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Kinematic analysis of flexible bipedal robotic systems

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Abstract

In spite of its intrinsic complexities, the passive gait of bipedal robots on a sloping ramp is a subject of interest for numerous researchers. What distinguishes the present research from similar works is the consideration of flexibility in the constituent links of this type of robotic systems. This is not a far-fetched assumption because in the transient (impact) phase, due to the impulsive forces which are applied to the system, the likelihood of exciting the vibration modes increases considerably. Moreover, the human leg bones that are involved in walking are supported by viscoelastic muscles and ligaments. Therefore, for achieving more exact results, it is essential to model the robot links with viscoelastic properties. To this end, the Gibbs-Appell formulation and Newton’s kinematic impact law are used to derive the most general form of the system’s dynamic equations in the swing and transient phases of motion. The most important issue in the passive walking motion of bipedal robots is the determination of the initial robot configuration with which the system could accomplish a periodic and stable gait solely under the effect of gravitational force. The extremely unstable nature of the system studied in this paper and the vibrations caused by the impulsive forces induced by the impact of robot feet with the inclined surface are some of the very serious challenges encountered for achieving the above-mentioned goal. To overcome such challenges, an innovative method that uses a combination of the linearized equations of motion in the swing phase and the algebraic motion equations in the transition phase is presented in this paper to obtain an eigenvalue problem. By solving this problem, the suitable initial conditions that are necessary for the passive gait of this bipedal robot on a sloping surface are determined. The effects of the characteristic parameters of elastic links including the modulus of elasticity and the Kelvin-Voigt coefficient on the walking stability of this type of robotic systems are also studied. The findings of this parametric study reveal that the increase in the Kelvin-Voigt coefficient enhances the stability of the robotic system, while the increase in the modulus of elasticity has an opposite effect.

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References

  1. KERIMOĞLU, D., MORGÜL, O., and SARANLI, U. Stability and control of planar compass gait walking with series-elastic ankle actuation. Transactions of the Institute of Measurement and Control, 39, 312–323 (2017)

    Article  Google Scholar 

  2. IIDA, F., MINEKAWA, Y., RUMMEL, J., and SEYFARTH, A. Toward a human-like biped robot with compliant legs. Robotics and Autonomous Systems, 57, 139–144 (2009)

    Article  Google Scholar 

  3. ZELIK, K. E., HUANG, T. W. P., ADAMCZYK, P. G., and KUO, A. D. The role of series ankle elasticity in bipedal walking. Journal of Theoretical Biology, 346, 75–85 (2014)

    Article  Google Scholar 

  4. DENG, K., ZHAO, M., and XU, W. Level-ground walking for a bipedal robot with a torso via hip series elastic actuators and its gait bifurcation control. Robotics and Autonomous Systems, 79, 58–71 (2016)

    Article  Google Scholar 

  5. WU, Y., YAO, D., and XIAO, X. The effects of ground compliance on flexible planar passive biped dynamic walking. Journal of Mechanical Science and Technology, 32, 1793–1804 (2018)

    Article  Google Scholar 

  6. FATHIZADEH, M., MOHAMMADI, H., and TAGHVAEI, S. A modified passive walking biped model with two feasible switching patterns of motion to resemble multi-pattern human walking. Chaos, Solitons & Fractals, 127, 83–95 (2019)

    Article  MathSciNet  Google Scholar 

  7. SHAFEI, A. M. and SHAFEI, H. R. Considering link flexibility in the dynamic synthesis of closed-loop mechanisms: a general approach. Journal of Vibration and Acoustics, 142, 021004 (2020)

    Article  Google Scholar 

  8. CHEN, B., HUANG, J., and JI, J. C. Control of flexible single-link manipulators having Duffing oscillator dynamics. Mechanical Systems and Signal Processing, 121, 44–57 (2019)

    Article  Google Scholar 

  9. KORAYEM, M. H., SHAFEI, A. M., DOOSTHOSEINI, M., ABSALAN, F., and KADKHODAEI, B. Theoretical and experimental investigation of viscoelastic serial robotic manipulators with motors at the joints using Timoshenko beam theory and Gibbs-Appell formulation. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 230, 37–51 (2016)

    Google Scholar 

  10. QIU, Z., LI, C., and ZHANG, X. Experimental study on active vibration control for a kind of two-link flexible manipulator. Mechanical Systems and Signal Processing, 11, 623–644 (2019)

    Article  Google Scholar 

  11. KORAYEM, M. H., SHAFEI, A. M., ABSALAN, F., KADKHODAEI, B., and AZIMI, A. Kinematic and dynamic modeling of viscoelastic robotic manipulators using Timoshenko beam theory: theory and experiment. The International Journal of Advanced Manufacturing Technology, 71, 1005–1018 (2014)

    Article  Google Scholar 

  12. SHANG, D., LI, X., YIN, M., and LI, F. Dynamic modeling and fuzzy compensation sliding mode control for flexible manipulator servo system. Applied Mathematical Modelling, 107, 530–556 (2022)

    Article  MathSciNet  Google Scholar 

  13. KORAYEM, M. H. and SHAFEI, A. M. A new approach for dynamic modeling of n-viscoelastic-link robotic manipulators mounted on a mobile base. Nonlinear Dynamics, 79, 2767–2786 (2015)

    Article  MathSciNet  Google Scholar 

  14. SHAFEI, H. R., BAHRAMI, M., and TALEBI, H. A. Design of adaptive optimal robust control for two-flexible-link manipulators in the presence of matched uncertainties. Journal of Vibration and Control, 27, 612–628 (2020)

    Article  MathSciNet  Google Scholar 

  15. SHEN, Y. and KUANG, Y. Transient contact-impact behavior for passive walking of compliant bipedal robots. Extreme Mechanics Letters, 42, 101076 (2021)

    Article  Google Scholar 

  16. SAFARTOOBI, M., DARDEL, M., and MOHAMMADI-DANIALI, H. Passive walking biped robot model with flexible viscoelastic legs. Nonlinear Dynamics, 109, 2615–2636 (2022)

    Article  Google Scholar 

  17. JIN, C. and SANKAR, T. S. A systematic method of dynamics for flexible robot manipulators. Journal of Robotic System, 9, 861–891 (1992)

    Article  Google Scholar 

  18. TANG, L. W., GOUTTEFARDE, M., SUN, H. N., YIN, L. R., and ZHOU, C. J. Dynamic modelling and vibration suppression of a single-link flexible manipulator with two cables. Mechanism and Machine Theory, 162, 104347 (2021)

    Article  Google Scholar 

  19. WEI, J., CAO, D., LIU, L., and HUANG, W. Global mode method for dynamic modeling of a flexible-link flexible-joint manipulator with tip mass. Applied Mathematical Modelling, 48, 787–805 (2017)

    Article  MathSciNet  Google Scholar 

  20. KHALIL, W., BOYER, F., and MORSLI, F. General dynamic algorithm for floating base tree structure robots with flexible joints and links. ASME Journal of Mechanisms and Robotics, 9, 031003 (2017)

    Article  Google Scholar 

  21. MATA, V., PROVENZANO, S., CUADRADO, J. I., and VALERO, F. Serial-robot dynamics algorithms for moderately large number large number of joints. Mechanism and Machine Theory, 37, 739–755 (2002)

    Article  Google Scholar 

  22. SHAFEI, A. M. and KORAYEM, M. H. Theoretical and experimental study of dynamic load-carrying capacity for flexible robotic arms in point-to-point motion. Optimal Control Applications and Methods, 38, 963–972 (2017)

    Article  MathSciNet  Google Scholar 

  23. REZAEI, V. and SHAFEI, A. M. Dynamic analysis of flexible robotic manipulators constructed of functionally graded materials. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 43, 327–342 (2019)

    Article  Google Scholar 

  24. SHAFEI, A. M. and RIAHI, M. M. The effects of mode shapes on the temporal response of flexible closed-loop linkages under the impulse excitation. Mechanical Systems and Signal Processing, 178, 109256 (2022)

    Article  Google Scholar 

  25. KORAYEM, M. H. and SHAFEI, A. M. Motion equations proper for forward dynamics of robotic manipulator with flexible links by using recursive Gibbs-Appell formulation. Scientia Iranica Transaction B-Mechanical Engineering, 16, 479–495 (2009)

    Google Scholar 

  26. SHAFEI, A. M. and SHAFEI, H. R. Oblique impact of multi-flexible-link systems. Journal of Vibration and Control, 24, 904–923 (2018)

    Article  MathSciNet  Google Scholar 

  27. SHAFEI, A. M. and MIRZAEINEJAD, H. A novel recursive formulation for dynamic modeling and trajectory tracking control of multi-rigid-link robotic manipulators mounted on a mobile platform. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 235, 1204–1217 (2021)

    Google Scholar 

  28. SHAFEI, A. M. and MIRZAEINEJAD, H. A general formulation for managing trajectory tracking in non-holonomic moving manipulators with rotary-sliding joints. Journal of Intelligent & Robotic Systems, 99, 729–746 (2020)

    Article  Google Scholar 

  29. KORAYEM, M. H., SHAFEI, A. M., and SEIDI, E. Symbolic derivation of governing equations for dual-arm mobile manipulators used in fruit-picking and the pruning of tall trees. Computers and Electronics in Agriculture, 105, 95–102 (2014)

    Article  Google Scholar 

  30. MIRZAEINEJAD, H. and SHAFEI, A. M. Modeling and trajectory tracking control of a two-wheeled mobile robot: Gibbs-Appell and prediction-based approaches. Robotica, 36, 1551–1570 (2018)

    Article  Google Scholar 

  31. SHAFEI, H. R. and BAHRAMI, M. Trajectory tracking control of a wheeled mobile robot in the presence of matched uncertainties via a composite control approach. Asian Journal of Control, 23, 2805–2823 (2021)

    Article  MathSciNet  Google Scholar 

  32. SHAFEI, A. M. and SHAFEI, H. R. Dynamic behavior of flexible multiple links captured inside a closed space. Journal of Computational and Nonlinear Dynamics, 11, 051016 (2016)

    Article  Google Scholar 

  33. SHAFEI, A. M. and SHAFEI, H. R. Dynamic modeling of tree-type robotic systems by combining 3 × 3 rotation and 4 × 4 transformation matrices. Multibody System Dynamics, 44, 367–395 (2018)

    Article  MathSciNet  Google Scholar 

  34. SHAFEI, A. M. and SHAFEI, H. R. Planar multibranch open-loop robotic manipulators subjected to ground collision. Journal of Computational and Nonlinear Dynamics, 12, 061003 (2017)

    Article  Google Scholar 

  35. ZAHEDI, A., SHAFEI, A. M., and SHAMSI, M. Application of hybrid robotic systems in crop harvesting: kinematic and dynamic analysis. Computers and Electronics in Agriculture, 209, 107724 (2023)

    Article  Google Scholar 

  36. ZAHEDI, A., SHAFEI, A. M., and SHAMSI, M. Kinetics of planar constrained robotic mechanisms with multiple closed loops: an experimental study. Mechanism and Machine Theory, 183, 105250 (2023)

    Article  Google Scholar 

  37. ZAHEDI, A., SHAFEI, A. M., and SHAMSI, M. On the dynamics of multi-closed-chain robotic mechanisms. International Journal of Non-Linear Mechanics, 147, 104241 (2022)

    Article  Google Scholar 

  38. SHAFEI, A. M. and SADEGHI, Z. The kinematics and kinetics of multi-closed-chain mechanisms in the impact and non-impact stages. Meccanica, 57, 2591–2608 (2022)

    Article  MathSciNet  Google Scholar 

  39. VALLEJOS, P., RUIZ-DEL-SOLAR, J., and SWETT, F. A new methodology for the design of passive biped robots: determining conditions on the robot’s parameters for the existence of stable walking cycles. Journal of Intelligent & Robotic Systems, 63, 503–523 (2011)

    Article  Google Scholar 

  40. OBAYASHI, I., AOI, S., TSUCHIYA, K., and KOKUBU, H. Common formation mechanism of basin of attraction for bipedal walking models by saddle hyperbolicity and hybrid dynamics. Japan Journal of Industrial and Applied Mathematics, 32, 315–332 (2015)

    Article  MathSciNet  Google Scholar 

  41. GRITLI, H., KHRAEIF, N., and BELGHITH, S. Period-three route to chaos induced by a cyclic-fold bifurcation in passive dynamic walking of a compass-gait biped robot. Communications in Nonlinear Science and Numerical Simulation, 17, 4356–4372 (2012)

    Article  MathSciNet  Google Scholar 

  42. DARDEL, M., SAFARTOOBI, M., PASHAEI, M. H., GHASEMI, M. H., and KAZEMI NAVAEI, M. Finite difference method to find period-one gait cycles of simple passive walkers. Communications in Nonlinear Science and Numerical Simulation, 20, 79–97 (2015)

    Article  Google Scholar 

  43. KORAYEM, M. H. and SHAFEI, A. M. Application of recursive Gibbs-Appell formulation in deriving the equations of motion of N-viscoelastic robotic manipulators in 3D space using Timoshenko beam theory. Acta Astronautica, 83, 273–294 (2013)

    Article  Google Scholar 

  44. AHMADIZADEH, M., SHAFEI, A. M., and JAFARI, R. Frictional impact-contacts in multiple flexible links. International Journal of Structural Stability and Dynamics, 21, 2150075 (2021)

    Article  MathSciNet  Google Scholar 

  45. KORAYEM, M. H. and SHAFEI, A. M. Motion equation of nonholonomic wheeled mobile robotic manipulator with revolute-prismatic joints using recursive Gibbs-Appell formulation. Applied Mathematical Modelling, 39, 1701–1716 (2015)

    Article  MathSciNet  Google Scholar 

  46. AHMADIZADEH, M., SHAFEI, A. M., and FOOLADI, M. Dynamic modeling of closed-chain robotic manipulators in the presence of frictional dynamic forces: a planar case. Mechanics Based Design of Structures and Machines, 51, 4347–4367 (2023)

    Article  Google Scholar 

  47. DU, X., CHEN, Y., ZHANG, J., GUO, X., LI, L., and ZHANG, D. Nonlinear coupling modeling and dynamics analysis of rotating flexible beams with stretching deformation effect. Applied Mathematics and Mechanics (English Edition), 44(1), 125–140 (2023) https://doi.org/10.1007/s10483-023-2951-9

    Article  MathSciNet  Google Scholar 

  48. CHEN, W., WANG, G., LI, Y., WANG, L., and YIN, Z. The quaternion beam model for hard-magnetic flexible cantilevers. Applied Mathematics and Mechanics (English Edition), 44(5), 787–808 (2023) https://doi.org/10.1007/s10483-023-2983-8

    Article  MathSciNet  Google Scholar 

  49. AHMADIZADEH, M., SHAFEI, A. M., and FOOLADI, M. Dynamic analysis of multiple inclined and frictional impact-contacts in multi-branch robotic systems. Applied Mathematical Modelling, 91, 24–42 (2021)

    Article  MathSciNet  Google Scholar 

  50. MEI, Z. and WANG, Z. Multiplicity-induced optimal gains of an inverted multiplicity-induced optimal gains of an inverted pendulum system under a delayed proportional-derivative-acceleration feedback. Applied Mathematics and Mechanics (English Edition), 43(11), 1747–1762 (2022) https://doi.org/10.1007/s10483-022-2921-8

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, Shahid Bahonar University of Kerman, Kerman, 7616913439, Iran

    R. Fazel & A. M. Shafei

  2. Departamento de Ingeniería de Sistemas y Automática, Escuela Tácnica Superior de Ingeniería, Universidad de Sevilla, Seville, 41092, Spain

    S. R. Nekoo

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  1. R. Fazel
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  2. A. M. Shafei
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  3. S. R. Nekoo
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Correspondence to A. M. Shafei.

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Fazel, R., Shafei, A.M. & Nekoo, S.R. Kinematic analysis of flexible bipedal robotic systems. Appl. Math. Mech.-Engl. Ed. 45, 795–818 (2024). https://doi.org/10.1007/s10483-024-3081-8

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  • DOI: https://doi.org/10.1007/s10483-024-3081-8

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