# Dynamic Parameters Identification of Serial Robot Based on Dual Quaternion
## 3 Materials and Methods
### Identification Method of Dynamic Parameters of Load for a Serial Robot
The equation for the force and torque exerted by the load on the six-dimensional force sensor during the motion of the robotic arm is  
\[\hat{\mathbf{F}}^{S}=\left(\hat{\mathbf{f}}^{S},\mathbf{\tau}^{S}\right)=\hat{\mathbf{q}}_{ load}^{S}*\hat{\mathbf{F}}_{load}*\hat{\mathbf{q}}_{load}^{S} \tag{9}\]  
\[\hat{\mathbf{f}}^{S}\in\mathbf{H}^{v} \tag{10}\]  
\[\mathbf{\tau}^{S}\in\mathbf{H}^{v} \tag{11}\]  
Figure 1: Dual-arm robot SDASF. (**a**). The dual-arm robot SDASF. (**b**). Schematic of dual-arm robot. where \(\hat{\mathbf{F}}^{S}\) is the relative force of the load to the sensor coordinate system, \(\hat{\mathbf{q}}_{load/s}\) is the dual quaternion representation of the load pose with respect to the sensor coordinate system, \(\hat{\mathbf{F}}_{load}\) is the dual quaternion representation of the relative inertia coordinate system of the force of load, \(\hat{\mathbf{f}}\) is the quaternion of the load force with respect to the sensor coordinate system, \(\mathbf{\tau}^{S}\) is the quaternion representation of the load torque with respect to the sensor coordinate system. \[\frac{d\hat{\mathbf{M}}_{I}^{B}}{dt}+\hat{\mathbf{V}}_{B/I}^{B}\times\hat{\mathbf{M}}_{I}^ {B}=\ \hat{\mathbf{F}}^{B} \tag{1}\]0  
where \(\hat{\mathbf{F}}_{load}\) is the dual quaternion representation of the relative inertia coordinate system between the force and torque generated by the load, \(m_{load}\) is the load mass, \(\hat{\mathbf{v}}_{load}^{S}\) is the quaternion of the load acceleration with respect to the sensor coordinate system. \[\frac{d\hat{\mathbf{M}}_{I}^{B}}{dt}+\hat{\mathbf{V}}_{B/I}^{B}\times\hat{\mathbf{M}}_{I}^ {B}=\ \hat{\mathbf{F}}^{B} \tag{1}\]1  
where \(\hat{\mathbf{v}}_{gravity}^{S}\) is the gravitational acceleration of the load, \(\hat{\mathbf{v}}_{se}\) is the sensor speed, \(\mathbf{w}_{load}^{S}\) is the angular velocity of the load, \(\hat{\mathbf{w}}_{load}^{S}\) is the angular acceleration of the load, \(\mathbf{p}_{load}^{s}\) is the load position. The parameters in Equation (13) are expressed in quaternion form in the sensor coordinate system. At this point, it can be concluded that  
\[\hat{\mathbf{f}}^{S}= \begin{bmatrix}\mathbf{V}_{gt}\mathbf{W}_{l}\end{bmatrix}\begin{bmatrix} m_{load}\\ m_{load}\cdot\mathbf{p}_{load}^{s}\end{bmatrix}  \tag{14}\]  
\[\mathbf{V}_{gt}= \begin{bmatrix}\hat{\mathbf{v}}_{gravity}^{S}+\hat{\mathbf{v}}_{se}\end{bmatrix}  \tag{15}\]  
\[\mathbf{W}_{l}= \begin{bmatrix}\begin{bmatrix}\hat{\mathbf{w}}_{load}^{S}\times\end{bmatrix}+ \begin{bmatrix}\mathbf{w}_{load}^{S}\times\end{bmatrix}\begin{bmatrix}\mathbf{w}_{load}^{ S}\times\end{bmatrix}\end{bmatrix}  \tag{16}\]  
The symbol \([\times]\) represents an antisymmetric matrix. \[\mathbf{\tau}^{S}=\mathbf{q}_{load}^{S}* \begin{pmatrix}\mathbf{I}_{load}\circ\hat{\mathbf{W}}_ {load}\end{pmatrix} *\mathbf{q}_{load}^{S}{}^{*}+\mathbf{p}_{load}^{s}\times \begin{pmatrix} m_{load}\cdot\hat{\mathbf{v}}_{load}^{S}\end{pmatrix}  \tag{17}\]  
where \(\mathbf{I}_{load}\) is the load inertia matrix, \(\hat{\mathbf{W}}_{load}\) is the acceleration matrix of the load. Furthermore, there is the equation  
\[\mathbf{q}_{load}^{S}* \begin{pmatrix}\mathbf{I}_{load}\circ\hat{\mathbf{W}}_{load}\end{pmatrix}  *\mathbf{q}_{load}^{S}{}^{*}= \begin{pmatrix}\mathbf{q}_{load}^{S}{}^{*}\end{pmatrix} ^ {-} \begin{pmatrix}\mathbf{q}_{load}^{S}\end{pmatrix} ^{+}M \begin{pmatrix}\hat{\mathbf{W} }\end{pmatrix} M \begin{pmatrix}\hat{\mathbf{I}}\end{pmatrix}  \tag{18}\]  
where \( \begin{pmatrix}\mathbf{q}_{load}^{S}{}^{*}\end{pmatrix} ^{-}\) is the right multiplication of a quaternion matrix, \( \begin{pmatrix}\mathbf{q}_{load}^{S}\end{pmatrix} ^{+}\) represents the left multiplied quaternion matrix, \(M \begin{pmatrix}\hat{\mathbf{W}}\end{pmatrix} \) represents the angular velocity matrix, \(M \begin{pmatrix}\hat{\mathbf{I}}\end{pmatrix} \) is the inertia parameter matrix. \[M \begin{pmatrix}\mathbf{I}\end{pmatrix} = \begin{bmatrix}M\begin{pmatrix}\mathbf{I}\mathbf{A} \end{pmatrix}M\begin{pmatrix}\mathbf{I}\mathbf{B}\end{pmatrix}\end{bmatrix} ^{T} \tag{19}\]  
\[M