## 1 Introduction
For instance, in reference [39], the dielectric elastomer actuator is embedded into the continuous cylindrical robot body, and silicone fiber is laid to avoid compression, and with this design method, the movement speed reaches 28 mm/min; in reference [40], reciprocating locomotion using different amplitudes and frequencies (of tilted bristles) induced by the dielectric elastomer actuators is generated to improve the forward speed. In order to supply a larger actuation force, dielectric elastomer films are stacked in layers [41]. Moreover, it is discovered that by attaching a prestretched dielectric elastomer to a polyethylene terephthalate frame, the dielectric elastomer with a polyethylene terephthalate frame can be considered as the body segment in a soft robot, which represents the structure-actuation integrated design [42]. [PERSON] et al. connect three dielectric elastomer actuators within a soft, locomotion robot with three body segments. For the structure-actuation integrated design, the average speed is 5.3 mm/s [43]. [PERSON] et al. improve the minimum energy structure with the dielectric elastomer actuator and increase the actuator body segments, which increases the movement speed to 11.5 mm/s [44]. The above researches mainly focus on the movement realization of earthworm-like locomotion robots with soft actuators without investigating the relationship between locomotion performance and gait. In reference [15], a gait theory for a bionic earthworm-like robot is proposed. It is discovered that, for an increase in the driven segments, the saturation phenomenon for the maximum average speed occurs. The soft actuator has the advantages of large deformation and low weight, which can greatly reduce the weight of the robot and improve the movement speed. Fortunately, since the dielectric elastomer actuator has the characteristics of low weight and fast actuation, the synchronous driven of multiple segments can be achieved. Thus, the dielectric elastomer is selected as the soft actuator in this work. Periodic electrical signals are applied to the soft segments to realize the reciprocating expansion of the segments. A segment cell is a vibration-driven system. The vibration-driven is used in the capsule robot, and the effective movement of the system is realized [45]. Based on the vibration-driven, the dielectric elastomer robot realizes bidirectional movement [46]. The cell phone vibrators are employed for the vibration-driven small robot Rizeh to reduce the cost of the robot [47]. A vibration-driven robot with a time-varying stiffness structure is proposed to reveal the nonlinear driven mechanism of a vibration-driven system [48]. The vibration-driven can also be applied to multisegment, earthworm-like locomotion robots. The earthworm-like robot can move forward under vibration excitation [49]. The driven mode determines the movement of the robot. Gait design can provide different movement modes. In order to improve the movement speed of the earthworm-like robot, it is very critical to investigate the gait of the earthworm-like robot under vibration-driven movement. By designing the gait, the robot can move effectively under ultra-low frequency excitation. Therefore, by combining the deformation characteristics of a soft actuator, a gait design for a soft, earthworm-like robot is proposed in this paper. The driven module modes in this study are bio-inspired by earthworms. In the driven module, a gait generation principle is proposed for different driven wave transmission modes. The modular gait methods improve the locomotion speed of the robot under ultra-low frequency excitation. This reveals that only the driven wave frequency and amplitude affect locomotion speed. Moreover, as the failure of an individual segment in a driven module occurs, locomotion can still be guaranteed due to the driven module design. Numerical simulations for different cases of generated gaits for different driven modules can be used to verify the modular driven design that can break through the saturation phenomenon of locomotion speed.