## 2 Experimental Methods
### Motion Trajectory  
In this part, the platform was divided into the upper platform and the motor output plane. The Cartesian coordinate was set to find the interrelationship of the initial upper platform coordinate \(U\), the upper platform coordinates after movement \(Ut\), and the motor output platform coordinate \(G\) as the basis of the platform motion trajectory. The coordinate relationship was schematically shown in Figure 5.  
Both the motor output platform and the upper platform were designed as equilateral triangles: the length of the motor output platform assumes \(\ell_{C}\), the length of the upper platform assumes \(\ell_{L}\), and the distance H from the center of mass of the upper platform to the motor output platform were assumed. The upper platform coordinate after movement \(U^{\prime}\) can be converted by the Euler angle transformation matrix. It was assumed that the coordinate \(U^{\prime}\) was the coordinate \(U\) turning \(\alpha\) angle to the \(X^{U}\) axis and turning \(I2\) angle to the \(Y^{U}\) axis as shown in Equation (3). Moreover, \(Z^{U}\) was shifted by h as shown in Equation (4). Then, the relationship of coordinate \(U^{\prime}\) and coordinate \(U\) was described by Equation (5). The distance from the center of mass to the three vertices was kept in the coordinate \(U^{\prime}\). The three vectors of the coordinate \(U\) could be superimposed to obtain the vector of the coordinate \(G\) and the final movement position \(\left\{U^{\prime}_{i}\right\}\). \[[R(\alpha)]=\left[ \begin{array}{cccc}1&0&0&0\\ 0&\cos(\alpha)&-\sin(\alpha)&0\\ 0&\sin(\alpha)&\cos(\alpha)&0\\ 0&0&0&1\end{array} \right],\ [R(\beta)]=\left[ \begin{array}{cccc}\cos(\beta)&0& \sin(\beta)&0\\ 0&1&0&0\\ -\sin(\beta)&0&\cos(\beta)&0\\ 0&0&0&1\end{array} \right] \tag{3}\]  
\[[d(h)]=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&h\\ 0&0&0&1\end{array} \right] \tag{4}\]  
\[\left\{U^{\prime}_{i}\right\}=[d(h)][R(\alpha)][R(\beta)]\{U_{i}\} \tag{5}\]  
After obtaining the position \(\left\{U^{\prime}_{i}\right\}\) of the upper platform, the position of the crank connecting rod joint \(\left\{C^{\prime}_{i}\right\}\) could be further derived. Assuming the position \(\left\{U^{\prime}_{i}\right\}\), position \(\left\{C^{\prime}_{i}\right\}\), crank length \(\ell_{C}\), and the connecting rod length \(\ell_{L}\) were a known condition. The dimension of the platform is shown in Figure 6.  
Figure 5: The basic Cartesian coordinate system of the 3-degrees of freedom (DoF) platform. According to Figure 7a, the contact relationship is shown in Equations (6) and (7), and through the top view of the platform, the angle \(\theta\) between the crank and the shaft X was a fixed value as shown in Figure 7b. The geometric relationship was shown in Equation (8). Finally, the \(\{C_{i}^{\prime}\}\) coordinate position can be calculated by Equations (6)-(8) to establish the motion trajectory. \[\left[(C_{i,X}-G_{i,X})^{2}+(C_{i,Y}-G_{i,Y})^{2}+(C_{i,Z}-G_{i,Z})^{2} \right]={\ell_{L}}^{2} \tag{6}\] \[\left[(U_{i,X}-C_{i,X})^{2}+(U_{i,Y}-C_{i,Y})^{2}+(U_{i,Z}-C_{i,Z} )^{2}\right]={\ell_{c}}^{2}\] (7) \[\tan\theta_{i}=\frac{C_{i,Y}-G_{i,Y}}{C_{i,X}-G_{i,X}} \tag{8}\]  
Figure 6: The dimension of the platform. Figure 7: (**a**) Platform architecture (**b**) Top view of the platform. After obtaining the position \(\{C_{i}^{\prime}\}\), the vector of the connection point between \(\{C_{i}^{\prime}\}\) and the upper platform was regarded as the connecting rod length \(\ell_{L}\) and the motor output platform. The contact connection vector was regarded as the crank length \(\ell_{C}\) and the vector relation, as shown in Equation (9). The crank length vector \(\ell_{C}\) was projected to the motor output platform to obtain the vector \(\ell\vec{\ell_{C}}_{i}\), as shown in Figure 8. Finally, the motor output angle could be obtained by Equation (10). \[\vec{\ell_{L}}_{i}=\{U_{i}^{\prime}\}-\{C_{i}^{\prime}\},\,\vec{\ell_{c,i}}= \{U_{i}^{\prime}\}-\{C_{i}^{\prime}\} \tag{9}\]  
Figure 6.0