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Variable stiffness control of series elastic actuated biped locomotion

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Abstract

This study investigates the problem of dynamic walking impact on a biped robot. Two online variable stiffness control algorithms, i.e., torque balance algorithm (TBA) and surface fitting algorithm (SFA), are proposed based on virtual spring leg to achieve compliant performance. These two algorithms target on solving the high nonlinearity commonly existing in legged robot actuators. A planar biped robot experiment platform is designed for testing the proposed variable stiffness control. The experiments compare the performance of TBA and SFA and verify that applying the variable stiffness control of a virtual spring leg is capable of effectively absorbing unforeseen ground impacts and thus improving stability and safety of walking biped robots.

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References

  1. Hogan N (1984) Impedance control: an approach to manipulation. In: IEEE American control conference, pp 304–313

  2. Hogan N (1987) Stable execution of contact tasks using impedance control. In: IEEE international conference on robotics and automation (ICRA), pp 1047–1054

  3. Zinn M, Khatib O, Roth B, Salisbury J (2004) Playing it safe [human-friendly robots]. IEEE Robot Autom Mag 11(2):12–21

    Article  Google Scholar 

  4. Luo J-w, Fu Y-l, Wang S-g (2017) 3D stable biped walking control and implementation on real robot. Adv Robot 31(12):634–649

    Article  Google Scholar 

  5. Jianwen L, Fu Y, Wang S, Qiao M (2017) How does the compliant legs affect walking stability. In: 2017 IEEE international conference on robotics and biomimetics (ROBIO). IEEE

  6. Ketelaar JG, Visser LC, Stramigioli S, et al (2013) Controller design for a bipedal walking robot using variable stiffness actuators. In: IEEE international conference on robotics and automation (ICRA). IEEE Press, Karlsruhe, pp 5650–5655

  7. Visser LC, Stramigioli S, Carloni R (2012) Robust bipedal walking with variable leg stiffness. In: 4th IEEE RAS & EMBS international conference biomedical robotics and biomechatronics (BioRob), pp 1626–1631

  8. Geyer H, Seyfarth A, Blickhan R (2006) Compliant leg behaviour explains basic dynamics of walking and running. Proc R Soc B Biol Sci 273(1603):2861–2867

    Article  Google Scholar 

  9. Rummel J, Blum Y, Maus HM et al (2010) Stable and robust walking with compliant legs. In: IEEE international conference on robotics and automation (ICRA). IEEE Press, Alaska, pp 5250–5255

  10. Vanderborght B, Albu-Schäffer A, Bicchi A et al (2013) Variable impedance actuators: a review. Robot. Auton. Syst. 61(12):1601–1614

    Article  Google Scholar 

  11. Visser LC, Carloni R, Stramigioli S (2011) Energy-efficient variable stiffness actuators. IEEE Trans Robot 27(5):865–875

    Article  Google Scholar 

  12. Visser LC, Carloni R, Klijnstra F et al (2010) A prototype of a novel energy efficient variable stiffness actuator. In: IEEE international conference on engineering in medicine and biology society (EMBC). IEEE Press, Buenos Aires, pp 3703–3706

  13. Pratt J (2000) Exploiting inherent robustness and natural dynamics in the control of bipedal walking robots. Ph.D. thesis, Massachusetts Institute of Technology

  14. Pratt G, Williamson M (1990) Series elastic actuators. In: IEEE international workshop on intelligent robots and systems, IROS, pp 399–406

  15. Boaventura T, Semini C, Buchli J, Frigerio M, Focchi M, Caldwell DG (2012) Dynamic torque control of a hydraulic quadruped robot. In: IEEE international conference on robotics and automation (ICRA). IEEE Press, Minnesota, pp 1889–1894

  16. Hyon S-H, Hale JG, Cheng G (2007) Full-body compliant human-humanoid interaction: balancing in the presence of unknown external forces. IEEE Trans Robot 23(5):884–898

    Article  Google Scholar 

  17. Pratt G, Matthew W (1995) Series elastic actuators. In: Proceedings of IEEE/RSJ international conference intelligent robots and systems 95. Human Robot Interaction and Cooperative Robots, vol. 1, pp 399–406

  18. Schepelmann A, Geberth K, Geyer H (2010) Compact nonlinear springs with user defined torque-deflection profiles for series elastic actuators. In: IEEE International conference on robotics and automation (ICRA). IEEE Press, Hong Kong, pp 3411–3416

  19. Jafari A, Tsagarakis NG, Vanderborght B, Caldwell DG (2010) A novel actuator with adjustable stiffness (AwAS). In: 2010 IEEE/RSJ international conference on intelligent robots and systems (IROS), pp 4201–4206

  20. Grioli G, Wolf S, Garabini M, Catalano M, Burdet E, Caldwell D, Bicchi A (2005) Variable stiffness actuators: the user’s point of view. Int J Robot Res 34(6):727–743

    Article  Google Scholar 

  21. Hopkins MA, Ressler SA, Lahr DF, Leonessa A, Hong DW (2015) Embedded joint-space control of a series elastic humanoid. In: IEEE/RSJ international conference on intelligent robots and systems, pp. 3358–3365

  22. Tagliamonte NL, Accoto D, Guglielmelli E (2014) Rendering viscoelasticity with series elastic actuators using cascade control. In: IEEE- RAS international conference on robotics and automation, pp 2424–2429

  23. Kim MJ et al (2017) Enhancing joint torque control of series elastic actuators with physical damping. 2017 IEEE international conference on robotics and automation (ICRA). IEEE

  24. Mehling JS, Holley J, O’Malley MK (2015) Leveraging disturbance observer based torque control for improved impedance rendering with series elastic actuators. In: 2015 IEEE/RSJ international conference on intelligent robots and systems (IROS). IEEE

  25. Sariyildiz E, Chen G, Haoyong Y (2016) An acceleration-based robust motion controller design for a novel series elastic actuator. IEEE Trans Ind Electron 63(3):1900–1910

    Article  Google Scholar 

  26. Boaventura T, Semini C, Buchli J, Frigerio M, Focchi M, Caldwell DG (2012) Dynamic torque control of a hydraulic quadruped robot. In: IEEE international conference on robotics and automation (ICRA). IEEE Press, Minnesota, pp 1889–1894

  27. Lu J et al (2015) Design and torque-mode control of a cable-driven rotary series elastic actuator for subject-robot interaction. In: 2015 IEEE international conference on advanced intelligent mechatronics (AIM). IEEE

  28. Agarwal P et al (2017) Design, control, and testing of a thumb exoskeleton with series elastic actuation. Int J Robot Res 36.3:355–375

    Article  Google Scholar 

  29. Laffranchi M, Chen L, Kashiri N, Lee J, Tsagarakis NG, Caldwell DG (2014) Development and control of a series elastic actuator equipped with a semi active friction damper for human friendly robots. Robot Auton Syst 62(12):1827–1836

    Article  Google Scholar 

  30. Oh S, Kong K (2017) High-precision robust force control of a series elastic actuator. IEEE/ASME Trans Mechatron 22(1):71–80

    Article  Google Scholar 

  31. Zhu Q et al (2016) Adaptive torque and position control for a legged robot based on a series elastic actuator. Int J Adv Rob Syst 13.1:26

    Article  Google Scholar 

  32. Zhao Y, Paine N, Jorgensen SJ, Sentis L (2017) Impedance control and performance measure of series elastic actuators. IEEE Trans Ind Electron 65:2817–2827

    Article  Google Scholar 

  33. Zhao Y, Nicholas P, Kim KS, Sentis L (2015) Stability and performance limits of latency-prone distributed feedback controllers. IEEE Trans Ind Electron 62(11):7151–7162

    Article  Google Scholar 

  34. Winter DA, Patla AE, Prince F, Ishac M, Gielo-Perczak K (1998) Stiffness control of balance in quiet standing. J Neurophysiol 80(3):1211–1221

    Article  Google Scholar 

  35. Zelik KE, Kuo AD (2010) Human walking isn’t all hard work: evidence of soft tissue contributions to energy dissipation and return. J Exp Biol 213(24):4257–4264

    Article  Google Scholar 

  36. Schmitt S, Günther M (2011) Human leg impact: energy dissipation of wobbling masses. Arch Appl Mech 81(7):887–897

    Article  MATH  Google Scholar 

  37. Dickinson MH et al (2000) How animals move: an integrative view. Science 288(5463):100–106

    Article  Google Scholar 

  38. Rummel J, Seyfarth A (2008) Stable running with segmented legs. Int J Robot Res 27(8):919–934

    Article  Google Scholar 

  39. Rummel J, Blum Y, Seyfarth A (2010) Robust and efficient walking with spring-like legs. Bioinspir Biomimet 5(4):046004

    Article  Google Scholar 

  40. Garofalo G, Ott C, Albu-Schaffer A (2012) Walking control of fully actuated robots based on the bipedal slip model. In: IEEE international conference on robotics and automation (ICRA). IEEE Press, Minnesota, pp 1456–1463

  41. Qiao M, Jindrich DL (2012) Task-level strategies for human sagittal-plane running maneuvers are consistent with robotic control policies. PLoS ONE 7(12):e51888. https://doi.org/10.1371/journal.pone.0051888

    Article  Google Scholar 

  42. Pratt J (1995) Virtual model control of a biped walking robot. M.Eng, thesis, Massachusetts Institute of Technology

  43. Torres A (1996) Virtual model control of a hexapod walking robot. B.S., thesis, Massachusetts Institute of Technology

  44. Pratt J, Torres A, Dilworth P et al (1996) Virtual actuator control, intelligent robots and systems. In: IEEE international conference on robotics and automation (IROS) 96. IEEE Press, Osaka, pp. 1219–1226

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Authors and Affiliations

  1. Key State Laboratory of Robotics and System, Harbin Institute of Technology, Harbin, 150000, China

    Jianwen Luo, Shuguo Wang & Yili Fu

  2. Agile Robotics Lab, Harvard University, Cambridge, MA, 02138, USA

    Ye Zhao

Authors
  1. Jianwen Luo
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  2. Shuguo Wang
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  3. Ye Zhao
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  4. Yili Fu
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Corresponding author

Correspondence to Jianwen Luo.

Appendix

Appendix

1. Given \(\alpha _\mathrm{knee} \) and \(x_d \), we solve \({\beta (}\alpha _\mathrm{knee}, x_d {)}\), \({T}_{\mathrm{sea}} \left( {\alpha _\mathrm{knee}, x_d } \right) \)

In \(\Delta EGH\),

$$\begin{aligned}&EG=f_{EG} \left( {x_d } \right) \text { and }\angle {EGF=}f_{\angle {EGF}} \left( {x_d } \right) \\&\angle {EGH}=\pi -\alpha _\mathrm{knee} -\angle {EGF}-\angle {HGI} = \pi \\&-\alpha _\mathrm{knee} -f_{\angle {EGF}} \left( {x_d } \right) -\angle {HGI}=f_{\angle {EGH}} \left( {\alpha _\mathrm{knee} , x_d } \right) \\&EH=\sqrt{EG^{2}+HG^{2}-2 \cdot EG \cdot HG\cdot \cos \left( {\angle {EGH}} \right) }\\&=\sqrt{f_{EG} \left( {x_d } \right) ^{2}+HG^{2}-2 \cdot f_{EG} \left( {x_d } \right) \cdot HG \cdot \cos \left( {f_{\angle {EGH}} \left( {\alpha _\mathrm{knee} , x_d } \right) } \right) }\\&=f_{{EH}} \left( {\alpha _\mathrm{knee}, x_d } \right) \\&\beta =\cos ^{-1}\left( {\frac{EH^{2}+HG^{2}-EG^{2}}{2\cdot EH \cdot HG}} \right) \\&=\cos ^{-1}\left( {\frac{f_{{EH}} \left( {\alpha _\mathrm{knee}, x_d } \right) ^{2}+HG^{2}-f_{EG} \left( {x_d } \right) ^{2}}{2 \cdot f_{{EH}} \left( {\alpha _\mathrm{knee}, x_d } \right) \cdot HG}} \right) =f_{\beta } \left( {\alpha _\mathrm{knee}, x_d } \right) \\&{T}_{\mathrm{sea}} ={T}_{\mathrm{sea}} \left( {\alpha _\mathrm{knee} , x_d } \right) =K_{\mathrm{spring}} \cdot \left( {x_{l0} -f_{{EH}} \left( {\alpha _\mathrm{knee}, x_d } \right) } \right) \\&\cdot HG \cdot {\sin }\left( {f_{\beta } \left( {\alpha _\mathrm{knee}, x_d } \right) } \right) \end{aligned}$$

where \(K_{\mathrm{spring}} \) is the spring stiffness.

figure a

2. Given \(x_d \) and \(x_l \), we solve \({\beta }\left( {x_l, x_d } \right) \), \({\varepsilon (}x_l , x_d {)}\) and \({l (}x_l, x_d {)}\)

The length of BA, BF, DF, DE, EF, FG, HG, GI are constants, which are determined by mechanical configuration. The angles \(\angle {DFE}\) and \(\angle {HGI}\) are also constant.

In \(\Delta ABF, \angle {BFA=}\sin ^{-1}{(BA/BF)}\)

In \(\Delta BDF, \angle {DFB=}\cos ^{-1}((DF^{2}+BF^{2}-BD(x_{s}))^{2})/(2\cdot DF\cdot BF))=f \angle DFB(x_{d})\)

where \(BD\left( {x_d}\right) =BC+x_d\).

$$\begin{aligned} \angle {EFG}=\pi -\angle {BFA}-\angle {DFB}-\angle {EFD}=f_{\angle {EFG}} \left( {x_d } \right) \end{aligned}$$

In \(\Delta EFG\),

$$\begin{aligned} EG= & {} \sqrt{EF^{2}+FG^{2}-2 \cdot EF \cdot FG \cdot cos\left( {\angle \text {EFG}} \right) }\\&=f_{{EG}} \left( {x_d}\right) \\ \angle EGF= & {} \cos ^{-1}\left( {\frac{EG^{2}+GF^{2}-EF^{2}}{2\cdot EG \cdot GF}} \right) =f_{\angle {EGF}} \left( {x_d }\right) \end{aligned}$$

So EG and \(\angle {EGF}\) are the functions of \(x_d\).

In \(\Delta EGH\),

$$\begin{aligned}&{\beta =}\cos ^{-1}\left( {\frac{EH\left( {x_l } \right) ^{2}+HG^{2}-EG^{2}}{2 \cdot EH\left( {x_l } \right) \cdot HG}} \right) \\&\quad =\cos ^{-1}\left( {\frac{EH\left( {x_l } \right) ^{2}+HG^{2}-f_{{EG}} \left( {x_d } \right) ^{2}}{2\cdot EH\left( {x_l } \right) \cdot HG}} \right) =f_{\beta } \left( {x_l , x_d } \right) \end{aligned}$$

where \(EH\left( {x_l } \right) =x_{l0} +x_l \)

$$\begin{aligned} \angle EGH= & {} cos^{-1}\left( \left( {EG^{2}+HG^{2}-EH\left( {x_l } \right) ^{2}} \right) /\right. \\&\left. \left( {2\cdot EG \cdot HG} \right) \right) \\ \alpha _\mathrm{knee}= & {} \pi -\angle {EGF}-\angle {EGH}-\angle {HGI} \end{aligned}$$

So \(\alpha _\mathrm{knee} =\alpha _\mathrm{knee} \left( {x_l, x_d } \right) \)

Considering that FG equals to IG, \(\epsilon =\epsilon (x_{l}, x_{d})=\alpha _\mathrm{knee}(x_{l}, x_{d}/2)\)

In \(\Delta FIG\)

$$\begin{aligned} l=l(x_{l}, x_{d})=2\cdot GI \cdot \sin (\epsilon (x_{l}, x_{d})) \end{aligned}$$

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Luo, J., Wang, S., Zhao, Y. et al. Variable stiffness control of series elastic actuated biped locomotion. Intel Serv Robotics 11, 225–235 (2018). https://doi.org/10.1007/s11370-018-0248-y

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