## 5 Vjm
### Multiple DOFs
#### 5.2.2 Six DOFs  
[PERSON] et al. considered the deformation of all components for stiffness identification. Each driving joint and connecting link were described by a 6 \(\times\) 6 stiffness matrix. Hence, 258 parameters were used for a traditional 6-axis robot. To solve this hypothetical model, a simplified model was proposed to reduce the number of identification parameters while maintaining the accuracy of the stiffness model identification [66]. In further research, [PERSON] et al. introduced the gravity \(G\) of a robot arm and an external load \(F\) into stiffness modelling (Figure 12). The relationship between the wrench and the displacement of the robot EE was linearized. Based on the static balance equation, the Cartesian stiffness \(\mathbf{K_{c}}\) was calculated. When this model was applied for deformation compensation control during the robot milling process, the static deviation was effectively improved [119]. Figure 11: Three DOFs stiffness model in [101]. Figure 12: VJM of an industrial robot with auxiliary loading [120]. In the general case, the stiffness matrix \(K_{\epsilon}\) can be expressed by (8), with Hessian \(H_{\theta\theta}\) expressed using (9). A small change in the Jacobian matrix of the robot is expressed by its second derivative \(\delta f_{\theta}^{(F)}=H_{\theta\theta}^{(F)}\cdot\delta\theta\), \(\delta f_{\theta}^{(G)}=H_{\theta\theta}^{(G)}\cdot\delta\theta\). Similarly, a slight change in the gravity load \(G\) is represented by its first derivative \(\delta G=\frac{\partial G}{\partial\theta}\delta\theta\). Among them, \(H_{\theta\theta}^{(G)}\) and \(H_{\theta\theta}^{(F)}\) are expressed in (9) and (11) [119]. \[K_{\epsilon}=\left(H_{\theta}^{(F)}\cdot(K_{\theta}-H_{\theta\theta})^{-1}J_{ \theta}^{(F)T}\right)^{-1} \tag{8}\]  
\[H_{\theta\theta}=H_{\theta\theta}^{(F)}+H_{\theta\theta}^{(G)}+J_{\theta}^{(G )T}\frac{\partial G}{\partial\theta} \tag{9}\]  
\[H_{\theta\theta}^{(G)}=\sum_{j=1}^{n}\frac{\partial^{2}S_{j}^{T}G_{j}}{ \partial\theta^{2}} \tag{10}\]  
\[H_{\theta\theta}^{(F)}=\frac{\partial^{2}S^{T}F}{\partial\theta^{2}} \tag{11}\]  
Based on this stiffness model, [PERSON] et al. further analysed the structure of a gravity compensator in heavy industrial robots. A gravity compensator was attached to joint 1 and joint 2 and formed a closed loop, which was defined as a linear spring structure and generated torque at joint 2. Combined with the stiffness of joint 2, the stiffness of the compensator was transferred to joint 2 to determine the equivalent stiffness of this joint. Thereafter, the geometric parameters of the compensator and the rotational stiffness were identified through experiments [30; 42]. Before and after loading, the deformation data of each joint were directly read by a secondary angle encoder [121], and a rotational stiffness model was established based on these data. Finally, three methods for position compensation feedback control were established, which used information from a single-angle encoder, a double-angle encoder, and a modified stiffness model. The analysis found that compared with traditional robot feedback control (angle feedback from the main angle encoder), the modified stiffness model effectively improved the position accuracy of the robot. The deformation error was reduced by more than 80%. However, the model was complex and computationally intensive. Therefore, the numerical iteration process of the static equilibrium structure took a long time to converge. For heavy industrial robots with gravity compensators, [PERSON] et al. proposed an efficient stiffness model. This model processed each joint/link as a virtual spring with six DOFs. In the model, all kinds of deformation were considered, including joint/link flexibility, link weights, and gravity compensators. First, the expressions for the reaction forces of each joint were derived from joint 6 to joint 1 for static equilibrium. Then, a stiffness model was established in conjunction with the CAD model and FEA method.