## 2 Constraint Analysis at the Velocity Level
### Analytic Constraint Equation
\begin{bmatrix}-s\xi_{i}&c\xi_{i}&0\end{bmatrix} \mathbf{v}+ \begin{bmatrix}a_{iz}c \xi_{i}&a_{iz}s\xi_{i}&-a_{ix}c\xi_{i}+a_{iy}s\xi_{i}\end{bmatrix} \mathbf{\omega}=0. \tag{23}\]  
Unpacking \(\mathbf{v}\) and \(\mathbf{\omega}\) in Equation (23) to further expanding the equation yields:  
\[-v_{x}s\xi_{i}+v_{y}c\xi_{i}+\omega_{x}a_{iz}c\xi_{i}+\omega_{y}a_{iz}s\xi_{i }+\omega_{z}(-a_{ix}c\xi_{i}-a_{iy}s\xi_{i})=0. \tag{24}\]  
Then, by rearranging the parasitic terms on one side and the independent terms on the other side as shown below:  
\[-v_{x}s\xi_{i}+v_{y}c\xi_{i}+\omega_{z}(-a_{ix}c\xi_{i}-a_{iy}s\xi_{i})=- \omega_{x}a_{iz}c\xi_{i}-\omega_{y}a_{iz}s\xi_{i}. \tag{25}\]  
Finally, rewriting Equation (25) in matrix form gives:  
\[ \begin{bmatrix}-s\xi_{i}&c\xi_{i}&(-a_{ix}c\xi_{i}-a_{iy}s\xi_{i}) \end{bmatrix}\begin{bmatrix}v_{x}\\ v_{y}\\ \omega_{z}\end{bmatrix} = \begin{bmatrix}-a_{iz}c\xi_{i}&-a_{iz}s\xi_{i} \end{bmatrix}\begin{bmatrix}\omega_{x}\\ \omega_{y}\end{bmatrix} , \tag{26}\]  
\[\mathbf{C}_{1i} \begin{bmatrix}v_{x}\\ v_{y}\\ \omega_{z}\end{bmatrix} =\mathbf{C}_{2i} \begin{bmatrix}\omega_{x}\\ \omega_{y}\end{bmatrix} . \tag{27}\]Writing \(C_{1i}\) and \(C_{2i}\) for \(i=1,2,\text{ and }3\) makes them a \(3\times 3\) and \(3\times 2\) mapping matrices, respectively. From the relation in Equation (27), we can obtain the parasitic motion of the moving plate as a function of the main motions, i.e.,  
\[ \begin{bmatrix}v_{x}\\ v_{y}\\ \omega_{z}\end{bmatrix} =\mathbf{C}_{1}^{-1}\mathbf{C}_{2} \begin{bmatrix}\omega_{x}\\ \omega_{y}\end{bmatrix} , \tag{28}\]  
where,  
\[\mathbf{C}_{1}= \begin{bmatrix}-s\xi_{1}&c\xi_{1}&-(a_{1x}c\xi_{1}+a_{1y}s\xi_{1}) \\ -s\xi_{2}&c\xi_{2}&-(a_{2x}c\xi_{2}+a_{2y}s\xi_{2})\\ -s\xi_{3}&c\xi_{3}&-(a_{3x}c\xi_{3}+a_{3y}s\xi_{3})\end{bmatrix} ,\text{ and }\mathbf{C}_{2}= \begin{bmatrix}-a_{1z}c\xi_{1}&-a_{1z}s\xi_{1}\\ -a_{2z}c\xi_{2}&-a_{2z}s\xi_{2}\\ -a_{3z}c\xi_{3}&-a_{3z}s\xi_{3}\end{bmatrix} .\]  
In Equation (28), we can freely define \(\omega_{x}\) and \(\omega_{y}\) and obtain \(v_{x},v_{y},\omega_{z}\). The z-axis linear velocity is zero if the manipulator is not moving along the z-axis. Equation (28), we can see that the z-axis linear motion is not related to the parasitic motion. Numerical simulation: A numerical simulation is presented to visualize the parasitic motion-induced for satisfying the structural constraint. Because the simulation result for all manipulators has shown the same property, a 3-RPS manipulator is chosen as a representative example. So, other manipulators also have similar outcomes regardless of the amplitude of the parasitic motion. The user-defined (desired) motion is the angular velocity in rad/sec shown in Equation (29). The motion along the z-axis is set to zero. The geometric parameter shown in Table 1 is used to obtain the solution shown in Figure 4. From Figure 4, we can see that our desired motion is different from what the manipulator can execute. If the parasitic motion obtained in Figure 4 is not incorporated in the task motion, the manipulator cannot function well or will be broken in the worst case. However, the parasitic motion is undesired from users' perspective. The result in Figure 4 demonstrates that the linear motion along the z-axis has no relation with the parasitic motion as in Equation (28). The user input variables \(\omega_{x}\) and \(\omega_{y}\) are shown in Figure 4d,e. As can be seen from the result, these terms are not entirely affected by the parasitic motion since they are freely chosen. In general, the task velocity in Figure 4 is constraint compatible motion that can be used to solve the joint rate. \[ \begin{bmatrix}\omega_{x}&\omega_{y}\end{bmatrix} ^{T}= \begin{bmatrix}0.2s2\pi t &-0.2c2\pi t\end{bmatrix} ^{T}. \tag{29}\]  
Note that \(s\) and \(c\) in Equation (29) stands for sine and cosine, respectively. From the simulation, we can see the motions that are undesired for the user but necessary for the manipulator to operate safely.