# The Optimal Locomotion of a Self-Propelled Worm Actuated by Two Square Waves

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Mathematical Description of Worm-Like Locomotion

## 3. Square Strain Waves (SSW)

## 4. Optimization

## 5. Verification and Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Expression of ${\mathit{b}}_{11}(\mathit{z},\mathit{\tau}),{\mathit{h}}_{1}(\mathit{z},\mathit{\tau}),{\mathit{q}}_{1}(\mathit{\tau})$ and ${\mathit{b}}_{12}(\mathit{z},\mathit{\tau})$

## Appendix B. Expression of ${\dot{\mathit{x}}}_{1\mathit{j}}(\mathit{\tau})$ for $\mathit{j}=1,2,\dots ,5,6$

- (1)
- When $\tau \in [0,\kappa ]$$${\dot{x}}_{11}\left(\tau \right)=\frac{{L}_{0}}{{\lambda}^{2}}\left(\left(1+\gamma +{\u03f5}_{1}+{\gamma}^{2}{\u03f5}_{1}\right)\lambda -{\u03f5}_{1}\kappa \left({\left(1+\gamma \right)}^{2}{\u03f5}_{1}\right)\right)$$When $\tau \in [\kappa ,{\lambda}_{1}]$$${\dot{x}}_{12}\left(\tau \right)=\frac{{L}_{0}}{{\lambda}^{2}}\left({\u03f5}_{1}\left(\lambda \left(\lambda -\tau +\gamma \left(2-\lambda +\tau \right)\right)+{\u03f5}_{1}\left(\gamma \left(\lambda -\tau -2\right)+\tau \right)\left(2\gamma +\left(\gamma -1\right)(\tau -{\lambda}_{1})\right)\right)\right)$$
- (2)
- When $\tau \in [{\lambda}_{1},\lambda -{\lambda}_{1}-1]$$${\dot{x}}_{13}\left(\tau \right)=\frac{{L}_{0}}{{\lambda}^{2}}\left(2\gamma {\u03f5}_{1}\left(\lambda +\gamma \left(\lambda -2\right){\u03f5}_{1}\right)\right)$$
- (3)
- When $\tau \in [\lambda -{\lambda}_{1}-1,\lambda -{\lambda}_{1}]$$${\dot{x}}_{14}\left(\tau \right)=\frac{{L}_{0}{\u03f5}_{1}}{{\lambda}^{2}}\left(\begin{array}{l}\lambda \left(1-2\lambda +\tau +\gamma \left(1+\lambda -\tau -{\lambda}_{1}\right)+{\lambda}_{1}\right)+{\gamma}^{2}{\u03f5}_{1}\left(1+\lambda -\tau -{\lambda}_{1}\right)\left(\tau +{\lambda}_{1}-1\right)-\\ {\u03f5}_{1}\left(2{\lambda}^{2}-3\lambda \left(1+\tau +{\lambda}_{1}\right)+{\left(1+\tau +{\lambda}_{1}\right)}^{2}-2\gamma \left({\left(\tau +{\lambda}_{1}-\lambda \right)}^{2}-1\right)\right)\end{array}\right)$$
- (4)
- When $\tau \in [\lambda -{\lambda}_{1},\lambda -1]$$${\dot{x}}_{15}\left(\tau \right)=\frac{{L}_{0}{\u03f5}_{1}}{{\lambda}^{2}}\left(\lambda \left(1+\gamma +{\u03f5}_{1}+{\gamma}^{2}{\u03f5}_{1}\right)-{\left(1+\gamma \right)}^{2}{\u03f5}_{1}\right)$$
- (5)
- When $\tau \in [\lambda -1,\lambda ]$$${\dot{x}}_{16}\left(\tau \right)=\frac{{L}_{0}}{{\lambda}^{2}}\left({\u03f5}_{1}\left(\kappa +\gamma \left(\lambda -{\lambda}_{1}-1\right)\right)\left({\u03f5}_{1}\left(\gamma -\kappa +{\lambda}_{1}\right)-\lambda \right)\right)$$

## References

- Boxerbaum, A.; Chiel, H.J.; Quinn, R.D. Softworm: A soft, biologically inspired worm-like robot. Neurosc. Abstr.
**2009**, 315, 44106. [Google Scholar] - Tyrakowski, T.; Kaczorowski, P.; Pawłowicz, W.; Ziółkowski, M.; Smuszkiewicz, P.; Trojanowska, I.; Marszaek, A.; Żebrowska, M.; Lutowska, M.; Kopczyńska, E.; et al. Discrete movements of foot epithelium during adhesive locomotion of a land snail. Folia Biol.
**2012**, 60, 99–106. [Google Scholar] [CrossRef] - Gray, J.; Lissmann, H. Studies In Animal Locomotion. J. Exp. Biol.
**1938**, 15, 506–517. [Google Scholar] - Chapman, G. The hydrostatic skeleton in the invertebrates. Biol. Rev.
**1958**, 33, 338–371. [Google Scholar] [CrossRef] - Boxerbaum, A.S.; Shaw, K.M.; Chiel, H.J.; Quinn, R.D. Continuous wave peristaltic motion in a robot. Int. J. Robot. Res.
**2012**, 31, 302–318. [Google Scholar] [CrossRef] - Agersborg, H.K. Notes on the locomotion of the nudibranchiate mollusk Dendronotus giganteus O’Donoghue. Biol. Bull.
**1922**, 42, 257–266. [Google Scholar] [CrossRef] - Jones, H. Observations on the locomotion of two British terrestrial planarians (Platyhelminthes, Tricladida). J. Zool.
**1978**, 186, 407–416. [Google Scholar] [CrossRef] - Yanagida, T.; Adachi, K.; Yokojima, M.; Na kamura, T. Development of a peristaltic crawling robot attached to a large intestine endoscope using bellows-type artificial rubber muscles. In Proceedings of the 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Vilamoura, Portugal, 7–12 October 2012. [Google Scholar]
- Omori, H.; Murakami, T.; Nagai, H.; Nakamura, T.; Kubota, T. Planetary subsurface explorer robot with propulsion units for peristaltic crawling. In Proceedings of the 2011 IEEE International Conference on Robotics and Automation (ICRA), Shanghai, China, 9–13 May 2011. [Google Scholar]
- Tanaka, Y.; Ito, K.; Nakagaki, T.; Kobayashi, R. Mechanics of peristaltic locomotion and role of anchoring. J. R. Soc. Interface
**2012**, 9, 222–233. [Google Scholar] [CrossRef] [PubMed] - Dobrolyubov, A.I. The Mechanism of Locomotion of Some Terrestrial Animals by Trveling Waves of Deformation. J. Theor. Biol.
**1986**, 119, 457–466. [Google Scholar] [CrossRef] - DeSimone, A.; Guarnieri, F.; Noselli, G.; Tatone, A. Crawlers in viscous environments: Linear vs non-linear rheology. Int. J. Non-Linear Mech.
**2013**, 56, 142–147. [Google Scholar] [CrossRef] - Noselli, G.; Tatone, A.; DeSimone, A. Discrete one-dimensional crawlers on viscous substrates: Achievable net displacements and their energy cost. Mech. Res. Commun.
**2014**, 58, 73–81. [Google Scholar] [CrossRef] - Gidoni, P.; Noselli, G.; DeSimone, A. Crawling on directional surfaces. Int. J. Non-Linear Mech.
**2014**, 61, 65–73. [Google Scholar] [CrossRef] - Fang, H.; Wang, C.; Li, C.; Wang, K.W.; Xu, J. A comprehensive study on the locomotion characteristics of a metameric earthworm-like robot: A. Modeling and gait generation. Multibody Syst. Dyn.
**2015**, 34, 391–413. [Google Scholar] [CrossRef] - Jung, K.; Koo, J.C.; Lee, Y.K.; Choi, H.R. Artificial annelid robot driven by soft actuators. Bioinspir. Biomim.
**2007**, 2, S42–S49. [Google Scholar] [CrossRef] [PubMed] - Boxerbaum, A.S.; Chiel, H.J.; Quinn, R.D. A New Theory and Methods for Creating Peristaltic Motion in a Robotic Platform. In Proceedings of the 2010 IEEE International Conference on Robotics and Automation (Icra), Anchorage, AK, USA, 3–7 May 2010; pp. 1221–1227. [Google Scholar]
- Onal, C.D.; Wood, R.J.; Rus, D. Towards printable robotics: Origami-inspired planar fabrication of three-dimensional mechanisms. In Proceedings of the 2011 IEEE International Conference on Robotics and Automation (ICRA), Shanghai, China, 9–13 May 2011. [Google Scholar]
- Quinn, R.D.; Boxerbaum, A.; Palmer, L.; Chiel, H.; Diller, E.; Hunt, A.; Bachmann, R. Novel Locomotion via Biological Inspiration. In Proceedings of the Unmanned Systems Technology Xiii, Orlando, FL, USA, 27–29 April 2011; p. 8045. [Google Scholar]
- Fang, H.; Li, S.; Wang, K.W.; Xu, J. Phase coordination and phase–velocity relationship in metameric robot locomotion. Bioinspir. Biomim.
**2015**, 10, 066006. [Google Scholar] [CrossRef] [PubMed] - Keller, J.B.; Falkovitz, M.S. Crawling of Worms. J. Theor. Biol.
**1983**, 104, 417–442. [Google Scholar] [CrossRef] - Gidoni, P.; DeSimone, A. Stasis domains and slip surfaces in the locomotion of a bio-inspired two-segment crawler. Meccanica
**2017**, 52, 587–601. [Google Scholar] [CrossRef] - DeSimone, A.; Tatone, A. Crawling motility through the analysis of model locomotors: Two case studies. Eur. Phys. J. E
**2012**, 35, 1–8. [Google Scholar] [CrossRef] [PubMed] - Jiang, Z.; Xu, J. Analysis of worm-like locomotion driven by the sine-squared strain wave in a linear viscous medium. Mech. Res. Commun.
**2017**, 85, 33–44. [Google Scholar] [CrossRef] - Bertsekas, D.P. Nonlinear Programming; Athena scientific Belmont: Belmont, MA, USA, 1999. [Google Scholar]
- Fang, H.; Wang, C.; Li, S.; Wang, K.W.; Xu, J. A comprehensive study on the locomotion characteristics of a metameric earthworm-like robot: B. Gait analysis and experiments. Multibody Syst. Dyn.
**2015**, 35, 153–177. [Google Scholar] [CrossRef]

**Figure 1.**A continuously deforming worm. (

**a**) The relaxed body in the reference configuration; (

**b**) The deformed body caused by the propagation of strain waves in the current configuration.

**Figure 2.**Four cases of worm-like locomotion based on the relationship between the dimensionless body length and the distance of two waves.

**Figure 3.**Two square strain waves with a dimensionless unit wave width propagating along the body axis of the worm at the constant speed c and the distance between two waves $\kappa $

**Figure 4.**A combination of the optimal conditions for four cases based on the relationship between $\lambda $ and $\kappa $.

**Figure 5.**The maximum average velocities related to $\kappa $ and $\gamma $ in four cases. (

**a**) Case I; (

**b**) Case II; (

**c**) Case III; (

**d**) Case IV.

**Figure 6.**A comparison of three different results in the case of contractive waves (

**a**

_{1}–

**a**

_{3}) and extensive waves (

**b**

_{1}–

**b**

_{3}). The approximate analytical prediction (dotted-dashed line), the numerical solutions from simulating the dynamic equation (solid coloured lines), and the numerical ones from simulating the quasi-static equation (dashed line).

**Figure 7.**Comparison of experimental results and theoretical results based on the relationship between average velocity and the actually deformed length of the driving modules in [26]. (

**a**) One driving module ($K=1$); (

**b**) Two driving modules ($K=2$).

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jiang, Z.; Xu, J.
The Optimal Locomotion of a Self-Propelled Worm Actuated by Two Square Waves. *Micromachines* **2017**, *8*, 364.
https://doi.org/10.3390/mi8120364

**AMA Style**

Jiang Z, Xu J.
The Optimal Locomotion of a Self-Propelled Worm Actuated by Two Square Waves. *Micromachines*. 2017; 8(12):364.
https://doi.org/10.3390/mi8120364

**Chicago/Turabian Style**

Jiang, Ziwang, and Jian Xu.
2017. "The Optimal Locomotion of a Self-Propelled Worm Actuated by Two Square Waves" *Micromachines* 8, no. 12: 364.
https://doi.org/10.3390/mi8120364