Bio-inspired Sky Polarized Orientation based on the atmospheric polarization symmetry mode and Inertial Sensors

This paper presents a sky polarized orientation method based on the symmetry of the atmospheric polarization mode and the inertial sensors. Firstly, a method of extracting solar meridians based on canny operator and Hough Transform is proposed by using the symmetry of atmospheric polarization pattern. Then using the horizontal Angle information of the inertial sensor and the ephemeris information of the sun, a method for solving the yaw Angle of the polarized light is designed. Finally, a single axis rotation experiment and a vehicle-mounted dynamic experiment are carried out to verify and compare the proposed method. Compared with the least square method, the positioning accuracy of the single-axis turntable experiment is improved by 22.61%~73.86%, and that of the vehicle-mounted dynamic experiment is improved by 47.71%, which shows that the method can provide a new orientation means for unmanned vehicles and unmanned aerial vehicles.


INTRODUCTION
In nature, there are many organisms that can use atmospheric polarized light for orientation.For example, insects such as bees, sand ants, and dragonflies can navigate a complex, tortuous foraging process to return to their nests in a near-straight line, with no visible reference, while being able to sense polarized light signals in the sky [1]- [2].Polarized light navigation is based on the perception mechanism of polarized light by ants' compound eyes to obtain the direction of the solar meridian and realize the calculation of carrier heading Angle.Bionic polarized light navigation uses the sky polarization information to navigate.The research focuses on the sky polarization algorithm modeling and the design of polarized light detection sensor [3].
Domestic and foreign teams have conducted research on bionic polarized light sensors.In 2011, Chu et al improved the polarized light sensor based on photoelectric secondary tube, realized the polarization information calculation through the design of the polarizer structure, and conducted in-depth research on the error model and compensation method of bionic polarized light compass [4].In 2016, SONY released the IMX250 series polarization camera, which realizes the CMOS polarization chip integration, and the camera has an ultra-high resolution of 5 million.These studies provide key theoretical and technical support for polarized light orientation and navigation, and greatly promote the development of this field.[5] At present, the research on bionic polarized light orientation algorithm is mainly divided into two parts, that is, polarized light orientation technology based on polarization vector orthogonal and orientation technology based on atmospheric polarization pattern symmetry.The polarized light orientation method based on polarization vector orthogonal takes advantage of the fact that the polarization direction of the sky observation light is perpendicular to the sun line of sight, and realizes the direction measurement by comparing the polarization vector of sunlight with the vector of the earth surface.In the adaptive orientation algorithm described by Yang et al. (2014), the algorithm realizes the estimation of the orthogonal vector field of polarized light by fitting the polarized light intensity model in the sky.By calculating the included Angle of the vector and the ray direction in the vector field, this method can achieve accurate azimuth estimation [6].The filtering algorithm proposed by Li et al. (2020) uses Kalman filtering to optimize the obtained data after measuring the polarization vector, so as to improve the orientation accuracy [7].
The orientation technique based on the symmetry of the atmospheric polarization pattern makes use of the symmetry of the atmospheric polarization pattern in the sky to determine the direction.The polarization pattern is related to the position of the sun, and the pattern is symmetrical under certain conditions.Galvez et al. (2006) detailed how organisms such as bees, birds, and fish use atmospheric polarization patterns for natural navigation [8].In 2015, Lu et al. proposed a solar meridian extraction method based on Hough transform, [9] and in 2020 Cui et al. compared it with LSM, and the accuracy was improved.[10] In 2017, Zhang et al. used polarization-light field camera to obtain atmospheric polarization patterns, proposed a symmetry evaluation method of polarization Angle, obtained the optimal solar meridian fitting results by minimizing symmetry errors, and conducted polarized light orientation experiments in a cloudy environment to verify the effectiveness of this method [11].
This paper mainly studies the orientation method of atmospheric polarization mode symmetry based on Hough transform.Firstly, the principle of polarized light sensor and Hough transform is introduced.Secondly, the orientation method of atmospheric polarization mode symmetry based on Hough transform is introduced in detail.The fixed-axis turntable experiment and vehicle navigation experiment are carried out, and the results are compared with the least square method.

PRINCIPLES OF POLARIZED LIGHT NAVIGATION
In this paper, the image obtained by polarization camera is used to realize the navigation and positioning.The orientation principle adopted is shown in Fig. 1.Firstly, the polarization Angle (AOP) and degree of polarization (DOP) are solved, and the polarization direction information is extracted.Then the feature threshold is set, the feature edge is extracted by canny operator, and the meridians are extracted by Hough transform.Finally, the yaw Angle is obtained by using the information of horizontal attitude Angle and solar ephemeris provided by inertial measurement unit(IMU).

Acquisition of sky polarization information
The polarization vision sensor used in this paper is mainly composed of four cameras, four wide-angle lenses and four Sony IMX250 polarization dedicated imaging chips, as shown in Fig. 2. The optical axes of the four cameras face the same direction and are distributed on the four vertices of the square.This arrangement makes the overlapping field of view between the cameras maximum and reduces the parallax between the cameras.The four polarizers are installed around the optical axis in the directions of 0°, 45°, 135° and 90°.[12][13] This installation mode can minimize the influence of noise when the polarization state is solved.

Fig. 2. Polarizing camera
For incident light in a certain direction, its basic parameters can be estimated from three unrelated measurements, so the optical axis of the polarized light needs to be distributed in several specific directions.The calculation of dense polarization information can be realized by traversing the whole polarization image through the sliding window.The polarized light measurement equation corresponding to the four cameras is as follows: According to Marius theorem, the incident light response equation of each pixel of the polarized light sensor can be established.For the pixel in the same grid window, the polarization information measurement unit can be regarded as 4 measurements.The non-homogeneous linear equations can be obtained by sorting out the four measurement equations:

AQ U (2)
Where A is the coefficient matrix of the equation and U is related to the camera response.
The polarization Angle and the polarization degree of the incident light can be obtained by the least square method, and the specific calculation process can be referred to [14].
The Angle of polarization () ) and degree of polarization ( ) 1 q , 2 q and 3 q are the three components of Q.

Solar meridian characteristic threshold definition :
The azimuth distribution model of solar polarized light is symmetrical about the solar meridian, and the azimuth of the solar meridian is 90 r q, which deviates from the original theoretical line.[15] According to the symmetry of the azimuth of polarized light in the sky, it is necessary to set the solar meridian feature threshold N to obtain the solar meridian feature region, where 0,90 N , the solar meridian feature region is an area formed by the polarization azimuth value of @ > 90 90 90 90

Feature point set acquisition based on canny operator:
The feature region of the solar meridian has a symmetric edge.In the edge detection of many images, canny operator has Gaussian smooth noise reduction, calculation of gradient amplitude and direction, non-maximum suppression of gradient amplitude, double threshold algorithm detection and connecting edge to detect the optimal edge.By using canny operator, the feature point set can be obtained quickly and effectively.[16] The two-dimensional projection of atmospheric polarization modes always shows a symmetrical distribution.In practical measurements, it has been found that the polarization angle characteristics are more stable than the polarization degree distribution.Therefore, the Angle of polarization distribution is used for symmetry axis extraction.First, for the polarization azimuth map, the following global processing is applied˖ 255, 90 , , 0, 90 , N is the feature threshold range.The pixel coordinate value that meets this condition is set to 255, and the pixel coordinate value is set to 0 if it does not meet this condition.At this time, the threshold segmentation method can separate the solar meridian from the background, and the original polarization azimuth becomes a binary image.

T U P
in the polar coordinate system.There are multiple accumulator units in the polar coordinates, and each accumulator corresponds to an accumulator with an initial value of 0. When the sine curve passes through the accumulator unit once, the accumulator value of the corresponding accumulator unit will be increased by one.The three points , 1 X , 2 X and 3 X on the line are converted to the sinusoidal curves corresponding to the polar coordinates respectively, and the point 1 , T U P is the intersection point, which means that the accumulator unit where the intersection point is located votes 3 times, then the value of the accumulator is increased by 3. Hough transform detects the line feature by finding the peak value of the accumulator of each accumulator unit in the polar coordinate system.

Obtain the heading Angle according to the solar meridian with the Inertial sensors
According to [19], when the polarized light sensor is tilted, the measured polarization information will no longer have symmetry.MIMU can provide two horizontal attitude angles through the accelerometer.
It is agreed that the north-East-geogeographical coordinate frame is the navigation frame( n -frame), the front-right-lower coordinate frame is the carrier coordinate frame( b -frame), and the right-front-top coordinate frame is the sensor coordinate frame( l -frame),.
The projection of the unit vector of the sun's line of sight direction in the n frame is:  (6) In the formula, s a is the sun's azimuth Angle, s h is the sun's altitude Angle, and the two are only related to time and geographical position.
The attitude transfer matrix from the b-frame to the l-frame is: Through the direction cosine matrix, a can be projected into the polarized light sensor coordinate system: The attitude transfer matrix b n C can be obtained from the rolling, pitch and yaw angles of the carrier.[20] By arranging equations ( 7)~( 9), the projection of the sun direction vector in the sensor coordinate system can be obtained.Then the projection of the E vector direction in the sensor coordinate system can be obtained as follows: u l e OP OS (9) Where, OP is the unit vector of the observation direction, and OS is the projection coordinate of the sun direction vector in the sensor coordinate system.Through the coordinate system transformation, the E vector in the incident optical coordinate system i can be obtained as: Where: i l C represents the rotation matrix from the carrier coordinate system to the incident optical coordinate system.It can be seen that the sensor coordinate system first rotates D around the Z axis and then J around the Y axis.
The Angle of polarization () ) is: ) e e (11) Where: ( )   i i e represents the i-th element of the direction of the E vector ( 1, 2, 3 i ).This paper adopts the method of obtaining yaw Angle according to the azimuth of the solar meridian.In order to more intuitively describe the principle of using the azimuth of the solar meridian to obtain yaw Angle, the position of the observation point is taken as the coordinate origin, the geographical due north direction is taken as the X-axis forward direction, and the geographical due east direction is taken as the Y-axis forward direction, and a two-dimensional plane coordinate system is established as shown in Fig. 3.The angles involved in yaw Angle acquisition are described.[18] Fig. 3. Yaw Angle acquisition principle In Fig. 3, E s represents the azimuth of the solar meridian, E h represents the heading Angle, and 0 E represents the Angle between the current orientation and the solar meridian.Thus, the formula for obtaining the heading Angle is: 0 E is the local solar azimuth, which is determined by time and local geographical position.

Introduction to single-axis rotation experiment and vehicle experiment
The experimental equipment of single-axis turntable is mainly composed of bionic polarized light compass, laptop computer, optical tablet and high-precision single-axis turntable, as shown in Figure 4.
Each of the four pixels of the camera lens is a measuring unit, each measuring unit has a grating with a polarization direction of 0°, 45°, 90° and 135°, and the camera lens pixels are.The turntable has 391 gear teeth, each tooth is about 0.9207°, and the repeated orientation accuracy can reach 0.01°.The polarizing vision camera is installed on a high-precision single-axis turntable for rotation.Sample the image every 30 frames.The single-axis turntable has a total of 391 bars in one circle, so the Angle of each rotation is 27.62°, and a total of 14 photos are taken in one circle.90 135 45 0 q q q q q q q q q q q q q q q q ª º « AOP infographics for sparse and dense measurements are obtained as Fig.Through the density measurement, the sparse information image of 1224×1024 can be obtained into the information image of 2447×2047 pixels, so that more dense information can be obtained.
After processing by canny operator, the region where the gradient change meets the threshold setting can be obtained, and a binary image can be obtained, as shown in Fig. 8.

Experimental results of single axis rotation:
The obtained heading angles are shown in Fig. 10.

Fig. 10. Carrier heading Angle estimation results
The mean square error of different feature thresholds N in dense and sparse cases is compared.In dense cases, N is 5 as the optimal value, and the mean absolute error (MAE) is 0.4610° and root mean square error (RMSE) is 0.0144°.
When N=5, compare the Hough transform with the absolute value of the least squares error, as shown in TABLE I. and Fig. 11.Hough Transform can obtain smoother error curve with smaller mean, which proves that Hough transform has higher precision and noise resistance than least squares fitting.

Results of on-board experimental orientation:
The experiment lasted about 10 minutes, and the polarized light sensor data and GNSS dual antennas were stored at 10Hz.Polarization images can be used to solve the polarization light orientation.Fig. 12 shows the polarization light orientation calculation results of the on-board experiment.The reference yaw was taken as the reference value and compared with the LSM method, and the orientation error was obtained as shown in Fig. 13 and TABLE II. .Compared with the LSM algorithm, the overall absolute error of the proposed method is smaller and the yaw output is more stable.The results show that the polarized light orientation algorithm proposed in this paper can effectively extract the position of solar meridians in polarized images, and the algorithm has high accuracy and robustness.

CONCLUSION
A bio-inspired sky polarized orientation method based on the atmospheric polarization mode and inertial sensors is proposed.The polarization image is collected by polarization camera, the feature threshold is set, the feature point set is extracted by Canny operator, the feature line is extracted by Hough transform, and the heading Angle is obtained by combining the horizontal attitude Angle obtained by local time, latitude and longitude and IMU.In addition, a turntable experiment and a dynamic on-board experiment are designed, in which the positioning accuracy of the turntable experiment is improved by 22.61%~73.86%,and that of the on-board dynamic experiment is improved by 47.71%, which verifies the feasibility of the algorithm.

Fig. 1 .
Fig. 1.Schematic diagram of polarized light orientation principle Transform: Hough transform is a technique that can recognize image features and can be used to identify straight line features in images.[17]It mainly detects the straight line feature through the voting algorithm, the principle is shown in the figure.The line in the cartesian coordinate system can be represented by the parametric equation U T T xcos ysin , which can be represented by the point 1 ,

TABLE I .
COMPARISON OF MAE AND RMSE OF THE TWO METHODS IN TURNTABLE EXPERIMENT

TABLE II .
COMPARISON OF MEAN ERROR AND MEAN SQUARE ERROR OF TWO METHODS IN VEHICLE-