The principle of measuring Area DSS is shown in Figure 1. The principle of multi-aperture is the same as that of a single aperture sun sensor. Calculating the mean centroid coordinate value of apertures can improve the measurement accuracy. The incident sun rays create an image spot on the image detector. (x_c, y_c) are the coordinate values of the sun spot center, l is the distance from the sun spot center to the origin of the coordinate system, h is the distance between the surface of the image detector and the bottom of the optical mask glass, θ is the incident angle, and α, β are two-axis incident angles, pitch angle and lead angle, respectively.

According to the measurement model, it is not difficult to summarize the formula as follows: (1) α = arctan ( x c h ) , β = arctan ( y c h ) , θ = arctan ( l h ) (2) l = x c 2 + y c 2 (3) tan θ = ( tan α ) 2 + ( tan β ) 2

The feature of theoretical measurement model: when either α or β angle is fixed, the corresponding coordinate value of x_c or y_c becomes constant. The compensation method is based on a multiple-apertures digital sun sensor, and the mask has 36 apertures with the same distance between every two apertures. In the cone FOV 64°, the sun sensor calculates the mean centroid coordinate value of all the apertures.

In reality, sun rays have to pass through air, quartz glass, air, protecting glass of the image detector surface and air to the image detector surface, as shown in Figure 2, due to the presence of the image detector protecting glass and the optical mask glass. According to the model of theoretical measurement, it is obvious to summarize the formula as follows: (4) l = ( h 2 + h 4 ) tan θ + h 3 tan θ 3 (5) n glass = sin θ sin θ 3

In formula (5), n_glass is the refractive index of the protecting glass of the image detector.

According to the geometric projection rule, the two-axis coordinates are: (6) x = l × cos φ , y = l × sin φ , φ = arctan ( tan β tan α )

Formulas (4), (5) and (6) are combined to calculate the change of x coordinate values using Matlab, with the condition that the angle α is fixed, and the angle β is altered. The shift trend is shown in Figure 3.

From the shift trend, the shift of x coordinate values follows the shift of the angle β when the angle α is fixed. The shift trend of x coordinate values is contrary with that of the angle β. From the shift values, the largest error reaches 10 pixels in the FOV 64°. Therefore, the major error factor of DSS is caused by optical refraction of the protecting glass on the image detector surface.

Because of the rotation error and optical refraction, the coordinate value of the angle α is not 0 when the incident angle α is 0° and the incident angle β is not 0°. As shown in Figure 6, the axis x is relative to the angle α, and the axis z is relative to the angle β.

From the optical refraction rule as shown in Figure 3, it's known that refraction doesn't bring about shift of the coordinate value of the angle α when the incident sun rays move along the axis y (the angle α is 0°). For the rotation angle, the coordinate value of angle α is not 0 when the incident sun rays move along the axis y (the angle α is 0°). Besides, the coordinate of the angle α is symmetrical concerning the origin of coordinates. So the rotation angle can be calculated through the two-axis coordinate values when the angle α is 0°, and the angle β is from 0° to +64°.

According to the rule of coordinate system rotation, the formula to calculate the rotation angle is as follows: (7) ε z = arctan ( | x 1 | | y 1 | )

In order to the rotation angle more accurate, the arithmetic mean value of the rotation angle ε_z is as the value for correlation.

The rotation angle ε_z serves to calculate the coordinates x₂, y₂ without rotation error. According to the rule of coordinate system rotation, the formula is as follows: (8) [ x 2 y 2 ] = [ cos ε z sin ε z − sin ε z cos ε z ] [ x 1 y 1 ]

In formula (8), x₁, y₁ are the coordinate values computed by the image processing chip of DSS, and x₂, y₂ are the coordinate values after the first rotation correlation.

Formula (9) is the equation used to compensate rotation error.

According to the analysis in Section 2.1.2, concerning the shift trend of coordinate values caused by refraction, the change of one-axis coordinate value depends on both the two-axis coordinate values. As a result, Dual-fit serves to map the coordinate values for correlation of optical refraction. The purpose of the map method is to make the coordinate values of the angle α under different values of the angle β equal to those of the angle α when the value of the angle β is 0. According to experimental results, the following Dual-fit map method can achieve high accuracy: (10) { x 3 = M 1 x 2 + M 2 y 2 + M 3 x 2 y 2 + M 4 y 3 = N 1 x 2 + N 2 y 2 + N 3 x 2 y 2 + N 4

In formula (10), x₂, y₂ are the coordinate values after the rotation correlation, and x₃, y₃ are the coordinate values after the coordinate map. In this paper, we collect 5° × 5° grid data points for calibration.

From formula (10), the error influencing factor of the two coordinate axis values changes following the incident angles. It is difficult to achieve high accuracy if one group of map factors is taken into consideration in the whole CFOV. Therefore, map zoning is used, which reduces some random errors as well as produces less error fluctuations due to the identical coordinate ratio (x/y) in each strip-shaped area trend.

Zoning in the first quadrant is shown in Figure 7, dual-fit is made in every strip-shaped area of four quadrants. Every strip-shaped area has one group of map factors.

The rotation angle is calculated through the coordinate values when angle α is 0° and angle β varies from 0 to 64° data in Table 1. The mean of the rotation angle ε z ¯ = 0.3226 °.

Based on a width of 10° of strip-shaped area, the map indexes of each strip-shaped area are listed in Tables 2 and 3.

The calibration experiment collects the relative coordinate values of the incident angle. Then the calibration data including the coordinate value and angle value serves to calculate the index of high order interpolation polynomials in Matlab. Formula (12) computes the tangent value of incident angle as follows: (12) tan α = ( 0.0002 ) x 5 + ( − 0.0026 ) x 4 + ( 0.0138 ) x 3 + ( − 0.0059 ) x 2 + ( 0.4487 ) x + ( − 0.0002 ) tan β = ( 0.0002 ) y 5 + ( − 0.0031 ) y 4 + ( 0.0154 ) y 3 + ( − 0.0077 ) y 2 + ( 0.4492 ) y + ( − 0.0001 )