Modeling, stability and control of biped robots - A general framework

Automatica

Volumn 40, Issue 10, 2004, Pages 1647-1664

  Hurmuzlu, Yildirim ^a   Génot, Frank ^b   Brogliato, Bernard ^c  

^a Southern Methodist University   (United States)

^b INRIA ROCQUENCOURT   (France)

^c INRIA RHÔNE ALPES   (France)

Author keywords

Biped robots; Complementarity conditions; Control synthesis; Gait stability; Hybrid system; Multiple impact laws; Non smooth mechanics; Unilateral constraints

Indexed keywords

COMPUTER SIMULATION; CONSTRAINT THEORY; DEGREES OF FREEDOM (MECHANICS); DESIGN; DISCRETE TIME CONTROL SYSTEMS; ENGINEERS; MATHEMATICAL MODELS; PERTURBATION TECHNIQUES; ROBOTIC ARMS; ROBOTICS; SUSTAINABLE DEVELOPMENT; TECHNOLOGY TRANSFER;

BIPED ROBOTS; COMPLEMENTARITY CONDITIONS; CONTROL SYNTHESIS; GAIT STABILITY; HYBRID SYSTEMS; MULTIPLE IMPACT LAWS; NON-SMOOTH MECHANICS; UNILATERAL CONSTRAINTS;

BIPED LOCOMOTION;

EID: 3843127457     PISSN: 00051098     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.automatica.2004.01.031     Document Type: Article

References (164)

1. Abadie, M. (2000). Dynamic simulation of rigid bodies: Modelling of frictional contact. In B. Brogliato (Ed.), Impacts in mechanical systems; Analysis and modelling, Lecture notes in Physics, Vol. 551 (pp. 61-144). Berlin: Springer.
2. Arakawa, T., & Fukuda, T. (1997). Natural motion generation of a biped robot using the hierarchical trajectory generation method consisting of GA, EP layers. Proceedings of the IEEE conference on robotics and automatation, Vol. 1, Washington, DC (pp. 211-216).
3. Automatica. (1999). A special issue on hybrid systems. Automatica, 35(3), 374-535.
4. Bainov D.D., Simeonov P.S. Systems with impulse effects; stability, theory and applications. Ellis Horwood series in Mathematics and its applications. 1989;Wiley, New York.
5. Basmajian, J. V. (1976). The human bicycle. In P. V. Komi (Ed.), Biomechanics V-A (pp. 297-302). Baltimore, MD: University Park Press.
6. Beletskii V.V. Dynamics of two legged walking, II. Izvestiya AN SSSR Mekhanika Tverdogo Tela. 10(4):1975;3-13.
7. Beletskii V.V., Berbyuk V.E., Samsonov V.A. Parametric optimization of motions of a bipedal walking robot. Izvestiya AN SSSR Mekhanika Tverdogo Tela. 17(1):1982;28-40.
8. Beletskii V.V., Bolotin Y.V. Model estimation of the energetics of bipedal walking and running. Mechanics of Solids. 18(4):1983;87-92.
9. Beletskii V.V., Chudinov P.S. Parametric optimization in the problem of bipedal locomotion. Izvestiya AN SSSR Mekhanika Tverdogo Tela. 12(1):1977;25-35.
10. Beletskii V.V., Chudinov P.S. The linear stabilization problem for two legged ambulation. Izvestiya AN SSSR Mekhanika Tverdogo Tela. 12(6):1977;65-74.
11. Beletskii V.V., Chudinov P.S. Control of motion of a bipedal walking robot. Izvestiya AN SSSR Mekhanika Tverdogo Tela. 15(3):1980;30-38.
12. Beletskii V.V., Kirsanova T.S. Plane linear models of biped locomotion. Izvestiya AN SSSR Mekhanika Tverdogo Tela. 11(4):1976;51-62.
13. Blajer W., Schielen W. Walking without impacts as a motion/force control problem. ASME Journal of Dynamic Systems, Measurement and Control. 114:1992;660-665.
14. Blickman R., Full R.J. Locomotion energetics of the ghost crab. II. Mechanics of the center of mass during walking and running. Journal of Experimental Biology. 130:1987;155-174.
15. Bolotin Y.V. Energetically optimal gaits of a bipedal walking robot. Mechanics of Solids. 19(6):1984;44-51.
16. Borovac B., Vukobratovic M., Surla D. An approach to biped control synthesis. Robotica. 7:1989;231-241.
17. Bourgeot, J. M., & Brogliato, B. (2003). Tracking control of complementarity Lagrangian systems. International Journal of Bifurcations and Chaos, special issue non-smooth dynamical systems: Recent trends and perspectives, to appear 2005.
18. Brogliato, B. (1999). Nonsmooth mechanics. Berlin: Springer, CCES (erratum and addendum at http://www.inrialpes.fr/bip/people/brogliato/brogli. html).
19. Brogliato, B. (2003). Some perspectives on the analysis and control of complementarity systems. IEEE Transactions on Automatic Control, 48(6), 918-935.
20. Brogliato B., Niculescu S.I., Monteiro-Marques M. On the tracking control of a class of complementarity-slackness hybrid mechanical systems. Systems and Control Letters. 39:2000;255-266.
21. Brogliato B., Niculescu S.I., Orhant P. On the control of finite dimensional mechanical systems with unilateral constraints. IEEE Transactions on Automatic Control. 42(2):1997;200-215.
22. Brogliato B., ten Dam A.A., Paoli L., Génot F., Abadie M. Numerical simulation of finite dimensional multibody nonsmooth mechanical systems. ASME Applied Mechanics Reviews. 55:2002;107-150.
23. Brogliato B., Zavala-Rio A. On the control of complementary slackness juggling mechanical systems. IEEE Transactions on Automatic Control. 45(2):2000;235-246.
24. Budd C., Dux F. Chattering and related behaviour in impact oscillators. Proceedings of the Royal Society of London A. 347:1994;365-389.
25. Cavagna G.A., Heglund N.C., Taylor C.R. Mechanical work in terrestrial locomotion. Two basic mechanisms for minimizing energy expandure American Journal of Physiology. 233:1977;R243-R261.
26. Chang T.H., Hurmuzlu Y. Sliding control without reaching phase and its application to bipedal locomotion. ASME Journal of Dynamic Systems, Measurement, and Control. 105:1994;447-455.
27. Channon P.H., Hopkins S.H., Pham D.T. Derivation of optimal walking motions for a bipedal walking robot. Robotica. 10:1992;165-172.
28. Channon, P. H., Hopkins, S. H., & Pham, D. T. (1996a). A gravity compensation technique for an n-legged robot. Proceedings of INSTN MECH ENGRS, ImechE, Vol. 210 (pp. 1-14).
29. Channon P.H., Hopkins S.H., Pham D.T. Optimal Control of an n-legged robot. Journal of Systems and Control Engineering. 210:1996;51-63.
30. Channon, P. H., Hopkins, S. H., & Pham, D. T. (1996c). A variational approach to the optimization of gait for a bipedal robot. Proceedings of INSTN MECH ENGRS, ImechE, Vol. 210 (pp. 177-186).
31. Chatterjee A., Garcia M. Small slope implies low speed in passive dynamic walking. Dynamics and Stability of Systems. 15(2):1998;139-157.
32. Chevallereau C. Time-scaling control of an underactuated biped robot. IEEE Transactions on Robotics and Automation. 19(2):2003;362-368.
33. Chevallereau, C., Abba, G., Aoustin, Y., Plestan, F., Westervelt, E. R., Canudas de Wit, C., & Grizzle, J. W. (2003). RABBIT: A testbed for advanced control theory. IEEE Control Systems Magazine, 23(5), 57-79.
34. Chevallereau C., Aoustin Y. Optimal reference trajectories for walking and running of a biped robot. Robotica. 19:2001;557-569.
35. Chevallereau, C., Formal'sky, A., & Perrin, B. (1997). Control of a walking robot with feet following a reference trajectory derived from ballistic motion. IEEE conference on robotics and automation, 20-25 April, Albuquerque, New Mexico, USA.
36. Chudinov P.S. One problem of angular stabilization of bipedal locomotion. Izvestiya AN SSSR Mekhanika Tverdogo Tela. 15(6):1980;49-54.
37. Chudinov P.S. Problem of angular stabilization of bipedal locomotion. Izvestiya AN SSSR Mekhanika Tverdogo Tela. 19(1):1984;166-169.
38. Collins S., Wisse M., Ruina A. A 3-d passive-dynamic walking robot with two legs and knees. International Journal of Robotics Research. 20(7):2001;607-615.
39. Cottle R.W., Pang J.S., Stone R.E. The linear complementarity problem. 1992;Academic Press, Boston, MA.
40. Dankowicz H., Adolfsson J., Nordmark A. 3D passive walkers. finding periodic gaits in the presence of discontinuities Nonlinear Dynamics. 24:2001;205-229.
41. Dankowicz H., Piiroinen P. Exploiting discontinuities for stabilisation of recurrent motions. Dynamical Systems: An International Journal. 17(4):2002;317-342.
42. Das S.L., Chatterjee A. An alternative stability analysis technique for the simplest walker. Nonlinear Dynamics. 28(3):2002;273-284.
43. Delassus E. Mémoire sur la théorie des liaisons finies unilatérales. Annales Scientifiques de L Ecole Normale Superieurs. 34:1917;95-179.
44. Dunn, E., & Howe, R. D. (1994). Toward smooth bipedal walking. IEEE international conferences on robotics and automation, Vol. 3, San Diego, CA (pp. 2489-2494).
45. El Hafi F., Gorce P. Behavioural approach for a bipedal robot stepping motion gait. Robotica. 17:1999;491-501.
46. Fallis, G. T. (1888). Walking toy. US Patents 376, 588, Washington, DC.
47. Ferris M.C., Pang J.S. Engineering and economic application of complementarity problems. SIAM Review. 39(6):1997;669-713.
48. François C., Samson C. A new approach to the control of the planar one-legged hopper. International Journal of Robotics Research. 17(11):1998;1150-1166.
49. Frank A. An approach to the dynamic analysis and synthesis of biped locomotion machines. Medical and Biological Engineering. 8:1970;465-476.
50. Fujiwara, K., Kanehiro, F., Kajita, S., Kaneko, K., Yokoi, K., & Hirukawa, H. (2002). UKEMI: Falling motion control to minimize damage to biped humanoid robot. Proceedings of the 2002 IEEE/RSJ international conference on intelligent robots and systems, EPFL, Lausanne, Switzerland (pp. 2521-2526).
51. Fukuda, T., Komota, Y., & Arakawa, T. (1997). Stabilization control of biped locomotion robot based learning with gas having self-adaptive mutation and recurrent neural networks. IEEE conference on robotics and automation, Albuquerque, New Mexico, USA (pp. 217-222).
52. Full R.J., Koditschek D.E. Templates and anchors. Neuromechanical hypotheses of legged locomotion on land The Journal of Experimental Biology. 202:1999;3325-3332.
53. Furusho J., Sano A. Sensor based control of a nine link biped. International Journal of Robotics Research. 9(2):1990;83-98.
54. Garcia M., Chatterjee A., Ruina A., Coleman M. The simplest walking model. stability, and scaling ASME Journal of Biomechanical Engineering. 120:1997;281-288.
55. Garcia E., Estremera J., Gonzales de Santos P. A comparative study of stability margins for walking machines. Robotica. 20:2002;595-606.
56. Génot, F. (1998). Contributions à la modélisation et à la commande des systèmes mécaniques de corps rigides avec contraintes unilatérales. Ph.D. thesis, Institut National Polytechnique de Grenoble, France, February.
57. Génot, F., Brogliato, B., & Hurmuzlu, Y. (1998). Control of sticking phases in mechanical systems with unilateral constraints. Seventh conference on nonlinear vibrations, stability, and dynamics of structures, July 26-30, Virginia Polytechnic Institute, Blacksburg, USA.
58. Génot, F., Brogliato, B., & Hurmuzlu, Y. (2001). Modeling, stability and control of biped robots. A general framework. INRIA research report RR-4290, http://www.inria.fr/rrrt/index.en.html.
59. Gienger, M., Löffler, K., & Pfeiffer, F. (2003). Practical aspects of biped locomotion. In B. Siciliono, & P. Dario (Eds.), Experimental robotics (pp. 95-104). Berlin, Heidelberg: Springer.
60. Golliday C.L., Hemami H. An approach to analyzing biped locomotion dynamics and designing robot locomotion control. IEEE Transactions on Automatic Control. 22(6):1977;963-972.
61. Goswami A. Postural stability of biped robots and the foot-rotation indicator (FRI) point. International Journal of Robotics Research. 18(6):1999;523-533.
62. Goswami, A., Thuilot, B., & Espiau, B. (1996). Compass like bipedal robot part I: Stability and bifurcation of passive gaits. INRIA research report, 2996, http://www.inria.fr/RRRT/RR-2996.html.
63. Goswami A., Thuilot B., Espiau B. A study of the passive gait of a compass-like biped robot. symmetry and chaos International Journal of Robotic Research. 17(12):1998;1282-1301.
64. Grishin A.A., Formal'sky A.M. Control of bipedal walking robot by means of impulses of finite amplitude. Izvestiya AN SSSR Mekhanika Tverdogo Tela. 25(2):1990;67-74.
65. Grishin A.A., Formal'sky A.M., Lensky A.V., Zhitomirsky S.V. Dynamic of a vehicle with two telescopic legs controlled by two drives. The International Journal of Robotics Research. 13(2):1994;137-147.
66. Grizzle, J. W., Abba, G., & Plestan, F. (1999). Proving asymptotic stability of a walking cycle for a five dof biped robot model. Second international conference on climbing and walking robots, September 1999, Portsmouth, UK (pp. 69-81).
67. Grizzle J.W., Abba G., Plestan F. Asymptotically stable walking for biped robots. Analysis via systems with impulse effects IEEE Transactions on Automatic Control. 46(1):2001;51-64.
68. Gubina F., Hemami H., McGhee R.B. On the dynamic stability of biped locomotion. IEEE Transactions on Biomedical Engineering. 21(2):1974;102-108.
69. Gunther M., Blickman R. Joint stiffness of the ankle and the knee in running. Journal of Biomechanics. 35:2002;1459-1474.
70. Han I., Gilmore B.J. Multi-body impact motion with friction - analysis, simulation and experimental validation. ASME Journal of Mechanical Design. 115:1993;412-422.
71. Heemels, W. P. M. H., & Brogliato, B. (2003). The complementarity class of hybrid dynamical systems. European Journal of Control, 9(2-3), 322-360.
72. Hemami H., Farnsworth R.L. Postural and gait stability of a planar five link biped by simulation. IEEE Transaction on Automatic Control. 22:1977;452-458.
73. Hemami H., Katbab A. Constrained inverted pendulum model for evaluating upright postural stability. ASME Journal of Dynamics Systems, Measurement, and Control. 104:1982;343-349.
74. Hemami H., Zheng Y.F., Hines M.J. Initiation of walk and tiptoe of a planar nine link biped. Mathematical Biosciences. 61:1982;163-189.
75. Hirai, K. (1997). Current and future perspective of Honda humanoid robots. Proceedings of IROS 1997 (pp. 500-508).
76. Hirai, K., Hirose, M., & Kenada, T. T. (1998). The development of Honda humanoid robot. Proceedings of IEEE international conference on robotics and automation, May 1998, Lewen, Belgium (pp. 1321-1326).
77. HONDA. US Patents number: 5,432,417; 5,311,109; 5,355,064; 5,349,277; 5,426,586; 5,459,659; 5,357,433; 5,402,050; 5,252,901.
78. Huang Q., Yokoi K., Kajita S., Kaneko K., Arai H., Koyachi N., Tanie K. Planning walking patterns for a biped robot. IEEE Transactions on Robotics and Automation. 17:2001;280-289.
79. Hurmuzlu Y. Dynamics of bipedal gait; Part I - objective functions and the contact event of a planar five link biped. Part II - stability analysis of a planar five link biped. ASME Journal of Applied Mechanics. 60(2):1993;331-344.
80. Hurmuzlu Y., Basdogan C. On the measurement of dynamic stability of human locomotion. ASME Journal of Biomechanical Engineering. 116(1):1994;30-36.
81. Hurmuzlu Y., Basdogan C., Stoianovici D. Kinematics and dynamic stability of the locomotion of polio patients. ASME Journal of Biomechanical Engineering. 118:1995;405-411.
82. Hurmuzlu Y., Marghitu D.B. Rigid body collisions of planar kinematic chains with multiple contact points. International Journal of Robotics Research. 13(1):1994;82-92.
83. Hurmuzlu Y., Moskowitz G.D. The role of impact in the stability of bipedal locomotion. Dynamics and Stability of Systems. 1(3):1986;217-234.
84. Hurmuzlu Y., Moskowitz G.D. Bipedal locomotion stabilized by impact and switching: I. Two and three dimensional, three elements models, II. Structural stability analysis of a four element bipedal locomotion model. Dynamics and Stability of Systems. 2(2):1987;73-112.
85. Igarashi E., Nogai T. Study of lower level adaptive walking in the sagittal plane by a biped locomotion robot. Advanced Robotics. 6(4):1992;441-459.
86. Jalics L., Hemami H., Clymer B. A control strategy for terrain adaptive bipedal locomotion. Autonomous Robots. 4:1997;243-257.
87. Kajita S., Tani K. Experimental study of biped dynamic walking. IEEE Control Systems. 16:1996;13-19.
88. Kajita S., Yamaura T., Kobayashi A. Dynamic walking control of a biped robot along a potential energy conserving orbit. IEEE Transactions on Robotics and Automation. 8(4):1992;431-438.
89. Kar D.C., Kurien I.K., Jayarajan K. Gaits and energetics in terrestrial legged locomotion. Mechanism and Machine Theory. 38:2003;355-366.
90. Katoh R., Mori M. Control method of biped locomotion giving asymptotic stability of trajectory. Automatica. 20(4):1994;405-414.
91. Khosravi B., Yurkovich S., Hemami H. Control of a four link biped in a back somersault maneuver. IEEE Transactions on Systems, Man, and Cybernetics. 17(2):1987;303-325.
92. Kobrinskii A.E. Dynamics of mechanisms with elastic connections and impact systems. 1965;Ilife books Ltd, London.
93. van der Kooij H., Jacobs R., Koopman B., van der Halm F. An alternative approach to synthesizing bipedal walking. Biological Cybernetics. 88(1):2003;46-59.
94. Kozlov V.V., Treshchev D.V. Billiards. A genetic introduction to the dynamics of systems with impacts. 1991;American Mathematical Society, Providence, RI.
95. Kuhn H.W., Tucker A.W. Nonlinear programming. Neyman J. Proceedings of the second Berkeley symposium on mathematical statistics and probability. 1951;University of California, Press, Berkeley, CA.
96. Kuo A.D. Stabilization of lateral motion in passive dynamic walking. International Journal of Robotics Research. 18(9):1999;917-930.
97. Lavrovskii E.K. Impact phenomena in problems of control of bipedal locomotion. Izvestiya AN SSSR Mekhanika Tverdogo Tela. 14(5):1979;41-47.
98. Lavrovskii E.K. Dynamics of bipedal locomotion at high velocity. Izvestiya AN SSSR Mekhanika Tverdogo Tela. 15(4):1980;50-58.
99. Lee, T. T., & Liao, J. H. (1988). Trajectory planning and control of a 3-link biped robot. Proceedings of IEEE conference on robotics and automation, New York (pp. 820-823).
100. Lötstedt P. Mechanical systems of rigid bodies subject to unilateral constraints. SIAM Journal of Applied Mathematics. 42(2):1982;281-296.
101. Lötstedt P. Numerical simulation of time-dependent contact and friction problems in rigid body mechanics. SIAM Journal of Scientific and Statistic Computation. 5(2):1984;370-393.
102. Lum H.K., Zribi M., Soh Y.C. Planning and control of a biped robot. International Journal of Engineering Science. 37:1999;1319-1349.
103. Mabrouk M. A unified variational model for the dynamics of perfect unilateral constraints. European Journal of Mechanics A/Solids. 17(5):1998;819-848.
104. Marghitu D.B., Hurmuzlu Y. Three dimensional rigid body collisions with multiple contact points. ASME Journal of Applied Mechanics. 62:1995;725-732.
105. Masri S.F., Caughey T.K. On the stability of the impact damper. ASME Journal of Applied Mechanics. 33:1966;586-592.
106. McGeer T. Passive dynamic walking. International Journal of Robotic Research. 9(2):1990;62-82.
107. Menini L., Tornambé A. Asymptotic tracking of periodic trajectories for a simple mechanical system subject to nonsmooth impacts. IEEE Transactions on Automatic Control. 46:2001;1122-1126.
108. Mitobe K., Capi G., Nasu Y. Control of walking robots based on manipulation of the zero moment point. Robotica. 18:2000;651-657.
109. Miura H., Shimoyama I. Dynamic walking of a biped. International Journal of Robotics Research. 3(2):1984;60-74.
110. Mochon S., McMahon T.A. Ballistic walking. Journal of Biomechanics. 13:1980;49-57.
111. Mochon S., McMahon T.A. Ballistic walking. An improved model Mathematical Bioscience. 52:1980;241-260.
112. Moreau J.J. Les liaisons unilatérales et le principe de Gauss. C.R. Academy of Sciences Paris. 256:1963;871-874.
113. Moreau J.J. Quadratic programming in mechanics. Dynamics of one sided constraints Journal of SIAM Control. 4(1):1966;153-158.
114. Moreau J.J. Standard inelastic shocks and the dynamics of unilateral constraints. Del Piero G., Maceri F. Unilateral problems in structural analysis. Vol. 288:1985;Springer, Berlin.
115. Moreau, J. J. (1986). Dynamique de systèmes à liaisons unilatérales avec frottement sec éventuel; essais numériques. Technical note 85-1, LMGC, Université du Languedoc, Montpellier, France.
116. Moreau, J. J., & Panagiotopoulos, P. D. (1988). In Unilateral contact and dry friction in finite freedom dynamics. Nonsmooth mechanics and applications, CISM courses and lectures, Vol. 302 (pp. 1-82). Berlin: Springer.
117. Novozhilov I.V. Control of three-dimensional motion of a bipedal walking robot. Izvestiya AN SSSR Mekhanika Tverdogo Tela. 19(4):1984;47-53.
118. Paoli L., Schatzman M. Penalty approximation for dynamical systems submitted to multiple non-smooth constraints. Multibody System Dynamics. 8(3):2002;345-364.
119. Park J.H. Impedance control for biped robot locomotion. IEEE Transactions on Robotics and Automation. 17:2001;870-883.
120. Perrin, B., Chevallereau, C., & Formal'sky, A. (1997). Control of a quadruped walking robot without feet for a gallop gait. IFAC symposium on robot control (SYROCO'97), 3-5 September, Nantes, France.
121. Pfeiffer, F., & Glocker, C. (1996). Multibody dynamics with unilateral contacts, Wiley Series in Nonlinear Science. New York: Wiley.
122. Piiroinen, P., & Dankowicz, H. (2002). Low-cost control of repetitive gait in passive bipedal walkers. International Journal of Bifurcation and Chaos, submitted for publication.
123. Piiroinen P., Dankowicz H., Nordmark A. On a normal-form analysis for a class of passive bipedal walkers. International Journal of Bifurcation and Chaos. 11(9):2001;2411-2425.
124. Piiroinen, P., Dankowicz, H., & Nordmark, A. (2003). Breaking symmetries and constraints: Transitions from 2D to 3D in passive walkers. Multibody System Dynamics, 10(12), 147-176.
125. Popovic M.R., Pappas I.P.I., Nakazawa K., Keller T., Morari M., Dietz V. Stability criterion for controlling standing in able-bodied subjects. Journal of Biomechanics. 33:2000;1359-1368.
126. Pratt G.A. Legged robots at MIT. What's new since Raibert IEEE Robotics and Automation Magazine. 7(3):2000;15-19.
127. Pratt J., Chew C.-M., Torres A., Dilworth P., Pratt G. An intuitive approach for bipedal locomotion. International Journal of Robotics Research. 20(2):2001;129-143.
128. Pratt G.A., Williamson M.M. Series elastic actuators. IEEE Conference on Intelligent Robots and Systems. 1:1995;399-406.
129. Quint van der Linde R. Passive bipedal walking with phasic muscle contraction. Biological Cybernetics. 81(3):1999;227-237.
130. Raibert M.H. Legged robots that balance. 1986;MIT Press, Cambridge, MA.
131. Rostami M., Bessonnet G. Sagittal gait of a biped robot during the single support phase. Part 2: Optimal motion. Robotica. 19:2001;241-253.
132. Rubanovich E.M., Formal'sky A.M. Some problems of dynamics of multiple element systems associated with impact phenomena, II. Izvestiya AN SSSR Mekhanika Tverdogo Tela. 16(3):1981;125-133.
133. Rutkovskii S.V. Walking, skipping and running of a bipedal robot with allowance for impact. Mechanics of Solids. 20(5):1985;44-49.
134. Saidouni T., Bessonnet G. Generating globally optimised sagittal gait cycles of a biped robot. Robotica. 21:2003;199-210.
135. Sardain P., Rostami M., Bessonnet G. An anthropomorphic biped robot. Dynamic concepts and technological design IEEE Transactions on Systems, Man and Cybernetics. 28:1998;823-838.
136. Sardain P., Rostami M., Thomas E., Bessonnet G. Biped robots. Correlation between technological design and dynamic behaviour Control Engineering Practice. 7:1999;401-411.
137. Saunders J.B., Inman V.T., Eberhart H.D. The major determinants in normal and pathological gait. Journal of Bone and Joint Surgery. 35(A):1953;543-558.
138. Seo Y.-J., Yoon Y.-S. Design of a robust dynamic gait of the biped using the concept of dynamic stability margin. Robotica. 13:1995;461-468.
139. Shaw J., Holmes P. A periodically forced piecewise linear oscillator. Journal of Sound and Vibration. 90:1983;129-155.
140. Shaw J., Shaw S. The onset of chaos in a two-degree-of-freedom impacting system. ASME Journal of Applied Mechanics. 56:1989;168-174.
141. Shih C.L. Ascending and descending stairs for a biped robot. IEEE Transactions on Systems, Man and Cybernetics - Part A: Systems and Humans. 29:1999;255-268.
142. Shih C.L., Gruver W.A., Lee T.T. Inverse kinematics and inverse dynamics for control of a biped walking machine. Journal of Robotic Systems. 10(4):1993;531-555.
143. Shih, C. L., Gruver, W. A., & Zhu, Y. (1991). Fuzzy logic force control for a biped robot. Proceedings of the 1991 IEEE international symposium on intelligent control (pp. 269-273).
144. Shih C.L., Klein C.A. An adaptive gait for legged walking machines over rough terrain. IEEE Transactions on Systems, Man, and Cybernetics. 23(4):1993;1150-1155.
145. van der Soest Knoek A.J., Heanen W.P., Rozendaal L.A. Stability of bipedal stance. The contribution of cocontraction and spindle feedback Biological Cybernetics. 88(4):2003;293-301.
146. Song J., Low K.H., Guo W. A simplified hybrid force/position controller method for the walking robots. Robotica. 17:1999;583-589.
147. Taga G. A model of the neuro-musculo-skeletal system for human locomotion. I. Emergence of basic gait. II. Real-time adaptability under various constraints. Biological Cybernetics. 73:1995;97-121.
148. Takanishi, A., Ishida, M., Ymazaki, Y., & Kato, I. (1985). The realization of dynamic walking by the biped robot WL-10RD. Proceedings of the international conference on advanced robotics, Tokyo (pp. 459-466).
149. Todd D.J. Walking machines: An introduction to legged robots. 1985;Kogan Page, London.
150. Vanel O., Gorce P. A new approach to dynamic posture control. Robotica. 15:1997;449-459.
151. Vaughan C.L. Theories of bipedal walking. An odyssey Journal of Biomechanics. 36:2003;513-523.
152. Vukobratovic, M. (1976). Walking robots and anthromopomorphic mechanics. Moskow: MIR Press (Russian translation).
153. Vukobratovic M., Borovac B., Surla D., Stokic D. Scientific fundamentals of robotics 7: Biped locomotion. 1990;Springer, New York.
154. Vukobratovic M., Juricic D. Contribution to the synthesis of biped gait. IEEE Transactions on Biomedical Engineering. 16(1):1969;1-6.
155. Werstervelt E.R., Grizzle J.W., Koditschek D.E. Hybrid zero dynamics of planar biped walkers. IEEE Transactions on Automatic Control. 48(1):2003;42-56.
156. Wieber, P. B. (2000). Modélisation et commande d'un robot marcheur anthropomorphe. Ph.D. thesis, INRIA Rhône-Alpes, France, December.
157. Wieber, P. B. (2002). On the stability of walking systems. Third IARP international workshop on humanoid and human friendly robotics, December 11-12, Tsukuba Research Center, Ibaraki, Japan.
158. Yagi M., Lumelsky V. Local on-line planning in biped robot locomotion amongst unknown obstacles. Robotica. 18:2000;389-402.
159. Yang J.S. A control study of a kneeless biped locomotion system. Journal of the Franklin Institute. 331b(2):1994;125-143.
160. Zarrugh, M. Y. (1976). Energy and Power in Human Walking. Ph.D. thesis, University of California, Berkeley.
161. Zavala-Rio A., Brogliato B. Direct adaptive control design for one-degree-of-freedom complementary-slackness jugglers. Automatica. 37:2001;1117-1123.
162. Zheng Y.F. Acceleration compensation for biped robot to reject external disturbances. IEEE Transactions on Systems, Man, and Cybernetics. 19(1):1989;74-84.
163. Zheng Y.F., Hemami H. Impacts effects of biped contact with the environment. IEEE Transactions on Systems, Man, and Cybernetics. SMC-14(3):1984;437-443.
164. Zheng Y.F., Sheng J. Gait synthesis for the SD-2 biped robot to climb sloping surface. IEEE Transactions on Robotics and Automation. 6(1):1990;86-96