Dynamic balance preservation and prevention of sliding for humanoid robots in the presence of multiple spatial contacts

Abstract

The main indicator of dynamic balance is the \(\mathit{ZMP}\). Its original notion assumes that both feet of the robot are in contact with the flat horizontal surface (all contacts are in the same plane) and that the friction is high enough so that sliding does not occur. With increasing capabilities of humanoid robots and the higher complexity of the motion that needs to be performed, these assumptions might not hold. Having in mind that the system is dynamically balanced if there is no rotation about the edges of the feet and if the feet do not slide, we propose a novel approach for testing the dynamic balance of bipedal robots, by using linear contact wrench conditions compiled in a single matrix (Dynamic Balance Matrix). The proposed approach has wide applicability since it can be used to check the stability of different kinds of contacts (including point, line, and surface) with arbitrary perimeter shapes. Motion feasibility conditions are derived on the basis of the conditions which the wrench of each contact has to satisfy. The approach was tested by simulation in two scenarios: biped climbing up and walking sideways on the inclined flat surface which is too steep for a regular walk without additional support. The whole-body motion was synthesized and performed using a generalized task prioritization framework.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Algorithm 1
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Notes

  1. 1.

    In this context “stable contact” represents the situation where there is no relative motion between the bodies in contact. In our opinion term “stable contact” is not the most suitable to describe the state of the contact when there is no sliding and no separation. It seems that terms such as “reliable, sustainable” are more descriptive.

  2. 2.

    One exception appears to be when the robot grasps the environment, so it can both push and pull. When looking at the grasp, it actually comprises several unilateral contacts, positioned in such a way that all directions of contact forces are possible. The only real exception of this assumption is when there is adhesion between the robot and the environment.

  3. 3.

    An on-line demo showing calculation of the DBM can be found at the author’s website www.milutinnikolic.info/dbm/.

References

  1. 1.

    Aagaard, P., Simonsen, E.B., Trolle, M., Bangsbo, J., Klausen, K.: Moment and power generation during maximal knee extensions performed at low and high speeds. Eur. J. Appl. Physiol. Occup. Physiol. 69(5), 376–381 (1994). doi:10.1007/BF00865398

    Article  Google Scholar 

  2. 2.

    Barber, C.B., Dobkin, D.P., Huhdanpaa, H.: The quickhull algorithm for convex hulls. ACM Trans. Math. Softw. 22(4), 469–483 (1996). doi:10.1145/235815.235821

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Caron, S., Pham, Q.C., Nakamura, Y.: Leveraging cone double description for multi-contact stability of humanoids with applications to statics and dynamics. In: Robotics: Science and System (2015)

    Google Scholar 

  4. 4.

    Caron, S., Pham, Q.C., Nakamura, Y.: Stability of surface contacts for humanoid robots: closed-form formulae of the contact wrench for rectangular support areas. In: 2015 IEEE International Conference on Robotics and Automation (ICRA). IEEE Press, New York (2015)

    Google Scholar 

  5. 5.

    Dai, H., Valenzuela, A., Tedrake, R.: Whole-body motion planning with centroidal dynamics and full kinematics. In: 2014 14th IEEE–RAS International Conference on Humanoid Robots (Humanoids), pp. 295–302 (2014). doi:10.1109/HUMANOIDS.2014.7041375

    Google Scholar 

  6. 6.

    DeVita, P., Hortobágyi, T.: Obesity is not associated with increased knee joint torque and power during level walking. J. Biomech. 36(9), 1355–1362 (2003). doi:10.1016/S0021-9290(03)00119-2

    Article  Google Scholar 

  7. 7.

    Escande, A., Kheddar, A., Miossec, S.: Planning contact points for humanoid robots. Robot. Auton. Syst. 61(5), 428–442 (2013). doi:10.1016/j.robot.2013.01.008

    Article  Google Scholar 

  8. 8.

    Fallon, M.F., Marion, P., Deits, R., Whelan, T., Antone, M., McDonald, J., Tedrake, R.: Continuous humanoid locomotion over uneven terrain using stereo fusion. In: 2015 IEEE–RAS 15th International Conference on Humanoid Robots (Humanoids), pp. 881–888 (2015). doi:10.1109/HUMANOIDS.2015.7363465

    Google Scholar 

  9. 9.

    Faraji, S., Pouya, S., Ijspeert, A.: Robust and agile 3D biped walking with steering capability using a footstep predictive approach. In: Robotics Science and Systems (RSS) (2014). http://www.roboticsproceedings.org/rss10/p28.html

    Google Scholar 

  10. 10.

    Ferreau, H., Kirches, C., Potschka, A., Bock, H., Diehl, M.: qpOASES: a parametric active-set algorithm for quadratic programming. Math. Program. Comput. 6(4), 327–363 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Fukuda, K., Prodon, A.: Double description method revisited. In: Deza, M., Euler, R., Manoussakis, I. (eds.) Combinatorics and Computer Science: 8th Franco–Japanese and 4th Franco–Chinese Conference Brest, France, July 3–5, 1995 Selected Papers, pp. 91–111. Springer, Berlin (1996). doi:10.1007/3-540-61576-8_77

    Google Scholar 

  12. 12.

    Goyal, S., Pinson, E.N., Sinden, F.W.: Simulation of dynamics of interacting rigid bodies including friction I: general problem and contact model. Eng. Comput. 10(3), 162–174 (1994)

    Article  Google Scholar 

  13. 13.

    Goyal, S., Pinson, E.N., Sinden, F.W.: Simulation of dynamics of interacting rigid bodies including friction II: software system design and implementation. Eng. Comput. 10(3), 175–195 (1994)

    Article  Google Scholar 

  14. 14.

    Harada, K., Hirukawa, H., Kanehiro, F., Fujiwara, K., Kaneko, K., Kajita, S., Nakamura, M.: Dynamical balance of a humanoid robot grasping an environment. In: Proceedings of the 2004 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2004), vol. 2, pp. 1167–1173. IEEE Press, New York (2004)

    Google Scholar 

  15. 15.

    Harada, K., Kajita, S., Kaneko, K., Hirukawa, H.: Dynamics and balance of a humanoid robot during manipulation tasks. IEEE Trans. Robot. 22(3), 568–575 (2006)

    Article  Google Scholar 

  16. 16.

    Herzog, A., Rotella, N., Mason, S., Grimminger, F., Schaal, S., Righetti, L.: Momentum control with hierarchical inverse dynamics on a torque-controlled humanoid. Auton. Robots 40(3), 473–491 (2015). doi:10.1007/s10514-015-9476-6

    Article  Google Scholar 

  17. 17.

    Herzog, A., Rotella, N., Schaal, S., Righetti, L.: Trajectory generation for multi-contact momentum control. In: 2015 IEEE–RAS 15th International Conference on Humanoid Robots (Humanoids), pp. 874–880 (2015). doi:10.1109/HUMANOIDS.2015.7363464

    Google Scholar 

  18. 18.

    Hirukawa, H., Hattori, S., Harada, K., Kajita, S., Kaneko, K., Kanehiro, F., Fujiwara, K., Morisawa, M.: A universal stability criterion of the foot contact of legged robots-adios ZMP. In: Proceedings 2006 IEEE International Conference on Robotics and Automation (ICRA 2006), pp. 1976–1983. IEEE Press, New York (2006)

    Google Scholar 

  19. 19.

    Kuindersma, S., Permenter, F., Tedrake, R.: An efficiently solvable quadratic program for stabilizing dynamic locomotion. In: 2014 IEEE International Conference on Robotics and Automation (ICRA), pp. 2589–2594 (2014). doi:10.1109/ICRA.2014.6907230

    Google Scholar 

  20. 20.

    Lengagne, S., Vaillant, J., Yoshida, E., Kheddar, A.: Generation of whole-body optimal dynamic multi-contact motions. Int. J. Robot. Res. 32(9–10), 1104–1119 (2013)

    Article  Google Scholar 

  21. 21.

    Lohmeier, S., Buschmann, T., Ulbrich, H., Pfeiffer, F.: Humanoid Robot LOLA—Research Platform for High-Speed Walking pp. 221–230. Springer, Dordrecht (2009). doi:10.1007/978-1-4020-9438-5_22.

    Google Scholar 

  22. 22.

    Mattingley, J., Boyd, S.: Cvxgen: a code generator for embedded convex optimization. Optim. Eng. 13(1), 1–27 (2011). doi:10.1007/s11081-011-9176-9

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Nikolić, M., Borovac, B., Raković, M.: Walking on slippery surfaces: generalized task-prioritization framework approach. In: Advances on Theory and Practice of Robots and Manipulators: Proceedings of Romansy 2014 XX CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators, pp. 189–196. Springer, Berlin (2014)

    Google Scholar 

  24. 24.

    Nikolić, M., Borovac, B., Raković, M., Savić, S.: A further generalization of task-oriented control through tasks prioritization. International Journal of Humanoid Robotics 10(03) (2013)

  25. 25.

    Nikolic, M., Savic, S., Borovac, B., Rakovic, M.: Task prioritization framework for kinesthetic teaching of a free-standing humanoid robot. In: 2015 IEEE 13th International Symposium on Intelligent Systems and Informatics (SISY), pp. 241–246 (2015). doi:10.1109/SISY.2015.7325387

    Google Scholar 

  26. 26.

    Pang, J.S., Trinkle, J.: Stability characterizations of rigid body contact problems with coulomb friction. Z. Angew. Math. Mech. 80(10), 643–663 (2000). doi:10.1002/1521-4001(200010)80:10<643::AID-ZAMM643>3.0.CO;2-E

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Sentis, L., Khatib, O.: Synthesis of whole-body behaviors through hierarchical control of behavioral primitives. Int. J. Humanoid Robot. 2(04), 505–518 (2005)

    Article  Google Scholar 

  28. 28.

    Sentis, L., Khatib, O.: A whole-body control framework for humanoids operating in human environments. In: Proceedings 2006 IEEE International Conference on Robotics and Automation (ICRA 2006), vol. 2006., pp. 2641–2648 (2006). doi:10.1109/ROBOT.2006.1642100

    Google Scholar 

  29. 29.

    Sentis, L., Park, J., Khatib, O.: Compliant control of multicontact and center-of-mass behaviors in humanoid robots. IEEE Trans. Robot. 26(3), 483–501 (2010)

    Article  Google Scholar 

  30. 30.

    Takao, S., Yokokohji, Y., Yoshikawa, T.: FSW (feasible solution of wrench) for multi-legged robots. In: Proceedings of the IEEE International Conference on Robotics and Automation, 2003 (ICRA’03), vol. 3, pp. 3815–3820. IEEE Press, New York (2003)

    Google Scholar 

  31. 31.

    Udwadia, F.E., Kalaba, R.E.: Analytical Dynamics: A New Approach. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  32. 32.

    Vukobratović, M., Borovac, B.: Zero-moment point—thirty five years of its life. Int. J. Humanoid Robot. 1(01), 157–173 (2004)

    Article  Google Scholar 

  33. 33.

    Vukobratović, M., Juricic, D.: Contribution to the synthesis of biped gait. IEEE Trans. Biomed. Eng. 1, 1–6 (1969)

    Article  Google Scholar 

  34. 34.

    Vukobratović, M., Potkonjak, V., Babković, K., Borovac, B.: Simulation model of general human and humanoid motion. Multibody Syst. Dyn. 17(1), 71–96 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

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

    Article  MATH  Google Scholar 

  36. 36.

    Wensing, P.M., Orin, D.E.: Generation of dynamic humanoid behaviors through task-space control with conic optimization. In: 2013 IEEE International Conference on Robotics and Automation (ICRA), pp. 3103–3109 (2013). doi:10.1109/ICRA.2013.6631008

    Google Scholar 

  37. 37.

    Whelan, T., Kaess, M., Johannsson, H., Fallon, M., Leonard, J.J., McDonald, J.: Real-time large-scale dense RGB-D slam with volumetric fusion. Int. J. Robot. Res. 34(4–5), 598–626 (2015). doi:10.1177/0278364914551008

    Article  Google Scholar 

  38. 38.

    Winter, D.A.: Biomechanics and Motor Control of Human Movement, 3rd edn. Wiley, New York (2004)

    Google Scholar 

Download references

Acknowledgements

This work was funded by the Ministry of Science and Technological Development of the Republic of Serbia in part under contract TR35003 and in part under contract III44008.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Milutin Nikolić.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nikolić, M., Borovac, B. & Raković, M. Dynamic balance preservation and prevention of sliding for humanoid robots in the presence of multiple spatial contacts. Multibody Syst Dyn 42, 197–218 (2018). https://doi.org/10.1007/s11044-017-9572-9

Download citation

Keywords

  • Humanoid robots
  • Contact stability
  • Whole-body motion