## 3 CMG Model and Control Law Design  
Low-Earth-orbit micro-satellites must continuously observe multiple targets within short orbital arc segments. Therefore, the satellite must have a high maneuvering speed, which requires an actuation system capable of providing high-precision control torques. Hence, we chose the Single Gimbal Control Moment Gyro (SGCMG) as the satellite's actuation mechanism. SGCMG comprises a low-speed gimbal and a high-speed rotor. The rotation axis of the low-speed gimbal is always perpendicular to that of the high-speed rotor. By imparting an angular velocity to the low-speed gimbal through the high-speed rotor's center of mass, the external gimbal's rotation can alter the internal rotor's angular momentum vector relative to the satellite. This produces an orthogonal output torque to both the rotor and gimbal axes. The relationship between the low-speed gimbal's angular velocity, the high-speed rotor's moment of inertia, and the output torque is described below. \[T_{cmg}=-(\dot{\delta}\mathbf{x}_{g})\times(h\mathbf{z}_{g})=-\dot{\delta}h\mathbf{y}_{g} \tag{7}\]  
where \(T_{cmg}\) represents the SGCMG's output torque, \(\dot{\delta}\) denotes the low-speed gimbal's angular velocity, \(\dot{\delta}\) is the angle through which the low-speed gimbal has rotated, \(h\) is the high-speed rotor's moment of inertia, and \(\mathbf{x},\mathbf{y},\mathbf{z}\) are the three axes of the SGCMG's body coordinate system, coinciding with the gimbal axis, rotor axis, and the direction of the output torque, respectively. When using CMG for attitude control, it is required that the designed control law has good maneuverability to drive the CMG group system to generate the desired torque. However, due to the presence of singular states, the design of the control law is more complex. In this study, low-orbit microsatellites continuously observe multiple targets within a short arc of passage, mainly maneuvering in the rolling axis direction. Therefore, satellites only need to have agile maneuverability in the rolling axis direction. The use of a full CMG configuration is not economically appropriate, which requires the design of a new control torque gyroscope configuration to achieve a maximum single-axis torque output, while the other two axes can be stably controlled through other actuators. The \"scissors\" configuration CMG is symmetrically installed through an even number of control moment gyroscopes, and the desired single-axis attitude maneuver control torque is reasonably coordinated between the CMGs. At the same time, the control law is used to keep the low-speed frame angle of the symmetrical CMG as synchronized as possible, avoiding disturbance torque in other directions. Therefore, in this study, we select two symmetrically installed scissor-configuration CMGs as the actuators, as illustrated in Figure 1.  
The torques generated by CMG1 along the \(x\)-axis and \(y\)-axis are as follows:  
\[\left\{ \begin{array}{l}T_{1x}=-\dot{\delta}_{1}h\cos\delta_{1}\\ T_{1y}=-\dot{\delta}_{1}h\sin\delta_{1}\end{array} \right. \tag{8}\]  
The torques generated by CMG2 along the \(x\)-axis and \(y\)-axis are as follows:  
\[\left\{ \begin{array}{l}T_{2x}=-\dot{\delta}_{2}h\cos\delta_{2}\\ T_{2y}=-\dot{\delta}_{2}h\sin\delta_{2}\end{array} \right. \tag{9}\]When both CMG frames have equal angles and angular velocities, the control torques in the \(y\)-axis direction have an equal magnitude but in opposite directions, canceling each other. This will only generate a net torque along the \(x\)-axis, with a magnitude of:  
\[T_{x}=-2\dot{\delta}_{1}h\cos\delta_{1} \tag{10}\]  
Therefore, the control law for a single set of scissors-configured CMG1 is:  
\[\dot{\delta}_{d1}=\frac{-T_{cmg}}{2h\cos\delta_{1}} \tag{11}\]  
To ensure that the low-speed gimbal angles of the paired CMGs remain synchronized, a Proportional-Integral (PI) control is introduced, guaranteeing that the nominal gimbal angular velocity allocation operates under desired conditions.