## 2 Forward Kinematics and Jacobian Matrix  
This section focuses on the geometric background of dual quaternion algebra and some relevant properties. Readers can find more detail on these concepts in the literature [24; 25; 26; 27; 28; 29; 30; 31]. The dual quaternion algebra offers a concise and efficient mathematical way to represent rotation and translation of the rigid body simultaneously. The motion of the EE of industrial robots can be expressed by dual quaternions as well. [PERSON] [24] compared dual quaternion algebra with the homogeneous transformation method (HTM) in terms of representation, storage, and computational efficiency. The author found that dual quaternions have an elegant representation and requires less memory locations. Moreover, the dual quaternion needs less addition, subtraction, multiplication, and division operations as the number of degree of freedom (DOF) increases. The formulations of the kinematic model and stiffness modeling are efficient and compact based on dual quaternion algebra. A dual quaternion is defined as  
\[\hat{q}=(q_{0}+\varepsilon q_{0}^{0})+(q_{1}+\varepsilon q_{1}^{0})i+(q_{2}+ \varepsilon q_{2}^{0})j+(q_{3}+\varepsilon q_{3}^{0})k. \tag{1}\]  
A dual quaternion can be separated into the real part \(q_{r}\) and the dual part \(q_{d}\)  
\[q_{r}=q_{0}+q_{1}\dot{\boldsymbol{i}}+q_{2}\dot{\boldsymbol{j}}+q_{3} \boldsymbol{k}, \tag{2}\]  
\[q_{d}=q_{0}^{0}+q_{1}^{0}\dot{\boldsymbol{i}}+q_{2}^{0}\dot{\boldsymbol{j}}+q_ {3}^{0}\boldsymbol{k}. \tag{3}\]  
In this paper, a convenient notation of a dual quaternion is given below:  
\[\hat{q}=q_{r}+\varepsilon q_{d}. \tag{4}\]  
The conjugate of a dual quaternion used in this paper is given by  
\[\overline{\hat{q}}^{*}=q_{r}^{*}-\varepsilon q_{d}^{*}. \tag{5}\]  
Define \"\(*\)\" as the multiplication operator of two dual quaternions. Then we get  
\[\hat{q}_{1}*\hat{q}_{2}=q_{r1}\circ q_{r2}+\varepsilon(q_{r1}\circ q_{d2}+q_{ d1}\circ q_{r2}) \tag{6}\]  
where \(q_{r1}\) and \(q_{r2}\) are the real parts of \(\hat{q}_{1}\) and \(\hat{q}_{2}\) respectively, \(q_{d1}\) and \(q_{d2}\) are the dual parts of \(\hat{q}_{1}\) and \(\hat{q}_{2}\), respectively. The operator \"\(\circ\)\" is an arithmetic product of two quaternions, and this product is given below:  
\[q_{1}\circ q_{2}=s_{1}s_{2}-\boldsymbol{v_{1}}\boldsymbol{\cdot}\boldsymbol{ v_{2}}+s_{1}\boldsymbol{v_{2}}+s_{2}\boldsymbol{v_{1}}+\boldsymbol{v_{1}} \boldsymbol{\times}\boldsymbol{v_{2}}, \tag{7}\]  
where \(s_{1}\) and \(s_{2}\) are scalar parts of \(q_{1}\) and \(q_{2}\) respectively, \(\boldsymbol{v_{1}}\) and \(\boldsymbol{v_{2}}\) are vector parts of \(q_{1}\)quaternion algebra is different from the Denavit-Hartenberg (DH) model, the DH notation can also be used by the dual quaternion algebra [24]. Considering the single link's parameters based on the DH notation, the forward kinematic model of an n-link serial robot with respect to the fixed base coordinate frame can be calculated as  
\[\hat{q}_{\text{fn}}=\hat{q}_{0,n}*\hat{q}_{i}*\overline{\hat{q}}_{0,n}^{*}, \tag{8}\]  
where \(\hat{q}_{0,n}=\hat{q}_{0,1}*\cdots*\hat{q}_{0,n}\). Equation (8) can formulate the forward kinematic equation of an 8-DOF serial industrial robot shown in Figure 1, which depicts one arm of the dual arm robot used next in the experiment. The relationship between joint velocity and the velocity of the EE in Cartesian space is represented by the Jacobian matrix. This well-known relationship is  
\[V=J(\theta)d\theta, \tag{9}\]  
where \(V\) is the velocity of the EE, and \(J(\theta)\) is the Jacobian matrix. In the dual quaternion space, the Jacobian matrix just relates the joint velocity vector \(d\theta\) and the time derivative of the dual quaternion. The relationship used in this paper is given below:  
\[d\hat{q}=J(\theta)d\theta, \tag{10}\]  
where \(\hat{q}\) is the dual quaternion calculated by Equation (8), which represents the position and orientation of the EE. Then, the Jacobian matrix in the dual quaternion space is given by  
\[J(\theta)=\frac{d\hat{q}}{d\theta}=\left[