## 3 Kinematics Analysis
### Inverse Kinematics  
The homogeneous transform \({}^{0}T_{5}\) of the coordinate system \(\left(xyz\right)_{5}\) relative to the global coordinate system \(\left(XYZ\right)_{O}\) is as follows:  
\[{}^{0}T_{5}=\prod\limits_{i=1}^{i=5}{{i}-1}T_{i} \tag{6}\]  
\[{}^{0}T_{5}=\left[\begin{array}{cccc}1&0&0&x\\ 0&1&0&y\\ 0&0&1&z\\ 0&0&0&1\end{array}\right] \tag{7}\]  
where: (\(x\), \(y\), \(z\)) are the coordinates of point \(P\), the origin of the end-effector coordinate system \(\left(xyz\right)_{5}\) in the global coordinate system.  
Equations (6) and (7) can be modified to obtain Equation (8):  
\[Rot(y,\alpha_{i})^{(-1)}\ {}^{0}T_{5}Rot(y,-\alpha_{i})^{(-1)}\]  
\[\left\{\begin{array}{l}\theta_{i3}=\arcsin(-t_{z}/c)\\ a\cos\theta_{i1}+b_{0}\cos\theta_{i2}=k_{0}\ \ \ \ \ (i=1,\ 2,\ 3)\\ a\sin\theta_{i1}+b_{0}\sin\theta_{i2}=-t_{y}\end{array}\right. \tag{8}\]  
where:  
\[t_{x}=\cos\alpha_{i}x-\sin\alpha_{i}x,t_{y}=y;\]  
\[t_{z}=\cos\alpha_{i}z+\sin\alpha_{i}x;\]  
\[b_{0}=c\cos\theta_{i3};\]  
\[k_{0}=t_{x}-r_{1}+r_{2};\]  
\(Rot(y,\alpha_{i})^{-1}\) is the inverse matrix of \(Rot(y,\alpha_{i})\), and the others follow the same rule.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Equivalent Link** & **Local Coordinate** & **Variable \(\theta_{1}\)** & **Transformation Matrix** & **Variation Range** \\ \hline
1 & \(\left(\mathrm{xyz}\right)_{1}\) & \(\theta_{11}\) & \({}^{0}T_{1}=\mathrm{Rot}(y,\alpha_{i})\mathrm{Trans}(x,r_{1})\mathrm{Rot}(z, -\theta_{11})\) & \(20^{\circ}\)–\(160^{\circ}\) \\
2 & \(\left(\mathrm{xyz}\right)_{2}\) & \(\theta_{2}\) & \({}^{1}T_{2}=\mathrm{Trans}(x,a)\mathrm{Rot}(z,\ \theta_{i_{1}})\mathrm{Rot}(z, -\theta_{2})\) & \(20^{\circ}\)–\(160^{\circ}\) \\
3 & \(\left(\mathrm{xyz}\right)_{3}\) & \(\theta_{i3}\) & \({}^{2}T_{3}=\mathrm{Rot}(y,\theta_{i3})\) & \(\pm 40^{\circ}\) \\
4 & \(\left(\mathrm{xyz}\right)_{4}\) & - & \({}^{3}T_{4}=\mathrm{Trans}(x,c)\mathrm{Rot}(y,\ -\theta_{i3})\mathrm{Rot}(z, \theta_{i2})\) & - \\
5 & \(\left(\mathrm{xyz}\right)_{5}\) & - & \({}^{4}T_{5}=\mathrm{Trans}(x,\ -r_{2})\mathrm{Rot}(y,\ -\alpha_{i})\) & - \\ \hline \end{tabular} Note: \(\mathrm{Trans}(x,r_{1})\) represents the homogeneous transform matrix for translating \(r_{1}\) units vector along the \(x\)-axis of the local coordinate system; \(\mathrm{Rot}(y,\alpha_{i})\) represents the homogeneous transformation matrix of rotating the \(a_{i}\) angle about the \(y\)-axis of the local coordinate system; the others follow the same rule. Three identical branches correspond to \(i=1\), \(2\), and \(3\), respectively.  
\end{table}
Table 1: Equivalent link parameters of improved Delta mechanism.  
Figure 3: Kinematic sketch of each branch.  
According to Equation (8), \(\theta_{i1}\)_and_\(\theta_{i2}\) can be expressed as:  
\[\theta_{i1} =2\arctan\left(\frac{-t_{i1}\pm\sqrt{t_{i1}{}^{2}-4t_{i2}t_{i0}}}{2t _{i2}}\right) \tag{9}\] \[\theta_{i2} =2\arctan\left(\frac{-n_{i1}\pm\sqrt{n_{i1}{}^{2}-4n_{i2}n_{i0}}}{2 n_{i2}}\right) \tag{10}\]  
where:  
\[t_{i2}=b_{0}^{2}-k_{0}^{2}-a^{2}-t_{y}^{2}-2 ak_{0}\]  
\[t_{i1}=-4 aty_{i}t_{i0}=b_{0}^{2}-k_{0}^{2}-a^{2}-t_{y}^{2}+2 ak_{0}\]  
\[n_{i2}=b_{0}^{2}+k_{0}^{2}-a^{2}+t_{y}^{2}+2b_{0}k_{0}\text{, }n_{i1}=4b_{0}t_{y}\]  
\[n_{i0}=b_{0}^{2}+k_{0}^{2}-a^{2}+t_{y}^{2}-2b_{0}k_{0}\]  
The values of \(\theta_{i1}\) is determined by Equation (9) and the inverse kinematics is completed. Moreover, \(\theta_{i2}\) and \(\theta_{i3}\) are obtained from Equations (10) and (8). Typically, there are two sets of solutions corresponding to \(\theta_{i1}\) and \(\theta_{i2}\). According to the rule of \"continuous variation\" of joint rotation angle, when the initial position and posture of the mechanism are given, the unique function expressions of \(\theta_{i1}\) and \(\theta_{i2}\) are easily determined from Equations (9) and (10).