Precise assembly of ring part with optimized hollowed finger
Fukukawa et al. Robomech J
Precise assembly of ring part with optimized hollowed finger
Tomoya Fukukawa 0
Junji Takahashi 2
Toshio Fukuda 1
0 Department of Mechanical Science and Engineering, Nagoya University , Furo-cho, Chikusa-ku, Nagoya 464-8603 , Japan
1 Faculty of Science and Engineering, Meijo University , 1-501, Shiogamaguchi, Tenpaku-ku, Nagoya 468-8502 , Japan
2 Department of Integrated Information Technology, Aoyama Gakuin University , 1-10-5, Fuchinobe, Chuo-ku, Sagamihara 252-5258 , Japan
We deal with a technical issue of assembling a ring part into a shaft part with the clearance of several micrometers by using a robotic manipulator. This issue is difficult because of deformation of a ring part compared with peg-in-hole assembly. We propose a precise assembly method of a ring part with finger shape to solve this issue. We also propose a method to decide design parameters of the finger by maximizing a closed area in Jamming diagram, which represents a successful condition of assembling rigid parts by quasi-static force analysis. Finally, availability of the proposed method is verified by an experiment of ring assembly with a robotic hand attached to the designed fingers.
Peg-in-hole; Ring assembly; Robotic manipulator
In recent years, many manufacturing companies have
adopted Factory Automation (FA). Introduction of FA
brings automation of various production processes,
which are conducted by human workers, by machines,
and can improve productivity, manufacturing cost, and
quality of products. Although a paint process, a weld
process and an inspection process have been automated
increasingly, most of the assembly processes have been
conducted manually by human workers even now. This is
because assembly processes by a robot has many issues
with regard to the precision and flexibility.
Assembly methods by a robot are divided into two
categories by focusing on a process of mating: a passive
assembly method and an active assembly method. The
passive assembly method is a method that adjusts
position and orientation of a part automatically by using
mechanical elastic elements or guiding jig. The passive
assembly has advantage of high speed assembly because
of sensor-less. However, the method is applied for only
In contrast, the active assembly method is a method
that controls impedance of an end effector properly with
sensor feedback. The method has advantage of versatility
and can be applied for difficult assembly such as
assembly of complex shape parts or flexible parts. However, the
assembly time of the method tends to be larger than the
passive assembly method.
Passive assembly methods have been studied since the
1980s. As a representative research of a passive
assembly method, Whitney proposed Jamming diagram, which
represents a successful condition in peg-in-hole task
. As a concrete example of passive assembly method,
Remote Center Compliance (RCC) device has been also
developed. Moreover, many researches have been
studied, and have extended application range of RCC devices
[2–4]. Mouri et al. dealt with narrow clearance assembly
and solved problems of high friction and jamming by
adding high-frequency vibration to a peg during
insertion process . These conventional approaches have
disadvantages: A mechanical element, such as a spring and a
rubber, need frequent maintenance due to their
degradation; the system tends to become large; the cost of design
and production tends to become large.
Active assembly methods have been proposed much in
recent years. Many researches of force sensor feedback
[6, 7] or vision sensor feedback have been studied. Today,
there are strong demand for assembly for flexible parts
such as a cable and rubber. For example, Nakagaki et al.
© 2016 Fukukawa et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
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tackled automatic insertions of linear and tubular
flexible parts based on shape detection by vision sensor .
Even though many active assembly methods have been
proposed, the methods that are used actually are few.
One reason of this is difficulty of developing the system.
Another reason is adaptability. Although human workers
can detect and recover an error state of assembly quickly
and succeed in the assembly smoothly, many assembly
automation system cannot perform as well as human
In this paper, we focus on a precise ring assembly task
as an example of difficult assembly. In the case of precise
assembly that a clearance between a ring part and a shaft
part is several micro meters, the assembly by a robotic
manipulator is difficult. This is because a ring part
deforms if the ring part is grasped by a robotic hand, and
the clearance disappears. When practicability is
considered, it is desired that such difficult assembly is realized
at high speed with minimal sensor system.
So far, the precise ring assembly task has been
hanndled by a method using a jig which is one of passive
methods. The assembly processes are described in Fig. 1.
In this conventional method, the precise ring assembly
becomes easy by guiding a ring part along an incline of
a jig (Fig. 1b). However, the method requires two
processes of attaching a jig to a shaft part and detaching a
jig from a shaft part as shown in Fig. 1a, c. Because the
Fig. 1 Conventional approach for ring assembly. a The robotic hand
picks up a jig and attaches it on a shaft. b The robotic hand picks up a
ring and mates it with the shaft. c The robotic hand grasps the jig and
remove it from the shaft
two processes make cycle time long, the processes should
be omitted. Originally, the function of the jig is to adjust
the position and orientation of the ring by guiding along
with the incline when the robotic hand pushes the ring
straight down. In order to realize this guiding effect
without using any jigs, we figured out a approach using a
hollowed finger with an incline. Our proposed hollowed
finger also solves the problem of ring deformation due to
The finger design approach for realizing the handling
an object including assembly have been studied long
time. Most pioneering research of gripper design was
done by Mason . He formulated a planar pose
problem and found a pushing plan to move a polygonal object
to a specified goal position and orientation. This concept
has become underlying of the self-alignment mechanism
on sensor-less gripper. Causey and Quinn summarized
gripper design guidelines  and mentioned the
importance of grasped part alignment as well as the importance
of part deformation avoidance for increasing
reliability. Zhang and Goldberg proposed unique parallel-jaw
grippers that can align an n-sided polygonal part in the
vertical plane as the jaw close . They also proposed
a numerical algorithm to design optimal gripper jaws
. Hirata et al. proposed design of handling device
for caging and aligning small circular objects . By
designing the triangular finger tips, their hand can grasp
a small object robustly at a unique position of the tips.
Although these conventional studies solve the problem
of self-alignment of a part with gripper, the problem of
alignment with other part is not considered. On the other
hand, our proposed hollowed finger can align centers of
a part and another part. The capability of self-alignment
between two parts by using the hollowed finger is our
Successful condition of ring assembly
Figure 2 and Table 1 show parameters and sizes of a ring
part and a shaft part dealt with in this research. As shown
in Table 1, diameters of a ring part and a shaft part have
size tolerance based on the Japanese Industrial Standards
Ring assembly has two states: mating and inserting.
Mating is a state that is most difficult in ring assembly.
A definition of the state is that length of insertion is less
than height of a ring. In the state, jamming, which is a
situation that assembly is fixed because a state of a ring
part and a shaft part is not desirable, occurs easily.
Inserting is the next state after mating. A definition of the state
is that length of insertion is longer than height of a ring.
Fig. 2 Parameters of ring and shaft parts
Table 1 Parameters of ring and shaft parts
Outside diameter of ring 51.944–51.990 (φ52g8)
Inside diameter of ring 50.000–50.039 (φ50H8)
Height of ring 10.000
Diameter of shaft 49.952–49.991 (φ50g8)
Chamfer of ring 0.050
Chamfer of shaft 0.050
Clearance between ring and shaft 0.009–0.087
We consider a geometrical successful condition of ring
assembly. An angle between axes of a ring part and a
shaft part should be small for proper mating. Let θm be a
maximum angular error of proper mating, then the
successful condition is described as Eq. (1).
We consider a mechanical successful condition of ring
assembly. Jamming diagram is useful for considering
a mechanical successful condition of mating .
Jamming diagram describes the successful condition on
two-dimensional plane. Although conventional Jamming
diagram is considered for the case of assembling a peg
(shaft part) into a hole (ring part), we convert the
previous case into the case of assembling a ring part into a
shaft part in this paper. Mechanical parameters for
Jamming diagram is shown in Fig. 3. The parameters
summarize in Table 2. By approximating θ ≈ 0 and assuming
that the radial thickness of the ring is zero, three
equations are given as follows,
Fig. 3 Mechanical parameters of ring assembly
Fz − µf 1 − µf 2 = 0
Fx − f1 + f2 = 0
M − µf 1Rri − f2l + µf 2Rri = 0
0 ≤ l ≤ 2L
Additionally, a condition of friction is given as follows,
According to Eqs. (5), (7) and (8), Jamming diagram
for ring assembly is obtained as shown in Fig. 4. If
relationship of mechanical parameters are in a closed area
(hatching area in Fig. 4) of Jamming diagram, it is
guaranteed that ring assembly is realized smoothly.
Fig. 4 Jamming diagram for ring assembly
Ring assembly with hollowed finger
Based on the successful conditions of ring assembly, this
section proposes a precise assembly method of a ring
with hollowed finger. A characteristic of the proposed
method is to assemble a ring part into a shaft part with
a hollow of each finger of a robotic hand. The proposed
method consists of three phases: approach,
adjustment of axes, and mating and insertion. Figure 5 shows
an overview of the proposed method. In the first phase,
the manipulator with having a ring part approaches a
shaft part and release and hang on the ring part to the
shaft part. The disappearing of grip force restores
circular shape of the ring from deformation. In the second
phase, the robotic hand goes down and holds a shaft part
with covering the ring part in the hollows of fingers to
adjust the axes of the robotic hand and the shaft part. In
the third phase, the robot goes down and the ring part
is assembled successfully by an effect of a space of the
hollow. The proposed method has four advantages as
Fig. 5 Ring assembly algorithm
• The proposed method does not have complicated
• The proposed method considers positional accuracy
of a robotic manipulator
• The proposed method considers size tolerance of
assembly parts (ring and shaft parts)
• The proposed method considers deformation of
assembly parts (ring part)
An objective of this step is rough positioning of a ring
part to a shaft part. Firstly, the robotic manipulator picks
up a ring part from a feeder. Then, the robotic
manipulator approaches the shaft part and releases the ring part
on the shaft part. In this case of precise assembly of a
ring part, the assembly cannot succeed because a
clearance between a ring part and a shaft part disappears by
deformation of the ring part. Therefore, the ring part is
released once. On the other hand, if the positional
difference between the ring part and the shaft part is large, the
ring part falls from a shaft part. Therefore, it is assumed
that the robotic manipulator is controlled with possible
Adjustment of axes
One problem of assembly by a robotic manipulator is
positional accuracy of the robot. In general, six-axis
manipulators have positional error of more than 20 μM.
Precise assembly cannot succeed with only position
control. An objective of this step is to reduce the positional
and angular errors between the robotic hand and the
A hand grasping motion makes the fingers move
toward the hand center with same width. When the
hand center is not aligned with the shaft center, a finger
contacts with the shaft in first even though others do
not. The left figure in Fig. 6 shows that only the finger 3
Fig. 6 Adjustment of axes
contacts with the shaft and the reaction force is
generated. The robotic arm moves the hand to the direction for
reducing the reaction force simultaneously with the hand
grasping motion. The arm motion is generated based
on Eq. (9). The simultaneous hand grasping motion and
the arm movement makes the centers of hand and shaft
yx == KK ×× FFyxwwrriisstt
where x, y are positional correction amount of the
robotic hand, Fxwrist , Fywrist are force detected at the
wrist of the robotic manipulator, K is a value of gain. At
the same time of holding the shaft part, the ring part is
trapped by the hollows of the fingers.
Mating and insertion
An objective of this step is to achieve mating and
inserting the ring part without jamming. The robotic hand goes
down in negative z direction along the shaft part.
Positional and angular errors between the ring part and the
shaft part are reduced through the second phase.
Therefore, the ring part is assembled successfully without
Advantages of the hollows on fingers are as follows:
prevention of ring deformation, geometrical
restriction of a ring part, giving optimal mating force to a ring
part. The first advantage is prevention of ring
deformation. A ring part held by a robotic hand is deformed by
grip force. This means that a clearance between a ring
part and a shaft part disappears. On the other hand, a
ring closed by hollows of fingers does not receive grip
force. Therefore, the proposed method can be applied to
any material rings. The second advantage is geometrical
restriction of a ring part. A position and an angle of a
ring part are restricted by the hollows of fingers. The
successful condition of ring assembly θ ≤ θm in the previous
section is satisfied by the hollows of fingers. This means
that a ring part can be assembled without jamming.
The final advantage is giving optimal mating force to a
ring part. A ring part closed by hollows receives mating
force from the top of the hollows. If the mating force
satisfies Jamming diagram (Fig. 4), the ring part can be
assembled without jamming.
Optimal design of finger shape
In this section, we determine parameters of the hollow.
One of the characteristics of the proposed method is to
assemble a ring part with hollows of fingers. The hollows
need to be designed to be able to assemble a ring part
successfully. The hollow has three parameters as follows:
width a, height b, incline angle φ.
Geometric condition of ring assembly is shown as Eq.
(1). Therefore, the hollow is designed as Eq. (1) is
satisfied. We consider a ring assembly with θ = θm (Fig. 7).
This is because the state is the most difficult state of ring
assembly. In Fig. 7, let t be a cosine of ring thickness, h be
difference between the highest point and the lowest point
of a ring. Two parameters t and h are led to as follows.
By using Eq. (10), the parameters of width a and height b
of hollows are determined.
Firstly, I determine a design condition of the width a
of the hollow. Equation (1) is always satisfied when a ≤ t
because of restriction in a horizontal direction. In
addition, the condition needs to satisfy (Rro − Rri) ≤ a not
to give external force in a radial direction of a ring part.
From the above, the design condition of the width a of
the hollow is as follows.
(Rro − Rri) ≤ a ≤ t
Next, I determine a design condition of the height b of
the hollow. Likewise, Eq. (1) is always satisfied when
b ≤ h because of restriction in a vertical direction. In
addition, the condition needs to satisfy 2L ≤ b because
of geometry. From the above, the design condition of the
height b of the hollow is as follows.
2L ≤ b ≤ h
Fig. 7 Design of paratmeters of the hollow and the force from the
upper side of hollow
We determine a design condition of the incline angle φ
of the hollow. The principle is that area surrounded by
Jamming diagram, which shows simplicity of assembly,
is maximum. Therefore, the incline angle φ is introduced
into the equations of Jamming diagram. Then, we
determine the optimal incline angle φ∗ by maximizing the area
of Jamming diagram.
Regarding ring assembly, equilibrium equations of force
and moment considering mating force from the hollow are
shown as follows. We assume that the radial thickness of a
ring is zero.
Fz′ − µf 1 − µf 2 + Fh sin φ = 0
M′ − µf 1Rri − f2l + µf 2Rri
where force from the hollow defines Fh, forces and a
moment from the hand which are not involving Fh are Fx,
Fy′, and M′. The fourth term of Eq. (14) means a force part
in a vertical direction for mating. The fourth term of Eq.
(15) means a force part in a horizontal direction for
adjusting a position of a ring. The fifth and sixth terms of Eq. (16)
means a moment part for adjusting an angle of a ring.
According to eliminating the contact force f1 and f2
from Eqs. (14), (15) and (16), Eq. (17) is led to.
M′ F ′
RriFz′ = µ( 1 − λ) Fxz′ + λ − cos(φ + ψ ) FFhz′ (17)
A condition that the area of Jamming diagram is
maximum is that an intercept in Eq. (17) is maximum. The
condition is shown as follows.
Equation (20) has a variable λ = l/2µR ri, and depends
on a length of insertion l. This means that the optimal
incline angle φ∗ is not determined uniquely. A
relationship between φ∗ and l is shown in Fig. 8. From Fig. 8, the
optimal incline angle φ∗ is an obtuse angle.
We modify Figs. 7, 8, 9 so that the Eqs. (14)–(16) can
be still applied even when the φ is obtuse angle. Here, we
define a new parameter c as the width of upper side of the
hollow. The limitation of c comes from the condition that
the upper side of the hollow contacts with the ring at the
inner edge. This limitation is expressed as follows,
In the case a < c, the insertion process needs to be
modified for avoiding a collision between the upper side of
hollow and the shaft. The modification is that the hand
opens little width when the insertion length l reaches
3/2L. This hand open motion has little effect on the
stability of the mating and insertion process, because it is
done after the ring is almost inserted. Besides, this
process takes virtually no time at all.
We determine an unique incline angle φd for
manufacturing. A condition is that mating and insertion force
is minimum in a process of assembly. The condition is
described as Eq. (22).
From Eq. (22), the incline angle is designed as φd = 120°.
The other parameters of the lower width, the height, and
upper width are determined by Eqs. (12), (13) and (21).
Actual design conditions are described as follows.
1.00 mm ≤ a ≤ 1.19 mm
10.00 mm ≤ b ≤ 11.00 mm
1.24 mm ≤ c
One of the above conditions only have to be satisfied to
give geometric restriction to a ring part. Therefore, the
lower width, height and upper width are designated as
ad = 1.1 (mm), bd = 12.0 (mm), and cd = 1.6 (mm). An
actual manufactured finger is shown in Fig. 10.
Results and discussion
Figure 11 shows an experimental system for ring
assembly. A robotic manipulator is an articulated robot RV-1A
(Mitsubishsi Electric Corp.). A robotic hand is a
threefingered gripper ESG1 (TAIYO, Ltd.). A force sensor is a
six-part force sensor (NITTA Corp.).
Figure 12 shows z-axis force: Fz from the force sensor
during mating process in the case of φ = 61°: out of the
design condition, 85°: within the design condition but not
optimal, and 120°: optimal design. In the case φ = 61°, the
Fz increases rapidly when the insertion length: l reaches
5 mm. This is because the jamming is occurred and the
Fig. 10 Manufacturing of a finger
Fig. 11 Experimental system
0.00 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Insertion length: l [mm]
Fig. 12 Reaction force of assembly
mating process results in failure. On the other hand, in
the case φ = 85 and 120° the insertion length exceeds
10 mm, which is the height of the ring, and it can be
confirmed that the mating and insertion process succeeded.
Comparing with the case 85° and the case 120°, the
Fz of the latter is lower than that of the former until
l = 8 mm. This results suggest that the optimal design of
the φ has better performance in self-alignment than
others. The reason why the Fz increases after l = 8 mm in
the case 120° is thought to be that the upper side of the
hollow collides against the shaft. This collision can be
avoided by opening the robotic hand more early timing.
Figure 13 shows a sequence of ring assembly in the case
of φ = 120°.
We also conducted multiple assembly experiments
to evaluate a success rate of the proposed method. The
number of success was 19 out of 20 trials and the success
ratio was 95 %. The failure in our experiments occurred
in the first step of the proposed method. This means that
the released ring fell from the shaft. Therefore, the effect
of the proposed hollows is verified although the proposed
method should be improved. To avoid the failure, the
error recovery method should be developed in the future.
In this study, we proposed a precise assembly method
of a ring part with hollowed fingers. Ring’s
deformation in a radial direction is a problem in ring assembly.
To solve the problem, the proposed method utilizes
hollowed fingers. In the proposed method, a ring part caged
in the space of hollows is assembled to a shaft part
without deformation. In addition, the proposed method can
achieve precise ring assembly without sensor feedback.
Fig. 13 a The robotic hand releases the ring. b The robotic hand grasp the shaft so as to align the centers. c The ring is mated without jamming by
hollowed finger effect. d The insertion has completed
We also discussed optimization of the shape of hollows
in this paper. The shape of hollows is optimized by
geometrical and mechanical conditions of ring assembly. We
verified that the proposed method can realize precise
ring assembly by assembly experiments.
TF formulated concepts and ideas and drafted this manuscript. JT and TF
contributed concepts and edited and revised this manuscript. All authors read
and approved the final manuscript.
The authors declare that they have no competing interests.
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