# 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 purpose of this study is to develop and validate a method for identifying the dynamic parameters of the load and linkages of a serial robot. Dual quaternion algebra is used as the mathematical framework to represent the relationship between force, torque, velocity, and acceleration in a unified way, which has practical applicability to the multi-functional robot platform. In publications [25; 26], the dynamic equation can be established by combining the load with the end-effector of the serial robot, and the real-time force and torque information under the load can be obtained using a six-dimensional force sensor to improve the quality of motion control. This part has studied the corresponding method by using the dual-arm robot SDA5F, which has two serial arms, as shown in Figure 1. Figure 1a,b shows the schematic of the dual-arm robot SDA5F based on dual quaternion, and the dynamic model can formulate the coordinates of each 7-DOF next. SDA5F is made by Motoman in Japan, and it has 15 axes, the maximum load is 5 kg, the workspace is 845 mm to 1118 mm, and the repeatability is 0.06 mm. It is used for assembly, packaging, handling, machine loading, transfer, and press automation. Figure 1b shows the schematic of the dual-arm robot based on a dual quaternion. The coordinate system [R1f] is set on the end-effector of the left arm, the coordinate system [R2r] is set on the end-effector of the right arm, and the base coordinate system [Base] is set on the basement of the dual-arm robot. According to [PERSON]. [PERSON] [9], in order to introduce an identification method of dynamic parameters of end load for a serial robot, first calculate the rigid body dynamics equation below:  
\[\frac{d\hat{\mathbf{M}}_{I}^{B}}{dt}+\hat{\mathbf{V}}_{B/I}^{B}\times\hat{\mathbf{M}}_{I}^ {B}=\ \hat{\mathbf{F}}^{B} \tag{1}\]  
\[\hat{\mathbf{M}}_{I}  
\[\mathbf{\tau}^{B} =\left(\mathbf{0},\overline{\tau^{B}}\right)\in\mathbf{H}^{v} \tag{5}\] \[\overline{f^{B}} =\left(f_{x}^{B},f_{y}^{B},f_{z}^{B}\right)\] (6) \[\overline{\tau^{B}} =\left(\tau_{x}^{B},\tau_{y}^{B},\tau_{z}^{B}\right) \tag{7}\]  
where \(\varepsilon\) is the Clifford operator, \(\varepsilon^{2}=0,\ \varepsilon\
eq 0,\hat{\mathbf{M}}_{I}^{B}\) is the representation of the dual momentum with respect to the inertia coordinate system in a rigid body coordinate system, \(\hat{\mathbf{V}}_{I}^{B}\) is the representation of the dual quaternion form of velocity with respect to the inertial coordinate system in a rigid body coordinate system, \(\hat{\mathbf{F}}^{B}\) is the dual force in a rigid coordinate system, \(\mathbf{w}_{I}^{B}\) is the representation of the quaternion of the angular velocity with respect to the inertial coordinate system in a rigid body coordinate system, \(\mathbf{v}_{I}^{B}\) is the representation of the quaternion of the linear velocity with respect to the inertial coordinate system in a rigid body coordinate system, \(\mathbf{I}^{B}\) is the inertia matrix in the rigid body coordinate system, \(\mathbf{H}^{v}\) is the set of vector quaternions, \(\overline{f^{B}}\) is the external force vector in the rigid body coordinate system, \(\overline{\tau^{B}}\) is the external torque vector in the rigid body coordinate system. Equation (1) is expressed in a rigid coordinate system and requires the use of its transformation form in the inertial coordinate system, and the equation is  
\[\frac{d\hat{\mathbf{M}}^{I}}{dt}=\hat{\mathbf{F}}^{I} \tag{8}\]  
where \(\hat{\mathbf{M}}^{I}\) is the dual momentum in the inertial coordinate system, \(\hat{\mathbf{F}}^{I}\) is the dual force in the inertial coordinate system. By using the force sensor installed on the end-effector of the robot to measure the six-dimensional force and torque information of the load, the load dynamic analysis formula can be derived based on Equation (3).