1. Introduction
Angle of arrival (AOA) measurement technologies for electromagnetic waves are employed in a wide range of applications. Both biomedical and military radar technology aim to measure not only the distance, but also the angle to objects reflecting electromagnetic waves. Wireless communication equipment, such as Wi-Fi routers, often estimate the angle of arrival to beamform their radiation to other network devices, and aircrafts use the AOA measurements to stationary beacons to localize themselves by means of automatic direction finders.
Traditionally, AOA measurement techniques require either a rotating directional antenna or multiple locations at which the signal is measured. These multiple measurement points are realized either by moving a single antenna to different locations while collecting signal measurements, or by multiple antennas, e.g., in the form of an antenna array. Often, these antenna arrays have a limited field from which they can estimate the AOA of a signal and therefore need to be additionally rotated. This leads to high performance, but also electronically and mechanically more complex and more costly devices, which has made AOA localization approaches less suitable for low-cost ultra-wideband (UWB) localization systems [
1].
The AOA measurement technique presented in this paper proposes to make use of the angle-dependent antenna transfer function, which manifests itself in the measured channel impulse response (CIR). Doing so, it enables estimation of the angle of arrival using a single, static antenna, requiring no additional hardware. Compared to our preliminary work [
2] we provide more detailed analytical and experimental results, showing
that the AOA estimation method is not bound to a specific environment,
that the method also works without reflective surfaces in the receiver antenna’s vicinity if the antenna is chosen accordingly, and
how the proposed method can be integrated with existing time of flight (TOF)-based localization methods, with which training data can be acquired on the go.
Furthermore, the data from the experiments presented in this paper are made available [
3], such that other estimation strategies could be tested on them.
1.1. Outline
This paper is structured as follows: In the remainder of this section, we review related work.
Section 2 discusses how the antenna transfer function and the environment influence the measured CIR between a transmitter and a receiver. Based on these insights, a machine learning approach mapping a measured CIR to an AOA probability distribution is presented in
Section 3. The data on which the method is tested is discussed in
Section 4 and results are shown in
Section 5. The method is then applied to a localization problem in
Section 6. Concluding remarks are made in
Section 7. Note that throughout the paper a two-dimensional problem setup is considered, if not stated otherwise.
1.2. Related Work
An overview of AOA estimation methods is given in [
4]. Common methods measure the phase difference of arrival of a signal with two or more antennas integrated into an antenna array (as for example done in [
5]) or employ beamforming techniques to steer the main radiation lobe(s) of the antenna array towards the angle of arrival [
6]. Algorithms fusing the outputs of such multi-port antenna arrays are discussed in [
7,
8]. Aside from conventional algorithms, recently deep learning has also been applied to process the output of antenna arrays for estimating the AOA. Its advantages compared to traditional methods when estimating multiple signal sources and their AOA are discussed in [
9]. Beamforming control by means of deep learning is discussed in [
10] where it is also shown how neural network quantization facilitates the deployment of such deep learning techniques in low-memory, low-overhead platforms such as mobile phones.
Modern Wi-Fi modules are often equipped with two or more antennas and provide the phase shifts of the different sensing elements in the channel state information (CSI). Impressive localization results based on such CSI measurements were obtained in [
11], where it is shown how localization can be achieved via multipath triangulation and time-of-flight-difference (TOFD) measurements using a single Wi-Fi module employing three antennas.
Alternatives to multi-port antenna arrays for beam steering and AOA estimation include single-port switched parasitic antenna arrays (as for example discussed in [
12]) or rotating directional antennas or antenna arrays, as employed by classical radars where each direction is scanned for an incoming signal [
13].
Instead of employing rotating antennas or antenna arrays, it is also possible to estimate the AOA by collecting signals from the same source sequentially at multiple locations or during movements similar to the synthetic aperture radar. This principle is utilized in [
14] to estimate the AOA using received signal strength measurements, and in [
15] using phase measurements, collected at different locations.
Compared to the previously discussed approaches to estimate the AOA, the method proposed herein only relies on CIR measurements acquired by a single static antenna at a single location using no actively controlled parasitic elements. It is based on the angle-dependent antenna transfer function, which leaves its mark in the measured CIR of a UWB propagation channel. UWB propagation channels in general are discussed in [
16], and in [
17,
18,
19] with a focus on the antenna transfer functions. Distortions of the measured CIR due to angle-dependent antenna transfer functions can lead to angle-dependent errors in the timestamps provided by leading edge detection algorithms. In turn, these angle-dependent errors in the timestamps lead to errors in the TOF or TOFD measurements. So far research in UWB localization has therefore focused either on tailored UWB antenna design [
20,
21] or on mitigating these effects via models predicting the systematic error. A neural network predicting the error in the TOF measurement based on CIR measurements was employed to this end in [
22] while in the authors’ previous work [
23] this error is predicted based on the AOA. Instead of compensating for effects of the angle-dependent transfer functions, this paper proposes to amplify them such that they can be exploited to estimate a signal’s AOA.
4. Acquiring the Datasets
In addition to the datasets used to qualitatively assess the AOA dependency of the measured CIR as discussed in
Section 2, datasets using multiple anchors and changing environments were also collected. To this end, a Roomba robot equipped with a UWB tag drove around in a random fashion in an area of
while recording the output of its wheel encoders with a frequency of 66
and sequentially collecting CIR and range measurements to five anchor modules with a frequency of 200
. Again the ranging protocol presented in [
27] was employed and the CIR of the last anchor reply was recorded. The anchor modules were equipped with Broadspec Time Domain antennas because of their constant antenna impulse response function over different angles as visible in
Figure A1. This limits the influence of the AOD on the measured CIR, which facilitates the CIR to AOA mapping problem. The antennas were placed
above ground around the area the Roomba robot was driving in, such that range measurements ranging from
to 9
were obtained. Different obstacles, i.e., a chair, a table, a wooden wall, a ladder and a tripod, were placed in the area. For each collected dataset containing roughly 200’000 CIR and range measurements, the Roomba robot was traveling a different, random trajectory and either the locations of the anchor modules, or the locations of obstacles within the area were changed as shown in
Figure 5. This figure also shows a picture of the floor, made of ceramic tiles and heavy metal plates, partially reflecting the UWB signals. Sport mattresses were placed on the floor to facilitate the Roomba robot’s locomotion. During the experiment, an overhead motion capture system recorded the ground truth position and orientation of the tag and anchor antennas with sub-centimeter and sub-degree accuracy. Synchronizing and processing all this data allows pairing of CIR measurements with the corresponding AOA to generate training and evaluation datapoints
for the previously presented neural network.
For ten such datasets, the UWB tag on the Roomba robot was equipped with the modified spline antenna; and for a further ten such datasets, the Partron dielectric chip antenna with mounted carbon plates in its vicinity was used, as shown in
Figure 2. These datasets are made publicly available here [
3].
5. Results
The neural network described in
Section 3 was trained and evaluated with the previously described datasets, separately for the modified spline antenna and for the Partron dielectric chip antenna with carbon plates in its vicinity. From the ten datasets originating from different setups, nine datasets were used for training and the evaluation was made on the remaining dataset. This was done ten times each time leaving out a different dataset in training for the following evaluation. The results of this leave-one-out cross-validation are presented by means of the error in the maximum a-posteriori AOA estimate
. It is given as the bin center of the bin with the highest probability as predicted by the neural network, i.e.,
As discussed in
Section 2, the resolution of the measured CIR is relatively coarse at approximately 1
. However, as it sampled at slightly different locations is each time, a more accurate maximum a-posteriori estimate can be found by collecting 10 consecutive CIR measurements
from the same transmitter, and choosing the estimated AOA bin as
Figure 6 shows the error distribution for the modified spline antenna. Averaged over the datasets, 58% of the maximum a-posteriori estimates have an error of less than 15
when using only one CIR measurement. When using 10 consecutive CIR measurements, this ratio is increased to about 64.5%, which is still significantly lower than the ratio for the AOA estimation modules based on multiple antennas created by Ubisense [
39] and Decawave [
5]. Almost 100% of their AOA measurements are reported to be within a 15
error bound.
However, the distribution predicted by the neural network is in general multimodal. Therefore, even if the maximum a-posteriori estimate might deviate by a large value, the probability of the bin corresponding to the ground-truth AOA
might still be high. This is visible in
Figure 7 where the average predicted a-posteriori AOA probability distribution for CIR measurements belonging to the datapoints
obtained with a ground truth AOA between
and
is shown, i.e.,
where
is used to denote the number of datapoints contained in set
. On the right, the average predicted a-posteriori AOA probability distribution is shown for the subset of these datapoints
, whose maximum a-posteriori AOA estimate deviates by more than 30
, i.e.,
It is evident that most maximum a-posteriori AOA assigned to these CIR measurements are at around
, even though the probability of the AOA corresponding to the actual AOA is still high. This can also explain the bump at around
in maximum a-posteriori error distribution, as seen in
Figure 6. Looking at
Figure 3, it is evident that the antenna impulse response function for the AOAs from
to
seems to be similar to the one for the AOAs from
to
. Therefore, the neural network has difficulty in mapping the measured CIR to the correct AOA. However, this uncertainty is also mirrored in the probability distribution given by the neural network.
Similar results were achieved with the Partron dielectric chip antenna with mounted carbon plates in its vicinity, as further discussed in
Appendix C, which shows that material in the antenna’s vicinity influencing its radiation pattern also helps to estimate the AOA with the proposed method.
Although it may at first appear contrived, in the majority of applications the antenna’s radiation pattern is distorted, either because the antenna is integrated into the device case, or because the device case and the device electronics reflect and dampen electromagnetic waves in different manners depending on the device orientation. However, these unintentional angle-dependent radiation patterns lead in general to multimodal probability distributions, as is also the case for the tested modified spline antenna and the Partron dielectric chip antenna with carbon plates. In order to improve accuracy of the maximum a-posteriori AOA estimate, the antenna design or the placement of the reflective surfaces should be optimized, which was not done in this work. Nevertheless, there are applications where multimodal distributions pose less of a problem, e.g., when it is possible to fuse multiple AOA distributions from different transmitters or receivers, as is the case for UWB localization problems, which is discussed in the next section.
6. Application to a Self-Localization Problem
In this section, the previously described method to estimate the AOA based on CIR measurements is used to localize a robot.
6.1. Self-Localization Problem
The datasets described in
Section 4 were collected with the help of a mobile robot (Roomba) and consist of CIR, range and odometry measurements along with the ground truth measurements provided by a motion capture system. Given the trained neural network as outlined in
Section 3, the CIR measurements with anchors at known locations provide sufficient information for the robot to estimate its state
in the inertial reference frame, where
and
are the robot’s Cartesian coordinates and
is the angle describing its orientation (see
Figure 8). The state estimate can be obtained via triangulation as visualized in
Figure 8, e.g., by maximizing the measurement likelihood
In order to self-localize, the robot does not need to move as long as its position and the position of the anchors cannot be circumscribed with a circle [
40]. This enables the robot to self-localize by only receiving the UWB signals from transmitters with a known location, which do not even need to be synchronized. It is clear that if time-of-flight measurements or time-of-flight-difference measurements with respect to the anchors are available, they significantly improve the performance of such a localization system and should therefore be fused with AOA measurements. The same applies to motion or process models, which should be used as well if available.
In the following, we will investigate the fusion of this information by means of a particle filter in order to assess the benefit of estimating the AOA with the proposed method in self-localization applications. Furthermore, such fusion approaches also allow the neural network to be trained without a motion capture system, as demonstrated in the following. A general comparison of time-of-flight, AOA, and received signal strength localization approaches is given in [
1].
6.2. Particle Filter
Two discrete-time process models for the mobile robot are considered, where is the system’s input and is the process noise at discrete time for a sampling period of 15 , which is equal to the period with which the Roomba robot’s wheel encoders can be recorded.
6.2.1. Random Walk Process Model
In the random walk process model, the system input
is assumed to be zero and the state is assumed to evolve solely based on the process noise, i.e.,
The process noise is assumed to have a zero mean normal distribution, i.
6.2.2. Roomba Process Model
In the more accurate Roomba process model, the robots state
is pushed forward by the robot’s odometry recordings
, where
and
are the measured distance travelled and the measured change in heading, respectively, during the sampling period. This process model is given as
For this process model, the process noise covariance is lowered to
6.2.3. Measurement Model
Measurement updates can be performed either with a-posteriori AOA probability distributions provided by the neural network, or with time-of-flight and the corresponding range measurements. These updates are further described in the following algorithm.
6.2.4. Particle Filter Algorithm
How these process and measurement models can be integrated in a particle filter is briefly outlined in the following summary, and the reader is referred to [
41] for a more in-depth introduction.
Initialization: The particle filter is initialized with particles whose initial x, y coordinates and headings are drawn from the uniform distributions , and .
Prediction step: At each iteration, the random walk (
15) and (
16) or the Roomba (
19) and (
20) process model is used to update each particle
as
Measurement update: When a UWB signal is received, the particle weights can be updated according to their likelihood given the current AOA a-posteriori probability distribution or the current range measurement. Using the AOA a-posteriori probability distribution, the particles weights are calculated as
where the expected AOA
of each particle p is
wherein
and
are the
x and
y coordinates of the anchor modules from which a signal is received. If the range measurement is used, the particle weights are calculated as
with
the measured range with a variance of
, and where the expected range
of each particle
p is calculated as
After the particle weights have been calculated, the particles are resampled to get
posterior particles, all with equal weights.
6.3. Training with Particle Filter AOA Data
So far, the training data for the neural network was obtained by means of a motion capture system, i.e., the AOA corresponding to a measured CIR was calculated based on motion capture data. With these training data, the neural network described in
Section 3 was trained. However, the AOA corresponding to a measured CIR can also be obtained by other means; namely, based on the estimated state by the particle filter fusing odometry and range measurements as outlined above. The data obtained might be of lower quality, i.e., the AOA assigned to a measured CIR might deviate if the estimated state also deviates. Nevertheless, as long as the data is unbiased, the neural network can be successfully trained with it.
To investigate this, ten new training datasets were generated for the modified spline antenna where the AOA was not provided by the motion capture system, but by the state estimate of the particle filter employing the Roomba process model to fuse odometry and range measurements. The particle filter’s position,
, and orientation,
, estimates are defined to be the particles’ average position and orientation, respectively. Using these estimates, the AOA corresponding to a CIR measurement obtained at time
k was calculated as
These ten new datasets were again used to train the neural network in a leave-one-out cross validation fashion. The error distribution of the maximum a-posteriori AOA estimate does not differ significantly from the distribution obtained with neural networks trained on motion capture training data, but the value of the maximum probability density is generally smaller. This can be explained by the additional noise now included in the training data, which acts as a regularizer and leads to more conservative, i.e., more uniform, a-posteriori AOA probability distributions predicted by the neural network.
6.4. Results
These newly trained neural networks were employed by the particle filter whose performance was evaluated for the two different process models, and the different measurement updates. A leave-one-out cross validation is again applied for the evaluation, in which the dataset not used for the neural network training was used for the evaluation.
Figure 9 shows the root mean square error (RMSE) in the heading,
, and position,
, estimates of the particle filter for the different configurations. The RMSE for each dataset is shown in a different color.
In case of the random walk process model, it is apparent that the orientation and position of the Roomba robot can be estimated solely based on AOA measurements in all ten datasets. When employing range measurement updates instead of AOA measurements, the error in the position estimate is significantly smaller. However, it is no longer possible to observe the orientation of the robot. The best performance is achieved when range and AOA measurements are combined. Note that two-way communication between the mobile robot and the anchors or clock synchronization is necessary in order to obtain range measurements, whereas the AOA estimation method presented only needs one-way communication and no clock synchronization.
In the case of the particle filters employing the Roomba process model, the orientation of the Roomba robot is also observable without AOA measurements. This allowed the neural network to be trained without motion capture data as described in
Section 6.3. Note that even though the RMSE in the heading of this particle filter employing the Roomba process model and range measurement updates was between 4
and 13
depending on the dataset (see column marked with * in
Figure 9), neural networks trained with these data and integrated in a particle filters lead to RMSE in the heading of below 4
as also visible in
Figure 9. Also for the Roomba process model, the best performance is achieved when range and AOA measurement updates are used.
The results of particle filters employing neural networks trained with motion capture data are included for completeness in
Appendix D.
7. Conclusions
This paper discusses a technique to estimate the AOA of a UWB signal based on CIR measurements. We identify that the antenna’s impulse response function is AOA dependent, and that objects in the antenna’s local environment create angle-dependent reflections that further affect the measured CIR. We use a neural network to learn the mapping between the CIR measurement and the AOA, and show that the UWB signal’s AOA can be estimated at no additional hardware cost, using just a single antenna, unlike conventional AOA estimation techniques. By combining AOA estimates to multiple fixed-location UWB anchors, we experimentally demonstrate the localization of a mobile robot, based only on AOA estimates obtained from CIR measurements (see
Supplementary Video), and in combination with range measurements. Given that in most real-world UWB applications the antennas’ impulse response functions are AOA dependent due to their integration into a device, we regard the AOA estimation method presented in this paper as a low-cost and software-only augmentation for any existing UWB TOF-based localization system, with which AOA-CIR training data can be collected on the go by data fusion approaches.
Outlook
The new AOA estimation principle presented in this paper should be investigated further to assess its full potential and its limitations. We regard the following topics as interesting to investigate in the future:
Hardware optimizations: In this paper, we changed the antenna’s impulse response function in a straightforward manner by modifying the antenna, or by placing carbon plates in its vicinity. Instead, better performance could be achieved if the antenna design or the placement of the reflective surfaces were optimized for the application at hand considering the selected carrier frequency and power settings. Such optimizations, performed via electromagnetic simulation software, could be aimed at rendering the a-posteriori AOA probability distribution unimodal, and at making the method more robust to non-line-of-sight conditions. Furthermore, as the CIR estimates provided by the DW1000 chips is also dependent on the clock speed of the receiver and transmitter, more stable clocks and shorter sampling periods could further help to improve the accuracy of the AOA estimation method, although this would again lead to increased hardware costs.
AOA of multi-path components: As shown in [
42,
43], if the locations of reflective surfaces in the environment are known, the timing of multi-path components can be used to localize a receiver. In this paper we trim multi-path components, and focus only on the first peak. However, these multi-path components are also affected by the receiving antenna’s AOA-dependent transfer function as shown in [
44], and it should be possible to also compute their AOAs using the techniques discussed in this paper.
Learning: The neural network applied to learn the mapping between the measured CIR and the AOA worked without significant tuning, however resulted in a binned probability distribution. It would be interesting to investigate whether accuracy could be improved using mixture-density networks, resulting in a continuous probability distribution, or using neural networks with complex weights. In the latter case, the complex envelope of the CIR could be fed directly to the neural network instead of feeding it via its magnitude and phase. This, together with a tailored network architecture could further improve the performance of this method.
AOD and AOA estimation: The CIR is affected by both the receiver’s and transmitter’s antenna, and by obstacles in their local environments. In order to minimize the influence of the transmitting antenna on the measured CIR and thus simplify measurement of the AOA, we outfitted transmitters with antennas having a very uniform transfer function (Time Domain Broadspec antenna). However, if both receiver and transmitter were equipped with antennas having strongly angle dependent impulse response functions, it should be possible to estimate both AOA and AOD from a single CIR measurement. In combination with a TOF method to estimate range, this would enable estimation of the full relative pose of the receiver with respect to the transmitter and thus localization of the receiver using just a single anchor.