## 5 Vjm
### Single Dof
#### 5.1.1 Linear Stiffness  
Linear stiffness simplifies the elastic deformation of each joint to a constant joint rotational stiffness coefficient. Therefore, \(\mathbf{K}_{\theta}\) in Equation (2) is a diagonal matrix and each joint stiffness coefficient in the \(\mathbf{K}_{\theta}\) matrix is a constant. [PERSON] et al. adopted this method to establish a stiffness model for the ABB IRB6400 robot. The measurement position was defined in the working space. After loading the motor spindle at the end of the robot, the 3D displacement and 6D wrench data were measured. In addition to these data, the stiffness of each axis joint was calculated using the least squares method, and an end-milling experiment on the cylinder head was carried out. The real-time deformation compensation based on the stiffness model was effectively reduced from the original surface dimensional accuracy of 0.9 mm to 0.3 mm [99; 100]. [PERSON] et al. [101] used two methods to obtain the Cartesian stiffness distribution of a five-joint industrial robot. The first method directly measured the stiffness of each joint. When measuring one joint, the other joints were fixed to prevent a coupling effect between all the joints. With Equation (3), the Cartesian stiffness was calculated using a rotational stiffness model. The second method was to directly measure the stiffness of the robot end. Four 800 mm \(\times\) 800 mm planes were selected in the \(Z\)-direction of the robot coordinate system (Figure 8), and 16 positions on each plane were selected to measure the X, Y, and \(Z\) stiffnesses of the robot EE. The stiffness of the entire plane was fitted through cubic interpolation. The experiment indicated that the second method estimated the stiffness better. The main reason was that the first method approximated the joint as a linear torsion spring. This stiffness estimation introduced errors due to other deformations. The second method, which directly measured the stiffness of the end, considered the deformation of the whole joint and the robot links [102]. Subsequently, this method was applied to model the stiffness of five-axis robots and PA-10 seven-axis robots, achieving good results in error compensation. However, this method requires a large quantity of experimental data, and the nonlinear difference estimation of the unmeasured position introduces certain errors [103; 104]. A further study on the traditional VJM method was conducted by [PERSON] and [PERSON]. They found that the traditional method included a nonconservative trans  
Figure 8: Experimental setup and result [102]: (**a**) Focused work space and \(Z\)-level. (**b**) Experimental direct\(-\)compliance h\({}_{\text{xx}}\) in the X-direction and h\({}_{\text{yy}}\) in the Y-direction. formation, and (3) was used on a robot without an external load. Therefore, based on the traditional method, a supplementary stiffness matrix:  
\[\mathbf{K}_{g}=\left[(\frac{\partial\mathbf{J}_{\theta}^{T}}{\partial\mathbf{\theta}_{1}}) \left(\frac{\partial\mathbf{J}_{\theta}^{T}}{\partial\mathbf{\theta}_{2}}\right)\ \dots\ \left(\frac{\partial\mathbf{J}_{\theta}^{T}}{\partial\mathbf{\theta}_{n}}\mathbf{J}\right)\right] \tag{4}\]  
Equation (4) was introduced to describe the structural change caused by the external load \(\mathbf{f}\) on the robot EE, where \(\mathbf{\theta}_{i}\) is the \(i\)th joint angle and \(\frac{\partial\mathbf{f}}{\partial\mathbf{f}}\mathbf{f}\) is an \(n\times 1\) column vector, with \(i=1,\ \dots,\ n\) and \(i\) is the number of the joint. Therefore, \(\mathbf{K}_{g}\) is an \(n\times n\) matrix. Besides, as shown in (5):  
\[\mathbf{J}_{\theta}^{T}\mathbf{K}_{c}\mathbf{\cdot}\mathbf{J}_{\theta}=\mathbf{K}_{\theta}-\mathbf{K} _{g} \tag{5}\]  
A conservative congruence transformation (CCT) was proposed between the rotational stiffness and Cartesian stiffness of the robot. Based on a digital simulation of the double-link planar manipulator, the work of the robot in the Cartesian space and the joint space was calculated.