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Predicted Step Viability: a stability criterion for biped gait

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Abstract

The formulation of a stability criterion is crucial to perform safe, versatile and efficient biped gaits for robots. We here present a stability criterion for non-cyclic gait synthesis, the Predicted Step Viability, inspired by human gait and N-Step Capturability. The Predicted Step Viability defines the constraints of the current step such that future steps will be able to guarantee convergence to a stable point in finite time. In this way, it is based on the prediction of future viable steps to ensure stability. The criterion was implemented using multiphase trajectory optimization on two biped models, the Compass Gait and the five-link model RABBIT. The Compass Gait was simulated with different model parameters and gait patterns including a random non-periodic one. The five-link model was tested with linear and random reference gait patterns. The Predicted Step Viability criterion successfully generated stable non-periodic gaits under a variety of conditions. Moreover, it is possible to prescribe any gait pattern completely uncoupled from the stability criterion. If it were impossible to follow the prescribed pattern without falling, the controller would give it up, maintaining stability.

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References

  1. Becerra VM (2010) Solving complex optimal control problems at no cost with psopt. In: 2010 IEEE international symposium on computer-aided control system design, IEEE, pp 1391–1396

  2. Chevallereau C, Abba G, Aoustin Y, Plestan F, Westervelt ER, Canudas-De-Wit C, Grizzle JW (2003) Rabbit: a testbed for advanced control theory. IEEE Control Syst 23(5):57–79. https://doi.org/10.1109/MCS.2003.1234651

    Article  Google Scholar 

  3. Collins S, Ruina A, Tedrake R, Wisse M (2005) Efficient bipedal robots based on passive-dynamic walkers. Science 307(5712):1082–1085. https://doi.org/10.1126/science.1107799

    Article  Google Scholar 

  4. Collins SH, Ruina A (2005) A bipedal walking robot with efficient and human-like gait. In: Proceedings of the 2005 IEEE international conference on robotics and automation, IEEE, pp 1983–1988

  5. de Boer T, Wisse M, Van der Helm F (2010) Mechanical analysis of the preferred strategy selection in human stumble recovery. J Biomech Eng 132(7):071012

    Article  Google Scholar 

  6. Duysens J, Forner-Cordero A (2018) Walking with perturbations: a guide for biped humans and robots. Bioinspir Biomim 13(6):061001

    Article  Google Scholar 

  7. Feng S, Whitman E, Xinjilefu X, Atkeson CG (2014) Optimization based full body control for the atlas robot. In: 2014 IEEE-RAS international conference on humanoid robots, IEEE, pp 120–127

  8. Forner Cordero A, Koopman H, van der Helm F (2003) Multiple-step strategies to recover from stumbling perturbations. Gait Posture 18(1):47–59

    Article  Google Scholar 

  9. Forner Cordero A, Koopman H, van der Helm FC (2004) Mechanical model of the recovery from stumbling. Biol Cybern 91(4):212–220

    Article  Google Scholar 

  10. Goswami A, Espiau B, Keramane A (1997) Limit cycles in a passive compass gaitbiped and passivity-mimicking control laws. Auton Robots 4(3):273–286

    Article  Google Scholar 

  11. Hobbelen D, Wisse M (2008) Controlling the walking speed in limit cycle walking. Int J Robot Res 27(9):989–1005

    Article  Google Scholar 

  12. Hobbelen DGE, Wisse M (2007) A disturbance rejection measure for limit cycle walkers: the gait sensitivity norm. IEEE Trans Robot 23(6):1213–1224. https://doi.org/10.1109/TRO.2007.904908

    Article  Google Scholar 

  13. Hof A, Gazendam M, Sinke W (2005) The condition for dynamic stability. J Biomech 38(1):1–8

    Article  Google Scholar 

  14. Hof AL (2008) The ‘extrapolated center of mass’ concept suggests a simple control of balance in walking. Hum Mov Sci 27(1):112–125. https://doi.org/10.1016/j.humov.2007.08.003

    Article  Google Scholar 

  15. Kajita S, Kanehiro F, Kaneko K, Fujiwara K, Yokoi K, Hirukawa H (2002) A realtime pattern generator for biped walking. In: Proceedings of the ICRA’02. IEEE international conference on robotics and automation, 2002, IEEE, vol 1, pp 31–37

  16. Kajita S, Kanehiro F, Kaneko K, Fujiwara K, Harada K, Yokoi K, Hirukawa H (2003) Biped walking pattern generation by using preview control of zero-moment point. In: Proceedings of the ICRA’03. IEEE international conference on robotics and automation, 2003, IEEE, vol 2, pp 1620–1626

  17. Koenemann J, Del Prete A, Tassa Y, Todorov E, Stasse O, Bennewitz M, Mansard N (2015) Whole-body model-predictive control applied to the hrp-2 humanoid. In: 2015 IEEE/RSJ international conference on intelligent robots and systems (IROS), IEEE, pp 3346–3351

  18. Koolen T, De Boer T, Rebula J, Goswami A, Pratt J (2012) Capturability-based analysis and control of legged locomotion, part 1: theory and application to three new simple gait models. Int J Robot Res 31(9):1094–1113

    Article  Google Scholar 

  19. Manchester IR, Umenberger J (2013) Real-time planning with primitives for dynamic walking over uneven terrain. CoRR arXiv:abs/1310.7062

  20. Mathworks (2016) 9.0.0.341360 (R2016a). The MathWorks Inc., Natick

  21. McGeer T (1990) Passive dynamic walking. Int J Robot Res 9(2):62–82

    Article  Google Scholar 

  22. Mordatch I, Lowrey K, Andrew G, Popovic Z, Todorov EV (2015) Interactive control of diverse complex characters with neural networks. In: Advances in neural information processing systems, pp 3132–3140

  23. Patla A, Rietdyk S (1993) Visual control of limb trajectory over obstacles during locomotion: effect of obstacle height and width. Gait Posture 1(1):45–60

    Article  Google Scholar 

  24. Posa M, Cantu C, Tedrake R (2014) A direct method for trajectory optimization of rigid bodies through contact. Int J Robot Res 33(1):69–81. https://doi.org/10.1177/0278364913506757

    Article  Google Scholar 

  25. Powell MJ, Hereid A, Ames AD (2013) Speed regulation in 3d robotic walking through motion transitions between human-inspired partial hybrid zero dynamics. In: 2013 IEEE international conference on robotics and automation, pp 4803–4810. https://doi.org/10.1109/ICRA.2013.6631262

  26. Powell MJ, Cousineau EA, Ames AD (2015) Model predictive control of underactuated bipedal robotic walking. In: 2015 IEEE international conference on robotics and automation (ICRA), IEEE, pp 5121–5126

  27. Pratt J, Tedrake R (2006) Velocity-based stability margins for fast bipedal walking. Springer, Berlin, pp 299–324. https://doi.org/10.1007/978-3-540-36119-0_14

    Book  MATH  Google Scholar 

  28. Pratt J, Carff J, Drakunov S, Goswami A (2006) Capture point: a step toward humanoid push recovery. In: 2006 6th IEEE-RAS international conference on humanoid robots, pp 200–207. https://doi.org/10.1109/ICHR.2006.321385

  29. Rossi LF, Rodrigues ST, Forner-Cordero A (2014) Do humans walk like robots when crossing an obstacle without visual information? In: 5th IEEE RAS/EMBS international conference on biomedical robotics and biomechatronics, pp 216–220. https://doi.org/10.1109/BIOROB.2014.6913779

  30. Schultz G, Mombaur K (2010) Modeling and optimal control of human-like running. IEEE/ASME Trans Mechatron 15(5):783–792. https://doi.org/10.1109/TMECH.2009.2035112

    Article  Google Scholar 

  31. Sherman MA, Seth A, Delp SL (2011) Simbody: multibody dynamics for biomedical research. Procedia IUTAM 2:241–261. https://doi.org/10.1016/j.piutam.2011.04.023

    Article  Google Scholar 

  32. Sreenath K, won Park H, Grizzle JW (2012) Design and experimental implementation of a compliant hybrid zero dynamics controller with active force control for running on. In: MABEL, proceedings of the 2012 IEEE international conference on robotics and automation, Saint Paul, MN, pp 51–56

  33. Stephens B (2011) Push recovery control for force-controlled humanoid robots. Ph.D. thesis, Carnegie Mellon University Pittsburgh, Pennsylvania, USA

  34. Tassa Y, Erez T, Todorov E (2012) Synthesis and stabilization of complex behaviors through online trajectory optimization. In: 2012 IEEE/RSJ international conference on intelligent robots and systems, pp 4906–4913. https://doi.org/10.1109/IROS.2012.6386025

  35. Tedrake R, Kuindersma S, Deits R, Miura K (2015) A closed-form solution for real-time zmp gait generation and feedback stabilization. In: 2015 IEEE-RAS 15th international conference on humanoid robots (humanoids), IEEE, pp 936–940

  36. Vukobratovic M, Borovac B (2004) Zero-moment point—thirty five years of its life. Int J Humanoid Robot 1(1):157–173

    Article  Google Scholar 

  37. Vukobratovic M, Juricic D (1969) Contribution to the synthesis of biped gait. IEEE Trans Biomed Eng BME 16(1):1–6. https://doi.org/10.1109/TBME.1969.4502596

    Article  Google Scholar 

  38. Vukobratovic M, Stepanenko J (1972) On the stability of anthropomorphic systems. Math Biosci 15(1–2):1–37

    Article  Google Scholar 

  39. Westervelt ER, Grizzle JW, Koditschek DE (2003) Hybrid zero dynamics of planar biped walkers. IEEE Trans Autom Control 48(1):42–56. https://doi.org/10.1109/TAC.2002.806653

    Article  MathSciNet  MATH  Google Scholar 

  40. Wieber PB (2002) On the stability of walking systems. In: Proceedings of the international workshop on humanoid and human friendly robotics, Tsukuba, Japan

  41. Wieber PB, Tedrake R, Kuindersma S (2016) Modeling and control of legged robots. In: Springer handbook of robotics, Springer, pp 1203–1234

  42. Wight DL, Kubica EG, Wang DW (2008) Introduction of the foot placement estimator: a dynamic measure of balance for bipedal robotics. J Comput Nonlinear Dyn 3(1):011009

    Article  Google Scholar 

  43. van Zutven P, Kostic D, Nijmeijer H (2012) Foot placement for planar bipeds with point feet. In: 2012 IEEE international conference on robotics and automation (ICRA), pp 983–988. https://doi.org/10.1109/ICRA.2012.6224823

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Acknowledgements

The present work was partially funded by National Research Council of Brazil (CNPq: 311055/2016-8, 458820/2013-0, 442216/2016-5. LFR acknowledges a Ph.D. Grant from the CNPq.

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Correspondence to Arturo Forner-Cordero.

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Technical Editor: Victor Juliano De Negri, D.Eng.

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Rossi, L.F., Parik-Americano, P., Simões, I.F.E. et al. Predicted Step Viability: a stability criterion for biped gait. J Braz. Soc. Mech. Sci. Eng. 41, 548 (2019). https://doi.org/10.1007/s40430-019-2052-9

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