# Study of Orientation Error on Robot End Effector and Volumetric Error of Articulated Robot
## 2 Orientation Test of Robot End Effector
### Radial Position Accuracy and Repeatability Test on a Hemisphere  
The positional accuracy of the robot at different orientations of the robot end effector, is tested on a hemisphere of 400 mm radius as defined and shown in Figure 12. The target positions are equivalent to 30 degree intervals on the hemisphere. Each of the target positions is defined by three different orientations of robot end effector at the same position of the robot end effector. The equipment setting and target positions are shown in Figure 12. The radial distance between the target position of the end effector and the base-ball was measured using a transducer and kinematic link at the different orientations.  
The experimental results are shown in the Figure 13. The experimental result, which is red color in the figure, are the mean value of the three different orientations at the same position of the robot end effector. The least squares sphere, which is black color on the Figure 13, is calculated from the experimental mean value using the least squares approximation method. The nominal radius is 400 mm, but the radius of the least squares sphere is 400.192 mm. The error band is seen to be just greater than 1 mm. The experimental data can also be seen to be distorted to one side when compared with the least squares sphere.  
The target angle positions of each of axes were found with the inverse kinematic solution. The gear transmission error of each of the axes is added to the theoretical target angle positions of each of the axes. The distance between the center of the hemisphere and the position of the end effector was calculated for these simulation results as shown in Figure 14. The radius of the least squares  
Figure 11: Simulation result of orientation error in the roll motion.  
Figure 12: Equipment setting and defined position.  
sphere is again larger than the nominal radius, and the error band just less than 1 mm at 0.922 mm. Again, the simulation data is distorted to one side when compared with the least squares sphere. Compared with the experimental result, the error tendency of the simulation data is almost identical. This distortion is due therefore to the gear transmission error and backlash of the robot axes since these are the only correction made to the simulation data.  
To test repeatability, the end effector is approached to the target position, and the distance between the two balls was measured with a transducer and kinematic link. The end effector is then moved to a position, which is in the same radial direction, but at a 3 mm greater nominal radius, then the end effector is returned to the target position in the same radial direction. The distance between two balls is again measured. Then the end effector is moved to a position, in the radial direction, which is 3 mm smaller than the nominal radius, and again the end effector returned to the target position in the same radial direction. The distance between the two balls is again measured. This procedure is repeated three times and six measurement data are acquired from one target point. The result of this repeatability test is shown in Figure 15. The uni-direction repeatability is found to be 0.261 mm, the bi-direction repeatability shows 1.567 mm. The bi-direction repeatability is seven times larger than uni-direction repeatability due to the backlash error of the theta and beta robot axes.  
Figure 14: Simulation result of radial accuracy.  
Figure 13: Experimental result of radial accuracy.