Biological Cybernetics

, Volume 112, Issue 4, pp 387–401 | Cite as

The effects of feedback on stability and maneuverability of a phase-reduced model for cockroach locomotion

  • J. L. Proctor
  • P. HolmesEmail author
Original Article


In previous work, we built a neuromechanical model for insect locomotion in the horizontal plane, containing a central pattern generator, motoneurons, muscles actuating jointed legs, and rudimentary proprioceptive feedback. This was subsequently simplified to a set of 24 phase oscillators describing motoneuronal activation of agonist–antagonist muscle pairs, which facilitates analyses and enables simulations over multi-dimensional parameter spaces. Here we use the phase-reduced model to study dynamics and stability over the typical speed range of the cockroach Blaberus discoidalis, the effects of feedback on response to perturbations, strategies for turning, and a trade-off between stability and maneuverability. We also compare model behavior with experiments on lateral perturbations, changes in body mass and moment of inertia, and climbing dynamics, and we present a simple control strategy for steering using exteroceptive feedback.


Exteroception Feedback control Hybrid systems Neuromechanics Proprioception Stability–maneuverability trade-off 



This work was partially supported by NSF EF-0425878 (Frontiers in Biological Research), NSF DMS-1430077 (CRCNS U.S.-German Collaboration) and Princeton’s J. Insley Blair Pyne Fund. We thank the anonymous reviewers for their useful suggestions and for helping us to correct several errors.


  1. Ahn A, Full R (2002) A motor and a brake: two leg extensor muscles acting at the same joint manage energy differently in a running insect. J Exp Biol 205:379–389PubMedGoogle Scholar
  2. Ahn A, Meijer K, Full R (2006) In situ muscle power differs without varying in vitro mechanical properties in two insect leg muscles innervated by the same motor neuron. J Exp Biol 209:3370–3382CrossRefPubMedGoogle Scholar
  3. Altendorfer R, Moore N, Komsuoglu H, Buehler M, Brown HB Jr, McMordie D, Saranli U, Full R, Koditschek D (2001) RHex: a biologically inspired hexapod runner. Auton Robots 11:207–213CrossRefGoogle Scholar
  4. Brown I, Scott S, Loeb G (1995) Preflexes—-programmable high-gain zero-delay intrinsic responses of perturbed musculoskeletal systems. Soc Neurosci Abstr 21(562):9Google Scholar
  5. Couzin-Fuchs E, Kiemel T, Gal O, Holmes P, Ayali A (2015) Intersegmental coupling and recovery from perturbations in freely-running cockroaches. J Exp Biol 218:285–297CrossRefPubMedPubMedCentralGoogle Scholar
  6. Cowan N, Lee J, Full R (2006) Task-level control of rapid wall following in the american cockroach. J Exp Biol 209:1617–1629CrossRefPubMedGoogle Scholar
  7. David I, Holmes P, Ayali A (2016) Endogenous rhythm and pattern generating circuit interactions in cockroach motor centers. Biol Open 5:1229–1240CrossRefPubMedPubMedCentralGoogle Scholar
  8. Delcomyn F (1980) Neural basis of rhythmic behaviors in animals. Science 210:492–498CrossRefPubMedGoogle Scholar
  9. Delcomyn F (2004) Insect walking and robotics. Annu Rev Entomol 149:51–70CrossRefGoogle Scholar
  10. Electronic Physics Auxiliary Publication Service E (2009) See document no. e-chaoeh-19-005992 for parameter values and code documentation. For more information on EPAPS, see
  11. Fuchs E, Holmes P, Kiemel T, Ayali A (2011) Intersegmental coordination of cockroach locomotion: adaptive control of centrally coupled pattern generator circuits. Front Neural Circuits 4:125PubMedPubMedCentralGoogle Scholar
  12. Fuchs E, Holmes P, David I, Ayali A (2012) Proprioceptive feedback reinforces centrally-generated stepping patterns in the cockroach. J Exp Biol 215:1884–1891CrossRefPubMedGoogle Scholar
  13. Full R, Koditschek D (1999) Templates and anchors: neuromechanical hypothesis of legged locomotion on land. J Exp Biol 202:3325–3332PubMedGoogle Scholar
  14. Full R, Tu M (1991) Mechanics of a rapid running insect: two-, four- and six-legged locomotion. J Exp Biol 156:215–231PubMedGoogle Scholar
  15. Full R, Kubow T, Schmitt J, Holmes P, Koditschek D (2002) Quantifying dynamic stability and maneuverability in legged locomotion. Integr Comp Biol 42:149–157CrossRefPubMedGoogle Scholar
  16. Ghigliazza R, Holmes P (2004a) A minimal model of a central pattern generator and motoneurons for insect locomotion. SIAM J Appl Dyn Syst 3(4):671–700CrossRefGoogle Scholar
  17. Ghigliazza R, Holmes P (2004b) Minimal models of bursting neurons: how multiple currents, conductances and timescales affect bifurcation diagrams. SIAM J Appl Dyn Syst 3(4):636–670CrossRefGoogle Scholar
  18. Goldman D, Chen T, Dudek D, Full R (2006) Dynamics of rapid vertical climbing in a cockroach reveals a template. J Exp Biol 209:2990–3000CrossRefPubMedGoogle Scholar
  19. Guckenheimer J, Holmes P (2002) Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, 6th edn. Springer, BerlinGoogle Scholar
  20. Guckenheimer J, Johnson S (1995) Planar hybrid systems. In: Lecture notes in computer science No. 999, Springer, Berlin, pp 202–225Google Scholar
  21. Hill A (1938) The heat of shortening and the dynamic constants of muscle. Proc R Soc Lond B 126:136–195CrossRefGoogle Scholar
  22. Holmes P, Full R, Koditschek D, Guckenheimer J (2006) The dynamics of legged locomotion: models, analyses and challenges. SIAM Rev 48(2):207–304CrossRefGoogle Scholar
  23. Hoover A, Burden S, Fu X, Sastry S, Fearing R (2010) Bio-inspired design and dynamic maneuverabiliity of a actuated six-legged robot. In: Proceedings of IEEE international conference on biomedical robotics and biomechatronics (BIOROB), pp 869–876Google Scholar
  24. Jayaram K, Mongeau JM, McRae B, Full R (2010) High-speed horizontal to vertical transitions in running cockroaches reveals a principle of robustness. In: Society for Integrative and Comparative Biology. abstractdetails.php3?id=1109
  25. Jindrich D, Full R (1999) Many-legged maneuverability: dynamics of turning in hexapods. J Exp Biol 202:1603–1623PubMedGoogle Scholar
  26. Jindrich D, Full R (2002) Dynamic stabilization of rapid hexapedal locomotion. J Exp Biol 205:2803–2823PubMedGoogle Scholar
  27. Kram R, Wong B, Full R (1997) Three-dimensional kinematics and limb kinetic energy of running cockroaches. J Exp Biol 200:1919–1929PubMedGoogle Scholar
  28. Kubow T, Full R (1999) The role of the mechanical system in control: a hypothesis of self stabilization in hexapedal runners. Philos Trans R Soc Lond B 354:849–861CrossRefGoogle Scholar
  29. Kukillaya R, Holmes P (2007) A hexapedal jointed-leg model for insect locomotion in the horizontal plane. Biol Cybern 97:379–395CrossRefPubMedGoogle Scholar
  30. Kukillaya R, Holmes P (2009) A model for insect locomotion in the horizontal plane: feedforward activation of fast muscles, stability, and robustness. J Theor Biol 261(2):210–226CrossRefPubMedGoogle Scholar
  31. Kukillaya R, Proctor J, Holmes P (2009) Neuro-mechanical models for insect locomotion: stability, maneuverability, and proprioceptive feedback. CHAOS Interdiscip J Nonlinear Sci 19(2):026107CrossRefGoogle Scholar
  32. Lee J, Sponberg S, Loh O, Lamperski A, Full R, Cowan N (2008) Templates and anchors for antenna-based wall following in cockroaches. IEEE Trans Robot 24(1):130–143CrossRefGoogle Scholar
  33. Mongeau JM, Alexander T, Full R (2012) Neuromechanical feedback during dynamic recovery after a lateral perturbation in rapid running cockroaches. In: Society for Integrative and Comparative Biology. ?id=555
  34. Moore T, Revzen S, Burden S, Full R (2010) Adding inertia and mass to test stability predictions in rapid running insects. In: Society for Integrative and Comparative Biology. 3?id=1290
  35. Pearson K (1972) Central programming and reflex control of walking in the cockroach. J Exp Biol 56:173–193Google Scholar
  36. Pearson K, Iles J (1970) Discharge patterns of coxal levator and depressor motoneurones in the cockroach Periplaneta americana. J Exp Biol 52:139–165PubMedGoogle Scholar
  37. Pearson K, Iles J (1971) Innervation of the coxal depressor muscles in the cockroach Periplaneta americana. J Exp Biol 54:215–232PubMedGoogle Scholar
  38. Pearson K, Iles J (1973) Nervous mechanisms underlying intersegmental co-ordination of leg movements during walking in the cockroach. J Exp Biol 58:725–744Google Scholar
  39. Proctor J, Holmes P (2008) Steering by transient destabilization in piecewise-holonomic models of legged locomotion. Regul Chaotic Dyn 13(4):267–282CrossRefGoogle Scholar
  40. Proctor J, Holmes P (2010) Reflexes and preflexes: on the role of sensory feedback on rhythmic patterns in legged locomotion. Biol Cybern 2:513–531CrossRefGoogle Scholar
  41. Proctor J, Kukillaya R, Holmes P (2010) A phase-reduced neuro-mechanical model for insect locomotion: feed-forward stability and proprioceptive feedback. Philos Trans R Soc A 368:5087–5104CrossRefGoogle Scholar
  42. Revzen S, Burden S, Moore T, Mongeau JM, Full R (2013) Instantaneous kinematic phase reflects neuromechanical response to lateral perturbations of running cockroaches. Biol Cybern 107:179–200CrossRefPubMedGoogle Scholar
  43. Schmitt J, Bonnono S (2009) Dynamics and stability of lateral plane locomotion on inclines. J Theor Biol 261:598–609CrossRefPubMedGoogle Scholar
  44. Schmitt J, Holmes P (2000) Mechanical models for insect locomotion: dynamics and stability in the horizontal plane—I. Theory Biol Cybern 83(6):501–515CrossRefPubMedGoogle Scholar
  45. Schmitt J, Holmes P (2003) Mechanical models for insect locomotion: active muscles and energy losses. Biol Cybern 89(1):43–55PubMedGoogle Scholar
  46. Schmitt J, Garcia M, Razo RC, Holmes P, Full RJ (2002) Dynamics and stability of legged locomotion in the horizontal plane: a test case using insects. Biol Cybern 86(5):343–353CrossRefPubMedGoogle Scholar
  47. Sefati S, Neveln I, Roth E, Mitchell T, Snyder J, MacIver M, Fortune E, Cowan N (2013) Mutually opposing forces during locomotion can eliminate the tradeoff between maneuverability and stability. Proc Natl Acad Sci 110(47):18798–18803CrossRefPubMedGoogle Scholar
  48. Seipel J, Holmes P, Full R (2004) Dynamics and stability of insect locomotion: a hexapedal model for horizontal plane motion. Biol Cybern 91(2):76–90CrossRefPubMedGoogle Scholar
  49. Sponberg S, Full R (2008) Neuromechanical response of musculo-skeletal structures in cockroaches during rapid running on rough terrain. J Exp Biol 211:433–446CrossRefPubMedGoogle Scholar
  50. Sponberg S, Spence A, Mullens C, Full R (2011) A single muscle’s multifunctional control potential of body dynamics for postural control and running. Philos Trans Roy Soc B 366:1592–1605CrossRefGoogle Scholar
  51. Ting L, Blickhan R, Full R (1994) Dynamic and static stability in hexapedal runners. J Exp Biol 197:251–269PubMedGoogle Scholar
  52. Zill S, Moran D (1981a) The exoskeleton and insect proprioception I. Responses of tibial campaniform sensilla to external and muscle-generated force in the American cockroach Periplaneta americana. J Exp Biol 91:1–24Google Scholar
  53. Zill S, Moran D (1981b) The exoskeleton and insect proprioception III. Activity of tibial campaniform sensilla during walking in the American cockroach Periplaneta americana. J Exp Biol 94:57–75Google Scholar
  54. Zill S, Moran D, Varela F (1981) The exoskeleton and insect proprioception II. Reflex effects of tibial campaniform sensilla in the American cockroach Periplaneta americana. J Exp Biol 94:43–55Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute for Disease ModelingBellevueUSA
  2. 2.Department of Mechanical and Aerospace Engineering, Program in Applied and Computational Mathematics and Princeton Neuroscience InstitutePrinceton UniversityPrincetonUSA

Personalised recommendations