A novel autonomous real-time position method based on polarized light and geomagnetic field

Many animals exploit polarized light in order to calibrate their magnetic compasses for navigation. For example, some birds are equipped with biological magnetic and celestial compasses enabling them to migrate between the Western and Eastern Hemispheres. The Vikings' ability to derive true direction from polarized light is also widely accepted. However, their amazing navigational capabilities are still not completely clear. Inspired by birds' and Vikings' ancient navigational skills. Here we present a combined real-time position method based on the use of polarized light and geomagnetic field. The new method works independently of any artificial signal source with no accumulation of errors and can obtain the position and the orientation directly. The novel device simply consists of two polarized light sensors, a 3-axis compass and a computer. The field experiments demonstrate device performance.

Many animals exploit polarized light in order to calibrate their magnetic compasses for navigation. For example, some birds are equipped with biological magnetic and celestial compasses enabling them to migrate between the Western and Eastern Hemispheres. The Vikings' ability to derive true direction from polarized light is also widely accepted. However, their amazing navigational capabilities are still not completely clear. Inspired by birds' and Vikings' ancient navigational skills. Here we present a combined real-time position method based on the use of polarized light and geomagnetic field. The new method works independently of any artificial signal source with no accumulation of errors and can obtain the position and the orientation directly. The novel device simply consists of two polarized light sensors, a 3-axis compass and a computer. The field experiments demonstrate device performance.
M odern navigation technologies such as GPS have made it straightforward for mankind to locate and track global positions. These techniques are classified as satellite navigation, radio navigation, inertia navigation or celestial navigation. Unfortunately each of these techniques has their own individual weakness. For example, satellite navigation 1 has the possibility of jamming and satellite signal loss. Radio navigation 2 cannot work without base stations. Inertia navigation 3 needs to correct its peculiar accumulated errors. Finally, celestial navigation 4 is complex and can be very expensive.
Technologies derived from ancient human beings and other organisms may overcome some of the weaknesses of modern technologies. For example, Viking navigators 5-7 were able to dominate the North Atlantic Ocean by using solar positioning instruments to determine the true direction. Sun-compasses were used when the sun was available. When the sun was obscured the sun position could be estimated by measuring two or more celestial points 5 due to the polarization pattern 8,9 . The polarization vector is almost perpendicular to the solar vector determined by the sun and the observer. Intense study has also been carried out upon insect's eyes that often utilize polarization to navigate [10][11][12][13][14] . However, most of their activities are on small scales and their abilities are limited to orientation. Bio-polarization 15-17 sensors derived from insect abilities have been constructed for orientation. The sensor consists of three polarization direction analyzers. Each polarization direction analyzer consists of a pair of POL sensors followed by a log-ratio amplifier. The information from two of three polarization direction analyzers is used to calculating the polarization angle. Long journeys can be observed in avian. Some migratory birds have position systems which can help them know where they are and which course is right for a goal 18 . In this case, both biological magnetic and celestial compasses are used as basic tools in order to help the birds to migrate. To date, these spectacular tools have not been directly imitated by humans.
Here we focus our attention on a novel global position system. The method combines the use of polarized light and geomagnetic field in order to determine the position and the orientation. This paper is organized as follows: Firstly, a novel device is constructed; secondly, the formula of the real-time position method is described in detail; thirdly, experiments are carried out. is perpendicular to the solar vector determined by the observer and the sun as shown in Fig. 2. The solar azimuth cannot be obtained directly as the true north is unknown. Therefore, two components of the solar vector, the fake solar azimuth (Fig. 3a) and the solar altitude angle are calculated below.
The projection of P 1 in the coordinate system O 1 X 1 Y 1 Z 1 and the projection of P 2 in the coordinate system O 2 X 2 Y 2 Z 2 are: Where P9 1 and P9 2 are the projections of the polarized-light vectors, whilst h 1 and h 2 are the measured values of the polarized-light sensors as shown in Fig. 3b-3c. Note that the coefficients A 1 and A 2 can take values of 1 or 21.
A movement transformation in the coordinate system is not necessary according to the property of the polarization pattern, though original points of the three coordinate systems are not coincident. By carrying out a rotation transformation in the coordinate system OXYZ we find: Then S o can be described by: Where S x , S y and S z are components of S o , k can take values of 1 or 21, and can be determined by naked eye judgment. The two values of k can be interpreted as vectors pointing opposite directions along a straight line. If A s 9 and h s are obtained when k is assumed to be 1, the solar azimuth and the solar altitude angle must be A s 9 1 180 and 2h s respectively when k is assumed to be 21. Then A s 9 and h s can be determined by: The relationship between the solar angles and the position. From the navigation triangle built inside a sphere, we get: Where d is the solar declination, w is the geographic latitude and v is the solar hour angle. v can be described by: Where g is the geographic longitude, E is the difference between the true solar hour and mean solar hour, and UT1 is the local time at the 0 meridian passing through Greenwich, England. The value of v ranges from 2180u to 1180u in theory. However it ranges from 2360u to 1360u in equation (6) if E is equal to 0. The periodicity of trigonometric functions means that the correct result can still be obtained. If the measuring time (Universal Time Coordinated), which can be obtained from the computer directly is known, we can get the exact value of d and E by inquiring the astronomical ephemeris based on the dynamical time 20 , see Supplementary Information SII for details.
The relationship between the fake solar azimuth and the solar azimuth. By using a 2-component magnetic compass, we can obtain equation (7) as shown in Fig. 3a: The relationship between D, w and g. In order to solve the problem of the parameter D, the IGRF (International Geomagnetic Reference Field) is used in our autonomous position method. The eleventh generation 21 (IGRF11), which was adopted in December 2009, is a linear predictive secular variation model for 2010.0-2015.0. D can be calculated if w, g, r and UTC are known according to the model. r is the geocentric altitude.
Calculation of the position. By substituting equation (6) and equation (7) for v and A s respectively into equation (5), and ignoring the little deviation between UTC and UT1 we obtain equations (8) and (9). The downloaded program can be considered as an independent equation (10) Therefore, the numerical solutions of D, w and g can be calculated by solving the combined equation. However, there are no analytical solutions of D, w and g in our method.
Field experiments performed at different times. We performed field experiments using our device (Fig. 4) in a local park (38u529450N, 121u319380E)

Results
Field experiments performed at different times. Two solutions were obtained from the combined equation at most measurement moments (Fig. 5). The k variable took the value of 21 for the first measurements and the value of 1 for the second measurements.
Field experiments with different H. Two solutions were obtained from the combined equation at each measurement moment (Fig. 6). The distribution of the correct solutions was tight and the distribution of the incorrect solutions was dispersive. The k variable took the value of 1 for measurements between 90.2u and 240.7u. It took the value of 21 for measurements between 270u and 361u as well as from 30.6u to 60.8u. For each angular range of H, 6 measurements were taken. See Supplementary Table S2 for the details of H and k.

Discussion
One or two solutions were obtained from the combined equation in most cases according to our simulation of the global positions at 10:50 AM May 28 2014 UTC and our field experiments. The cases of three or more solutions were very few. When there were more than one solution, the values of A s 9 and h s were the same and the values of A s and D were different. In other words, the different valves of A s and D were the main differences between the correct and incorrect solutions. We suggest that the distribution of D produced the wrong solutions.
It is hard to exclude incorrect or useless solutions when there is only one measurement. The vertical component of the geomagnetic field may help under this condition. However, a continuous trace exists in general when the device is in actual use. It's easy to exclude incorrect solutions in most cases as the former and the latter correct positions are very close based upon our simulation of a continuous trace. In contrast, the former and latter wrong positions were dispersive. Furthermore, the distributions of the correct and incorrect solutions were different (Fig. 6). Further research may exclude incorrect solutions by taking advantage of this property.
The set of positions obtained from the simulated data along with the random error associated with the polarized-light sensors and the compass have been calculated to discuss the impact of time on our method. The maximum error in the polarized-light sensor is 0.2u 16 . According to the datasheet, the maximum error in the magnetic north is 0.5u. The ideal value of H is assumed to be 92u. The time of each area is assumed to be the time of the first measurement in each period. A very accurate model of the geomagnetic field will be obtained when the device is in actual use, so the simulation has  ignored the errors caused by the IGRF11. The errors caused by the pitch and roll angles have been ignored since their small errors have been merged into the error of the magnetic north. The areas from simulated data take geometry similar to a rhombus apart from the one after-sunrise that is the largest. This may be caused by the fact that two possible solutions are too close that only one solution is obtained. The three areas of the sunset periods show that the errors are smaller when the sun is going down.
Compared with a solar sensor that tests the sunlight directly, our device can work well at twilight and even when the sun is largely below the horizon. A potential advantage is the redundant information from different combinations of endless celestial points. It costs about 0.1 s to calculate a single position by solving a combined equation that only includes data measured at one moment. The computer program is running on a common computer (Computer Memory is 8 GB and CPU Clock Speed is 3.0 GHz). Therefore the measurement is essentially performed in real time.
Despite the many advantages associated with our novel measurement method, there are a number of weaknesses. The polarization pattern is not the same in all weather conditions. A research shows that migratory Savannah sparrows use polarized light cues from the region of sky near the horizon to recalibrate the magnetic compass at both sunrise and sunset 22 . The polarization pattern at sunset or sunrise is suitable for our device. The device cannot work at noon or in the night and the neutral points 8 must be evaded. Much more work is needed to expand the working time of the device. More experiments at special places such as the poles or the equator are necessary to support the method in the further research. The polarization pattern might be changed partially by the complex geomagnetic field at the poles. What's more, the method is useless in polar nights at the poles.
k is a parameter that cannot be determined by the device itself. The field experiments performed at different times show that k points in a single direction along a straight line. The field experiments with different H show that the distribution of k may follow some rules. We suggest that the meridian of the sun is the boundary. k takes the value of 21 when the line OX as shown in Fig. 1 is on one side of the boundary and the value of 1 when the line OX is on the other side. The two polarized-light vectors may be parallel when the solar meridian is parallel to the axis Y. Consequently nothing useful can be obtained.
The device built by us is the simplest example of a real-time position method. Changing the spatial relationship of the three planes, the device can have many different types. By using a simple transformation of coordinates, whilst utilizing further information from the geomagnetic field, a device can be constructed without the need to adjust plane A to the local horizontal. However, the magnetic compass may have a lower accuracy if the angle of pitch or roll is large. It's obvious that the orientation can be obtained by using this paper's method without the help of the magnetic compass if the latitude and other easy-getting parameters d are obtained. The calibration of magnetic orientation by celestial rotation 23 , which is not very clear to human beings, can tremendously simplify the method.
If the accuracies of polarized-light sensors and compasses can be improved, the accuracies of our device can be improved significantly. We are going to build better devices with higher accuracy, smaller sizes 24,25 and the ability to evade the neutral points. Simple and useful ways to find the best solution from the possibilities are significant too. What's more, the simulation of accuracy performed in a global grid is also a necessary work.

Conclusion
A novel real-time position method has been proposed that works independently of any artificial signal source and has no accumulation of errors. The algorithm has been presented in detail and a navigation device based on the method has been constructed. The experimental data indicate that the method is reliable. The method opens up new possibilities since it is a promising way to elucidate the means of avian positional navigation.