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

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

Nomenclature

Abbreviations
PD	Proportional-derivative
PI	Proportional-integral
RENS Q1	Robot with Embodied Neural System
SLIP	Spring-loaded inverted pendulum
Variables
Bl	Moments of damping of external loads equivalent to the values inside the motor
Bm	Moments of damping of the rotor in the motor
\mathit{C}(\theta ,\dot{\theta })	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
FN	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
Jl	Moments of inertia of external loads equivalent to the values inside the motor
Jm	Moments of inertia of the rotor in the motor
J	Jacobi matrix
J(q)	Joint inertia matrix
KS	Spring’s elastic stiffness
Kv	Virtual rotational stiffness
Kd	Damping coefficient matrices
Kp	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
Lx	Length of the line from the center of the foot end to the center of the output end of the calf joint motor
Ly	Projection of Lx on the YZ plane
m	Mass of the leg bar
MA	Knee torque value
MB	Hip torque value
MS	Spring torque
n	Number of working coils
N	Reduction ratio
\ddot{\mathit{q}}	Angular acceleration of joint motion
td	Delay time
v	Speed of jumping at the end of the foot
WM	Work performed by the motor
WS	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
θd	Rotating angle of the hip motor relative to the initial angle
θf	Feedback angle of the knee joint
θk	Initial angle of the knee joint
θout	Motor output angle
θref	Reference joint angle
{\dot{\mathit{\theta }}}_{\mathrm{r}\mathrm{e}\mathrm{f}}	Reference joint angular velocity
\dot{\mathit{\theta }}	Angular velocity of joints
\ddot{\mathit{\theta }}	Angular acceleration of joints
τm	Motor torque
τ	Single-leg dynamic control torque
τout	Input motor torque
τref	Reference torque
τs	Support-related joint moment

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).

Conflict of Interest

The authors declare that they have no conflict of interest.

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