Biological Cybernetics

, Volume 105, Issue 5–6, pp 355–370 | Cite as

Towards a theoretical foundation for morphological computation with compliant bodies

  • Helmut HauserEmail author
  • Auke J. Ijspeert
  • Rudolf M. Füchslin
  • Rolf Pfeifer
  • Wolfgang Maass
Open Access
Original Paper


The control of compliant robots is, due to their often nonlinear and complex dynamics, inherently difficult. The vision of morphological computation proposes to view these aspects not only as problems, but rather also as parts of the solution. Non-rigid body parts are not seen anymore as imperfect realizations of rigid body parts, but rather as potential computational resources. The applicability of this vision has already been demonstrated for a variety of complex robot control problems. Nevertheless, a theoretical basis for understanding the capabilities and limitations of morphological computation has been missing so far. We present a model for morphological computation with compliant bodies, where a precise mathematical characterization of the potential computational contribution of a complex physical body is feasible. The theory suggests that complexity and nonlinearity, typically unwanted properties of robots, are desired features in order to provide computational power. We demonstrate that simple generic models of physical bodies, based on mass-spring systems, can be used to implement complex nonlinear operators. By adding a simple readout (which is static and linear) to the morphology such devices are able to emulate complex mappings of input to output streams in continuous time. Hence, by outsourcing parts of the computation to the physical body, the difficult problem of learning to control a complex body, could be reduced to a simple and perspicuous learning task, which can not get stuck in local minima of an error function.


Morphological computation Embodiment Analog computation Volterra series Nonlinear mass-spring systems 



Written under partial support by the European Union projects # FP7-216593 (SECO), # 216886 (PASCAL2), # 248311 (AMARSi), and by the Austrian Science Fund FWF, project # P17229- N04. We also want to thank the anonymous reviewers for their very helpful suggestions and comments.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Supplementary material

422_2012_471_MOESM1_ESM.pdf (294 kb)
ESM 1 (PDF 295 kb)


  1. Atiya AF, Parlos AG (2000) New results on recurrent network training: unifying the algorithms and accelerating convergence. IEEE Trans Neural Networks 11(3): 697–709CrossRefGoogle Scholar
  2. Bartlett PL, Maass W (2003) Vapnik-Chervonenkis dimension of neural nets. In: Arbib MA (eds) The handbook of brain theory and neural networks, 2nd edn. MIT Press, Cambridge, pp 1188–1192Google Scholar
  3. Boyd S (1985) Volterra series: engineering fundamentals. PhD thesis, UC BerkeleyGoogle Scholar
  4. Boyd S, Chua L (1985) Fading memory and the problem of approximating nonlinear operators with volterra series. IEEE Trans Circuits Syst 32(11): 1150–1161CrossRefGoogle Scholar
  5. Collins S, Ruina A, Tedrake R, Wisse M (2005) Efficient bipedal robots based on passive-dynamic walkers. Science 307: 1082–1085PubMedCrossRefGoogle Scholar
  6. Ferris DP, Louie M, Farley CT (1998) Running in the real world: adjusting leg stiffness for different surfaces. Proc Biol Sci 265(1400): 989–994PubMedCrossRefGoogle Scholar
  7. Hochreiter S, Schmidhuber J (1997) Long short-term memory. Neural Comput 9:1735–1780. ISSN 0899-7667Google Scholar
  8. Hornik K, Stinchcombe M, White H (1989) Multilayer feedforward networks are universal approximators. Neural Networks 2(5): 359–366CrossRefGoogle Scholar
  9. Iida F, Pfeifer R (2006) Sensing through body dynamics. Robot Auton Syst 54(8): 631–640CrossRefGoogle Scholar
  10. Khalil HK (2002) Nonlinear Systems, 3rd edn, Prentice Hall, Upper Saddle RiverGoogle Scholar
  11. Legenstein R, Chase SA, Schwartz AB, Maass W (2010) Functional network reorganization in motor cortex can be explained by reward-modulated Hebbian learning. In: Proceedings of NIPS 2009: advances in neural information processing systems, vol 22. MIT Press, pp 1105–1113.Google Scholar
  12. Maass W, Sontag ED (2000) Neural systems as nonlinear filters.. Neural Comput 12(8): 1743–1772PubMedCrossRefGoogle Scholar
  13. Maass W, Natschlaeger T, Markram H (2002) Real-time computing without stable states: A new framework for neural computation based on perturbations. Neural Comput 14(11): 2531–2560PubMedCrossRefGoogle Scholar
  14. McGeer T (1990) Passive dynamic walking. Int J Robot Res 9(2):62–82 ISSN 0278-3649Google Scholar
  15. Palm WJ III (1999) Modeling, analysis, and control of dynamic systems, 2nd edn. Wiley, New York. ISBN 0-471-07370-9Google Scholar
  16. Paul C (2006) Morphological computation: a basis for the analysis of morphology and control requirements. Robot Auton Syst 54(8): 619–630CrossRefGoogle Scholar
  17. Paul C, Valero-Cuevas FJ, Lipson H (2006) Design and control of tensegrity robots for locomotion. IEEE Trans Robot 22(5):944–957. ISSN 1552-3098. doi: 10.1109/TRO.2006.878980 Google Scholar
  18. Pfeifer R, Bongard JC (2007) How the body shapes the way we think. MIT Press, Cambridge. ISBN 0262162393Google Scholar
  19. Pfeifer R, Lungarella M, Iida F (2007) Self-organization, embodiment, and biologically inspired robotics. Science 318: 1088–1093PubMedCrossRefGoogle Scholar
  20. Shim Y, Husbands P (2007) Feathered flyer: integrating morphological computation and sensory reflexes into a physically simulated flapping-wing robot for robust flight manoeuvre. In: Almeida e Costa F et al (eds) ECAL. Springer, Berlin, pp 756–765Google Scholar
  21. Slotine J-J E, Li J-J W (1991) Applied nonlinear control, 1st edn. Prentice Hall, Englewood CliffsGoogle Scholar
  22. Tedrake R, Zhang TW, Seung HS (2005) Learning to walk in 20 minutes. In: Proceedings of the fourteenth yale workshop on adaptive and learning systems. Yale University, New Haven, CT, 2005Google Scholar
  23. Vapnik VN (1998) Statistical learning theory. Wiley, New YorkGoogle Scholar
  24. Wiskott L, Sejnowski TJ (2002) Slow feature analysis: unsupervised learning of invariances. Neural Comput 14(4):715–770. ISSN 0899-7667Google Scholar
  25. Wisse M, Van Frankenhuyzen J (2003) Design and construction of MIKE; a 2D autonomous biped based on passive dynamic walking. In: Proceedings of international symposium of adaptive motion and animals and machines (AMAM03)Google Scholar
  26. Wood RJ (2007) Design, fabrication, and analysis of a 3DOF, 3cm flapping-wing MAV. IEEE/RSJ international conference on intelligent robots and systems, 2007. IROS 2007, 29 Oct–Nov 2 2007, pp 1576–1581. doi: 10.1109/IROS.2007.4399495
  27. Ziegler M, Iida F, Pfeifer R (2006) “Cheap” underwater locomotion: roles of morphological properties and behavioural diversity. In: CLAWARGoogle Scholar

Copyright information

© The Author(s) 2012

Open AccessThis is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  • Helmut Hauser
    • 1
    Email author
  • Auke J. Ijspeert
    • 3
  • Rudolf M. Füchslin
    • 1
    • 2
  • Rolf Pfeifer
    • 1
  • Wolfgang Maass
    • 4
  1. 1.Artificial Intelligence Laboratory, Department of InformaticsUniversity of ZurichZurichSwitzerland
  2. 2.ZHAW Zurich University of Applied Sciences, Center for Applied Mathematics and Physics ZAMPWinterthurSwitzerland
  3. 3.École Polytechnique Fédérale de Lausanne, Biorobotics Laboratory BIOROBLausanneSwitzerland
  4. 4.Graz University of Technology, Institute for Theoretical Computer ScienceGrazAustria

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