# Locomotion Dynamics for Bio-inspired Robots with Soft Appendages: Application to Flapping Flight and Passive Swimming

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## Abstract

In animal locomotion, either in fish or flying insects, the use of flexible terminal organs or appendages greatly improves the performance of locomotion (thrust and lift). In this article, we propose a general unified framework for modeling and simulating the (bio-inspired) locomotion of robots using soft organs. The proposed approach is based on the model of Mobile Multibody Systems (MMS). The distributed flexibilities are modeled according to two major approaches: the Floating Frame Approach (FFA) and the Geometrically Exact Approach (GEA). Encompassing these two approaches in the Newton–Euler modeling formalism of robotics, this article proposes a unique modeling framework suited to the fast numerical integration of the dynamics of a MMS in both the FFA and the GEA. This general framework is applied on two illustrative examples drawn from bio-inspired locomotion: the passive swimming in von Karman Vortex Street, and the hovering flight with flexible flapping wings.

## Keywords

Soft robotics Locomotion Newton–Euler dynamics MAVs Fish-like robots## Mathematics Subject Classification

68T40## Notes

### Acknowledgments

The authors would like to thank all their colleagues and PHD students (whose names appear as coauthors of the referenced papers) who contributed over the last years to develop the specific cases of this general framework.

## Supplementary material

Supplementary material 1 (mp4 5181 KB)

## References

- Ahlborn, B.K., Blake, R.W., Megill, W.M.: Frequency tuning in animal locomotion. Zoology
**109**, 43–53 (2006)CrossRefGoogle Scholar - Albu-Schäffer, A., Eiberger, O., Grebenstein, M., Haddadin, S., Ott, C., Wimböck, T., Wolf, S., Hirzinger, G.: Soft robotics: from torque feedback-controlled lightweight robots to intrinsically compliant systems. IEEE Robot. Autom. Mag.
**15**(3), 20–30 (2008)CrossRefGoogle Scholar - Alexander, R.M.: Elastic Mechanisms in Animal Movement, 1st edn. Cambridge University Press, Cambridge (1988)Google Scholar
- Alexander, R.M.: Models and the scaling of energy costs for locomotion. J. Exp. Biol.
**208**, 1645–1652 (2005)CrossRefGoogle Scholar - Antman, S.S.: Nonlinear Problems of Elasticity (Mathematical Sciences vol 107), 2nd edn. Springer, New York (2005)Google Scholar
- Beal, D.N., Hover, F.S., Triantafyllou, M.S., Liao, J.C., Lauder, G.V.: Passive propulsion in vortex wakes. J. Fluid Mech.
**549**, 385–402 (2006)CrossRefGoogle Scholar - Bennet-Clark, H.C.: The energetics of the jump of the locust schistorcerca gregaria. J. Exp. Biol.
**63**, 53–83 (1975)Google Scholar - Bergou, A.J., Ristroph, L., Guckenheimer, J., Cohen, I., Wang, Z.J.: Turning maneuver in free flight. Phys. Rev. Lett.
**104**(14), 148,101 (2010)CrossRefGoogle Scholar - Boyer, F., Ali, S.: Recursive inverse dynamics of mobile multibody systems with joints and wheels. IEEE Trans. Robot.
**27**(2), 215–228 (2011). doi: 10.1109/TRO.2010.2103450 CrossRefGoogle Scholar - Boyer, F., Ali, S., Porez, M.: Macro-continuous dynamics for hyper-redundant robots: application to kinematic locomotion bio-inspired by elongated body animals. IEEE Trans. Robot.
**28**(2), 303–317 (2012). doi: 10.1109/TRO.2011.2171616 CrossRefGoogle Scholar - Boyer, F., Coiffet, P.: Generalization of Newton–Euler model for flexible manipulators. J. Robot. Syst.
**13**(1), 11–24 (1996)CrossRefMATHGoogle Scholar - Boyer, F., Glandais, N., Khalil, W.: Flexible multibody dynamics based on a non-linear Euler–Bernoulli kinematics. Int. J. Numer. Methods Eng.
**54**(1), 27–59 (2002)CrossRefMATHGoogle Scholar - Boyer, F., Khalil, W.: An efficient calculation of flexible manipulator inverse dynamics. Int. J. Robot. Res.
**17**(3), 282–293 (1998)CrossRefGoogle Scholar - Boyer, F., Porez, M.: Multibody system dynamics for bio-inspired locomotion: from geometric structures to computational aspects. Bioinspir. Biomim.
**10**(2), 025007 (2015)CrossRefGoogle Scholar - Boyer, F., Porez, M., Khalil, W.: Macro-continuous computed torque algorithm for a three-dimensional eel-like robot. IEEE Trans. Robot.
**22**(4), 763–775 (2006). doi: 10.1109/TRO.2006.875492 CrossRefGoogle Scholar - Boyer, F., Porez, M., Leroyer, A.: Poincaré–Cosserat equations for the Lighthill three-dimensional large amplitude elongated body theory: Application to robotics. J. Nonlinear Sci.
**20**, 47–79 (2010). doi: 10.1007/s00332-009-9050-5 CrossRefMATHGoogle Scholar - Boyer, F., Porez, M., Leroyer, A., Visonneau, M.: Fast dynamics of an eel-like robot—comparisons with Navier–Stokes simulations. IEEE Trans. Robot.
**24**(6), 1274–1288 (2008)CrossRefGoogle Scholar - Boyer, F., Primault, D.: Finite element of slender beams in finite transformations: a geometrically exact approach. Int. J. Numer. Methods Eng.
**59**(5), 669–702 (2004)MathSciNetCrossRefMATHGoogle Scholar - Canavin, J., Likins, P.: Floating reference frames for flexible spacecraft. J. Spacecr. Rockets
**14**(12), 724–732 (1977)CrossRefGoogle Scholar - Candelier, F., Boyer, F., Leroyer, A.: Three-dimensional extension of Lighthill’s large-amplitude elongated-body theory of fish locomotion. J. Fluid Mech.
**674**, 196–226 (2011)MathSciNetCrossRefMATHGoogle Scholar - Candelier, F., Porez, M., Boyer, F.: Note on the swimming of an elongated body in a non-uniform flow. J. Fluid Mech.
**716**, 616–637 (2013)MathSciNetCrossRefMATHGoogle Scholar - Damaren, C., Sharf, I.: Simulation of flexible-link manipulators with inertial and geometric nonlinearities. J. Dyn. Syst. Meas. Control
**117**(1), 74–87 (1995)CrossRefMATHGoogle Scholar - D’Eleuterio, G.M.T.: Dynamics of an elastic multibody chain: part C—recursive dynamics. Dyn. Stab. Syst.
**7**(2), 61–89 (1992)MATHGoogle Scholar - Dhatt, G., Lefrançois, E., Touzot, G.: Finite Element Method, 1st edn. Wiley, Hoboken (2012)Google Scholar
- Dickinson, M.H., Farley, C.T., Full, R.J., Koehl, M.A., Kram, R., Lehman, S.: How animals move: an integrative view. Science
**288**(5463), 100–106 (2000)CrossRefGoogle Scholar - Dickinson, M.H., Lehmann, F.O., Sane, S.P.: Wing rotation and the aerodynamic basis of insect flight. Science
**284**(5422), 1954–1960 (1999). http://www.sciencemag.org/content/284/5422/1954.abstract - Dombre, E., Khalil, W.: Modélisation et commande des robots. Hermes, Paris (1988)MATHGoogle Scholar
- Featherstone, R.: The calculation of robot dynamics using articulated-body inertias. Int. J. Robot. Res.
**2**(1), 13–30 (1983)CrossRefGoogle Scholar - Featherstone, R.: Robot Dynamics Algorithms. Springer, Boston (1987)Google Scholar
- Featherstone, R.: Rigid Body Dynamics Algorithms, 1st edn. Springer, New-York (2008)Google Scholar
- Full, R.J., Koditschek, D.E.: Templates and anchors: neuromechanical hypotheses of legged locomotion on land. J. Exp. Biol.
**202**, 3325–3332 (1999)Google Scholar - Hatton, R.L., Choset, H.: Geometric motion planning: the local connection, stokes’ theorem, and the importance of coordinate choice. Int. J. Robot. Res.
**30**(8), 988–1014 (2011). doi: 10.1177/0278364910394392. http://ijr.sagepub.com/content/30/8/988.abstract - Hughes, P.C., Sincarsin, G.B.: Dynamics of elastic multibody chains: part B—global dynamics. Dyn. Stab. Syst.
**4**(3–4), 227–243 (1989)MATHGoogle Scholar - Kelly, S.D., Murray, R.M.: Geometric phases and robotic locomotion. J. Robot. Syst.
**12**(6), 417–431 (1995)CrossRefMATHGoogle Scholar - Khalil, W., Boyer, F., Morsli, F.: General dynamic modeling of floating base tree structure robots with flexible joints and links. IEEE Transactions on Man Systems and Cybernetics (2016)Google Scholar
- Khalil, W., Gallot, G., Boyer, F.: Dynamic modeling and simulation of a 3-D serial eel like robot. IEEE Trans. Syst. Man Cybern. Part C Appl. Rev.
**37**(6), 1259–1268 (2007)CrossRefGoogle Scholar - Lighthill, M.J.: Note on the swimming of slender fish. J. Fluid Mech.
**9**(2), 305–317 (1960)MathSciNetCrossRefGoogle Scholar - Lighthill, M.J.: Large-amplitude elongated-body theory of fish locomotion. Proc. R. Soc. Lond. Ser. B Biol. Sci.
**179**(1055), 125–138 (1971)CrossRefGoogle Scholar - Liu, H.: Integrated modeling of insect flight: from morphology, kinematics to aerodynamics. J. Comput. Phys.
**228**(2), 439–459 (2009)MathSciNetCrossRefMATHGoogle Scholar - Long, J.H., Nipper, K.S.: The importance of body stiffness in undulatory propulsion. Am. Zool.
**36**(6), 678–694 (1996)CrossRefGoogle Scholar - Marsden, J.E., Ostrowski, J.: Symmetries in Motion: Geometric Foundations of Motion Control. Motion, Control, and Geometry: Proceedings of a Symposium. National Academies Press, Washington, DC (1998)Google Scholar
- McMichael, J.M., Francis, M.S.: Micro air vehicles\(---\)toward a new dimension in flight. Technical report DARPA (1997). http://www.fas.org/irp/program/collect/docs/mavauvsi.htm
- Meirovitch, L.: Methods of Analytical Dynamics. McGraw-Hill, New York (1970)MATHGoogle Scholar
- Murray, R.M., Sastry, S.S., Zexiang, L.: A Mathematical Introduction to Robotic Manipulation, 1st edn. CRC Press Inc., Boca Raton (1994)MATHGoogle Scholar
- Nakata, T., Liu, H.: A fluid-structure interaction model of insect flight with flexible wings. J. Comput. Phys.
**231**(4), 1822–1847 (2012)MathSciNetCrossRefMATHGoogle Scholar - Ostrowski, J., Burdick, J.: Gait kinematics for a serpentine robot. In: Proceedings of IEEE International Conference on Robotics and Automation, pp. 1294–1299 (1996)Google Scholar
- Pfeifer, R., Lungarella, M., Iida, F.: Self-organization, embodiment, and biologically inspired robotics. Science
**318**(5853), 1088–1093 (2007)CrossRefGoogle Scholar - Porez, M., Boyer, F., Belkhiri, A.: A hybrid dynamic model for bio-inspired robots with soft appendages - application to a bio-inspired flexible flapping-wing micro air vehicle. In: Proceeding of IEEE International Conference on Robotics and Automation (ICRA’2014), pp. 3556–3563 (2014)Google Scholar
- Porez, M., Boyer, F., Ijspeert, A.: Improved lighthill fish swimming model for bio-inspired robots: modeling, computational aspects and experimental comparisons. Int. J. Robot. Res.
**33**(10), 1322–1341 (2014)CrossRefGoogle Scholar - Reissner, E.: On a one-dimensional large displacement finite-strain beam theory. Stud. Appl. Math.
**52**(2), 87–95 (1973)CrossRefMATHGoogle Scholar - Roberts, T.J., Azizi, E.: Flexible mechanisms: the diverse roles of biological springs in vertebrate movement. J. Exp. Biol.
**214**(3), 353–361 (2011)CrossRefGoogle Scholar - Shabana, A.A.: Dynamics of Multibody Systems. Wiley, New York (1989)MATHGoogle Scholar
- Simo, J.C., Vu-Quoc, L.: On the dynamics of flexible beams under large overall motions—the plane case: part I. J. Appl. Mech.
**53**(4), 849–854 (1986)CrossRefMATHGoogle Scholar - Simo, J.C., Vu-Quoc, L.: On the dynamics in space of rods undergoing large motions—a geometrically exact approach. Comp. Methods Appl. Mech. Eng.
**66**(2), 125–161 (1988). doi: 10.1016/0045-7825(88)90073-4. http://www.sciencedirect.com/science/article/pii/0045782588900734 - Tallapragada, P.: A swimming robot with an internal rotor as a nonholonomic system. In: Proceeding of American Control Conference (ACC’2015), pp. 657–662 (2015)Google Scholar
- Walker, J.A.: Rotational lift: something different or more of the same? J. Exp. Biol.
**205**, 3783–3792 (2002)Google Scholar - Walker, M.W., Luh, J.Y.S., Paul, R.C.P.: On-line computational scheme for mechanical manipulator. Transaction ASME. J. Dyn. Syst. Meas. Control
**102**(2), 69–76 (1980)CrossRefGoogle Scholar - Whitney, J., Wood, R.: Aeromechanics of passive rotation in flapping flight. J. Fluid Mech.
**660**, 197–220 (2010)MathSciNetCrossRefMATHGoogle Scholar - Wood, R.J.: Design, fabrication, and analysis of a 3DOF, 3cm flapping-wing MAV. In: International conference on intelligent robots and systems, 2007. IROS 2007. IEEE/RSJ, pp. 1576–1581 (2007)Google Scholar
- y Alvarado, P.V., Youcef-Toumi, K.: Design of machines with compliant bodies for biomimetic locomotion in liquid environments. J. Dyn. Syst. Meas. Control
**128**(1), 3–13 (2006)CrossRefGoogle Scholar