## 5 Conclusions  
In this paper, driven modules and modular gaits that are realized by using soft actuators due to their low weight and flexibility are proposed. Two driven modules and two locomotion transmission modes are analyzed. For the different driven modules, we give the gait generation algorithms. When compared to rigid-body earthworm-like robots, the soft, earthworm-like robots with a modular driven mode has great advantages in locomotion performance. The soft, earthworm-like robots can locomote under ultra-low frequency excitation with appropriate shifts. Based on the gait analysis and simulation results, the main contributions of this study are summarized as follows:  
1. A locomotion algorithm is proposed to simulate the locomotion of the soft, earthworm-like robot. The locomotion of a 12-segment earthworm-like robot is numerically  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**The Number of Body Segments** & **Driven Modules** & **Wave Transmission Modes** & **Average Speeds** \\ \cline{3-4}
**in a Driven Module (\(n\))** & & MG-I & \(V=3\ \overline{V}\). \\ \cline{3-4} \multirow{2}{*}{\(n\)=3} & U-DM & MG-II & \(n_{T}=1:V_{1}=\overline{V}_{i}\) \\  & & MG-II & \(n_{T}=2:V_{2}=2\ \overline{V}\). \\ \hline \multirow{6}{*}{\(n\)=4} & \multirow{2}{*}{U-DM} & MG-I & \(V=4\ \overline{V}\). \\ \cline{3-4}  & & MG-II & \(n_{T}=1:V_{1}=\overline{V}_{i}\) \\  & & MG-II & \(n_{T}=2:V_{2}=2\ \overline{V}\); \\  & & & \(n_{T}=3:V_{3}=3\ \overline{V}\). \\ \cline{2-4}  & & MG-I & \(V=3\ \overline{V}\). \\ \cline{2-4}  & & B-DM & \(n_{T}=1:V_{1}=\overline{V}_{i}\) \\  & (\(n_{pq}=3\), \(n_{pr}=1\)) & MG-II & \(n_{T}=2:V_{2}=2\ \overline{V}\); \\  & & & \(n_{T}=3:V_{3}=3\ \overline{V}\). \\ \hline \hline \end{tabular}
\end{table}
Table 2: The average speeds when \(k=1\).  
simulated based on different modes. The locomotion speed of the earthworm-like robot is positively correlated with the frequency and amplitude of the receding peristaltic wave. The robot can obtain higher speeds when the proportion of the actuated segments increases. For the driven module containing \(n\) segments, the maximum speed is \(kn\overline{V}\);
2. The average speeds for different locomotion modes are obtained. For the dielectric-elastomer-driven robot, the proportion of the number of actuated segments strongly affects the average speed. As the proportion of the number of actuated segments increases, the average speed increases;
3. When there are failed segments in the robot, a modular gait mode can be designed to improve the locomotion speed of the robot. For the driven module, \(n_{pa}=3\) and \(n_{pr}=1\), there are three different speeds: \(\overline{V}\), \(2\overline{V}\), and \(3\overline{V}\).  
Based on the results of the modular gait generation method and the corresponding average speed, the modular driven mode has excellent locomotion performance. The theoretical analysis and simulation results of this paper provide the necessary theoretical support for the flexibility and low weight of earthworm-like locomotion robots.  
Conceptualization, X.S. and Z.Q.; methodology, Z.Q.; software, Z.Q.; validation, X.S. and Z.Q.; formal analysis, Z.Q.; investigation, Z.Q.; resources, Z.Q.; data curation, Z.Q.; writing--original draft preparation, Z.Q.; writing--review and editing, X.S.; visualization, X.S. and Z.Q.; supervision, X.S.; project administration, X.S.; funding acquisition, X.S. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Natural Science Foundation of China grant number 12122208, 11972254, and 11932015 and the Fundamental Research Funds for Central Universities.  
Not applicable.  
All data generated or analyzed during this study are included in this published article.  
The authors would like to thank the reviewers and editors for improving this manuscript.  
The authors declare that they have no conflict of interest.