###### Abstract  
Kinematic modeling is essential for planning and controlling continuum robot motion. The traditional Denavit Hartenberg (DH) model involves complex matrix multiplication operations, resulting in computationally intensive inverse solutions and trajectory planning. Solving position and orientation changes in continuum robots using the double quaternion rule can reduce computational complexity. However, existing dual quaternion methods are direct equational transformations of DH rules and do not give a complete modeling process. They usually require more interpretability when applying continuum robot kinematic modeling. This paper uses the dual quaternion method to establish a kinematic model of a continuum robot. It uses a two-section continuum robot model to compare the advantages of dual quaternion and traditional modeling methods. In addition, this paper proposes a five-polynomial interpolation algorithm based on the dual quaternion method for trajectory planning of continuum robots. This  
of rigid elements resulting in only changing themselves at discrete points in their structure, continuum robots can theoretically change any position in their system [18; 19], which leads to challenging kinematic and dynamic modeling of continuum robots and further leads to difficulties in real-time dynamic control. Continuum robots have widespread use in quasi-static environments where dynamic models may not be applicable [20]. Researchers have employed various approaches to solving the kinematics of continuum robots, such as utilizing motion combinations of fake rigid manipulators to simulate their motion and applying the Denavit-Hartenberg (DH) model, which was initially developed for rigid manipulators' accuracy in emulating continuum robot behavior; kinematic and shape correspondence between super-redundant manipulators and desired spatial profiles have been introduced [21; 22]. Recently, the incorporation of continuum curvature into a modified DH modeling procedure using differential geometry has provided a comprehensive approach to modeling continuum robots [23]. Building upon this work, the researcher has proposed the variable reality of the central axis, associating the driving variable with the central axis curve to modify and enhance existing ideas [11]. The Jacobian matrix of the model and the corresponding kinematic control method have also been discussed. However, special numerical treatment is required when approaching these models' straight (zero-curvature) cross-section configuration. The researcher expanded the driver variables, employing the Taylor series to address this issue and, thus, preventing model invalidity at zero curvature [24; 25; 26]. Nevertheless, modeling a multi-system or multi-joint manipulator arm using the above modeling approach becomes difficult, as the method of obtaining the end pose by multiplying the pose matrix places a significant workload on the system, and the relationship between each part of the system and the global coordinate system must be constantly considered. To address the challenge of dealing with sections with nearly straight deformation, the researcher has proposed using dual quaternions to solve this problem. Although the dual quaternion method offers increased efficiency in representing changes in the position and spatial elements of the robot, the existing approach is directly converted from the DH rule based on mathematical rules without considering the perspective of manipulator motion. This leads to limited interpretability of the dual quaternion method when applied to the kinematic modeling of manipulators in scientific journal articles. Building upon previous research, this paper explores the dual quaternion method from the standpoint of kinematics in order to tackle continuum manipulator problems. The solution is established based on the definition, and the merits of the dual quaternion method are emphasized by comparing its computational efficiency with traditional DH model-based algorithms. The paper is structured as follows.