## 4 Simulation Results and Discussion  
To evaluate the accuracy of the dual quaternion model compared to traditional kinematic models, specifically the DH and DH Taylor expansion models, we conducted a comparative analysis using the same driving variable. As demonstrated in Figure 5, our findings reveal that the error computed by the dual quaternion model aligns closely with those of the DH and DH Taylor expansion models. This corroborates the precision and reliability of our proposed methodology. These results underscore the potential of dual quaternions for enhancing the accuracy of kinematic models for continuum robots, laying the groundwork for future research in robotics and related fields. To prove the improvement of calculation speed using dual quaternion modeling, we use the same solution algorithm to solve the same target and compare the calculation time of dual quaternions: DH and DH Taylor. We anticipate the endpoint of the robotic arm to traverse from its initial position, P1 (0, 0, 960 mm), ultimately arriving at the desired position, P2 (369.8146 mm, 345.8315 mm, 702.9017 mm), as shown in Table 1. We used the optimization toolbox in MATLAB, and the CPU was an Intel(R) Xeon(R) W-2245 CPU @ 3.90 GHz 3.91 GHz processor for calculation. From Table 1, we can see that, in the numerical method, the dual quaternion model solves the target position with high precision and a short calculation time of 0.45 s; compared with the traditional DH method of 1.15 s, the calculation time is doubled. In standard meta-heuristic algorithms, the dual quaternion models are shortened by more than half.  
Figure 5: The two-section driven manipulator contains four driving variables: \(\varphi_{1}\in[-\pi,\pi]\), \(\delta_{1}\in[0,\pi/2]\), \(\varphi_{2}=0\), and \(\delta_{2}\
eq 0\). The spatial coordinates solved by the DH Taylor expansion series model are used as standard results to compare the errors in the results of the dual quaternion model and the DH model. (**a**) The dual quaternion model and the standard result solve the error in the x-coordinate direction between the coordinates. (**b,c**) are the errors in the \(y\)-axis and \(z\)-axis with the standard result, respectively. (**d**) The errors between the DH model solution coordinates and the standard result in the x-coordinate direction, (**e,f**), are the errors on the \(y\)-axis and \(z\)-axis, respectively.  
To achieve smooth angular velocity and acceleration changes at the end of the robotic arm during operation, a quintic polynomial interpolation algorithm based on dual quaternions is proposed for motion planning of the robotic arm. The simulated movement of the robotic arm end, as illustrated in Figure 6, demonstrates that the velocity and acceleration of the variables are continuum and smooth during the robotic arm's movement from point P1 to point P2, without any abrupt changes. This indicates that the robotic arm's motion is not subject to speed distortion and can operate seamlessly.  
Figure 6: Use the fifth-order polynomial interpolation algorithm to plan the trajectory of the robotic arm and solve the problem of the end of the two-section continuous robot moving from point \(P_{1}\) to point \(P_{2}\) in space. (**a**) Schematic diagram of the motion trajectory of the two-section continuous robot. The path is planned through the fifth-order polynomial interpolation algorithm; that is, the continuous robot needs to move 50 steps from point \(P_{1}\) to point \(P_{2}\) according to the interpolation sequence. (**b**) The error between the continuous robot’s actual path and the algorithm’s path (**c**–**e**), respectively, represents the changes in the angle, angular velocity, and angular acceleration of the driving amount when the continuous robot moves from point \(P_{1}\) to point \(P_{2}\) and sequentially moves 50 interpolation trajectory points. The \(x\)-axis represents the 50 trajectory points.