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A Novel Impedance Control Based on Equivalent Stiffness for Hydraulic Single-leg Robot

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  • Robot and Applications
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Abstract

This paper presents a proposed strategy for improving the position-tracking accuracy of force-based impedance control in hydraulic single-leg robots. Initially, the mechanical structure and drive system of the single-leg robot are introduced. Subsequently, a kinetic and dynamic model is developed to determine the desired position and force for each joint based on the given action. The proposed strategy, called equivalent stiffness impedance control, is then presented. It combines a penalty function and the stiffness of each joint near the desired position to calculate the equivalent stiffness. Simulations and experiments are conducted to evaluate the performance of the control strategy. The results demonstrate that the proposed strategy achieves fast response speed and high position tracking accuracy. Moreover, the mechanical characteristics near the desired position are comparable to traditional impedance control. This research provides valuable insights for impedance control in bionic-legged robots.

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Abbreviations

A p :

Piston area of cylinder

B c :

Damping matrix of hydraulic cylinder

B x, B z :

Horizontal and vertical damping coefficient of leg

C tp :

Leakage coefficient of cylinder

F a, F h_a, F k_a :

Actuator force vector, actuator force of hip and knee joint

F a_com, F h_com, F k_com :

Actuator force vector, actuator force of hip and knee joint to provide compensation force

F act :

Actual output force of actuator

F a_d :

Desired cylinder output force

F a_e :

Equivalent actuator force

F a_f, F h_f, F k_f :

Actuator force vector, actuator force of hip and knee joint to provide foot output force

F f_f, F f_x, F f_z :

Foot output force vector, foot output force in horizontal and vertical direction

G :

Gravity matrix

G sv (s):

Transfer function of servo valve

g :

Gravitational acceleration

H :

Centripetal force and Coriolis force matrix

h h, h h_d :

Height of hip, desired height of hip

Δh h :

Height difference between desired and actual height of hip

I :

Output current of servo valve amplifier

I zz_t, I zz_c, I zz_z :

Rotational inertia of thigh, calf and actuator

J :

Jacobian matrix

K a, K sv :

Magnification of servo valve amplifier and servo valve

K c_p :

Proportionality coefficient of PI controller

K d, K h_d, K k_d :

Desired actuator stiffness vector, desired actuator stiffness of hip and knee joint

K e :

Equivalent stiffness

k h_max, k k_max :

Maximum penalty actuator stiffness of hip and knee joint

K p, K h_p, K k_p :

Penalty actuator stiffness vector, penalty actuator stiffness of hip and knee joint

K q, K c :

Flow gain and flow-pressure coefficient of servo valve

K x, K z :

Horizontal and vertical stiffness of leg

l a, l ah_h, l ak_k :

Moment arm vector, moment arm of hip and knee actuator

l h, l t, l c, l a :

Length of hip, thigh, calf and actuator

l mc :

Distance between calf mass center and hip joint

M :

Inertial force matrix

m h, m t, m c, m a :

Mass of hip, thigh, calf and actuator

m L, B L, K L :

Equivalent mass, damping and stiffness of load

p L :

Load pressure of hydraulic system

Q :

Flow rate

T c_i :

Integration time constant of PI controller

U c :

Input voltage of servo valve amplifier

V p :

Penalty coefficient

V t :

Volume of cylinder

x hj_g, x kj_g :

X-coordinate of hip joint and knee joint (In leg global coordinate system)

x hm, x tm, x cm, x am :

Mass center x-coordinate of hip, thigh, calf and actuator (In local coordinate system)

x tm_g, x cm_g, x akm_g :

Mass center x-coordinate of hip, thigh, calf and knee actuator (In leg global coordinate system)

x f :

X-coordinate of foot (In leg global coordinate system)

X p :

Displacement of cylinder piston

X v :

Spool displacement of servo valve

z hj_g, z kj_g :

Z-coordinate of hip joint and knee joint (In leg global coordinate system)

z hm, z tm, z cm, z am :

Mass center z-coordinate of hip, thigh, calf, and actuator (In local coordinate system)

z tm_g, z cm_g, z akm_g :

Mass center z-coordinate of hip, thigh, calf, and knee actuator (In leg global coordinate system)

z f :

Z-coordinate of foot (In leg global coordinate system)

β e :

Hydraulic oil bulk modulus of elasticity

φ c :

Angle between calf axis and the connection line of hip joint and calf mass center

η :

Ratio of actual joint angle error and desired maximum joint angle error

θ, θ h, θ k :

Angle vector, angle of hip joint and knee joint

Δθ, Δθ h, Δθ k :

Difference angle vector, difference angle of hip joint and knee joint

θ ah_h, θ ak_t :

Angle between hip joint actuator and hip, angle between knee actuator and thigh

θ h_d, θ k_d :

Desired angle of hip joint and knee joint

τ, τ h, τ k :

Joint torque vector, hip and knee joint torque

τ com, τ h_com, τ k_com :

Compensation torque vector, compensation hip and knee joint torque

τ f, τ h_f, τ k_f :

Torque vector, torque of hip joint and knee joint to provide foot output force

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Authors and Affiliations

  1. Zhejiang Lab, 1818 West Wenyi Road, Hangzhou, Zhejiang, 311121, China

    Pengyu Zhao, Anhuan Xie, Shiqiang Zhu, Lingkai Chen, Lingyu Kong & Dan Zhang

  2. Zhejiang University, 38 Zheda Road, Hangzhou, Zhejiang, 310027, China

    Anhuan Xie & Shiqiang Zhu

  3. York University, Toronto, Canada

    Dan Zhang

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  1. Pengyu Zhao
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Correspondence to Pengyu Zhao.

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The authors declare that there is no competing financial interest or personal relationship that could have appeared to influence the work reported in this paper.

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The work presented in this paper is supported by the National Natural Science Foundation of China (Grant No. 52205076), the Key Research Project of Zhejiang Lab (No. G2021NB0AL03), the Leading Innovation and Entrepreneurship Team of Zhejiang Province of China (Grant No. 2018R01006) and the Ten Thousand Talents Program of Zhejiang Province (Grant No. 2019R51010).

Pengyu Zhao received his Ph.D. degree in mechatronic engineering from Zhejiang University, Hangzhou, China, in 2018. He is currently a researcher at Intelligent Robot Research Center, Zhejiang Lab. His research interests include biped robot drive and control system, fluid power transmission and control.

Anhuan Xie received his M.S. degree in mechatronic engineering from Zhejiang University, Hangzhou, China, in 2011. He is currently working at Intelligent Robot Research Center, Zhejiang Lab and Zhejiang University. His research interests include biped robot and low altitude manned vehicle.

Shiqiang Zhu received his Ph.D. degree in mechatronic engineering from Zhejiang University, Hangzhou, China, in 1995. He is currently the chief of Zhejiang Lab and a professor in Zhejiang University.

Lingkai Chen received his M.S. degree in mechanical manufacturing and automation from Zhejiang University, Hangzhou, China, in 2011. He is currently an engineer at Intelligent Robot Research Center, Zhejiang Lab. His research interests include biped robot mechanical structure design and optimization.

Lingyu Kong received his Ph.D. degree in Vehicle Engineering from Shanghai Jiao Tong University, China, in 2018. He is currently a researcher at Intelligent Robot Research Center, Zhejiang Lab. His research interests include mechatronics engineering, robotics and manipulator calibration.

Dan Zhang received his Ph.D. degree in robotics and mechatronics from Laval University, Quebec City, Canada, in 2000. Dr. Zhang was the Director of the Board of Directors of Durham Region Manufacturing Association, Canada, and is the Director of the Board of Directors of Professional Engineers Ontario, Lake Ontario Chapter, Canada. He is a Registered Professional Engineer of Canada, a Fellow of Canadian Society for Mechanical Engineering (CSME), a Senior Member of Society of Manufacturing Engineers (SME), and a member of American Society of Mechanical Engineers (ASME). He was the recipient of the MMO Industrial Fellowship.

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Zhao, P., Xie, A., Zhu, S. et al. A Novel Impedance Control Based on Equivalent Stiffness for Hydraulic Single-leg Robot. Int. J. Control Autom. Syst. 22, 1636–1653 (2024). https://doi.org/10.1007/s12555-022-0264-8

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  • DOI: https://doi.org/10.1007/s12555-022-0264-8

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