# Transmission Angle Analysis of a Type of Parallel Manipulators
## 4 Comparison Analysis of Condition Number and ITA/CTA
### Angle Analysis of Delta Robot  
In the \(O-xyz\) coordinate system shown in Figure 4b, substituting \(n=3\) into Equation (2) and considering the assembly mode give:  
\[\theta_{i}=2\text{arctan}\frac{-E_{i}-\sqrt{E_{i}^{2}-G_{i}^{2}+F_{i}^{2}}}{G_{i }-F_{i}},\,i=1,2,3 \tag{24}\]  
where \(\theta_{i}\) is the position angle of the active links, \(E_{i}=2l_{1}(\mathbf{r}-\mathbf{e}_{i})^{T}\mathbf{z},F_{i}=-2l_{1}(\mathbf{r}-\mathbf{e}_{i})^{T}\)  
\(\left(\cos\beta_{i}\mathbf{x}+\sin\beta_{i}\mathbf{y}\right)\), \(G_{i}=\left(\mathbf{r}-\mathbf{e}_{i}\right)^{T}(\mathbf{r}-\mathbf{e}_{i})+l_{1}^{2}-l_{2}^{2}\), \(\mathbf{e}_{i}=e\big{(}\cos\beta_{i}\quad\sin\beta_{i}\quad 0\big{)}^{T}\) and \(\beta_{i}=(i-1)\frac{2\pi}{6}-\frac{\pi}{6}\). Here, \(\mathbf{x},\mathbf{y}\) and \(\mathbf{z}\) are unit vectors along the \(x\)-, \(y\)- and \(z\)-axes. Thus, \(\mathbf{u}_{i}\) and \(\mathbf{w}_{i}\) can be determined by \(\mathbf{u}_{i}=(\cos\beta_{i}\cos\theta_{i}\,\sin\beta_{i}\cos\theta_{i}\,-\sin \theta_{i})^{T}\), \(\mathbf{w}_{i}=(\mathbf{r}-\mathbf{e}_{i}-l_{1}\mathbf{u}_{i})/l_{2}\). Substituting \(n=3\) into Equation (8) and rewriting the Jacobian matrix in the \(O-xyz\) coordinate system lead to:  
\[\dot{\theta}_{1}=\mathbf{J}_{1}^{-1}\mathbf{J}_{2}\mathbf{v}=\mathbf{J}\mathbf{v},\,i=1,2,3 \tag{25}\]  
\[\text{where}\mathbf{J}_{1}=diag\big{[}l_{1}\mathbf{w}_{1}^{T}(\mathbf{v}_{1}\times\mathbf{u} _{1})\quad l_{1}\mathbf{w}_{2}\mathbf{v}^{T}(\mathbf{v}_{2}\times\mathbf{u}_{2})\quad l_{1}\bm {w}_{3}\mathbf{}^{T}(\mathbf{v}_{3}\times\mathbf{u}_{3})\big{]}\text{ and}\mathbf{J}_{2}=\big{[}\mathbf{w}_{1}\quad\mathbf{w}_{2}\quad\mathbf{w}_{3} \big{]}^{T}.\]  
By examining Equations (11) and (18), the ITA and CTA of the Delta robot can be given as:  
\[\cos\alpha_{i}=\frac{\mathbf{w}_{i}^{T}(\mathbf{v}_{i}\times\mathbf{u}_{i})}{|\mathbf{v}_{i} \times\mathbf{u}_{i}|},\,i=1,2,3 \tag{26}\]\[\cos\beta_{23}=\frac{\mathbf{w_{1}}^{T}(\mathbf{w}_{2}\times\mathbf{w}_{3})}{|\mathbf{w}_{2} \times\mathbf{w}_{3}|},\cos\beta_{31}=\frac{\mathbf{w_{2}}^{T}(\mathbf{w}_{3}\times\mathbf{w}_{ 1})}{|\mathbf{w}_{3}\times\mathbf{w}_{1}|},\cos\beta_{12}=\frac{\mathbf{w_{3}}^{T}(\mathbf{w}_{ 1}\times\mathbf{w}_{2})}{|\mathbf{w}_{1}\times\mathbf{w}_{2}|} \tag{27}\]  
The normalized geometrical parameters of the Delta robot are \(l_{1}=1\), \(e=0.4\), \(H=1.9\) and \(h=0.7\). Since the workspace of the Delta robot is a space volume rather than a plane, it is better to investigate the distribution algorithm of \(\kappa\), the maximum ITA (\(\alpha_{\max}=\max(\alpha_{1}\quad\alpha_{2}\quad\alpha_{3})\)) and the maximum CTA (\(\beta_{\max}=\max(\beta_{12}\quad\beta_{23}\quad\beta_{31})\)) with a given \(H\), by which the moving platform is restrained to have motion only on the horizontal plane. As shown in Figure 7, the variation in \(\kappa\) is extremely similar to that in \(\beta_{\max}\), no matter what value \(H\) may take, which is similar to the relationship analyzed in the Diamond robot. However, it can be seen from Figure 7b that the variation in \(\alpha_{\max}\) changes violently with the decrease in \(H\). Furthermore, it can be concluded that a bad ITA may still be taken even if the condition number is fine throughout the workspace. In other words, the condition number cannot thoroughly depict the distance from the singularity of parallel robots. Therefore, it may not be very suitable for merely taking the condition number or its modified versions as a performance index of parallel robots for performance evaluation or optimal design. Then, it is necessary to introduce the ITA and CTA into the optimal design of parallel robots. Figures 8 and 9 exactly justify the above analysis and prove that all of these characteristic parameters nearly play the same role in determining a better-conditioned workspace. Figure 8: Contours of \(\kappa\) and of the maximum ITA and CTA in bounded workspace.