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Abstract
To address the needs of polarized light navigation for accurate position information of feature points in the sky, an accurate solar position detection method based on an all-sky polarization pattern imaging system is proposed. Unlike the traditional spot-based solar position detection method, this method uses the polarization information inherent in the atmosphere to accurately measure solar position. This approach is characterized by simple detection, high accuracy, and wide application range. The optical acquisition system is composed of three miniature large-field camera modules and polarizers, which enables a more compact structure, smaller size, and lesser height. Based on this principle, the solar position solution algorithm was simulated and then verified in various weather environments using the optical acquisition system built as part of this study. Solar position was detected at different moments on the same day in clear weather, and the accuracy of the measured solar altitude and azimuth angles was 0.024° and 0.03°, respectively. The accuracy of the measured solar altitude and azimuth angles was 0.08° and 0.05°, respectively, when the sun was shielded by high-rise buildings and 0.3° and 0.1° when the sun was shielded by branches and tree leaves. Aerosol concentrations exceeding a certain amount destroyed the Rayleigh distribution pattern of polarized light, thus affecting solar position detection accuracy. It is concluded that this novel detection method can not only meet the needs of polarized light navigation for solar position, but also provide a new exploration idea for enthusiasts who are eager to explore the mysteries of the universe.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
How to obtain the precise position of the sun, which is one of the most prominent features in the sky, has gradually become one of the key issues in navigation. Traditional methods to obtain solar position include latitude positioning tracking, sunlight intensity tracking, and five-point tracking [1–2]. These methods can be divided into two main categories. The first tracks and calculates solar position based on the astronomical calendar. However, this method highly depends on the accuracy of the time information captured and is limited because it can only collect information at a fixed point and time and will obtain the solar position with a large error once the geographical location or time changes. Moreover, the solar time angle must also be repeatedly calibrated with any change in location. Therefore, this method cannot be applied to navigation. The second method detects solar spots visually [3–5]. Although not limited by time and position, the visual method has strict requirements for information on the solar spot and needs to process a large amount of disordered data to distinguish ideal spots from other information. This means that it must perform several filtering processes, which not only destroy the original information, but also are cumbersome. Accordingly, these two methods to obtain solar position in space are no longer suitable for the development of navigation today.
Sunlight is polarized by particle scattering in the atmosphere and reflection from the ground after it enters the atmosphere through irradiation, forming an atmospheric polarization pattern throughout the air [6–7]. This atmospheric polarization pattern has received more attention from researchers recently due to its rich polarization information [8–10], which not only has fixed attributes and is not interfered with by humans, but also provides very important and stable reference information for polarized light navigation. Therefore, increasing numbers of experts and researchers around the world are involved in the study of atmospheric polarization patterns [11–12]. Among them, Lanbrinos successfully calculated the atmospheric polarization pattern and measured the azimuth angle of the Sahabot polarization navigation robot developed in the same study, thus demonstrating for the first time the feasibility of using the atmospheric polarization pattern [13]. Yang et al. proposed polarized light compass-aided inertial navigation under discontinuous observations environment [14]. Chu et al. first developed a polarized light sensor based on a power supply, which successfully realized accurate navigation for robots with an accuracy of 0.2° [15–16]. Based on the principles of bionics,Gao et al. proposed to apply all-sky atmospheric polarization patterns for polarized light navigation [17] and obtained an all-sky atmospheric polarization pattern using an atmospheric polarization pattern generation method based on the pattern’s continuous distribution [18]. Later, the two research teams respectively conducted a exploration of computational orientation model based on an artificial neural network and brain-inspired navigation model based on absolute heading [19–20].
Based on the above analysis, a solar spatial position detection method for polarized light navigation was proposed in this study. Firstly, the atmospheric polarization model was established. Spatial position information for the sun was obtained by inverse solution and multiple iterations of the spherical triangle composed of the angle of polarization (AOP) and the sun based on Rayleigh theory, after which the algorithm was verified in various weather environments using the optical acquisition system as built. Notably, compared with traditional methods, the solar position detection method proposed in this paper is not only free from constraints of position and time, but also maintains high measurement accuracy, which satisfies the high accuracy requirements of polarized light navigation for solar position.
2. Atmospheric polarization information measurement method
The testing system built in this paper consists mainly of a micro-miniature optical charge-coupled device (CCD), fish-eye lens, model RP-L220; three polarization plates with detection directions of 0°, 45°, and 90°; and a supercomputer. As shown in Fig. 1, polarized light intensity in different directions can be obtained by three polarizers in different directions and then sent to the supercomputer by the CCD for processing to obtain sensitivity vectors in different directions. In the system as constructed, the three CCDs use a synchronous control module to collect images simultaneously to ensure real-time performance. Besides, the aperture and exposure parameters of the three fish-eye lenses must be consistent to avoid errors caused by differences in field of view. In addition, the polarizers in the three directions must be calibrated to avoid errors caused by manual rotation.
Any light wave in atmospheric polarization measurement can be described using the Stokes vector [I, Q, U, V], where I indicates the total intensity of partially polarized light, Q and U indicate the linearly polarized light components with reference directions of 0° and 45°, respectively, and V indicates circularly polarized light, which is usually neglected except when studying the polarization characteristics of scattered light.
The light intensities collected by the detector through the polarizers in the three directions are denoted by I1, I2, and I3, and the degree of polarization (DOP) can be calculated using Eq. (1):
The atmospheric polarization distribution in the whole sky can be obtained by calculating the DOP.
3. Solar position solution method
An atmospheric polarization pattern model as shown in Fig. 2 was developed, with the center of the circle as the observer's observation position. Point P denotes the observed point, with the altitude and azimuth angles denoted by ${h_p}$ and ${\alpha _p}$, respectively. S represents the position of the sun, with its altitude and azimuth angles denoted by ${h_s}$ and ${\alpha _s}$, respectively. It can be assumed that a beam of sunlight arrives at the observer’s eye through Rayleigh scattering. Then the DOP, $P(\theta )$, at point P can be calculated by Eq. (2), where ${P_{max}}$ represents the maximum DOP in the atmospheric polarization pattern:
Next, the spherical triangle formed by Z, P, and S is observed. According to the spherical cosine theorem, Eq. (3) can be obtained:
The DOP, $P = DOP({{h_s},{\alpha_s},{h_p},{\alpha_p}} )$, of the scattered light is calculated by solving Eqs. (2) and (3).
The angle between the polarization vector (E) direction of the measured point and the meridian is defined as the AOP, expressed by $e = \nu \cos \gamma + \textrm{h}\sin \gamma $. According to the Rayleigh scattering law, the E-vector at point P is:
where $\nu = ({ - \textrm{cos}{h_p}\textrm{cos}{\alpha_p}, - \textrm{cos}{h_p}\textrm{sin}{\alpha_p},\textrm{sin}{h_p}} )$, ${h_p} = ({ - \textrm{sin}{\alpha_p},\textrm{cos}{\alpha_p},0} )$. When the solar position in space is $({{h_s},{\alpha_s}} )$, AOP is $\gamma = {\tan ^{ - 1}}\left( {\frac{{\sinh {{\cosh }_s} - \cosh {{\sinh }_s}\cos ({{a_s} - a} )}}{{\sin ({{a_s} - a} ){{\sinh }_s}}}} \right)$:The AOP and DOP are represented by the Rayleigh model as $\gamma = AOP({S,{h_p},{\alpha_p}} )$ and $P = DOP({S,{h_p},{\alpha_p}} )$, respectively, where S denotes the solar vector as $S = {({{h_s},{\alpha_s}} )^T}$. Then the azimuth and polarization information of the measured point can be substituted to obtain Eqs. (6) and (7):
The sun vector S contains two unknowns. Because the number of measured information points is much greater than 2, these equations are over-determined and cannot be solved directly. In this study, numerical methods were used to solve these over-determined equations. Taking Eq. (6) as an example, the evaluation function, Eq. (8), constructed by using the least-squares principle on Eq. (6), is:
Setting $F(S )= {[{{f_1}(S ){f_2}(S )\cdots {f_n}(S )} ]^T}$, the above equation can be solved by the Gauss-Newton method, and the resulting iterative equation is:
where $J({{S_k}} )$ is the Jacobian matrix of $F({{S_k}} )$, i.e.,The specific process of multiple iteration is:
The flow diagram of algorithm as shown in Fig. 3. The spatial position $S = ({{h_s},{a_s}} )$ of the sun can be calculated by the algorithm.
4. Experimental design
4.1 Simulation of the solar position solution algorithm
Figure 4 illustrates the effect of the initial value of the algorithm iteration on its accuracy. Figures 4(a) and 4(b) respectively indicate the solar altitude and azimuth angles based on the DOP distribution information, and Figs. 4(c) and 4(d) respectively indicate the solar altitude and azimuth angles based on the AOP distribution information. According to the results, the accuracy of the algorithms based on the AOP and DOP was very high in their respective convergence intervals, with both solar altitude angle and solar azimuth errors of less than 10−6.
Fig. 4. Effect of optimized initial value on the algorithm. (a) Solar altitude angle error distribution based on the DOP; (b) solar azimuth error distribution based on the DOP; (c) Solar altitude angle error distribution based on the AOP; (d) solar azimuth error distribution based on the AOP.
Furthermore, comparing the two models, the solar vector calculated by polarization information possessed a larger convergence region than that calculated by polarization angle information, indicating that the algorithm has a relatively loose restriction on the initial value. In contrast, the algorithm based on the AOP information can converge only when the initial value of the solar azimuth is very close to the theoretical value. However, in the actual atmospheric environment, the AOP distribution is more stable than that of the DOP. This study analyzed possible problems in the AOP information and found that they were caused by the discontinuous AOP near the solar meridian. Subsequently, the relationship between the solar vector and the evaluation functions of the AOP and DOP was analyzed, as shown in Fig. 5. The minimum point of the AOP information evaluation function in the distribution process is not clear, and the change in the function gradient is not continuous, which is not favorable for algorithm convergence.
In this research, based on the characteristic that the AOP is symmetric about the solar meridian, the absolute value of AOP was taken in the evaluation function, making the AOP distribution continuous, and the evaluation function is given as Eq. (12):
Figure 6 shows a diagram of the improved AOP evaluation function distribution. The continuous function gradient distribution near the theoretical value is beneficial to optimal solution of the function. Next, the algorithm was simulated under different initial iteration values (the simulated ranges of initial iteration values were a solar altitude angle of 0°-60° and an azimuth angle of 10°-200°). According to Fig. 7, except for the edge region of the theoretical value, the altitude angle and azimuth errors of the sun are as low as 10−10 in the vast majority of the initial value region, with high accuracy. The simulation not only exhibits that the algorithm itself has very low error, which can be neglected, but also demonstrates that the algorithm greatly expands the convergence interval for the improved evaluation function.
Fig. 7. Simulation results of the algorithm after improving the evaluation function. (a) Solar altitude angle error; (b) solar azimuth error.
4.2 Solar position measurement experiment
The experiment was carried out with a measurement system composed of a three-channel polarization imaging device, horizontal platform, and supercomputer (Fig. 8).
To avoid the influence of haze and dust, the experiment described in this paper was carried out on a mountaintop at high altitude in Taiyuan, Shanxi Province (longitude 118.443°E, latitude 38.015°N). The day was sunny with very high sky visibility, which ensured a consistent sky polarization pattern using the Rayleigh model. Measurements were taken from 8 A.M. to 5 P.M., with data collected every hour. Figure 9 shows plots of the acquired experimental data, where column A represents the original image, column B represents the DOP distribution map obtained by calculation, and column C represents the AOP distribution map obtained by calculation.
Figure 10 displays the distribution of the experimental data calculated for solar spatial position. The measured solar spatial location is compared with the theoretical position in Fig. 10(a), and the distribution of the absolute value of the error is shown in Fig. 10(b). The accuracy of the solar position obtained by the algorithm in this paper is as high as 10−2 degree. The errors in azimuth and altitude angles are distributed around 0.03° and 0.024°, respectively, as analyzed from linear fitting and are distributed uniformly, which is of a certain practical value.
Fig. 10. Experimental data analysis diagram in sunny weather.(a) Experimental data plots of solar altitude angle and azimuth angle; (b) experimental error plots of solar altitude and azimuth angles.
In this study, experimental measurements in a building occlusion environment were carried out with tall buildings in Taiyuan City (longitude 113.443°E, latitude 33.015°N), which was a sunny day with very high sky visibility. The experimental results in the building occlusion environment are shown in Fig. 11(a). The sky is available only in a narrow window area due to the influence of building occlusion, which affects the solar position calculation, mainly because a large part of the meridian is occluded by the building. However, because the meridian is a straight line with a span equal to 180°, the remaining part can still provide sufficiently accurate meridian direction information. The solar altitude and azimuth angles calculated under building occlusion are illustrated in Fig. 12. The calculation error of solar altitude angle is about 0.08°, but the calculation error of the AOP is relatively large. The solar azimuth error is around 0.05°, which is relatively more accurate. The decrease in the angular accuracy of solar altitude was considered to be caused by the shift of the neutral point from the solar position to the zenith due to multiple scattering of sunlight in the atmosphere.
Fig. 12. Experimental data analysis diagram under building occlusion.(a) Experimental data plot of solar altitude angles; (b) experimental data plot of solar azimuth angles.
Experimental measurements in a tree occlusion environment were carried out in a park with dense woods in Taiyuan City (longitude 113.443°E, latitude 33.015°N), which was a sunny day with very high sky visibility. The experimental results in the tree occlusion environment are shown in Fig. 11(b). The color images show that the area of the sky blocked by dense branches and leaves is more than 80%. However, according to the distribution image results of the AOP and DOP, the distribution of polarized light in the sky is far less affected than expected by occlusion by branches and leaves. Only the partial occlusion of the thick trunk demonstrates a significant effect on the polarization pattern, with the polarization patterns of other parts remaining almost unaffected, especially near the maximum polarization region, which is conducive to navigation using polarization images. The solar altitude and azimuth angles calculated under tree occlusion are shown in Fig. 13. The calculation error of the solar altitude angle is about 0.3°, but the calculation error of the AOP is relatively large. The solar azimuth error is about 0.1°, which is relatively more accurate. The accuracy is still stable, although it is slightly lower than in the sunny environment without building occlusion, as well as in the sunny and building-occluded environment.
Fig. 13. Experimental data analysis diagram under tree occlusion. (a) Experimental data plot of the solar altitude angle; (b) experimental data plot of the solar azimuth angle.
In sunny weather, the atmosphere contains a small amount of aerosol, which will enhance polarized light when it is within a certain range, but will destroy the Rayleigh distribution pattern of polarized light if it exceeds a certain concentration threshold (such as in clouds, haze, dust, and other weather). The aerosol concentrations (mainly PM2.5 (Particulate Matter 2.5)) in this study were obtained by the local environmental protection monitoring department and were detected using physical and chemical methods, with high reliability in terms of collection density as well as remote-sensing equipment. Figures 14 and 15 illustrates the experimental results in the atmospheric aerosol environment. The figures show that the number of times of scattering is greatly increased due to aerosols and that the vibration intensity in all directions tends to be uniform, resulting in depolarization, or in other words, some parts of the DOP will be reduced. The calculated solar altitude angle and azimuth angle errors range from 1.5° to 3.7° at aerosol concentrations of 20-110 mg/m3. Note that the error of the solar altitude angle calculated in the aerosol environment diverges from that of the fitted curve, but that the error of the calculated solar azimuth angle converges to that of the fitted curve. The calculated solar altitude angle and azimuth angle errors range from 1.5° to 6.6° at aerosol concentrations of 20-560 mg/m3. The calculated solar altitude angle error in the aerosol environment is also relatively divergent, and the calculated solar azimuth error is also relatively convergent. The experiment revealed that an increase in aerosol concentration leads to a certain degree of distortion in the AOP data, which affects solar position detection accuracy and limits the navigation accuracy of polarized light in specific applications. Subsequently, it was found that the effect of aerosol on solar position calculation and polarized light navigation can be weakened or eliminated by reconstructing polarization information using a neural network algorithm.
Fig. 15. Experimental data analysis diagram under aerosol conditions.(a) Effect of quantitative aerosol concentration on calculation error of the solar altitude angle and azimuth angle.(b) Effect of high aerosol concentration on the calculation errors of the solar altitude angle and azimuth angle.
The solar information calculation error may be attributed to the following factors: (1) the polarization camera cannot obtain complete all-sky atmospheric polarization information due to the influence of field of view and focal length, which has a certain impact on solar position information, (2) installation error of the polarized light compass may also affect accurate acquisition of solar position information, (3) image and circuit noise in the polarized light compass also has a certain impact on accurate acquisition of solar position, and (4) aerosols also produce a depolarization phenomenon, resulting in a decrease in the DOP in some areas. Therefore, the cause of solar information calculation error will be further analyzed to weaken or even eliminate the impact of errors on solar position.
5. Conclusions
In an effort to address the needs of polarized light navigation for accurate position information for feature points in the sky, the atmospheric polarization pattern based on the Rayleigh model was analyzed, and an accurate solar position detection method based on an all-sky polarization pattern imaging system was proposed. The all-day solar positions in three environments were measured with a three-channel polarization measurement system that was built as part of the study, and the results were compared with theoretical solar positions. The accuracy of the measured solar altitude and azimuth angles was found to be 0.024° and 0.03°, respectively, in a sunny weather environment with no occlusions, 0.08° and 0.05°, respectively, in a building occlusion environment, and 0.3° and 0.1°, respectively, in a tree occlusion environment. This accuracy can fully satisfy the needs of a polarized light system for solar position for the purpose of navigation. The increase in aerosol concentration in bad weather will lead to a certain degree of distortion of AOP data, which will affect solar position detection accuracy and limit the navigation accuracy of polarized light in specific applications.
Funding
The General Project of Shanxi Provincial Basic Research Program (202303021211169); Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2021L295); National Natural Science Foundation of China (61905172), Science and Technology Innovation Teams of Shanxi Province (202304051001035).
Acknowledgments
We thank International Science Editing (http://www.international.science.editing.com) for editing this manuscript. We acknowledge the financial support from the Natural Science Foundation of China (61905172), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2021L295), the General Project of Shanxi Provincial Basic Research Program (202303021211169), and the special fund for Science and Technology Innovation Teams of Shanxi Province (202304051001035).
Disclosures
The authors declare no competing financial interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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