## 5 Vjm
### Single Dof
#### 5.1.1 Linear Stiffness
The results indicated that the CCT method was closer to the actual situation than the traditional method [70; 105; 106]. [PERSON] and [PERSON] used experimental methods to calculate the axis joint stiffness and Cartesian stiffness of a six-axis continuous articulated robot, further verifying the effectiveness of the CCT method [107]. Based on the CCT method, [PERSON] et al. identified the rotational stiffness \(\mathbf{K}_{\theta}\) of the KUKA KR240-2 robot. The effect of the supplementary stiffness matrix \(\mathbf{K}_{g}\) on \(\mathbf{K}_{c}\) was analysed. It was found that the larger the torque applied to the robot EE, the greater the effect of \(\mathbf{K}_{g}\) on \(\mathbf{K}_{c}\). Therefore, when EE is subjected to a large force and torque, the stiffness of the robot will have a large deviation from the actual situation. At a specific location, however, \(\mathbf{K}_{g}\) could be ignored relative to \(\mathbf{K}_{\theta}\). For this reason, the conditional number \(K_{\text{F}}(M)\) based on the Frobenius norm (6):  
\[\mathbf{K}_{\text{F}}(M)=\frac{\mathbf{1}}{m}\sqrt{\mathbf{tr}\big{(}\mathbf{M}^{T}\mathbf{M}\big{)} \mathbf{tr}\big{[}\big{(}\mathbf{M}^{T}\mathbf{M}\big{)}^{-1}\big{]}} \tag{6}\]  
was defined to identify and determine the measuring position where the effect of \(\mathbf{K}_{g}\) on \(\mathbf{K}_{c}\) could be ignored [72; 108]. Therefore, Equation (7) let \(\mathbf{J}_{N}\) become a standard Jacobian matrix:  
\[\mathbf{J}_{N}=\left[ \begin{array}{cc}\mathbf{\frac{1}{L}}\mathbf{I}_{3\times 3}&\mathbf{O}_{3 \times 3}\\ \mathbf{O}_{3\times 3}&\mathbf{I}_{3\times 3}\end{array} \right]\mathbf{J}_{\theta} \tag{7}\]  
\(\mathbf{I}_{3\times 3}\) is a \(3\times 3\) identity matrix and \(\mathbf{O}_{3\times 3}\) is a \(3\times 3\) zero matrix. Moreover, \(\mathbf{L}\) is the characteristic length, obtained by means of the methodology proposed in [16], and it normalizes the units in the Jacobian matrix. Then, the inverse condition number of \(\mathbf{J}_{N}\) is illustrated in Figure 9 [72]. The lower \(\mathbf{K}_{\text{F}}(\mathbf{J}_{N})^{-1}\) is, the closer the robot is to singularities. As the robot pose is close to the singularity, the determinant of \(\mathbf{J}_{\theta}\) approaches zero, which leads to abnormally high Cartesian stiffness \(\mathbf{K}_{c}\). Therefore, the VJM method is not suitable for the estimation of the Cartesian stiffness near the singular position [109]. By applying the CCT method and using Equation (4), [PERSON] et al. identified the rotational stiffness of a robot in the entire workspace. The results of the study indicated that the effect of \(\mathbf{K}_{g}\) on \(\mathbf{K}_{c}\) was greater than 10% in a small part of the workspace, whereas its effect in the rest of the workspace was less than 5%. This result was consistent with [PERSON]'s conclusion [98]. [PERSON] et al. also used the CCT method to establish a stiffness model of the ABB IRB 6640 robot and applied the model to milling processing. Given the characteristics of milling, only five DOFs were necessary to satisfy the positioning requirement. Hence, the DOF of rotation about the tool axis became functionally redundant. Due to the redundant DOF, the robot's posture can be changed without changing its position. In the milling experiment, the 2D plane of the working space was evenly divided into positions 1 cm apart. While keeping the direction of the tool axis unchanged, the tool axis was planned in 0.01 rad increments in the interval \((0,2\pi)\). Based on the stiffness model, the stiffness of each robot posture was identified. According to a comparative analysis, the change rate of the robot Cartesian stiffness at each position in the working plane was between 58% and 161% [19]. For a six-DOF industrial robot, in general, the first three joints determine the robot's position in joint space, while the last three joints determine the robot's posture. [PERSON] et al. defined the first three joints as arm joints and the last three joints as wrist joints. In their research, two bottlenecks were observed.