Dynamic bipedal walking under stick-slip transitions. (English) Zbl 1342.70018
Summary: This paper studies the hybrid dynamics of bipedal robot walking under stick-slip transitions. We focus on two simple planar models with point feet: the rimless wheel and the compass biped. Unlike most of the existing works in the literature that assume sticking contact between the foot and the ground, we explore the case of insufficient friction, which may induce foot slippage. Numerical simulations of passive dynamic walking reveal the onset of stable periodic solutions involving stick-slip transitions. In the case of the compass biped with controlled joint torque actuation, we demonstrate how one can exploit kinematic trajectories of passive walking in order to induce and stabilize gaits with slipping impact.
MSC:
| 70E18 | Motion of a rigid body in contact with a solid surface |
| 34A38 | Hybrid systems of ordinary differential equations |
| 68T40 | Artificial intelligence for robotics |
| 70E60 | Robot dynamics and control of rigid bodies |
Software:
OpenSimReferences:
| [1] | G. Aguirre-Ollinger, J. E. Colgate, M. A. Peshkin, and A. Goswami, {\it Inertia compensation control of a one-degree-of-freedom exoskeleton for lower-limb assistance: Initial experiments}, IEEE Trans. Neural Systems Rehab. Engrg., 20 (2012), pp. 68-77. |
| [2] | A. D. Ames, E. A. Cousineau, and M. J. Powell, {\it Dynamically stable robotic walking with NAO via human-inspired hybrid zero dynamics}, in Proceedings of the 15th ACM International Conference on Hybrid Systems: Computation and Control, ACM, New York, 2012, pp. 135-144. · Zbl 1364.70017 |
| [3] | M. Anitescu and F. A. Potra, {\it Formulating dynamic multi-rigid-body contact problems with friction as solvable linear complementarity problems}, Nonlinear Dynam., 14 (1997), pp. 231-247. · Zbl 0899.70005 |
| [4] | S. Aoi and K. Tsuchiya, {\it Stability analysis of a simple walking model driven by an oscillator with a phase reset using sensory feedback}, IEEE Trans. Robotics, 22 (2006), pp. 391-397. |
| [5] | A. Back, J. Guckenheimer, and M. Myers, {\it A dynamical simulation facility for hybrid systems}, in Hybrid Systems, Lecture Notes in Comput. Sci. 736, Springer, Berlin, 1993, pp. 255-267. |
| [6] | M. D. Berkemeier and R. S. Fearing, {\it Sliding and hopping gaits for the underactuated acrobot}, IEEE Trans. Robotics and Automation, 14 (1998), pp. 629-634. |
| [7] | P. Birkmeyer, K. Peterson, and R. S. Fearing, {\it DASH: A dynamic 16g hexapedal robot}, in Proceedings of the IEEE/RSJ Conference on Intelligent Robots and Systems (IROS), St. Louis, MO, 2009, pp. 2683-2689. |
| [8] | R. Blickhan and R. J. Full, {\it Similarity in multilegged locomotion–bouncing like a monopode}, J. Comparative Physiology A, 173 (1993), pp. 509-517. |
| [9] | H. F. Bohn and W. Federle, {\it Insect aquaplaning: Nepenthes pitcher plants capture prey with the peristome, a fully wettable water-lubricated anisotropic surface}, Proc. Natl. Acad. Sci. USA, 101 (2004), pp. 14138-14143. |
| [10] | B. Brogliato, {\it Nonsmooth Mechanics}, Springer-Verlag, Berlin, 1999. · Zbl 0917.73002 |
| [11] | T. H. Chang and Y. Hurmuzlu, {\it Sliding control without reaching phase and its application to bipedal locomotion}, ASME J. Dyn. Syst. Measurement Control, 105 (1994), pp. 447-455. · Zbl 0800.93208 |
| [12] | A. Chatterjee and A. Ruina, {\it A new algebraic rigid body collision law based on impulse space considerations}, ASME J. Appl. Mech., 65 (1998), pp. 939-951. |
| [13] | A. J. Clark and T. E. Higham, {\it Slipping, sliding and stability: locomotor strategies for overcoming low-friction surfaces}, J. Experimental Biol., 214 (2011), pp. 1369-1378. |
| [14] | M. J. Coleman, {\it Dynamics and stability of a rimless spoked wheel: A simple \(2\) D system with impacts}, Dyn. Syst. Int. J., 25 (2010), pp. 215-238. · Zbl 1379.70028 |
| [15] | M. J. Coleman, A. Chatterjee, and A. Ruina, {\it Motions of a rimless spoked wheel: A simple \(3\) D system with impacts}, Dyn. Stability Syst., 12 (1997), pp. 139-160. · Zbl 0932.70007 |
| [16] | S. Collins, A. Ruina, R. Tedrake, and M. Wisse, {\it Efficient bipedal robots based on passive dynamic walkers}, Science, 307 (2005), pp. 1082-1085. |
| [17] | C. A. Coulomb, {\it Théorie des machines simples, en ayant égard au frottement de leur partie, et a la raideur des cordages, avec 5 planches}, Recueil des savants étrangers de l’Académie Royale des Sciences, 10 (1781), pp. 161-332. |
| [18] | C. Canudas de Wit, {\it On the concept of virtual constraints as a tool for walking robot control and balancing}, Ann. Rev. Control, 28 (2004), pp. 157-166. |
| [19] | S. L. Delp, F. C. Anderson, A. S. Arnold, P. Loan, A. Habib, C. T. John, E. Guendelman, and D. G. Thelen, {\it Opensim: Open-source software to create and analyze dynamic simulations of movement}, IEEE Trans. Biomed. Engrg., 54 (2007), pp. 1940-1950. |
| [20] | M. H. Dickinson, C. T. Farley, R. J. Full, M. A. R. Koehl, R. Kram, and S. Lehman, {\it How animals move: an integrative view}, Science, 288 (2000), pp. 100-106. |
| [21] | A. Esquenazi, M. Talaty, A. Packel, and M. Saulino, {\it The ReWalk powered exoskeleton to restore ambulatory function to individuals with thoracic-level motor-complete spinal cord injury}, Amer. J. Phys. Med. Rehab., 91 (2012), pp. 911-921. |
| [22] | C. D. Fitzgibbon, {\it Why do hunting cheetahs prefer male gazelles?}, Animal Behaviour, 40 (1990), pp. 837-845. |
| [23] | L. B. Freidovich, A. S. Shiriaev, and I. R. Manchester, {\it Stability analysis and control design for an underactuated walking robot via computation of a transverse linearization}, in Proceeding of the 17th IFAC World Congress, Seoul, Korea, 2008, pp. 10166-10171. |
| [24] | R. J. Full and D. Koditschek, {\it Templates and anchors: Neuromechanical hypotheses of legged locomotion on land}, J. Experimental Biol., 202 (1999), pp. 3325-3332. |
| [25] | B. Gamus and Y. Or, {\it Analysis of dynamic bipedal robot locomotion with stick-slip transitions}, in Proceedings of the IEEE International Conference on Robotics and Automation, Karlsruhe, 2013, pp. 3348-3355. |
| [26] | M. Garcia, A. Chatterjee, and A. Ruina, {\it Efficiency, speed, and scaling of two-dimensional passive-dynamic walking}, Dyn. Stability Syst., 15 (2000), pp. 75-99. · Zbl 0964.92004 |
| [27] | M. Garcia, A. Chatterjee, A. Ruina, and M. Coleman, {\it The simplest walking model: Stability, complexity and scaling}, ASME J. Biomech. Engr., 120 (1998), pp. 281-288. |
| [28] | R. Goebel, R. G. Sanfelice, and A. R. Teel, {\it Hybrid dynamical systems}, IEEE Control Systems Mag., 29 (2009), pp. 28-93. · Zbl 1395.93001 |
| [29] | A. T. Goswami, {\it A study of the passive gait of a compass-like biped robot: Symmetry and chaos}, Int. J. Robotics Res., 17 (1998), pp. 1282-1301. |
| [30] | R. D. Gregg and M. W. Spong, {\it Reduction-based control of three-dimensional bipedal walking robots}, Int. J. Robotics Res., 29 (2010), pp. 680-702. |
| [31] | J. W. Grizzle, C. Chevallereau, A. D. Ames, and R. Sinnet, {\it \(3\) D bipedal robotic walking: Models, feedback control, and open problems}, in Proceedings of the 8th IFAC Symposium on Nonlinear Control Systems (NOLCOS), Bologna, Italy, 2010, pp. 505-532. · Zbl 1297.93120 |
| [32] | M. Gruhn, O. Hofmann, M. Duebbert, H. Scharstein, and A. Bueschges, {\it Tethered stick insect walking: A modified slippery surface setup with optomotor stimulation and electrical monitoring of tarsal contact}, J. Neurosci. Methods, 158 (2006), pp. 195-206. |
| [33] | M. Gruhn, L. Zehl, and A. Bueschges, {\it Straight walking and turning on the slippery surface}, J. Experimental Biol., 212 (2009), pp. 194-209. |
| [34] | J. Guckenheimer and P. Holmes, {\it Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields}, Springer-Verlag, New York, 1983. · Zbl 0515.34001 |
| [35] | H. Hemami and Y.-F. Zheng, {\it Dynamics and control of motion on the ground and in the air with application to biped robots}, J. Robotic Syst., 1 (1984), pp. 101-116. |
| [36] | K. Hirai, M. Hirose, Y. Haikawa, and T. Takenaka, {\it The development of Honda humanoid robot}, in Proceedings of the IEEE International Conference on Robotics and Automation, Leuven, vol. 2, 1998, pp. 1321-1326. |
| [37] | P. Holmes, R. J. Full, D. Koditschek, and J. Guckenheimer, {\it The dynamics of legged locomotion: Models, analysis and challenges}, SIAM Rev., 48 (2006), pp. 207-304. · Zbl 1100.34002 |
| [38] | Y. Hurmuzlu, F. Génot, and B. Brogliato, {\it Modeling, stability and control of biped robots–a general framework}, Automatica, 40 (2004), pp. 1647-1664. · Zbl 1155.93393 |
| [39] | S. Kajita, K. Kaneko, K. Harada, F. Kanehiro, K. Fujiwara, and H. Hirukawa, {\it Biped walking on a low friction floor}, in Proceedings of the IEEE/RSJ Conference on Intelligent Robots and Systems (IROS), IEEE, Piscataway, NJ, 2004, pp. 3546-3552. |
| [40] | K. Kaneko, K K. Harada, F. Kanehiro, G. Miyamori, and K. Akachi, {\it Humanoid robot HRP-3}, in Proceedings of the IEEE/RSJ Conference on Intelligent Robots and Systems (IROS), IEEE, Piscataway, NJ, 2008, pp. 2471-2478. |
| [41] | K. Kaneko, F. Kanehiro, S. Kajita, M. Morisawa, K. Fujiwara, K. Harada, and H. Hirukawa, {\it Slip observer for walking on a low friction floor}, in Proceedings of the IEEE/RSJ Conference on Intelligent Robots and Systems (IROS), IEEE, Piscataway, NJ, 2005, pp. 634-640. |
| [42] | K. Kato and S. Hirose, {\it Development of the quadruped walking robot, TITAN-IX — mechanical design concept and application for the humanitarian de-mining robot}, Adv. Robotics, 15 (2001), pp. 191-204. |
| [43] | H. K. Khalil, {\it Nonlinear Systems}, 2nd ed., Prentice Hall, Englewood Cliffs, NJ, 1996. |
| [44] | E. Krotkov and R. Simmons, {\it Perception, planning, and control for autonomous walking with the Ambler planetary rover}, Int. J. Robotics Res., 16 (1996), pp. 155-180. |
| [45] | A. D. Kuo, {\it Energetics of actively powered locomotion using the simplest walking model}, J. Biomech. Engrg., 124 (2002), pp. 113-120. |
| [46] | D. W. Marhefka and D. E. Orin, {\it Gait planning for energy efficiency in walking machines}, in Proceedings of the IEEE International Conference on Robotics and Automation, Albuquerque, NM, vol. 1, 1997, pp. 474-480. |
| [47] | M. T. Mason and Y. Wang, {\it On the inconsistency of rigid-body frictional planar mechanics}, in Proceedings of the IEEE International Conference on Robotics and Automation, Philadelphia, vol. 1, 1988, pp. 524-528. |
| [48] | T. McGeer, {\it Passive dynamic walking}, Int. J. Robotics Res., 9 (1990), pp. 62-82. |
| [49] | B. E. Moyer, A. J. Chambers, M. S. Redfern, and R. Cham, {\it Falls, injuries due to falls, and the risk of admission to a nursing home}, New England J. Med., 337 (1997), pp. 1279-1284. |
| [50] | B. E. Moyer, A. J. Chambers, M. S. Redfern, and R. Cham, {\it Gait parameters as predictors of slip severity in younger and older adults}, Ergonomics, 49 (2006), pp. 329-343. |
| [51] | M. P. Murphy, A. Saunders, C. Moreira, A. A. Rizzi, and M. Raibert, {\it The LittleDog robot}, International J. Robotics Res., 30 (2011), pp. 145-149. |
| [52] | R. M. Murray, Z. Li, and S. Sastry, {\it A Mathematical Introduction to Robotic Manipulation}, CRC Press, Boca Raton, FL, 1993. · Zbl 0858.70001 |
| [53] | T. Nomura, K. Kawa, Y. Suzuki, M. Nakanishi, and T. Yamasaki, {\it Dynamic stability and phase resetting during biped gait}, Chaos, 19 (2009), 026103. |
| [54] | Y. Or, {\it Painlevé’s paradox and dynamic jamming in simple models of passive dynamic walking}, Reg. Chaotic Dynam., 19 (2014), pp. 64-80. · Zbl 1353.70006 |
| [55] | Y. Or and A. D. Ames, {\it Stability and completion of Zeno equilibria in Lagrangian hybrid systems}, IEEE Trans. Automat. Control, 56 (2011), pp. 1322-1336. · Zbl 1368.93476 |
| [56] | Y. Or and E. Rimon, {\it Computation and graphical characterization of robust multiple-contact postures in two-dimensional gravitational environments}, Int. J. Robotics Res., 25 (2006), pp. 1071-1086. |
| [57] | Y. Or and E. Rimon, {\it Investigation of Painlevé’s paradox and dynamic jamming during mechanism sliding motion}, Nonlinear Dynam., 67 (2012), pp. 1647-1668. · Zbl 1311.70016 |
| [58] | Y. Or and A. R. Teel, {\it Zeno stability of the set-valued bouncing ball}, IEEE Trans. Automat. Control, 56 (2011), pp. 447-452. · Zbl 1368.93606 |
| [59] | P. Painlevé, {\it Sur les lois du frottement de glissement}, C. R. Acad. Sci., 121 (1895), pp. 112-115. · JFM 26.0916.04 |
| [60] | J.-S. Pang and J. C. Trinkle, {\it Complementarity formulations and existence of solutions of dynamic multi-rigid-body contact problems with Coulomb friction}, Math. Programming, 73 (1996), pp. 199-226. · Zbl 0854.70008 |
| [61] | F. Pfeiffer and C. Glocker, {\it Multibody Dynamics with Unilateral Contacts}, John Wiley \(\&\) Sons, New York, 1996. · Zbl 0922.70001 |
| [62] | M. Raibert, K. Blankespoor, G. Nelson, R. Playter, and the Bigdog Team, {\it BigDog, the rough-terrain quadruped robot}, in Proceeding of the 17th IFAC World Congress, Seoul, Korea, 2008, pp. 10822-10825. |
| [63] | E. Rimon, R. Mason, J. W. Burdick, and Y. Or, {\it A general stance stability test based on stratified morse theory with application to quasi-static locomotion planning}, IEEE Trans. Robotics, 24 (2008), pp. 626-641. |
| [64] | A. Ruina, J. E. A. Bertram, and M. Srinivasan, {\it A collisional model of the energetic cost of support work qualitatively explains leg sequencing in walking and galloping, pseudo-elastic leg behavior in running and the walk-to-run transition}, J. Theoret. Biol., 237 (2005), pp. 170-192. · Zbl 1445.92028 |
| [65] | U. Saranli, M. Buehler, and D.E. Koditschek, {\it RHex: a simple and highly mobile hexapod robot}, Int. J. Robotics Res., 20 (2001), pp. 616-631. |
| [66] | J. Schmitt and P. Holmes, {\it Mechanical models for insect locomotion: Dynamics and stability in the horizontal plane - I. Theory}, Biol. Cybernet., 83 (2000), pp. 501-515. · Zbl 1033.92005 |
| [67] | J. Seipel and P. Holmes, {\it A simple model for clock-actuated legged locomotion}, Reg. Chaotic Dynam., 12 (2007), pp. 502-520. · Zbl 1229.70028 |
| [68] | M. Sheen, B. Kogan, J. Hasaneini, and A. Ruina, {\it Evaluating simulation tools for use in walking robot design and controls optimization}, in Dynamic Walking, Zurich, Switzerland, 2014. |
| [69] | F. M. Silva and J. A. T. Machado, {\it Energy analysis during biped walking}, in Proceedings of IEEE International Conference on Robotics and Automation, Detroit, MI, vol. 1, 1999, pp. 59-64. |
| [70] | J. A. Smith, I. Sharf, and M. Trentini, {\it PAW: A hybrid wheeled-leg robot}, in Proceedings of IEEE International Conference on Robotics and Automation, Orlando, FL, 2006, pp. 4043-4048. |
| [71] | R. Smithers, {\it The Mammals of Botswana}, National Museums of Rhodesia, Rhodesia, South Africa, 1971. |
| [72] | A. Spence, K. Parsons, M. Ferrari, T. Pfau, A. Wilson, and A. Thurman, {\it Effects of substrate properties on equine locomotion}, Comparative Biochem. Physiol. A Molecular Integrative Physiol., 146 (2007), Suppl., p. S109. |
| [73] | M. W. Spong and F. Bullo, {\it Controlled symmetries and passive walking}, IEEE Trans. Automat. Control, 50 (2005), pp. 1025-1031. · Zbl 1365.93329 |
| [74] | M. W. Spong, J. K. Holm, and D. Lee, {\it Passivity-based control of bipedal locomotion}, IEEE Robotics Automat. Mag., 14 (2007), pp. 30-40. |
| [75] | L. Strandberg and H. Lanshammar, {\it The dynamics of slipping accidents}, J. Occupational Accidents, 3 (1981), pp. 153-162. |
| [76] | W. J. Stronge, {\it Impact Mechanics}, Cambridge University Press, Cambridge, UK, 1979. · Zbl 0961.74002 |
| [77] | N. Suzuki and Y. Yamamoto, {\it Pursuing entertainment aspects of SONY AIBO quadruped robots}, in 4th International Conference on Modeling, Simulation and Applied Optimization (ICMSAO), Jalan Semarak, Kuala Lumpur, Malaysia, 2011, pp. 1-5. |
| [78] | A. Tavakoli and Y. Hurmuzlu, {\it Robotic locomotion of three generations of a family tree of dynamical systems, part I: Passive gait patterns}, Nonlinear Dynam., 73 (2013), pp. 1969-1989. |
| [79] | J. A. Vazquez and M. Velasco-Villa, {\it Numerical analysis of the sliding effects of a \(5\)-dof biped robot}, in Proceedings of the 8th International Conference on Electrical Engineering Computing Science and Automatic Control, Merida City, Mexico, 2011, pp. 1-6. |
| [80] | J. A. Vazquez and M. Velasco-Villa, {\it Experimental estimation of slipping in the supporting point of a biped robot}, J. Appl. Res. Tech., 11 (2013), pp. 348-359. |
| [81] | Y. Wang and M. T. Mason, {\it Two-dimensional rigid body collisions with friction}, J. Appl. Mech., 10 (1993), pp. 292-352. |
| [82] | E. R. Westervelt, J. W. Grizzle, C. Chevallereau, J. H. Choi, and B. Morris, {\it Feedback Control of Dynamic Bipedal Robot Locomotion}, CRC Press, Boca Raton, FL, 2007. |
| [83] | E. R. Westervelt, J. W. Grizzle, and D. E. Koditschek, {\it Hybrid zero dynamics of planar biped walkers}, IEEE Trans. Automat. Control, 48 (2003), pp. 42-56. · Zbl 1364.70015 |
| [84] | B. H. Wilcox, T. Litwin, J. Biesiadecki, J. Matthews, M. Heverly, J. Morrison, J. Townsend, N. Ahmed, A. Sirota, and B. Cooper, {\it ATHLETE: A cargo handling and manipulation robot for the moon}, J. Field Robotics, 24 (2007), pp. 421-434. |
| [85] | D. Zarrouk and R. S. Fearing, {\it Cost of locomotion of a dynamic hexapedal robot}, in Proceedings of the IEEE International Conference on Robotics and Automation, Karlsruhe, Germany, 2013, pp. 2548-2553. |
| [86] | J. Zhang, K. H. Johansson, J. Lygeros, and S. Sastry, {\it Zeno hybrid systems}, Int. J. Robust Nonlinear Control, 11 (2001), pp. 435-451. · Zbl 0977.93047 |
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