SUGISAKAMASANORI

Journal of Advances in Artificial Life Robotics
Vol. 1(3); December (2020), pp. 144–152
ON LINE ISSN 2435-8061; ISSN-L 2435-8061
https://alife-robotics.org/jallr.html

Research Article

A new rotary actuator capable of rapid motion using anantagonistic cam mechanism
Katsuaki Suzuki1, Yuya Nishida1, Takashi Sonoda2, Kazuo Ishii1

Kyushu Institute of Technology, 2-4 Hibikino, Wakamatsu, Kitakyushu, Fukuoka 808-0196, Japan
Nishinippon Institute of Technology, 1-11 Aratsu, Kanda, Miyako, Fukuoka 800-0394, Japan

1
2

ARTICLE INFO
Article History
Received 18 November 2019
Accepted 28 August 2020
Keywords
High-power joint
Link mechanism
Cam mechanism
Design principle

ABSTRACT

Animals can achieve agile behaviors such as jumping and throwing in addition to flexible
behaviors with the same musculoskeletal systems, and those movements can extend the range of
their activities. We have been working on actuators capable of rapid and flexible motions learning
the musculoskeletal systems. In this paper, we propose a new rotary actuator using a pair ofmotors, springs, and cams to perform three functions, namely, normal motion, rapid or
instantaneous motion, and rigidity control using an antagonistic cam mechanism, and describe theoperating principle of the proposed mechanism, the mathematical model of the mechanism during
rapid motion, and the design principle of the cam, which is a key mechanical element in thismechanism. Finally, we present an analysis of the error between the theoretical the measurement
results during rapid motion.

© 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/).

motions using energy charging system and proposed Inertia
Actuator and Cam Charger [7], [8].

1. Introduction

For jumping, it is necessary to accelerate the upper
body first and then the entire body, including the actuator.
The output part must be passively moved when an
external force is applied to it to achieve flexible contact.
It is difficult to realize these functions in a robot because
a high-output general motor and reducer inevitably tend
to be heavy. In addition, actuators that combine a motorand speed reducer generally do not exhibit backdrivability. For robots to achieve rapid acrobatic
movement and flexible contact movements, lightweight,
high-power actuators and flexible actuators are necessary[9],[10],[11],[12]. The authors developed a high-power
joint mechanism that mimics the rear leg of a locust. Thismechanism is operated using a motor to realize an
actuator with high stand-alone properties and various
functions [13],[14]. The mechanism can realize normal
motion and rapid motion and can control instantaneous
force. When jumping, the locust stores energy in theelastic elements in the exoskeleton of its hind legs by

Robots are generally designed for necessary target
tasks such as picking and walking with the negotiation of
actuator power, structural strength, computer power,
battery and so on. Designing a robot with additional
functions such as jumping, throwing, and flexible contact
movements causes the design conflict because ofavailable actuators and materials, however, is one way of
expanding the range of robot activity. Marc Raibert et al.
developed one-, two-, and four-leg that can balance in theplane and in 3D in 1980s [1], [2] and currently thetechnologies are taken over to famous quadruped robots
Big Dog and Spot. H. Okubo designed a jumper activatedby small actuators using self-energizing springs [3]. S.
Hyon and T. Mita developed robots able to jump, hop,and execute other gymnastic movements [4],[5],[6], andtheir robots realized biologically inspired hopping,
multiple-DOF jumping, and multilink gymnastics. We
have been working on actuators capable of rapid and flexible

Corresponding author’s E-mail: suzuki.katsuaki448@mail.kyutech.jp, ynishida@lsse.kyutech.ac.jp, ishii@brain.kyutech.ac.jp

144

proposed mechanism is that it has three functions:
Normal motion, rapid motion, and rigidity control of themain joint. For normal motion, the output link moves
with the same output (output = torque × speed) as thebuilt-in motor. The output link is moved by rotating two
cams that are coaxial with the motor in the same direction.
In the case of rapid motion, the output link can move
momentarily to generate more power than the built-in
motor. When performing rapid motion, first the two cams
are rotated in different directions to store energy in thespring. Next, the cam follower, which is in contact withthe cam, moves away from the cam when it reaches therelease-operation point. Thereafter, the stored energy isreleased. At this time, since the joint of the passive link
is not affected by the viscous resistance of the motor
reducer, the release operation is performed with minimalloss. After release, the slider performs translationalmotion using the force stored in the spring.

simultaneously contracting the high-strength extensor
muscles and the small flexor muscles [15],[16]. Then, by
relaxing only the extensor muscle, it releases the stored
power and jumps. During the contraction of the muscles,
the flexor muscles use a portion called a ramp to increase
the moment arm to compensate for the difference in force
between the flexors and extensors. The mechanism
developed by the authors comprises a link mechanism forthe exoskeleton, a spring for the elastic element, a
reducer with different reduction ratios, and a motor andwire for the extensor and flexor muscles to closely mimic
the rear leg structure of the locust. The structure of theramp is expressed by making the link ratio asymmetric.
In rapid motion, the link corresponding to the leg is
moved rapidly by removing the voltage applied to themotor on the flexor side after spring compression.
However, at this time, the flexor side motor has astructure that rotates passively, and hence, the viscous
resistance of the reducer affects it, and a large
instantaneous force cannot be obtained. Furthermore,since the link ratio is asymmetric and the reduction ratiosof the two motors are different, the rotation direction of
the link corresponding to the leg is limited to one
direction during the rapid motion.
In the present research, we proposed a new actuator
locust-mimicking mechanism to realize functions such as
normal motion, more powerful rapid motion, and thestiffness control of the main joint. The cam can set anarbitrary reduction ratio curve according to its shape.
Since intermittent motion can be realized by abruptly
changing the reduction ratio, the release mechanism withthe cam realizes a release motion that is unaffected by
viscosity [17],[18]. In addition, by adopting asymmetrical link ratio of the mechanism and using
identical specifications for the two motors and reducers,
it is possible to eliminate the restrictions on the rotation
direction during rapid motion. In this paper, we describe
the operating principle of the high-power joint
mechanism with cams, the design principle of the cam,and the results of evaluation of the validity of applying
an instantaneous force using an actual machine. The
cams play an important role in this mechanism.

Output link
Main Joint

Counterclockwise

Clockwise

Linear shaft

wire

Constant

Passive link
SpringSliderCam Cam follower Motor(a) Overview(b) Normal motion
𝑇𝑇𝑜𝑜Compress

𝑭𝑭𝒘𝒘𝒍𝒍
𝑇𝑇𝑐𝑐1

𝑇𝑇𝑠𝑠1

̇
𝜃𝜃𝑐𝑐1

𝑇𝑇𝑠𝑠2
̇
𝜃𝜃𝑐𝑐2

(c) Charging

𝑭𝑭𝒘𝒘𝒍𝒍
𝑇𝑇𝑐𝑐2

𝜃𝜃𝑜𝑜̇

𝐹𝐹𝑠𝑠𝑠𝑠𝑣𝑣𝑙𝑙(d) Rapid motion

Fig.1 Model of the proposed mechanism

2. Overview of the proposed mechanism
The proposed mechanism is shown in Fig. 1. The
mechanism consists of two motors, cams, cam followers,a passive link, wires, a slider, a linear shaft, a spring, amain joint, and an output link. The main feature of the

145

𝐹𝐹𝑤𝑤2

Table 1 Variables used for analyzing the joint mechanism

𝒀

𝚺𝑮

𝒁

𝑿

𝒍𝒍𝒘𝒘𝒍𝒍𝑙𝑙𝑠𝑠𝑙𝑙1

𝒍𝒍𝒍𝒍𝒍𝒍𝜃𝜃𝑤𝑤1 𝑙𝑙𝑙𝑙𝑙𝑙1
𝑘𝑘𝑠𝑠𝑠𝑠

𝜃𝜃𝑜𝑜

𝒍𝒍𝒙𝒙

∅

𝑙𝑙𝑙𝑙𝑙𝑙∗Scalar projection of 𝒍𝒍𝒍𝒍∗∅

Angle between 𝒍𝒍𝒍𝒍𝒍𝒍 and 𝒍𝒍𝒍𝒍𝒍𝒍

𝑭𝑭𝒔𝒔𝒔𝒔Force vector of spring

𝑇𝑇𝑐𝑐∗

Torque of motor

𝑙𝑙𝑝𝑝𝑝𝑝∗𝜃𝜃𝑜𝑜𝑙𝑙𝑙𝑙𝑙𝑙2𝜃𝜃𝑤𝑤∗

𝒍𝒍𝒍𝒍𝒍𝒍𝑘𝑘𝑠𝑠𝑠𝑠𝒍𝒍𝒔𝒔𝒔𝒔 𝜃𝜃𝑤𝑤2

Link vector (x = 𝑙𝑙 ∗,𝑤𝑤 ∗,𝑝𝑝 ∗,𝑐𝑐𝑐𝑐 ∗,𝑐𝑐𝑐𝑐 ∗,𝑝𝑝𝑝𝑝 ∗,𝑠𝑠 ∗,sp)
Scalar projection of 𝒍𝒍𝒑𝒑∗

Absolute angle of output link

Absolute angle of wire
Spring constant𝑭𝑭𝒘𝒘∗

Tension vector of wire

𝒍𝒍𝒔𝒔𝒍𝒍𝑇𝑇𝑝𝑝∗Torque of passive link

Fig.2 Link vectors in the proposed mechanism𝜓𝜓∗

Angle between 𝒍𝒍𝒑𝒑∗ and 𝒍𝒍𝒑𝒑𝒑𝒑∗

𝒍𝒍𝒔𝒔𝒍𝒍𝜓𝜓1

𝒍𝒍𝒄𝒄𝒔𝒔𝒍𝒍

𝜃𝜃𝑐𝑐𝑠𝑠1

𝒍𝒍𝒔𝒔𝒍𝒍𝒍𝒍
𝒍𝒍𝒄𝒄𝒍𝒍𝒍𝒍

𝒍𝒍𝒔𝒔𝒍𝒍𝒍𝒍𝒔𝒔𝒍𝒍𝑙𝑙𝒍𝒍𝒔𝒔𝒍𝒍𝒍𝒍 𝑠𝑠𝑙𝑙2

𝒍𝒍𝒄𝒄𝒍𝒍𝒍𝒍𝜓𝜓2

𝒍𝒍𝒄𝒄𝒔𝒔𝒍𝒍

𝒍𝒍𝒘𝒘𝒍𝒍𝜃𝜃𝑐𝑐𝑠𝑠2

𝑭𝑭𝒔𝒔𝒔𝒔 = 𝑘𝑘𝑠𝑠𝑠𝑠 𝜹𝜹𝒔𝒔𝒔𝒔

Torque of output link

𝒗𝒗𝒍𝒍
𝜃𝜃𝑜𝑜̇

Velocity vector of slider𝜃𝜃𝑐𝑐𝑐𝑐∗∗Then, the translational movement of the slider isconverted to the rotational movement of the output link
via the wire. Thus, swift rotary motion is realized. For
stiffness control, the magnitude of stiffness of the main
joint can be changed. In a fully compressed spring, asshown in Fig. 1 (c), the two wires will experience moretension than if the spring were not compressed. When thetwo cams are maintained in this state and a force isapplied to the output link from the outside, the reaction
force of the output link becomes larger than that in thestate in which the spring is not compressed. Owing tothese characteristics, the mechanism can change theamount of compression of the spring and the rigidity ofthe main joint by controlling the postures of the two cams.
We derived a mathematical model for rapid motion.
First, consider the torque generated in the output linkfrom a static point of view. Fig. 2 shows the link vectorof the proposed mechanism, and Table 1 lists thevariables considered in the analysis. The force 𝑭𝑭𝒔𝒔𝒔𝒔
applied to the slider is determined by 𝜹𝜹𝒔𝒔𝒔𝒔 and 𝑘𝑘𝑠𝑠𝑠𝑠 :𝜹𝜹𝒔𝒔𝒔𝒔 = 𝑳𝑳𝒔𝒔𝒔𝒔𝒔𝒔 − 𝒍𝒍𝒔𝒔𝒔𝒔

𝑇𝑇𝑜𝑜

Angular velocity of output link
Absolute angle of 𝒍𝒍𝒄𝒄𝒄𝒄∗
Index 1 or 2

The thrust 𝑭𝑭𝒔𝒔𝒔𝒔 of the slider is transmitted to the output
link via the wire. The force 𝐹𝐹𝑤𝑤∗ generated on the wire isexpressed as shown in the following equation.
𝐹𝐹𝑤𝑤∗ =

�𝑭𝑭𝒔𝒔𝒔𝒔 �𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃𝑤𝑤∗(3)

where 𝜃𝜃𝑤𝑤∗ is the angle of the wire.
By using the Jacobian matrix 𝑱𝑱𝒔𝒔𝒔𝒔∗ = (𝒍𝒍𝒍𝒍∗ × 𝒆𝒆𝒛𝒛 ) derived
from the outer product of the force 𝑭𝑭𝒘𝒘∗ generated on thewire and the rotation direction vector 𝒆𝒆𝒛𝒛 , the torque 𝑇𝑇𝑜𝑜
generated in the output link is obtained.
𝑇𝑇𝑜𝑜 = 𝑱𝑱𝑻𝑻𝒔𝒔𝒔𝒔∗ 𝑭𝑭𝒘𝒘∗(4)

Further, the reduction ratio, which represents therelationship between the input and output when the force
𝑭𝑭𝒔𝒔𝒔𝒔 generated by the spring is converted into the torque
𝑇𝑇𝑜𝑜 of the output link, is expressed as follows.
𝐺𝐺𝑠𝑠𝑠𝑠 =

(1)

𝑇𝑇𝑜𝑜

�𝑭𝑭𝒔𝒔𝒔𝒔 �(5)During rapid motion, the slider, two wires, and an
output link are the elements that move. Among theseelements, the inertial force of the slider and the output
link is large and cannot be ignored. We calculated thevelocity from the viewpoint of energy loss. The energybalance during the rapid motion is expressed as follows:(2)

where δsp is the spring displacement vector; Lspn; lsp is theslider position vector; and ksp is the spring constant. Lspn
represents the natural length of the spring.
146

1
1
1
2
2
𝑚𝑚𝑙𝑙 𝜈𝜈𝑙𝑙 2 + 𝐼𝐼𝑜𝑜 𝜃𝜃𝑜𝑜̇ = 𝐸𝐸 − 𝑘𝑘𝑠𝑠𝑠𝑠 �𝜹𝜹𝒔𝒔𝒔𝒔 �2
2
2

Further, 𝜃𝜃𝑝𝑝𝑝𝑝̇ ⋆ needs to be adjusted if the motor does notsatisfy the constraints.
After determining the four abovementioned
parameters, the spring constant and the cam should bedesigned.

(6)

where the mass of the slider is 𝑚𝑚𝑙𝑙 ; velocity, 𝑣𝑣𝑙𝑙 ; inertia of
the output link, 𝐼𝐼𝑜𝑜 ; angular velocity, 𝜃𝜃𝑜𝑜̇ ; and energy
completely accumulated in the spring, 𝐸𝐸
Further, 𝑣𝑣𝑙𝑙 can be expressed using 𝐺𝐺𝑠𝑠𝑠𝑠 from equations(5), and 𝜃𝜃𝑜𝑜̇ as follows.
𝑣𝑣𝑙𝑙 = 𝐺𝐺𝑠𝑠𝑠𝑠 𝜃𝜃𝑜𝑜̇

3.1. Design of the spring constant
In designing the cam, the characteristic of the
reduction ratio curve required for the cam is greatly
affected by the magnitude of the spring constant. We
determined the spring constant by modeling the fullycompressed spring using the four abovementioned
parameters.
In the fully compressed state of spring, torque 𝑇𝑇𝑝𝑝𝑝𝑝∗
generated in the passive link can be expressed using
motor torque 𝑇𝑇𝑐𝑐∗ , angular velocity 𝜃𝜃𝑐𝑐∗̇ , and angularvelocity of passive link 𝜃𝜃𝑝𝑝𝑝𝑝̇ ⋆ as follows.

(7)

From equations (6) and (7), the angular velocity 𝜃𝜃𝑜𝑜̇ ofthe output link can be expressed as follows.
𝜃𝜃𝑜𝑜̇ = �22𝐸𝐸 − 𝑘𝑘𝑠𝑠𝑠𝑠 �𝜹𝜹𝒔𝒔𝒔𝒔 �𝑚𝑚𝑙𝑙 𝐺𝐺𝑠𝑠𝑠𝑠 2 + 𝐼𝐼𝑜𝑜

(8)

3. Design of the cam mechanism

𝑇𝑇𝑝𝑝𝑝𝑝∗ =The design method of the cam mechanism that cancompress the spring while maintaining the posture of theoutput link and the operating points related to the motor
torque and angular velocity as constant as possible isdescribed in this section. When designing the cammechanism using this method, the designer should
proceed in the following order.

𝑇𝑇𝑐𝑐∗ 𝜃𝜃𝑐𝑐∗̇𝜃𝜃𝑝𝑝𝑝𝑝̇ ⋆(9)

The tension 𝐹𝐹𝑤𝑤𝑤𝑤∗ generated on the wire on the releaseside is expressed as follows.
𝐹𝐹𝑤𝑤𝑤𝑤∗ =𝑇𝑇𝑝𝑝𝑝𝑝∗𝑙𝑙𝑝𝑝𝑝𝑝∗(10)

When the spring is fully compressed, the output link
has to maintain the posture immediately before starting
the rapid motion. To achieve this, the resultant force ofthe output torque, which is exerted by the tension of theleft and right wires, must be 0. That is, to maintain the
antagonistic state, the following formula should besatisfied:

1. Basic machine parameters, such as motor and link ratio
2. Design of the spring constant of the linear motion
spring used
3. Cam design
For basic machine parameters, the link ratio of link unit,
instantaneous angular velocity 𝜃𝜃𝑝𝑝𝑝𝑝̇ ⋆ of the passive link in
the fully compressed state of spring, motor torque 𝑇𝑇𝑐𝑐∗
during spring compression, and motor angular velocity𝜃𝜃𝑐𝑐∗̇ must be selected. For deciding the link ratio, it isnecessary to determine 𝑙𝑙𝑙𝑙∗ ,φ, 𝑙𝑙𝑤𝑤 , 𝑙𝑙𝑝𝑝 𝑙𝑙𝑏𝑏 , 𝑙𝑙𝑠𝑠𝑠𝑠𝑠𝑠 , and 𝑙𝑙𝑠𝑠𝑠𝑠𝑠𝑠
based on the output characteristics required for theapplication. Further, 𝜃𝜃𝑝𝑝𝑝𝑝̇ ⋆ needs to be determined fromthe time that can be spent compressing the spring. 𝑇𝑇𝑐𝑐∗ and𝜃𝜃𝑐𝑐∗̇ during spring compression are steady values. For
example, it is necessary to select an operating point or
limit operating point that is efficient for the actuator.
These parameters are also the parameters that are to bechanged if the designed cam does not meet theconstraints such as the size of the mechanism. The

𝐹𝐹𝑤𝑤𝑤𝑤1 𝑙𝑙𝑙𝑙𝑙𝑙1 − 𝐹𝐹𝑤𝑤𝑤𝑤2 𝑙𝑙𝑙𝑙𝑙𝑙2 = 0(11)

From equation (11), 𝐹𝐹𝑤𝑤𝑤𝑤1 and 𝐹𝐹𝑤𝑤𝑤𝑤2 are related asfollows.
𝐹𝐹𝑤𝑤𝑤𝑤2 =𝑙𝑙𝑙𝑙𝑙𝑙1𝐹𝐹
𝑙𝑙𝑙𝑙𝑙𝑙2 𝑤𝑤𝑤𝑤1(12)Thus, it is seen that the magnitudes of the forces
generated on the two wires are different. From therelationship between the force applied to the spring andthe amount of displacement of the spring, the spring
constant 𝑘𝑘𝑠𝑠𝑠𝑠 can be calculated as follows using equation
(13).

147

𝑘𝑘𝑠𝑠𝑠𝑠 =𝐹𝐹𝑤𝑤𝑤𝑤1 cos 𝜃𝜃𝑤𝑤1 + 𝐹𝐹𝑤𝑤𝑤𝑤2 cos 𝜃𝜃𝑤𝑤2
�𝜹𝜹𝒔𝒔𝒔𝒔 �Movement area of the release side passive link(13)

𝜃𝜃𝑐𝑐𝑐𝑐𝑟𝑟

From equations (9), (10), and (13), it can be seen that𝑘𝑘𝑠𝑠𝑠𝑠 greatly contributes to the torque 𝑇𝑇𝑐𝑐∗ and angularvelocity 𝜃𝜃𝑐𝑐∗̇ of the motor, the link ratio, and the angulaṙ of the passive link.
velocity 𝜃𝜃𝑝𝑝𝑝𝑝∗
3.2.

𝜃𝜃𝑐𝑐𝑐𝑐𝑐𝑐

Design of the cam

In the proposed mechanism, the cam has two types of
movement areas and one stop area. Fig. 3 shows theconfiguration for cam allocation. The movement areas
are used to move the release side passive link andcontraction side passive link. The stop area is forreceiving followers after the release operation. In
designing the cam curve, first the torque generated in thepassive link during the spring compression process
should be considered. The torque 𝑇𝑇𝑝𝑝∗ generated in thepassive link can be expressed from the tension 𝐹𝐹𝑤𝑤∗
generated in the wire during the spring contraction
process as follows.
𝑇𝑇𝑝𝑝∗ = 𝐹𝐹𝑤𝑤∗ 𝑙𝑙𝑝𝑝𝑝𝑝∗

𝑙𝑙𝑙𝑙𝑙𝑙∗ 𝑘𝑘𝑠𝑠𝑠𝑠 �𝜹𝜹𝒔𝒔𝒔𝒔 �𝑙𝑙𝑙𝑙𝑙𝑙2 cos 𝜃𝜃𝑤𝑤1 + 𝑙𝑙𝑙𝑙𝑙𝑙1 cos 𝜃𝜃𝑤𝑤2

Fig.3 Configuration for cam allocation

𝑇𝑇𝑝𝑝∗
𝑇𝑇𝑐𝑐∗

𝜆𝜆⋆

𝜃𝜃𝑝𝑝𝑝𝑝∗ =�𝜃𝜃𝑐𝑐∗ =∆𝜃𝜃𝑐𝑐𝑐𝑐⋆𝜆𝜆⋆
𝐺𝐺 (𝜃𝜃 )∆𝜃𝜃𝑐𝑐𝑐𝑐⋆
𝜃𝜃𝑐𝑐∗ 𝑐𝑐∗ 𝑐𝑐∗(17)Here, 𝜃𝜃𝑝𝑝𝑝𝑝∗ indicates the movement amount of thepassive link at the time of initial movement, and ∆𝜃𝜃𝑐𝑐𝑎𝑎⋆
indicates the minute rotation amount of the cam. Further,𝜆𝜆⋆ is a constant for adjusting the reduction ratio. The total
movement of the moment arm during the compression
process of the spring 𝜃𝜃𝑝𝑝𝑝𝑝∗ is expressed using 𝜃𝜃𝑝𝑝𝑝𝑝∗ and cam
allocation angle 𝜃𝜃𝑐𝑐𝑐𝑐⋆ as follows:(14)𝜃𝜃𝑐𝑐𝑐𝑐⋆𝜃𝜃𝑝𝑝𝑝𝑝∗ = � 𝐺𝐺𝑐𝑐∗ (𝜃𝜃𝑐𝑐∗ )𝛥𝛥𝜃𝜃𝑐𝑐𝑐𝑐⋆ + 𝜃𝜃𝑝𝑝𝑝𝑝∗

(15)

(18)

𝜃𝜃𝑐𝑐∗ =𝜆𝜆⋆The designer needs to optimize the parameters 𝜆𝜆⋆ and𝜃𝜃𝑝𝑝𝑝𝑝̇ ⋆ to equalize the movement amount of the required
specifications with 𝜃𝜃𝑝𝑝𝑝𝑝∗ calculated using equation (18).
The displacement of 𝜃𝜃𝑝𝑝∗ with respect to the displacement
amount of 𝜃𝜃𝑐𝑐∗ can be expressed as follows:

For the motor and cam to generate the torque 𝑇𝑇𝑝𝑝∗ in thepassive link, the cam should have a reduction ratio 𝐺𝐺𝑐𝑐∗ asfollows.
𝐺𝐺𝑐𝑐∗ =

𝜃𝜃𝑐𝑐𝑠𝑠Movement area of the contraction side passive link

Here, 𝐹𝐹𝑤𝑤∗ can be expressed by using the following
formula using the relation between equations (12) and(13).
𝐹𝐹𝑤𝑤∗ =

Stop area(16)

Equation (16) does not take into consideration thetorque required to accelerate each machine element,
Coulomb friction, viscous friction, etc. At the time ofinitial movement, the force stored in the spring is verysmall, and hence, the reduction ratio 𝐺𝐺𝑐𝑐∗ of the cam
becomes extremely small, and the angular velocity of thepassive link diverges. Hence, the following equation isapplied in the design to prevent this phenomenon.

𝜃𝜃𝑝𝑝∗ =

𝜃𝜃𝑝𝑝∗ =𝑛𝑛�𝜃𝜃𝑐𝑐∗ =𝛥𝛥𝜃𝜃𝑐𝑐𝑐𝑐⋆𝜆𝜆⋆
𝐺𝐺 (𝜃𝜃 )𝛥𝛥𝜃𝜃𝑐𝑐𝑐𝑐⋆
𝜃𝜃𝑐𝑐∗ 𝑐𝑐∗ 𝑐𝑐∗𝑛𝑛�𝜃𝜃𝑐𝑐∗ =𝛥𝛥𝜃𝜃𝑐𝑐𝑐𝑐⋆(𝛥𝛥𝜃𝜃𝑐𝑐𝑐𝑐⋆ ≤ 𝑛𝑛 ≤ 𝜆𝜆⋆ )(19)

𝐺𝐺𝑐𝑐∗ (𝜃𝜃𝑐𝑐∗ )𝛥𝛥𝜃𝜃𝑐𝑐𝑐𝑐⋆ + 𝜃𝜃𝑝𝑝𝑝𝑝∗ (𝜆𝜆⋆ < 𝑛𝑛 ≤ 𝜃𝜃𝑐𝑐𝑐𝑐⋆ )(20)

By substituting an arbitrary 𝜃𝜃𝑐𝑐∗ for the variable 𝑛𝑛 inequations (19) and (20), an arbitrary 𝜃𝜃𝑝𝑝∗ can be known.
From equations (19) and (20), 𝑙𝑙𝑐𝑐𝑐𝑐∗ becomes clear fromthe geometrical relation. As shown in Fig. 4, the vector𝒄𝒄𝒍𝒍𝒄𝒄𝒄𝒄∗ for determining the coordinate point of the cam148curve can be calculated by the following formula in termsof the rotation matrix 𝑹𝑹(−𝜃𝜃𝑐𝑐∗ ):
𝒄𝒄

40
35

(21)

𝒍𝒍𝒄𝒄𝒄𝒄∗ = 𝑹𝑹(−𝜃𝜃𝑐𝑐∗ )𝒍𝒍𝒄𝒄𝒄𝒄∗

30
25

𝒍𝒍𝒔𝒔𝒍𝒍∗𝒍𝒍𝒄𝒄𝒔𝒔∗𝜃𝜃𝑠𝑠∗𝒍𝒍𝒄𝒄𝒍𝒍∗

𝚺𝒄𝒄

𝒁

𝑿

𝒍𝒍𝒔𝒔∗
𝜃𝜃𝑐𝑐∗
𝒀

𝒍𝒍𝒔𝒔𝒍𝒍∗Δ𝜃𝜃𝑐𝑐𝑐𝑐𝑐𝑐

𝒍𝒍𝒄𝒄𝒔𝒔∗𝜃𝜃𝑠𝑠∗𝒍𝒍𝒄𝒄𝒍𝒍∗

𝑿

𝚺𝒄𝒄

𝒁

𝒀

𝜃𝜃𝑠𝑠 [deg]Fig. 5 shows the contour curve of the cam created by
calculating the coordinate points of the cam curve fromequation (21) and using 3DCAD. The contour curve of
the cam is obtained using the corresponding parameterslisted in Table 2.The contour curve is drawn bycomplementing the coordinate points with the spline
curve and is offset outward by the size of the radius of
the follower. Fig. 6 shows the movement of the passive
link with respect to the movement of the cam. When 𝜃𝜃𝑐𝑐
is displaced from 0° to 160°, the slope of the curve
becomes gentle. This means that the reduction ratio isincreased with respect to the increase in the compression
amount of the spring to counter the gradually increasing
reaction force of the spring.

20
15

Release side cam

5

Contraction side cam

0

0

20

40

60

80

𝜃𝜃𝑜𝑜 [deg]

100

120

140

160

Fig.6 Theoretical cam curves

Table 2. Specifications of the device𝑙𝑙𝑙𝑙∗

Length of 𝒍𝒍𝒍𝒍∗

0.0599[m]𝑙𝑙𝑝𝑝∗

Length of 𝒍𝒍𝒑𝒑∗

0.08[m]Length of 𝒍𝒍𝒔𝒔∗

0.0285[m]Length of 𝜹𝜹𝒔𝒔𝒔𝒔

0.0460[m]Length of 𝒍𝒍𝒘𝒘∗

0.215[m]𝑙𝑙𝑝𝑝𝑝𝑝∗Length of 𝒍𝒍𝒑𝒑𝒑𝒑∗

0.02[m]𝑙𝑙𝑠𝑠𝑠𝑠nLength of 𝒍𝒍𝒔𝒔𝒔𝒔𝐧𝐧

𝑙𝑙𝑤𝑤∗
𝑙𝑙𝑠𝑠∗

𝒍𝒍𝒔𝒔∗

𝛿𝛿𝑠𝑠𝑠𝑠𝜃𝜃𝑐𝑐∗

Fig.4 Design of the pitch curve

Coordinate point

Spline curve
Cam profile

Fig.5 Designed cam shape149

0.211[m]∅Direction of 𝒍𝒍𝒍𝒍∗

115°𝜃𝜃𝑐𝑐𝑐𝑐∗Angle between 𝒍𝒍𝒑𝒑∗ and 𝒍𝒍𝒑𝒑𝒑𝒑∗

105°

𝜆𝜆𝑐𝑐

Absolute angle of 𝒍𝒍𝒄𝒄𝒄𝒄∗

Constant for adjusting of the contraction side

38.9°

𝜆𝜆𝑟𝑟

Constant for adjusting of the release side

57.4°

𝜃𝜃𝑐𝑐𝑐𝑐𝑐𝑐

Contraction side cam allocation angle

160°𝜃𝜃𝑐𝑐𝑐𝑐𝑟𝑟

Release side cam allocation angle

160°𝜃𝜃𝑐𝑐𝑐𝑐Stop area angle

40°

Δ𝜃𝜃𝑐𝑐𝑐𝑐∗

Minute rotation amount of cam

0.350°

𝜃𝜃𝑚𝑚𝑚𝑚𝑚𝑚

Maximum movement range+22.5°

𝜃𝜃𝑚𝑚𝑚𝑚𝑚𝑚

Minimum movement range-22.5°

𝑘𝑘𝑠𝑠𝑠𝑠Spring constant3810[N/m]

𝑚𝑚𝑙𝑙

Mass of Slider

1.03[kg]

𝐼𝐼𝑜𝑜

Inertia of Output link

𝑇𝑇𝑐𝑐∗

Motor torque0.65[Nm]𝜃𝜃𝑐𝑐∗̇

Motor angular velocity

2.72[rad/s]

̇𝜃𝜃𝑝𝑝𝑝𝑝𝑝𝑝

Contraction side angular velocity of passive

0.224[rad/s]

̇𝜃𝜃𝑝𝑝𝑝𝑝𝑝𝑝

Release side angular velocity of passive link𝜓𝜓∗Δ𝜃𝜃𝑐𝑐𝑐𝑐𝑐𝑐

Coordinate points of the follower𝜃𝜃𝑐𝑐∗ Absolute angle of the cam𝜃𝜃𝑐𝑐∗ Absolute angle of the passive link
Pitch curve

10

155°

0.00598[kgm2]

link in the fully compressed state of springin the fully compressed state of spring

0.338[rad/s]

4. Performance evaluation test

Motor

To compare the theoretical value and the measured
value for torque and angular velocity during rapid motion,
we evaluated the performance using an experimental
device. The appearance of the device is shown in Fig. 7,
and the design parameters are listed in Table 2. Silicon
oil is applied to the linear shaft to improve the sliding
performance of the actual machine. Fig. 8 shows theangle of the output link with respect to the measured
elapsed time in the real machine. This actual
measurement value was calculated based on the pixel
value of the captured image depicting the motion of theoutput link.
Fig. 9 shows the results of comparing the theoreticaland measured values for the angular velocity of theoutput link. The theoretical value was calculated using
equation (8). The measured value was calculated bygenerating a three-dimensional approximation curve
based on the angle data in Fig. 8 and by numericallydifferentiating the approximation curve. As a result,although the two waveforms show the same tendency, it
can be confirmed that the experimental value is slightly
below the theoretical value and the error gradually
increases. This factor is considered to be the loss due to
Coulomb friction and viscous friction generated between
the slider and the shaft, which is not theoreticallymodeled. Furthermore, one of the factors could be thatthe compression amount of the spring does not reach thedesign value due to the elastic deformation of the wires
and links. Further, since the speed increases in the latter
half of the operation, the influence of viscous friction
proportional to the square of the speed is considered to
be dominant.
Fig. 10 shows a comparison of the theoretical and theactual measured values of the output link torque. The
theoretical value was calculated using equation (4). The
measured value was obtained by numericallydifferentiating the approximate curve in Fig. 8 to
calculate the angular acceleration and was calculatedusing equation (22).
𝑇𝑇𝑜𝑜 = 𝐼𝐼𝑜𝑜 𝜃𝜃𝑜𝑜̈

SliderSpringRuler
Protractor
Output Link

Cam
Passive Link

Wire

Fig.7 Experimental device
25
y = -217076x3 + 31597x2 - 278.48x - 22.032

𝜃𝜃𝑜𝑜 [deg]

12.5

0

-12.5

Sampling time:0.00416[s]
-25

0

0.02

Time [s]

0.04

0.06

Fig.8 Measurement angle and approximate curve30𝜃𝜃𝑜𝑜̇ [rad/s]

20

simulation

10

real machine

0

-25

-12.5

0

𝜃𝜃𝑜𝑜 [deg]

12.5

25

Fig.9 Angular velocity of the output link
10
simulation

𝑇𝑇𝑜𝑜 [Nm]real machine
5

(22)
0

As seen from Fig. 10, the error is larger than the
angular velocity because the mass of the slider and theloss due to viscous friction is not considered in equation
(4).

-25

-12.5

0

𝜃𝜃𝑜𝑜 [deg]12.5

Fig.10 Torque of the output link

150

25

5. Conclusion

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pp. 533–540, 2008.
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14. Yuya Nishida, et. al., “Design Principle of High Power
Joint Mechanism Possible to Walking and Jumping
Imitating Locust Leg Structure,” Journal of Robotics andMechatronics, Vol.23, No.2, pp.225–230, 201115. H. C. Bannet-Clark, “The energetic of the Locust
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Physiology A: Neuroethology, Sensory, Neural, andbehavioral physiology, Vol.89, No.1, pp. 93–104, 1974.
17. A.A.F Nassiraei, et al., Realization of the Rapid
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In this paper, we propose a special mechanism withthree functions: normal motion, rapid motion, andrigidity control of the main joint. The motion principle ofthe mechanism and the design principle of the cam are
discussed in this paper. Finally, we built a real machineand discussed the effectiveness of the proposedmechanism. It was shown that in the cam design principle,
by dividing the cam area appropriately, different
reduction ratio curves can be set arbitrarily on thecontraction and release sides. By using the designprinciple, the posture of the output link can be maintained,
and the spring can be compressed efficiently. In theperformance evaluation of the rapid motion of the realmachine, we found that the measured value was lower
than the theoretical value because of the effect of viscous
friction between the shaft and slider. However, the error
was small, and the validity of the mathematical model
was confirmed. As future research subjects, we will
evaluate the effectiveness of normal motion and rigidity
control of the main joint by using an actual machine.
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Authors Introduction
Mr. Katsuaki Suzuki

151

He received his Master's degreefrom the Department of HumanIntelligence Systems, Graduate
School of Life Science andSystems Engineering, Kyushu
Institute of Technology, Japan
in 2017. He is currently adoctoral student in Kyushu
Institute of Technology, Japan.

Dr. Yuya Nishida

Dr. Takashi Sonoda

Prof. Kazuo Ishii

He is an Associate Professor at
Graduate School of Life
ScienceandSystemEngineering, Kyushu Instituteof Technology, Japan. His
research interests include filed
robotics, its application, anddata processing.

He is an Associate Professor at
Department Integrated System
Engineering,Nishi-Nippon
InstituteofTechnology,
Fukuoka, Japan. His researchareunderwaterinterests
robotics and robot manipulator
systems.

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

152