Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-10T09:10:55.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Lateral undulation of snake robots: a simplified model and fundamental properties

Published online by Cambridge University Press:  15 April 2013

Pål Liljebäck ,
Kristin Y. Pettersen ,
Øyvind Stavdahl  and
Jan Tommy Gravdahl
Pål Liljebäck*
Affiliation:
Norwegian University of Science and Technology (NTNU), Department of Engineering Cybernetics, NO-7491 Trondheim, Norway E-mails: Kristin.Y.Pettersen@itk.ntnu.no, Oyvind.Stavdahl@itk.ntnu.no, Tommy.Gravdahl@itk.ntnu.no. SINTEF ICT, Department of Applied Cybernetics, NO-7465 Trondheim, Norway
Kristin Y. Pettersen
Affiliation:
Norwegian University of Science and Technology (NTNU), Department of Engineering Cybernetics, NO-7491 Trondheim, Norway E-mails: Kristin.Y.Pettersen@itk.ntnu.no, Oyvind.Stavdahl@itk.ntnu.no, Tommy.Gravdahl@itk.ntnu.no.
Øyvind Stavdahl
Affiliation:
Norwegian University of Science and Technology (NTNU), Department of Engineering Cybernetics, NO-7491 Trondheim, Norway E-mails: Kristin.Y.Pettersen@itk.ntnu.no, Oyvind.Stavdahl@itk.ntnu.no, Tommy.Gravdahl@itk.ntnu.no.
Jan Tommy Gravdahl
Affiliation:
Norwegian University of Science and Technology (NTNU), Department of Engineering Cybernetics, NO-7491 Trondheim, Norway E-mails: Kristin.Y.Pettersen@itk.ntnu.no, Oyvind.Stavdahl@itk.ntnu.no, Tommy.Gravdahl@itk.ntnu.no.
*
*Corresponding author. E-mail: Pal.Liljeback@sintef.no.
Get access Rights & Permissions [Opens in a new window]

Summary

This paper considers the lateral undulation motion of snake robots. The first contribution of the paper is a model of lateral undulation dynamics developed for control design and stability analysis purposes. The second contribution is an analysis of the simplified model that shows that any asymptotically stabilizing control law for the snake robot to an equilibrium point must be time varying. Furthermore, the analysis shows that a snake robot (with four links) is strongly accessible from almost any equilibrium point, except for certain singular configurations, and that the robot does not satisfy sufficient conditions for small-time local controllability. The third contribution is based on using averaging theory to prove that the average velocity of the robot during lateral undulation will converge exponentially fast to a steady-state velocity which is given analytically as a function of the gait pattern parameters. From the averaging analysis, we also derive a set of fundamental relationships between the gait parameters of lateral undulation and the resulting forward velocity of the snake robot. The paper presents simulation results and results from experiments with a physical snake robot that support the theoretical findings.

Keywords

Snake robotLateral undulationModellingDynamics
Type
Articles
Information
Robotica , Volume 31 , Issue 7 , October 2013 , pp. 1005 - 1036
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Gray, J., “The mechanism of locomotion in snakes,” J. Exp. Biol. 23 (2), 101–120 (1946).CrossRefGoogle ScholarPubMed
2.Hirose, S., Biologically Inspired Robots: Snake-Like Locomotors and Manipulators (Oxford University Press, Oxford, UK, 1993).Google Scholar
3.Chirikjian, G. S., “Theory and applications of hyper-redundant robotic manipulators,” Ph.D. dissertation (Pasadena, CA: California Institute of Technology, 1992).Google Scholar
4.Chirikjian, G. and Burdick, J., “The kinematics of hyper-redundant robot locomotion,” IEEE Trans. Robot. Autom. 11 (6), 781–793 (1995).CrossRefGoogle Scholar
5.Ostrowski, J. P., “The mechanics and control of undulatory robotic locomotion,” Ph.D. dissertation (Pasadena, CA: California Institute of Technology, 1996).Google Scholar
6.Prautsch, P. and Mita, T., “Control and Analysis of the Gait of Snake Robots,” In: Proceedings of the IEEE International Conference on Control Applications, Kohala Coast, HI, USA (1999) pp. 502–507.Google Scholar
7.Ma, S., “Analysis of creeping locomotion of a snake-like robot,” Adv. Robot. 15 (2), 205–224 (2001).Google Scholar
8.Date, H., Sampei, M. and Nakaura, S., “Control of a Snake Robot in Consideration of Constraint Force,” In: Proceedings of the IEEE International Conference on Control Applications, Mexico City (Sep. 5–7, 2001) pp. 966–971.Google Scholar
9.Saito, M., Fukaya, M. and Iwasaki, T., “Serpentine locomotion with robotic snakes,” IEEE Contr. Syst. Mag. 22 (1), 64–81 (Feb. 2002).Google Scholar
10.Hicks, G. P., “Modeling and control of a snake-like serial-link structure,” Ph.D. dissertation (North Carolina: North Carolina State University, 2003).Google Scholar
11.Nilsson, M., “Serpentine Locomotion on Surfaces with Uniform Friction,” In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Sendai, Japan (Sep 28–Oct. 2, 2004) pp. 1751–1755.Google Scholar
12.Matsuno, F. and Sato, H., “Trajectory Tracking Control of Snake Robots Based on Dynamic Model,” In: Proceedings of the IEEE International Conference on Robotics and Automation, Barcelona, Spain (April 18–22, 2005) pp. 3029–3034.Google Scholar
13.Transeth, A. A., van de Wouw, N., Pavlov, A., Hespanha, J. P. and Pettersen, K. Y., “Tracking Control for Snake Robot Joints,” In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, San Diego, CA, USA (Oct.–Nov. 2007) pp. 3539–3546.Google Scholar
14.Li, J. and Shan, J., “Passivity Control of Underactuated Snake-Like Robots,” In: Proceedings of the Seventh World Congress on Intelligent Control and Automation, Chongqing, China (June 2008) pp. 485–490.Google Scholar
15.Hu, D., Nirody, J., Scott, T. and Shelley, M., “The mechanics of slithering locomotion,” PNAS, 106, 10081–10085 (2009).CrossRefGoogle ScholarPubMed
16.Hatton, R. and Choset, H., “Approximating Displacement with the Body Velocity Integral,” In: Proceedings of the Robotics: Science and Systems, Seattle, Washington (2009).Google Scholar
17.Liljebäck, P., Pettersen, K. Y., Stavdahl, Ø. and Gravdahl, J. T., “Controllability and stability analysis of planar snake robot locomotion,” IEEE Trans. Autom. Control 56 (6), 1365–1380 (2011).CrossRefGoogle Scholar
18.Vela, P. A., Morgansen, K. A. and Burdick, J. W., “Underwater Locomotion from Oscillatory Shape Deformations,” In: Proceedings of the IEEE Conference on Decision and Control, vol. 2 (Dec. 2002) pp. 2074–2080.CrossRefGoogle Scholar
19.McIsaac, K. and Ostrowski, J., “Motion planning for anguilli form locomotion,” IEEE Trans. Robot. Autom. 19 (4), 637–652 (2003).CrossRefGoogle Scholar
20.Morgansen, K., Triplett, B. and Klein, D., “Geometric methods for modeling and control of free-swimming fin-actuated underwater vehicles,” IEEE Trans. Robotics 23 (6), 1184–1199 (Dec. 2007).CrossRefGoogle Scholar
21.Liljebäck, P., Pettersen, K. Y., Stavdahl, Ø. and Gravdahl, J. T., Snake Robots – Modelling, Mechatronics, and Control, Series: Advances in Industrial Control (Springer-Verlag, London, 2013).CrossRefGoogle Scholar
22.Vela, P. A., Morgansen, K. A. and Burdick, J. W., “Second Order Averaging Methods for Oscillatory Control of Underactuated Mechanical Systems,” In: Proceedings of the American Control Conference, vol. 6, Anchorage, Alaska, USA (2002) pp. 4672–4677.Google Scholar
23.Melli, J. B., Rowley, C. W. and Rufat, D. S., “Motion planning for an articulated body in a perfect planar fluid,” SIAM J. Appl. Dyn. Syst. 5 (4), 650–669 (2006).CrossRefGoogle Scholar
24.Yu, J., Wang, L., Shao, J. and Tan, M., “Control and coordination of multiple biomimetic robotic fish,” IEEE Trans. Control Syst. Technol. 15 (1), 176–183 (2007).CrossRefGoogle Scholar
25.Liljebäck, P., Pettersen, K. Y., Stavdahl, Ø. and Gravdahl, J. T., “A Simplified Model of Planar Snake Robot Locomotion,” In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Taipei, Taiwan (2010) pp. 2868–2875.Google Scholar
26.Liljebäck, P., Pettersen, K. Y., Stavdahl, Ø. and Gravdahl, J. T., “Stability Analysis of Snake Robot Locomotion Based on Averaging Theory,” In: Proceedings of the IEEE Conference on Decision and Control, Atlanta, GA, USA (2010) pp. 1977–1984.Google Scholar
27.Liljebäck, P., Pettersen, K. Y., Stavdahl, Ø. and Gravdahl, J. T., “Fundamental Properties of Snake Robot Locomotion,” In: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Taipei, Taiwan (2010) pp. 2876–2883.Google Scholar
28.Liljebäck, P., Pettersen, K. Y., Stavdahl, Ø. and Gravdahl, J. T., “Experimental Investigation of Fundamental Properties of Snake Robot Locomotion,” In: Proceedings of the IEEE International Conference Control, Automation, Robotics, and Vision (ICARCV), Singapore (2010) pp. 187–194; finalist for the Best Paper Award.Google Scholar
29.Khalil, H. K., Nonlinear Systems, 3rd ed. (Prentice Hall, Upper Saddle, NJ, 2002).Google Scholar
30.Sfakiotakis, M. and Tsakiris, D., “Biomimetic centering for undulatory robots,” Int. J. Robot. Res. 26, 1267–1282 (2007).CrossRefGoogle Scholar
31.Pettersen, K. Y. and Egeland, O., “Exponential Stabilization of an Underactuated Surface Vessel,” In: Proceedings of the 35th IEEE Conference on Decision and Control, vol. 1, Kobe, Japan (Dec. 1996) pp. 967–972.CrossRefGoogle Scholar
32.Brockett, R., “Asymptotic stability and feedback stabilization,” Differ. Geom. Control Theory 181–191, vol. 27 (1983).Google Scholar
33.Coron, J.-M. and Rosier, L., “A relation between continuous time-varying and discontinuous feedback stabilization,” J. Math. Syst. Estim. Control 4 (1), 67–84 (1994).Google Scholar
34.Nijmeijer, H. and Schaft, A. v. d., Nonlinear Dynamical Control Systems (Springer-Verlag, New York, 1990).CrossRefGoogle Scholar
35.Sussmann, H. J., “A general theorem on local controllability,” SIAM J. Control Optim. 25 (1), 158–194 (1987).CrossRefGoogle Scholar
36.Bianchini, R. M. and Stefani, G., “Graded approximations and controllability along a trajectory,” SIAM J. Control Optim. 28 (4), 903–924 (1990).CrossRefGoogle Scholar
37.Sanders, J. A., Verhulst, F. and Murdock, J., Averaging Methods in Nonlinear Dynamical Systems, 2nd ed., Series: Applied Mathematical Sciences, vol. 59 (Springer, London, 2007).Google Scholar
38.Liljebäck, P., Pettersen, K. Y. and Stavdahl, Ø., “A Snake Robot with a Contact Force Measurement System for Obstacle-aided Locomotion,” In: Proceedings of the IEEE International Conference on Robotics and Automation, Anchorage, AK, USA (2010) pp. 683–690.Google Scholar
39.Lochmatter, T., Roduit, P., Cianci, C., Correll, N., Jacot, J. and Martinoli, A., “Swistrack – a Flexible Open Source Tracking Software for Multi-agent Systems,” In: IEEE/RSJ International Conference on Intelligent Robots and Systems, Nice, France (2008) pp. 4004–4010.Google Scholar