Vehicle-to-vehicle relative navigation of a network of vehicles travelling in an urban canyon is assessed using least-squares and Kalman filtering covariance simulation techniques. Between-vehicle differential GPS is compared with differential GPS augmented with between-vehicle ultrawideband range and bearing measurements. The three measurement types are combined using both least-squares and Kalman filtering to estimate the horizontal positions of a network of vehicles travelling in the same direction on a road in a simulated urban canyon. The number of vehicles participating in the network is varied between two and nine while the severity of the urban canyon was varied from 15-to 65-degree elevation mask angles. The effect of each vehicle’s azimuth being known a priori, or unknown is assessed. The resulting relative positions in the network of vehicles are then analysed in terms of horizontal accuracy and statistical reliability of the solution. The addition of both range and bearing measurements provides protection levels on the order of 2 m at almost all times where DGPS alone only rarely has observation redundancy and often exhibits estimated accuracies worse than 200 m. Reliability is further improved when the vehicle azimuth is assumed to be known a priori.
While single point GPS positioning may be sufficient for user navigation and guidance on most highways, other applications require a more precise intervehicle relative position. This paper assesses the utility of integrating between-vehicle differential GPS with direct measurements of intervehicle range using ultrawideband ranging (UWB) radios and direct observation of intervehicle bearing.
Relative positioning between vehicles, or vehicle-to-vehicle (V2V) navigation, using GPS has been a topic of interest for at least a decade [
While relative GPS can provide V2V navigation in many environments, in urban canyons and under dense foliage GPS alone may not be able to provide a reliable solution estimate [
UWB is a promising technology for direct ranging between vehicles. This ability would be even more valuable if the angle-of-arrival (or bearing from one vehicle to another as measured in the body frame of the first vehicle) could also be measured. Being able to measure both of the distances and bearings to a network of nearby vehicles would result in good overall network geometry and the possibility of reliable relative navigation even in the total absence of GPS signals. While systems to measure bearing using UWB ranging signals have been developed [
The purpose of this paper is to assess the usefulness of UWB ranging and bearing measurements to augment GPS as a means of determining the relative positions of a network of moving vehicles. Specifically this paper analyzes, through covariance analysis, the effect of adding direct between-vehicle UWB range and bearing measurements to networks of vehicles separated by less than 300 m that are using code (pseudorange) DGPS methods to establish their relative horizontal positions. It is noted that the use of carrier phase data would theoretically provide considerably higher accuracy than using pseudoranges (e.g., [
The remainder of this paper is organized as follows: Section
This section presents the mathematical models for the various measurements used. The integration algorithm is then described.
The mathematical models for GPS positioning are well known [
The largest spatially correlated differential error source is the ionosphere, which has a typical spatial correlation of about 3 ppm [
Equation (
The second measurement considered in this work is the UWB range measurement,
The final measurement considered herein is the bearing measurement. Bearing is an angular measure of the horizontal direction of another vehicle relative to the
Graphical representation of a bearing measurement.
From Figure
As mentioned in the previous section, the primary GPS measurements used are the DD pseudoranges. Correspondingly, it is apparent from (
The covariance analysis presented in this work focuses primarily on the use of least-squares (LS). This is done because the relative benefit of the UWB and bearing data on the relative position solution can be more easily assessed
The elements of the estimated state vector vary according to what type of measurements are being used. In all cases the state vector includes a
Only pseudorange data is used from the GPS receiver in all cases. The inclusion of UWB and/or bearing measurements is used to assess their relative benefit to the GPS-only solution. For simplicity, GPS pseudorange measurement accuracy is assumed to be independent of satellite elevation.
For vehicle networks consisting of more than two vehicles, only linearly independent observations were used to avoid a singular solution with least-squares. The algorithm described by Saalfeld [
This section discusses the covariance simulations conducted and the metrics used to assess the performance of the different scenarios.
The simulated environment models an urban canyon with all vehicles moving along a road with a constant azimuth and with buildings on either side. All buildings have the same dimension (specifiable) and are assumed to each occupy an entire city block. Buildings continue to infinity in the forward and reverse directions. A diagram of the urban canyon environment used in this work is shown in Figure
Diagram of the simulated urban canyon environment.
To assess the impact of the size of the vehicle network on performance, up to nine vehicles were simulated traveling in the same direction. The relative position of the vehicles is shown in Figure
Relative locations of vehicles driving in simulated urban canyon environment.
Throughout the simulation, the vehicles maintain the
The buildings were simulated to move at a rate of 0.2 m/s. This allows the building geometry to repeat every ten minutes (i.e., 120 m/0.2 m/s = 600 s), over which time the satellite geometry (in the absence of obstructions) does not change very much. In terms of the data analysis, the assumption is that the satellite geometry is approximately constant over ten minutes and so any variations in performance within that time interval are attributed mostly to the location of the vehicles within the urban canyon. Furthermore, the slow motion of the buildings means that the location of the vehicles within the urban canyon is sampled with sufficiently high resolution. In other words, the simulation should capture a representative range of satellite geometries resulting from the combination of satellite motions and locations of the vehicles within the urban canyon.
The height of the building and the azimuth of the road were left as specifiable parameters in the simulations. The road azimuth was selected as 0° or 90°. The heights of the buildings were computed based on the maximum elevation angle they would produce for a vehicle located in the centre of the road. The maximum elevation angles were varied from 15° to 60° in 15° increments. An elevation mask of 10° was used for the forward and reverse directions (since there are no simulated buildings in those directions).
The simulations were run over 24 hours with measurements every 5 s (0.2 Hz). The GPS almanac used was taken from GPS week 1587 and consisted of 30 healthy satellites. It is noted that most GPS receivers allow measurements at a rate of 1 Hz or higher, which is higher than the rate of 0.2 Hz used in the simulations. Recall that the main objective of the covariance simulation is to assess the benefit of including UWB and bearing data, particularly when GPS availability is limited. As such, the “absolute” simulated data rate itself was not considered a critical parameter in the simulation. Instead, the results are most affected by GPS satellite geometry changes due to satellite motion and the local signal masking environment. With this in mind, the use of a 0.2 Hz measurement rate will still give a very high “resolution” in terms of satellite motion. As mentioned above, the effect of local signal masking is addressed by selecting a suitably appropriate velocity for the buildings.
The GPS measurement simulation is typical of similar studies. The satellite geometry is evaluated and satellites that are in view (above the elevation mask and not blocked by the simulated urban canyon) are assumed to be available for positioning.
UWB ranges are assumed to be available between all vehicles in the network, but each distance is only observed one way (e.g., from vehicle 1 to 2 but not the other way). If measurements made in the other direction were included, the net effect would be the same as the one-way case but with the measurement accuracy reduced by
The bearing sensor is simulated as a device that is capable of measuring the angle between the “forward” axis of the vehicle body frame and the line of sight to the other vehicle. Bearing measurements are simulated from each vehicle to all the vehicles “in front” of it. In this context, “in front” refers to the vehicle number, not necessarily their relative position. For example, vehicle 4 measures a bearing to vehicle 3, even though it is behind it, and similarly, vehicle 3 does not observe the bearing to vehicle 4. This was done to limit the number of observations and also to ensure that each nonlead vehicle is making at least one bearing observation. This enables each nonlead vehicle to have an observable azimuth in the case where the azimuth is assumed to be unknown, thus simplifying software development.
The measurements were assigned the (undifferenced) measurement accuracy shown in Table
Standard measurement accuracies used for covariance simulations.
Measurement | Accuracy (1 |
---|---|
GPS pseudorange | 5 m |
UWB range | 0.5 m |
Bearing | 0.5° |
The UWB error of 0.5 metres is based on previous tests conducted with UWB radios under the assumption that nothing is being done to estimate or calibrate the various systematic errors present in current UWB radios [
The bearing sensor accuracy was selected to represent what is expected on commercial vehicles in the near future [
The performance metrics used herein include availability, accuracy, and reliability.
Availability refers to the percentage of the time a system is able to provide the user with navigation solutions [
The accuracy of the solution is assessed using the estimated uncertainties in the horizontal directions, as obtained from the state covariance matrix of the least-squares or Kalman filter solution. These uncertainties are rotated from the local level frame into the frame defined by the forward direction of the vehicle (using the known road azimuth) in order to show the along- and across-track uncertainties. The motivation for this is that along- and across-track uncertainties are generally more useful for making vehicle-centric decisions such as “is the vehicle ahead in the same lane as I am?” or “is there a vehicle in the lane next to me?”
Finally, reliability refers to the ability to detect blunders in the measurements and to estimate the effects of undetected blunders on the navigation solution. Reliability is further divided into internal and external reliability, both of which are based on assumed probabilities for rejecting a good observation (5% in this work) and accepting a biased observation (10% in this work). Internal reliability refers to the smallest measurement bias—often called the marginal detectable bias (MDB)—that can occur and still be detected. In contrast, external reliability is the worst case error in the estimated parameters that result from an MDB occurring on a single measurement at a given epoch and is typically called a protection level (PL). Similar to the accuracy assessment, the protection level is computed in the along- and across-track directions for convenience.
Details regarding how to compute reliability parameters can be found, for example, in Leick [
As mentioned earlier, most of the analysis focuses on the least-squares results, as this allows for the most obvious assessment of the benefit of UWB and bearing measurements. Kalman filtering results are discussed only briefly towards the end to give an idea of the possible improvements that could be achieved through modelling of vehicle dynamics and use of observations from multiple epochs.
Also, given the number of parameters at play, it is impossible to show all results in a concise manner. As such, only the most interesting findings are included explicitly. Whenever possible, however, comments regarding other findings are described in the text.
Before continuing, unless stated otherwise, inclusion of bearing measurements assumes the azimuth of the vehicle making the measurements is known.
Figure
GPS-only availability for different network sizes as a function of maximum elevation angle for road azimuths of 0° and 90°.
Another important point to note in this figure is that the 0° road azimuth is considerably worse than for 90°. This is because the GPS satellite orbit inclination angle of 55° results in GPS satellites rarely being visible directly to the north for users in Calgary (simulated location). In a north-south urban canyon, there are effectively no satellites directly to the north, few to the south, and the satellites on either side (east or west of the user) are obstructed by buildings. In contrast, when the canyon is oriented east-west, there is good along-track visibility (satellites to the east and west of the user) while the canyon blocks the north and south, where there are fewer satellites to begin with. The effect is particularly visible for the steepest canyon (60°) where there is very little availability for the 0° azimuth case and over 60% in all cases for the 90° azimuth canyon.
If other satellite-based navigation systems were incorporated into the solution (e.g., GLONASS, Galileo, Beidou/Compass, etc.), the availability would improve, perhaps significantly. This is not considered here in order to keep the analysis more tractable and because currently, GPS is the only system available on most vehicles—although this is changing. In addition, if the availability benefits of adding other systems can be quantified, the results in this paper can still be used to infer the relative benefits of UWB and bearing sensors. For example, consider the left plot in Figure
To better illustrate the solution availability differently, Figure
Solution availability of each vehicle as a function of the number of vehicles in the network for a 0° road azimuth and a 45° maximum elevation angle.
Having shown the GPS-only availability, Figure
Solution availability for the 0° road azimuth for GPS + UWB, GPS + bearing, and GPS + UWB + bearing as a function of maximum elevation angle and number of vehicles.
Also, with both observation types added there is effectively 100% availability except for the case with the most severe elevation mask. This is not surprising since a range and bearing between two vehicles are sufficient to determine the relative
Finally, unlike the GPS-only case, as more vehicles are added the availability tends to improve. This is because each vehicle that is added now also results in several additional UWB and bearing measurements (recall that each vehicle is ranging to all others and each new vehicle added has bearing measured to each existing vehicle in the network).
Since the estimated accuracy will vary over time due to changes satellite geometry and also because of a vehicle’s location relative to the simulated buildings, it is impractical to show all results here. Instead, the 95th percentile uncertainties (standard deviations) are plotted. Furthermore, these percentile values are computed across all vehicles in the network.
Figures
95th percentile along-track and across-track accuracy using GPS as a function of the number of vehicles in the network and the maximum elevation angle for a road azimuth of 0°.
95th percentile along-track and across-track accuracy using GPS as a function of the number of vehicles in the network and the maximum elevation angle for a road azimuth of 90°.
The other point to note from Figures
Before moving on, given that the 0° road azimuth case is the worst-case scenario, it follows that this is when UWB and/or bearing data will be most important, that is, offer the most improvement. As such, for the rest of the paper, only results for the 0° road azimuth case are shown with the understanding that the 90° road azimuth results are better.
Figure
95th percentile along-track and across-track accuracy using GPS + UWB (left) and GPS + bearing (right) as a function of maximum elevation angle for road azimuths of 0°.
Adding bearing data results in a similar overall position accuracy improvement as when adding UWB data, but this time mainly in the across-track direction (instead of the along-track direction). The across-track improvement using bearing data is better than the along-track improvement due to UWB range because the assumed measurement accuracy of the bearing. Specifically, the “across-track ranging accuracy” for bearing is 0.5° × 15 m = 0.17 m compared to 0.5 m for UWB in the along-track direction. The 15 m values used in the bearing calculation is the largest distance from any one vehicle to its nearest neighbour (actually 14.14 m) and in many cases the distance is smaller resulting in an even smaller error.
Finally, although not shown, including both UWB and bearing data yields horizontal solutions that are accurate to better than one metre, regardless of the elevation mask of the urban canyon. This is because a horizontal position solution can be determined using
Similar to the accuracy results presented above, the reliability results are presented as 95th percentile values as a function of maximum elevation angle and number of vehicles in the network. The top two plots in Figure
95th percentile along-track and across-track protection levels and protection level availability using GPS as a function of maximum elevation angle for a road azimuth of 0°.
Similar plots for when UWB or bearing observations are used in the solution are shown in Figure
95th percentile along-track and across-track protection levels and protection level availability using GPS + UWB (left) and GPS + bearing (right) as a function of maximum elevation angle for a road azimuth of 0°.
Finally, Figure
95th percentile along-track and across-track protection levels and protection level availability using GPS + UWB + bearing as a function of maximum elevation angle for a road azimuth of 0°.
In the above results, it was assumed that the azimuth of any vehicle making a bearing measurement was known. In this section, the case where the azimuth(s) of bearing-measuring vehicle(s) need to be estimated is considered. Figure
Effect of unknown vehicle azimuth on the 95th percentile accuracy (left) and protection levels (right) in the along- and across-track directions using GPS + bearing as a function of the number of vehicles in the network for a road azimuth of 0°.
For the reliability, the along-track performance is effectively the same regardless of whether the vehicle azimuth is known or not. This follows from the fact that bearing observations primarily affect the across-track solution. By extension, the across-track protection levels when the vehicle azimuth is unknown are observed to increase by a factor of approximately two compared to the azimuth-known case.
All of the previous results were based on the use of a least-squares estimation algorithm even though most practical systems are expected to use some form of Kalman filtering. Therefore, some results obtained using a Kalman filter are shown here to give an idea of the level of improvement that may be possible relative to the least-squares case. However, given the additional parameters/factors associated with Kalman filtering—primarily the type of stochastic model(s) selected and the level of process noise—a comprehensive assessment is beyond the scope of this paper. The results presented should therefore be interpreted as an
Kalman filter model parameters.
State | Model | Parameters |
---|---|---|
Position | Integrated velocity | N/A |
GPS clock bias | Integrated clock drift | N/A |
Horizontal velocity | First-order Gauss-Markov | ( |
Vertical Velocity | First-order Gauss-Markov | ( |
GPS clock drift | First-order Gauss-Markov | ( |
Although not shown, for the GPS-only case, the least-squares accuracy (95th percentile) was always greater than 400 m regardless of the number of vehicles in the network. However, when a Kalman filter was used, the accuracy improved to approximately 25 m. Accuracy results for the case where UWB or bearing data is used are shown in Figure
Least-squares and Kalman filter 95th percentile along- and across-track accuracy using GPS + UWB (left) and GPS + Bearing (right) for a 0° road azimuth and a 45° maximum elevation.
In terms of reliability, the GPS-only results are largely unaffected when using a Kalman filter because of the relatively loose dynamics constraints in Table
Least-squares and Kalman filter 95th percentile along- and across-track protection levels using GPS + UWB (left) and GPS + bearing (right) for a 0° road azimuth and a 45° maximum elevation.
This paper investigated the effect of adding UWB range and bearing measurements to differential GPS as a means of determining the relative horizontal positions of various sizes of networks of moving vehicles.
The main conclusions that can be drawn from the results are as follows. Solution availability benefits considerably from the inclusion of UWB and/or bearing data. In addition, larger benefits are observed as more vehicles are added to the network. UWB measurements provide the largest improvement in the along-track direction while bearing measurements provide the largest improvement in the across-track direction. For scenarios with elevation mask angles 45° or less, augmenting GPS with either sensor provides along- and across-track positioning accuracies (95th percentile) of less than 100 m while the GPS-only solution often exceeds 200 m. When both UWB and bearing data are used, the solution accuracy was always at the metre level due to the high quality of these measurements and the fact that a range and bearing can determine the relative horizontal position without and GPS data at all. Although the GPS-only protection levels were reasonable, the number of epochs where reliability checking was possible in the simulated urban canyon was highly compromised. Including UWB or bearing observations not only improved the protection levels, but also provided a much higher probability of having a redundant set of observations. In addition, using either sensor with GPS showed that reliably continued to improve as more vehicles were added to the network. When GPS was augmented with both UWB and bearing data, as long as there are at least three vehicles in the network, the protection levels are below 2 m and the availability of redundant solutions is nearly 100% even with a GPS elevation mask angle of 45°. If the azimuth of bearing-observing vehicles needs to be estimated, the accuracy and reliability of the solution decreased by approximately a factor of two.
This paper presents covariance simulation results only. Preliminary field tests have been conducted to test a three-vehicle V2V scenario [
The research presented in this paper was conducted as part of a collaborative research and development grant between the University of Calgary, General Motors of Canada, and Natural Sciences and Engineering Research Council of Canada. Additional funding was also provided by Alberta Innovates Technology Futures (formerly Alberta Ingenuity Fund).