I Introduction
With the growing interest in the development of autonomous driving systems, accurate positioning of vehicles in urban environments becomes the fundamental requirement to accomplish such applications. In order to achieve the current goal of autonomous driving, namely the Level 4 systems which should enable driverless operations in a restricted domain, decimeter to centimeterlevel positioning accuracy is required to maintain a vehicle on the road within its lane [12]. It has been suggested that for typical road geometries and driving scenarios, the positioning accuracy of a passenger vehicle on urban roads should be at least 10 cm, 10 cm and 48 cm in the lateral (across the road), longitudinal (along the road) and vertical directions, respectively [26]. Realizing and maintaining such high accuracy does therefore demand a rigorous integration of multiple measuring sensors. As a result, an array of multiple measuring sensors including, but not limited to, Global Navigation Satellite System (GNSS) receivers and Light Detection and Ranging (lidar) devices, is commonly found on modern vehicles. While GNSS is considered an essential positioning technology as it provides globally referenced solutions, its signals suffer from blockages and multipath effects in cities due to the high density of buildings [10, 39]. This is the more so as highprecision GNSS positioning requires the utilization of multifrequency carrier phase signals transmitted by a modest number of visible satellites, a condition that is challenging to be met in urban areas [40]. In contrast, lidar is not subject to the aforementioned error sources, providing abundant measurements in cities due to the existence of rich geometric features, yet it only offers locally referenced positioning solutions [41]. In this contribution, we therefore aim to exploit the complimentary advantages of GNSS and lidar for urban positioning with a particular focus on the ultraprecise carrier phase measurements.
Although GNSS code (pseudorange) measurements are easily accessible and served for standard positioning services, it is their carrier phase counterparts that can deliver precise parameter solutions [1, 27, 36, 10, 8, 13]. GNSS carrier phase measurements are approximately two orders of magnitude more precise than the corresponding code versions, and are crucial for achieving centimeterlevel positioning. The challenge of using carrier phase measurements, however, is that they are biased by a) unknown integervalued ambiguities and b) instrumental phase delays [27]. The latter can be eliminated by the widely used positioning technique of realtime kinematic (RTK). In RTK, observations of a nearby reference GNSS station are subtracted from those of the tobepositioned receiver to remove or largely reduce common sources of GNSS errors like clock offsets, instrumental biases and atmospheric delays [22]
. However, the former, i.e. the unknown integer ambiguities, have to be estimated as realvalued float solutions first. Depending on the precision of the float solutions, Integer ambiguity resolution (IAR) is then employed to map such float ambiguities to their correct integers. If performed successfully, IAR yields ambiguityresolved carrier phase measurements that can act like ultraprecise code measurements for positioning.
Whether or not IAR is successful is determined by the probability of correct integer estimation, the socalled ambiguity ‘success rate’
[29]. The success rate is driven by the float ambiguity variance matrix, which on its turn, is governed by the number and precision of the measurements. In the event that the ambiguity success rate is low, one must refrain from fixing the float ambiguities as it often leads to unacceptably large positioning errors. On the other hand, for the vehicle positioning case where the location of the moving GNSS receiver is highly varying in time, successful IAR is required to be carried out in an instantaneousmanner so as to maintain continuous centimeterlevel positioning. Instantaneous or singleepoch IAR is only made possible when a large number of measurements from multiple satellite systems/frequencies are available
[32], which is often not the case in densely builtup areas due to signal blockage. In such environments, complementary sensing devices are needed for additional measurement.As a prominent example of such complementary devices, lidar is capable of collecting featurerich point clouds of the surrounding environments in urban canyons, providing 3D point maps which can be used for mitigation of GNSS errors, see, e.g., [37, 38, 3]. In particular, lidar’s aiding role in GNSS IAR has been investigated by the recent studies [24, 25, 17, 18]
, which rely on features described by geometric characteristics that can be unavailable in complex environments. Meanwhile, recent advances in deep learning has offered datadriven approaches for producing point feature descriptors which are more detailed and invariant than those from the traditional methods, thus improving the chance of successful point cloud registration
[43, 7]. The goal of the present contribution is to leverage such learningbased lidar features for highprecision positioning and present a GNSSlidar integration method so as to realize successful instantaneous IAR. To enable an a priori prediction of the method’s IAR performance, we develop closedform analytical expressions for both the lidaraided float ambiguity variance matrix and the associated Ambiguity Dilution of Precision (ADOP) [28, 21]. As an intrinsic measure for the average precision of the float ambiguities, ADOP indicates the extent to which the mapping of the float ambiguities to their integers is successful. The applicability of analytical IAR measures presented are supported by experimental results, showing how lidar measurements provide conditions under which instantaneous IAR becomes successful even for the singlefrequency GNSS data of lowcost receivers.The remainder of this paper proceeds as follows. In section II, a lidarbased method using a highdefinition (HD) map for aiding GNSS ambiguity resolution is proposed. Accordingly, lidar measurements are generated by registering the measured point cloud with the HD map, which contains georeferenced scans of the road environment collected from a previous time, via a learningbased keypoint extraction strategy [42]. Such learningbased point cloud registration method is used to produce lidar observations for positioning. Section III presents the mixed measurement model with which one can integrate the stated lidar observations with their GNSS counterparts. The lidar observations take the form of matched keypoints’ coordinates, aiding GNSS observations to estimate the float ambiguities. Analytical measures for the precision of the lidaraided float ambiguities are then presented in Section IV. We thereby show how the precision of the integrated solution competes with that of the lidaronly solution to reduce the ADOP, thus improving the corresponding IAR performance. Section V presents the configurations of the simulated experiment that is followed by Section VI showing the corresponding numerical results. A discussion on the results is given in Section VII. Finally, concluding remarks are drawn in Section VIII.
Ii Lidar observations generated by point cloud registration
The key to enabling GNSS carrier phase ambiguity resolution is to improve the precision of float solutions with redundant measurements. In this paper, lidar measurements are integrated with GNSS code observations by employing the Weighted Least Squares (WLS) method to obtain the float solutions. To achieve the ‘minimumvariance’ float solutions, the weight matrix underlying WLS is taken as the inverse of the measurements’ variance matrix [30]. The measurements are produced by registering rover (online) scans collected by the laser scanner onboard a vehicle with reference (offline) scans obtained from a prebuilt HD map. The corresponding keypoints are extracted to estimate a rigid transformation which provides the position of the origin of a rover scan, i.e., the vehicle, provided that the laser scanner is calibrated to align with the center of the vehicle. This lidar positioning method has been shown to be effective in terms of availability and accuracy of positioning in our previous work [42]. Fig. 1 depicts the workflow of the proposed lidaraided ambiguity resolution method, in which the lidar observation generation discussed in this section is reflected in the steps in the top left part.
Several coordinate systems are used throughout the following sections. We define eframe as the geocentric WGS84 frame, which is used for all GNSS measurements and the computations of the positioning solutions. The point clouds are originally collected in a Cartesian frame with the sensor at its center, namely the lframe. To integrate the generated lidar measurements with GNSS, reference scans in the HD map are transformed and aligned to a local frame cframe with an arbitrary origin first, then to the eframe.
Iia HD map definition
An HD map includes information needed for vehicle positioning and can take various forms [19]. In this research, we use an HD map that utilizes accurately georeferenced point clouds of the road environment collected from a previous time. Each point cloud is stored in its original lframe with the laser scanner as the origin, as well as a georeferencing matrix to transform it to eframe so that the positioning solutions are in this coordinate system. Such point clouds that make up of the HD map are referred to as reference scans and must be available prior to the vehicle positioning tasks. A procedure for preparing an HD map using KITTI dataset [6] is provided in Section VB.
IiB Learningbased keypoint extraction
The registration between a pair of rover and reference scans requires correctly corresponded keypoints. For fast and accurate keypoint extraction, we use MSSVConv, a multiscale deep neural network that outputs feature vectors from point clouds
[11]. A model needs to be pretrained on a large number of point clouds with ground truth alignments before it can be used to compute feature vectors.Assuming that for one epoch a rover scan is collected by the lidar sensor and a nearby reference scan which shares an overlap is identified from the HD map using position estimates from less demanding techniques such as Standard Point Positioning (SPP), the pretrained MSSVConv model is applied to produce the feature vectors per point, based on which the keypoint matching is performed. Since keypoints extracted by MSSVConv often contain incorrect correspondences, a subset of keypoints with their feature vectors are selected to register the two point clouds using Random Sample Consensus (RANSAC) [4]
, which finds an outlierfree set of keypoint correspondences to estimate a transformation matrix that georeferences the rover scan from
lframe to eframe. Positions of successfully matched keypoints are then used as lidar observations for vehicle positioning.IiC Lidar observation equations
The lidar observations are constructed by the estimation of the transformation from the rover scan to the reference scan, in which the translational parameters are identical to the vehicle position. The coordinates of the keypoints from the rover scan in lframe serve as the measurements, corresponding with the known coordinates of their matched counterparts from the reference scan in eframe. Thus, each keypoint provides 3 observations. For pairs of matched keypoints, the observation equation of lidar measurements is therefore as follows
(1) 
in which denotes the Kronecker product [9], and is a matrix of ones. is the vector of the estimated vehicle position in eframe and is a matrix of the unknown rotational parameters. The Jacobian matrix is given by . By defining as the vector of the measured coordinates of one keypoint in lframe and concatenating all the keypoints, the vectors of measured keypoint coordinates, measurement residuals and known keypoint coordinates, namely , and , are formed as
For integration with GNSS observations in the WLS method which will be discussed in the next section, the uniform weight matrix of lidar measurements is defined by
(2) 
with the root mean squared residual distance , in which is the residual distance between a registered keypoint and its correspondence in the reference scan in the RANSAC estimation.
Iii GNSSlidar integration and ambiguity resolution
In this section, the model integrating lidar and doubledifferenced (DD) GNSS observations to deliver the float solutions including ambiguities and using them for IAR is discussed. It is also reflected in the top right and bottom parts of Fig. 1. We base our method on the assumptions that the collected lidar and GNSS measurements are timesynchronized and are both referenced from the center of the vehicle.
Iiia Doubledifferenced GNSS observation equations
Using to denote the vectors of DD observedminuscomputed code and carrier phase measurements for visible satellites and frequencies collected in one epoch, respectively, the two can be concatenated as . Assuming that the baseline between the GNSS receiver installed on the vehicle and the reference station is short enough so that the DD atmospheric delays can be ignored, the linearized GNSS observation equation is given as [14, 13]
(3) 
where with diagonal matrix links the DD ambiguities with the GNSS measurements by the wavelength of each observed frequency. The DD satellitetoreceiver range vector can be linearized as , with being its approximate version. The coefficient matrix has the dimensions of , with the matrix containing the satellitetoreceiver direction unit vectors, and the matrix forming the betweensatellite differences. Therefore, is a vector of the increments to the unknown vehicle position. Similar to the lidar observation equation (1), the vector contains the residuals of the GNSS code and carrier phase measurements.
The GNSS measurements are weighted according to the satellite elevation angles . The dimensionless weight matrix for undifferenced GNSS measurements can hence be constructed as
(4) 
Accordingly, the weight matrices of DD code and carrier phase measurements are given as and , respectively, where and
denote the zenithreferenced standard deviations of the undifferenced code and phase observations
[16]. The phase observations are assumed to be 100 times more precise than their code counterparts in our implementation. Thus , with being the phasetocode variance ratio. The factor in both expressions indicates that the variances of the measurements are doubled due to differencing.IiiB Float solution by GNSSlidar integration
With the observation equations (1) and (3) in place, which share the common unknown vehicle position , we can estimate the float solution using the WLS principle. Due to the intertwining of the unknown rotational parameters and the measurements in the lidar observation equations, the mixed measurement model is employed to perform the estimation [42, 30]. The mixed model combining measurements from the two sensors can be formed as follows
(5) 
where the vector of unknown parameters , with being the vector form of a close approximation of , contains the DD ambiguities, the vehicle position and the rotational parameters of the rover scan. and are the vectors of measurements and residuals. The mixed model is linearized using the firstorder Taylor expansion of about the point to give the expression on the right hand side of (5), in which is the approximated version of in each iteration, giving the unknown increment vector . Hence, . The Jocabian matrices and for and are given as follows
(6) 
with denoting a block diagonal matrix. The Jacobian submatrices are given by , and , in which matrix is formed by the lidar measurements (). Application of WLS to the mixed model (5) gives the float increment solution and their variance matrix as
(7)  
with . The weight matrix is constructed as . The float solution is iteratively computed as , in which is replaced by the solution from the previous iteration. The solution convergence is considered reached by a stopping criterion, for instance, when the magnitude of is smaller than a specified threshold.
IiiC Integer ambiguity resolution
The float solution obtained from (7) contains the estimated vehicle position (), DD ambiguities which are realvalued () and the rotational parameters for the georeferencing of the rover scan (). For simplicity, we use to denote the nonambiguity unknown parameters here. In order to utilize the highly precise carrier phase measurements for positioning, the float ambiguities need to be mapped to the correct integers using an ambiguity resolution method to yield the fixed solution [32]
, provided that the estimated float parameters follow the normal distribution:
(8) 
In this paper, we use the well known LAMBDA method for IAR, which employs the Integer LeastSquares (ILS) ambiguity estimator [27]. By assessing the float and , LAMBDA outputs the fixed integer ambiguities , as well as the evaluated formal success rate [35]. In order to determine whether the fixed ambiguities should be accepted, an acceptance test is needed. For example, the formal success rate can be required to be higher than a given threshold such as 99.9%. Otherwise, the float solution is retained. Once is accepted, the fixed solution of the remaining parameters, say and its variance matrix , can be obtained by [27]
(9) 
where the first 3 elements of are the fixed position .
Iv Theoretical performance assessment using ADOP
So far we have discussed the model to integrate GNSS and lidar measurements for IAR. To provide a prediction of the method’s IAR performance before taking any measurements, one needs the variance matrix of the float ambiguity solutions. Such variance matrix enables us to a priori evaluate the ambiguity success rate and quantify the IAR performance using its associated ADOP, which indicates the upper bound of the ambiguity success rate [28, 36]. In this section, we therefore present closedform expressions for both the ambiguity variance matrix and its ADOP. Given such closedform analytical measures, we will then evaluate the formal ambiguity success rates and ADOP of the proposed lidaraided model with the aid of PsLAMBDA software [34].
Iva Variance matrix of the float solution
Firstly, the expressions of the precision of float ambiguities and positioning solution are developed to support the evaluation. Applying the WLS principle yields the normal matrix of the estimated parameters for lidar as [30, 42]
(10) 
with . The reduced normal matrix of the estimated position is therefore , which leads to the variance matrix of the lidaronly positioning solution as follows
(11) 
On the other hand, the variance matrix of the float solution obtained with GNSS code observations is given as
(12) 
with the projection matrix . Therefore, (11) and (12) can be combined to give the variance matrix of the integrated float solution:
(13) 
Likewise, the variance matrix of float ambiguities computed using both code and lidar measurements takes the following form
(14) 
IvB ADOP of lidaraided ambiguity resolution
Next to to formal ambiguity success rates, one can also evaluate the ADOP of one’s measurement model to assess the underlying IAR performance. ADOP is defined on the basis of the ambiguity variance matrix as follows [28]
(15) 
where denotes the determinant of a matrix. The smaller the ADOP, the higher the ambiguity success rate becomes [13, 31]. For a minimum success rate of 99.9%, ADOP should be smaller than 0.12 cycles, whereas for 99%, it should be smaller than 0.14 cycles [21]. For the singleepoch GNSSonly model (3), ADOP can be expressed as [21]:
(16) 
with , and , where is the diagonal entries of . In order to compare the ADOP of GNSSonly (16), denoted by , with that of the lidaraided method, say , one can express their ratio as follows (see Appendix)
(17) 
where the eigenvalues
() are the roots of the characteristic equations(18) 
The ADOPratio (17) tells us the extent to which the ADOP of the lidaraided model is smaller than that of the GNSSonly model given in (16). The first expression of (17) indicates that the precision of the integrated solution in (13) competes with that of the lidaronly solution in (11) to reduce the ADOPratio. Consider the hypothetical scenario where the precision of the lidar measurements is extremely poor (i.e., ), the ADOPratio reduces to 1, that is, . This would imply that the lidar measurements do not contribute to the integrated solutions. Now consider another extreme case where the lidar measurements are significantly more precise than the GNSS code measurements so that the precision of the integrated solutions becomes almost identical to that of the lidaronly solutions, i.e., or . Given the small phasetocode variance ratio , such extreme case would therefore make the ADOPratio close to zero. This again makes sense, showing the more precise lidar measurements the smaller the ADOPratio (17).
The second expression of (17) reveals the link between the IAR performance of the GNSSlidar integrated solution and the eigenvalues defined in (18). These eigenvalues are in fact the stationary values of the objective function [28], with being an arbitrary unit direction vector. The smallest one, i.e., , indicates the minimum reduction in the variance of the lidaronly positioning solution when GNSS code data is integrated with the lidar data. Likewise, the largest eigenvalue indicates the maximum reduction in the variance of the positioning solution.
Each of these three eigenvalues can be served to ‘approximate’ the ADOPratio (17). For instance, by assuming that the eigenvalues are equal to one another, i.e. (), three different approximate versions of (17) are made as follows
(19) 
Fig. 2 presents the ADOPratio (17) (solid lines) and its three approximate versions (19) (dashed lines) as functions of the number of visible satellites when a singlefrequency GNSS receiver is aided with 44 correctly matched lidar keypoints, which is the empirical minimum number of keypoints per epoch found in our experiment (cf. Section VI). The phasetocode variance ratio is assumed to be . In Fig. (a)a, precise lidar measurements are considered with m, which is the mean value obtained from our experiment, whereas Fig. (b)b shows the results for less precise lidar measurements with m, which is approximately the maximum point spacing in point clouds used in the experiment collected by a Velodyne HDL64E scanner [33]. In either case, it is observed that the ADOPratio (17) gets closer to 1 the higher the number of satellites, meaning that the lidarintegration cannot contribute to the GNSSonly IAR performance by much when the receiver tracks a rather large number of satellites (e.g., ). Interestingly however, the closetozero ADOPratios for show that the lidarintegration can be indeed instrumental when not too many satellites are tracked. This is often the case in urban canyons where GNSS receivers frequently lose the tracking of GNSS signals. For the cases where standalone GNSSonly IAR is not possible, i.e. when , the integrated solution becomes almost identical to that of the lidaronly solutions, i.e., . That is why the ADOPratio is shown to be almost zero in the figure. In the following, this notion will be made more precise by comparing the underlying ADOPs and ambiguity success rates of the GNSSonly and lidaraided models under various scenarios.
IvC Ambiguity resolution performance compared
We now analyze the IAR performance using (16) and (17), as well as the ambiguity success rates under various configurations for both the GNSSonly and lidaraided models to study how many satellites they require for successful IAR. We first present and discuss the ambiguity resolution performance of the GNSSonly model in a generalized scenario without considering satellite elevation angles by simplifying to equal weights, i.e., . The precision of undifferenced code and phase data, namely and , are assumed to be 0.2 m and 0.002 m, respectively, for highgrade receivers. Hence, the precision of phase observations is about 1% of their wavelengths (). Fig. 3 depicts the ADOP and ambiguity success rates (evaluated by PsLAMBDA, denoted by ) computed in different setups with respect to the number of tracked satellites. It is shown in Fig. (a)a that with dualfrequency GNSSonly data (), and the corresponding success rate can reach 0.097 cycles and 99.9% with only 5 satellites, meaning that successful ambiguity resolution with singlesystem, dualfrequency observations is indeed possible if the corresponding GNSS code data are not imprecise (e.g., m). In this case, there is no need for the inclusion of lidar data. In comparison, when only singlefrequency data from 5 satellites is available () for the GNSSonly model, and the success rate are evaluated as about 0.547 cycles and 11.2%, respectively, which are insufficient for IAR. In order to constrain to 0.12 cycles, has to be increased to 8, while PsLAMBDA only reports a success rate higher than 99.9% when .
Note that the use of equal weights in this approximation ignores the elevationdependent effects on the GNSS signals, an assumption which cannot always hold in practice. This highlights that the ambiguity resolution performance can be expected to be even poorer in practice. Moreover, lowgrade GNSS receivers can have much less precise code observations. For example, assuming that code data is severely affected by the low quality of the receiver and/or antenna by setting m and the same as above, using singlefrequency data becomes 1.247 cycles for 5 satellites and at least 10 satellites are needed for cycles, as illustrated in Fig. (b)b. To ensure an evaluated success rate of 99.9%, at least 11 satellites are required. Therefore, it is not feasible to pursue instantaneous IAR using the GNSSonly model with singlefrequency data, unless a large number of satellites from multiple systems can be tracked.
We now examine the lidar measurements for the float solutions to show their impact on ambiguity resolution. Similar to the optimistic configuration above, with m and m, each lidar observation is assumed to be slightly more precise than a code observation with m. Using the same number of keypoints as that in Fig. 2, and () obtained using (18), and with respect to the number of satellites are also presented in Fig. (a)a. It is indicated that the integrated data consistently achieves similar or even better ambiguity resolution performance than dualfrequency GNSSonly data, with the two quantities always being around 0.02 cycles and 100%. This advantage is more evident when there are only a few tracked satellites, as lidar becomes the main contributor to the float solutions. Notably, in the case of tracking only 2 or 3 satellites, the GNSSonly model fails to resolve integer ambiguities due to the lack of measurements, whereas the lidaraided model can still enable IAR since the lidar measurements devote to the estimation of the three positional unknown parameters. Furthermore, as lidar provides a large number of measurements, the difference between the ambiguity resolution performances of using singlefrequency or dualfrequency GNSS data becomes negligible.
On the other hand, lidar measurements can suffer from observational noise, environmental changes, dynamic objects, etc., leading to a lower precision. In a pessimistic scenario with m, as well as m for imprecise code measurements, , and their corresponding are shown in Fig. (b)b. Notably, due to the lower precision of code observations, the IAR performance of GNSSonly model decreases, requiring more satellites to be tracked for an ambiguity success rate of 99.9%. In contrast, the lidaraided counterpart is similar to previous results, with and consistently being around 0.05 cycles and 99.9%, suggesting that successful ambiguity resolution can still be achieved with any number of satellites and/or frequencies under such pessimistic assumptions.
IvD ADOP evaluation of lidaraided ambiguity resolution
So far we have learned that with the tracking of a sufficient number of GNSS satellites, IAR can be realized without lidar contribution, and the same holds true when there are limited GNSS observations but a number of lidar keypoints. To study the requirement of lidar precision for successful IAR, which is the other factor affecting the performance of the lidaraided model, let us now consider the situation in which the number of corresponding keypoints is either 4 or 44, which are the theoretical minimum number to make (1) solvable and the empirical minimum used in previous comparison, respectively. and are maintained as m and m in this analysis. In contrast to the previous investigation, this analysis is established in an elevationweighted scenario using GNSS satellites with high elevation angles to study the ambiguity resolution performance for vehicle positioning in urban canyons, where satellites with low elevation angles can be blocked by buildings. The satellite skyplot in Fig. 4
indicates the geometry of the satellites used in this analysis with respect to the receiver, where the satellite elevation angles are above 40°. These satellites are included by descending order of their elevation angles for the evaluation. Since the IAR performance can be sensitive to the geometric distributions of the keypoints when their number is low, each ADOP value is computed as the average of 100 trials with keypoints randomly selected around the receiver.
Fig. 5 presents the values for numerous combinations of the numbers of satellites () and lidar precision () for the lidaraided instantaneous ambiguity resolution method. For comparison, to obtain values below 0.12 cycles computed using (16), the GNSSonly model requires at least 9 highelevation satellites for singlefrequency GNSS data (), which can be difficult to access in densely builtup areas for vehicle positioning, while 5 highelevation satellites with dualfrequency GNSS data () are needed to produce the similar . With the lidaraided model, on the other hand, by using singlefrequency GNSS observations and the theoretical minimum of 4 keypoints, the number of highelevation satellites needed for the value of 0.12 cycles reduces to 7, provided that m, as shown in Fig. (a)a. Similarly, the lidar precision requirement can be relaxed to 0.77 m for 0.14 cycles or the success rate upper bound of 99%. It is therefore evident that 4 matched keypoints only can already reduce the required satellites by 2 for successful singlefrequency IAR. In addition, Fig. (b)b shows that by introducing dualfrequency GNSS observations, is always below 0.12 cycles for 3 and more satellites, whereas for 2 satellites, would increase above 0.12 cycles but is still below 0.14 cycles for poor lidar precision ( m). Nonetheless, it is observed in Fig. (c)c that once the number of keypoints increases to the empirical minimum of 44, can be kept below 0.12 cycles for any number of satellites from 2 even with the lowest lidar precision m, which corresponds with the results in Fig. (b)b. It should be remarked that using the proposed keypoint extraction strategy, the quantity of corresponded keypoints can be expected to be at least equal to the empirical minimum upon successful registration. Conceivably, lidaraided IAR is even more accessible by using dualfrequency GNSS data with this configuration, hence the results demonstrating the values for dualfrequency GNSS observations and 44 correctly matched keypoints are omitted. In summary, the theoretical ADOP analysis has established that even with the theoretical minimum number of keypoints (i.e., 4), successful IAR is possible for any number of satellites using the lidaraided method when dualfrequency GNSS data is available. More importantly, when a reasonable amount of lidar keypoints are present (e.g., 44), the proposed method can substantially improve the ambiguity resolution performance to the extent that even singlefrequency IAR is made feasible for only a few highelevation satellites without requiring highly precise lidar measurements. This is a great advantage for vehicle positioning since the GNSSonly model can fail IAR in urban canyons due to restricted satellite visibility.
V Experimental setup
The proposed instantaneous lidaraided ambiguity resolution method is evaluated in an experiment simulated using GNSS and lidar measurements from two real datasets. In this section, the data collection and preprocessing details are presented. To verify the performance predictions made earlier in Section IV, we collected GNSS data from a 30minute session of observations on a stationary point in a controlled environment, while the lidar data was obtained from the KITTI dataset [6] and simulated around the same point to build the HD map.
Va GNSS data collection
The GNSS raw observations used in the experiment were collected using a lowcost ublox F9P dualfrequency receiver with an ANN–MB patch antenna between 1:56:29 AM and 2:26:28 AM, GPST (GPS time) on 29 June 2021, at the sampling rate of 1 Hz. The antenna was placed on a fixed point in an opensky environment in Melbourne, Australia. A surveygrade GNSS receiver Leica GS16 was also used to measure the same point simultaneously to provide the ground truth coordinates as reference. From now on, this point will be referred to as Target. The equipment configuration is shown in Fig. 6.
For data obtained from both receivers, differential code and phase observations are derived using a nearby Continuously Operating Reference Station (CORS), namely EMEL, which is equipped with a Trimble NETR9 receiver. Therefore, a short baseline Target–EMEL illustrated in Fig. 7 is formed with the approximate distance of 1470 m. By defining empirical success rate as the proportion of positioning epochs with correctly fixed integer ambiguities, we take the DD ambiguities computed using GPS+QZSS (L1/L2) solutions obtained with ublox F9P and EMEL as the benchmark ambiguities to evaluate the IAR capability of the proposed method, as the formal ambiguity success rates are assessed higher than 99.9% for all epochs using this configuration.
VB KITTI lidar data preprocessing
In Section II we presented the process of producing lidar measurements by learningbased point cloud registration. Here we provide the preprocessing of lidar data obtained from the KITTI dataset for simulating the prebuilt and georeferenced HD map in eframe, so that the generated lidar measurements can be integrated with their GNSS counterparts. A total of 3600 point clouds collected in Karlsruhe, Germany with Velodyne HDL64E are split into equal numbers of rover and reference scans. In other words, we assume that the rover scans are collected by the lidar sensor onboard of the rover vehicle, whereas the reference scans are used for the HD map. Due to the lack of SPP positions in the dataset, the roverreference scan matches are preassigned with a 3 s time interval to ensure that each pair of rover and reference scans contains a reasonable amount of overlap to enable registration.
The point clouds are originally in lframe, they need to be transformed to cframe, which is equivalent to the lframe of the first scan in the sequence, then to eframe to produce lidar measurements that can be combined with their GNSS counterparts. For the () point cloud in the sequence, the transformation which transforms it to cframe is given in the ground truth poses of KITTI. Hence, for each pair of rover and reference scans we have transformations and , and the transformation to align the two can be obtained as
(20) 
Note that this transformation is not used for positioning, but rather simulating the reference scans at locations near Target so that the estimated vehicle positions should coincide with Target. A transformation matrix that georeferences the rover scan is defined with only a translation that moves its origin (i.e., ) to the ground truth coordinates of Target measured by the surveygrade receiver. The reference scan is therefore georeferenced in eframe using:
(21) 
which is applied to the matched keypoints found in the reference scans upon successful registration so that the computed lidar measurements are in eframe.
Vi Results
In this section, we present the experimental results in terms of positioning accuracy and IAR performance of the proposed lidaraided ambiguity resolution method. Using the GNSS and lidar data prepared in Section V, the proposed method is tested by resolving the position of Target for 1800 epochs using GPS (L1)+lidar data to simulate the limited satellite visibility in urban environments. In addition, other positioning approaches using GPS (L1), GPS (L1/L2), GPS+QZSS (L1/L2) and Lidaronly measurements are used for comparison. In terms of the precision of GNSS code and carrier phase measurements, and are chosen as 0.2 m and 0.002 m, respectively, as the GNSS data was collected in a controlled environment despite a lowcost receiver was used. On the other hand, for the weighting of lidar measurements is computed as approximately 0.15 m on average, therefore .
Via Keypoint matching accuracy
Keypoint matching to produce lidar measurements is the first step of the proposed positioning method. Feature vectors are computed for all points in the scans using MSSVConv and 3000 pairs of them per epoch are randomly selected to estimate the transformation to align the rover scan to the reference scan using RANSAC [4]. Instead of training and testing the deep learning model on the same dataset, we use the model pretrained with ETH dataset[23] and perform inference on the KITTI lidar data to test the transferability of MSSVConv. In order to numerically evaluate the accuracy of the matched keypoints, we use SRE (scaled registration error)[5], a scalar measure of registration accuracy which considers both rotational and translational errors. For registered point cloud and its ground truth counterpart with corresponded points, namely and , given that is the geometric centroid of , the SRE is computed with
(22) 
where denotes the Euclidean norm. Since the registration error of each point is scaled by its distance to the centroid, one can infer that SRE reflects the phenomenon that rotational errors have a larger impact on points further from the laser scanner. As a result, registration is successfully conducted for all of the 1800 pairs of rover and reference scans, averaging 134 keypoints per epoch, with the minimum number of 44. Fig. 8 shows the distribution of the SRE values, in which 1044 epochs have SRE smaller than 0.005, while the maximum is 0.027. In comparison, Fontana et al. [5] obtained a mean SRE of 0.408 using the Iterative Closest Point (ICP) algorithm on multiple public datasets. Therefore, all of the registrations are considered accurate, and MSSVConv exhibits a good transferability on point clouds from different environments.
ViB Positioning accuracy
To demonstrate the accuracy of the tested positioning approaches, we compute the RMSE (root mean squared error) of the solutions with respect to the ground truth. Lidaronly positioning solutions are produced using the generated lidar measurements since the estimated translational parameters are equivalent to the sought unknown positions. For GNSSinvolved methods, we use an acceptance test that integer ambiguities are fixed when the formal ambiguity success rate is evaluated as equal to or above 99.9%, otherwise the float solutions are retained.
Positioning  Horizontal  Vertical  3D 

Method  RMSE [m]  RMSE [m]  RMSE [m] 
GPS (L1)  0.648  1.190  1.355 
GPS (L1/L2)  0.052  0.065  0.083 
GPS+QZSS (L1/L2)  0.009  0.018  0.020 
Lidaronly  0.031  0.021  0.038 
GPS (L1)+lidar  0.008  0.015  0.017 
Positioning errors per epoch, time series of the numbers of satellites and ADOP of the tested methods are provided in Fig. 9. In addition, Table I illustrates the horizontal, vertical and 3D RMSE of the 5 positioning approaches. It is shown that for the positioning duration, 6 to 7 GPS satellites and 3 QZSS satellites can be tracked. According to the analysis using ADOP in Section IV, at least 8 satellites are required to achieve the success rate of 99.9% with one frequency. This corresponds with the GPS (L1) results in Fig. (a)a, in which all of the solutions are float because of high ADOP (or low success rates). Fig. (a)a also suggested that the minimum number of satellites needed for ADOP 0.12 cycles with two frequencies is 5, which agrees with Fig. (b)b, in which all solutions are fixed for GPS (L1/L2) since sufficient satellites are observed. However, two epochs are found with wrongly fixed solutions when the numbers of satellites decrease from 7 to 6. Unsurprisingly, the accuracy of GPS (L1) is the lowest among all, giving a meterlevel 3D RMSE of 1.355 m, whereas GPS (L1/L2) improves it to 0.083 m because of having more observations to enable IAR. In comparison, GPS+QZSS (L1/L2) can successfully fix integer ambiguities for all epochs due to the additional satellites, significantly improving the precision of the positioning results. The horizontal and 3D RMSE also dramatically decrease to 0.009 m and 0.02 m, offering millimeter to centimeterlevel accuracy.
Moving on to the positioning methods with the integration of lidar measurements, although lidar registration is considered highly accurate in this experiment (Fig. 8), the horizontal, vertical and 3D RMSE of the Lidaronly positioning solutions are 0.031 m, 0.021 m and 0.038 m, respectively, which are less accurate than GPS+QZSS (L1/L2). However, as predicted in Fig. (c)c, with abundant lidar keypoints with decimeterlevel precision and more than two GNSS satellites, singlefrequency IAR is feasible. Fig. (e)e shows that by integrating lidar and GPS (L1) observations, ADOP is always below 0.12 cycles and all ambiguities are correctly fixed, giving the positioning accuracy that is even higher than GPS+QZSS (L1/L2) thanks to the contribution of lidar, with the horizontal, vertical and 3D RMSE of 0.008 m, 0.015 m and 0.017 m.
Fig. 10
presents the cumulative distribution function (CDF) of the horizontal and 3D errors of the tested methods. The superiority of the proposed lidaraided method in terms of positioning accuracy is clearly shown, outperforming all the other positioning methods. Moreover, to demonstrate the precision improvement of the GPS (L1)+lidar solutions brought by the lidaraided ambiguity resolution, we evaluate the squareroot precision gain in the EastNorthUp directions in Fig.
11, which shows how many times the precision of positioning solutions increases by fixing integer ambiguities. Due to the considerably larger number of measurements and higher precision of lidar than the code observations, the float solutions are almost the same as the Lidaronly ones, which can be observed from Fig. (d)d and (e)e. However, by correctly fixing the integer ambiguities, the carrier phase observations further improve the positioning precision by around 6 times horizontally and 2 times vertically, which explains the higher accuracy and precision of GPS (L1)+lidar results than those of Lidaronly as shown in Table I and Fig. (e)e. Note that the precision of Lidaronly solutions in 3 directions are homogeneous in this experiment, and the lower vertical precision gain is caused by the lower precision of GNSS observations in the Up direction.ViC Ambiguity resolution performance
In order to assess the ambiguity resolution performance in terms of the proportion of correctly fixed epochs, full ambiguity resolution is applied by removing the acceptance test and forcing the resolved integer ambiguities to be fixed for all epochs for the demonstrated positioning methods. Fig. (f)f shows positioning errors per epoch, numbers of tracked satellites and ADOP time series of GPS (L1) results, with the empirical success rate computed as 22.7% and the mean formal success rate evaluated as 48.2%. Note that GPS+QZSS (L1/L2) and Lidaronly results are not applicable for empirical success rate evaluation since the former provides the benchmark ambiguities and the latter does not utilize GNSS observations. The results of GPS (L1/L2) and the proposed method, namely GPS (L1)+lidar, are also omitted as they are identical to those in Fig. (c)c and (e)e since all fixed solutions are already accepted when the acceptance test is present. The empirical success rates of these two methods are 99.9% and 100%, respectively, while their formal success rates are both computed as above 99.9%. Again, we have previously concluded that IAR with singlefrequency GNSSonly observations from fewer than 8 satellites is not feasible, which is reflected here by the low empirical success rate of GPS (L1). In comparison, the integration of lidar data substantially increases the empirical success rate without requiring additional GNSS measurements and correctly fixes all the integer ambiguities, while keeping ADOP below 0.12 cycles and achieving comparable ambiguity resolution performance as the GNSSonly model using dualfrequency data.
Vii Discussion
Viia Quality of lidar measurements and HD map
Although we have shown analytically that decimeterlevel precision of lidar measurements can enable successful IAR (Fig. 5), one limitation of our experiment is that we have used highly accurate lidar data, as the SRE of the registered keypoints suggested (Fig. 8). In terms of the HD map, the reference scans are georeferenced with the validated ground truth information provided in the KITTI dataset. In practice, for largescale HD map products of urban road environments, such accurate georeferencing can be difficult to achieve and they may be produced with larger errors, decreasing the accuracy of the derived lidar measurements. On the other hand, since the reference scans are meant to be acquired from a previous time, the registration accuracy may be influenced by the environmental differences between the rover and reference scans if the HD map is not uptodate, which is not reflected in our experiment. Furthermore, due to the lack of GNSS raw observations synchronized with the lidar data in KITTI, the roverreference scan pairs are preassigned with a 3 s interval, which is equivalent to the distance of a few meters. In reality, the nearest reference scan from the HD map should be identified using position estimates of the vehicle in real time from less demanding techniques such as SPP.
ViiB Number of lidar keypoints and empirical success rate
It has been demonstrated in Section IV that the number of correctly matched keypoints is not required to be large to ensure the ambiguity success rate upper bound of 99.9%, provided that the precision of the lidar measurements is at the decimeterlevel, or that a few satellites can be tracked. In order to determine a recommended number of lidar keypoints for consistently successful IAR in a practical environment, we have repeated the experiment using the GPS (L1)+lidar positioning setup with the number of lidar keypoints limited to a range between 5 and 45 by applying full ambiguity resolution. The empirical success rates against different numbers of keypoints are shown in Fig. 12, which indicates that the empirical success rate is beyond 99% with only 10 keypoints, and increases to above 99.9% when 35 or more keypoints are used. It should be remarked that this quantity of correspondences between registered rover and reference scans can be easily obtained, as the empirical minimum number of keypoints per epoch in our experiment is 44.
ViiC Runtime efficiency
The keypoint matching and positioning stages of the proposed method are experimented with the TorchPoints3D
[2] implementation of MsSVConv and MATLAB [20], respectively. On a platform consisting of AMD Ryzen 3800XT CPU and NVIDIA RTX 3070 GPU, the former approximately takes 0.85 s and the latter takes 0.05 s to complete the computation for each epoch. Therefore, the proposed instantaneous lidaraided ambiguity resolution method has the potential of realtime positioning for vehicles.Viii Concluding remarks
In this contribution we proposed an instantaneous lidaraided ambiguity resolution method focusing on vehicle positioning in urban canyons, where GNSS signals are prone to blockage and multipath. The lidar measurements are generated by a keypoint extraction strategy via learningbased point cloud registration between rover scans and reference scans from a prebuilt HD map. A mixed measurement model is employed to integrate such lidar measurements with their DD GNSS counterparts to obtain precise float solutions and enable instantaneous IAR. Closedform expressions of the ambiguity variance matrix and the corresponding ADOP are developed to provide a priori evaluation of the ambiguity resolution performance using the numbers of available satellites and keypoints, as well as the precision of the measurements (cf. 17).
Our analytical study has shown when limited GNSS satellites and/or frequencies are accessible, which is often the case in urban environments, the proposed lidaraided method can significantly reduce the ADOP value comparing with the GNSSonly approach, thus enabling successful instantaneous IAR (Fig. 2). Moreover, a moderate number of lidar keypoints can reduce the required satellites to track, to the extent that IAR is feasible with singlefrequency data from only 2 or 3 satellites (Fig. 5). The numerical results from a simulated experiment illustrate that the proposed method achieves the empirical ambiguity success rate of 100% and 3D positioning RMSE of 0.017 m using GPS L1 and lidar measurements, thereby outperforming both the Lidaronly and GPS+QZSS (L1/L2) positioning methods (Fig. 9). Future work will undertake realworld experiments to further examine the performance of the proposed method in GNSSchallenging urban canyons.
Acknowledgments
This research did not receive any specific grant from funding agencies in the public, commercial, or notforprofit sectors. The first author acknowledges the financial support from The University of Melbourne through the Melbourne Research Scholarship.
[Supplementary proofs] An application of the determinant factorization rule [15, 21] to the ambiguity variance matrix (14) gives
(23) 
Substitution of the identities , [21], and the phasetocode variance ratio into (23) yields
(24) 
The above expression for the determinant of the ambiguity variance matrix, together with the ADOP definition (15), gives
(25) 
For the GNSSonly case, we have . This simplifies the last term in (25) to , from which the ADOP of the GNSSonly model (16) follows. For the integrated GNSSlidar case however, we have instead (cf. 13). The first expression of the ADOPratio (17) follows then by substituting into , showing that
(26) 
Finally, the second expression of the ADOPratio (17) follows from the defintion of the generalized eigenvalues (18), that is
(27) 
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