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Topological duality in humanoid robot dynamics

Published online by Cambridge University Press:  17 February 2009

V. Ivancevic  and
C. E. M. Pearce
V. Ivancevic
Affiliation:
Torson Productions Pty Ltd, Adelaide SA 5034, Australia.
C. E. M. Pearce
Affiliation:
Applied Mathematics Department, The University of Adelaide, Adelaide SA 5005, Australia.
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Abstract

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A humanoid robot system may be viewed as a collection of segments coupled at rotational joints which geometrically represent constrained rotational Lie groups. This allows a study of the dynamics of the motion of a humanoid robot. Several formulations are possible. In this paper, dual invariant topological structures are constructed and analyzed on the finite-dimensional manifolds associated with the humanoid motion. Both cohomology and homology structures are examined on the tangent (Lagrangian) as well as on the cotangent (Hamiltonian) bundles on the manifold of the humanoid motion configuration. represented by the toral Lie group. It is established all four topological structures give in essence the same description of humanoid dynamics. Practically this means that whichever of these approaches we use, ultimately we obtain the same mathematical results.

Type
Research Article
Information
The ANZIAM Journal , Volume 43 , Issue 2 , October 2001 , pp. 183 - 194
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Abraham, R., Marsden, J. and Ratiu, T., Manifolds, tensor analysis and applications, Appl. Math. Sci. (New York, Springer, 1988).CrossRefGoogle Scholar
[2]Arnold, V. I., Mathematical methods of classical mechanics, 2nd ed., Graduate Texts in Math. (New York, Springer, 1989).CrossRefGoogle Scholar
[3]Bot, R. and Tu, L. W.. Differential forms in algebraic topology (New York, Springer, 1982).CrossRefGoogle Scholar
[4]Bullo, F., Nonlinear control of mechanical systems: a Riemannian geometry approach, Technical Report CDS 98–010 for Control and Dynamical System Option, 1998.Google Scholar
[5]Channon, P., Hopkins, S. and Pham, D., “A variational approach to the optimization of gait for a bipedal robot”, J. Mech. Eng. Sci. 210 (2) (1996) 177–186.CrossRefGoogle Scholar
[6]Chiel, H. J., Beer, R. D., Quinn, R. D. and Espenschied, K. S., “Robustness of a distributed neural network controller for locomotion in a hexapod robot”, IEEE Trans. Robotics Auto. 8 (3) (1992) 293–303.CrossRefGoogle Scholar
[7]Choquet-Bruhat, Y., DeWitt, M. C. and Dillard-Blieck, M., Analysis, manifolds and physics, Rev. ed., (Amsterdam, North-Holland, 1982).Google Scholar
[8]De Rham, G., Differentiable manifolds (transl. from French), Comprehensive Studies in Math. (Berlin, Springer, 1984).CrossRefGoogle Scholar
[9]Dorigo, M. and Schnepf, U., “Genetics-based machine learning and behavior-based robotics: a new synthesis”, IEEE Trans. Syst. Man and Cybernetics 23 (1993) 141–154.CrossRefGoogle Scholar
[10]Hashimoto, S., “Humanoid robots in Waseda University: Hadaly-2 and Wabian.” In IARP First International Workshop on Humanoid and Human Friendly Robotics, 1–2, 1998.Google Scholar
[11]Hatsopoulos, N. G.. “Coupling the neural and physical dynamics in rhythmic movements”, Neural Computation 8 (1996) 567–581.CrossRefGoogle ScholarPubMed
[12]Hurmuzlu, Y., “Dynamics of bipedal gait”, J. Appl. Mech., 60 (1993) 331–343.CrossRefGoogle Scholar
[13]Igarashi, E. and Nogai, T., “Study of lower level adaptive walking in the saggital plane by a biped locomotion robot”, Advanced Robotics 6 (4) (1992) 441–459.CrossRefGoogle Scholar
[14]Ivancevic, V., Introduction to biomechanical systems: modeling, control and learning (in Serbian) (Belgrade, Scientific Book, 1991).Google Scholar
[15]Ko, H. and Badler, N., “Animating human locomotion with inverse dynamics”, IEEE Computer Graphics Appl. 16 (1996) 50–59.Google Scholar
[16]Lieh, J., “Computer oriented closed-form algorithm for constrained multibody dynamics for robotics applications”, Mechanism and Machine Theory 29 (3) (1994) 357–371.CrossRefGoogle Scholar
[17]Mac Lane, S., Categories for the working mathematician, Graduate Texts in Math. (New York, Springer, 1971).CrossRefGoogle Scholar
[18]Marsden, J. E. and Ratiu, T. S., Introduction to mechanics and symmetry, Texts in Appl. Math. 17 (New York, Springer, 1994).CrossRefGoogle Scholar
[19]McGeer, T., “Passive dynamic walking”, Internat. J. Robotics Res. 9 (2) (1990) 62–82.CrossRefGoogle Scholar
[20]Pratt, J. and Pratt, G., “Exploiting natural dynamics in the control of a planar bipedal walking robot”, Proc. 36th Annual Allerton Conf. on Comm., Control and Comp. (1998) 739–748.Google Scholar
[21]Pribe, C., Grossberg, S. and Cohen, M. A., “Neural control of interlimb oscillations II. Biped and quadrupled gaits and bifurcations”, Biological Cybernetics 77 (1997) 141–152.CrossRefGoogle Scholar
[22]Puta, M., Hamiltonian mechanical systems and geometric quantization (Dordrecht, Kluwer, 1993).CrossRefGoogle Scholar
[23]Sardain, P., Rostami, M. and Bessonnet, G., “An anthropomorphic biped robot: dynamic concepts and technological design”, IEEE Trans. Syst. Man and Cybernetics 28a (6) (1999) 823–838.CrossRefGoogle Scholar
[24]Schaal, S., “Is imitation learning the route to humanoid robots?”, Trends in Cognitive Sci. 3 (6) (1999) 233–242.CrossRefGoogle ScholarPubMed
[25]Schaal, S. and Atkeson, C. G., “Constructive incremental learning from only local information”, Neural Comp. 10 (1998) 2047–2084.CrossRefGoogle ScholarPubMed
[26]Seraji, H., “Configuration control of redundant manipulators: theory and implementation”, IEEE Trans. Robotics Auto. 5 (4) (1989).Google Scholar
[27]Seward, D., Bradshaw, A. and Margrave, F., “The anatomy of a humanoid robot”, Robotica 14 (1996) 437–443.CrossRefGoogle Scholar
[28]Shih, C. L., Gruver, W. and Lee, T., “Inverse kinematics and inverse dynamics for control of a biped walking machine” J. Robotic Syst. 10 (1993) 531–555.CrossRefGoogle Scholar
[29]Shih, C. L. and Klein, C. A., “An adaptive gait for legged walking machines over rough terrain”, IEEE Trans. Syst. Man and Cybernetics 23 (1993) 1150–1154.CrossRefGoogle Scholar
[30]Vukobratovic, M., “On the stability of biped locomotion”, IEEE Trans. Biomed. Eng. 17 (1970) 25–36.CrossRefGoogle ScholarPubMed
[31]Vukobratovic, M., Legged locomotion, robots and anthropomorphic mechanisms (Belgrade, Mihailo Pupin, 1975).Google Scholar
[32]Vukobratovic, M., Borovac, B., Surla, D. and Stokic, D., Biped locomotion: dynamics, stability, control and applications (Berlin, Springer, 1990).CrossRefGoogle Scholar
[33]Vukobratovic, M., Juricic, D. and Frank, A., “On the control and stability of one class of biped locomotion systems”, ASME J. Basic Eng. 92 (1970) 328–332.CrossRefGoogle Scholar
[34]Vukobratovic, M. and Stepanenko, Y., “On the stability of anthropomorphic systems”, Math. Biosci. 15 (1972) 1–37.CrossRefGoogle Scholar
[35]Vukobratovic, M. and Stepanenko, Y., “Mathematical models of general anthropomorphic systems”, Math. Biosci. 17 (1973) 191–242.CrossRefGoogle Scholar
[36]Yoshikawa, T., “Analysis and control of robot manipulators with redundancy”, in Robotics Res. (Eds Brady, M. and Paul, R., Cambridge MA, MIT Press, 1984) 735–747.Google Scholar