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

doi:10.1007/s11465-024-0784-4

Frontiers of Mechanical Engineering

2024, Vol. 19

Issue (2): 13 https://doi.org/10.1007/s11465-024-0784-4

本期目录

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

Yaguang ZHU^1,²(

), Minghuan ZHANG¹, Xiaoyu ZHANG¹, Haipeng QIN¹

¹. Key Laboratory of Road Construction Technology and Equipment of Ministry of Education, School of Engineering Machinery, Chang’an University, Xi’an 710064, China
². State Key Laboratory of Robotics and Systems, Harbin Institute of Technology, Harbin 150001, China

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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.

Key words： dynamic responsiveness energy dissipation legged locomotion parallel joints quadruped robot

收稿日期: 2023-09-25 出版日期: 2024-05-30

Corresponding Author(s): Yaguang ZHU

引用本文:

. [J]. Frontiers of Mechanical Engineering, 2024, 19(2): 13.
Yaguang ZHU, Minghuan ZHANG, Xiaoyu ZHANG, Haipeng QIN. Dynamic compliance of energy-saving legged elastic parallel joints for quadruped robots: design and realization. Front. Mech. Eng., 2024, 19(2): 13.

链接本文:

https://academic.hep.com.cn/fme/CN/10.1007/s11465-024-0784-4
https://academic.hep.com.cn/fme/CN/Y2024/V19/I2/13

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Abbreviations
PD	Proportional-derivative
PI	Proportional-integral
RENS Q1	Robot with Embodied Neural System
SLIP	Spring-loaded inverted pendulum
Variables
B_l	Moments of damping of external loads equivalent to the values inside the motor
B_m	Moments of damping of the rotor in the motor
$C (θ, θ ˙)$	Velocity multiplication term containing Coriolis and centrifugal forces
D	Spring-loaded inverted pendulum system damping
D₁	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
h₁	Hip height for a single leg
h₂	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	Damping 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
L₁	Root joint link length
L₂	High rod length
L₃	Calf bar length
L₄	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 L_x 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
$q ¨$	Angular acceleration of joint motion
t_d	Delay time
v	Speed of jumping 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
θ₀	Initial angle of the spring
θ₁	Heel joint
θ₂	Hip angle
θ₃	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
$θ ˙ r e f$	Reference joint angular velocity
$θ ˙$	Angular velocity of joints
$θ ¨$	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

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