# 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  
When the robot generated the circular contouring using the main three axes, i.e., Theta W and U axes, only the position of the end effector is controlled, and the orientation of end effector is constantly changing from the center of the circle. During the six axes operation, the robot was able to control precisely both the position and orientation of the robot end effector as shown in Figure 1. To measure the orientation and position of the robot end effector simultaneously is difficult due to the limitations of the robot working space, and transducer length. Achieving a robot end effector whose position pose is parallel to the XY plane, is limited to a maximum circle radius of 70 mm. Special equipment was made using a six points kinematic concept to measure the yaw motion of the robot end effector as shown Figure 2. Three small balls at angular spacing of 120\({}^{\circ}\) were fixed to faces of two circular disks. A linear bearing was fixed onto one of the circular disks to facilitate a sliding motion between the disks and positive contact maintained between the measurement spheres by a spring force. Two transducers, which touch on the first circular disk, are mounted at the robot end effector; the distance between the two transducers being set to 50 mm. The equipment is self-supporting by the spring force between the base-ball and the end effector as shown in Figure 2. The difference value of the two transducers was measured whilst the robot operated in a manner to maintain constant orientation in the radial direction during the circular contouring. The yaw motion angle error of the orientation was calculated from this difference value between the two transducers. Figure 1: Configuration of circular contouring using six axes of articulated robot. Figure 2: Equipment setting for the yaw motion test. Figure 3a,b show the results in both the clockwise and anticlockwise directions and the error shape displays a sine curve function. This shape results from an eccentric setting between the center of the base-ball and the center of the circular contouring. This eccentricity situation is shown in Figure 4. If the base-ball is located at the center of the circle, and there is no yaw motion error in the robot axes, the circular disk is always square to the robot end effector during the robot generated circular contouring, in the radial direction. If eccentricity between the center of the base-ball and the circle exists, then the error shape is a sine curve function [22], as shown in Figure 3. The amplitude of the sine curve is a function of the magnitude of eccentricity and the circle radius. For circle contouring with a radius R, the amplitude of sine curve error E based on the its eccentricity is [23]. \[E_{max}[arcsec]=\pm 206e[\mu m]/R[mm]\]  
where \(e\) is the eccentricity between the center of the base-ball and the circle. Depending on the 70 mm radius of the circular contouring, an eccentricity of 14.142 mm results in measuring deviation of \(\pm\)11.56 degrees as shown in Figure 4. Figure 4 shows the angle error due to eccentricity, and the resultant error is seen to be a sine curve. Therefore, the experimental sine curve wave seen in Figure 3a,b is due to eccentricity. The least squares sine curve was calculated and also plotted in Figure 3a,b. The equation of least squares sin curve is shown in the below. Assume that in the Figure 5  
\[y_{k}=A+r\sin(\theta_{k}+\alpha), \tag{1}\]  
Figure 2.0 The angle error due to eccentricity. Figure 2.1 Experimental result of yaw motion test: (**a**) clockwise direction and (**b**) anticlockwise direction. where, r: Amplitude, A: Offset of amplitude, \(\alpha\): Shift-angle, and \(\theta_{k}\): Rotation-angle. According to least squares definition  
Figure 2.2  
is minimized. To illustrate least squares error approximation, consider the important problem of fitting a sin curve. This guess function for this rather special fit will be denoted by  
\[