# Modeling and Compensation of Motion Errors for 6-DOF Robotic Manipulators
## 5 Experimental Studies
### Compensation of Position Error under Different Loads  
In this section, the end position of the manipulator arm under two conditions with a load of 10.6 N and 18.04 N was measured. Since the positioning precision of the manipulator at a load of 0 N load was much higher than the absolute positioning accuracy, the end position of the manipulator at a load of 0 N was considered the reference. A total of 11 points were randomly selected in the Cartesian coordinate space of the manipulator in the experiment, as shown in Figure 12.  
According to the error-joint angle model (Equation (14)),  
\[\Delta s=J(q)dq=J(q)K_{q}^{-1}\tau=J(q)K_{q}^{-1}J^{T}(q)F\]  
where \(J(q)=\left[ \begin{array}{cccc}J_{1}(q)&J_{2}(q)&J_{3}(q)&J_{4}(q)&J_{5}(q)&J_ {6}(q)\end{array} \right]\). Figure 12: Selected 11-point trajectory. According to the Jacobian construction method in Section 3.1, \(J_{i}(q)\) can be obtained:  
\[J_{i}(q)=\left[ \begin{array}{c}-n_{ix}p_{iy}+n_{iy}p_{ix}\\ -o_{ix}p_{iy}+o_{iy}p_{ix}\\ -a_{ix}p_{iy}+a_{iy}p_{ix}\\ n_{iz}\\ o_{iz}\\ a_{iz}\end{array} \right]\ (i=1,2,\ldots 6) \tag{45}\]  
where the homogeneous transformation parameters of joint \(i\): \(n_{ix}\), \(n_{iz}\), \(o_{ix}\), \(o_{iz}\), \(a_{ix}\), \(a_{iz}\), \(p_{ix}\), and \(p_{iy}\) can be obtained according to Equation (46):  
\[ \begin{array}{c}{}^{0}_{i}T=\prod_{i=1}^{n}A_{i}=\left[\begin{array}{cccc} n_{ix}&o_{ix}&a_{ix}&p_{ix}\\ n_{iy}&o_{iy}&a_{iy}&p_{iy}\\ n_{iz}&o_{iz}&a_{iz}&p_{iz}\\ 0&0&0&1\end{array}\right]\ (n=1,2,\ldots 6)\end{array}  \tag{46}\]  
where \(A_{i}\) is the homogeneous transformation matrix for joint \(i\). \(K_{q}\) can be obtained from Section 4.3, and \(F\) is the six-dimensional generalized force applied at the end of the manipulator, as shown in Equation (28). The compensated angle \(\Delta q\) can be obtained as  
\[\Delta q=J^{-1}(q)\Delta s \tag{47}\]  
The first point and the last point coincide, which is not only for the verification of the repetition accuracy of the same point but also for the observation of the position error of joint stiffness superposition. Tables 2 and 3 show the compensation angles of the six joint angles under a load of 10.6 N and 18.0 N, respectively. Figure 13 shows the position errors before and after compensation, where X_error, Y_error, and Z_error represent the position errors in the X direction, Y direction, and Z direction, respectively, and X_SM, Y_SM, and Z_SM represent the position errors after compensation based on the stiffness model in the X direction, Y direction, and Z direction, respectively. As shown in Figure 13, the position error in the X and Y directions was generally less than 0.01 mm because the external force was mainly applied in the Z direction. After the joint angles were compensated, the displacement errors in these two directions were reduced to less than 0.005 mm. In the Z direction, the maximum error reached \(-0.3\) mm, which was basically reduced to less than 0.01 mm, and the minimum value was only 0.0015 mm after compensation. The compensation effect of the displacement errors in the X, Y, and Z directions reached 95.2%, 98.7%, and 96.2%, respectively. The experimental results of this paper indicate that the error compensation method by the stiffness model