# Integrated Magnetic Management of Stored Angular Momentum in Autonomous Attitude Control Systems
## 4 Velocity Dumping Control  
The integrated control for the magnetic management of stored angular momentum can not only be applied to continuously desaturate the reaction wheels during three-axis attitude control modes, but can be easily modified to perform a fast and robust spacecraft's velocity dumping, which can be exploited, for example, during the detumbling mode in the launch and early operations phase (LEOP) of a typical space mission. Despite the effectiveness and the simplicity of the well-consolidated \"b-dot\" control law [10], there are situations in which the spacecraft's velocity dumping is too slow. In these cases, the detumbling phase can be sped up with the use of reaction wheels. However, dumping  
Figure 11: MIL simulation of nadir-pointing control without integrated magnetic angular momentum management.  
Figure 10: Stored angular momentum evolution in MIL simulation of nadir-pointing control.  
the spacecraft's velocity with momentum exchange actuators results in a fast build-up of the stored angular momentum, which soon leads to the actuator's saturation. The integration of a simple velocity dumping controller with the proposed method for the magnetic management of the stored angular momentum allows solving this problem.  
For this application, the main control law is a simple velocity proportional control:  
\[\mathbf{t}^{DMP}(t)=-k_{DMP}\,\boldsymbol{\omega}(t), \tag{24}\]  
where the gain parameter \(k_{DMP}\) is used to tune the rapidity of the controller in dumpingout the angular velocity of the spacecraft. Note that this parameter should be also tuned according to the maximum actuation capacity of the available actuators. The velocity dumping control is then split between the reaction wheels and magnetic torquers as in Equation (9), and it is integrated with the desaturation torque as:  
\[\mathbf{t}_{C_{V}}=(1-k)\,\mathbf{t}_{R}^{DMP}+k\,\mathbf{t}_{M}^{DMP}+ \mathbf{t}_{R}^{DES}, \tag{25}\]  
where the desaturation torque \(\mathbf{t}_{R}^{DES}\) is computed as explained in Equation (19). Note that in this application, the velocity dumping and the desaturation controls should have approximately the same bandwidth to avoid a quickly stored angular momentum accumulation. Empirically, this is achieved by designing the gain parameter as:  
\[k_{DMP}\sim I_{R}\omega_{R}^{Max}k_{DES}. \tag{26}\]  
Exploiting the same simulation approach described in the previous section and considering a similar case study scenario, the results of the proposed velocity dumping method are reported in Figure 12, with the spacecraft initial tumbling rates in the order of \(10^{1}\) deg/s. It can be clearly seen how the proposed method is capable of dumping out the initial spacecraft angular velocity by accumulating it in the reaction wheels. At the same time, the application of integrated magnetic angular momentum management avoids reaching the saturation levels and quickly dissipates the stored angular momentum. The detumbling phase can be considered completed in less than 30 min, which is a good value compared with typical magnetic \"b-dot\" control performance. This did not happen in the simulation with the integrated control left inactive, shown in Figure 13. As expected from classical control methods, the velocity dumping with angular exchange actuators is not effective and fails as soon as the reaction wheels reach the angular momentum saturation. In this case, the spacecraft begins to chaotically rotate, exchanging angular momentum with the internal rotors.  
Figure 12: MIL simulation of velocity dumping control.