# A Structure for Accurately Determining the Mass and Center of Gravity of Rigid Bodies
## 3 Parametric Representation of the Structure  
Here, for a convenient and simple explanation, the platform with the load cells is called the base-frame, and the upper platform is called the platform-frame [17]. The base-frame (O\({}_{\text{B}}\),X\({}_{\text{B}}\),Y\({}_{\text{B}}\),Z\({}_{\text{B}}\)) and the platform-frame (O\({}_{\text{P}}\),X\({}_{\text{P}}\),Y\({}_{\text{P}}\),Z\({}_{\text{P}}\)) are two right-handed orthonormal coordinate systems with (X\({}_{\text{B}}\),Y\({}_{\text{B}}\),Z\({}_{\text{B}}\)) and (X\({}_{\text{P}}\),Y\({}_{\text{P}}\),Z\({}_{\text{P}}\)) as their bases respectively, as shown in Figure 7. The three centers of ball grooves, which are held by load cells, always rest on the X\({}_{\text{B}}\)Y\({}_{\text{B}}\)-plane. The origin O\({}_{\text{B}}\) of the base-platform is the intersection of the bisectors of the three angles of the base frame triangle. Coordinates of the centers of the ball grooves formulate the locations of the ball grooves. The origin O\({}_{\text{P}}\) of the platform-frame is the center of the ball groove shown in Figure 7, and the X\({}_{\text{P}}\) axis is parallel with the axis of the cylindrical groove. The plane is parallel with the X\({}_{\text{P}}\)Y\({}_{\text{P}}\)-plane. In the neutral configuration, the platform-frame has a  
The rotation matrix \(R_{BP}\) is expressed by:  
\[R_{BP}=\left[ \begin{array}{ccc}q_{11}&q_{12}&q_{13}\\ q_{21}&q_{22}&q_{23}\\ q_{31}&q_{32}&q_{33}\end{array} \right] \tag{4}\]  
where:  
\[ \begin{array}{l}q_{11}=\cos\gamma\cdot\cos\beta\\ q_{12}=\cos\gamma\cdot\sin\beta\cdot\sin\alpha-\sin\gamma\cdot\cos\alpha\\ q_{13}=\cos\gamma\cdot\sin\beta\cdot\cos\alpha+\sin\gamma\cdot\sin\alpha\\ q_{21}=\sin\gamma\cdot\cos\beta\\ q_{22}=\sin\gamma\cdot\sin\beta\cdot\sin\alpha+\cos\gamma\cdot\cos\alpha\\ q_{23}=\sin\gamma\cdot\sin\beta\cdot\cos\alpha-\cos\gamma\cdot\sin\alpha\\ q_{31}=-\sin\beta\\ q_{32}=\cos\beta\cdot\sin\alpha\\ q_{33}=\cos\beta\cdot\cos\alpha\end{array}  \tag{5}\]  
\(\overset{\rightarrow}{l_{i}}\) is the direction of \(B_{i}^{\rightarrow}P_{i}\) (\(i\) = 1, 2, 3), which can be described by Equation (6) [17]:  
\[\overset{\rightarrow}{l_{i}}=[P_{i}]_{B}-B_{i}=(T_{BP}+R_{BP}\cdot P_{i})-B_{i} \tag{6}\]  
Ignore the effects of friction, the direction of \(B_{i}^{\rightarrow}P_{i}\) always passes through the centers of the balls. As a result, the magnitudes of \(\overset{\rightarrow}{l_{i}}\) are constant, which equal to the diameter of the balls, as shown in Equation (7):  
\[\overset{\rightarrow}{\|\overset{\rightarrow}{l_{i}}\|}=D \tag{7}\]  
\(F_{i}\) are the forces loading on the load cells, and the direction of \(F_{i}\) expressed in O\({}_{\text{B}}\)-X\({}_{\text{B}}\)Y\({}_{\text{B}}\)Z\({}_{\text{B}}\) can be expressed by the unit vector \(\overset{\rightarrow}{n_{Bi}}\), as shown in Equation (8):  
\[\overset{\rightarrow}{n_{Bi}}=\frac{\overset{\rightarrow}{l_{i}}}{\| \overset{\rightarrow}{l_{i}}\|}=\frac{1}{D}\cdot\overset{\rightarrow}{l_{i}} \tag{8}\]  
While the direction of \(F_{i}\) expressed in O\({}_{\text{P}}\)-X\({}_{\text{P}}\)Y\({}_{\text{P}}\)Z\({}_{\text{P}}\) can be expressed by the unit vector \(\overset{\rightarrow}{n_{Pi}}\), as shown in Equation (9):  
\[\overset{\rightarrow}{n_{Bi}}=R_{BP}\cdot\overset{\rightarrow}{n_{Pi}} \tag{9}\]  
Figure 8: The vector loop between ball and ball groove. The stability of the platform-frame must meet two conditions, namely, the force balance theorem and equilibrium of couples [18], which can be formulated in Equation (10):  
\[\left[ \begin{array}{ccc}\overrightarrow{n}_{p_{1}}&\overrightarrow{n}_{p_{2}}& \overrightarrow{n}_{p_{3}}\\ \overrightarrow{r}_{1}\times\overrightarrow{n}_{p_{1}}&\overrightarrow{r}_{2 }\times\overrightarrow{n}_{p_{2}}&\overrightarrow{r}_{3}\times \overrightarrow{n}_{p_{3}}\\ \end{array} \right]\left[ \begin{array}{c}F_{1}\\ F_{2}\\ F_{3}\\ \end{array} \right]=\left[ \begin{array}{c}F\\ M\\ \end{array}