## 2 The Model of the Planar 3-RRR Parallel Robot
### The Kinematic Model of the 3-RRR Planar Parallel Robot  
According to the geometric relationships shown in Figure 2, the coordinates of the motor in the static coordinate system are shown as follows,  
\[A=\left[ \begin{array}{ccc}A_{1}&A_{2}&A_{3}\end{array} \right]=\left[ \begin{array} []{ccc}0&\sqrt{3}R&\frac{\sqrt{3}}{2}R\\ 0&0&\frac{3}{2}R\end{array} \right] \tag{1}\]  
The coordinates of the center of the moving platform in the moving coordinate system are shown as follows,  
\[P=\left[ \begin{array}{ccc}C_{1}&C_{2}&C_{3}\end{array} \right]=\left[ \begin{array} []{ccc}-\frac{\sqrt{3}}{2}r&\frac{\sqrt{3}}{2}r&0\\ -\frac{1}{2}r&-\frac{1}{2}r&r\end{array} \right] \tag{2}\]  
The spatial transformation matrix \(T\) for transforming the moving coordinate system into the static coordinate system is shown as follows,  
\[T=\left[ \begin{array}{ccc}\cos\beta\cos\gamma&\sin\beta\cos\gamma\sin\alpha- \sin\gamma\cos\alpha&\sin\beta\cos\gamma\cos\alpha+\sin\gamma\sin\alpha&x\\ \sin\gamma\cos\beta&\sin\gamma\sin\beta\sin\alpha+\cos\gamma\cos\alpha&\sin \gamma\sin\beta\cos\alpha-\cos\gamma\sin\alpha&y\\ -\sin\beta&\cos\beta\sin\alpha&\cos\beta\cos\alpha&\rm z\\ 0&0&0&1\end{array} \right] \tag{3}\]  
where \(\alpha,\beta,\gamma\) are the rotation angles around \(X\), \(Y\) and \(Z\), respectively. \(x\), \(y\), \(z\) are the distances along the \(X\), \(Y\) and \(Z\) axes, respectively. Due to the parallel robot studied in this paper, they can only be translated along the \(x\), \(y\) axis and rotated around the \(z\) axis,  
\[\alpha~{}=~{}0,\beta~{}=~{}0,z~{}=~{}0 \tag{4}\]  
\(T\) can be expressed as follows,  
\[T=\left[ \begin{array}{ccc}\cos\gamma&-\sin\gamma&0&x\\ \sin\gamma&\cos\gamma&0&y\\ 0&0&1&0\\ 0&0&0&1\end{array} \right] \tag{5}\]  
The coordinates of the moving platform center in the static coordinate system can be expressed as follows,  
\[\left[ \begin{array}{c}C\\ 1\end{array} \right]=T\times\left[ \begin{array}{c}P\\ 1\end{array} \right] \tag{6}\]  
Combining Equations (2) and (5), \(C\) can be expressed as follows,  
\[C=\left[ \begin{array}{cc}C_{1}&C_{2}&C_{3}\end{array} \right]=\left[ \begin{array} []{cc}x-r\cos\left(\frac{\pi}{6}-\gamma\right)&x+r\cos\left(\gamma-\frac{\pi} {6}\right)&x-r\sin\gamma\\ y-r\sin\left(\frac{\pi}{6}+\gamma\right)&y+r\sin\left(\gamma-\frac{\pi}{6} \right)&y+r\cos\gamma\end{array} \right] \tag{7}\]  
For any one branch \(A_{i}B_{i}C_{i}\), the parallel robot satisfies the following motion constraint of the closed-loop vector. The vector equation is expressed as follows [8],  
\[\vec{\vec{OC}_{i}}=\vec{OA}_{i}+\vec{A_{i}B_{i}}+\vec{B_{i}C_{i}} \tag{8}\]  
where \(i=1\), \(2\), \(3\). Combined with Equation (8), \(\theta_{Bi}\) is the angle between the link \(B_{i}C_{i}\) and the positive direction of the X axis. \[\left\{ \begin{array}{l}x_{Ci}=x_{Ai}+l_{1}\cos\theta_{Ai}+l_{2}\cos\theta_{Bi}\\ y_{Ci}=y_{Ai}+l_{1}\sin\theta_{Ai}+l_{2}\sin\theta_{Bi}\end{array} \right. \tag{9}\]  
Equation (9) can be expressed as follows,  
\[f_{i1}\sin\theta_{Ai}+f_{i2}\cos\theta_{Ai}+f_{i3}=0 \tag{10}\]  
where  
\[f_{i1}=2l_{1}(y_{Ai}-y_{Ci})\]  
\[f_{i2}=2l_{1}(x_{Ai}-x_{Ci})\]  
\[f_{i3}=x_{Ci}^{2}+y_{Ci}^{2}+x_{Ai}^{2}+y_{Ai}^{2}-2x_{Ci}x_{Ai}-2y_{Ci}y_{Ai} +l_{1}^{2}-l_{2}^{2}\]  
Let \(t_{i}=\tan(\theta_{i}/2)\), substituting Equation (10), according to the universal formula of the trigonometric function, Equation (10) can be expressed as follows,  
\[(f_{i3}-f_{i2})t_{i}^{2}+2f_{i1}t_{i}+(f_{i3}+f_{i2})=0 \tag{11}\]  
According to the rooting formula, the roots can be expressed as follows,  
\[\left\{ \begin{array}{l}\theta_{Ai}^{1}=2\text{arctan}\bigg{(}\frac{-f_{i1} +\sqrt{f_{i1}^{2}+f_{i2}^{2}-f_{i2}^{2}}}{f_{i3}-f_{i2}}\bigg{)}\\ \theta_{Ai}^{2}=2\text{arctan}\bigg{(}\frac{-f_{i1}-\sqrt{f_{i1}^{2}+f_{i2}^{ 2}-f_{i2}^{2}}}{f_{i3}-f_{i2}}\bigg{)}\end{array} \right. \tag{12}\]  
Similarly, \(\theta_{Bi}\) can be obtained. In order to build the genetic algorithm controller, it is necessary to obtain the transformation matrix of the motor physical model.