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 [21], for which C([22]) or Matlab (http://www.mathworks.com/matlabcentral/fileexchange/4605-sunposition-m) versions are available. This accuracy degrades with larger zenithal angle due to atmospheric refraction, which depends on local meteorological conditions (for an accurate and wavelength dependent refraction formula, see [23]). Irregular variations in Earth rotation also limit the accuracy of ephemerids independently of refraction. On the other hand, the absolute accuracy of commercial rotation stages used in Sun trackers, e.g., Newport RV-160, is only 0.01°. This reduces the interest of using an accurate but complex algorithm for ephemeris's calculation, which is anyway not necessary providing a closed-loop control is performed on the mirrors' position. In this case, the lowest acceptable accuracy is thus determined by the field of view of the 4-quadrant diode: once the Sun image hits the quadrant, the tracking can be performed in closed-loop.

Figure 2 shows the typical optical scheme between the tracker and the quadrant diode, the optical axis has been aligned for the sake of clarity. The distance L is measured from the first mirror which reflects the sunlight, i.e., M2 in Figure 2. Close to the spectrometer, a part of the beam from the tracker, with diameter Φ_T corresponding to the small axis of M1 and M2 on Figure 2, is deflected by a mirror of diameter Φ_M to a lens which focuses the beam onto the quadrant. The maximum field-of-view seen by the diode (FOV₁) depends on the focal length of the lens (f) and on the diameter of the quadrant (Φ_D), according to FOV 1 = arctan Φ D f. The lens L1 is unlikely to reduce the FOV assuming its size superior to the mirror's one. Indeed, the beam is parallel before the lens which implies that distance D can be reduced if necessary. The tracker aperture, in our case defined by the azimuthal stage free aperture, is more important, especially since the diameter of a rotation stage is limited and the distance between the tracker and the deflecting mirror (M4) depends on the observatory configuration. The mirror's FOV due to the tracker is FOV 2 = arctan Φ T L.

Considering our set-up in Brussels, which is a typical FTIR station, the tracker's mirrors are 10 cm (Φ_T) wide and the optical path to the spectrometer (L) 5 m long, which leads to a FOV₂ of 20 mrad. On the other hand, the quadrant diameter (Φ_D) is 6 mm while the focal length of the lens ((f)) is 200 mm, which gives 30 mrad for FOV₁. 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 Figure 3, as consecutive rotations around three orthogonal axes needed to account for the pitch, roll and yaw of this observatory Converting the solar altitude and azimuth to the observatory frame requires thus to compute the multiplication of three rotation matrices along the different axes, i.e., R_x(γ), R_y(β) and R_z(α). The resulting matrix M_offset expresses the transformation of coordinates due to the Euler angles. (1) M offsets = [ 1 0 0 0 cos γ − sin γ 0 sin γ cos γ ] × [ cos β 0 sin β 0 1 0 − sin β 0 cos β ] × [ cos α − sin α 0 sin α cos α 0 0 0 1 ]

The calculations leads to: (2) M offsets = [ cos α cos β − sin α cos β sin β cos α sin β sin γ + sin α cos ( γ ) cos α cos γ − sin α sin β sin γ − cos β sin γ sin α sin γ − cos α sin β cos γ cos α sin γ + sin α sin β cos γ cos β cos γ ]

In Cartesian coordinates, the unit vector (x_t,y_t,z_t) giving the direction of the Sun in the observatory frame will thus be related to the solar spherical coordinates(az₀,alt₀) in the altazimuthal system: (3) [ x t y t z t ] = M offsets × [ cos alt 0 cos az 0 cos alt 0 sin az 0 sin alt 0 ]

Substituting M_offset with Equation (2) we get the following expressions for those coordinates: (4) { x t = cos ( α + az 0 ) cos β cos alt 0 + sin β sin alt 0 y t = ( cos ( α + az 0 ) sin β sin γ + sin ( α + az 0 ) cos γ ) cos alt 0 − cos β sin γ sin alt 0 z t = ( sin ( α + az 0 ) sin γ − cos ( α + az 0 ) sin β cos γ ) cos alt 0 + cos β cos γ sin alt 0

These new Cartesian coordinates can then be converted to altitude (alt_t) and azimuth (αz_t) angles relative to the tracker: (5) { ρ t = x t 2 + y t 2 alt 0 = atan 2 ( z t , ρ t ) a z t = atan 2 ( y t , x t )

In the above equation, atan₂(y, x), available in many programming languages, stands for the argument of the complex number x + iy. It is closely related to the arctangent of y/x but it indicates unambiguously the quadrant of this angle on the trigonometric circle.

Determining Euler angles accurately by measurements is not easy. An analytical method to estimate them is given in [17] which basically consists of recording the position of the tracker at three different times and solving Equation (4). This is appropriate for the studied case, i.e., a collector for solar energy application installed outside with only one mirror and no closed-loop control. With our considered two-mirror tracker, which does not collect light but directs it toward a spectrometer, other sources of misalignments appear. Indeed, the tracker is also likely to be misaligned compared to the spectrometer, and mirror themselves can be tilted. Other angles can be considered in M_offsets and is done in [10]. In practical applications, despite the three Euler angles, the calculated mode is likely able to reach the Sun within the FOV of the 4-quadrant diode. With the closed-loop control it is easy to track the Sun during a whole clear-sky day providing an operator correctly sets the Sun tracker initially. Euler angles can then be fitted using all the recorded positions of the mirrors during the day. It has the advantage that other sources of misalignment are included: even if only three angles are fitted which may not exactly be the Euler angles, they minimize simultaneously the effects of all offsets. We implement this method in Section 4. This requires the closed-loop control of the tracker on the Sun position.

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: (6) M = I − 2 n n Twhere I is the identity matrix and n the normal vector to the mirror surface. At reference position, the mirror are parallel and thus their normal is the same, given by the vector ( 0 , 1 2 , − 1 2 ). The transformation matrix for the two mirror is thus the same, M_R, which is, from Equation (6): M R = [ 1 0 0 0 0 1 0 1 0 ]

Figure 4 presents the tracker pointing to an azimuth θ₁ and a zenith angle of θ₂. The reference frames ℜ₁ and ℜ₂ are respectively attached to the mirrors M₁ and M₂, with the x axes in the direction of their small semi-axes and the y axes along the line joining the two mirrors. The optical system inside the frame, with only mirrors M₁ and M₂, can be expressed as a transformation whose matrix M_tracker is: (7) M tracker = R z ( θ 1 ) × R y ( θ 2 ) × M R × R y ( − θ 2 ) × M R × R z ( − θ 1 )

The above formula is derived as follow: (a) the reflection on M₂ (M_R) is expressed in the reference frame of ℜ₁ with a change of basis involving R_y(−θ₂); (b) this product of three matrices is multiplied on its right side by the preceding (seen from the spectrometer) reflection on M₁(M_R); (c) another change of basis is performed to express the transformation in ℜ₀, involving R_z(−θ₁). i.e., (8) M tracker = [ cos θ 1 − sin θ 1 0 sin θ 1 cos θ 1 0 0 0 1 ] × [ cos θ 2 0 sin θ 2 0 1 0 − sin θ 2 0 cos θ 2 ] × [ 1 0 0 0 0 1 0 1 0 ] × [ cos θ 2 0 − sin θ 2 0 1 0 sin θ 2 0 cos θ 2 ] × [ 1 0 0 0 0 1 0 1 0 ] × [ cos θ 1 sin θ 1 0 − sin θ 1 cos θ 1 0 0 0 1 ]

Developing the matrix product yields the matrix of the tracker optical system as a function of the tracker position (θ₁,θ₂): M tracker = [ cos θ 1 cos θ 2 cos ( θ 1 − θ 2 ) + sin θ 1 sin ( θ 1 − θ 2 ) cos θ 1 cos θ 2 sin ( θ 1 − θ 2 ) − sin θ 1 cos ( θ 1 − θ 2 ) cos θ 1 sin θ 2 sin θ 1 cos θ 2 cos ( θ 1 − θ 2 ) − cos θ 1 sin ( θ 1 − θ 2 ) cos θ 1 cos ( θ 1 − θ 2 ) + sin θ 1 cos θ 2 sin ( θ 1 − θ 2 ) sin θ 1 cos θ 2 − sin θ 2 cos θ 1 − θ 2 − sin θ 2 sin θ 1 − θ 2 cos θ 2 ]

The transformation expressed by M_tracker can now be applied to a vector corresponding to the Sun light beam direction on the spectrometer side of the tracker. It will lead to the position of the Sun in Cartesian coordinates. The vector is built from the 4 diode signals (VA,VB,VC,VD), as represented in Figure 5. Basically an offset position (ε₁,ε₂) is computed for the Sun spot on the diode plane compared to its center by: (9) { ɛ 1 = ( V B + V C ) − ( V A + V D ) ɛ 2 = ( V A + V B ) − ( V C + V D )

The spot offset (ε₁,ε₂) defines 2 coordinates of the beam vector. The last one, Λ, should represent the distance from the diode to mirror M1. Multiplying M_tracker by the quadrant vector (ε₁,ε₂,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 (x_s,y_s,z_s) hereafter is thus not absolute but is sufficient to get the sign of the rotations to apply on the axes. Λ can be chosen arbitrarily as long as its absolute value is large enough compared to ε₁ and ε₂. The solar pseudo-coordinates are then: (10) [ x s y s z s ] = M tracker × [ ɛ 1 ɛ 2 Λ ]

In practice, the quadrant vector may differ from (ε₁,ε₂,Λ) due to reflections such as on the mirrors M0 and M4q on Figure 1, necessary to deviate a part of the beam to the 4-quadrant photodiode. Defining the position vector thus requires to pay attention to the optical path from M1 to the photodiode. In section 4, we explain how we deal with the problem in our particular case.

Developing Equation (10) yields: (11) { x s = ( cos θ 1 cos θ 2 sin ( θ 1 − θ 2 ) − sin θ 1 cos ( θ 1 − θ 2 ) ) ɛ 2 + ( sin θ 1 sin ( θ 1 − θ 2 ) + cos θ 1 cos θ 2 cos ( θ 1 − θ 2 ) ) ɛ 1 + Λ cos θ 1 sin θ 2 y s = ( sin θ 1 cos b sin ( θ 1 − θ 2 ) + cos θ 1 cos ( θ 1 − θ 2 ) ) ɛ 2 + ( sin θ 1 cos θ 2 cos ( θ 1 − θ 2 ) − cos θ 1 sin ( θ 1 − θ 2 ) ) ɛ 1 + Λ sin θ 1 sin θ 2 z s = − sin θ 2 sin ( θ 1 − θ 2 ) ɛ 2 − sin θ 2 cos ( θ 1 − θ 2 ) ɛ 1 + Λ cos θ 2

It is then possible to calculate roughly an altitude(θ_2S) and azimuth(θ_1S) for the Sun applying the Cartesian to spherical coordinates conversion (Equation (5)). This position is approximate and relative to the tracker since it does not take into account the Euler angles described in the last section, but what matters are the signs of the differences between these calculated values and the current altitude and azimuth relative to the tracker, defined by θ₁ and θ₂. The angular corrections to apply on the two axes are then: (12) { d θ 1 = sgn ( θ 1 S − θ 1 ) k 1 d θ 2 = sgn ( θ 2 S − θ 2 ) k 2where k₁ and k₂ are the tracking angle steps that should be small to have a smooth tracking, yet large enough for the mechanical resolution of the rotation stages and the apparent movement of the Sun. The azimuth changes for instance at a rate of 15° per hour, assuming 1 second between the steps, k₁ should not be under 0.004°.