Optimal Design for Flexible Passive Biped Walker Based on Chaotic Particle Swarm Optimization

  • Wu, Yao (School of Power and Mechanical Engineering, Wuhan University) ;
  • Yao, Daojin (School of Power and Mechanical Engineering, Wuhan University) ;
  • Xiao, Xiaohui (School of Power and Mechanical Engineering, Wuhan University)
  • Received : 2017.08.09
  • Accepted : 2018.06.30
  • Published : 2018.11.01


Passive dynamic walking exhibits humanoid and energy efficient gaits. However, optimal design of passive walker at multi-variable level is not well studied yet. This paper presents a Chaotic Particle Swarm Optimization (CPSO) algorithm and applies it to the optimal design of flexible passive walker. Hip torsional stiffness and damping were incorporated into flexible biped walker, to imitate passive elastic mechanisms utilized in human locomotion. Hybrid dynamics were developed to model passive walking, and period-one gait was gained. The parameters global searching scopes were gained after investigating the influences of structural parameters on passive gait. CPSO were utilized to optimize the flexible passive walker. To improve the performance of PSO, multi-scroll Jerk chaotic system was used to generate pseudorandom sequences, and chaotic disturbance would be triggered if the swarm is trapped into local optimum. The effectiveness of CPSO is verified by comparisons with standard PSO and two typical chaotic PSO methods. Numerical simulations show that better fitness value of optimal design could be gained by CPSO presented. The proposed CPSO would be useful to design biped robot prototype.


Supported by : National Natural Science Foundation of China (NSFC)


  1. Hayder FN Al-Shuka, Burkhard Corves, Wen-Hong Zhu, and Bram Vanderborght, "Multi-level control of zero-moment point-based humanoid biped robots: a review," Robotica, vol. 34, no. 11, pp. 2440-2466, 2016.
  2. Bum-Joo Lee and Kab Il Kim, "Modifiable walking pattern generation handling infeasible navigational commands for humanoid robots," Journal of Electrical Engineering & Technology, vol. 9, no. 1, pp. 344-351, 2014.
  3. Yoshiaki Sakagami, RyujinWatanabe, Chiaki Aoyama, Shinichi Matsunaga, Nobuo Higaki, and Kikuo Fujimura, "The intelligent asimo: System overview and integration," In Intelligent Robots and Systems, 2002. IEEE/RSJ International Conference on, pp. 2478-2483, 2002.
  4. Kenji Kaneko, Kensuke Harada, Fumio Kanehiro, Go Miyamori, and Kazuhiko Akachi, "Humanoid robot hrp-3," In Intelligent Robots and Systems, 2008. IROS 2008. IEEE/RSJ International Conference on, pp. 2471-2478, 2008.
  5. Scott Kuindersma, Robin Deits, Maurice Fallon, Andres Valenzuela, Hongkai Dai, Frank Permenter, Twan Koolen, Pat Marion, and Russ Tedrake, "Optimization-based locomotion planning, estimation, and control design for the atlas humanoid robot," Autonomous Robots, vol. 40, no. 3, pp. 429-455, 2016.
  6. Miomir Vukobratovic and J Stepanenko, "On the stability of anthropomorphic systems," Mathematical biosciences, vol. 15, no. 1-2, pp. 1-37, 1972.
  7. Hong, Young-Dae, Ki-Baek Lee, and Bumjoo Lee, "Dynamic simulation of modifiable walking pattern generation to handle infeasible navigational commands for humanoid robots," Journal of Electrical Engineering & Technology, vol. 11, no. 3, pp. 1921-1928, 2016.
  8. Seungchul Lim and Young Ik Son, "Discrete-time circular walking pattern for biped robots," Journal of Electrical Engineering & Technology, vol. 11, no. 5, pp. 1395-1403, 2016.
  9. Woong-Ki Lee, Dongkyoung Chwa, and Young-Dae Hong, "Control strategy for modifiable bipedal walking on unknown uneven terrain," Journal of Electrical Engineering & Technology, vol. 11, no. 6, pp. 1787-1792, 2016.
  10. Ian R Manchester, Uwe Mettin, Fumiya Iida, and Russ Tedrake, "Stable dynamic walking over uneven terrain," The International Journal of Robotics Research, vol. 30, no. 3, pp. 265-279, 2011.
  11. McGeer Tad, "Passive Dynamic Walking," The International Journal of Robotics Research, vol. 9, no. 2, pp. 62-82, 1990.
  12. Fumihiko Asano, Masaki Yamakita, and Katsuhisa Furuta, "Virtual passive dynamic walking and energy-based control laws," In Intelligent Robots and Systems, 2000. (IROS 2000). Proceedings.2000 IEEE/RSJ International Conference on, pp. 1149-1154, 2000.
  13. Richard Quint Van Der Linde, "Active leg compliance for passive walking," In Robotics and Automation, 1998, Proceedings, IEEE International Conference on, pp. 2339-2344, 1998.
  14. Fumiya Iida, Yohei Minekawa, Jurgen Rummel, and Andre Seyfarth, "Toward a human-like biped robot with compliant legs," Robotics and Autonomous Systems, vol. 57, no. 2, pp. 139-144, 2009.
  15. Ben Whittington, Amy Silder, Bryan Heiderscheit, and Darryl G Thelen, "The contribution of passiveelastic mechanisms to lower extremity joint kinetics during human walking," Gait & posture, vol. 27, no. 4, pp. 628-634, 2008.
  16. Narukawa, Terumasa, Masaki Takahashi, and Kazuo Yoshida. "Efficient walking with optimization for a planar biped walker with a torso by hip actuators and springs," Robotica, vol. 29, no. 4, pp. 641-648, 2011.
  17. Wu, Yao, Daojin Yao, and Xiaohui Xiao, "The effects of ground compliance on flexible planar passive biped dynamic walking," Journal of Mechanical Science and Technology, vol. 32, no. 4, pp. 1793-1804, 2018.
  18. Joachim Hass, J Michael Herrmann, and Theo Geisel, "Optimal mass distribution for passivity-based bipedal robots," The International Journal of Robotics Research, vol. 25, no. 11, pp. 1087-1098, 2006.
  19. Maxine Kwan and Mont Hubbard, "Optimal foot shape for a passive dynamic biped," Journal of theoretical biology, vol. 248, no. 2, pp. 331-339, 2007.
  20. S. Anbarasi, and S. Muralidharan, "Hybrid bfpso approach for effective tuning of pid controller for load frequency control application in an interconnected power system," Journal of Electrical Engineering & Technology, vol. 12, no. 3, pp. 1027-1037, 2017.
  21. Seong-In Kang, Koon-Tae Kim, Seung-Jae Lee, Jeong-Phill Kim, Kyung Choi, and Hyeong-Seok Kim, "A study on a gain-enhanced antenna for energy harvesting using adaptive particle swarm optimization," Journal of Electrical Engineering & Technology, vol. 10, no. 4, pp. 1780-1785, 2015.
  22. Riccardo Poli, James Kennedy, and Tim Blackwell, "Particle swarm optimization," Swarm intelligence, vol. 1, no. 1, pp. 33-57, 2007.
  23. Premalatha Kandasamy and A M Natarajan. "Combined Heuristic Optimization Techniques for Global Minimization," International Journal of Advances in Soft Computing and Its Applications, vol. 2, no. 1, pp. 85-99, 2010.
  24. Yang Dixiong, Zhenjun Liu and Ping Yi. "Computational efficiency of accelerated particle swarm optimization combined with different chaotic maps for global optimization," Neural Computing and Applications, vol. 28, no. 1, pp. 1245-1264, 2017.
  25. Yan Danping, et al. "Empirically characteristic analysis of chaotic PID controlling particle swarm optimization," PloS one, vol. 12, no. 5, pp. e0176359, 2017.
  26. Chuang Li-Yeh, Cheng-Hong Yang, and Jung-Chike Li. "Chaotic maps based on binary particle swarm optimization for feature selection," Applied Soft Computing, vol. 11, no. 1, pp. 239-248, 2011.
  27. Cheng-Hong Yang, Sheng-Wei Tsai, Li-Yeh Chuang and Cheng-Huei Yang. "An improved particle swarm optimization with double-bottom chaotic maps for numerical optimization," Applied Mathematics and Computation, vol. 219, no. 1, pp. 260-279, 2012.
  28. Hong-ji Meng, Peng Zheng, Rong-Yang Wu, Xiao-Jing Hao and Zhi Xie. "A hybrid particle swarm algorithm with embedded chaotic search," Cybernetics and Intelligent Systems, 2004 IEEE Conference on, pp. 367-371, 2004.
  29. Liu Ning, Li Junfeng, and Wang Tianshu, "The effects of parameter variation on the gaits of passive walking models: simulations and experiments," Robotica, vol. 27, no. 4, pp. 511-528, 2009.
  30. Westervelt E R, Grizzle J W, Chevallereau C, et al. Feedback control of dynamic bipedal robot locomotion. CRC press, 2007.
  31. C. S. Hsu, Cell to Cell Mapping A Method of Global Analysis for Nonlinear Systems. Springer Science+ Business Media, 1987.
  32. Gyebroszki, Gergely, and Gabor Csernak. "Clustered Simple Cell Mapping: An extension to the Simple Cell Mapping method," Communications in Nonlinear Science and Numerical Simulation, vol. 42, pp. 607-622, 2017.
  33. Jonathan B Dingwell, Hyun Gu Kang, and Laura C Marin, "The effects of sensory loss and walking speed on the orbital dynamic stability of human walking," Journal of biomechanics, vol. 40, no. 8, pp. 1723-1730, 2007.
  34. B. Morris and J. W. Grizzle, "A restricted poincare map for determining exponentially stable periodic orbits in systems with impulse effects: Application to bipedal robots," In IEEE Conference on Decision and Control, pp. 4199-4206, 2006.
  35. Jie Zhao, Xiaoguang Wu, Xizhe Zang, et al, "The analysis on period doubling gait and chaotic gait of the compass-gait biped model," In Robotics and Automation, 2011 IEEE International Conference on, pp. 2015-2020, 2011.
  36. Daniel D Frey and Hungjen Wang, "Adaptive one-factor-at-a-time experimentation and expected value of improvement," Technometrics, vol. 48, no. 3, pp. 418-431, 2006.
  37. Asanga Ratnaweera, Saman K Halgamuge, and Harry CWatson, "Self-organizing hierarchical particle swarm optimizer with time-varying acceleration coefficients," IEEE Transactions on evolutionary computation, vol. 8, no. 3, pp. 240-255, 2004.
  38. ChunhuaWang, Xiaoming Liu, and Hu Xia, "Multipiecewise quadratic nonlinearity memristor and its 2n-scroll and 2n+1-scroll chaotic attractors system," Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 27, no. 3, pp. 033114, 2017.
  39. Simin Yu, Jinhu Lu, Henry Leung, and Guanrong Chen, "Design and implementation of n-scroll chaotic attractors from a general jerk circuit," IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 52, no. 7, pp. 1459-1476, 2005.
  40. Yan Huang, Qining Wang, Guangming Xie, et al. "Optimal mass distribution for a passive dynamic biped with upper body considering speed, efficiency and stability. " Humanoid Robots, 2008. Humanoids 2008. 8th IEEE-RAS International Conference on, pp. 515-520, 2008.