# Dynamic Parameters Identification of Serial Robot Based on Dual Quaternion
## 3 Materials and Methods
### Identification Method of Dynamic Parameters of Linkages  
Based on the rigid body dynamics model, it is necessary to establish the linkage dynamics equation from the robot base to the end linkage. Firstly, consider combining the \(i\)-th and \(n\)-th connected linkages as a whole part for the dynamic model, where the dynamic parameters of the \(i+1\)-th to \(n\)-th connected linkages are known, and the dynamic parameters of the \(i\)-th connected linkage are unknown. The dynamic equation of the connecting combination of linkages is as follows:  
\[\frac{d\hat{\mathbf{M}}_{in}^{I}}{dt}=\ \hat{\mathbf{F}}_{in}^{J}=(\mathbf{f}_{in}^{J},\mathbf{ \tau}_{in}^{I}) \tag{21}\]  
where \(\hat{\mathbf{M}}_{in}^{I}\) represents the relative dual momentum of the \(i\)-th joint coordinate system formed by the combination of the \(i\)-th and \(n\)-th connected linkages with respect to the \(i\)-th joint coordinate system. \(\hat{\mathbf{F}}_{in}^{I}\) is the dual quaternion representation of the overall dual force formed by the combination of the \(i\)-th connected linkage to the \(n\)-th connected linkage at the end with respect to the \(i\)-th joint coordinate system. \(\mathbf{f}_{in}^{J}\) is the quaternion representation of the force formed by the combination of the \(i\)-th connected linkage to the \(n\)-th connected linkage at the end with respect to the \(i\)-th joint coordinate system. \(\mathbf{\tau}_{in}^{I}\) is the quaternion representation of the moment with respect to the \(i\)-th joint coordinate system formed by the combination of the \(i\)-th connected linkage and the \(n\)-th connected linkage at the end. Considering the equation for \(\mathbf{\tau}_{in}^{I}\) as  
\[\mathbf{\tau}_{in}^{I}= \mathbf{f}_{I}^{I}+\mathbf{\tau}_{I}^{I}+\mathbf{q}_{i+1}^{I}*\mathbf{\tau}_{i+1, n}^{I}*\mathbf{q}_{i+1}^{I}*+\mathbf{p}_{i+1}^{I}\times\left(\mathbf{q}_{i+1}^{I}*\mathbf{f}_{i+1, n}^{J}*\mathbf{q}_{i+1}^{I}*\right) \tag{22}\] \[\mathbf{f}_{I}^{I}+\mathbf{\tau}_{I}^{I}=m_{I}\cdot\hat{\mathbf{v}}_{ig}^{I}+ \mathbf{p}_{ci}^{J}\times\left(\mathbf{m}_{I}\cdot\hat{\mathbf{v}}_{ci}^{J}\right)+\mathbf{q}_ {ci}^{I}*\left(\mathbf{I}_{I}\diamond\hat{\mathbf{W}}_{I}\right)*\mathbf{q}_{ci}^{I}* \tag{23}\]  
where \(\mathbf{f}_{I}^{I}\) is the representation of the quaternion of the force of the \(i\)-th connected linkage with respect to the \(i\)-th joint coordinate system, \(\mathbf{\tau}_{I}^{I}\) is the representation of the quaternion of the moment of the \(i\)-th connected linkage with respect to the \(i\)-th joint coordinate system, \(\mathbf{q}_{i+1}^{I}\) is the representation of the quaternion of the \(i+1\)-th connected linkage coordinate system with respect to the \(i\)-th connected linkage coordinate system with respect to the \(i\)-th connected linkage coordinate system. \(\mathbf{\tau}_{i+1,n}^{I}\) is the quaternion representation of the moment with respect to the \(n\)-th joint coordinate system formed by the combination of the \(i+1\)-th connected linkage and the \(n\)-th connected linkage at the end with respect to the \(n\)-th joint coordinate system.