SUGISAKAMASANORI

Journal of Advances in Artificial Life Robotics
Vol. 4(3); December (2023), pp. 122–127
ON LINE ISSN 2435-8061; ISSN-L 2435-8061
https://alife-robotics.org/jallr.html

Research Article

Realizing Variable Stiffness through Structures Designed for
Quick Motion
Katsuaki Suzuki1, Yuya Nishida2, Kazuo Ishii2

Kumamoto Industrial Research Institute, 3-11-38 Higashi machi, Higashi-ku, Kumamoto 862-0901, Kumamoto, Japan
Life Science and Systems Engineering, Kyushu Institute of Technology, 2-4 Hibikino, Wakamatsu-ku, Kitakyushu 808-0196, Fukuoka, Japan

1
2

ARTICLE INFO
Article History
Received 25 November 2022
Accepted 15 November 2023
Keywords
Variable stiffness
Link mechanism

ABSTRACTTo expand a robot’s range of activities, we aim to realize three functions: normal motion, quick
motion, and variable stiffness. In this study, we developed a method for realizing variable stiffness
using a joint mechanism that can achieve normal and quick motions. Our analysis results
confirmed that stiffness could be changed by adjusting the initial displacement of the spring
incorporated in the joint mechanism. Furthermore, we manufactured a machine containing the
proposed mechanism and evaluated the proposed method by comparing our theoretical andexperimental results considering errors. Our findings showed similar trends between the measured
and theoretical values with an error caused by friction between machine components.

© 2022 The Author. Published by Sugisaka Masanori at ALife Robotics Corporation Ltd.
This is an open access article distributed under the CC BY-NC 4.0 license(http://creativecommons.org/licenses/by-nc/4.0/)

combine actuators with springs to achieve quick motion
while suppressing the increase in mass [1], [2]. However,these mechanisms do not include variable stiffness andare thus difficult to use for applications that require
variable stiffness. Compliance control is commonly
performed using force sensors to achieve stiffness change,
but this approach has certain disadvantages, such as thedifficulty of responding to external forces that exceed thecontrol cycle [3]. Methods of achieving mechanicalvariable stiffness are also being investigated to solve theabovementioned problems [4], [5], [6]. However, thesemechanisms do not discuss agile movements and are
therefore hard to use for applications that need quick
movements. In 2013, Nakamura et al. reported a joint
mechanism that can achieve quick motion and variable
stiffness using pneumatic artificial muscles [7], [8].
Although this joint mechanism can achieve two functions,
it requires two pneumatic artificial muscles and one
magneto rheological fluid brake and utilizes air pressure
as its power source. Therefore, when these joints are usedin autonomous field robots that investigate extreme

1. Introduction
Various objectives can possibly be accomplishedusing robots with multiple functions, such as improving
industrial productivity and robotic resource sampling in
the deep sea. For multiple functions, our focus is on
equipping robots with the ability to change the flexibilityof their joints and be able to move quickly in the same
way as humans do. Robots with these capabilities may beable to sample important marine resources, such as lowhardness minerals and fast-moving organisms, in
extreme environments, such as the deep sea.
Actuators and joint mechanisms are core technologies
for realizing these functions. The joint mechanism
installed in an industrial robot generally combines areducer with a single reduction ratio and a motor. In thispaper, the motion achieved by this reducer–motor
combination is defined as normal motion. Quick motion
can be achieved using these mechanisms by increasing
the motor output, but this increases the motor mass.
Many studies are being conducted on mechanisms that

Corresponding author’s E-mail: k-suzuki@kumamoto-iri.jp, y-nishida@lsse.kyutech.ac.jp, ishii@brain.kyutech.ac.jp122environments, installing a pneumatic tank or compressor
is necessary, causing the robot to become larger.
We developed a joint mechanism that can perform
normal and instantaneous movements to expand a robot’s
range of activities [9]. In the current paper, we add afeature to the abovementioned joint mechanism bydeveloping a method of achieving variable stiffness using
two motor drive methods while maintaining the existingmechanism structure.

(a)

Cam
Linear guideCam follower
Natural length
Main Joint0

Slider

2. Variable-stiffness function of proposedmechanism

Initial displacement of spring

Fig. 1 shows the process of changing the stiffness ofan end effector. The proposed mechanism consists of anend effector, a main joint, a linear guide, a slider, a spring,
two wires, two passive links, two cams, and two motors.
Assume that the positions of the linear guide and main
joint are fixed in space. Additionally, the linear guide andslider are a prismatic pair.
First, assume that the end effector angle 𝜃𝜃𝑜𝑜 of the state(a) is 0° and initial displacement of the spring is 0 (naturallength). Further, assume that the brakes are acting on thetwo motors. When a load torque 𝑇𝑇𝑞𝑞 is applied to the end
effector and the end effector is rotated clockwise, theslider translates to the right via the wire, resulting in thestate shown in (b). As the distance between the slider andmain joint reduces, the spring is compressed to 𝛿𝛿𝑠𝑠𝑠𝑠 and acompressive force 𝐹𝐹𝑠𝑠𝑠𝑠 is applied to the spring. When the
X-direction forces of 𝐹𝐹𝑠𝑠𝑠𝑠 and 𝐹𝐹𝑤𝑤 balance each other, themovement of each mechanical element stops; here, 𝐹𝐹𝑤𝑤 is
the tension applied by 𝑇𝑇𝑞𝑞 .
Further, in the 𝜃𝜃𝑜𝑜 = 0° orientation, we consider asituation in which the two motors are rotated by the sameangle but in different directions and initial spring
displacement is 𝜀𝜀𝑠𝑠𝑠𝑠 . Subsequently, brakes are applied tothese two motors in a similar manner. Meanwhile, in state(c), 𝐹𝐹𝑠𝑠𝑠𝑠 acts on the spring and 𝐹𝐹𝑤𝑤 acts on each of the two
wires so that the forces are balanced. When the same
magnitude of 𝑇𝑇𝑞𝑞 as in the (b) state is applied to the end
effector, the slider undergoes translational motion to theright, resulting in the (d) state. In the (c) state, because
𝐹𝐹𝑠𝑠𝑠𝑠 is already acting on the spring, compared to the (b)state, the slider only experiences slight displacement inthe (d) state, enabling a balance between the X-direction
force components of 𝐹𝐹𝑤𝑤 and 𝐹𝐹𝑠𝑠𝑠𝑠 . Hence, the angulardisplacement of 𝜃𝜃𝑜𝑜 in state (d) is smaller compared tostate (b), indicating that the stiffness of the end effector
can be changed by adjusting 𝜀𝜀𝑠𝑠𝑠𝑠 .

Motor

(b)
Brake

Brake

SpringEnd effector
Wire
Passive link

Large

(c)

(d)

Small
Fig. 1 Variable-stiffness function of proposedmechanism1233. Analysis of output characteristics related tostiffness
In the case of variable stiffness, we consider the load
torque and the rotational stiffness of the end effector,
which change with the initial displacement of the spring.
Fig. 2 shows the link vectors of the proposed mechanism.
The load torque applied to rotate the end effector should
be of the same magnitude as the torque generated aroundthe end effector due to internal force, such as wire tension.
The relation between the load torque 𝑇𝑇𝑞𝑞 and the internal
force acting inside the mechanism is expressed as follows
based on their geometric relation:
𝑇𝑇𝑞𝑞 = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃𝑜𝑜 )𝑘𝑘𝑠𝑠𝑠𝑠 𝛿𝛿𝑠𝑠𝑠𝑠 𝑙𝑙𝑙𝑙 (− sin 𝛽𝛽 + tan 𝜃𝜃𝑤𝑤 cos 𝛽𝛽).

(1)Here, 𝛿𝛿𝑠𝑠𝑠𝑠 is expressed by the following equation that
employs the distance 𝑙𝑙𝑠𝑠𝑠𝑠𝑠𝑠 from the slider to the main joint
when the spring possesses natural length and distance 𝑙𝑙𝑠𝑠𝑠𝑠
after displacement:

Fig. 2 Link vectors of the proposed mechanism

0.6

(2)

𝛿𝛿𝑠𝑠𝑠𝑠 = 𝑙𝑙𝑠𝑠𝑠𝑠𝑠𝑠 − 𝑙𝑙𝑠𝑠𝑠𝑠 .

0.4

In Eq. (1), β is expressed as follows:
𝛽𝛽 = �

𝜃𝜃𝑜𝑜 −−𝜃𝜃𝑜𝑜 −∅

2∅2(𝜃𝜃𝑜𝑜 ≥ 0)

0.2

(3)

.
(𝜃𝜃𝑜𝑜 < 0)0
-0.2

Furthermore, the wire angles 𝜃𝜃𝑤𝑤 and passive link angle
𝜃𝜃𝑚𝑚 are expressed as follows based on their geometrical
relation:−1 𝑙𝑙𝑠𝑠 +𝑙𝑙𝑚𝑚 sin 𝜃𝜃𝑚𝑚 −𝑙𝑙𝑙𝑙 sin 𝛽𝛽𝜃𝜃𝑤𝑤 = sin (𝑙𝑙𝑤𝑤

𝜀𝜀𝑠𝑠𝑠𝑠𝜃𝜃𝑚𝑚 = cos −1 (cos 𝜃𝜃𝑚𝑚𝑚𝑚 − ).
𝑙𝑙
𝑚𝑚

),

=0mm

=10mm

-0.4
-0.6

(4)

=20mm

-60

-40

-20

0

20

40

60

Fig. 3 Relation between the rotational angle of end
effector and load torque(5)

Fig. 3 shows the profile of the load torque 𝑇𝑇𝑞𝑞 and angle𝜃𝜃𝑜𝑜 of the end effector at initial spring displacements 𝜀𝜀𝑠𝑠𝑠𝑠
of 0, 10, and 20 mm (Eq. (1)) [10]. Further, Table 1 shows
the values of each parameter used in the analysis. When𝜀𝜀𝑠𝑠𝑠𝑠 is 0 mm, the magnitude of 𝑇𝑇𝑞𝑞 changes continuouslywith respect to 𝜃𝜃𝑜𝑜 ; however, when 𝜀𝜀𝑠𝑠𝑠𝑠 is 10 and 20 mm,
the magnitude of 𝑇𝑇𝑞𝑞 changes quickly around 0°. This isbecause the compressive force preset in the springchanges with 𝜀𝜀𝑠𝑠𝑠𝑠 . Therefore, stiffness can be changed
considerably and instantaneously around 𝜃𝜃𝑜𝑜 = 0° bychanging the magnitude of 𝜀𝜀𝑠𝑠𝑠𝑠 .

Table 1. Design parameters
𝑘𝑘𝑠𝑠𝑠𝑠
𝑙𝑙𝑙𝑙
𝑙𝑙𝑠𝑠
𝑙𝑙𝑤𝑤
𝑙𝑙𝑠𝑠𝑠𝑠𝑠𝑠
𝑙𝑙𝑚𝑚
𝑙𝑙𝑐𝑐𝑐𝑐∅𝑚𝑚𝑚𝑚𝑚𝑚(𝜃𝜃𝑜𝑜 )
𝑚𝑚𝑚𝑚𝑚𝑚(𝜃𝜃𝑜𝑜 )𝜃𝜃𝑚𝑚𝑚𝑚124

Spring constant
Length of 𝒍𝒍𝒍𝒍
Length of 𝒍𝒍𝒔𝒔
Length of 𝒍𝒍𝒘𝒘
Length of 𝒍𝒍𝒔𝒔𝒔𝒔𝒔𝒔
Length of 𝒍𝒍𝒎𝒎
Length of 𝒍𝒍𝒄𝒄𝒄𝒄
Direction of 𝒍𝒍𝒍𝒍
Maximum movement range
Minimum movement range
Absolute angle of 𝒍𝒍𝒎𝒎
when 𝜃𝜃𝑜𝑜 = 0,𝑙𝑙𝑠𝑠𝑠𝑠 = 𝑙𝑙𝑠𝑠𝑠𝑠𝑠𝑠

500 N/m
0.025 m0.012 m0.068 m0.092 m0.03 m
0.0181 m
180°
+60°
−60°
36.3°

4. Design of experimental equipment

Conceptual modelCam

We built an experimental device to evaluate whether
changing the initial displacement of the spring changes
the rotational stiffness of the end effector of a realmachine. Fig. 4 shows the designed and manufactured
joint mechanism. The design parameters of the realmachine are shown in Table 1. The real machine wasdesigned to have a gripper-type structure in anticipationof future experiments related to object grasping andsimilar tasks. Its main structure is divided into four parts:
a slider, the base, an output link and the end effector. The
output link and end effector are connected via a spur gear
mechanism. The reduction ratio of the spur gear
mechanism is 1:1. Disregarding the friction between the
spur gear and the bearing, the torque generated in the
output link and end effector remains equivalent.
Additionally, a worm gear is installed between the motorand cam. The worm gear is included to reduce power
consumption when compressing the spring to maintainrigidity. The correspondence between the conceptualmodel of the joint mechanism and actual machine isshown in the cross-sectional view A in Fig. 4. The end
effector in the conceptual model serves the same purposethe output link in the experimental equipment. The
bottom image (side view) shown in Fig. 4 is the proposedmechanism mounted on and under the base. In otherwords, two mechanisms were built into the actualmachine, and the two end effectors were controlled
independently.
To evaluate the variable stiffness of the joint
mechanism, we designed a device known as the load
torque generator (Fig. 5) for applying a load torque to theend effector. The load torque generator comprises a link,
motor, and fixed part. The fixed part is fastened to thebase of the real machine such that the motor output shaftof the load torque generator and the rotary joint of end
effector are coaxial. Moreover, the link of the load torquegenerator is attached to the end effectors.
Fig. 6 shows the experimental device system. First, 24
V DC power is supplied to a Maxon EPOS2 motor driverand the manufactured machine. The pulse values of theencoders mounted on the motors are sent to a PC via theEPOS2. The encoder pulse values obtained by the PC areused to solve the kinematics of the joint mechanism.
Thereafter, a pulse width modulation signal is sent to themotor driver to control the angle of the motor’s output
shaft.

Main Joint
SliderEnd effector

SpringWire
Passive link

Motor
Cam follower

Linear guides

Output Link
End effector 1

SpringEnd effector 2
Cam Slider

Base
SECTION A-A
Worm Wheel Worm Screw

Motor

A

Output Link

A

Wire

SpringCam

End effector 2

Wire

Output Link
Side view

End effector 1

End effector 2

Fig. 4 Designed and manufactured machine

End effector
Load torque
motor

Motor

Side view

Front viewFig. 5 Load torque generator

125

5. Experiments

PC

Machines contain various component pairs, such as
rolling pairs in linear guides and bearings and sliding
pairs in spur gears, so the effects of friction between thesepairs should be considered. However, frictional force isdifficult to model and reflect in a theoretical formula.
Therefore, to eliminate the effect of friction between
component pairs, we applied load torque to theexperimental device with the compression spring
removed, and the load caused by friction was measured.
Fig. 7 shows the relation between the load torque applied
to the mechanism and the angle of the end effector. The
results confirmed that the torque increased with the initialdisplacement of the spring. The reason for this was thatthe force acting on the linear guide increased in the thrust
direction as the initial displacement of the spring
increased because the angle between the wire and thepassive link became acute. The measured results werelinearly approximated, and a first-order approximation
formula was derived (Fig. 7).
Next, we evaluated the variable stiffness of the joint
mechanism with the springs by applying load torque tothe end effectors. Fig. 8 shows four photos taken at 1second intervals, while Fig. 9 shows the experimental
results. The experiment was conducted by changing theinitial displacement of the spring to 0, 10, and 20 mm.
The rotational speed of the load motor was set to 500 rpm.
The load torque caused by friction was subtracted fromthe measured value using the approximation formula
shown in Fig. 7 because it is not considered in thetheoretical formula. Consequently, the torque profiles ofthe experimental and calculated values exhibited roughly
similar trends. On the contrary, the torque magnitude waslarger in the experimental value than in the calculated
value. This was because the compressive force acting on
the spring enlarged the frictional force generated in eachcomponent pair when load torque was applied to themechanism and the spring was displaced. In addition, theexperimental results confirmed that a larger initialdisplacement of the spring led to a larger error between
the calculated value and the real machine. Thisphenomenon was due to the same factors affecting theresults in Fig. 7.

Power supply (24V)

Serial communication

DC24V

Motor driver

Motor driver

Encoder Motor
Load torque generator

Encoder Motor
Motor
Encoder
Encoder
Motor
Encoder
Motor
Joint mechanism

Digital signal PWM

Digital signal PWM

Fig. 6 Device system
0.5Linear Approximation=20mm)
Experiment=20mm)
Linear Approximation=10mm)
Experiment=10mm)
Linear Approximation=0mm)
Experiment
=0mm)

0.4

0.3

y = 0.0045x + 0.1148

0.2

y = 0.0026x + 0.1217

0.1

0

y = 0.0004x + 0.0987
0

10

20

30

40

50

60

Fig. 7 Load torque and joint angle without internal
spring1

2

3

4

Fig. 8 Experiment situation
1.2

Calculation
ExperimentCalculation
ExperimentCalculation
Experiment1

0.8=20mm)=20mm)=10mm)=10mm)
=0mm)
=0mm)

0.6

0.4

0.2

0

0

10

20

30

40

50

Fig. 9 Load torque and joint angle

126

60

6. Conclusion
In this study, we demonstrated a method for realizing
variable stiffness using a joint mechanism that canperform normal and quick motions. Additionally, wederived a relational expression between the joint angle
and the load torque applied to the end effector. The
analysis results showed that the magnitude of the load
torque required to rotate the end effector changed
according to the initial displacement of the spring, thuschanging the stiffness. Furthermore, we created anexperimental device and measured the load torque andjoint angle at different initial spring displacements. A
comparison of the measured and theoretical values
revealed that despite the friction-induced error in themagnitudes of the numerical values, the measured andtheoretical values exhibit the same trends. Therefore,variable stiffness was achieved using the proposedmechanism by adjusting the initial displacement of thespring. One of the challenges here is to obtain a link
length parameter that allows for considerable changes in
stiffness. Another important issue is to reduce thefrictional force acting on the prismatic and revolute pairs
inside the mechanism. Further, improving the structureof the link mechanism is necessary for changing 𝑇𝑇𝑞𝑞
continuously from 𝜃𝜃𝑜𝑜 = 0°, which will enable it to be usedin various applications.

7.

8.

9.

10.

Authors Introduction
Dr. Katsuaki Suzuki

References
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2.

3.
4.

5.
6.

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Dr. Yuya Nishida

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Prof. Kazuo Ishii

127

He is a Researcher at the
Kumamoto Industrial ResearchInstitute, Japan. He received his
Ph.D. degree from the Kyushu
Institute of Technology in 2021.
His research interests include
joint mechanisms and theirapplications.

He is an Associate Professor at
the Graduate School of Life
ScienceandSystemEngineering, Kyushu Institute ofTechnology, Japan. His researchinterests include filed robotics,
its application, and dataprocessing.

He is a Professor at the Graduate
School of Life Science andSystem Engineering, Kyushu
Institute of Technology, Japan.
His research interests include
field robots and intelligent robot
systems.