SpringerLink
Log in

Dynamic compliance of energy-saving legged elastic parallel joints for quadruped robots: design and realization

  • Research Article
  • Published:
Frontiers of Mechanical Engineering Aims and scope Submit manuscript

Abstract

Achieving dynamic compliance for energy-efficient legged robot motion is a longstanding challenge. Although recent predictive control methods based on single-rigid-body models can generate dynamic motion, they all assume infinite energy, making them unsuitable for prolonged robot operation. Addressing this issue necessitates a mechanical structure with energy storage and a dynamic control strategy that incorporates feedback to ensure stability. This work draws inspiration from the efficiency of bio-inspired muscle–tendon networks and proposes a controllable torsion spring leg structure. The design integrates a spring-loaded inverted pendulum model and adopts feedback delays and yield springs to enhance the delay effects. A leg control model that incorporates motor loads is developed to validate the response and dynamic performance of a leg with elastic joints. This model provides torque to the knee joint, effectively reducing the robot’s energy consumption through active or passive control strategies. The benefits of the proposed approach in agile maneuvering of quadruped robot legs in a realistic scenario are demonstrated to validate the dynamic motion performance of the leg with elastic joints with the advantage of energy-efficient legs.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Abbreviations

PD:

Proportional-derivative

PI:

Proportional-integral

RENS Q1:

Robot with Embodied Neural System

SLIP:

Spring-loaded inverted pendulum

B l :

Moments of dam** of external loads equivalent to the values inside the motor

B m :

Moments of dam** of the rotor in the motor

\(\boldsymbol{C}(\theta,\dot{\theta})\) :

Velocity multiplication term containing Coriolis and centrifugal forces

D :

Spring-loaded inverted pendulum system dam**

D 1 :

Average helix diameter

d :

Diameter of the spring wire

E :

Tensile modulus of elasticity

F N :

Foot-end position force during the stabilization of the support phase

F :

Foot-end force

g :

Gravitational acceleration

G ( θ ) :

Gravity term

h :

Height of the leg

h 1 :

Hip height for a single leg

h 2 :

Height of the foot-end jump

h :

Moment vector incorporating Coriolis, centrifugal, gravitational, and spring-loaded forces

H ( θ ) :

Leg mass matrix

I :

Moment of inertia of the leg

J l :

Moments of inertia of external loads equivalent to the values inside the motor

J m :

Moments of inertia of the rotor in the motor

j :

Jacobi matrix

J ( q ) :

Joint inertia matrix

K S :

Spring’s elastic stiffness

K v :

Virtual rotational stiffness

K d :

Dam** coefficient matrices

K p :

Stiffness coefficient matrices

L :

Distance from the end of the foot to the center of the output of the calf joint

L 1 :

Root joint link length

L 2 :

High rod length

L 3 :

Calf bar length

L 4 :

Length of the follower

L x :

Length of the line from the center of the foot end to the center of the output end of the calf joint motor

L y :

Projection of Lx on the YZ plane

m :

Mass of the leg bar

M A :

Knee torque value

M B :

Hip torque value

M S :

Spring torque

n :

Number of working coils

N :

Reduction ratio

\(\ddot{\boldsymbol{q}}\) :

Angular acceleration of joint motion

t d :

Delay time

v :

Speed of jum** at the end of the foot

W M :

Work performed by the motor

W S :

Work done by the potential energy of the spring

(x, y, z):

Spatial position of the foot end relative to the hip joint coordinate system

μ :

Left- and right-leg sign variables

θ :

Actual joint angle

θ 0 :

Initial angle of the spring

θ 1 :

Heel joint

θ 2 :

Hip angle

θ 3 :

Knee angle

θ d :

Rotating angle of the hip motor relative to the initial angle

θ f :

Feedback angle of the knee joint

θ k :

Initial angle of the knee joint

θ out :

Motor output angle

θ ref :

Reference joint angle

\(\dot{\boldsymbol{\theta}}_{\text{ref}}\) :

Reference joint angular velocity

\(\dot{\boldsymbol{\theta}}\) :

Angular velocity of joints

\(\ddot{\boldsymbol{\theta}}\) :

Angular acceleration of joints

τ m :

Motor torque

τ :

Single-leg dynamic control torque

τ out :

Input motor torque

τ ref :

Reference torque

τ s :

Support-related joint moment

References

  1. Raibert M, Blankespoor K, Nelson G, Playter R. Bigdog, the rough-terrain quadruped robot. IFAC Proceedings Volumes, 2008, 41(2): 10822–10825

    Article  Google Scholar 

  2. Liu X. Harnessing compliance in the design and control of running robots. Dissertation for the Doctoral Degree. Newark: The University of Delaware, 2017

    Google Scholar 

  3. Hutter M, Gehring C, Höpflinger M A, Blösch M, Siegwart R. Toward combining speed, efficiency, versatility, and robustness in an autonomous quadruped. IEEE Transactions on Robotics, 2014, 30(6): 1427–1440

    Article  Google Scholar 

  4. Wensing P M, Wang A, Seok S, Otten D, Lang J, Kim S. Proprioceptive actuator design in the MIT Cheetah: impact mitigation and high-bandwidth physical interaction for dynamic legged robots. IEEE Transactions on Robotics, 2017, 33(3): 509–522

    Article  Google Scholar 

  5. Hutter M, Gehring C, Lauber A, Gunther F, Bellicoso C D, Tsounis V, Fankhauser P, Diethelm R, Bachmann S, Bloesch M, Kolvenbach H, Bjelonic M, Isler L, Meyer K. Anymal-toward legged robots for harsh environments. Advanced Robotics, 2017, 31(17): 918–931

    Article  Google Scholar 

  6. Koco E, Mirkovic D, Kovačić Z. Hybrid compliance control for locomotion of electrically actuated quadruped robot. Journal of Intelligent & Robotic Systems, 2019, 94(3–4): 537–563

    Article  Google Scholar 

  7. Liu X, Rossi A, Poulakakis I. A switchable parallel elastic actuator and its application to leg design for running robots. IEEE/ASME Transactions on Mechatronics, 2018, 23(6): 2681–2692

    Article  Google Scholar 

  8. Sharbafi M A, Yazdanpanah M J, Ahmadabadi M N, Seyfarth A. Parallel compliance design for increasing robustness and efficiency in legged locomotion–theoretical background and applications. IEEE/ASME Transactions on Mechatronics, 2021, 26(1): 335–346

    Google Scholar 

  9. Mazumdar A, Spencer S J, Hobart C, Salton J, Quigley M, Wu T F, Bertrand S, Pratt J, Buerger S P. Parallel elastic elements improve energy efficiency on the STEPPR bipedal walking robot. IEEE/ASME Transactions on Mechatronics, 2017, 22(2): 898–908

    Article  Google Scholar 

  10. Ashtiani M S, Aghamaleki Sarvestani A, Badri-Spröwitz A. Hybrid parallel compliance allows robots to operate with sensorimotor delays and low control frequencies. Frontiers in Robotics and AI, 2021, 8: 645748

    Article  Google Scholar 

  11. Yin X C, Yan J C, Wen S, Zhang J T. Spring-linkage integrated mechanism design for jum** robots. Robotics and Autonomous Systems, 2022, 158: 104268

    Article  Google Scholar 

  12. Ruppert F, Badri-Spröwitz A. Series elastic behavior of biarticular muscle-tendon structure in a robotic leg. Frontiers in Neurorobotics, 2019, 13: 64

    Article  Google Scholar 

  13. Heim S, Millard M, Le Mouel C, Badri-Spröwitz A. A little dam** goes a long way: a simulation study of how dam** influences task-level stability in running. Biology Letters, 2020, 16(9): 20200467

    Article  Google Scholar 

  14. Grimminger F, Meduri A, Khadiv M, Viereck J, Wüthrich M, Naveau M, Berenz V, Heim S, Widmaier F, Flayols T, Fiene J, Badri-Spröwitz A, Righetti L. An open torque-controlled modular robot architecture for legged locomotion research. IEEE Robotics and Automation Letters, 2020, 5(2): 3650–3657

    Article  Google Scholar 

  15. Rond J J, Cardani M C, Campbell M I, Hurst J W. Mitigating peak impact forces by customizing the passive foot dynamics of legged robots. Journal of Mechanisms and Robotics, 2020, 12(5): 051010

    Article  Google Scholar 

  16. Yesilevskiy Y, Gan Z Y, Remy C D. Optimal configuration of series and parallel elasticity in a 2d monoped. In: Proceedings of 2016 IEEE International Conference on Robotics and Automation. Stockholm: IEEE, 2016: 1360–1365

    Google Scholar 

  17. Ambrose E, Ames A D. Improved performance on moving-mass hop** robots with parallel elasticity. In: Proceedings of 2020 IEEE International Conference on Robotics and Automation. Paris: IEEE, 2020: 2457–2463

    Google Scholar 

  18. Piovan G, Byl K. Approximation and control of the slip model dynamics via partial feedback linearization and two-element leg actuation strategy. IEEE Transactions on Robotics, 2016: 399–412

  19. Yang J J, Sun H, An H, Wang C H. Impact mitigation for dynamic legged robots with steel wire transmission using nonlinear active compliance control. In: Proceedings of 2021 IEEE International Conference on Robotics and Automation. **’an: IEEE, 2021: 2525–2531

    Google Scholar 

  20. Zhu Y G, Zhou S J, Gao D X, Liu Q. Synchronization of nonlinear oscillators for neurobiologically inspired control on a bionic parallel waist of legged robot. Frontiers in Neurorobotics, 2019, 13: 59

    Article  Google Scholar 

  21. Zhu Y G, Zhang L, Manoonpong P. Generic mechanism for waveform regulation and synchronization of oscillators: an application for robot behavior diversity generation. IEEE Transactions on Cybernetics, 2022, 52(6): 4495–4507

    Article  Google Scholar 

  22. Sharbafi M A, Yazdanpanah M J, Ahmadabadi M N, Seyfarth A. Parallel compliance design for increasing robustness and efficiency in legged locomotion-theoretical background and applications. IEEE/ASME Transactions on Mechatronics, 2021, 26(1): 335–346

    Google Scholar 

  23. Park H W, Wensing P M, Kim S. High-speed bounding with the MIT Cheetah 2: control design and experiments. The International Journal of Robotics Research, 2017, 36(2): 167–192

    Article  Google Scholar 

  24. Kolvenbach H, Hampp E, Barton P, Zenkl R, Hutter M. Towards jum** locomotion for quadruped robots on the moon. In: Proceedings of 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems. Macau: IEEE, 2019: 5459–5466

    Google Scholar 

  25. Zhang C, Zou W, Ma L P, Wang Z Q. Biologically inspired jum** robots: a comprehensive review. Robotics and Autonomous Systems, 2020, 124: 103362

    Article  Google Scholar 

  26. Kau N, Schultz A, Ferrante N, Slade P. Stanford doggo: an open-source, quasi-direct-drive quadruped. In: Proceedings of 2019 International Conference on Robotics and Automation. Montreal: IEEE, 2019: 6309–6315

    Google Scholar 

  27. Roberts S, Koditschek D E. Mitigating energy loss in a robot hop** on a physically emulated dissipative substrate. In: Proceedings of 2019 International Conference on Robotics and Automation. Montreal: IEEE, 2019: 6763–6769

    Google Scholar 

  28. Haldane D W, Plecnik M, Yim J K, Fearing R S. A power modulating leg mechanism for monopedal hop**. In: Proceedings of 2016 IEEE/RSJ International Conference on Intelligent Robots and Systems. Daejeon: IEEE, 2016: 4757–4764

    Google Scholar 

  29. Liu X, Rossi A, Poulakakis I. A switchable parallel elastic actuator and its application to leg design for running robots. IEEE/ASME Transactions on Mechatronics, 2018, 23(6): 2681–2692

    Article  Google Scholar 

  30. Mo A, Izzi F, Haeufle D F B, Badri-Spröwitz A. Effective viscous dam** enables morphological computation in legged locomotion. Frontiers in Robotics and AI, 2020, 7: 110

    Article  Google Scholar 

  31. Lakatos D, Friedl W, Albu-Schäffer A. Eigenmodes of nonlinear dynamics: definition, existence, and embodiment into legged robots with elastic elements. IEEE Robotics and Automation Letters, 2017, 2(2): 1062–1069

    Google Scholar 

  32. Picardi G, Chellapurath M, Iacoponi S, Stefanni S, Laschi C, Calisti M. Bioinspired underwater legged robot for seabed exploration with low environmental disturbance. Science Robotics, 2020, 5(42): eaaz1012

    Article  Google Scholar 

  33. Ruina A, Bertram J E A, Srinivasan M. A collisional model of the energetic cost of support work qualitatively explains leg sequencing in walking and gallo**, pseudo-elastic leg behavior in running and the walk-to-run transition. Journal of Theoretical Biology, 2005, 237(2): 170–192

    Article  MathSciNet  Google Scholar 

  34. Zheng J H, Niu J C, Jiang M S, Li M, Rong X W. Dynamic analysis and simulation of spring legs in quadruped robot based on trot gait. Journal of Central South University, 2015, 46(8): 2877–2883

    Google Scholar 

  35. Zhang M H, Zhu YG, Cao A, Wei Q B, Liu Q. Body trajectory optimisation of walking gait for a quadruped robot. IET Cyber-Systems and Robotics, 2023, 5(3): e12094

    Article  Google Scholar 

  36. Ding Y R, Park H W. Design and experimental implementation of a quasi-direct-drive leg for optimized jum**. In: Proceedings of 2017 IEEE/RSJ International Conference on Intelligent Robots and Systems. Vancouver: IEEE, 2017: 300–305

    Google Scholar 

  37. Fahmi S, Mastalli C, Focchi M, Semini C. Passive whole-body control for quadruped robots: experimental validation over challenging terrain. IEEE Robotics and Automation Letters, 2019, 4(3): 2553–2560

    Article  Google Scholar 

  38. Qin H P, Zhu Y G, Zhang Y Q, Wei Q B. Terrain estimation with least squares and virtual model control for quadruped robots. Journal of Physics: Conference Series, 2022, 2203(1): 012005

    Google Scholar 

Download references

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant No. 62373064), in part by the State Key Laboratory of Robotics and Systems, Harbin Institute of Technology, China (Grant No. SKLRS-2023-KF-05), and in part by the Fundamental Research Funds for Central Universities, China (Grants Nos. 300102259308 and 300102259401).

Author information

Authors and Affiliations

  1. Key Laboratory of Road Construction Technology and Equipment of Ministry of Education, School of Engineering Machinery, Chang’an University, **’an, 710064, China

    Yaguang Zhu, Minghuan Zhang, **aoyu Zhang & Haipeng Qin

  2. State Key Laboratory of Robotics and Systems, Harbin Institute of Technology, Harbin, 150001, China

    Yaguang Zhu

Authors
  1. Yaguang Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Minghuan Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. **aoyu Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Haipeng Qin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yaguang Zhu.

Ethics declarations

Conflict of Interest The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhu, Y., Zhang, M., Zhang, X. et al. Dynamic compliance of energy-saving legged elastic parallel joints for quadruped robots: design and realization. Front. Mech. Eng. 19, 13 (2024). https://doi.org/10.1007/s11465-024-0784-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11465-024-0784-4

Keywords

Search

Navigation