## 3 Joint Stiffness Modeling and Identification  
The stiffness transformation that relates the joint stiffness of a robot to the stiffness characteristics in a workspace is essential to the improvement of the EE localization, and the stiffness characteristics are important for the force improvement of industrial robots. Thus, the stiffness transformation is critical to the stiffness modeling. The CCT theory is a valid and complete stiffness-mapping algorithm in the stiffness analysis of industrial robots [23], and dual quaternion algebra can also be used to formulate the compliance model with respect to the CCT theory. Before the formulation of the stiffness modeling below, it is necessary to make three assumptions. First, the EE of the robot is rigid, and the weak stiffness of the robot is the major cause of the deformations. Second, the elastic deformation of the joints causes the displacement, and the links of the robot are rigid. Third, the operation of robot task is under stable conditions. In addition, the external load is the sole factor that influences the deformation. The external force and torque applied to the EE can cause the EE to experience a small change. The infinitesimal displacement \(\delta\hat{q}_{\theta i}\) of the EE of the industrial robot represented by the infinitesimal joint angle \(\delta\theta_{i}\) is given by  
\[\delta\hat{q}_{\theta i}=\left(cos\frac{\delta\theta_{i}}{2},0,0,0,sin\frac{ \delta\theta_{i}}{2},0,0,0\right). \tag{12}\]  
The principle of virtual work is considered in the formulation, and the dual quaternion form of virtual work of the serial robot that has n revolute joints is given by  
\[Ta=\text{sc}\left(\sum_{i=0}^{n}\hat{q}_{ri}\bullet\delta\hat{q}_{\theta i} \right), \tag{13}\]  
where \(\bullet\) is the circle product [29], \(Ta\) is the work represented in joint space, \(\hat{q}_{ri}\) is the \(i\)-th dual force applied to \(i\)-th joint of an industrial robot, \"sc\" means the scalar part of the dual quaternion. In addition, the dual quaternion form of the virtual work of the EE is given by  
\[Tc=\text{sc}\left(\hat{F}\bullet\hat{D}\right), \tag{14}\]  
where \(Tc\) is the work done by the EE, \(\hat{F}\) is the dual force applied to the EE of the industrial robot, \(\hat{D}\) is the dual displacement of the EE. A dual quaternion has a real part and a dual part. In addition, both of these two parts have a scalar part and a vector part, respectively. Extract the scalar parts from \(Ta\) and \(Tc\) and we have  
\[\text{sc}(Ta_{r})+\text{sc}(Ta_{d})=\text{sc}(Tc_{r})+\text{sc}(Tc_{d}), \tag{15}\]  
where \(Ta_{r}\) and \(Ta_{d}\) are the real part and the dual part of \(Tg\), respectively, \(Tc_{r}\) and \(Tc_{d}\) are the real part and the dual part of \(Tc\), respectively. The linear algebra can be applied to the calculation of dual quaternion algebra. Thus, Equation (15) can be rewritten in the form of matrices for convenient calculation, and the equation is given by  
\[\tau_{q}{}^{T}d\theta=TF^{T}D, \tag{16}\]  
where \(\tau_{q}\) is the modified torque vector, \(d\theta\) is the joint angle vector, \(F\) is the force and torque applied to the EE, \(D\) denotes small displacement, \(T=[1\ 0\ 0\ 0\ 1\ 0\ 0\ 0]\), and all of these elements are represented in dual quaternion space by matrix form. The displacement \(D\) in Equation (16) can be represented by the Jacobian matrix and the corresponding infinitesimal elements. The relationship is given by  
\[D=Jd\theta, \tag{17}\]where \(J\) is the Jacobian matrix, and \(d\theta\) contains the variables of the EE. Substitute Equation (17) into Equation (16), and the following relationship is obtained  
\[{\tau_{q}}^{T}=TF^{T}J. \tag{18}\]  
The relationship between the torque \(\Gamma\) and joint angle \(\Delta\Theta\) is given by  
\[\Gamma=K_{\theta}\Delta\Theta, \tag{19}\]  
where \(K_{\theta}\) is the diagonal matrix which represents the joint stiffness matrix. Dual quaternion algebra can also be used to represent \(K_{\theta}\) and \(\Delta\Theta\).