS algorithms used iteration in order to minimize the error between the measure ranges and the distance from the anchors to the estimated position, and indeed the NLLS has the second lowest spatial MAE as seen in table But since the anchors are spaced so closely together, the range to each anchor will be very similar, meaning that an estimate 180 from the actual position of the tag may be just as correct to the NLLS algorithm as an estimate that is 0 from the actual position. This causes the initial guess to heavily influence the estimate, and since the initial guess of the NLLS algorithm was the result of the LLS method, the difference in angular error between the two was very small. 5.2 Wall Interference The effect of the wall blocking the LOS in cases 3 and 4 seems to have had a minimal impact on both the angular and spatial accuracy and precision. The small impact on the angular error can perhaps be explained by the fact that since all the signals from the different anchors take almost the exact same path they will all suffer from similar interference from the wall, causing only a longer ToF but no major increase in noise. Thus the angular estimate will not be impacted much by the wall. The spatial estimate however is affected more by the wall, since even if the only effect of the wall is to make the ToF longer, the spatial estimate will be put further away from the anchors, resulting in a larger error. Comparing the numerical values in tables and for the LLS Kalman algorithm, the wall increases the angular accuracy value by 53% in the worst case, while the spatial accuracy value is increased by 480% in the worst case. The WBN and WCBN algorithms show similar results. Comparing instead the values of the LLS or NLLS algorithms, the angular accuracy and precision is actually better in cases 3 and 4 than case 1, while the spatial accuracy and precision are still worse. This shows that using an angular estimate in cases where NLOS is expected may be more effective with this configuration than using spatial estimates. 5.3 Human Body Interference In case 2 only the Linear Least Squares algorithm using Kalman filtered data was able to accurately find the direction to the tag, indicating that the human body attenuates much of the signal, causing the received signal in the antennae to be much less clear. This is of great interest since many useful applications of a small UWB device could be as a device held or attached to a human user where human body interference would always be present. Some studies have been made regarding this; Welch et al. [12] and Kiliç et al. [13] both found that the human body strongly attenuates the signals, potentially resulting in loss of direct-path signal, but that this effect is mitigated by the excellent performance of UWB using multipath signals, especially in smaller rooms and by proper choice of detection threshold. Further study may be needed to determine how severe this 29

37 problem is for the setup used in this thesis, and whether using different hardware or using other UWB such as a different channel or data rate may have an impact. 5.4 Future work This paper only used stationary tags and anchors, and only in one configuration, with the tag along the y-axis. The results may be different when the tag, the anchor or both are moving, or when the tag is in different angular positions. Different antennas have varying performance depending on the orientation of transmitter and receiver, and thus it may be of interest to test different antennas as well. Also, as mentioned above, further study of the effects of human body interference could be necessary if aiming to produce applications that would be hand-held or carried on ones person. 6 Conclusion All algorithms were found to perform well enough to find the direction to the tag within 15 in both LOS and NLOS cases, except in the case of human body interference. The Linear Least Squares method using measurements filtered using a simple Kalman filter was found to be the most accurate and precise, finding the direction of the tag within 0.5 in 95% of estimates, and within 2 in 100% of estimates. In the case of human body interference, the same algorithm found the direction of the tag within 2.3 in 95% of estimates, and within 5 in 100% of estimates. The disadvantage of the Kalman Filter method is that it has to filter every measurement, and as such may take a long time to produce a single estimate than just using the Linear Least Squares method without filtering. While performing worst in all cases, if the lower accuracy and precision is sufficient, LLS without filtering may be superior to the other algorithms due to its speed and simplicity since it does not rely on iteration. The use of this setup in hand-held applications may be complicated by the interference of the human body, as well as performance of the algorithms in moving and rotating scenarios. If those problems can be solved or deemed sufficiently small, the performance in NLOS and multipath environments means that placing the anchors closely together should be a good solution for hand-held devices. Using the setup on drones may be possible, there would be no interference from human bodies, though as with hand-held devices the effects of a moving system needs to be investigated. When using Kalman filtered measurements the majority of estimates stay within 20 centimeters in LOS conditions, and within a meter in NLOS, which may be sufficient depending on the requirements of the system. 30

38 References [1] Haosheng Huang and Georg Gartner. A Survey of Mobile Indoor Navigation Systems. In: (Jan. 2010), pp [2] Abdulrahman Alarifi et al. Ultra Wideband Indoor Positioning Technologies: Analysis and Recent Advances. In: Sensors(Basel, Switzerland) 16.5 (2016). [3] Chee-Hyun Park and Joon-Hyuk Chang. Adaptive robust time-of-arrival source localization algorithm based on variable step size weighted block Newton method. In: J Wireless Com Network (2017). doi: s [4] William S. Murhpy Jr. and Willy Herman. Determination of a Position in Three Dimensions Using Trilateration and Approximate Distances [5] André G. Ferreira et al. Performance Analysis of ToA-Based Positioning Algorithms for Static and Dynamic Targets with Low Ranging Measurements. In: Sensors 17.8 (2017). [6] Pablo Cotera et al. Indoor Robot Positioning Using an Enhanced Trilateration Algorithm. In: International Journal of Advanced Robotic Systems 13.3 (2016). [7] S. Roy et al. Ultrawideband radio design: the promise of high-speed, short-range wireless connectivity. In: Proceedings of the IEEE 92.2 (2004), pp issn: doi: /JPROC [8] M. Chiani and A. Giorgetti. Coexistence Between UWB and Narrow-Band Wireless Communication Systems. In: Proceedings of the IEEE 97.2 (2009), pp issn: doi: /JPROC [9] D.E. Manolakis. Efficient solution and performance analysis of 3-D position estimation by trilateration. In: IEEE Transactions on Aerospace and Electronic Systems 32.4 (1996), pp doi: / [10] Bilgin Esme. Kalman Filter for Dummies. url: http : / / bilgin. esme. org / BitsAndBytes/KalmanFilterforDummies. [11] DWM1001 System Overview And Performance. decawave. [12] T. B. Welch et al. The effects of the human body on UWB signal propagation in an indoor environment. In: IEEE Journal on Selected Areas in Communications 20.9 (2002), pp issn: doi: /JSAC [13] Y. Kiliç et al. The effect of human-body shadowing on indoor UWB TOA-based ranging systems. In: th Workshop on Positioning, Navigation and Communication. 2012, pp