New nonlinear stiffness actuator with predefined torque‒deflection profile

doi:10.1007/s11465-022-0721-3

Front. Mech. Eng.

2023, Vol. 18

Issue (1) : 5 https://doi.org/10.1007/s11465-022-0721-3

RESEARCH ARTICLE

New nonlinear stiffness actuator with predefined torque‒deflection profile

Wenjie JU¹, Hexi GONG¹, Keke QI¹, Rongjie KANG¹, Jian S. DAI^1,^2,³, Zhibin SONG¹(

)

¹. School of Mechanical Engineering, Tianjin University, Tianjin 300350, China
². College of Engineering, Department of Mechanical and Energy Engineering, Southern University of Science and Technology, Shenzhen 518055, China
³. King’s College London, University of London, WC2R 2LS London, UK

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Abstract

A nonlinear stiffness actuator (NSA) could achieve high torque/force resolution in low stiffness range and high bandwidth in high stiffness range, both of which are beneficial for physical interaction between a robot and the environment. Currently, most of NSAs are complex and hardly used for engineering. In this paper, oriented to engineering applications, a new simple NSA was proposed, mainly including leaf springs and especially designed cams, which could perform a predefined relationship between torque and deflection. The new NSA has a compact structure, and it is lightweight, both of which are also beneficial for its practical application. An analytical methodology that maps the predefined relationship between torque and deflection to the profile of the cam was developed. The optimal parameters of the structure were given by analyzing the weight of the NSA and the mechanic characteristic of the leaf spring. Though sliding friction force is inevitable because no rollers were used in the cam-based mechanism, the sliding displacement between the cam and the leaf spring is very small, and consumption of sliding friction force is very low. Simulations of different torque‒deflection profiles were carried out to verify the accuracy and applicability of performing predefined torque‒deflection profiles. Three kinds of prototype experiments, including verification experiment of the predefined torque‒deflection profile, torque tracking experiment, and position tracking experiment under different loads, were conducted. The results prove the accuracy of performing the predefined torque‒deflection profile, the tracking performance, and the interactive performance of the new NSA.

Keywords compliant actuator nonlinear stiffness actuator nonlinear spring predefined torque−deflection profile

Corresponding Author(s): Zhibin SONG

Just Accepted Date: 18 July 2022 Issue Date: 17 March 2023

Cite this article:

Wenjie JU,Hexi GONG,Keke QI, et al. New nonlinear stiffness actuator with predefined torque‒deflection profile[J]. Front. Mech. Eng., 2023, 18(1): 5.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-022-0721-3
https://academic.hep.com.cn/fme/EN/Y2023/V18/I1/5

Fig.1 Configuration of nonlinear stiffness actuator: (a) principle block diagram and (b) side view.

Fig.2 Configuration of nonlinear stiffness actuator: (a) undeflected nonlinear spring and (b) nonlinear spring deflected for α.

Fig.3 Static analysis diagram of the cam-based mechanism. The dull red shape denotes the cam shaft, and the bold red line represents the cam profile. The blue shape denotes the leaf spring. The green shape is the leaf spring base.

Fig.4 Schematic used for deriving the minimum stiffness of nonlinear stiffness actuator.

Fig.5 Flowchart of the design process of cam profile curve.

Fig.6 Influence of R and h₁ on (a) the weight of cam and (b) the strength of leaf spring.

Tab.1 Design parameters of nonlinear spring

Fig.7 Influence of sliding friction force on nonlinear spring: (a) accumulated sliding distance of leaf spring and (b) powers of the sliding friction force and external torque and their percentage.

Fig.8 Three-dimensional model of the nonlinear stiffness actuator. 1: motor and reducer, 2: nonlinear spring, 2-1: leaf spring base, 2-2: cam shaft, 2-3: leaf spring, 2-4: output, 2-5: static disk of encoder, 2-6: kinetic disk of encoder, and 2-7: gasket.

Fig.9 Simulations of nonlinear springs with different torque–deflection profiles: (a) stiffness curves, (b) calculated cam profile curves, comparison between simulated curve with predefined profile for (c) (0.687α³ + 0.213α² + 0.18α), (d) (0.415α⁴ ? 2.22α³ + 5.19α² + 0.18α), and (e) (?0.141α³ + 2.7α² + 0.18α).

Tab.2 Specifications of nonlinear spring

Fig.10 Prototype of nonlinear stiffness actuator (NSA): (a) NSA prototype, (b) overall structure of nonlinear spring, and (c) internal structure of nonlinear spring.

Fig.11 Experimental setup of the verification experiment of the torque–deflection profile.

Fig.12 Results of the verification experiment of the predefined torque–deflection profile.

Fig.13 Experimental results of sinusoidal tracking: (a) torque tracking for an amplitude of 15 N?m at 1 Hz, (b) position tracking for an amplitude of 100° under no load at 1 Hz, and (c) position tracking for an amplitude of 100° under a load of 2 kg at 1 Hz.

Fig.14 Experimental results of the zero-torque tracking.

Tab.3 Comparison of different nonlinear stiffness actuators

Abbreviations
NSA	Nonlinear stiffness actuator
PID	Proportion integration differentiation
RMS	Root mean square
SEA	Series elastic actuator
VSA	Variable stiffness actuator
Variables
A_z	Area of the cross section
b	Width of the leaf spring
c	Count of uniform intervals
D_cam	Weight of the cam in the cam shaft
E	Young’s modulus of the leaf spring
F	Contacting force between the cam and the leaf spring
F_a	Axial component of F
F_n	Normal force
F_s	Sliding friction force
F_t	Tangential component of F
G	Shear modulus of the leaf spring
h	Height of the leaf spring
h₁	Height of the free end of the leaf spring
h₂	Height of the fixed end of the leaf spring
I_z	Moment of inertia of the cross section
k_min	Minimum stiffness of the NSA
l	Effective length of the leaf spring
l₀	Effective length of the leaf spring on the first contact point
l_m	Effective length of the leaf spring in the mth interval
l_m?1	Effective length of the leaf spring in the (m ? 1)th interval
l_min	Minimum value of l
L	Length of the leaf spring
m	Sequence number of intervals
M_max	Maximum bending moment in the leaf spring
n	Count of leaf springs in a single rotational direction
P_s	Power of the sliding friction force
P_τ	Power of the external torque
R	Distance between points O and P
s_cm	Length of the cam profile curve in the mth interval
s_lm	Effective length variation of the leaf spring in the mth interval
s_rm	Sliding distance of the leaf spring in the mth interval
S_r	Accumulated sliding distance of leaf spring
u	Axial offset of the leaf spring
v	Tangential deformation of the leaf spring
v_s	Sliding velocity
v_M	Velocity of point M
W	Section modulus on the fixed end of the leaf spring
x₀	Abscissa of the first point on the cam profile curve
x_m	Abscissa of the mth point on the cam profile curve
x_m?1	Abscissa of the (m ? 1)th point on the cam profile curve
x_N	Abscissa of point N
x_N″	Abscissa of point N''
$x N f$	Abscissa of point N_f
y₀	Ordinate of the first point on the cam profile curve
y_m	Ordinate of the mth point on the cam profile curve
y_m?1	Ordinate of the (m ? 1)th point on the cam profile curve
y_N	Ordinate of point N
y_N″	Ordinate of point N''
$y N f$	Ordinate of point N_f
z	Distance of a point to the free end on the leaf spring
α	Deflection of the nonlinear spring
β	Angle between F_a and F_t
θ	Deflection angle of the leaf spring with length l
θ_e	Rotation of output
θ_m	Rotation of motor
μ	Coefficient of sliding friction force
ξ	Coefficient related to the uniform distribution of the shear stress
ρ	Density of cam shaft
σ	Maximum normal stress in the leaf spring
[σ]	Permissible bending stress of the leaf spring
τ	External torque
τ_max	Maximum torque applied to the NSA
$ω$	Angular velocity of the spring base

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