## 3 Kinematic Model of Continuum Robotics
### Coordinate Systems and Dual Quaternion Transformations of Points and Lines  
A single-segment continuous robot is used to model using the dual quaternion method. The forward kinematics of the robot are to solve its end pose after driving. The physical model of the single-section robotic arm when driven is shown in Figure 3a.  
As shown in Figure 3, if {F} and {E} are two reference frames, while \(\hat{\mathbf{q}}_{E},\hat{\mathbf{q}}_{F}\) are the dual quaternion of those reference frames relative to a fixed coordinate system in space, then the relative position relationship between these two coordinate systems is called \(\hat{\mathbf{q}}_{EF}\), which is  
Figure 3: **(a)** Physical model of a single-section robotic arm in the driving state. Among them, {F}[1][2][3][4][E] are the coordinate systems, respectively, and \(L_{E}\) and \(L_{F}\) represent the straight lines where the _z_-axis of the {F} and {E} coordinate systems are located, respectively. **(b)** Mathematical model of a single-section robotic arm in the driving state. It describes the _z_-axis and _y_-axis in the coordinate system {F}, that is, the straight lines \(L_{F}\) and \(P_{F}\), which after coordinate transformation become the straight lines \(L_{E}\) and \(P_{E}\) in the coordinate system {E}.  
Figure 2: The Structure diagram of the single-section mechanical arm is driven and not driven. **(a)** Schematic diagram of the driving line of the single-section robotic arm when it is not driven, where \(L_{ij}^{0}\) represents the rope length. **(b)** Schematic diagram of the driving line of the robotic arm after driving time \(t\), where \(L_{ij}^{t}\) represents the rope length.  
represented by (10) and can be obtained from (4); a more detailed derivation process can be found in Appendix C.  
\[\hat{\mathbf{q}}_{EF}=\hat{\mathbf{q}}_{E}\otimes\hat{\mathbf{q}}_{F}=\left[cos\frac{\varphi} {2},-sin\delta sin\frac{\varphi}{2},cos\delta sin\frac{\varphi}{2},0\delta \epsilon,0,0,0,\rho\cdot sin\frac{\varphi}{2}\right] \tag{10}\]  
Let the lines in the front-end coordinate system \(\{\text{F}\}\) where the \(y\)-axis and \(z\)-axis lie be \(L_{F},P_{F}\) and the direction vectors be \(I_{F}\) and \(\mathbf{p}_{F}\), respectively. By [PERSON]'s theorem, the line \(L_{F}\) in the coordinate system \(\{\text{F}\}\) becomes \(L_{1}\) after a rotation around the \(z\)-axis and a translation \(\rho\) along the axis \(x_{1}\), and then \(L_{2}\) around the axis \(y_{2}\), before a translation \(\rho\) along the new \(x_{3}\) axis becomes \(L_{E}\) in the coordinate system \(\{\text{E}\}\). Similarly, \(\text{P}_{F}\) can become \(P_{E}\). The moment vectors are \(\mathbf{m}_{\text{L}},\mathbf{m}_{P}\), respectively, which are expressed in the Plucker coordinate system as \(\mathbf{L}_{F}=(I_{F},\mathbf{m}_{L}),\mathbf{P}_{F}=(\mathbf{p}_{F},\mathbf{m}_{F})\). They can be expressed as \(\mathbf{L}_{F}=I_{F}+\epsilon\mathbf{m}_{L},\mathbf{P}_{F}=\mathbf{p}_{F}+\epsilon\mathbf{m}_{F}\) by a dual quaternion. Substituting (10) into (3), we can obtain the relationship between the straight line \(L_{F}\) and \(L_{E}\) on the coordinate system \(\{\text{F}\}\) and \(\{\text{E}\}\) as (11).  
\[\hat{\mathbf{L}}_{F}=\hat{\mathbf{q}}_{EF}^{*}\otimes\hat{\mathbf{L}}_{E}\otimes\hat{\mathbf{ q}}_{EF} \tag{11}\]  
Similarly, the relationship between \(\text{P}_{\text{F}}\) and \(\text{P}_{E}\) is (12).  
\[\hat{\mathbf{P}}_{F}=\hat{\mathbf{q}}_{EF}^{*}\otimes\hat{\mathbf{P}}_{E}\otimes\hat{\mathbf{ q}}_{EF} \tag{12}\]  
According to [PERSON]'s law, the intersection points of the two are the position of the end coordinate system, and the pose can be expressed as \([I_{F},\mathbf{p}_{F},I_{F}\times\mathbf{p}_{F}]\).