real time uwb error estimation in a tightly- coupled gps ...€¦ · consider band-limited signals;...

ION ITM 2009: International Technical Meeting of the Institute of Navigation Anaheim, CA, 26-28 Jan, Session B2.

Real Time UWB Error Estimation in a Tightly-Coupled GPS/UWB Positioning System

Glenn D. MacGougan and Kyle O’Keefe

Department of Geomatics Engineering, Schulich School of Engineering, The University of Calgary

BIOGRAPHY

Glenn D. MacGougan is a PhD candidate in Geomatics Engineering at the University of Calgary in Calgary, Alberta, Canada. He attained a BSc and MSc in Geomatics Engineering from the same department in 2000 and 2003. Glenn has worked in GNSS R&D since 1998 both in industry and at the University of Calgary. His current research focuses on extending GNSS RTK using UWB augmentation.

Kyle O’Keefe is an Assistant Professor of Geomatics Engineering at the University of Calgary. He completed BSc and PhD degrees in the same department in 2000 and 2004. He has worked in positioning and navigation research since 1996 and in satellite navigation since 1998. His major research interests are GNSS system simulation and assessment, space applications of GNSS, carrier phase positioning, and local and indoor positioning with ground based ranging systems.

ABSTRACT

This paper presents a tightly-coupled integration of GPS observations and ultra-wideband ranges that accounts for systematic UWB errors in real-time using additional error states to augment an extended Kalman filter. Results of range measurements obtained using commercially available UWB radios in optimal line-of-sight conditions are compared to surveyed ranges using conventional methods to assess UWB error characteristics.

UWB range measurements from multiple UWB reference stations as well as differential GPS measurements (i.e. pseudorange, Doppler, and carrier phase) are collected during a kinematic field test to assess the ability to estimate UWB bias and scale factor states and then assess the ability to perform UWB augmented RTK surveying in a degraded GPS signal environment. The collected measurements are post-processed with a simulated real-time approach with an extended Kalman filter that utilizes differential GPS pseudorange and carrier phase measurements as well as UWB ranges

The accuracy of the ‘real-time’ bias and scale factor estimation is compared to post-processed best line fit analysis. The positioning output of the estimation filter is compared to surveyed reference points and the accuracy

and geometry (i.e. DOP) improvement is quantified with and without the addition of UWB.

INTRODUCTION

Real-time kinematic (RTK) positioning using GPS provides centimeter-level accuracy with good quality signal conditions and high satellite availability. This technique is now common in industry but is limited in application primarily due to signal masking, attenuation and multipath in hostile environments. Urban canyons, forests and congested construction sites are prime examples of environments where GPS RTK surveying fails to operate well. At a minimum, GPS RTK requires 4 satellites with good positioning geometry. In fact, many commercial systems often fail to fix carrier phase ambiguities unless 5 satellites are present. Hence, in order to maintain centimeter-level accuracies in such environments, a method to augment GPS RTK under sub-optimal signal conditions is required.

Increasing the number of available satellites can be achieved by utilizing satellite based augmentation systems (SBAS, e.g. WAAS) and other Global Navigation Satellite Systems (GNSS, e.g. GLONASS). For example Wanninger and Wallstab-Freitag (2007) recently investigated the current use of GPS, GLONASS and SBAS for RTK. The additional signal processing requirements add complexity and cost to the RTK receiver used for surveying. In deep urban canyons, high buildings block satellite signals with low to medium elevation angles and significantly degrade the solution geometry, or dilution of precision (DOP), drastically reducing the improvement of using additional satellite systems.

GPS RTK can be successfully augmented using pseudolites, which are ground based in-band GPS-like (i.e. pseudo-satellite) transmitters (Cobb, 1997). The GPS receiver requires software modifications to enable pseudolites usage but generally no additional receiver hardware is required. The near/far problem applies to pseudolites because as the receiver approaches the transmitter the pseudolite signal becomes strong enough to jam the relatively weak signals from the distant GPS satellites. This can be somewhat mitigated using pulsed signals (Cobb, 1997). The use of pseudolites is


constrained by the need for licenses to transmit within the protected GPS frequency bands. Pseudolites also require timing synchronization with GPS. Due to these constraints pseudolites are not suitable for widespread use but are well suited for RTK applications where fixed infrastructure is available such as deep, open pit mining (Stone and Powell, 1998), deformation monitoring (Dai et al, 2002) and for precision approach and landing systems for aircraft (Bartone and Kiran, 2001).

Augmentation with wideband ground based ranging systems that operate in unlicensed bands (e.g. ISM) such as presented in Zimmerman et al (2005) and in Barnes et al (2006) extend RTK capabilities using similar carrier phase processing techniques as used in GPS. These are time-of-arrival systems and synchronized timing is required. Centimeter to decimeter level performance is achieved using specialized carrier phase processing techniques.

The unambiguous (non-phase based) range accuracy attainable using a wideband system is fundamentally limited. This limit is assessed mathematically using the Cramer-Rao lower bound (CRLB), a statistical measure described well by Kay (1993) that states that the variance of any unbiased estimator is at least as high as the inverse of the Fisher information. The CRLB provides a benchmark for performance comparison with any unbiased estimator. For ranging methods, it provides a method to assess the theoretical best performance of a ranging estimator. The CRLB for a time delay estimate, with any pulse waveform, is given by Kay (1993) and Poor (1988) as

22

0

*1

2/

1)ˆvar(ββ

τSNR

NE

=≥ (1)

where β is called effective bandwidth or sometimes RMS bandwidth, E is the energy of the pulse, N0 is noise spectral density, and E/(N0/2) is signal-to-noise ratio (SNR) (Van Trees, 1968). Effective bandwidth is given by

( )

∫

∫∞

∞−

∞

∞−=dffS

dffSf

2

22

2

)(

)(2πβ

(2)

where f is frequency (Hz). Note that effective bandwidth is not related to half-power bandwidth or noise bandwidth

Equation 1.0 indicates that ranging precision is a function of the bandwidth of the system and the received SNR. As an example, these equations are available for GPS delay estimates in Parkinson and Spilker (1996). Clearly, larger

effective bandwidth directly implies better ranging accuracy. SNR which is very subject to operating conditions and transmit energy has less effect but is still important to obtain accurate ranges.

Wideband ground-based radio navigation systems have bandwidth up to 100 MHz (e.g. 2.4-2.5 GHz ISM band, 5.725–5.875 GHz ISM band). Most are less than 20 MHz in general and ranging accuracy is meter level. Using the CRLB measure, the ranging accuracy of wideband systems can be assessed. Using a Gaussian pulse, the CRLB is available analytically as

SNRBc

SNRBcerfinv

bbr 2

99

2

299

2

222

ˆ 95.52))99.0((

≈≥π

σ (3)

where 2r̂σ is the variance of a one-way range estimate

(m2), Bb99 is the bandwidth (Hz) containing 99% of the pulse energy at baseband, and c is light speed (m/s). The derivation of Equation 3 is intensive and will be available soon in the first author’s dissertation (contact the author for more information). The CRLBs for wideband ranging estimators using Gaussian pulses with Bb99=10 MHz and Bb99=50 MHz bandwidth at baseband (i.e. 20 MHz and 100 MHz bandwidth when carrier modulated) are ~2.5m and ~0.5m at one standard deviation respectively with a SNR of 14 dB. Keep in mind that the CRLB does not consider band-limited signals; however, by describing the signal in terms of 99% energy bandwidth, band limiting at that threshold should provide similar performance.

Imagine now a system with Bb99 = 3.75 GHz (99% energy bandwidth at baseband). The CRLB with an SNR of 14 dB is now 6.5 mm! The spectrum for this ultra-wideband (UWB) signal, a term commonly referring to large relative or very large absolute bandwidth systems, is now available unlicensed.

Of late, there is intense interest in UWB due to the release of a tremendous 7.5GHz of unlicensed spectrum by the United States Federal Communications Commission (FCC) in 2002 (FCC, 2002). UWB has many advantages including signal robustness (to interference), high communications capacity (e.g. 400 Mbps), resistance to frequency selective fading (i.e. multipath), and fine time resolution (e.g. cm level).

The FCC’s Report and Order allows unlicensed use of spectrum mainly between 3.1 and 10.6 GHz albeit at extremely low power (maximum of -41 dBm/MHz). It is intended primarily for high data rate (e.g. 400 Mbps), short range (e.g. 10m), wireless communications and also for low data rate, very low power, communications such as for low cost sensor networks (e.g. IEEE-802.15.4a (2007)). Ranging sensors well fit the latter category and benefit from centimeter level ranging precision and also from the ability to readily distinguish between the line of


sight signal and multipath signals as frequency selective fading due to multipath is highly mitigated. Ranging through building walls and obstacles is also possible with power spread over such a large bandwidth. Intuitively, some frequencies will be better than others in traveling through different materials and detection of a direct signal may be feasible depending on the material composition and thickness. MacGougan et al (2008) contains the detailed definitions for UWB signals and for the interested reader, a detailed and thorough history of UWB is given by Barrett (2001).

UWB is of particular interest to positioning and navigation applications because of the huge bandwidth available for time transfer (i.e. ranging). UWB offers the potential of centimeter level range measurements. For this reason, the technology is potentially very suitable to augment high precision surveying equipment such as GPS RTK.

With the advent of commercially available pulse-based UWB ranging devices, study of GPS RTK augmented with multiple UWB ranges became feasible. MacGougan et al (2008) investigated tightly-coupled estimation using GPS and UWB for RTK positioning. The tightly-coupled navigation estimation approach combines GPS and UWB at the measurement level as inputs to an extended Kalman filter. The ability to perform RTK improved significantly when GPS signal conditions and GPS satellite availability are degraded. Utilizing two-way time-of-flight UWB ranges with GPS RTK provided better accuracy and enhanced fixed ambiguity solution availability.

UWB measurements suffer from bias and scale factor errors which generally degrade measurements from 0.5m to 1.0m. For low precision (e.g. meter level) applications, there is generally no need to complicate the estimation approach with bias and scale factor states. To achieve RTK level positioning accuracy, it is important that UWB ranges are compensated for bias and for scale factor error. This was achieved in MacGougan et al (2008) using calibration in post-processing. This is not possible for real-time application.

The objective of this paper is to develop a tightly-coupled estimation approach for combining GPS and UWB measurements to perform RTK surveying. A combined filter using state augmentation to estimate UWB bias and scale factor states ‘on-the-fly’ is presented. The error levels are stable enough during a typical survey (while continuous power is maintained) to result in centimeter to decimeter level range accuracy after error compensation.

The high positioning accuracy of GPS RTK (e.g. 2cm) with nominal conditions is used to facilitate the estimation of the UWB bias and scale factor states. Once these states are well estimated, the corrected UWB range

measurements enable and extend RTK accuracy into conditions that are hostile to GPS alone.

Line-of-sight UWB range measurements have the potential for centimeter level accuracy. However, if the line-of-sight signal is not measured (i.e. too weak to be detected) large range biases from non-line-of-sight signals are likely. The inclusion of such observation blunders (e.g. meter level) is very harmful to the accuracy of the estimation filter and the ability to fix GPS ambiguities correctly. The secondary objective of this paper is to develop suitable reliability techniques to detect and isolate UWB measurement blunders and adapt the estimation process accordingly.

BACKGROUND

Real-Time Kinematic GPS

Real-time kinematic positioning with GPS utilizes double differenced carrier phase observations and solves the associated unknown integer valued carrier phase ambiguities. This is well described in GPS texts such as Misra and Enge (2006).

Measurements are first differenced with a reference receiver at a known location thereby reducing spatially correlated errors such as unmodelled tropospheric and ionospheric delay. The measurements are then differenced between satellites and the effect of the receiver clock offset is mitigated. The resulting double differenced measurements are modeled by

ελρ +Δ∇+Δ∇=ΔΦ∇ N (4)

where ∇ΔΦ is the double differenced carrier phase measurement, ∇Δρ is the double differenced geometric distance between the estimated position and the satellite positions, ∇ΔλN is an integer valued ambiguity multiplied by the signal’s carrier wavelength, and ε includes multipath, noise, and unmodelled error effects.

GPS RTK generally uses extended Kalman filtering, an efficient recursive filter that estimates the error states of a linearized dynamic system (Brown and Hwang, 1997), to estimate unknown parameters.


Ultra-wideband

Ultra-wideband (UWB) signals are signals with large relative or very large absolute bandwidths. On the other hand in the time domain, UWB signals have very fine time resolution. For intuitive comparison with wideband signals, a wideband and UWB Gaussian impulse signal is shown in Figure 1.

Figure 1 – Wideband and UWB Signals

The First Report and Order released by the U.S. Federal Communications Commission (FCC) in 2002 (FCC, 2002) brought UWB into the spotlight with the release of 7.5 GHz of unlicensed spectrum. The FCC also issued a Second Report and Order in 2005 (FCC, 2005). These documents provide the most pertinent definition of an UWB signal. Signals with center frequencies above 2.5 GHz are considered UWB emitters if they have 10dB bandwidth greater than 500 MHz. UWB systems with center frequencies less than 2.5 GHz have fractional bandwidth equal to or greater than 0.20. For more introductory details refer to MacGougan et al (2008).

IMPULSE UWB

Historically UWB signals are synonymous with impulse radio, also known as baseband radio, but large bandwidth multi-carrier methods, such as OFDM, can also be considered UWB based on the FCC definition. The reader is invited to consult Reed (2005) for further general information regarding UWB signals.

A few studies have examined using multi-carrier methods for ranging. Parikh and Michalson (2008) is one such study. Ranging based on impulse UWB is much more prevalent in the literature and commercial systems are already available that use this technique.

No commercial systems that use the multi-carrier method for ranging are yet known. This paper focuses solely on impulse based UWB ranging.

Impulse UWB uses pulses which are very short. They range from a few tens of picoseconds to a few

nanoseconds. Pulses may be modulated using a carrier to fit the available spectrum but energy is present only for a few cycles of an RF carrier wave. Since the pulses are short, the energy is spread across a large bandwidth and results in a very low power density. For example, a Gaussian pulse which maximizes the use of the available FCC spectrum when modulated with a 6.85 GHz carrier is shown in Figure 2.

Figure 2 – Best fit Gaussian Pulse UWB Signal

ULTRA WIDEBAND AND MULTIPATH

Signals that can be classified as UWB have the ability to distinguish well between the line-of-sight response and most multipath. This is demonstrated graphically in Figure 3 and Figure 4. A Gaussian pulse is given by

643.3)99.0(2

,)(

9999

)2/()( 22

pp

t

terfinv

tetp

≈=

= −−

σ

σμ

(5)

where μ is the location of the pulse midpoint in time, and tp99 describes the 99% energy pulse width. The baseband bandwidth corresponding to the 99% energy pulse width is given by

9999

2

99056.1)99.0(

ppb tt

erfinvB ≈=π

(6)

where Bb99 is the 99% energy bandwidth. Two signals are shown, a line-of-sight signal with a pulse width of 100 ns (B99=11 MHz) in Figure 3 and a line-of-sight signal with a pulse width of 3 ns (B99=352 MHz) in Figure 4. Both signals were combined with a multipath with 20 ns delay and a multipath with 2ns delay. Multipath distorts the received pulse waveform considerably for the 100 ns pulse. The multipath signals cannot be distinguished from the direct signal. For the UWB signal with 3 ns pulse width, the 20 ns pulse can be clearly distinguished, while the 2 ns multipath slightly distorts the line of sight response.


Figure 3 – GPS-Like Wideband Signal with Multipath

Figure 4 – UWB Signal with Multipath

In general, any multipath that occurs with a delay less than the width of the line-of-sight pulse will result in distortion of the received line-of-sight pulse shape and the multipath cannot be distinguished. Maximum delay estimate error induced by multipath is one half of the pulse width (99% energy pulse width).

While UWB is very good at distinguishing between the line-of-sight response and multipath when the line-of-sight signal is detectable, there is a significant danger of measuring the first strongest multipath otherwise. This is demonstrated in Figure 5. This figure shows a weak line-of-sight signal, a multipath signal with 7 ns delay, and a multipath signal with 20 ns delay. If the line-of-sight signal is too weak to be detected, the first strongest multipath will be measured and a meter level measurement blunder is likely.

Figure 5 – Non-line-of-sight measurement

UWB RANGING METHODS

Ranging observations cannot be produced directly from time-of-arrival (TOA) measurements unless both the transmitter and receiver are synchronized in time. This is not generally the case. UWB ranging radios are used in TOA systems such as one available commercially from Multispectral Solutions Inc (MSSI) which is described by US patent 6054950 (Fontana, 2000).

Asynchronous ranging, ranging in the absence of clock synchronization, is a method of obtaining a range measurement wherein the requester device uses knowledge of its own clock and a known turn-around-time to measure a two-way range as shown Figure 6. This method and associated measurement error effects are described in the IEEE 802.15.4a specification, Appendix D1.3, (IEEE-802.15.4a, 2007).

Figure 6 – Two-Way Time-of-Flight Ranging

It is important to note that the speed of light used for UWB ranging is light speed ‘in air’ and is a function of the temperature, pressure, and water vapour pressure (Rüeger, 2002). This value can differ from light speed in a vacuum by up to 300 ppm.


TWO-WAY TIME-OF-FLIGHT BIAS

Due to frequency bias error in the oscillators (frequency standards) used by the requester and responder UWB transceivers there are two induced ranging error effects. Ranging error is given by

)(2

)( BABAaroundturn

Aflight eeeetet +++= −ε (6)

where ε is ranging error, tflight is time-of-flight, tturn-around is the fixed turn–around time, eA is the frequency standard bias for the requester and eB is the frequency standard bias for the responder. The first term in Equation 6 is a small scale factor error which for oscillators currently used ranges from 2 ppm to 20 ppm (e.g. 0.2 mm to 2 mm for a 100 m distance). The second term is independent of the distance measured and is thus a bias term which can reach the meter level. Turn-around times are much greater than the time-of-flight. For example for devices with 20 ppm oscillators and a 205 μs turn-around-time, the turn-around-time bias can be as much as 1.23 m. Fortunately, oscillator stability is relatively constant and this bias term is likely well estimated as an additional filter state modeled as a random walk process with suitable process noise to compensate for oscillator frequency bias instability (e.g. due to temperature).

GEOMETRIC WALK ERROR

The main sources of inaccuracy in pulse based ranging systems are noise-generated timing jitter and geometric walk timing error. Jitter in timing determines the precision of the range measurement. This is primarily a function of the received SNR. Pulse amplitude and shape variations create timing error in the fine time-pickoff circuit and this error is called geometric walk error (Amann et al, 2001).

The fine time-pickoff pickoff circuit utilizes some form of time discriminator. The task of the discriminator is to observe fine time information from the electric pulse signal received. In pulse based laser ranging, commonly used discriminator designs include leading edge timing (constant amplitude), zero crossing timing (derivation, first moment timing (integration), and constant fraction timing (Amann et al, 2001).

The UWB receiver used in this paper employs a constant threshold energy detection method to detect pulses and estimates fine time using a leading edge discriminator.

Threshold detection receivers, also known as leading edge detection receivers, set a threshold signal value and any incoming pulse that crosses the threshold is detected and demodulated. This receiver design requires calibration to set the threshold such the number of false alarms (false pulses) that are noise spikes which happen to cross the

threshold is within a desirable operation range. In radar, this type of receiver is often called a constant false alarm rate receiver. A tunnel diode is often used as a pulse detector in these receivers (Reed, 2005). The UWB receivers used in this study use this type of architecture with a tunnel diode pulse detector and a threshold value based on internal receiver noise only, which is generally set once shortly after turning on the receiver (Fontana, 1999).

Threshold detection receivers suffer from a geometric walk error that is a function of the signal amplitude with respect to the threshold value selected. This is discussed in Amman et al (2001). Figure 7 provides an intuitive graphic illustration of the effect the geometric ‘walk’ of the leading edge with the set threshold value as signal strength decreases.

Figure 7 – Geometric ‘walk’ and threshold energy

detection


COMBINED GPS AND UWB

UWB-GPS Co-Axial Mount

To integrate the UWB radio with the GPS receiver, a UWB-GPS antenna mount was built. The mount design is such that the phase centers of the GPS receiver and the UWB antenna are co-linear vertically. The UWB radio mounted beneath the GPS receiver is shown in Figure 8.

Multispectral Solutions UWB Radios

The University of Calgary obtained four ranging radios from Multispectral Solutions Inc (MSSI) suitable for testing and analysis. The form factor for the radios, as shown in the following figure, is rugged and suitable for field applications with a built-in rechargeable battery. Data collection software was written to configure the radios and collect ranging data using the RS232 protocol. Measurements from the radios are collected using a PC which is synchronized to GPS time to within 20 ms or better.

(from MSS, 2007)

Figure 8 – Multispectral Solutions Ranging Radios and a radio mounted beneath a Trimble R8 receiver

The MSSI ranging radios use a Gaussian-like pulse signal of approximately 3 ns duration modulated with a 6.35 GHz C-Band carrier. The radios are capable of providing range measurements at more than 200 Hz. It operates using two-way ranging and threshold energy detection to detect the leading edge of the transmit pulse. The receiver uses on/off keying for pulse modulation and hence utilizes one pulse for each bit. The pulse repetition rate is 1 pulse every 1000 ns and the gating period is comparable to the transmit pulse width (Fontana, 2002). This technique results in high SNR values but the receiver likely requires at least 16 dB SNR in order to operate. The radio design is described well by the US patent 5901172 (Fontana, 1999). The operational range of the radio is up to 200 m.

The turn-around time for this radio is 205 μs and worst case oscillator performance is 20 ppm. The worst case turn-around time bias is expected to be 1.23 m.

The MSSI radios have range measurements that are limited by the output measurement resolution to 0.5 ns or about 15 cm.

LINE-OF-SIGHT TESTING

A test was performed June 12, 2008, using the MSSI ranging radios in Sholdice Park, Calgary, Alberta, Canada (51°04’17.7”N, 114°10’08.0”W, 1082 m). The testing location was a soccer field with a very level surface. There were few if any sources of signal reflection other than the operators, the equipment and the ground. Figure 9 provides a photo of the test site and equipment. Horizontal distances were measured using a total station (i.e. using electronic distance measurement) and points were established every 5 m to 30 m and every 10 m from 40 m to 100 m.

Figure 9 – Line-of-sight UWB Range Testing

Each point was occupied for a minute or more and a few thousand UWB range measurements were collected. All distances were measured using all possible radio pairs.

The mean ranging errors based on independent tests for six MSSI requester-responder ranging radio pairs are shown in Figure 10 as a function of measurement distance. The bias error which is ±0.2 m can most likely be attributed to turn-around-time-bias. There is significant scale factor error present and ranges from 4900 ppm to nearly 12000 ppm. The range values produced used vacuum light speed. This accounts for up to 300 ppm scale factor error. Oscillator drift during time-of-flight accounts for another 20 ppm. The rest of the scale factor error is likely attributed to geometric walk error due to the use of a threshold energy pulse detector using a constant threshold leading edge discriminator for fine-time resolution.

The threshold used by the detector is set only once after the radio is turned on based on internal noise only. Thus, for MSSI receivers with similar internal noise characteristics the geometric walk error will be similar. MSSI radio number 8 likely has higher internal noise as it is consistently associated with high scale factor errors on the order of 10000 ppm to 12000 ppm. Any pair which


excludes number 8 has scale factor error between 4000 ppm and 6000 ppm.

Figure 10 – UWB range error vs. distance

The linear trends observed in line-of-sight testing are stable during the tests performed. Subsequent testing on a following day exhibited substantially different bias and scale factor values. If scale factor error is due to geometric walk and the use of a threshold energy detector, the in-run stability of the scale factor error is likely attributed to the threshold value which is set once when the radio is turned on. Thus, it is likely that scale factor error is stable while a unit is turned on.

MULTIPLE UWB RANGE AUGMENTED GPS RTK

Given the ‘in-run’ stability of the bias and scale factor errors determined in line-of-sight testing, it is likely that bias and scale factor errors can be estimated in real-time as additional states in a tightly-coupled GPS+UWB extended Kalman filter.

The software developed for this research utilizes an extended Kalman filtering approach. The filter can use 5 Hz UWB range measurements as well as 5 Hz differential GPS L1 pseudorange and carrier phase measurements. A two stage estimation approach is used. Firstly, the unknown position, receiver clock offset, and single difference ambiguities are estimated (i.e. between-receiver differencing only). This estimator is augmented to include bias and scale factor states for each UWB range pair. The single difference float solution ambiguity estimates are then differenced between satellites and the LAMBDA method is used to obtain double difference integer ambiguity estimates.

This study utilizes GPS L1 measurements with relatively short baselines of less than 1 km (i.e. strong spatially correlated errors). Utilizing GPS L2 phase measurements aids the ability to estimate fixed integer ambiguities but the focus of this research is the examination of the impact of UWB range augmentation and this impact is assessed well with L1 only (for short baselines).

The Kalman filter models the position states as random walk processes with process noise suitable for mildly dynamic operations (i.e. land surveying). The receiver clock offset state is also treated as random walk; however, it is given huge process noise with each prediction step so that the clock offset is fully estimated at each update (i.e. it is not filtered).

An UWB bias state is modeled as a random walk process with process noise that allows the bias to change slowly over time. The bias may change over time since the two-way time-of-flight bias is a function of the stability of the UWB oscillators which may vary with temperature.

An UWB scale factor state is modeled as a random walk process with very little process noise as it is not expected to change while the UWB radios are powered on. This assumes that the threshold used for the leading edge detector is constant during the test.

As the position of the phase center of the GPS antenna is estimated in the estimation filter, the lever arm between the UWB antenna and the GPS antenna phase center must be monitored. A tilt sensor with an accuracy of about 2° is used to monitor this lever arm. Additional noise is added to the UWB measurement standard deviation used in the filter based on the tilt angle. The approximate lever arm between the GPS antenna and the UWB antenna is 12 cm. At a tilt of 20°, this adds approximately 4 cm of measurement ‘noise’.

Since the range of the UWB bias is known based on the quality of the oscillators used (e.g. for 20 ppm, the bias is ±1.23 m), and the range of the scale factor error is well known from line-of-sight testing (e.g. 4000-12000 ppm), inequality constraints are used in the filter. This method is well described by Richards (1995).

Innovation testing was employed to detect and then exclude UWB range blunders in the filter. This method is well described by Teunissen (1990).

Testing

Three UWB reference stations (labeled 7, 8, and 9 below) were setup within 200 m of a GPS reference station located on the roof of the University of Calgary Engineering building as shown in Figure 11. A photo of the survey pole mounted GPS and UWB system is shown beside a photo of one of the UWB reference stations in Figure 12.


Figure 11 – Map of Test Area

Figure 12 – Test Setup Photos

The locations of the UWB reference stations were determined using static differential GPS with Trimble R8 GPS receivers. NovAtel OEM3 GPS receivers were used for the GPS reference station and pole-mounted survey receivers.

In order to observe the UWB bias and scale factor states, a range of motion is required with good quality GPS conditions. Once these states are sufficiently estimated, a survey may proceed in degraded GPS signal conditions with the benefit of the corrected UWB measurements.

The test consists of a pre-survey initialization walk following by walking a circular route on which there are 3 static test points which were pre-surveyed. This is illustrated in Figure 13. For the test system, an elevation mask of 40° is applied when entering the survey area.

To obtain a reference trajectory during the entire survey, GPS only results were obtained for the entire test without the 40° elevation mask. Fixed ambiguity GPS only RTK solutions were obtained using 7 satellites for the duration of the test. The number of double difference ambiguities used in the GPS-only solution is shown in Figure 14 along with corresponding dilution of precision (DOP) values. The test trajectory is shown in Figure 15.

Figure 13 - The Test

Figure 14 – RTK Truth Number of Observations and DOP

Figure 15 – Test Trajectory


Assessing UWB Ranging

The ranges measured by the UWB pairs can be compared to the RTK solutions obtained using GPS only. This allows assessment of the actual range errors as the RTK derived ranges are accurate to a few centimetres. Bias and scale factor estimates are obtained using a best line fit of the UWB range errors versus the GPS RTK derived range. This is shown for all three ranging pairs in Figure 16.

Figure 16 – UWB Ranging Errors

Some interesting behaviour is observable in Figure 16. The UWB biases are not constant and change over time. For UWB7, the first two static periods occur at the same point and the apparent UWB bias changed by about 6 cm. The temperature during testing varied from -5°C to -10°C and the radios were turned on just as the test began. It is likely, that the radios warmed up as the test proceeded and that the onboard oscillators exhibit frequency bias as a function of temperature. There are also clear multipath blunders occurring especially for range pair marked UWB9.

Tightly-Coupled Test Results

The number of double difference ambiguities used in the GPS+UWB solution is shown in Figure 17 along with corresponding dilution of precision (DOP) values. The epoch when the 40° elevation mask is applied is clearly evident in the VDOP and PDOP plots. Note that the HDOP degrades only slightly because of good horizontal observability due to the UWB measurements. A GPS only solution fails completely in this case due to the lack of satellite observations. The GPS+UWB solution was able to maintain fixed ambiguity solutions for the duration of the test.

Figure 17 – GPS+UWB Observations and DOP

The trajectory of the solution matches that of the RTK truth solution well as shown in Figure 18. The ability to measure the static survey points corresponds reasonably well with the RTK truth solutions, especially considering the 40° elevation mask, as shown in Figure 19, Figure 20, and Figure 21. The GPS+UWB solutions differ by less than 6cm compared to the ‘truth’ solution.

Figure 18 – GPS+UWB Trajectory

Figure 19 – Survey Point 1 (GPS+UWB 40° mask)




The overall accuracy of the combined GPS+UWB solution is compared to the GPS only ‘truth’ solution by comparing the 3D baselines obtained. The GPS+UWB system performs with 1cm of the truth solution with similar GPS conditions with the exception of an error spike due to the inclusion of a UWB range blunder. This is shown in Figure 22. When the GPS conditions are degraded using a 40° elevation mask and only 3 satellites are used, the performance of the system is typically better than 5cm and better than 10cm most of the time.

Figure 22 – Baseline Differences

The filter estimation of the UWB bias values agree well with the post-processed line fit values. This is shown for the 3 UWB range pairs in Figure 23, Figure 24, and Figure 25. It is clear from the bottom left subplot in Figure 23 that the UWB bias requires the pre-survey initialization walk to estimate the UWB bias with sufficient accuracy.

The ability of estimate the UWB scale factor value also agrees reasonably well with the post-processed line fit values. This is shown in Figure 26, Figure 27, and Figure 28.

Figure 23 – UWB7 Bias Estimate




Figure 26 – UWB7 Scale Factor Estimate




Numerous UWB blunders were detected in testing as shown in Figure 29. For example, the UWB9 range pair exhibited multiple blunders with errors ranging from a few metres to tens of metres. The filter performed well to detect most of these blunders and exclude them from the filter. However, as shown in the baseline difference figure, a short delay multipath blunder went undetected and affected the solution accuracy.

Figure 29 – UWB Ranges and Blunders

Comparing the raw range error with the range error corrected with the filter bias and scale factor values assesses the ability of the filter to ‘correct’ the UWB range measurements. This is shown in Figure 30 and it is clear that the filter reduces the UWB measurement error but there is room for improvement. Performing a longer pre-survey initialization walk should improve performance. Allowing the UWB transceivers warm-up time so that the temperature of their oscillators is stable should also improve performance.

Figure 30 – UWB Range Errors

CONCLUDING REMARKS

Utilizing tightly-coupled two-way time-of-flight UWB ranges with differential GPS measurements for RTK provides better extended RTK capabilities, better accuracy and enhanced fixed ambiguity solution availability. This was clearly demonstrated in MacGougan et al. (2008). To achieve RTK level positioning accuracy, it is important that UWB ranges are compensated for turn-around-time bias and for scale factor error. This work demonstrated that UWB errors can be successfully estimated in a real-time filter.

The precision achievable using UWB is centimeter level as shown using CRLB analysis. The UWB ranging technology used provided decimeter level measurement precision and is expected to reach centimeter levels soon.

The simplicity of utilizing two-way UWB range measurements and the moderate cost of UWB ranging


devices make augmenting GPS RTK with UWB ranges very suitable for many applications.

In terms of future work, there is need for further testing of the combined solution in practical hostile environments.

ACKNOWLEDGEMENTS

This research was sponsored by the Natural Science and Engineering Research Council of Canada (NSERC) and the Alberta Ingenuity Fund (AIF).

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