# Study of Orientation Error on Robot End Effector and Volumetric Error of Articulated Robot
## 2 Orientation Test of Robot End Effector
### Orientation Test of Yaw Motion in the Robot End Effector
\begin{array}{rl}E(y)&=\sum\limits_{k=1}^{m}[A+r\sin(\theta_{k}+\alpha)-y_{ k}]^{2}\\ &=[A+r\sin(\theta_{1}+\alpha)-y_{1}]^{2}+\cdots+[A+r\sin(\theta_{m}+\alpha)-y_{ m}]^{2}\end{array}  \tag{3}\]  
But  
\[ \begin{array}{rl}y_{k}&=A+r\sin(\theta_{k}+\alpha)\\ &=A+[(r\cos\alpha)\sin\theta_{k}+r\sin\alpha\cos\theta_{k}]\\ &=A+a\sin\theta_{k}+b\cos\theta_{k}\end{array}  \tag{4}\]  
Let \(r\cos\alpha=a\,r\sin\alpha=b\)  
i.e.,  
\[E(y)=\sum\limits_{k=1}^{m}[A+a\sin\theta_{k}+b\cos\theta_{k}-y_{k}]^{2}\]  
We know from calculus that this will occur when  
\[ \begin{array}{rl}\frac{\partial E(y)}{\partial\lambda}=0,\ \frac{\partial E(y)}{ \partial a}=0,\ \frac{\partial E(y)}{\partial b}=0\\ \frac{\partial E(y)}{\partial A}=2\sum\limits_{k=1}^{m}(A+a\sin\theta_{k}+b \cos\theta_{k}-y_{k})=0,\end{array}  \tag{5}\]  
\[\frac{\partial E(y)}{\partial a}=2\sum\limits_{k=1}^{m}\Bigl{[}A\sin\theta_{k }+a\sin^{2}\theta_{k}+b\cos\theta_{k}\sin\theta_{k}-y_{k}\sin\theta_{k}\Bigr{]} =0, \tag{6}\]  
\[\frac{\partial E(y)}{\partial b}=2\sum\limits_{k=1}^{m}\Bigl{[}A\cos\theta_{k }+a\sin\theta_{k}\cos\theta_{k}+b\cos^{2}\theta_{k}-y_{k}\cos\theta_{k}\Bigr{]} =0, \tag{7}\]  
From the Equations (5-7)  
Figure 5: Least square error approximation—curve fitting for sine error curve. \[\left| \begin{array}{ccc}m&\sum\sin\theta_{k}&\sum\cos\theta_{k}\\ \sum\sin\theta_{k}&\sum\sin^{2}\theta_{k}&\sum\cos\theta_{k}\sin\theta_{k}\\ \sum\cos\theta_{k}&\sum\sin\theta_{k}\cos\theta_{k}&\sum\cos^{2}\theta_{k}\\ \end{array} \right|\left| \begin{array}{c}A\\ a\\ b\end{array} \right|=\left| \begin{array}{c}\sum y_{k}\\ \sum y_{k}\sin\theta_{k}\\ \sum y_{k}\cos\theta_{k}\\ \end{array} \right|, \tag{8}\]  
\[\left| \begin{array}{c}A\\ a\\ b\end{array} \right|=\frac{1}{\det(Q)}\left| \begin{array}{cccc}b_{11}&b_{12}&b_{ 13}\\ b_{21}&b_{22}&b_{23}\\ b_{31}&b_{32}&b_{33}\\ \end{array} \right|\left| \begin{array}{c}\sum y_{k}\\ \sum y_{k}\sin\theta_{k}\\ \sum y_{k}\cos\theta_{k}\\ \end{array} \right|, \tag{9}\]  
\[ \begin{array}{c}A=\frac{1}{\det(Q)}(b_{11}\sum y_{k}+b_{12}\sum y_{k}\sin \theta_{k}+b_{13}\sum y_{k}\cos\theta_{k})\\ a=\frac{1}{\det(Q)}(b_{21}\sum y_{k}+b_{22}\sum y_{k}\sin\theta_{k}+b_{23} \sum y_{k}\cos\theta_{k})\\ b=\frac{1}{\det(Q)}(b_{31}\sum y_{k}+b_{32}\sum y_{k}\sin\theta_{k}+b_{33} \sum y_{k}\cos\theta_{k})\\ \end{array} \right., \tag{10}\]  
From the Equation (1)  
\[ \begin{array}{c}y_{k}\quad=A+a\sin\theta+b\cos\theta\\ \quad=A+r\sin(\theta+\alpha)\\ \end{array} , \tag{11}\]  
where  
\[ \begin{array}{c}r=\sqrt{a^{2}+b^{2}}\ \alpha=\tan^{-1}\!\left(\frac{b}{a} \right)\\ r=\frac{1}{\det(Q)}\sqrt{\left(b_{21}\!\sum y_{k}+b_{22}\!\sum y_{k}\sin \theta_{k}+b_{23}\!\sum y_{k}\cos\theta_{k}\right)^{2}+\left(b_{31}\!\sum y_{ k}+b_{32}\!\sum y_{k}\sin\theta_{k}+b_{33}\!\sum y_{k}\cos\theta_{k}\right)^{2}}\\ \alpha=\tan^{-1}\!\left(\!\frac{b_{31}\sum y_{k}+b_{32}\!\sum y_{k}\sin\theta_ {k}+b_{33}\!\sum y_{k}\cos\theta_{k}}{b_{11}\sum y_{k}+b_{22}\sum y_{k}\sin \theta_{k}+b_{23}\!\sum y_{k}\cos\theta_{k}}\!\right)\\ \end{array} \]  
The deviations between the real data and least squares sine curve are real orientation errors of yaw motion in the robot end effector, as shown Figure 6a,b. The error band is about \(0.53^{\circ}\) in the clockwise direction and \(0.85^{\circ}\) anticlockwise. The error shape is slightly different due to the backlash error of the robot axes. The fluctuation of the error is due to the gear transmission error of the robot axes. The angle change of the robot axes is found with the inverse kinematic solution for the robot generated circular contouring. Figure 7 shows this angle change during the circular contouring. All the robot axes, except the gamma axis, show inflexion points during the circle contouring. The backlash points occur at the direction change of the angle in each axes. The rectangular mark shows the direction change point in Figure 7. If backlash exists in the robot axes, then angle error will suddenly be apparent during the robot operation.