# Equivalent Dynamic Analysis of a Cable-Driven Snake Arm Maintainer
## 6 Conclusions  
In this paper, an underactuated SAM was designed, and the attendant kinematics were analyzed based on the application requirements of the narrow spaces of a nuclear power plant. Based on various force balance and torque balance algorithms, the static characteristics of each joint of the SAM in the initial pose were analyzed. Based on the principle of equivalent transformation and the Lagrangian equation, a decoupling algorithm for the strong coupling dynamics of the SAM was proposed. The traction force of the cable coupled between the joints was equivalent to the joint torque, and the real-time equivalent joint torque was obtained. The decoupling calculation of the cable traction was then realized via inverse transformation of the joint torque. To verify the correctness of the equivalent dynamics, we designed a model-based dynamic controller based on the equivalent dynamics. The simulation results demonstrate the effectiveness of SAM motion control based on equivalent dynamics. The maximum tracking error at a steady state is less than \(1\times 10^{-3}\) rad. In addition, a SAM prototype with a single-joint group and a 1500 mm arm length was created, and experimental data were collected. Accuracy tests on a 1500 mm arm-length SAM validate the rationality of the underactuated design. Using the open-loop control trajectory control experiment of the SAM single-joint group, the cable force, joint angle tracking error, and equivalent joint torques were analyzed to verify the accuracy of the equivalent dynamic model. The theoretical dynamic model will be further optimized in future studies. The main areas to be optimized are as follows: (1) The equivalent friction model of the cable in the circular hole will be developed to identify non-linear friction. (2) The linear elastic deformation model of the cable will be developed to identify the dynamics of flexible errors. (3) Neural network and machine learning will be used to identify and compensate for the unknown dynamic parameters. Figure 13: Comparative analysis of theoretical and experimental data of equivalent joint torques. (**a**) Theoretical and experimental data of \(\mathbf{T_{1}}\) under vertical sinusoidal trajectory. (**b**) Theoretical and experimental data of \(\mathbf{T_{2}}\) under horizontal sinusoidal trajectory. (**c**) \(\mathbf{T_{1}}\) and \(\mathbf{T_{2}}\) errors. (**d**) Analysis of the error characteristics of \(\mathbf{T_{1}}\) and \(\mathbf{T_{2}}\). **Supplementary Materials:** The following supporting information can be downloaded at: [[https://www.youtube.com/channel/UCOJ1](https://www.youtube.com/channel/UCOJ1) DxVz8 QaT1 IO5 XOKpuGg]([https://www.youtube.com/channel/UCOJ1](https://www.youtube.com/channel/UCOJ1) DxVz8 QaT1 IO5 XOKpuGg), Video S1: Left and right.mp4, Video S2: Up and down.mp4 (accessed on 14 July 2022). And [[https://www.mdpi.com/article/10.3](https://www.mdpi.com/article/10.3)]([https://www.mdpi.com/article/10.3](https://www.mdpi.com/article/10.3)) 390/app12157494/s1, File S1: Supplementary materials.docx. **Author Contributions:** G.Q. : methodology, validation, software, data curation, writing--original draft; A.J. : conceptualization, formal analysis, funding acquisition, investigation; H.W. : validation, writing--review and editing, project administration. All authors have read and agreed to the published version of the manuscript. **Funding:** This work was carried out within the framework of the EUROfusion Consortium, funded by the European Union via the Euratom Research and Training Programme (Grant Agreement No. 101052200--EUROfusion). Views and opinions expressed are those of the author(s) only and do not necessarily reflect those of the European Union or the European Commission. Neither the European Union nor the European Commission can be held responsible for them. This work is supported by the National Key R&D Program of China (Grant No. 2019 YFB1309600), the National Natural Science Foundation of China (Grant Nos.