# An Effective Sensor Architecture for Full-Attitude Determination in the HERMES Nano-Satellites
## 3 Sensor Selection
### Sun Sensors
#### 3.2.2 Photodiodes and Coarse Sun Sensor  
Complete Sun detection coverage is achieved with the usage of photodiodes configured to cover all the satellite's faces, in a way that composes a coarse Sun sensor (CSS). In fact, a single photodiode is a simple Sun angle sensor, whose output is an electric current proportional to the cosine of the incident sunlight's angle, \(\alpha\), as  
\[I=I_{0}\cos(\alpha), \tag{9}\]  
where \(I_{0}\) is the maximum current produced when the Sun is directly above the photodiode. In the proposed sensor architecture, each of the six spacecraft faces is equipped with two hot redundant identical photodiodes. Their individual measurements are averaged, unless a failure is detected in one of the two components, as described in [7]. Then, for each face, a single current value is available, and it can be associated with a Sun angle with respect to that face:  
\[\alpha_{n}=\cos^{-1}\left(\frac{\text{mean}(I_{1_{n}},I_{2_{n}})}{I_{0}} \right), \tag{10}\]  
\begin{table}
\begin{tabular}{l c c l} \hline \hline
**Specification** & **Value** & **Units** & **Notes** \\ \hline Typical Measurement Error & \(\sim\)0.5 & deg & Calibrated \\ Scale Factor Error & 12,500 & ppm & Calibrated \\ Typical Measurement Error & 1.5 & deg & Uncalibrated \\ Maximum Measurement Error & 3.5 & deg & Uncalibrated; Worst Axis Value \\ Scale Factor Error & 50,000 & ppm & Uncalibrated; Worst Axis Value \\ \hline \hline \end{tabular}
\end{table}
Table 4: Fine sun sensor specifications.  
Figure 8: Light measurements with the FSS.  
where \(n\) denotes the normal direction versor of each face in the body reference frame, \(\mathbf{\hat{n}}_{n}\). The complete Sun direction can be computed from Equation (9) as:  
\[\mathbf{\hat{s}}_{i}^{B}=\frac{\mathbf{s}_{i}^{B}}{\left|\mathbf{s}_{i}^{B} \right|},\ \text{and}\ \mathbf{s}_{i}^{B}=\sum_{n=1}^{6}\cos(\alpha_{n})\mathbf{\hat{n}}_{n}, \tag{11}\]  
where \(\mathbf{\hat{s}}_{i}^{B}\) is the Sun direction in the body reference frame.  
The important performance for each single photodiode is the accuracy with respect to the ideal cosine law, the stability of the maximum current value, and the spectral response with respect to the incoming light sources. The latter is very important to implement proper albedo-filtering functions because, knowing the position of the Earth, it allows estimating the current intensity due to the Earth's radiation. The HERMES nano-satellites use the Osram SFH 2716 A01 photodiodes. They have a spectral sensitivity very similar to the human eye, with low influence from infrared radiation, which is positive for reducing the influence of the Earth's irradiance. Moreover, they have a very small surface-mounted package, with an extremely low weight of 4 mg.  
The directional response is smooth and symmetric, allowing an easy calibration with respect to the cosine law, as is evident from the test results reported in Figure 9. These data were acquired with a sampling rate of 20 Hz, exploiting the ADC associated with each photodiode and the microcontroller that manages all these 12 light sensing elements on the spacecraft. In this case also, the test is performed by rotating a known light source along an arc of 120 deg. Calibrating the incoming light intensity with respect to the Sun intensity in LEO, the maximum output current was in the order of 67 uA, and the error with respect to the cosine law can be parametrized with a single coefficient as:  
\[I=I_{0}\cos(k\alpha). \tag{12}\]  
In the proposed example case, the calibration parameter results were approximately equal to \(k\simeq 0.8\). The calibrated formula in Equation (12) shall be used in place of the one in 17. Then, Equations (10) and (11) shall be modified accordingly, if the photodiode calibration needs to be applied.