## 3 Kinematic Model of Continuum Robotics
### Control the Motion of the Robotic Arm through the End Position  
In the previous section, the forward kinematic equations of the continuum arms were derived using the dual quaternion method. The Jacobian matrix of each part of the manipulator is first solved to solve the inverse kinematics numerically. Let the expression of the Plucker form of the coordinates of a point \(P_{0}\) in the coordinate system \([0]\) be \(\hat{\boldsymbol{P}}_{0}\), and the dual quaternion relationship between the coordinate system \([0]\) and the coordinate system \([1]\) is \(\hat{\boldsymbol{q}}_{i}\). Let the expression of the Pluck form of the coordinates of a point \(P_{0}\) in the coordinate system \([0]\) be \(\hat{\boldsymbol{P}}_{0}\), and the dual quaternion relationship between the coordinate system \([0]\) and the coordinate system \([1]\) is \(\hat{\boldsymbol{q}}_{i}\), then the point can be expressed in the coordinate system \([1]\) for (17). \[\hat{\boldsymbol{P}}_{1}=\boldsymbol{Q}_{1}=\hat{\boldsymbol{q}}_{i}^{*}\otimes \hat{\boldsymbol{P}}_{0}\otimes\hat{\boldsymbol{q}}_{i} \tag{17}\]  
Writing (18) as a vector pattern gives (18). \[\boldsymbol{Q}^{1}=\boldsymbol{Q}_{1}=[1,\hat{\boldsymbol{q}}_{1}]+[0,\varepsilon \boldsymbol{x}_{1}] \tag{18}\]  
The complete forward kinematics of \(i\)th section relative to \(\{0\}\), denoted by \(Q_{i}\), is given by (19). \[\hat{\boldsymbol{P}}_{i}=\boldsymbol{Q}^{i}=\hat{\boldsymbol{q}}_{i}^{*} \otimes\cdots\hat{\boldsymbol{q}}_{1}^{*}\otimes\hat{\boldsymbol{P}}_{0} \otimes\hat{\boldsymbol{q}}_{1}\otimes\cdots\hat{\boldsymbol{q}}_{i} \tag{19}\]  
Write (19) as a vector, as shown in (20). Here \(\hat{\boldsymbol{\theta}}\in\mathbb{R}^{4}\) is the rotation and \(\hat{\boldsymbol{x}}\in\mathbb{R}^{4}\) is the displacement. \[\boldsymbol{Q}^{i}=\boldsymbol{Q}_{1}\cdots\otimes\boldsymbol{Q}_{i}=[1, \boldsymbol{\theta}_{1,2\ldots i}]+[0,\varepsilon\boldsymbol{x}_{1,2\ldots i}] \tag{20}\]  
\[\boldsymbol{Q}^{i}=\boldsymbol{Q}_{1}\otimes\boldsymbol{Q}_{2}\otimes\cdots \otimes\boldsymbol{Q}_{i}=\boldsymbol{Q}^{i-1}\otimes\boldsymbol{Q}_{i} \tag{21}\]  
Putting \(\hat{\boldsymbol{q}}=[\boldsymbol{q}_{r},\varepsilon\boldsymbol{q}_{d}]\) into (21), we can obtain (22) and (23), which are the dual quaternion representations of the position and pose matrix of the \(i\)th robotic arm. \[\hat{\boldsymbol{q}}_{1,2\ldots i}=\hat{\boldsymbol{q}}_{1,2\ldots i-1} \otimes\hat{\boldsymbol{q}}_{i,r} \tag{22}\]  
\[\hat{\boldsymbol{x}}_{1,2\ldots i}=\hat{\boldsymbol{x}}_{1,2\ldots i-1} \otimes\hat{\boldsymbol{q}}_{i,r}+\hat{\boldsymbol{\theta}}_{1,2\ldots i-1} \otimes\hat{\boldsymbol{q}}_{i,d} \tag{23}\]  
We need to use the obtained position and pose to obtain partial derivatives of the joint variables, so as to obtain the velocity Jacobian matrix of the manipulator vector. Let \(\boldsymbol{J}_{i}^{\theta}\) and \(\boldsymbol{J}_{i}^{x}\), respectively, be the position and pose quaternion Jacobians from (24) and (25). \[\boldsymbol{J}_{i}^{\phi}=\frac{\partial(\hat{\boldsymbol{q}}_{1,2\ldots i})} {\partial(\delta,\theta)}\in\mathbb{R}^{4\times 2i} \tag{24}\]\[\mathbf{J}_{i}^{x}=\frac{\partial(x_{1,2,...i})}{\partial(\delta,\theta)}\in\mathbb{R}^ {4\times 2i} \tag{25}\]  
Solve the derivatives of joint variables for (22) and (23) to obtain (26) and (24), where the formula for derivation is (24) and (25). \[\mathbf{J}_{i}^{\phi}=\mathbf{J}_{i-1}^{\phi}\otimes\mathbf{q}_{i,r} \tag{26}\]  
\[\boldsymbol{Q}^{1}=\boldsymbol{Q}_{1}=[1,\hat{\boldsymbol{q}}_{1}]+[0,\varepsilon \boldsymbol{x}_{1}] \tag{18}\]0  
The Jacobians derived in (26) and (27) are only valid in the Pluck coordinate system, so we need to transform the Pluck coordinate system into the inertial coordinate system. The Cartesian angular velocity, \(\mathbf{\omega}_{x}\in\mathbb{R}^{3}\) relative to \(\{0\}\), can be recovered from the quaternion velocities as (28).