## 3 Solar Arrays Pointing Algorithms
### Control Performance  
The performance of the ADCS is assessed at first with numerical simulations on an ideal system. In the ideal case, the dynamic of the spacecraft is realistic, with all the disturbance environmental torques. However, the current measurements are ideal, with no errors or noise. Similarly, the actuators are ideal and capable to apply the desired torque commanded by the ADCS. In fact, the control torques discussed in this section are directly experienced by the satellite in the simulations. The performance is assessed in terms of the specific electrical current generated by the solar arrays, \(I_{S}/I_{0}\), which corresponds to the Sun angle, \(\alpha_{S}\), according to Equation (8). The numerical simulations account for the discrete-time nature of the ADCS algorithms, which are integrated with continuous-time dynamics, kinematics, and environment simulator. Figures 3 and 4 show a simulation of 2 orbital periods in a Sun-synchronous orbit with local time of ascending node (LTAN) at 06:00. Thus, the orbit is not affected by eclipses and the solar arrays can generate electrical power continuously. The simulation is initialized in a random attitude state, with random initial angular rates of the spacecraft. The angular rates are dispersed on the three-body axes in a way that the magnitude of \(\omega\) is ten times the orbital angular rate at 3\(\sigma\). The ADCS is designed to work at 10 Hz, with \(K=100\) and \(W=50\) number samples. Hence, each ADCS cycle lasts for 200 sample along 20 s, and each pulse of the proportional control action is active for 5 s. In this way, the duty cycle of the proportional control is 25 %. The solar arrays alignment is achieved in less than 2000 s and then maintained with very little control pulses. The control modulation experienced by the satellite is evident in Figure 4, both in terms of the pulsed control axis sequence and of the scaling coefficient term, \(f(\alpha_{S})\). The lack of any measurement or actuator noise and error makes the control very sharp, with just a short Sun acquisition period at the beginning of the simulation. During this period, the Sun is also lost from the front of the solar arrays, but eventually, the control is capable to get to the correct target. Note that \(I_{S}/I_{0}\) goes down to \(-1\), which is associated to \(\alpha_{s}=180\) deg when the Sun is not in front of the solar arrays. Figures 5 and 6 show a simulation of 2 orbital periods in an equatorial orbit, which has an eclipse period of \(t_{ecl}\simeq 30\,\mathrm{min}\). The simulation is initialized as in the previous case, with the ADCS that shall acquire and point the Sun with the solar arrays. Analogously, the ADCS is the same as the previous simulation. Also, in this case, the ideal determination and control are capable of sharply orienting the spacecraft and maintaining the pointing across the different eclipse windows. Along with the eclipses, the control with respect to the angular rates only maintains a constant spacecraft attitude in the inertial reference frame, which is approximately equal to Sun pointing for short time periods. This allows a fast re-pointing of the panels from a quasi-aligned condition that is typically completed in less than \(\sim 500\,\mathrm{s}\), as shown in Figure 5. Further simulation results supporting this claim will be discussed in the followings. In Figure 6, reporting the control torques applied to the satellite in these ideal system simulations, the alternation between the full control and the purely derivative control is evident, in particular during the eclipses. The typical behavior of the proposed control algorithm can be appreciated in the first simulation time, during the Sun acquisition phase, for both simulations in Figures 3 and 5. The Sun angle decreases (i.e., \(I_{5}/I_{0}\) increases) and approaches the perfect alignment with some small wobbles. In fact, when the Sun angle increases (i.e., \(I_{5}/I_{0}\) decreases), the algorithm starts looking back again for the correct rotation axis to point to the Sun.