ORIGINAL ARTICLE

Springback Mechanism Analysis and Experiments on RoboticBending of Rectangular Orthodontic Archwire

Jin-Gang Jiang1 • Ying-Shuai Han1 • Yong-De Zhang1 • Yan-Jv Liu1,2 •

Zhao Wang1 • Yi Liu3

Received: 21 May 2016 / Revised: 9 April 2017 / Accepted: 20 April 2017 / Published online: 15 May 2017

� Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2017

Abstract Fixed-appliance technology is the most common

and effective malocclusion orthodontic treatment method,

and its key step is the bending of orthodontic archwire. The

springback of archwire did not consider the movement of the

stress-strain-neutral layer. To solve this problem, a spring-

back calculation model for rectangular orthodontic archwire

is proposed. A bending springback experiment is conducted

using an orthodontic archwire bending springback mea-

surement device. The springback experimental results show

that the theoretical calculation results using the proposed

model coincide better with the experimental testing results

than when movement of the stress-strain-neutral layer was

not considered. A bending experiment with rectangular

orthodontic archwire is conducted using a robotic

orthodontic archwire bending system. The patient expriment

result show that the maximum and minimum error ratios of

formed orthodontic archwire parameters are 22.46% and

10.23% without considering springback and are decreased to

11.35% and 6.13% using the proposed model. The proposed

springback calculation model, which considers the move-

ment of the stress-strain-neutral layer, greatly improves the

orthodontic archwire bending precision.

Keywords Robotic bending � Rectangular orthodontic

archwire � Springback mechanism � Stress-strain-neutral

layer

1 Introduction

Malocclusion is a common oral disease, and is currently

the third-largest oral disease responsible for harming peo-

ple’s health. Its main features are irregular tooth alignment,

abnormal tooth-jaw relations between the upper and lower

dental arches, and abnormal size, morphology, and location

of the jaw. In China, the malocclusion incidence in ado-

lescents and children is up to 52.8%–72.9%. This not only

affects appearance, dentofacial growth, craniofacial mor-

phology, and oral function, but also generates negative

effects when those affected apply for jobs, search for

mates, and vie for promotions. Moreover, it is likely to

cause indigestion and gastrointestinal disease owing to

reduced masticatory function, and it also causes psycho-

logical damage [1–4]. Fixed-appliance technology is the

most common and effective malocclusion orthodontic

treatment method, and its key step is the bending of

orthodontic archwire [5, 6]. Because of archwire hypere-

lasticity, complexity of formed archwire shapes, and the

uncertainty of manual operation, it is difficult to realize

personalized archwire bending. Orthodontic specialists

often require a long period of time for to be trained for

archwire bending in order to achieve a high standard of

orthodontic treatment [7, 8]. Different orthodontic arch-

wires have different elastic moduli. Nonlinear stress-strain

Supported by National Natural Science Foundation of China (Grant

Nos. 51205093, 61403222), China Postdoctoral Science Foundation

(Grant No. 2016M591538), Heilongjiang Postdoctoral Science

Foundation (Grant No. LBH-Z16091), and Science Funds for the

Young Innovative Talents of Harbin University of Science and

Technology (Grant No. 201509)

& Jin-Gang Jiang

jiangjingang@hrbust.edu.cn

1 Robotics and Its Engineering Research Center, Harbin

University of Science and Technology, Harbin 150080, China

2 Computing Center, Qiqihar University, Qiqihar 161000,

China

3 Department of Orthodontics, Peking University Hospital of

Stomatology, Beijing 100081, China

123

Chin. J. Mech. Eng. (2017) 30:1406–1415

https://doi.org/10.1007/s10033-017-0142-0

archwires are difficult to be bent accurately by manual

operation. Additionally, the bending process is time-con-

suming, and the archwires have low bending accuracy. The

traditional manual method for bending archwire using

personal medical experience will be changed once robotic

orthodontic archwire bending systems are manufactured;

this will greatly improve the efficiency and accuracy of

orthodontic archwire bending [9, 10]. Suresmile Corpora-

tion has taken the lead in automatic orthodontic archwire

bending, and the orthodontic archwire bending robot is

emerging at the right moment [11, 12]. In the process of

archwire bending using a robot, springback of the

orthodontic archwire is generated because of the hypere-

lasticity and high-strength of the orthodontic archwire.

Bending springback seriously affects the accuracy of

orthodontic archwire bending.

Numerous domestic and foreign research institutions

have performed research into archwire bending springback

and achieved useful results [13–16]. The bending proper-

ties of Ti-Ni-Cu alloy castings is investigated in a three-

point bending test for orthodontic applications in relation to

phase transformation. A special three-point bending fixture

is invented to determine superelastic properties in simu-

lated clinical conditions, where wire samples is held in a

fixture similar to an oral cavity [17]. Static and cyclic load-

deflection characteristics of Ni-Ti orthodontic archwires

are studied through three-point bending tests. Superelastic

behavior is investigated by focusing on bending time,

temperature, and the number of cycles, which affects the

energy dissipation capacity [18]. Surface morphology and

mechanical properties have been studied for various sizes

and tempers of Australian archwires [19]. Torque charac-

teristics of Ni-Ti and steel orthodontic archwires sustaining

a pure bending moment have been compared [20]. Three-

dimensional finite element models of superelastic nickel-

titanium orthodontic wires with a Gibbs-potential-based

formulation and thermodynamic principles have been

established, and a bending test was simulated to study the

force variation of an orthodontic Ni-Ti archwire when

loaded up to the deflection of 3 mm [21]. The load/de-

flection ratios of the heat-activated Ni-Ti orthodontic wires

of dimensions 0.014 inch 9 0.025 inch and 0.016 inch 9

0.022 inch have been compared [22, 23]. The rigidity,

elasticity, strength, and fatigue properties of five

orthodontic wires (Australian orthodontic wire, Elgiloy

yellow wire from RMO, Chinese orthodontic wire, Chinese

stainless steel wire, and heated Chinese stainless steel wire)

were tested and compared [24]. The elemental components,

surface characteristics, and mechanical properties of beta-

titanium alloy wires, Ni-Ti wires, and stainless steel wires

were compared. EDAX and SUPRATM 55 SEM equip-

ment were used to analyze components and surface char-

acteristics. A Nano Indenter XP was used to test the

elasticity modulus and hardness of the wires. Three-point

bending tests were performed to compare mechanical

properties using Instron 5848 equipment [25–27]. How-

ever, all of the above–mentioned efforts studied mechani-

cal properties and bending springback of orthodontic

archwire, while none have considered the effects of neutral

plane movement on bending springback.

Focusing on the springback problem of bending

orthodontic archwire using robotics, mechanism analysis

and calculation of springback were studied here considering

neutral plane movement in the process of bending rectan-

gular orthodontic archwire. A bending springback experi-

ment was conducted using a springback measurement

device for orthodontic archwire. The oral parameters of one

patient were selected, and a bending experiment on rectan-

gular orthodontic archwire was conducted using the pro-

posed robotic orthodontic archwire bending system.

Research on the springback mechanism of orthodontic

archwire will greatly improve the orthodontic archwire

bending precision when using robotic bending. The tradi-

tional manual method for bending archwire using personal

medical experience will be improved upon by the intro-

duction of robotic orthodontic archwire bending systems,

which not only meet physiological function and aesthetic

requirements for malocclusion patients, but also realize

quantification, standardization, and automation of forming

orthodontic archwire. This will improve bending efficiency

and accuracy and promote the development of orthodontics.

2 Springback mechanism analysis for rectangularorthodontic archwire

2.1 Springback analysis of the bending process

The bending process of orthodontic archwire is as shown in

Fig. 1. In the bending process, one end of the orthodontic

archwire is clamped by a fixture in order to prevent its

rotation and movement. The fixed mold is kept stationary,

and the rotation mold rotates clockwise around the gyration

center of the fixed mold to enable the orthodontic archwire

Fig. 1 Process analysis of archwire bending

Springback Mechanism Analysis and Experiments on Robotic Bending 1407

123

to take shape. After the rotation mold rotates counter-

clockwise and the load is removed, springback of the

orthodontic archwire is generated due to elastic deforma-

tion. The process of archwire bending consists of parts BC

and AB. The BC part forms the orthodontic archwire,

which is in an elastic-plastic deformation state. The AB

part is affected by the bending moment of the BC part,

inevitably generating bending deformation, and is in an

elastic deformation state.

In order to simplify the calculation process, on the

premise of guaranteed theoretical analysis validity, when

analyzing the bending springback of orthodontic archwire,

the following basic assumptions were adopted.

(1) The material of orthodontic archwire is composed of

a continuous medium. It is uniformly distributed and

isotropic, and the Bauschinger effect of the material

is ignored.

(2) In the bending process, the cross section perpendic-

ular to the center axis of the archwire is a plane

before and after bending, and the cross section is not

subject to distortion.

(3) In the bending process, the strain-hardening law for

material between the lateral tensile part and the inner

compression part is the same. Moreover, the plastic

deformation of the material does not cause the

volume to change.

On the basis of the classical formation theory, in the

initial phase of bending, the bending angle and the spring-

back angle increase with the increase of the bending moment

because of the small load torque. At this stage, the elastic

deformation is clearly greater than the plastic deformation.

Hence, this is the elastic bending stage. When the bending

moment increases to a certain extent, the bending angle no

longer changes with the bending moment. That is, while the

bending angle increases, the bending moment remains

unchanged. The plastic deformation is then clearly greater

than the elastic deformation; hence, this is the plastic

bending stage.

2.1.1 Springback analysis of the elastic bending stage

In the elastic bending stage, deformation is mainly elastic.

Therefore, it is reasonable to simplify the bending model of

orthodontic archwire to a simply-supported beam model. A

simplified model of the elastic bending phase is shown in

Fig. 2. Assuming that the model is subjected to a bending

moment, the bending moment of the model can be con-

verted into a concentrated force acting at the distance n

from point B. As can be seen in Fig. 2, hib is the bending

angle generated by the orthodontic archwire’s bending, and

his is the springback angle. According to knowledge of

material mechanics, the deflection equations of simply-

supported beams l and n are, respectively, as follows:

vn ¼ �Fðx� lÞ6EI

½nð3x� lÞ � ðx� lÞ2�; ð1Þ

vl ¼Fnx

6EIl½l2 � x2�: ð2Þ

The bending angle hib and the springback angle his are,

respectively, as follows:

hib ¼ dvn

dxx¼lþnj ¼ � Fn

6EI½2lþ 3n� ¼ � Mi

6EI½2lþ 3n�; ð3Þ

his ¼dvl

dxx¼lj ¼ � Fnl

3EI¼ �Mil

3EI: ð4Þ

With the continuous increase of the bending moment,

the bending angle and the springback angle also change.

When the bending moment increases enough to cause the

archwire to experience plastic deformation, the bending

moment remains constant. We assumed that the bending

moment was ML at this instant. In the elastic bending stage,

the formed angle of the orthodontic archwire is as follows:

h0i ¼ jhibj � jhisj ¼Mi

6EI½2lþ 3n� � Mil

3EI; 0�Mi �ML:

ð5Þ

2.1.2 Springback analysis of the plastic bending stage

In the plastic bending stage, the moment load remains

constant. Bending deformation is mainly focused on the

BC part, and is pure bending deformation. First, the rela-

tive bending radius is much smaller than 5, where the

relative bending radius is the ratio of the bending gyration

radius and the thickness of the orthodontic archwire. This

value indicates typical large-curvature bending. Hence, the

stress-strain-neutral layer does not coincide with the geo-

metric center layer in the bending process, and it is at a

certain distance from the bending center. The relationship

of stress and strain considering the movement of the stress-

strain-neutral layer is shown in Fig. 3. The movement of

the stress-strain neutral layer seriously affects the precision

of the calculation of bending springback [28]. Therefore,

the effects of stress-strain-neutral layer movement on

bending springback must be considered. Then, as a result

Fig. 2 Simplified model of elastic bending phase

1408 J. Jiang et al.

123

of the large strain of the orthodontic archwire in the

bending process, real stress and strain data must be used to

calculate the bending springback.

We assumed that a is the deflection distance between the

stress-strain-neutral layer and the geometric center layer

and that y is the distance between any fibrous layer and the

geometric center layer in the archwire. h is the height of

archwire. R is the curvature radius of geometric center

layer before the removal of the load, R1 is the curvature

radius of the inner side of archwire, R2is the curvature

radius of the outer side of archwire, ML is the loading

bending moment. The engineering strain of the fibrous

layer before the removal of the load is as follows:

�e ¼ l2 � l1

l1¼ hðqþ yÞ � hq

hq¼ y

q; ð6Þ

where �e is the strain of the fibrous layer before the removal

of the load, l2 is the length of the stress-strain-neutral layer

before the removal of the load, l1 is the length of the fibrous

layer before the removal of the load, q is the curvature

radius of the stress-strain-neutral layer before the removal

of the load, and h is the bending angle.

Similarly, the engineering strain of the fibrous layer

after the removal of the load is as follows:

�e0 ¼ l02 � l01l01

¼ h0ðq0 þ yÞ � h0q0

h0q0¼ y

q0; ð7Þ

where �e0 is the strain of the fibrous layer after the removal

of the load, l02 is the length of stress-strain-neutral layer

after the removal of the load, l01 is the length of the fibrous

layer after the removal of the load, q0 is the curvature

radius of stress-strain neutral layer after the removal of the

load, h0 is the formed angle after the springback of the

orthodontic archwire.

The strain difference De of the fibrous layer before and

after the removal of the load is as follows:

De ¼ �e� �e0 ¼ y � 1

q� 1

q0

� �: ð8Þ

De is a small strain with large deformation, and its

stress-strain relationship behaves according to Hooke’s

law. Therefore, the stress of the fibrous layer with the

removal of the load is as follows:

r ¼ E � De ¼ E � y � 1

q� 1

q0

� �: ð9Þ

The unloading bending moment is as follow:

MU ¼ZA

r � ydA ¼ZA

E � y2 � 1

q� 1

q0

� �dA: ð10Þ

In the loading procedure, on the basis of material

stretching tests, the stress-strain relationship for rectangular

orthodontic archwire can be obtained using the polynomial

fitting method. Therefore, the loading bending moment is

as follows:

ML ¼ZA

rð�eÞ � ydA: ð11Þ

Springback and bending are reverse processes.

Orthodontic archwire is in a free state after the removal of

the load. That is, orthodontic archwire is in a state without

a bending moment. Therefore,

MU ¼ ML: ð12Þ

That isZA

E � y2 � 1

q� 1

q0

� �dA ¼

ZA

rð�eÞ � ydA: ð13Þ

In the bending process, according to the fixed length prin-

ciple of the stress-strain-neutral layer, the following holds:

qh ¼ q0h0: ð14Þ

The springback angle and the formed angle in the plastic

bending stage can be calculated by the synthesis of

Eqs. (10)–(14). The key to the solution is the calculation of

the curvature radius, the unloading bending moment, and

Fig. 3 Relationship of stress and strain considering the movement of

the stress-strain-neutral layer

Springback Mechanism Analysis and Experiments on Robotic Bending 1409

123

the loading bending moment of the stress-strain-neutral

layer.

2.2 Curvature radius of the stress-strain-neutral

layer

In the bending process, the stress-strain-neutral layer

moves a certain distance towards to the bending center.

Hence, the curvature radius of the stress-strain-neutral

layer is not equal to the sum of the radii of the fixed mold

and the half-thickness of orthodontic archwire; rather, it is

less than this sum. According to curved beam theory, we

assumed that the orthodontic archwire is bent by a certain

angle, and the calculation principle of the stress-strain-

neutral layer’s curvature radius is as shown in Fig. 4.

For the study of a fraction of orthodontic archwire dh, the

arc length of the fibrous layer for which the distance

from the stress-strain-neutral layer is y can be expressed as

ð15Þ

At this stage, we assumed that the dh section generates a

tiny bending deformation for which the bending angle is

ddh. This enables to stretch by . The extension

quantity is as follows:

ð16Þ

Hence, the strain of is as follows:

ð17Þ

According to the generalized Hooke law, the stress is as

follows:

r ¼ E � �e ¼ E � y

yþ q� ddh

dh: ð18Þ

The tensile force acting on the dh section is as follows:

T ¼ZA

rdA ¼ E0 � ddhdh

ZA

y

yþ qdA: ð19Þ

The dh section is subjected only to a bending moment,

so the tensile force is T ¼ 0. That is

E0 � ddhdh

ZA

y

yþ qdA ¼ 0; ð20Þ

where E0 � ddhdh

6¼ 0, so

ZA

y

yþ qdA ¼ 0: ð21Þ

Let yþ q ¼ R, Based on simplification of Eq. (21), the

curvature radius of the stress-strain-neutral layer can be

expressed as:

q ¼

RA

dA

RA

1RdA

¼ ARA

1RdA

: ð22Þ

2.3 Calculation of bending moment

For the rectangular orthodontic archwire, its cross section

and dimension are as shown in Fig. 5. Under the condition

of considering the movement of the stress-strain-neutral

layer, the cross-section differential of rectangular

orthodontic archwire is as follows:

dA ¼ bdy; ð23Þ

where b is the width of archwire.

Eq. (23) is substituted into Eq. (22), and then the cur-

vature radius of the stress-strain-neutral layer of rectan-

gular orthodontic archwire is as follows:

q ¼ h

ln R3þhR3

; ð24Þ

where R3 is the radius of the bending gyration center of the

fixed mold.

Fig. 4 Calculation principle of the stress-strain-neutral layer’s cur-

vature radius Fig. 5 Cross section shape of rectangular orthodontic archwire

1410 J. Jiang et al.

123

Eq. (23) is substituted into Eq. (10), and then the

unloading bending moment of rectangular orthodontic

archwire is as follows:

MU ¼ Ebh3 þ 12Eba2h

12

1

q� 1

q0

� �: ð25Þ

Similarly, Eq. (23) is substituted into Eq. (11), and then

the loading bending moment of rectangular orthodontic

archwire is as follows:

ML ¼ b

Z h2þa

�ðh2�aÞ

rðeÞ � ydy: ð26Þ

a can then be calculated as follows:

a ¼ R1 þ h=2 � q: ð27Þ

Under the condition of considering the real stress-strain

relationship, the calculation of the real strain is as follows:

e ¼ lnð1 þ �eÞ; ð28Þ

where �e is as follows:

�e ¼ y

q: ð29Þ

Therefore

y ¼ ðee � 1Þq; ð30Þdy ¼ qeede: ð31Þ

We assumed that the constitutive model of rectangular

orthodontic archwire is as follows:

r ¼ miei þ mi�1e

i�1 þ � � � þ m1e1 þ m0 ; i 2 N: ð32Þ

On the basis of the synthesis of Eqs. (26)-(32), the

loading bending moment of rectangular orthodontic arch-

wire is as follows:

ML ¼ q2b

Zlnð1þh=2þa

q Þlnð1þ�h=2þa

q ÞrðeÞ � ðe2e � eeÞde: ð33Þ

2.4 Calculation of springback angle and the formed

angle

On the basis of the synthesis of Eqs. (12), (14), (24), (25),

and (33), the calculation equations for the springback angle

and the formed angle of rectangular orthodontic archwire

are as follows:

hps ¼12MLq

Ebh3 þ 12Eba2hh; ð34Þ

h0p ¼ h� 12MLqEbh3 þ 12Eba2h

h: ð35Þ

Hence, when 0� h� hib; 0�Mi �ML, the calculation

equation for the total formed angle of rectangular

orthodontic archwire is as follows:

h0 ¼ Mi

6EI½2lþ 3n� � Mil

3EI: ð36Þ

When h� hib , he calculation equation for the total

formed angle of rectangular orthodontic archwire is as

follows:

h0 ¼ h� 12MLqEbh3 þ 12Eba2h

h� ML

6EI½2lþ 3n� �MLl

3EI

� �

þ K:

ð37Þ

where K is a correction coefficient from the elastic bending

stage to the plastic bending stage. In the bending process of

orthodontic archwire, it is continuous process from the

elastic bending stage to the plastic bending stage. When the

maximum bending moment is ML, and then the forming

angle and bending angle between the elastic bending stage

to the plastic bending stage is equal, therefore K is

determined.

3 Bending experiment for rectangular orthodonticarchwire

3.1 Bending springback experiment

3.1.1 Material performance parameter measurement

of rectangular orthodontic archwire

According to the wire material tensile testing method of the

national standard of the People’s Republic of China, tensile

tests of rectangular orthodontic archwire were conducted

using an Instron 5569 electronic universal material testing

machine at Harbin Institute of Technology. The Instron

5569 electronic universal material testing machine is as

shown in Fig. 6. The material of the rectangular

orthodontic archwire is stainless steel. Its dimensions are as

Fig. 6 Intron 5569 electronic universal material testing machine

Springback Mechanism Analysis and Experiments on Robotic Bending 1411

123

shown in Table 1. Tensile tests of rectangular orthodontic

archwire were conducted three times, and the testing elastic

modulus, yield strength and tensile strength of rectangular

orthodontic archwire are 191.070 GPa, 1.934 GPa and

2.409 GPa, respectively. Based on MATLAB, polynomial

fitting was conducted using the least squares method. The

test results and fitted stress-strain curve for the rectangular

orthodontic archwire are as shown in Fig. 7. As can been

seen from Fig. 6, the fitting effect of the stress-strain curve

is good. Its standard deviation is 6.0918. The fitted con-

stitutive model of rectangular orthodontic archwire is as

follows:

r ¼ �2:837 7e6e3 � 0:109 2e6e2 þ 0:045 4e6e1: ð38Þ

3.1.2 Springback experiment for rectangular orthodontic

archwire

Bending springback measurement experiments with rect-

angular orthodontic archwire were conducted using an

orthodontic archwire bending springback measuring

device designed in-house. The measuring equipment for

the formed angle of the orthodontic archwire adopted was

the OLYMPUS BX51M reflected/transmitted light dual-

use microscope, as shown in Fig. 8. The experimental

testing results, theoretical calculation results considering

the movement of the stress-strain-neutral layer, and the-

oretical calculation results without considering the

movement of the stress-strain-neutral layer for springback

of rectangular orthodontic archwire were compared and

analyzed. The relationship of the bending angle and the

formed angle of the rectangular orthodontic archwire is

shown in Fig. 9.

The theoretical calculation results considering the

movement of the stress-strain-neutral layer coincide better

with the experimental testing results than when movement

of the stress-strain-neutral layer was not considered. In

the initial bending stage, the main part of the total

deformation was elastic deformation. Therefore, the

springback angle occupied a large proportion of the

bending angle. With the increase of bending angle, plastic

deformation gradually became more obvious. The

springback angle increased with the increase of the

bending angle. However, its portion of the total bending

angle was gradually reduced. Finally, the bending angle

approximated a linear relationship with the springback

angle. The theoretical calculation results considering the

movement of the stress-strain-neutral layer were superior

to those in Ref. [29].

Table 1 Dimensions of rectangular orthodontic archwire

Length l/mm Cross-section

dimension S/(mm�mm)

Extensometer

gauge length l/mm

10.00 0.4490.54 25.00

Fig. 7 Testing and fitted stress-strain curve of rectangular orthodon-

tic archwire

Fig. 8 Measuring equipment for the formed angle of orthodontic

archwire

Fig. 9 Relationship of the bending angle and the formed angle of the

rectangular orthodontic archwire

1412 J. Jiang et al.

123

3.2 Springback analysis of the bending process

3.2.1 Experimental robotic system for bending orthodontic

archwire

The proposed experimental robotic system for bending

orthodontic archwire is shown in Fig. 10, and consists of

the following components: a prototype of the orthodontic

archwire bending robot, a central control system (com-

puter) with orthodontic archwire bending software for

motion planning and control, and a set of PMAC multi-axis

controllers. The bending robot has five degrees of freedom,

consisting of rotation and a feed structure for orthodontic

archwire as well as a bending structure. The orthodontic

archwire bending process using the bending robot is shown

in Fig. 11.

3.2.2 Patient selection for the bending experiment

with rectangular orthodontic archwire

To ensure that our experiments were representative for a

majority of patients, we selected a patient with no clinical

experience of orthodontic treatment as a test case. Dental

arch treatment parameters of the patient were as shown in

Table 2. Respectively, L1 and W1 are the height and width

of the canine teeth offset, L2 and W2 are the height and

width of the molar offset, S3, L3 and W3 are the length,

height and width of the dental arch.

3.2.3 Experimental results for rectangular orthodontic

archwire bending

The bending experiment with rectangular orthodontic

archwire was conducted using the experimental robotic

orthodontic archwire bending system. The cross section

dimensions of the rectangular orthodontic archwire were

0.44 mm 9 0.54 mm. Experimental results for rectangular

orthodontic archwire without considering springback were

as shown in Fig. 12. Experimental results for rectangular

orthodontic archwire considering springback were as

shown in Fig. 13.

The error ratio is the quotient, whose the numerator is

the absolute value of difference between the patient’s

dental arch parameter and real forming archwire’s param-

eter, and the denominator is the patient’s dental arch

parameter. As seen in Fig. 12, while not considering

springback, the error ratios for the height of the dental arch,

width of the dental arch, height of the molar offset, width

of the molar offset, height of the canine teeth offset, and

width of the canine teeth offset, were 10.23%, 22.46%,

10.34%, 19.68%, 10.27%, and 14.75%, respectively. As

seen from Fig. 13, while considering springback, the error

ratios for the height of the dental arch, width of the dental

arch, height of molar offset, width of molar offset, height

Fig. 10 Experimental system of orthodontic archwire bending robot

Fig. 11 Orthodontic archwire bending process using the bending

robot

Table 2 Dental arch treatment parameters of the patient

Parameter Length S/mm Height L/mm Width W/mm

Dental arch 92.37 34.03 49.50

Molar offset - 24.58 42.81

Canine teeth offset - 4.68 19.97

Fig. 12 Experimental results for rectangular orthodontic archwire

without considering springback

Springback Mechanism Analysis and Experiments on Robotic Bending 1413

123

of the canine teeth offset, and width of the canine teeth

offset were 6.17%, 11.35%, 7.64%, 10.87%, 6.13%, and

7.36%, respectively. Hence, the experimental results for

rectangular orthodontic archwire considering springback

are superior to those without considering springback.

4 Conclusions

(1) The springback process of rectangular orthodontic

archwire is analyzed. The neutral plane’s radius of

curvature and bending moment for rectangular

orthodontic archwire are calculated considering the

movement of the stress-strain-neutral layer.

(2) A springback calculation model is established based

on the constitutive equation obtained from tensile

testing. A bending springback experiment is con-

ducted using an orthodontic archwire bending

springback measuring device. The bending spring-

back experimental results verified the correctness of

the springback calculation model.

(3) The robotic system could automatically generate a

formed orthodontic archwire that fitted a patient

according to the patient’s dental arch parameters.

While not considering springback, the maximum and

minimum error ratios of the formed orthodontic

archwire parameters were 22.46% and 10.23%,

respectively. In contrast, when considering spring-

back, the maximum and minimum error ratios for the

height of the dental arch were 11.35% and 6.13%,

respectively, demonstrating clear improvement in

accuracy.

(4) A novel system that can generate orthodontic

archwire is developed. The traditional method of

manually determining orthodontic archwire may

become obsolete if robotic systems can be used to

assist in generating more standardized and person-

alized orthodontic archwires. This research lays a

foundation for the successful application of robots in

orthodontic treatment, and also provides a way to

standardize the process of orthodontic archwire

bending.

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Jin-Gang Jiang, born in 1982, is currently an associate professor and

M. S candidate supervisor at Robotics & its Engineering Research

Center, Harbin University of Science and Technology, China. He

received his BS degree, MS degree and PhD degree from Harbin

University of Science and Technology, China, in 2005, 2008 and

2013, respectively. His main research interests include medical robot

and biomimetic robot. Tel: ?86-18846445528; E-mail:

jiangjingang@hrbust.edu.cn

Ying-Shuai Han, born in 1993, is currently a master candidate at

Robotics & its Engineering Research Center, Harbin University of

Science and Technology, China. E-mail: gljs1993@163.com

Yong-De Zhang, born in 1965, is currently a professor and a PhD

candidate supervisor at Robotics & its Engineering Research Center,

Harbin University of Science and Technology, China. His main

research interests include medical robot, education robot and

biomimetic robot. E-mail: zhangyd@hrbust.edu.cn

Yan-Jv Liu, born in 1974, is currently a professor and M. S candidate

supervisor at Computing Center, Qiqihar University, China. His main

research interests include medical robot and image processing.

E-mail: 15146692464@163.com

Zhao Wang, born in 1990, is currently a master candidate at

Robotics & its Engineering Research Center, Harbin University of

Science and Technology, China. E-mail: zhaowang1117@163.com

Yi Liu, born in 1973, is currently a chief physician, associate

professor and master candidate supervisor at Peking University

Hospital of Stomatology, China. His main research interests include

Cone-beam CT and oral three-dimensional digital technology,

orthodontic treatment of occlusion function disorder and temporo-

mandibular joint disease E-mail: lyortho@163.com.

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