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Controlled Recurrence of a Biped with Torso

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Cyber Physical Systems. Model-Based Design (CyPhy 2018, WESE 2018)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 11615))

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Abstract

We have recently used a symbolic reachability method for controlling the stability of special hybrid systems called “sampled switched systems”. We show here how the method can be extended in order to control the stability of more general hybrid systems with guard conditions and state resets. We illustrate the method through the example of a biped robot with 6 state variables, using a proportional-derivative (PD) controller. More specifically, we isolate a state region R such that, starting from a state located in R just after a footstep, the PD-control makes the robot state return to R at the end of the following footstep.

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Notes

  1. 1.

    Actually, the boxes are not all of the same size, but are generated according to an adaptative tiling procedure (see Sect. 3.3).

  2. 2.

    Condition (1) is true a first time when the legs are parallel, but we ignore such a “scuffing” and assume the swing leg to continue without collision.

  3. 3.

    The expression \(e_{\max }\) differs for each k, and the notation should be \(e_{\max }^k\), but the upper index k is dropped for the sake of simplicity.

  4. 4.

    In Figs. 2, 3, 4 and 5, we did not plot all the images \(Post^k(T)\), and \(Post^k(T')\), for \(1\le k\le N^+\), but only some of them for the sake of readability of the pictures.

References

  1. Agrawal, A., et al.: First steps towards translating HZD control of bipedal robots to decentralized control of exoskeletons. IEEE Access 5, 9919–9934 (2017)

    Article  Google Scholar 

  2. Althoff, M., Stursberg, O., Buss, M.: Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization. In: Proceedings of the 47th IEEE Conference on Decision and Control, CDC 2008, Cancún, Mexico, pp. 4042–4048 (2008)

    Google Scholar 

  3. Chen, X., Ábrahám, E., Sankaranarayanan, S.: Flow*: an analyzer for non-linear hybrid systems. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 258–263. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39799-8_18

    Chapter  Google Scholar 

  4. Alexandre dit Sandretto, J.A., Chapoutot, A., Mullier, O.: Tuning PI controller in non-linear uncertain closed-loop systems with interval analysis. In: 2nd International Workshop on Synthesis of Complex Parameters (SynCoP 2015). OpenAccess Series in Informatics (OASIcs), Dagstuhl, Germany, vol. 44, pp. 91–102 (2015)

    Google Scholar 

  5. Feng, S., Amur, S.A.Y., Sun, Z.: Biped walking on level ground with torso using only one actuator. Sci. China Inf. Sci. 56(11), 1–9 (2013)

    Article  Google Scholar 

  6. Frehse, G., et al.: SpaceEx: scalable verification of hybrid systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 379–395. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22110-1_30

    Chapter  Google Scholar 

  7. Fribourg, L., Kühne, U., Soulat, R.: Finite controlled invariants for sampled switched systems. Formal Methods Syst. Des. 45(3), 303–329 (2014)

    Article  Google Scholar 

  8. Fribourg, L., Soulat, R.: Control of Switching Systems by Invariance Analysis: Application to Power Electronics, 144 p. Wiley-ISTE (2013)

    Google Scholar 

  9. Girard, A.: Reachability of uncertain linear systems using zonotopes. In: Morari, M., Thiele, L. (eds.) HSCC 2005. LNCS, vol. 3414, pp. 291–305. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-31954-2_19

    Chapter  MATH  Google Scholar 

  10. Girard, A., Le Guernic, C.: Zonotope/hyperplane intersection for hybrid systems reachability analysis. In: Egerstedt, M., Mishra, B. (eds.) HSCC 2008. LNCS, vol. 4981, pp. 215–228. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78929-1_16

    Chapter  MATH  Google Scholar 

  11. Hussien, O., Ames, A.D., Tabuada, P.: Abstracting partially feedback linearizable systems compositionally. IEEE Control Syst. Lett. 1(2), 227–232 (2017)

    Article  Google Scholar 

  12. Kühn, W.: Zonotope dynamics in numerical quality control. In: Hege, H.C., Polthier, K. (eds.) Mathematical Visualization, pp. 125–134. Springer, Heidelberg (1998). https://doi.org/10.1007/978-3-662-03567-2_10

    Chapter  Google Scholar 

  13. Kühne, U., Soulat, R.: MINIMATOR 1.0 (2015). https://bitbucket.org/ukuehne/minimator/overview

  14. McGeer, T.: Passive dynamic walking. Int. J. Rob. Res. 9(2), 62–82 (1990)

    Article  Google Scholar 

  15. Tabuada, P.: Verification and Control of Hybrid Systems: A Symbolic Approach, 1st edn. Springer, New York (2009). https://doi.org/10.1007/978-1-4419-0224-5

    Book  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, Aalborg University, Selma Largerløfs Vej 300, 9220, Aalborg, Denmark

    Adrien Le Coënt

  2. LSV - ENS Paris-Saclay and CNRS and INRIA, University Paris-Saclay, 91 Avenue du Président Wilson, 94230, Cachan, France

    Laurent Fribourg

Authors
  1. Adrien Le Coënt
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  2. Laurent Fribourg
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Correspondence to Adrien Le Coënt .

Editor information

Editors and Affiliations

  1. Computer Science and Engineering, Washington University, St. Louis, MO, USA

    Roger Chamberlain

  2. School of Information Technology, Halmstad University, Halmstad, Sweden

    Walid Taha

  3. Department of Machine Design, KTH Royal Institute of Technology, Stockholm, Sweden

    Martin Törngren

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Le Coënt, A., Fribourg, L. (2019). Controlled Recurrence of a Biped with Torso. In: Chamberlain, R., Taha, W., Törngren, M. (eds) Cyber Physical Systems. Model-Based Design. CyPhy WESE 2018 2018. Lecture Notes in Computer Science(), vol 11615. Springer, Cham. https://doi.org/10.1007/978-3-030-23703-5_8

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  • DOI: https://doi.org/10.1007/978-3-030-23703-5_8

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  • Print ISBN: 978-3-030-23702-8

  • Online ISBN: 978-3-030-23703-5

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