## 2 The Measurement Principle of Mass Characteristic
### Test Principle for Determining the Moment of Inertia  
According to the angular momentum theorem [25], the time derivative of the angular momentum of the particle system at any fixed point \(O\) is equal to the vector sum of the moments of the external forces acting on that point \(O\). This can be expressed as:  
\[M=\lim_{t\to 0}\frac{\Delta L}{\Delta t}=\frac{dL}{dt} \tag{7}\]  
where \(M\) is the vector sum of the external moments of particle system, \(L\) is the angular momentum, and \(t\) is the time. Angular momentum can be expressed as:  
\[L=J\omega \tag{8}\]  
where \(J\) is the moment of inertia, and \(\omega\) is the angular velocity. According to Equations (7) and (8), \(M\) can be expressed as:  
\[M=J\alpha \tag{9}\]  
where \(\alpha\) is the angular acceleration. When the different particle systems are identical to the external moment (\(M\)) of the same fixed point \(O\), \(M\) can be expressed as:  
\[M=J_{1}\alpha_{1}=J_{2}\alpha_{2} \tag{10}\]  
When integration is used on both sides of Equation (10), it can be expressed as:  
\[J_{1}\int_{0}^{t}\alpha_{1}d_{t}=J_{2}\int_{0}^{t}\alpha_{2}d_{t} \tag{11}\]  
According to Equation (11), the final expression is:  
\[J_{1}\omega_{1}=J_{2}\omega_{2} \tag{12}\]  
In the actual measurement procedure, the external moment of the measurement system contains the output torque (\(M_{e}\)) of the motor, the friction torque (\(M_{f}\)) produced by the friction force of the reduction drive and bearings, and the air resistance torque (\(M_{a}\)) generated by the air. The sum of the external moments (\(M\)) can be expressed as:  
\[M=M_{e}+M_{f}+M_{a} \tag{13}\]Because \(M_{f}<<M_{e}\) and \(M_{a}<<M_{e}\), the total moment of the actual measurement system is approximately equal to \(M_{e}\). The moment of inertia of the fixture tools and worktable is \(J_{0}\), the motor starting acceleration time is \(t_{0}\), the rated angular velocity is \(\omega_{e}\), and the angular velocity at time \(t_{1}\) (\(t_{1}\leq t_{0}\)) is \(\omega_{0}\). Hence the relationship of the angular velocity (\(\omega\)) and the start time of (\(t\)) can be expressed as:  
\[\left\{ \begin{array}{l}\omega=\dfrac{\omega_{0}}{t_{1}}t,t\leq t_{0}\\ \omega_{e},t\geq t_{0}\end{array} \right. \tag{14}\]  
Assuming the moment of inertia of the pico-satellite in the direction of rotation is \(J_{1}\), when the pico-satellite is equipped with the measurement system, the motor starting time is \(t^{\prime}_{0}\), the rated angular velocity is \(\omega_{e}\), and the angular velocity at time \(t_{2}\) (\(t_{2}\leq t^{\prime}_{0}\)) is \(\omega^{\prime}{}_{0}\). Then, the relationship of the angular velocity (\(\omega^{\prime}\)) and the start time of \(t^{\prime}\) can be expressed as:  
\[\left\{ \begin{array}{l}\omega^{\prime}=\dfrac{\omega^{\prime}{}_{0}}{t_{2}} t^{\prime},t^{\prime}\leq t^{\prime}{}_{0}\\ \omega_{e},t^{\prime}\geq t^{\prime}{}_{0}\end{array} \right. \tag{15}\]  
When Equations (14) and (15) are in the same time \(T\) (\(T\leq t_{0}\)), the angular velocities of Equations (14) and (15) are \(\frac{\omega_{0}T}{I_{1}}\) and \(\frac{\omega^{\prime}{}_{0}T}{I_{2}}\), respectively. According to Equation (12), it can be expressed as:  
\[\dfrac{\omega_{0}T}{t_{1}}J_{0}=(J_{0}+J_{1})\dfrac{\omega^{\prime}{}_{0}T}{t _{2}} \tag{16}\]  
Then, according to Equation (16), the moment of inertia of the pico-satellite (\(J_{1}\)) in the direction of rotation can be expressed as:  
\[J_{1}=\bigg{(}\dfrac{\omega_{0}t_{2}}{\omega^{\prime}{}_{0}t_{1}}-1\bigg{)}J_{0} \tag{17}\]  
After the measurement system rotates \(90^{\circ}\), the moments of inertia of the pico-satellite (\(J_{2}\) and \(J_{3}\)) in the other two directions can also be obtained. Finally, the moments of inertia of the pico-satellite on each axis of the coordinate system (\(J_{1}=J_{z}\), \(J_{2}=J_{y}\), \(J_{3}=J_{x}\)) can be obtained.