# Transmission Angle Analysis of a Type of Parallel Manipulators
## 4 Comparison Analysis of Condition Number and ITA/CTA
### Angle Analysis of Diamond Robot  
With the inverse kinematic analysis discussed in Section 2 and the transmission angle defined in Section 3, in the \(O-xy\) coordinate system in Figure 4a, taking \(n=2\) into Equation (2) and considering the assembly mode give:  
\[\theta_{i}=2\text{arctan}\frac{-B-\sqrt{B^{2}-C^{2}+A^{2}}}{C-A} \tag{20}\]  
where \(\theta_{i}\) is the position angle of the active links, respectively, and:  
\[A=-2l_{1}(\mathbf{r}-\mathbf{a}_{i})^{T}\mathbf{e}_{1}\,C=(\mathbf{r}-\mathbf{a}_{i})^{T}(\mathbf{r}- \mathbf{a}_{i})+{l_{1}}^{2}-{l_{2}}^{2},\ \mathbf{e}_{2}=(0\quad 1)^{T}\]  
Thus, \(\mathbf{u}_{i}\) and \(\mathbf{w}_{i}\) can be obtained as:  
\[\mathbf{u}_{i}=(\cos\theta_{i}\quad\sin\theta_{i})^{T},\mathbf{w}_{i}=(\mathbf{r}-\mathbf{a}_ {i}-l_{1}\mathbf{u}_{i})/l_{2}\]  
By examining Equation (8), rewriting the Jacobian matrix in the planar \(O-xy\) coordinate system leads to:  
\[\dot{\theta}_{1}={J_{1}}^{-1}{J_{2}}\mathbf{v}=\mathbf{J}\mathbf{v} \tag{21}\]  
where \(\mathbf{J}\) is the Jacobian matrix, and \({J_{1}}\) and \({J_{2}}\) denote the direct and indirect Jacobian matrices for the parallel manipulator, respectively:  
\[\mathbf{J}_{1}=diag\big{[}{l_{1}}\mathbf{w}_{1}{}^{T}\mathbf{Q}\mathbf{u}_{1}\quad{l_{1}}\mathbf{ w}_{2}{}^{T}\mathbf{Q}\mathbf{u}_{2}\big{]},\mathbf{J}_{2}=\big{[}\mathbf{w}_{1}\quad\mathbf{w}_{2} \big{]}^{T},Q=\begin{bmatrix}0&-1\\ 1&0\end{bmatrix}\]  
By examining Equations (11) and (17), in order to obtain the ITA and CTA of the Diamond robot, planar vectors \(\mathbf{u}_{i}\) and \(\mathbf{w}_{i}\) should be rewritten in the Cartesian space coordinate system as follows:  
Figure 4: Schematic diagram of the Diamond robot and Delta robot. **(a)** Diamond robot; **(b)** Delta robot.  
\[\mathbf{uu}_{i}=\begin{bmatrix}\mathbf{u}_{i}{}^{T}&0\end{bmatrix}^{T},\mathbf{ww}_{i}= \begin{bmatrix}\mathbf{w}_{i}{}^{T}&0\end{bmatrix}^{T}\]  
Then, the ITA and CTA of the Diamond robot can be given as:  
\[\cos\alpha_{i}=\frac{\mathbf{ww}_{i}{}^{T}(\mathbf{v}_{i}\times\mathbf{uu}_{i})}{|\mathbf{v}_{ i}\times\mathbf{uu}_{i}|},\quad i=1,2 \tag{22}\]  
\[\cos\beta_{12}=|\mathbf{ww}_{1}\times\mathbf{ww}_{2}| \tag{23}\]  
Referring to [20], we give the normalized geometrical parameters of the Diamond robot: \(l_{1}=1\), \(l_{2}=2\) and \(|\mathbf{a}_{i}|=0.25\).  
In order to obtain a full insight of the effect of the angle on the kinematic performance of the Diamond robot, it is necessary to analyze the distribution algorithm of the ITA and CTA and its relationship with the condition number in space. Figure 5 shows the variations in condition number \(\kappa\), the maximum ITA (\(\alpha_{\max}=\max(\alpha_{1}\quad\alpha_{2})\)) and CTA in the work space bounded by a given condition number. It can be seen that the distribution pattern of the maximum ITA and CTA appears to be closely similar to that of \(\kappa\). Meanwhile, the maximum ITA and CTA always take the maximum value on the upper bound of \(\kappa\). Furthermore, the amplitude fluctuation in the maximum ITA is more severe than that in \(\kappa\), which is more similar to that in the CTA. It signifies that the ITA and the CTA are still different from \(\kappa\), though they are alike on a large proportion. Figure 6 shows the contours of \(\kappa\) and the maximum ITA and CTA in the workspace bounded by a given condition number. It can provides us with a more profound understanding of the similarity of these three characteristic parameters. Indeed, it can be easily noted that a better-conditioned robot must be achieved at the cost of the size of the workspace. In this study, the condition number is used to achieve a better workspace for a parallel robot. We can see that the maximum ITA and CTA have nearly the same function for determining the workspace with the condition number.