CN114440895A - Atmospheric pressure assisted Wi-Fi/PDR indoor positioning method based on factor graph - Google Patents

Atmospheric pressure assisted Wi-Fi/PDR indoor positioning method based on factor graph Download PDF

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CN114440895A
CN114440895A CN202210209003.0A CN202210209003A CN114440895A CN 114440895 A CN114440895 A CN 114440895A CN 202210209003 A CN202210209003 A CN 202210209003A CN 114440895 A CN114440895 A CN 114440895A
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姚子扬
尚俊娜
施浒立
李方州
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Hangzhou Dianzi University
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Abstract

The invention discloses a factor graph-based atmospheric pressure assisted Wi-Fi/PDR indoor positioning method. The pedestrian track estimation is improved: and performing step detection by using a peak value detection and filtering method based on the hypothesis of step points, calculating a logarithmic model parameter by using a Gaussian-Newton iteration method in step length estimation, and processing course angle data by using a jump elimination method and IIR filtering in direction estimation. A mutation filtering method based on the generalized continuation extrapolation method is provided for filtering and replacing the pressure data mutation value; and combining a CSI correction positioning method of a dense connection network, a PDR and a difference barometric pressure altimetry result construction factor graph model, fusing measurement information of different update rates, and flexibly and efficiently plugging and playing. The method can reduce the error of the pedestrian track result, complete detection with lower cost by a mutation filtering method, fully utilize various measurement information and improve the accuracy and stability of positioning.

Description

Atmospheric pressure assisted Wi-Fi/PDR indoor positioning method based on factor graph
Technical Field
The invention belongs to the technical field of indoor navigation positioning, relates to a multi-source information fusion method based on a factor graph, and particularly relates to an air pressure assisted Wi-Fi/PDR indoor positioning method based on the factor graph.
Background
Because a single positioning source cannot meet the requirements of positioning precision and stability of indoor positioning, combining multiple positioning information by adopting a multi-source information fusion method is one of the main means for enhancing the robustness and accuracy of positioning in the current indoor positioning technology. By virtue of the advantages of low calculation amount and high accuracy, the filtering method represented by the Kalman filter occupies a place in the multi-source fusion positioning field with high real-time requirement.
Unlike the estimation based on the markov assumption of kalman filtering, which relies only on the state at the previous time, the graph optimization method can perform nonlinear optimization on the whole or part of previous data, and as long as the calculation resources allow, the optimization precision higher than that of the filter technology can be obtained. In recent years, with the rise of cloud computing and the internet of things, large-scale computing becomes possible, and an optimization method represented by a factor graph is gradually applied to the fields of navigation positioning, map construction and the like, and plays an invisible role in aspects such as unmanned driving and intelligent logistics. The factor graph is an information transfer model that can be factored into a global function in the form of a product of several local functions by a bipartite graph. The factor graph-based indoor multi-source fusion positioning method is essentially maximum posterior estimation of the Bayesian network, can complete dynamic fusion of various positioning methods, can cope with the conditions of different output frequencies of sensors and measurement information loss, realizes plug and play, and is very flexible.
Most of the prior multi-source information fusion methods are based on a federal filter, and when the measurement is carried out on the measurement with different sampling frequencies, a proper amount of measurement information is usually left out in order to ensure the time consistency of subsystems; on the other hand, a series of filters represented by EKF are based on the establishment of markov assumption, and although the computational complexity is greatly reduced, many important information is discarded. Therefore, the factor graph can fully utilize the measurement information to improve the positioning precision while solving the problem of time synchronization.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a factor graph-based air pressure assisted Wi-Fi/PDR indoor positioning method, Channel State Information (CSI) is used as fingerprint data, each step of Pedestrian trajectory Reckoning (PDR) is improved, the influence of inertial device error accumulation on positioning is weakened, a barometer is used, height variation is obtained through a self differential air pressure height measurement method, and therefore the positioning accuracy is improved, and the positioning robustness is enhanced.
The factor graph-based air pressure assisted Wi-Fi/PDR indoor positioning method specifically comprises the following steps:
step S1: a positioning device is arranged within the positioning region. The positioning device comprises Wi-Fi equipment, a wireless network card, an inertial device module and a digital air pressure sensor.
The signal coverage positioning area of the Wi-Fi equipment is used for constructing a wireless sensor network, meshing the positioning area in an off-line stage, and transmitting acquired channel state information through a wireless network card. And constructing a CSI fingerprint database according to the data acquired by the Wi-Fi equipment and the corresponding position coordinates, training and positioning a neural network to predict the position coordinates, and then expanding the CSI fingerprint database by using a generalized continuation interpolation method to obtain an interpolation fingerprint database. After the positioning neural network corrects the estimated position output according to the CSI signal at the moment i by the interpolation fingerprint database, the coordinate measurement data used for inputting the factor graph are obtained
Figure BDA0003532376220000021
The inertial device module comprises an accelerometer and a gyroscope and is used for continuously outputting three-axis acceleration and a heading angle in the positioning process. The digital air pressure sensor is used for continuously outputting the air pressure value and the air temperature of the current position.
Step S2: the method comprises the following steps of improving PDR, constructing a step size model and outputting track calculation measurement information:
step S21: obtaining the three-axis acceleration a ═ a in the carrier coordinate system by the inertial device modulex ay az]TAnd the acceleration a' under a relative coordinate system when the z axis of the carrier coordinate system is vertically downward is obtained by being multiplied by the rotation matrix R by the left side:
a′＝[a′x a′y a′z]T＝Ra
Figure BDA0003532376220000022
a'zG as the quantity to be detected for step detection, where g is the local gravitational acceleration, ax、ay、azThe components of the acceleration along the directions of the x-axis, the y-axis and the z-axis in the carrier coordinate system are respectively a'x、a′y、a′zThe components of the acceleration along the directions of the x-axis, the y-axis and the z-axis under the relative coordinate system are respectively. α, β and γ are heading angle, pitch angle and roll angle, respectively.
To all time to be detected quantity a'z-g, carrying out peak value detection, filtering the step point moments lower than a set threshold value to obtain a step point moment set ts1. Since a large peak caused by a lot of jitter is also calculated only by peak judgment, it is necessary to set t to the time of the step points1And (5) carrying out secondary screening. Generally, there is an upper limit to the step frequency of a person during normal walking, and the maximum instantaneous acceleration in the vertical direction during travel is caused by the landing of a step, and two reasonable assumptions are made from this:
firstly, only one step point is needed in the shortest step time;
and secondly, if a plurality of step points appear in the shortest step using time, the only step point with the largest amount to be detected is taken as the step point in the interval.
From the two assumptions above, t is set for the step pointss1Further screening, when the tau is recorded as the shortest pace,
Figure BDA0003532376220000023
the j-th step point moment in the tau time, the step point set t after the secondary screenings2The following conditions are satisfied:
Figure BDA0003532376220000024
step S22: establishing a logarithmic model of step length and acceleration, wherein w and b are model coefficients, and the model coefficients are shown as follows:
f(w，b)＝w·log(a′z(t)-g)+b，t∈ts2
collecting data output by the inertial device module under different speeds and different routes for multiple times, and constructing a trajectory calculation training data set
Figure BDA0003532376220000031
Wherein
Figure BDA0003532376220000032
For the number of walking steps, SiiFor the walking distance, M is the data volume.
And optimizing parameters in the step length and acceleration logarithmic model by adopting a Gaussian-Newton iteration method to ensure that the sum of squared path residuals is minimum, wherein the optimization formula is as follows:
Figure BDA0003532376220000033
wherein the content of the first and second substances,
Figure BDA0003532376220000034
the step size is estimated for the jj step of the ii training data.
Step S23: the heading angle value range of the gyroscope is generally [0, 360 degrees ], and in the process of filtering heading angle data, the gyroscope is easy to jump back and forth at the boundary of the range, and only one jump across the boundary can also cause serious influence on the filtering result. Therefore, the course angle data needs to be processed, and the course angle alpha at the later momentkMinus heading of previous timeCorner
Figure BDA0003532376220000035
The absolute value of the value is more than 180 degrees, the jump can be considered to occur, and the specific processing method is as follows:
Figure BDA0003532376220000036
Figure BDA0003532376220000037
wherein the content of the first and second substances,
Figure BDA0003532376220000038
indicating the processed heading angle,
Figure BDA0003532376220000039
designing an IIR filter by adopting a bilinear transformation method, and processing the processed course angle
Figure BDA00035323762200000310
Low pass filtering is performed.
Step S24: notation stepLeniObtaining PDR measurement information for the input factor graph for the estimated step length at time i
Figure BDA00035323762200000311
Figure BDA00035323762200000312
Figure BDA00035323762200000313
Step S3: in practical application, the positioning accuracy is reduced due to sudden changes of air pressure measurement caused by unsmooth air circulation and the like in the digital air pressure sensor, and a sudden change filtering method based on a generalized continuation extrapolation method is provided for detecting and replacing a sudden change point in the air pressure measurement.
Step S31: collecting a set of time series barometric pressure data
Figure BDA00035323762200000314
Calculating L generalized continuation extrapolation models:
Figure BDA00035323762200000315
in the formula (I), the compound is shown in the specification,
Figure BDA00035323762200000316
for the coefficients of the fitting function, t is used for the l-th modelm-lTo tn-lCalculating fitting coefficients of data of time intervals, wherein the data quantity of each model is fixed n-m +1, and estimating t by utilizing the L generalized continuation extrapolation modelsn+1Air pressure value at a moment
Figure BDA0003532376220000041
Figure BDA0003532376220000042
Figure BDA0003532376220000043
Step S32: the standard deviation of the measurement data output by the digital air pressure sensor is sigma, the significance level is a, a is 0.05, and the test statistic z is obtained through a standard normal distribution tablea/2By extending the predicted value in a generalized way
Figure BDA0003532376220000044
Estimating barometric truth value
Figure BDA0003532376220000045
Then passes through the set confidence regionWorkshop
Figure BDA0003532376220000046
Judging whether the measurement data output by the digital air pressure sensor is a mutation point or not, considering that the measurement data has a mutation when the measurement data exceeds a confidence interval, and using an estimated air pressure true value
Figure BDA0003532376220000047
The value is replaced.
Step S33: calculating the differential height value at the time i according to a differential air pressure conversion height formula
Figure BDA0003532376220000048
Differential barometric altitude measurement information as input factor graph
Figure BDA0003532376220000049
Figure BDA00035323762200000410
Wherein the content of the first and second substances,
Figure BDA00035323762200000411
is the current air temperature and is the air temperature,
Figure BDA00035323762200000412
and
Figure BDA00035323762200000413
the pressure values at two adjacent time points after being filtered in step S32 are divided.
Step S4: measuring the values obtained in steps S1-S3
Figure BDA00035323762200000414
And inputting a factor graph model. The state quantity of the factor graph is X ═ X y z α']TIncluding three-axis coordinate values x, y, z and course angle alpha', constructing new factor nodes according to the input new measurement information type, and sequentially performing expansion structure and incrementAnd (3) eliminating new variables and variables, and then carrying out graph optimization, wherein an optimization objective function is as follows:
Figure BDA00035323762200000415
Figure BDA00035323762200000416
Figure BDA00035323762200000417
wherein the content of the first and second substances,
Figure BDA00035323762200000418
to measure the error covariance matrix, | · | | non-calculationRIn order to obtain the Mahalanobis distance, delta represents the state quantity difference value of variable nodes at adjacent time points.
And finally, obtaining the optimized factor graph state quantity X, and taking the three-axis coordinate value as an output positioning result.
The invention has the following beneficial effects:
1. the method has the advantages that the Wi-Fi equipment widely distributed in most rooms, inertial devices and barometers which are easily obtained from intelligent terminals are utilized to conduct multi-source fusion navigation positioning, and the factor graph is adopted to ensure that the measurement information can be still fully utilized to improve the accuracy and stability of positioning under the condition that the updating frequency of the measurement information output by each equipment is inconsistent.
2. The method is characterized by improving a pedestrian track calculation method, detecting steps by using a peak detection method and a step point filtering method, calculating logarithmic model parameters by using a Gaussian-Newton iteration method to realize step length estimation, processing course angle data by using a jump elimination method and IIR filtering, and further reducing the error of a track calculation result.
3. And a mutation filtering method based on a generalized continuation extrapolation method is introduced, so that the judgment robustness is enhanced, and a mutation value in the air pressure data is filtered and replaced.
Drawings
FIG. 1 is a block diagram of a factor graph-based barometric-assisted Wi-Fi/PDR indoor positioning method;
FIG. 2 is a graph of Wi-Fi/PDR and barometer assisted factor graphs;
FIG. 3 is a graph of a Wi-Fi/PDR and barometer assisted indoor positioning method experimental trajectories based on a factor graph;
FIG. 4 is a cumulative probability distribution of localization errors for different fusion methods.
Detailed Description
The invention is further explained below with reference to the drawings;
as shown in fig. 1, the method relates to Wi-Fi, PDR and barometric pressure, and constructs a factor graph model using the results of the CSI correction positioning method, PDR and differential barometric pressure altimetry method combined with a dense connection network, as shown in fig. 2, the structure of the factor graph enables high expandability, sensors with different measurement frequencies can be flexibly combined, and the method can also cope with the situation that the sensors are temporarily unavailable or new sensor information is introduced, and realize the function of plug and play.
Since a large number of Wi-Fi devices are deployed in most indoor environments, a positioning technology based on Wi-Fi signals can achieve good positioning effect at low cost, and compared with RSSI (Received Signal Strength), the positioning technology based on CSI is used as fingerprint data and has better fine granularity and stronger stability.
The PDR generally comprises three steps, namely step detection, step length estimation and course estimation, and the method of each step can be improved to weaken the influence of error accumulation of an inertial device on positioning and further improve the positioning accuracy of the PDR. The method firstly provides two hypotheses, and performs step detection by using a peak detection and step point filtering method on the basis; calculating a logarithmic model parameter by using a Gaussian-Newton iteration method in the step length estimation problem; and in the direction estimation, a jump elimination method and an IIR filtering method are adopted to process course angle data, and the obtained pedestrian track result has smaller error than that before improvement.
Atmospheric pressure is a key index of atmospheric state, the atmospheric pressure can reflect the height under the corresponding pressure to a certain extent, in order to avoid periodic atmospheric pressure fluctuation with the time scale of day or year and random atmospheric pressure fluctuation closely related to weather to weaken the accuracy and stability of barometric height measurement, a mutation filtering method based on generalized continuation extrapolation is used for enhancing the judgment robustness by using multiple extrapolation results, filtering and replacing mutation values in atmospheric data, and the self differential barometric height measurement method is adopted to obtain the height variation, so that the positioning robustness can be enhanced.
In the embodiment, equipment department and data acquisition are carried out in the areas of four buildings, five buildings and staircases of an office building, so that indoor positioning is realized. The used receiving terminal is a Huashuo FX50JX notebook computer provided with an IWL 5300NIC wireless network card, the Wi-Fi equipment is a millet router 4C, the model of the digital air pressure sensor is BMP388, and the model of the inertial device module is LINS 16460.
Step one, belonging to a positioning device in a positioning area, dividing grids, then collecting CSI at each training node, recording corresponding coordinates to build a CSI fingerprint base, and training a neural network according to the CSI fingerprint base to perform position classification. And then carrying out average processing on the CSI fingerprint database, expanding the CSI fingerprint database by using a generalized continuation interpolation method, and constructing an interpolation fingerprint database for correction. And then acquiring data output by the inertial device module at different speeds and different routes to form a training data set, and calculating step size model parameters by adopting a Gaussian-Newton iteration method to obtain a step size estimation model.
And step two, arranging the experimenter to carry the inertial device module to walk at a constant speed, wherein the walking route is that the experimenter walks in a four-floor corridor in an anticlockwise direction and walks in a clockwise direction after arriving at a five-floor corridor through stairs, and walking tracks in the corridor are all rectangular.
And s2.1, inputting the CSI acquired in real time into the neural network trained in the first step to obtain an estimated coordinate, correcting the estimated coordinate by adopting an interpolation fingerprint library and K neighbors, and then inputting the corrected positioning coordinate into a factor graph model.
And s2.2, step point extraction is carried out on the triaxial acceleration output by the inertial device module, and a logarithmic model of the step length and the acceleration is established to estimate the step length. And then, processing and low-pass filtering the jumped course angle data, and inputting the PDR measurement information into a factor graph model.
And s2.3, processing data output by the digital air pressure sensor by adopting a mutation filtering method based on the generalized continuation extrapolation method, converting the data into differential height, and inputting the differential height into the factor graph model.
Step three, according to the measurement information processed in the step two, establishing a corresponding factor node in a factor graph model, then performing structure expansion, incremental updating and variable elimination, performing nonlinear optimization on the factor graph to solve a current variable state, and outputting a positioning coordinate, wherein the final result is shown in fig. 3, a black solid line in the graph is a reference track, a dotted line with circles is a factor graph positioning track, the direction of the dotted line with circles is a left arrow pointing to a starting direction, and the direction of the dotted line with circles is a right arrow pointing to an ending direction.
Compared with a Kalman filter and a federal filter replacing factor graph, the positioning error accumulation probability distribution of different fusion methods is shown in figure 4, the average positioning error finally obtained by the method is 0.291m, and compared with a multi-source fusion algorithm using the Kalman filter and the federal filter, the average positioning error is respectively reduced by 42.9% and 40.3%.
Finally, it should be noted that: the above embodiments are only for illustrating the technical solutions of the present invention and not for limiting the same, and those of ordinary skill in the art should understand; the invention can be modified or partially replaced, and the protection scope of the invention is not limited by the patent protection scope of the invention, and the protection scope of the invention is subject to the claims; all structural changes made by the contents of the drawings or the specification of the invention are intended to be covered by the protection scope of the patent.

Claims (4)

1. The factor graph-based air pressure assisted Wi-Fi/PDR indoor positioning method is characterized by comprising the following steps: the method comprises the following steps:
step S1: arranging a positioning device in the positioning area; the positioning device comprises Wi-Fi equipment, a wireless network card, an inertial device module and a digital air pressure sensor;
the letter of the Wi-Fi deviceCovering a positioning area by a number, transmitting acquired channel state information through a wireless network card, constructing a CSI fingerprint base according to coordinates of corresponding positions, training a positioning neural network, and then expanding the CSI fingerprint base by utilizing a generalized continuation interpolation method to obtain an interpolation fingerprint base; the positioning neural network outputs an estimated position coordinate according to the CSI signal at the moment i, and the estimated position coordinate is used as coordinate measurement data of an input factor graph after being corrected by an interpolation fingerprint database
Figure FDA0003532376210000011
The inertial device module comprises an accelerometer and a gyroscope and is used for continuously outputting three-axis acceleration and a course angle in the positioning process; the digital air pressure sensor is used for continuously outputting the air pressure value and the air temperature of the current position;
step S2: the method comprises the following steps of improving the PDR, constructing a step size model and outputting track calculation measurement information, and specifically comprises the following steps:
step S21: obtaining the three-axis acceleration a ═ a in the carrier coordinate system according to the inertial device modulex ay az]TAnd converting the acceleration a ' to [ a ' when the z-axis of the carrier coordinate system is vertically downward 'x a′ y a′z]TThen a'z-g is used as the amount to be detected for step detection, peak detection is carried out on the amount to be detected at all moments, the step point moments lower than a set threshold value are filtered out, and a step point moment set t is obtaineds1(ii) a Then, the step point sets t are collecteds1Secondary screening is carried out, jitter is eliminated, and the step point set t after secondary screening is carried outs2Comprises the following steps:
Figure FDA0003532376210000012
wherein, when tau is used at the shortest pace,
Figure FDA0003532376210000013
the j step point moment in the tau time, and g is the gravity acceleration;
step S22: establishing a logarithmic model of step length and acceleration:
f(w,b)＝w·log(a′z(t)-g)+b,t∈ts2
wherein w and b are model coefficients, data output by the inertial device module under different speeds and different routes for M times are collected, and a trajectory calculation training data set is constructed
Figure FDA0003532376210000014
Wherein
Figure FDA0003532376210000015
For the number of walking steps, SiiFor the walking path, aiming at the minimum sum of squared path residuals, optimizing parameters in a logarithmic model f (w, b) by adopting a Gaussian-Newton iteration method, wherein the optimization formula is as follows:
Figure FDA0003532376210000016
wherein the content of the first and second substances,
Figure FDA0003532376210000021
estimating a step size for the jj step of the ii piece of training data;
step S23: obtaining the course angle output by the inertial device module, and obtaining the course angle alpha at the current momentkAngle of course with last moment
Figure FDA0003532376210000022
When the absolute value of the difference is larger than 180 degrees, the jump processing is carried out:
Figure FDA0003532376210000023
Figure FDA0003532376210000024
wherein the content of the first and second substances,
Figure FDA0003532376210000025
indicating the processed heading angle,
Figure FDA0003532376210000026
then the processed course angle is processed
Figure FDA0003532376210000027
Carrying out low-pass filtering;
step S24: notation stepLeniObtaining PDR measurement information for the input factor graph for the estimated step length at time i
Figure FDA0003532376210000028
Figure FDA0003532376210000029
Figure FDA00035323762100000210
Step S3: the method for processing the air pressure data output by the digital air pressure sensor specifically comprises the following steps:
step S31: collecting a set of time series barometric pressure data
Figure FDA00035323762100000211
Calculating L generalized continuation extrapolation models:
Figure FDA00035323762100000212
wherein the content of the first and second substances,
Figure FDA00035323762100000213
for the coefficients of the fitting function, t is used for the l-th modelm-lTo tn-lCalculating fitting coefficients of data of time intervals, wherein the data quantity of each model is n-m +1, and estimating t by utilizing the L generalized continuation extrapolation modelsn+1Air pressure value at a moment
Figure FDA00035323762100000214
Figure FDA00035323762100000215
Figure FDA00035323762100000216
Step S32: calculating standard deviation sigma, significance level a and test statistic z of measurement data output by digital air pressure sensora/2By extending the predicted value in a generalized way
Figure FDA00035323762100000217
Estimating barometric truth value
Figure FDA00035323762100000218
Then setting confidence interval
Figure FDA00035323762100000219
When the measured data output by the digital air pressure sensor exceeds the confidence interval, the estimated air pressure true value is used
Figure FDA00035323762100000220
Replacing the measurement data;
step S33: calculating the differential height value at the time i according to a differential air pressure conversion height formula
Figure FDA00035323762100000221
Differential barometric altitude measurement as input factor graphInformation processing device
Figure FDA00035323762100000222
Figure FDA00035323762100000223
Wherein the content of the first and second substances,
Figure FDA00035323762100000224
is the current air temperature and is the air temperature,
Figure FDA00035323762100000225
and
Figure FDA00035323762100000226
dividing the air pressure values of two adjacent moments after the filtering in the step S32;
step S4: measuring the values obtained in steps S1-S3
Figure FDA0003532376210000031
Inputting a factor graph model; the state quantity of the factor graph is X ═ X y z α']TThe method comprises the following steps of establishing a new factor node according to an input new measurement information type, sequentially performing extension structure, increment updating and variable elimination, and then performing graph optimization, wherein the three-axis coordinate values x, y, z and a course angle alpha', and the optimization objective function is as follows:
Figure FDA0003532376210000032
Figure FDA0003532376210000033
Figure FDA0003532376210000034
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003532376210000035
to measure the error covariance matrix, | · | | non-calculationRIn order to solve the Mahalanobis distance, delta represents the state quantity difference of variable nodes at adjacent moments;
and finally, obtaining the optimized factor graph state quantity X, and taking the three-axis coordinate value as an output positioning result.
2. The factor graph-based barometric-assisted Wi-Fi/PDR indoor positioning method of claim 1, wherein: obtaining the three-axis acceleration a ═ a in the carrier coordinate system by the inertial device modulex ay az]TAnd the acceleration a' under a relative coordinate system when the z axis of the carrier coordinate system is vertically downward is obtained by being multiplied by the rotation matrix R by the left side:
a′＝[a′x a′y a′z]T＝Ra
Figure FDA0003532376210000036
ax、ay、azthe components of the acceleration along the directions of the x-axis, the y-axis and the z-axis in the carrier coordinate system are respectively a'x、a′y、a′zThe components of the acceleration along the directions of an x axis, a y axis and a z axis under a relative coordinate system are respectively; α, β and γ are heading angle, pitch angle and roll angle, respectively.
3. The factor graph-based barometric-assisted Wi-Fi/PDR indoor positioning method of claim 1, wherein: in step S23, an IIR filter is designed by using a bilinear transformation method to process the processed heading angle
Figure FDA0003532376210000037
Low pass filtering is performed.
4. The factor graph-based barometric-assisted Wi-Fi/PDR indoor positioning method of claim 1, wherein: and setting the significance level a of the measurement data output by the digital air pressure sensor to be 0.05.
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