# Locomotion of Self-Excited Vibrating and Rotating Objects in Granular Environments
## 4 Conclusions and Future Research  
Inspired by the biological experiments of sand lizard diving into desert sand and _Escherichia coli_ propelling themselves forward by rotating helically in a viscous fluid, we used DEM numerical simulations to design a rising-sinking device in granular materials, driven by a combination of vibration and rotation. The simulations allowed us to test the vertical motion of the SO in response to periodic same-frequency rotational and translational transverse oscillations. The transverse oscillation was in the \(y\) direction, \(y=Asin(2\pi ft)\), and the rotation was around the \(x\) axis, \(\omega=\Omega cos(2\pi ft)\). We have found that the SO rose to the bed surface when \(\Omega<0\) and sink to approximately \(Z=-0.3\) m when \(\Omega>0\). The saturation in the sinking is likely an effect of the bottom boundary. To understand the effects of the key SO actuation parameters: \(A\), \(f\), and \(\Omega\), on its rising and sinking rates, we varied these parameters systematically. Our results show that the rising rate is linearly proportional to the \(A\) and \(\Omega\) and that there is an optimal operation region for \(f\). These findings can inform locomotion biology in understanding how animals appear to move effectively across a diversity of complex substrates. This work also has the potential to facilitate the design, control and development of a new type of bio-physically inspired robots on or within complex seabed or desert terrains.  
This research can be extended in several directions, some of which we intend to explore. The range of motion and material parameters can be broadened, including the density, size and friction coefficient, so as to mimic living organisms, which often operate optimally in variable environments [7]. For example, the cross-sectional shape of the sandfish has been shown to aid rapid burial into granular media [37]. This also suggests  
Figure 8: The displacement of the SO in the \(z\) direction as a function of the frequency \(f\) for \(A=0.03\) m for (**a**) \(\Omega=-80\) rad/s and (**b**) \(\Omega=80\) rad/s. The rising rate as a function of \(f\) at the same amplitude for (**c**) \(\Omega=-80\) rad/s and (**d**) \(\Omega=80\) rad/s.  
that it would be interesting to explore the effect of this morphology, combined with head tapering on performance. Other intruder shapes should also be investigated. Another interesting direction would be to simulate non-spherical, e.g., polyhedral, bed particles, to understand the effect on the SO dynamics. We believe that in such environments the SO vertical motion would be slower, but this remains to be investigated. Another important direction is theoretical--a mathematical predictive model for this phenomenon does not yet exist. Such a theoretical model could be then tested quantitatively against our numerical results.  
Conceptualization: P.L., X.R., W.T. and R.B.; data collection: J.Z. and P.L.; methodology, P.L. and X.R.; Resources, Q.C.; Supervision, X.R. and R.B.; original draft, P.L.; review and editing, W.T. and R.B. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant No. 11002162) and General Scientific Research Projects of NUDT (Grant No. ZK16-03-01).  
The authors acknowledge the administrative support given by the Physics Department of our college. The authors are also grateful to the Sunway-TaihulLight supercomputer Center for the technical support given. P.L. is grateful for the hospitality of the Cavendish Laboratory, Cambridge university, where this paper was prepared.  
The authors declare no conflict of interest.