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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Direct sunlight absorption by trace gases can be used to quantify them and investigate atmospheric chemistry. In such experiments, the main optical apparatus is often a grating or a Fourier transform spectrometer. A solar tracker based on motorized rotating mirrors is commonly used to direct the light along the spectrometer axis, correcting for the apparent rotation of the Sun. Calculating the Sun azimuth and altitude for a given time and location can be achieved with high accuracy but different sources of angular offsets appear in practice when positioning the mirrors. A feedback on the motors, using a light position sensor close to the spectrometer, is almost always needed. This paper aims to gather the main geometrical formulas necessary for the use of a widely used kind of solar tracker, based on two 45° mirrors in altazimuthal set-up with a light sensor on the spectrometer, and to illustrate them with a tracker developed by our group for atmospheric research.

Spectroscopic analyses of direct incident sunlight are commonly used in atmospheric research. Such experiments make use of the Sun as a light source to quantify molecular absorptions in the atmosphere and then retrieve trace gas abundances. Stratospheric ozone [

Several kinds of trackers, sometimes referred to as heliostats, are used for atmospheric spectrometry, based on setups of one or several rotating mirrors. Some of them are equatorially mounted, like in Table Mountain Facility [

Because developing a solar tracker is typically a master's thesis work [

Calculating the Sun position in the sky given the time of observation and the geographical coordinates is well documented. A reference algorithm is given by Jean Meeus in [

_{T}_{M}_{1}) depends on the focal length of the lens (_{D}

Considering our set-up in Brussels, which is a typical FTIR station, the tracker's mirrors are 10 cm (Φ_{T}_{2} of 20 mrad. On the other hand, the quadrant diameter (Φ_{D}_{1}. The actual FOV is the minimum, 20 mrad, limited by the tracker size. This value is superior to the apparent diameter of the Solar disk which is important to track the center of the Sun. A simple algorithm can achieve such an accuracy for the ephemeris calculation, like the one given in the appendix. It is however necessary to take into account the orientation of the tracker, which can lead to pointing errors superior to the FOV.

One source of error, when pointing to the calculated Sun position, is the orientation of the baseplate of the tracker compared to the altazimuthal system. Depending on the observatory configuration, it may be difficult or impossible to align accurately the tracker along the North-South direction. If it was the only problem, the remaining constant offset could be simply fitted and added in the calculated azimuth. However, because the baseplate is never completely leveled either, other offsets are added to the calculated positions, affecting both azimuth and elevation in a way that depends on the pointing direction of the tracker. Some tracker uses an active search method to solve this problem. In practice they reach the calculated position and achieve spiral motion around this point to set the sun spot in the field-of-view of the sensor. Misalignment effect can on the other hand be taken into account in the calculation requires determining the Euler angles of the observatory and the tracker baseplate, respectively, compared to the azimuthal system. We discuss the Euler angles before describing our way to determine them. For the sake of simplicity, we only mention the observatory in the following, considering that the baseplate to be part of it.

Euler angles of an observatory may be seen, as in _{x}_{y}_{z}_{offset}

The calculations leads to:

In Cartesian coordinates, the unit vector (_{t},y_{t},z_{t}_{0},_{0}) in the altazimuthal system:

Substituting _{offset}

These new Cartesian coordinates can then be converted to altitude (_{t}_{t}

In the above equation, _{2}(

Determining Euler angles accurately by measurements is not easy. An analytical method to estimate them is given in [_{offsets}

The photodiode signal indicates that the Sun beam is tilted compared to the optical axis of the spectrometer. The photodiode signals must be converted into angular movements of the altitude and azimuth axes of the tracker to correct the misalignment. If the photodiode was placed on the reference frame of the mirror M2 this conversion would be straightforward, but due to its position after the tracker it depends on the position of the tracker mirrors. A trial-and-error method to correct the misalignment is theoretically possible using analogue electronics without a computer but a smoother tracking can be achieved if the conversion is understood.

The conversion can be expressed once again by a matrix, which transforms in this case a vector hitting mirror M1 to a vector pointing to a direction in the sky given by its altitude and azimuth. It is the opposite of the light direction but is simpler to figure out, and considering Fermat principle, yields the same information.

The rotation of the two motorized stages can be accounted for using rotation matrices as described in the previous section. The reflection on the two mirrors is modeled using another matrix which takes the form:
_{R}

_{1} and a zenith angle of _{2}. The reference frames ℜ_{1} and ℜ_{2} are respectively attached to the mirrors _{1} and M_{2}, with the _{1} and _{2}, can be expressed as a transformation whose matrix _{tracker}

The above formula is derived as follow: (a) the reflection on _{2} (_{R}_{1} with a change of basis involving _{y}_{2}); (b) this product of three matrices is multiplied on its right side by the preceding (seen from the spectrometer) reflection on _{1}(_{R}_{0}, involving _{z}_{1}).

Developing the matrix product yields the matrix of the tracker optical system as a function of the tracker position (_{1}_{2}):

The transformation expressed by _{tracker}_{1},_{2}) is computed for the Sun spot on the diode plane compared to its center by:

The spot offset (_{1},_{2}) defines 2 coordinates of the beam vector. The last one, Λ, should represent the distance from the diode to mirror M1. Multiplying _{tracker}_{1},_{2},Λ) would yield accurate Sun angles after conversion to spherical coordinates, but is not practically possible with a diode, contrary to an imaging sensor. The calculated position (_{s},y_{s},z_{s}_{1} and _{2}. The solar pseudo-coordinates are then:

In practice, the quadrant vector may differ from (_{1},_{2},Λ) due to reflections such as on the mirrors M0 and M4q on

Developing

It is then possible to calculate roughly an altitude(_{2}_{S}_{1}_{S}_{1} and _{2}. The angular corrections to apply on the two axes are then:
_{1} and _{2}_{1} should not be under 0.004°.

From a control theory perspective, the altazimuthal tracker and its feedback is a non-linear multi-input multi-output (MIMO) system. Indeed, two outputs defining the pointing direction (_{1} and _{2}_{1} and _{2}), and the relationship between the inputs and the outputs varies with the position of the tracker. However, having modeled this relationship in the previous section, it is possible to change the feedback scheme while tracking. In control theory, this is an example of adaptive control.

The correction of the azimuth and altitude angles discussed earlier only takes into account the current error,

_{1}_{c}_{2}_{c}_{1}(_{2}(

Considering the latitude of the Reunion Island observatory where the tracker is installed (20.9°S) it is worth considering an issue occurring with the altazimuthal geometry,

Our group has been doing FTIR measurements at Reunion Island for several years ([

A second FTIR station has been installed in Saint-Denis in September 2012. This station, which is also automated, is based on a Bruker 125 HR spectrometer, more appropriate to measure CO_{2} atmospheric loading, in the framework of the new Total Carbon Column Observing Network (TCCON). The geometry of the Sun tracker is altazimuthal. It was built at our institute and used to validate the methods described in the last section.

This new solar tracker uses a Newport RV-160 rotation stage for the azimuth rotation and a Vexta stepping motor with a gear box for the altitude. Both rotations are driven by a Newport XPS controller, linked to the controlling PC. The tracker mirrors are elliptical with a 10 cm minor axis. The photodiode setup was purchased from Bruker with the spectrometer and is installed at the input window of the spectrometer. It consists of a 1 cm mirror which reflects a small portion of the incoming light to a 18 cm focal length lens which focuses the beam onto the 4-quadrant photodiode. The optical path is shown in

We have derived the geometrical formulas needed to track the Sun with a kind of altazimuthal tracker widely used in atmospheric remote sensing. The setup is based on two rotating 45° mirrors facing each other and a 4-quadrant photodiode involved in a closed-loop control of the tracker. After discussing the required accuracy for the calculated mode and calculating the FOV of the sensor, we described how to take into account and estimate the Euler angles, representing the orientation of the tracker compared to the ground. These sections can actually be applied to other tracking setups. On the other hand, even if the method is general, the formula for the active tracking depends strongly on the optical configuration and may not be used for other trackers' geometries. We have proposed a control loop with PID to achieve a smooth tracking while reducing overshoot and the residual part of the error. Finally, we have tested the formulas with a custom-built solar tracker that has been installed together with a FTIR spectrometer at Reunion Island in September 2011.

Among the future work will be the improvement of the tracking smoothness, and particularly the tuning of the six parameters of the PID controllers. We will also implement the solution presented in Section 3 and check whether the measurement dead time can be reduced.

A characteristic of the Maido observatory is the very regular cloud cycle. At noon, the clouds reach the observatory almost every day. This is very convenient for clouds studies but less for solar occultation trace gases measurements. On the other hand, the nights are so clear up there that the observatory was first supposed to be dedicated to astronomical research. This is thus a good place to try Moon tracking and we plan to work on that in the future.

This work was funded by the Belgian Science Policy (BELSPO). The authors wish to thank Thomas Blumenstock for his advices and for having sent him the work of M. Huster. They also thank Filip Desmet, Bart Dils and Sébastien Henrotin for useful discussions.

For the sake of completeness, we reproduce here the simple algorithm we use to calculate the solar coordinates given a date and a position, taken from [

We first compute the fractional year (γ) in radians:

Then we derive the equation of time (Δ

The equation of time represents the difference between apparent solar time and mean solar time, which can be as large as 16 min. It is due to the obliquity of the ecliptic and the elliptical form of the earth orbit.

From the fractional year, we also get the solar declination (_{⊙}) in radians:

The declination is the equivalent of the latitude on the celestial sphere.

The offset _{off}

The true solar time (

The solar hour angle, in degrees, comes from the true solar time as:

For a given latitude, the hour angle and the declination are converted to horizontal coordinates, i.e., solar zenith angle (

Finally, the effect of refraction in arcminutes can be approximated using Sæmundsson's formula [

Comparison between the formulas reproduced in appendix and JPL ephemerides.

Geometrical setup of the considered solar tracker, using two 45-degree mirrors, M1 and M2, rotating along orthogonal axes. Mirror M0 directs the Sun light into a spectrometer. A fraction of the light beam is deflected toward a 4-quadrant photodiode enabling a closed-loop control of the mirrors position.

Optical scheme from the tracker mirrors to the 4-quadrant diode. The field of view seen by the diode depends on the different aperture sizes and path lengths. The green and red beams represent the Sun light path when the tracker is not perfectly aligned on the Sun.

Illustration of the Euler Angles of an observatory compared to the altazimuthal coordinate system.

The tracker mirrors and their rotation can be modeled as rotation matrices in their reference frames, which apply to the beam vector. Note that the _{0}, ℜ_{1}) and (ℜ_{1}, ℜ_{2}).

Sun spot hitting the quadrant, not to scale.

Control loop for an altazimuthal tracker.

Optical paths of our particular set-up from the tracker to the 4-quadrant diode (not to scale).

Fit of the Euler Angles to take into account the alignment offsets in the calculation mode. The track was performed on 12 September 2011.