# An Approach to Improve the Indoor Positioning Performance of Pseudolite/UWB System with Ambiguity Resolution

Authors: Xingli Gan; Zhihui Huo; Lu Sun; Shan Yang; Kaihui Liu
Journal: Journal of Sensors
 (2022)

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## Abstract

Indoor localization is still an open challenge, and some pseudolites have been developed to achieve seamless positioning service based on some commercialized GNSS chips. However, most of these indoor localization technologies often fail in a reasonable solution to the key problems such as low cost and highly accurate and efficient for users. In this paper, we propose an indoor location method based on integrated pseudolite and UWB; the virtual pseudo-range measurements of UWB are used to replace the pseudo-range measurements of pseudolite to solve the indoor multipath problem, which is tightly coupled with the corrected carrier phase measurements of pseudolite. Then, the channel-difference observation equation and UWB-aided ambiguity resolution are proposed for precise positioning. In order to test the proposed method, several experiments are conducted. The results show that the virtual pseudo-range errors from UWB are smaller than that of GNSS, and such a small bias will be better for the fast fixing of ambiguity. In addition, the positioning accuracy of the proposed indoor location method is improved from cm-level for the float solution to mm-level for the fixed solution; these performances would be more convincing to users than that given in the most pseudolite and UWB.


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## Body

## 1. Introduction

Global Satellite Navigation System (GNSS) is unable to provide the indoor location service because their signals can be blocked by buildings, and indoor localization is still an open problem. Some indoor localization approaches based on different techniques, such as ultra-wideband (UWB) [1, 2], pseudolite [3], bluetooth [4, 5], Wi-Fi [6], and vision-based on cameras [7, 8], have been developed for location-based services (LBS). The most difficult challenge for indoor positioning is to find an accurate location system, which can use the same receiver from outdoor to indoor. Therefore, many researchers are developing an indoor pseudolite to achieve indoor and outdoor seamless positioning services, such as Indoor Messaging System (IMES) and Locata [9, 10], the in-band GNSS-like signals of which can be received by some commercialized GNSS chips and output pseudo-range and carrier phase measurements.Some pseudolites for indoor high-precision positioning have also been developed. The multichannel pseudolite with array antenna at a spacing of half a wavelength is proposed [11], which does not require the time synchronization and indoor multipath problems and the positioning accuracy varies from centimeter- to meter-level according to the geometric relation between the antenna array and the receiver. However, the multichannel pseudolite is difficult to support dynamic positioning and has only a 4 m ×4 m positioning coverage area. A combined approach of Doppler and carrier-based hyperbolic positioning with a multichannel Global Positioning System- (GPS-) pseudolite for indoor localization has also been proposed [12], a state equation with three-dimensional (3D) position and orientation, and ambiguity is established, a nonlinear observation equation for carrier phase difference between pseudolite is used to estimate ambiguity, but it is difficult to overcome such problem as ambiguity resolution. A new indoor multichannel pseudolite system is introduced [13], which overcomes the problem of time synchronization, base stations, and ambiguity resolution of the traditional indoor pseudolite; the multichannel transmitters have an identical clock source and the clock drift of those can also be the same. The high-precision Doppler velocity measurement and positioning method is developed without ambiguity resolution in this work, and the dynamic positioning accuracy is better than 0.3 m. Locata pseudolite consists of a network (LocataNet) of time-synchronized transceivers, which has the potential to allow point positioning with sub-centimeter (cm) precision (using carrier phase and ambiguity resolution) for a mobile unit [14]. To achieve synchronization among all the pseudolite’s clock, TimeLoc technology can provide an autonomously synchronized network, which requires the additional ranging signals and the visibility to each other. Therefore, LocataNet is very expensive and difficult to apply in the indoor environment.Due to the influence of indoor multipath on the pseudo-range measurement for above-mentioned pseudolite, the maximum measurement error of tens of meters may be generated and in different statistics in different indoor scenarios [15], which cannot be used to solve and fix the integer ambiguity based on Least-square Ambiguity Decorrelation Adjustment (LAMBDA) method. Therefore, the Known Point Initialization (KPI) method [16, 17] has been generally adopted to solve ambiguities, but it has three disadvantages which is unacceptable or unusable for most users [18]: One is to survey a large number of known points with precise coordinates; the other is to start positioning at a known point; to make it worse, once the positioning error is more than 2 m, the positioning results using the KPI algorithm will continue to diverge, which needs to start from a new known point, and the positioning process is not continuous; in addition, even if the points of KPI are centimeter-level precision, the ambiguity validation still cannot be passed.Recently, the ultrawide bandwidth technology has attracted great interest in outdoor/indoor position application. Many UWB systems are now available at commercial level and a set with four anchors and one tag costs only a few hundred dollars. Recent studies have discussed the great potential of tightly coupling UWB with GPS [19]. The effectiveness of tightly coupling GPS and UWB range measurements is used to demonstrate to yield an improved ability to fix integer ambiguity during both kinematic and static applications [20, 21], the accuracy improvement of the float solution was noticeable when UWB measurements were included, and the LAMBDA method was employed to fix solution. UWB is integrated into the Real-time Kinematic (RTK) algorithm to achieve a highly precise positioning with two GPS receivers and reduce the ambiguity resolution search space of LAMBDA method [22]. The positioning strategy is proposed with UWB, low-cost GPS and MEMS onboard sensors, and an unscented Kalman filter is used for the measurement model without any linearization [23]. These researches proposed that the additional measurements of UWB can be directly used to assist ambiguity resolution of phase measurements; as such, the indoor pseudolite system can also be tightly coupled with UWB. Considering that the pseudo-range measurement error of pseudolite is much larger than that of GNSS due to multipath effect; then, the pseudo-range measurements of UWB need to be converted into that of pseudolite.An indoor precision positioning algorithm is proposed in this paper, which use the carrier phase measurements of pseudolite and the pseudo-range measurements of UWB. The contribution is summarized as follows:(1)
An indoor location system of pseudolite/UWB is proposed, which uses UWB tag and GNSS chip as the receiver to realize precise point positioning. Thus, the proposed system is highly efficient, low cost, and highly accurate for users(2)
With the fusion of carrier phase of pseudolite and virtual pseudo-ranges of UWB, the channel-difference observation equation is proposed, which can effectively reduce the clock offset, and hardware phase delay(3)
To improve the traditional KPI method of indoor pseudolite, the UWB-aided ambiguity resolution is proposed in order to use carrier phase of pseudolite for centimeter-level positioning, and which does not need a known point initializationThe paper is organized as follows: Section2 provides discussion on the basic terms including the composition of pseudolite/UWB indoor location system and positioning strategy. Section 3 describes the pseudolite integrated navigation model to show the system observation equation, virtual pseudo-range model, phase measurement model, and integer ambiguity resolution. Section 4 provides discussion on the experimental results of the proposed algorithm. Section 5 describes the results and concludes the paper.


## 2. Indoor Location System Based on Pseudolite and UWB

### 2.1. Indoor Location System of Pseudolite/UWB

The pseudolite/UWB indoor location system consists of three parts: the indoor synchronous pseudolite, the low-cost UWB system, and the pseudolite/UWB receiver, as shown in Figure1.
(1)
The indoor synchronous pseudolite integrates a multichannel signal transmitter and multiple antennas, the time synchronization of indoor pseudolite is different from that of the outdoor pseudolite, Locata, or GNSS, which uses the same 1 Pulse Per Second (1PPS) to generate the multichannel signals, and the clock offset of each channel can be equal, but the hardware delay is different and must be corrected(2)
The low-cost UWB system uses the DecaWave DW1000 chip with IEEE802.15.4-2011 compliant, which is based on the time of flight ranging measurement to obtain an accurate position, and 95% of the 3D positions have an error equal to or smaller than 50 cm [24–27].(3)
The pseudolite/UWB receiver includes a UWB tag, a GNSS receiver chip (such as ublox M8T/F9P), and their respective antennas. The GNSS chip tracks signals of the indoor pseudolite and provides the carrier phase observation in each epoch, and the UWB tag provides the ranges and the estimated position. The data from each was collected separately for postprocessing and analysis by a laptop, which is synchronized to GPS timeFigure 1
Composition of pseudolite/UWB indoor location system.


### 2.2. Pseudolite and UWB-Aided Location Strategies

Based on the composition of pseudolite/UWB indoor location system, the improved indoor positioning strategy is discussed; the use of UWB-aided ambiguity resolution for pseudolite needs to overcome the following problems:(1)
Carrier phase ambiguity resolution is the key problem, and the indoor multipath on pseudo-range measurement error is much greater than that of outdoor, which may be tens of meters; therefore, the pseudo-range measurement of pseudolite cannot be used to the ambiguity resolution. Now, UWB is not only cheap but also can provide centimeter-level ranging accuracy, which can be used to solve the ambiguity for pseudolite and generate a virtual pseudo-range measurement from the pseudolite/UWB receiver to the pseudolite(2)
Hardware delay of receiver and transmitter is also an important influence factor for GNSS or pseudolite positioning [28–30]; the method to measure and calibrate the phase delay is using a reference receiver, and phase measurement models of pseudolite-based positioning need to be modified by inter-channel hardware delay biasesThe proposed strategy includes the following steps, as shown in Figure2: Firstly, we need to estimate the unknown position of the user receiver by UWB, then the virtual pseudo-range measurements are computed from the user receiver to the pseudolite. Secondly, the hardware phase delay of the pseudolite can also be obtained based on the carrier phase measurements of a reference receiver, which is used to correct the carrier phase measurements from the user receiver. Thirdly, the observation equation for indoor high-precision positioning will be established; the corrected carrier phase measurements of pseudolite are tightly coupled with the virtual pseudo-range measurements of UWB. Finally, the carrier phase precise positioning is performed on the user receiver in each epoch, which is based on integrity ambiguity resolution and validation.Figure 2
Flowchart for indoor high-precision positioning using pseudolite and UWB.


## 2.1. Indoor Location System of Pseudolite/UWB

The pseudolite/UWB indoor location system consists of three parts: the indoor synchronous pseudolite, the low-cost UWB system, and the pseudolite/UWB receiver, as shown in Figure1.
(1)
The indoor synchronous pseudolite integrates a multichannel signal transmitter and multiple antennas, the time synchronization of indoor pseudolite is different from that of the outdoor pseudolite, Locata, or GNSS, which uses the same 1 Pulse Per Second (1PPS) to generate the multichannel signals, and the clock offset of each channel can be equal, but the hardware delay is different and must be corrected(2)
The low-cost UWB system uses the DecaWave DW1000 chip with IEEE802.15.4-2011 compliant, which is based on the time of flight ranging measurement to obtain an accurate position, and 95% of the 3D positions have an error equal to or smaller than 50 cm [24–27].(3)
The pseudolite/UWB receiver includes a UWB tag, a GNSS receiver chip (such as ublox M8T/F9P), and their respective antennas. The GNSS chip tracks signals of the indoor pseudolite and provides the carrier phase observation in each epoch, and the UWB tag provides the ranges and the estimated position. The data from each was collected separately for postprocessing and analysis by a laptop, which is synchronized to GPS timeFigure 1
Composition of pseudolite/UWB indoor location system.


## 2.2. Pseudolite and UWB-Aided Location Strategies

Based on the composition of pseudolite/UWB indoor location system, the improved indoor positioning strategy is discussed; the use of UWB-aided ambiguity resolution for pseudolite needs to overcome the following problems:(1)
Carrier phase ambiguity resolution is the key problem, and the indoor multipath on pseudo-range measurement error is much greater than that of outdoor, which may be tens of meters; therefore, the pseudo-range measurement of pseudolite cannot be used to the ambiguity resolution. Now, UWB is not only cheap but also can provide centimeter-level ranging accuracy, which can be used to solve the ambiguity for pseudolite and generate a virtual pseudo-range measurement from the pseudolite/UWB receiver to the pseudolite(2)
Hardware delay of receiver and transmitter is also an important influence factor for GNSS or pseudolite positioning [28–30]; the method to measure and calibrate the phase delay is using a reference receiver, and phase measurement models of pseudolite-based positioning need to be modified by inter-channel hardware delay biasesThe proposed strategy includes the following steps, as shown in Figure2: Firstly, we need to estimate the unknown position of the user receiver by UWB, then the virtual pseudo-range measurements are computed from the user receiver to the pseudolite. Secondly, the hardware phase delay of the pseudolite can also be obtained based on the carrier phase measurements of a reference receiver, which is used to correct the carrier phase measurements from the user receiver. Thirdly, the observation equation for indoor high-precision positioning will be established; the corrected carrier phase measurements of pseudolite are tightly coupled with the virtual pseudo-range measurements of UWB. Finally, the carrier phase precise positioning is performed on the user receiver in each epoch, which is based on integrity ambiguity resolution and validation.Figure 2
Flowchart for indoor high-precision positioning using pseudolite and UWB.


## 3. Methods

### 3.1. Virtual Pseudo-Range Measurement Model with UWB

The user receiver consists of a UWB module and a GNSS/pseudolite receiver module, once the position of the receiver is calculated by the UWB system, which can be used to calculate the geometric distance between the receiver and the pseudolite. At the same time, this distance is used to replace the pseudo-range measurement of pseudolite system. The positioning error equation of UWB can be written as:(1)xU=xu+εUxyU=yu+εUyzU=zu+εUz,where xU, yU, and zU are the coordinates of the user receiver, which is calculated by UWB system; xu, yu, and zu are the true coordinates of the user receiver u; εUx, εUy, and εUz are the virtual measurement errors.The virtual pseudo-range measurement is built as:(2)ρuPL,i=xU−xPL,i2+yU−yPL,i2+zU−yPL,i2=xu−xPL,i2+yu−yPL,i2+zu−yPL,i2+εUPL,i=RuPL,i+εUPL,i,where ρrPL,i is the virtual pseudo-range measurement between receiver u and pseudolite i; RrPL,i is the geometric range between receiver u and channel i of the pseudolite; xPL,i, yPL,i, and zPL,i are the transmitting antenna coordinates of the pseudolite, i for channel number; εUPL,i is the noise of virtual pseudo-range measurement.


### 3.2. Phase Measurement Model of BDS/GPS Pseudolite

#### 3.2.1. Phase Measurement Model

We consider the carrier phase measurements from pseudolite to receiver; the carrier phase measurement models are described as:(3)ϕuPL,i=RuPL,i+ctu−tPL+λNuPL,i+τuPL,i+εuPL,iϕuPL,j=RuPL,j+ctu−tPL+λNuPL,j+τuPL,j+εuPL,j,where ϕuPL,i and ϕuPL,j are the carrier phase measurements in meters for channel i and j of the pseudolite, respectively; RuPL,j is the geometric range between receiver u and channel j of the pseudolite; tu is the receiver clock offset, tPL is the pseudolite clock offset, and c is the speed of light; NuPL,i and NuPL,j are the phase ambiguity, and λ is the carrier wavelength; τuPL,i is the phase delay for receiver u and channel i of pseudolite, and τuPL,j is the phase delay for receiver u and channel j of pseudolite; εuPL,i and εuPL,j are the noises of the carrier phase measurement.Consider that the clock offset of the receiver and the pseudolite is the same, a single difference between channeli and j of pseudolite that cancels the clock offset is built as:
(4)ΔϕuPL,ij=ΔRuPL,ij+λΔNuPL,ij+ΔτPL,ij+ΔεuPL,ij,where ΔϕuPL,ij is the difference of phase measurement; ΔRuPL,ij is the geometric range difference, and ΔNuPL,ij is the ambiguity difference; ΔτPL,ij is the hardware phase delay difference; by forming Equation (4), the phase delay due to the user receiver can be eliminated.The geometric distance can be calculated by:(5)RuPL,i=xu−xPL,i×xu+yu−yPL,i×yu+zu−zPL,i×zuxu−xPL,i2+yu−xPL,i2+zu−xPL,i2=exieyiezi×xuyuzu,where exieyiezi is the geometry matrix.Then, the geometric range difference equation is denoted by:(6)ΔRuPL,ij=exj−exieyj−eyiezj−ezi×xuyuzu=ΔexijΔeyijΔezij×xuyuzu,where ΔexijΔeyijΔezij is the difference geometry matrix.


#### 3.2.2. Hardware Phase Delay

The hardware phase delay can be calculated in real time by a reference receiver [31]. When the antenna of the reference receiver and the transmitting antenna of the pseudolite are accurately surveyed by the total station, the distance between the receiver and the pseudolite is considered to be a known parameter; then, the difference equation of phase delay can be represented as:
(7)Δτ∧PL,ij=ΔϕrPL,ij−ΔR^rPL,ij+λΔN^rPL,ij−ΔεrPL,ij,where ΔR^rPL,ij is the true geometric distance difference; ΔN^rPL,ij is the true ambiguity difference, which is the integral part of ΔR^rPL,ij. We can obtain the equations as follows:
(8)Δϕ^rPL,ij=ΔR^rPL,ij+λΔN^rPL,ij−ΔN^rPL,ij=RoundΔR^rPL,ijλ,where Δϕ^rPL,ij is the fractional part of ΔR^rPL,ij; Roundx is use to separate the integral part and fractional part of a real number.The true geometric distance difference can be written as:(9)ΔR^rPL,ij=x∧r−xPL,i2+y∧r−yPL,i2+z∧r−yPL,i2−x∧r−xPL,j2+y∧r−yPL,j2+z∧r−yPL,j2,where x^r, y^r, and z^r are the true coordinates of the reference receiver r.


### 3.3. Pseudolite/UWB Observation Equation

The observation equations for the user receiver can be expressed as follows:(10)ρuPL,1=xu−xPL,12+yu−yPL,12+zu−yPL,12+εUPL,1⋮ρuPL,n=xu−xPL,n2+yu−yPL,n2+zu−yPL,n2+εUPL,nΔϕ^uPL,21=ΔϕuPL,21−Δτ∧PL,21=ΔRuPL,21+λΔNuPL,21+ΔεuPL,21⋮Δϕ^uPL,n1=ΔϕuPL,n1−Δτ∧PL,n1=ΔRuPL,n1+λΔNuPL,n1+ΔεuPL,n1,where Δϕ^uPL,21, ⋯, Δϕ^uPL,n1 are the differenced term of phase measurement with the hardware phase delay correction; n is the number of transmission channel.The observation equation can be written as follows:(11)ρuPL,1⋮ρuPL,nΔϕ^uPL,21⋮Δϕ^uPL,n1=ex1ey1ez1000⋮⋱⋮⋮⋱⋮exneynezn000Δex21Δey21Δez21λ00⋮⋱⋮⋮⋱⋮Δexn1Δeyn1Δezn10⋯λxuyuzuΔNuPL,21⋮ΔNuPL,n1+εUPL,1⋮εUPL,nΔεuPL,21⋮ΔεuPL,n1.


### 3.4. Integer Ambiguity Resolution

The LAMBDA method is adopted for rapid ambiguity resolution [32–34], then Equation (10) can be described by the following form:
(12)y=abxn+ε=ax+bn+ε,where y are the differenced observations; and a and b are the corresponding coefficients; x is the vector of the unknown coordinate; n is the integer-valued unknown ambiguities; ε is the measurement noise.The LAMBDA method has the minimization criterium for solving Equation (12):
(13)minx,ny−ax+bnQy−12,withx∈Rn∈Z,where ·Qy2=·TQy−1· and Qy−1 is the covariance matrix of observables.The procedure can be divided into three steps: the first step for float solution, by means of a common least-squares method, which takes as starting point forx^ and n^ as real values.The second step for integer solution with the minimization problem:(14)minnn^−nQn^−12,withn∈Zn∈Z,where Qn−1 is covariance matrix of integer-valued ambiguities; the LAMBDA method with ambiguity decorrelation (Z-transform) and the actual integer ambiguity estimation are utilized.In the third step, the fixed ambiguitiesn˘ to correct the float parameters x^ and the corresponding variance-covariance matrix:
(15)x˘=x^−Qx^n^Qn^−1n^−n˘,Qx˘=Qx^−Qx^n^Qn^−1Qn^x^.The least-squares estimatesx˘ and n˘ are the solution to Equation (13).


## 3.1. Virtual Pseudo-Range Measurement Model with UWB

The user receiver consists of a UWB module and a GNSS/pseudolite receiver module, once the position of the receiver is calculated by the UWB system, which can be used to calculate the geometric distance between the receiver and the pseudolite. At the same time, this distance is used to replace the pseudo-range measurement of pseudolite system. The positioning error equation of UWB can be written as:(1)xU=xu+εUxyU=yu+εUyzU=zu+εUz,where xU, yU, and zU are the coordinates of the user receiver, which is calculated by UWB system; xu, yu, and zu are the true coordinates of the user receiver u; εUx, εUy, and εUz are the virtual measurement errors.The virtual pseudo-range measurement is built as:(2)ρuPL,i=xU−xPL,i2+yU−yPL,i2+zU−yPL,i2=xu−xPL,i2+yu−yPL,i2+zu−yPL,i2+εUPL,i=RuPL,i+εUPL,i,where ρrPL,i is the virtual pseudo-range measurement between receiver u and pseudolite i; RrPL,i is the geometric range between receiver u and channel i of the pseudolite; xPL,i, yPL,i, and zPL,i are the transmitting antenna coordinates of the pseudolite, i for channel number; εUPL,i is the noise of virtual pseudo-range measurement.


## 3.2. Phase Measurement Model of BDS/GPS Pseudolite

### 3.2.1. Phase Measurement Model

We consider the carrier phase measurements from pseudolite to receiver; the carrier phase measurement models are described as:(3)ϕuPL,i=RuPL,i+ctu−tPL+λNuPL,i+τuPL,i+εuPL,iϕuPL,j=RuPL,j+ctu−tPL+λNuPL,j+τuPL,j+εuPL,j,where ϕuPL,i and ϕuPL,j are the carrier phase measurements in meters for channel i and j of the pseudolite, respectively; RuPL,j is the geometric range between receiver u and channel j of the pseudolite; tu is the receiver clock offset, tPL is the pseudolite clock offset, and c is the speed of light; NuPL,i and NuPL,j are the phase ambiguity, and λ is the carrier wavelength; τuPL,i is the phase delay for receiver u and channel i of pseudolite, and τuPL,j is the phase delay for receiver u and channel j of pseudolite; εuPL,i and εuPL,j are the noises of the carrier phase measurement.Consider that the clock offset of the receiver and the pseudolite is the same, a single difference between channeli and j of pseudolite that cancels the clock offset is built as:
(4)ΔϕuPL,ij=ΔRuPL,ij+λΔNuPL,ij+ΔτPL,ij+ΔεuPL,ij,where ΔϕuPL,ij is the difference of phase measurement; ΔRuPL,ij is the geometric range difference, and ΔNuPL,ij is the ambiguity difference; ΔτPL,ij is the hardware phase delay difference; by forming Equation (4), the phase delay due to the user receiver can be eliminated.The geometric distance can be calculated by:(5)RuPL,i=xu−xPL,i×xu+yu−yPL,i×yu+zu−zPL,i×zuxu−xPL,i2+yu−xPL,i2+zu−xPL,i2=exieyiezi×xuyuzu,where exieyiezi is the geometry matrix.Then, the geometric range difference equation is denoted by:(6)ΔRuPL,ij=exj−exieyj−eyiezj−ezi×xuyuzu=ΔexijΔeyijΔezij×xuyuzu,where ΔexijΔeyijΔezij is the difference geometry matrix.


### 3.2.2. Hardware Phase Delay

The hardware phase delay can be calculated in real time by a reference receiver [31]. When the antenna of the reference receiver and the transmitting antenna of the pseudolite are accurately surveyed by the total station, the distance between the receiver and the pseudolite is considered to be a known parameter; then, the difference equation of phase delay can be represented as:
(7)Δτ∧PL,ij=ΔϕrPL,ij−ΔR^rPL,ij+λΔN^rPL,ij−ΔεrPL,ij,where ΔR^rPL,ij is the true geometric distance difference; ΔN^rPL,ij is the true ambiguity difference, which is the integral part of ΔR^rPL,ij. We can obtain the equations as follows:
(8)Δϕ^rPL,ij=ΔR^rPL,ij+λΔN^rPL,ij−ΔN^rPL,ij=RoundΔR^rPL,ijλ,where Δϕ^rPL,ij is the fractional part of ΔR^rPL,ij; Roundx is use to separate the integral part and fractional part of a real number.The true geometric distance difference can be written as:(9)ΔR^rPL,ij=x∧r−xPL,i2+y∧r−yPL,i2+z∧r−yPL,i2−x∧r−xPL,j2+y∧r−yPL,j2+z∧r−yPL,j2,where x^r, y^r, and z^r are the true coordinates of the reference receiver r.


## 3.2.1. Phase Measurement Model

We consider the carrier phase measurements from pseudolite to receiver; the carrier phase measurement models are described as:(3)ϕuPL,i=RuPL,i+ctu−tPL+λNuPL,i+τuPL,i+εuPL,iϕuPL,j=RuPL,j+ctu−tPL+λNuPL,j+τuPL,j+εuPL,j,where ϕuPL,i and ϕuPL,j are the carrier phase measurements in meters for channel i and j of the pseudolite, respectively; RuPL,j is the geometric range between receiver u and channel j of the pseudolite; tu is the receiver clock offset, tPL is the pseudolite clock offset, and c is the speed of light; NuPL,i and NuPL,j are the phase ambiguity, and λ is the carrier wavelength; τuPL,i is the phase delay for receiver u and channel i of pseudolite, and τuPL,j is the phase delay for receiver u and channel j of pseudolite; εuPL,i and εuPL,j are the noises of the carrier phase measurement.Consider that the clock offset of the receiver and the pseudolite is the same, a single difference between channeli and j of pseudolite that cancels the clock offset is built as:
(4)ΔϕuPL,ij=ΔRuPL,ij+λΔNuPL,ij+ΔτPL,ij+ΔεuPL,ij,where ΔϕuPL,ij is the difference of phase measurement; ΔRuPL,ij is the geometric range difference, and ΔNuPL,ij is the ambiguity difference; ΔτPL,ij is the hardware phase delay difference; by forming Equation (4), the phase delay due to the user receiver can be eliminated.The geometric distance can be calculated by:(5)RuPL,i=xu−xPL,i×xu+yu−yPL,i×yu+zu−zPL,i×zuxu−xPL,i2+yu−xPL,i2+zu−xPL,i2=exieyiezi×xuyuzu,where exieyiezi is the geometry matrix.Then, the geometric range difference equation is denoted by:(6)ΔRuPL,ij=exj−exieyj−eyiezj−ezi×xuyuzu=ΔexijΔeyijΔezij×xuyuzu,where ΔexijΔeyijΔezij is the difference geometry matrix.


## 3.2.2. Hardware Phase Delay

The hardware phase delay can be calculated in real time by a reference receiver [31]. When the antenna of the reference receiver and the transmitting antenna of the pseudolite are accurately surveyed by the total station, the distance between the receiver and the pseudolite is considered to be a known parameter; then, the difference equation of phase delay can be represented as:
(7)Δτ∧PL,ij=ΔϕrPL,ij−ΔR^rPL,ij+λΔN^rPL,ij−ΔεrPL,ij,where ΔR^rPL,ij is the true geometric distance difference; ΔN^rPL,ij is the true ambiguity difference, which is the integral part of ΔR^rPL,ij. We can obtain the equations as follows:
(8)Δϕ^rPL,ij=ΔR^rPL,ij+λΔN^rPL,ij−ΔN^rPL,ij=RoundΔR^rPL,ijλ,where Δϕ^rPL,ij is the fractional part of ΔR^rPL,ij; Roundx is use to separate the integral part and fractional part of a real number.The true geometric distance difference can be written as:(9)ΔR^rPL,ij=x∧r−xPL,i2+y∧r−yPL,i2+z∧r−yPL,i2−x∧r−xPL,j2+y∧r−yPL,j2+z∧r−yPL,j2,where x^r, y^r, and z^r are the true coordinates of the reference receiver r.


## 3.3. Pseudolite/UWB Observation Equation

The observation equations for the user receiver can be expressed as follows:(10)ρuPL,1=xu−xPL,12+yu−yPL,12+zu−yPL,12+εUPL,1⋮ρuPL,n=xu−xPL,n2+yu−yPL,n2+zu−yPL,n2+εUPL,nΔϕ^uPL,21=ΔϕuPL,21−Δτ∧PL,21=ΔRuPL,21+λΔNuPL,21+ΔεuPL,21⋮Δϕ^uPL,n1=ΔϕuPL,n1−Δτ∧PL,n1=ΔRuPL,n1+λΔNuPL,n1+ΔεuPL,n1,where Δϕ^uPL,21, ⋯, Δϕ^uPL,n1 are the differenced term of phase measurement with the hardware phase delay correction; n is the number of transmission channel.The observation equation can be written as follows:(11)ρuPL,1⋮ρuPL,nΔϕ^uPL,21⋮Δϕ^uPL,n1=ex1ey1ez1000⋮⋱⋮⋮⋱⋮exneynezn000Δex21Δey21Δez21λ00⋮⋱⋮⋮⋱⋮Δexn1Δeyn1Δezn10⋯λxuyuzuΔNuPL,21⋮ΔNuPL,n1+εUPL,1⋮εUPL,nΔεuPL,21⋮ΔεuPL,n1.


## 3.4. Integer Ambiguity Resolution

The LAMBDA method is adopted for rapid ambiguity resolution [32–34], then Equation (10) can be described by the following form:
(12)y=abxn+ε=ax+bn+ε,where y are the differenced observations; and a and b are the corresponding coefficients; x is the vector of the unknown coordinate; n is the integer-valued unknown ambiguities; ε is the measurement noise.The LAMBDA method has the minimization criterium for solving Equation (12):
(13)minx,ny−ax+bnQy−12,withx∈Rn∈Z,where ·Qy2=·TQy−1· and Qy−1 is the covariance matrix of observables.The procedure can be divided into three steps: the first step for float solution, by means of a common least-squares method, which takes as starting point forx^ and n^ as real values.The second step for integer solution with the minimization problem:(14)minnn^−nQn^−12,withn∈Zn∈Z,where Qn−1 is covariance matrix of integer-valued ambiguities; the LAMBDA method with ambiguity decorrelation (Z-transform) and the actual integer ambiguity estimation are utilized.In the third step, the fixed ambiguitiesn˘ to correct the float parameters x^ and the corresponding variance-covariance matrix:
(15)x˘=x^−Qx^n^Qn^−1n^−n˘,Qx˘=Qx^−Qx^n^Qn^−1Qn^x^.The least-squares estimatesx˘ and n˘ are the solution to Equation (13).


## 4. Implementations and Evaluation

In this section, several experiments are designed to verify the pseudolite and UWB-aided location algorithm. One is a static test to analyze the characterization of pseudolite’s carrier phase measurements, UWB’s virtual pseudo-ranges, and static positioning results. The other is a dynamic testing for Only-UWB, Only-pseudolite, and pseudolite/UWB.


### 4.1. Experiment Setup

To evaluate our methods, we conduct field experiments in Figure3. In order to obtain high-precision carrier phase positioning of pseudolite/UWB, it is necessary in a line-of-sight (LOS) environment. Our field test system included four UWB anchors, one indoor synchronous pseudolite, one receiver with UWB tag and GNSS chip. The indoor synchronous pseudolite has eight signal transmission channels, each of which is connected with a right-handed polarized transmission antenna. At the same time, four UWB anchors are installed in the square test area; it can provide three-dimensional location. A GNSS chip (ublox F9P) with four-arm spiral antennas is used to receive pseudolite’s signals and outputs pseudo-range and carrier phase measurements according to UBX protocol through serial port. At the beginning of the test, all antenna coordinates of pseudolite and UWB need to be known. GNSS receiver and UWB tag run at 1 Hz; the frequency of our ambiguity resolutions and position solutions is in accordance with that of GNSS receiver as 1 Hz.Figure 3
Experimental environment of indoor location system based on pseudolite and UWB.


### 4.2. Static Testing and Analysis

#### 4.2.1. Carrier Phase Characterization of Pseudolite

In the static test situations, the data quality of the carrier phase measurements is analyzed by the epoch-by-epoch difference of Equation (4), which is called CCD-EED method and can be written as:
(16)ΔϕuPL,ijk+1−ΔϕuPL,ijk=ΔεuPL,ijk+1−ΔεuPL,ijk,where k is the epoch count.Figure4 is the data quality of the carrier phase measurements using CCD-EED method, Cm represents the channel m of pseudolite, C1 is the reference channel. First, using the carrier phase measurement data, the difference between each channel and channel 1 is calculated. Table 1 shows the statistical results obtained by CCD-EED method are more stable, the max error is 0.0034 m (0.018 cycles), the minimum error is -0.0036 m (-0.019 cycles), the mean error is in the range of 5.71 × 10-7 m to 1.90 × 10-6 m, the standard error is in the range of 7.32 × 10-4 m to 9.39 × 10-4 m, and these represent an acceptable in phase measurement error of pseudolite for precise point positioning.Figure 4
Data quality of the carrier phase measurements using CCD-EED method: (a) C2-C1; (b) C3-C1; (c) C4-C1; (d) C5-C1; (e) C6-C1; (f) C7-C1.
(a)
(b)
(c)
(d)
(e)
(f)Table 1
Statistical results of CCD-EED method.
C2-C1
C3-C1
C4-C1
C5-C1
C6-C1
C7-C1
Max (m)
0.0034
0.0029
0.003
0.0027
0.0032
0.0025
Min (m)
-0.003
-0.0027
-0.0032
-0.0036
-0.0034
-0.0027
Mean (m)
1.14 × 10-6
1.33 × 10-6
3.8 × 10-7
1.52 × 10-6
1.90 × 10-6
5.71 × 10-7
STD (m)
8.01 × 10-4
8.54 × 10-4
8.24 × 10-4
8.41 × 10-4
9.39 × 10-4
7.32 × 10-4


#### 4.2.2. Virtual Pseudo-Range Characterization of UWB

According to Equation (2), the virtual measurement errors can be written as
(17)εUPL,i=ρuPL,i−RuPL,i.Figure5 is the UWB virtual pseudo-range testing. There were few if any sources of signal reflection other than the equipment and the ground. The coordinates of the eleven points were established every 1 m to 11 m starting at UWB anchor; each point was occupied for ten minutes or more and a few thousand UWB range measurements were collected.Figure 5
UWB virtual pseudo-range testing.The UWB ranging bias is well calibrated by a polynomial curve fitting method, as shown in Figure6. The mean ranging errors based on independent tests from the UWB anchor to the UWB tag are shown in Figure 7 as a function of true distance. It can be seen that the UWB range bias may be 0.6 m before initiating a calibration, but which is less than 0.02 m after the calibration is finished.Figure 6
UWB range bias and calibration.Figure 7
UWB range error vs. epoch: (a) UWB anchor 0; (b) UWB anchor 1; (c) UWB anchor 2.
(a)
(b)
(c)After positioning with UWB measurement data, we can obtain the virtual pseudo-ranges from the user receiver to the pseudolite. Figure8 is the UWB virtual pseudo-range errors, to evaluate the characteristics of the UWB virtual pseudo-range errors; statistical results are performed in Table 2. Results show that the max error of virtual pseudo-range is about 0.051 m, the minimum error is from 0.001 m to 0.011 m, the mean error is from 0.001 m to 0.020 m, and the standard deviation is from 0.002 m to 0.009 m, which means that the UWB virtual pseudo-range may have a deviation of smaller than 1 cycle (GPS L1). Compared to the GNSS pseudo-range, the data quality is much improved; therefore, such a small bias will not affect the ambiguity fixing, but be better for the fast fixing of ambiguity.Figure 8
UWB virtual pseudo-range errors: (a) C1_u; (b) C2_u; (c) C3_u; (d) C4_u; (e) C5_u; (f) C6_u; (g) C7_u; (h) C8_u.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Table 2
Statistical results of UWB virtual pseudo-range errors.
C1_u
C2_u
C3_u
C4_u
C5_u
C6_u
C7_u
C8_u
Max (m)
0.039
0.031
0.030
0.009
0.019
0.040
0.044
0.051
Min (m)
0.008
0.001
0.004
0.007
0.011
0.001
0.002
0.011
Mean(m)
0.016
0.015
0.013
0.001
0.004
0.018
0.020
0.020
STD (m)
0.007
0.006
0.005
0.002
0.004
0.007
0.008
0.009


#### 4.2.3. Static Positioning Results

Pseudolite and UWB range data were collected at a static point, and the float solution positioning errors are shown in Figure9. The fixed solution positioning errors are shown in Figure 10, and the statistical results are shown in Table 3. The average three-dimensional positioning error for the float solution is 84 mm, 34 mm, and 121 mm, respectively; and the max three-dimensional positioning error is 211 mm, 231 mm, and 137 mm, respectively. The average positioning error for integer-fixed solutions is 1 mm in the X-axis, 0.5 mm in the Y-axis, and 28 mm in the Z-axis. In summary, with these experimental results, we conclude that the static positioning accuracy of the indoor location system based on pseudolite and UWB is improved from cm-level for the float solution to mm-level for the fixed solution; these performance specifications would be more convincing to users than the specifications given in the most pseudolite systems [11–14].Figure 9
Float solution in the static test: (a) errors ofX-axis; (b) errors of Y-axis; (c) errors of Z-axis.
(a)
(b)
(c)Figure 10
Fixed solution in the static test: (a) errors of X-axis; (b) errors of Y-axis; (c) errors of Z-axis.
(a)
(b)
(c)Table 3
Statistical results of float solution and fixed solution in the static test.
Float solution
Fixed solution
X-axis
Y-axis
Z-axis
X-axis
Y-axis
Z-axis
Max (mm)
211
231
137
4
3
42
Min (mm)
23
2
83
1
0. 1
22
Mean (mm)
84
34
121
1
0. 5
28
STD (mm)
42
70
12
0.5
0. 4
5.4


### 4.3. Dynamic Testing and Analysis

A testbed is set up in the indoor open space, as shown in Figure11, UWB anchors are placed on four corners of the rectangular area, and the test size of the test area is about 6 m long and 5 m wide. The experiment set two routes, one is a linear route from point 3 to point 5, and the other is a rectangular route composed of point 3, point 4, point 5, and point 6. Due to the lack of a reference instrument for millimeter (mm)-level positioning accuracy, we can only evaluate the positioning accuracy at the start point and the end point, the consistency between the estimated trajectory and the true route, and whether it can be closed-loop in the rectangular trajectory.Figure 11
Dynamic positioning with the rectangular route.Figure12 shows the dynamic positioning results of the two routes, the black line is the true trajectory, the pink line is the estimated trajectory of the fixed solution using pseudolite and UWB-aided location method, the green line is the estimated trajectory using the KPI method of Only-pseudolite [18, 35], and the red line is the estimated trajectory of Only-UWB. Table 4 shows the statistical results at the start/end point in the dynamic test. For the integer-fixed solution, the average position error is 5.4 mm in the X-axis, and 6.2 mm in the Y-axis; the max position error is 7.5 mm in the X-axis, and 10.1 mm in the Y-axis; the minimal position error is 0.3 mm in the X-axis, and 0.4 mm in the Y-axis; the standard deviation in the X-axis is 2.2 mm, and that in Y-axis is 2.4 mm, respectively. For the KPI method, the average position error is 160.1 mm in the X-axis, and 20.8 mm in the Y-axis; the max position error is 167.9 mm in the X-axis, and 26.7 mm in the Y-axis; the minimal position error is 154.4 mm in the X-axis, and 13.2 mm in the Y-axis; the standard deviation in the X-axis is 3 mm, and that in Y-axis is 4.3 mm, respectively. For the UWB method, the average position error is 38.5 mm in the X-axis, and 155.8 mm in the Y-axis; the max position error is 66.2 mm in the X-axis, and 185.4 mm in the Y-axis; the minimal position error is 20.1 mm in the X-axis, and 119 mm in the Y-axis; the standard deviation in the X-axis is 12 mm, and that in Y-axis is 19 mm, respectively. It can be seen that the proposed fixed solution method compared with the KPI method of Only-pseudolite and Only-UWB, the average position accuracy is increased from decimeter-level or cm-level to millimeter-level.Figure 12
Dynamic positioning results: (a) linear route; (b) rectangular route; (c) closed-loop of the rectangular route.
(a)
(b)
(c)Table 4
Statistical results at the start/end point in the dynamic test.
Fixed solution
KPI method
UWB
X-axis
Y-axis
X-axis
Y-axis
X-axis
Y-axis
Max (mm)
7.5
10.1
167.9
26.7
66.2
185.4
Min (mm)
0.3
0.4
154.4
13.2
20.1
119
Mean (mm)
5.4
6.2
160.1
20.8
38.5
155.8
STD (mm)
2.2
2.4
3
4.3
12
19Figure12(c) shows the closed-loop of the rectangular route, the pink ellipse is the closed-loop of the proposed fixed solution method, the green ellipse is the closed-loop with the KPI method of Only-pseudolite, and the red ellipse is the closed-loop of UWB. Both the KPI method of Only-pseudolite and Only-UWB has a closed-loop deviation, but the proposed fixed solution method can almost be closed-loop.


## 4.1. Experiment Setup

To evaluate our methods, we conduct field experiments in Figure3. In order to obtain high-precision carrier phase positioning of pseudolite/UWB, it is necessary in a line-of-sight (LOS) environment. Our field test system included four UWB anchors, one indoor synchronous pseudolite, one receiver with UWB tag and GNSS chip. The indoor synchronous pseudolite has eight signal transmission channels, each of which is connected with a right-handed polarized transmission antenna. At the same time, four UWB anchors are installed in the square test area; it can provide three-dimensional location. A GNSS chip (ublox F9P) with four-arm spiral antennas is used to receive pseudolite’s signals and outputs pseudo-range and carrier phase measurements according to UBX protocol through serial port. At the beginning of the test, all antenna coordinates of pseudolite and UWB need to be known. GNSS receiver and UWB tag run at 1 Hz; the frequency of our ambiguity resolutions and position solutions is in accordance with that of GNSS receiver as 1 Hz.Figure 3
Experimental environment of indoor location system based on pseudolite and UWB.


## 4.2. Static Testing and Analysis

### 4.2.1. Carrier Phase Characterization of Pseudolite

In the static test situations, the data quality of the carrier phase measurements is analyzed by the epoch-by-epoch difference of Equation (4), which is called CCD-EED method and can be written as:
(16)ΔϕuPL,ijk+1−ΔϕuPL,ijk=ΔεuPL,ijk+1−ΔεuPL,ijk,where k is the epoch count.Figure4 is the data quality of the carrier phase measurements using CCD-EED method, Cm represents the channel m of pseudolite, C1 is the reference channel. First, using the carrier phase measurement data, the difference between each channel and channel 1 is calculated. Table 1 shows the statistical results obtained by CCD-EED method are more stable, the max error is 0.0034 m (0.018 cycles), the minimum error is -0.0036 m (-0.019 cycles), the mean error is in the range of 5.71 × 10-7 m to 1.90 × 10-6 m, the standard error is in the range of 7.32 × 10-4 m to 9.39 × 10-4 m, and these represent an acceptable in phase measurement error of pseudolite for precise point positioning.Figure 4
Data quality of the carrier phase measurements using CCD-EED method: (a) C2-C1; (b) C3-C1; (c) C4-C1; (d) C5-C1; (e) C6-C1; (f) C7-C1.
(a)
(b)
(c)
(d)
(e)
(f)Table 1
Statistical results of CCD-EED method.
C2-C1
C3-C1
C4-C1
C5-C1
C6-C1
C7-C1
Max (m)
0.0034
0.0029
0.003
0.0027
0.0032
0.0025
Min (m)
-0.003
-0.0027
-0.0032
-0.0036
-0.0034
-0.0027
Mean (m)
1.14 × 10-6
1.33 × 10-6
3.8 × 10-7
1.52 × 10-6
1.90 × 10-6
5.71 × 10-7
STD (m)
8.01 × 10-4
8.54 × 10-4
8.24 × 10-4
8.41 × 10-4
9.39 × 10-4
7.32 × 10-4


### 4.2.2. Virtual Pseudo-Range Characterization of UWB

According to Equation (2), the virtual measurement errors can be written as
(17)εUPL,i=ρuPL,i−RuPL,i.Figure5 is the UWB virtual pseudo-range testing. There were few if any sources of signal reflection other than the equipment and the ground. The coordinates of the eleven points were established every 1 m to 11 m starting at UWB anchor; each point was occupied for ten minutes or more and a few thousand UWB range measurements were collected.Figure 5
UWB virtual pseudo-range testing.The UWB ranging bias is well calibrated by a polynomial curve fitting method, as shown in Figure6. The mean ranging errors based on independent tests from the UWB anchor to the UWB tag are shown in Figure 7 as a function of true distance. It can be seen that the UWB range bias may be 0.6 m before initiating a calibration, but which is less than 0.02 m after the calibration is finished.Figure 6
UWB range bias and calibration.Figure 7
UWB range error vs. epoch: (a) UWB anchor 0; (b) UWB anchor 1; (c) UWB anchor 2.
(a)
(b)
(c)After positioning with UWB measurement data, we can obtain the virtual pseudo-ranges from the user receiver to the pseudolite. Figure8 is the UWB virtual pseudo-range errors, to evaluate the characteristics of the UWB virtual pseudo-range errors; statistical results are performed in Table 2. Results show that the max error of virtual pseudo-range is about 0.051 m, the minimum error is from 0.001 m to 0.011 m, the mean error is from 0.001 m to 0.020 m, and the standard deviation is from 0.002 m to 0.009 m, which means that the UWB virtual pseudo-range may have a deviation of smaller than 1 cycle (GPS L1). Compared to the GNSS pseudo-range, the data quality is much improved; therefore, such a small bias will not affect the ambiguity fixing, but be better for the fast fixing of ambiguity.Figure 8
UWB virtual pseudo-range errors: (a) C1_u; (b) C2_u; (c) C3_u; (d) C4_u; (e) C5_u; (f) C6_u; (g) C7_u; (h) C8_u.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Table 2
Statistical results of UWB virtual pseudo-range errors.
C1_u
C2_u
C3_u
C4_u
C5_u
C6_u
C7_u
C8_u
Max (m)
0.039
0.031
0.030
0.009
0.019
0.040
0.044
0.051
Min (m)
0.008
0.001
0.004
0.007
0.011
0.001
0.002
0.011
Mean(m)
0.016
0.015
0.013
0.001
0.004
0.018
0.020
0.020
STD (m)
0.007
0.006
0.005
0.002
0.004
0.007
0.008
0.009


### 4.2.3. Static Positioning Results

Pseudolite and UWB range data were collected at a static point, and the float solution positioning errors are shown in Figure9. The fixed solution positioning errors are shown in Figure 10, and the statistical results are shown in Table 3. The average three-dimensional positioning error for the float solution is 84 mm, 34 mm, and 121 mm, respectively; and the max three-dimensional positioning error is 211 mm, 231 mm, and 137 mm, respectively. The average positioning error for integer-fixed solutions is 1 mm in the X-axis, 0.5 mm in the Y-axis, and 28 mm in the Z-axis. In summary, with these experimental results, we conclude that the static positioning accuracy of the indoor location system based on pseudolite and UWB is improved from cm-level for the float solution to mm-level for the fixed solution; these performance specifications would be more convincing to users than the specifications given in the most pseudolite systems [11–14].Figure 9
Float solution in the static test: (a) errors ofX-axis; (b) errors of Y-axis; (c) errors of Z-axis.
(a)
(b)
(c)Figure 10
Fixed solution in the static test: (a) errors of X-axis; (b) errors of Y-axis; (c) errors of Z-axis.
(a)
(b)
(c)Table 3
Statistical results of float solution and fixed solution in the static test.
Float solution
Fixed solution
X-axis
Y-axis
Z-axis
X-axis
Y-axis
Z-axis
Max (mm)
211
231
137
4
3
42
Min (mm)
23
2
83
1
0. 1
22
Mean (mm)
84
34
121
1
0. 5
28
STD (mm)
42
70
12
0.5
0. 4
5.4


## 4.2.1. Carrier Phase Characterization of Pseudolite

In the static test situations, the data quality of the carrier phase measurements is analyzed by the epoch-by-epoch difference of Equation (4), which is called CCD-EED method and can be written as:
(16)ΔϕuPL,ijk+1−ΔϕuPL,ijk=ΔεuPL,ijk+1−ΔεuPL,ijk,where k is the epoch count.Figure4 is the data quality of the carrier phase measurements using CCD-EED method, Cm represents the channel m of pseudolite, C1 is the reference channel. First, using the carrier phase measurement data, the difference between each channel and channel 1 is calculated. Table 1 shows the statistical results obtained by CCD-EED method are more stable, the max error is 0.0034 m (0.018 cycles), the minimum error is -0.0036 m (-0.019 cycles), the mean error is in the range of 5.71 × 10-7 m to 1.90 × 10-6 m, the standard error is in the range of 7.32 × 10-4 m to 9.39 × 10-4 m, and these represent an acceptable in phase measurement error of pseudolite for precise point positioning.Figure 4
Data quality of the carrier phase measurements using CCD-EED method: (a) C2-C1; (b) C3-C1; (c) C4-C1; (d) C5-C1; (e) C6-C1; (f) C7-C1.
(a)
(b)
(c)
(d)
(e)
(f)Table 1
Statistical results of CCD-EED method.
C2-C1
C3-C1
C4-C1
C5-C1
C6-C1
C7-C1
Max (m)
0.0034
0.0029
0.003
0.0027
0.0032
0.0025
Min (m)
-0.003
-0.0027
-0.0032
-0.0036
-0.0034
-0.0027
Mean (m)
1.14 × 10-6
1.33 × 10-6
3.8 × 10-7
1.52 × 10-6
1.90 × 10-6
5.71 × 10-7
STD (m)
8.01 × 10-4
8.54 × 10-4
8.24 × 10-4
8.41 × 10-4
9.39 × 10-4
7.32 × 10-4


## 4.2.2. Virtual Pseudo-Range Characterization of UWB

According to Equation (2), the virtual measurement errors can be written as
(17)εUPL,i=ρuPL,i−RuPL,i.Figure5 is the UWB virtual pseudo-range testing. There were few if any sources of signal reflection other than the equipment and the ground. The coordinates of the eleven points were established every 1 m to 11 m starting at UWB anchor; each point was occupied for ten minutes or more and a few thousand UWB range measurements were collected.Figure 5
UWB virtual pseudo-range testing.The UWB ranging bias is well calibrated by a polynomial curve fitting method, as shown in Figure6. The mean ranging errors based on independent tests from the UWB anchor to the UWB tag are shown in Figure 7 as a function of true distance. It can be seen that the UWB range bias may be 0.6 m before initiating a calibration, but which is less than 0.02 m after the calibration is finished.Figure 6
UWB range bias and calibration.Figure 7
UWB range error vs. epoch: (a) UWB anchor 0; (b) UWB anchor 1; (c) UWB anchor 2.
(a)
(b)
(c)After positioning with UWB measurement data, we can obtain the virtual pseudo-ranges from the user receiver to the pseudolite. Figure8 is the UWB virtual pseudo-range errors, to evaluate the characteristics of the UWB virtual pseudo-range errors; statistical results are performed in Table 2. Results show that the max error of virtual pseudo-range is about 0.051 m, the minimum error is from 0.001 m to 0.011 m, the mean error is from 0.001 m to 0.020 m, and the standard deviation is from 0.002 m to 0.009 m, which means that the UWB virtual pseudo-range may have a deviation of smaller than 1 cycle (GPS L1). Compared to the GNSS pseudo-range, the data quality is much improved; therefore, such a small bias will not affect the ambiguity fixing, but be better for the fast fixing of ambiguity.Figure 8
UWB virtual pseudo-range errors: (a) C1_u; (b) C2_u; (c) C3_u; (d) C4_u; (e) C5_u; (f) C6_u; (g) C7_u; (h) C8_u.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Table 2
Statistical results of UWB virtual pseudo-range errors.
C1_u
C2_u
C3_u
C4_u
C5_u
C6_u
C7_u
C8_u
Max (m)
0.039
0.031
0.030
0.009
0.019
0.040
0.044
0.051
Min (m)
0.008
0.001
0.004
0.007
0.011
0.001
0.002
0.011
Mean(m)
0.016
0.015
0.013
0.001
0.004
0.018
0.020
0.020
STD (m)
0.007
0.006
0.005
0.002
0.004
0.007
0.008
0.009


## 4.2.3. Static Positioning Results

Pseudolite and UWB range data were collected at a static point, and the float solution positioning errors are shown in Figure9. The fixed solution positioning errors are shown in Figure 10, and the statistical results are shown in Table 3. The average three-dimensional positioning error for the float solution is 84 mm, 34 mm, and 121 mm, respectively; and the max three-dimensional positioning error is 211 mm, 231 mm, and 137 mm, respectively. The average positioning error for integer-fixed solutions is 1 mm in the X-axis, 0.5 mm in the Y-axis, and 28 mm in the Z-axis. In summary, with these experimental results, we conclude that the static positioning accuracy of the indoor location system based on pseudolite and UWB is improved from cm-level for the float solution to mm-level for the fixed solution; these performance specifications would be more convincing to users than the specifications given in the most pseudolite systems [11–14].Figure 9
Float solution in the static test: (a) errors ofX-axis; (b) errors of Y-axis; (c) errors of Z-axis.
(a)
(b)
(c)Figure 10
Fixed solution in the static test: (a) errors of X-axis; (b) errors of Y-axis; (c) errors of Z-axis.
(a)
(b)
(c)Table 3
Statistical results of float solution and fixed solution in the static test.
Float solution
Fixed solution
X-axis
Y-axis
Z-axis
X-axis
Y-axis
Z-axis
Max (mm)
211
231
137
4
3
42
Min (mm)
23
2
83
1
0. 1
22
Mean (mm)
84
34
121
1
0. 5
28
STD (mm)
42
70
12
0.5
0. 4
5.4


## 4.3. Dynamic Testing and Analysis

A testbed is set up in the indoor open space, as shown in Figure11, UWB anchors are placed on four corners of the rectangular area, and the test size of the test area is about 6 m long and 5 m wide. The experiment set two routes, one is a linear route from point 3 to point 5, and the other is a rectangular route composed of point 3, point 4, point 5, and point 6. Due to the lack of a reference instrument for millimeter (mm)-level positioning accuracy, we can only evaluate the positioning accuracy at the start point and the end point, the consistency between the estimated trajectory and the true route, and whether it can be closed-loop in the rectangular trajectory.Figure 11
Dynamic positioning with the rectangular route.Figure12 shows the dynamic positioning results of the two routes, the black line is the true trajectory, the pink line is the estimated trajectory of the fixed solution using pseudolite and UWB-aided location method, the green line is the estimated trajectory using the KPI method of Only-pseudolite [18, 35], and the red line is the estimated trajectory of Only-UWB. Table 4 shows the statistical results at the start/end point in the dynamic test. For the integer-fixed solution, the average position error is 5.4 mm in the X-axis, and 6.2 mm in the Y-axis; the max position error is 7.5 mm in the X-axis, and 10.1 mm in the Y-axis; the minimal position error is 0.3 mm in the X-axis, and 0.4 mm in the Y-axis; the standard deviation in the X-axis is 2.2 mm, and that in Y-axis is 2.4 mm, respectively. For the KPI method, the average position error is 160.1 mm in the X-axis, and 20.8 mm in the Y-axis; the max position error is 167.9 mm in the X-axis, and 26.7 mm in the Y-axis; the minimal position error is 154.4 mm in the X-axis, and 13.2 mm in the Y-axis; the standard deviation in the X-axis is 3 mm, and that in Y-axis is 4.3 mm, respectively. For the UWB method, the average position error is 38.5 mm in the X-axis, and 155.8 mm in the Y-axis; the max position error is 66.2 mm in the X-axis, and 185.4 mm in the Y-axis; the minimal position error is 20.1 mm in the X-axis, and 119 mm in the Y-axis; the standard deviation in the X-axis is 12 mm, and that in Y-axis is 19 mm, respectively. It can be seen that the proposed fixed solution method compared with the KPI method of Only-pseudolite and Only-UWB, the average position accuracy is increased from decimeter-level or cm-level to millimeter-level.Figure 12
Dynamic positioning results: (a) linear route; (b) rectangular route; (c) closed-loop of the rectangular route.
(a)
(b)
(c)Table 4
Statistical results at the start/end point in the dynamic test.
Fixed solution
KPI method
UWB
X-axis
Y-axis
X-axis
Y-axis
X-axis
Y-axis
Max (mm)
7.5
10.1
167.9
26.7
66.2
185.4
Min (mm)
0.3
0.4
154.4
13.2
20.1
119
Mean (mm)
5.4
6.2
160.1
20.8
38.5
155.8
STD (mm)
2.2
2.4
3
4.3
12
19Figure12(c) shows the closed-loop of the rectangular route, the pink ellipse is the closed-loop of the proposed fixed solution method, the green ellipse is the closed-loop with the KPI method of Only-pseudolite, and the red ellipse is the closed-loop of UWB. Both the KPI method of Only-pseudolite and Only-UWB has a closed-loop deviation, but the proposed fixed solution method can almost be closed-loop.


## 5. Conclusions

In this paper, a pseudolite/UWB integrated method is proposed to improve the indoor position accuracy, which is based on the fusion of carrier phase measurements of pseudolite and the virtual pseudo-ranges of UWB. The performance of the proposed method has been verified, the data quality of the carrier phase measurements is analyzed by CCD-EED method, and the mean error is in the range of 5.71 × 10-7 m to 1.90 × 10-6 m, which is acceptable for high-precision indoor positioning. The virtual pseudo-range errors of UWB are from 0.001 m to 0.020 m, which means that the UWB virtual pseudo-range may have a deviation of smaller than 1 cycle (GPS L1), and such a small bias will be better for the fast fixing of ambiguity. According to the static and kinematic test, the results show that the positioning accuracy of the indoor location system based on pseudolite and UWB is improved from cm-level for the float solution to mm-level for the fixed solution, and compared with the KPI method of Only-pseudolite and Only-UWB, the average position accuracy of the proposed method is increased from decimeter-level or centimeter-level to millimeter-level.In the future, the study of pseudolite/UWB integrated method will be focused on the MIMO pseudolite system [36, 37], pseudolite/UWB integrated navigation, and deep learning-aided cycle slip detection and repair [38, 39]. It is foreseen that the precise positioning results will be benefit to the high accuracy, low-cost indoor localization.



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*Source: 3962014-2022-07-01.xml*