designandvalidationofanovelcable-drivenhyper-redundant

Research ArticleDesign and Validation of a Novel Cable-Driven Hyper-RedundantRobot Based on Decoupled Joints

Long Huang12 Bei Liu 1 Lairong Yin 1 Peng Zeng1 and Yuanhan Yang1

1School of Automotive and Mechanical Engineering Changsha University of Science and Technology Changsha 410114 China2Hunan Provincial Key Laboratory of Intelligent Manufacturing Technology for High-performance Mechanical EquipmentChangsha University of Science and Technology Changsha 410114 China

Correspondence should be addressed to Bei Liu 17871947856163com and Lairong Yin yinlairongcsusteducn

Received 15 July 2021 Revised 26 August 2021 Accepted 9 September 2021 Published 2 November 2021

Academic Editor Yaoyao Wang

Copyright copy 2021 Long Huang et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In most of the prior designs of conventional cable-driven hyper-redundant robots the multiple degree-of-freedom (DOF)bendingmotion usually has bending coupling effects It means that the rotation output of each DOF is controlled by multiple pairsof cable inputs e bending coupling effect will increase the complexity of the driving mechanism and the risk of slack in thedriving cables To address these problems a novel 2-DOF decoupled joint is proposed by adjusting the axes distribution of theuniversal joints Based on the decoupled joint a 4-DOF hyper-redundant robot with two segments is developed e kinematicmodel of the robot is established and the workspace is analyzed To simplify the driving mechanism a kinematic fitting approachis presented for both proximal and distal segments and the mapping between the actuator space and the joint space is significantlysimplified It also leads to the simplification of the driving mechanism and the control system Furthermore the cable-drivenhyper-redundant robot prototype with multiple decoupled joints is establishede experiments on the robot prototype verify theadvantages of the design

1 Introduction

In recent years due to the advantages of compact structureand flexible bending motion in the confined environmenthyper-redundant robots have received high attention in thefield of minimally invasive surgery maintenance and testing[1ndash3] Various kinds of cable-driven hyper-redundant ro-bots have been reported by researchers [4ndash11] Generally thecable-driven hyper-redundant robots are usually composedof several segments driven by external actuators and mul-tiple cables Every segment consists of several identical jointsin serial Joint types for the cable-driven hyper-redundantrobot mainly can be classified into the 1-DOF joint and 2-DOF joint [12] e 1-DOF joints mainly include the rev-olute joint [13 14] the flexible beam [15 16] and the cy-lindrical rolling joint [17 18] while the multi-DOF jointsmainly include the universal joint [19 20] the flexiblebackbone [21 22] and the spherical rolling joint Based onthe flexible backbone Li et al developed a 2-DOF flexible

endoscope driven by multiple cables which is more dex-terous than rigid endoscopes [23] Dong et al proposed acontinuum robot with a low ratio between diameter andlength based on compliant joints which can apply to theinspection and maintenance of aero engines [24] Based onuniversal joints Jin et al designed cable-driven snake-like 4-DOF surgical forceps [25] Kim et al designed a novel rollingjoint with a block mechanism to develop a snake-like robotfor minimally invasive surgery [26]

For most of the prior cable-driven hyper-redundantrobots the bending motion in all directions is usuallycoupled It means the rotation output of each DOF iscontrolled by multiple pairs of cable inputs [27] Besides themapping between the actuator space and joint space of therobot is complex which causes the driving mechanism andthe control system difficult [28] Some researchers have triedto address the drawbacks of the coupling bending motionthrough special joint structure design and further simplifythe mapping between actuator space and joint space [29 30]

HindawiJournal of RoboticsVolume 2021 Article ID 5124816 16 pageshttpsdoiorg10115520215124816

Based on multiple cylindrical rolling joints Kim et aldesigned a cable-driven hyper-redundant robot which di-minishes the bending coupling effect by enlarging the spacefor the passage of the center cable [17]

In this paper a novel 2-DOF decoupled joint is proposedby adjusting the distribution of two rotation axes For thedecoupled joint a pair of antagonistic cable inputs onlycontrols a 1-DOF rotational output and the rotationaloutput of each DOF is only determined by a single pair ofantagonistic cable inputs By connecting two 2-DOF seg-ments in serial a cable-driven hyper-redundant robot ispresented Each segment consists of multiple identical 2-DOF decoupled joints in serial A kinematics linear fittingapproach is presented to simplify the mapping betweenactuator space and joint space Based on the linear fittingerror analysis each pair of antagonistic cables is driven by amotor through a circular pulley which can simplify thedesign of the driving mechanism e proposed robot isverified by the bending motion experiments the cabletension test and the load experiments

e rest of this paper is organized as follows Section 2introduces the challenges of the existing robots Moreover anovel cable-driven hyper-redundant robot is also introducedin Section 2 e kinematics and the robot workspace areanalyzed in Section 3e presented kinematics linear fittingand error analysis of the robot are discussed in Section 4erobot prototype through some experiments is verified inSection 5 Section 6 presents the conclusion

2 Robot Design

21 Challenges of the Existing Robots

211 Coupling Effect For most of the prior designs of cable-driven hyper-redundant robots the 2-DOF bending motionin each segment of the robot is coupled e rotationaloutput of each DOF of the joint is controlled by multiplepairs of antagonistic cable inputs Consequently the cou-pling effect will increase the complexity of the robotrsquos drivingmechanism and the risk of slack in the driving cables

e relationship between the cable length and the ro-tation angles is determined by the joint types and theirstructural parameters For instance the coupling effect of therevolute joints is determined by the cable distribution circleradius the number of cables and the distribution of therotation axes Since the cable distribution circle radius andthe number of cables are confined by the practical factorsthe distribution of joint rotation axes is a significant factor toavoid the coupling effect Figure 1 shows a conventionaluniversal joint with two intersecting rotation axes located atthe middle of two disks [31] When the upper disk rotatesaround rotation axesw1 through the releasing of cable A andtightening of cable C cable B and cable Dmust be tightenedsimultaneously to avoid slack as shown in Figure 1(b)Similarly the rotation around axis w2 requires the control ofall four cables Otherwise cable A and cable B will becomeslack It can be proved that the rotational output of eachDOF requires the control of multiple pairs of antagonisticcables regardless of the location of the two intersecting axes

in the joint For the universal joint with two nonintersectingrotation axes located between the two disks [32] it can bealso proved that the coupling effect exists in the joint when atleast one rotation axis is located at the middle place of twodisks which is similar to the universal joint as shown inFigure 1

Based on the above analysis we tried to change thedistribution of the joint rotation axes to avoid the couplingeffect erefore this paper proposes a novel joint withoutcoupling effects as shown in Figure 2 e two axes in thejoint are respectively coincident with the upper surface ofthe lower disk and the lower surface of the upper disk Sincepoints A2 and B2 are located at the axis w1 points A2 and B2achieve circular motion around points A1 and B1 when theupper disk rotates around the axis w1 erefore the lengthof cable A and cable B will not change while the length ofcable C and cable D will also change Since the points C2 andD2 are located at the axis w2 the points C2 and D2 can beconsidered as the fixed points when the upper disk rotatesaround the axisw2 In consequence the length of cableC andcableDwill not change while the length of cable A and cableB will changeis indicates that the coupling effect does notexist in the proposed joint with the special distribution oftwo axes positions

212 Challenge of the Driving Mechanism Design e re-lationship between cable length and bending angles shouldbe considered to design the driving mechanism of the cable-driven hyper-redundant robot [33 34] For most cable-driven hyper-redundant robots the relationship between thelength inputs of each cable and bending angles is a nonlinearfunction It means the tightened amount on one cable is notequal to the released amount on the antagonistic cable whenthe robot bends to an arbitrary configuration It is difficultfor cable-driven hyper-redundant robots to design a com-pact and simple driving mechanism

e following driving mechanism design approaches areadopted e first approach is that each cable is driven by aseparate motor and a circular cable pulley It is convenient todesign the driving mechanism However this approachincreases the complexity of the control system as shown inFigure 3(a) e second approach is that noncircular cablepulleys are designed to realize a motor driving a pair ofcables based on the nonlinear function as shown inFigure 3(b)is method significantly reduces the number ofthe motor but the fabrication and assembly of noncircularcable pulleys require high accuracy e third approach isthat a pair of cables is driven by a separate motor and acircular cable pulley as shown in Figure 3(c) is methodrequires the releasing amount of one cable is equal to thetightening amount of the antagonistic cable which cansimplify the driving mechanism

22 A Novel Robot Design with the Decoupled Joints issection proposes a cable-driven hyper-redundant robotbased on the multiple decoupled joints as shown in Figure 4e robot is composed of a proximal segment a distalsegment and a drivingmechanism Each segment consists of

2 Journal of Robotics

A2 A2

A1 A1

A AB B

DDC C

B2

B2

w2w2

r

h

w1

w1

B1 B1

D1 D1

D2

D2

C1 C1

C2

C2Disk

Disk

Driving mechanism(a) (b)

Driving mechanism

Tighten

TightenRelease

Joint

Figure 1 e traditional universal joint with two intersecting rotation axes

A2 A2

A1 A1

A AB B

D

DCC

B2

B2

B1 B1

D2D2

D1 D1

C2

C2

C1 C1

w2

w2

w1 w1

Upper disk

Lower disk


Driving mechanism

Tighten

TightenRelease Release

Joint

Figure 2 A novel joint without coupling effects

Tighten Release

(a)

Tighten Release

(b)

Tighten Release

(c)

Figure 3 Classic driving system types

Journal of Robotics 3

six identical 2-DOF joints Each joint contains two disks andone spatial linkage e cylindrical bulge surface on the diskand the cylindrical concave surface on the spatial linkagecooperate to form two rotating pairs e axisw1 and axis w2coincide respectively with the upper surface of the lowerdisk and the lower surface of the upper disk Joint structureparameters are defined as shown in Table 1

e 2-DOF bending motion of each segment is achievedby two motors controlling a pair of antagonistic cablesthrough the circular pulleys Cable A cable B cable C andcable D control the 2-DOF bending motion of the proximalsegment as shown in Figure 5(a) while cable E cable Fcable G and cable H control the 2-DOF bending motion ofthe distal segment as shown in Figure 5(b) Since the tworotation axes of each joint are coincident with the endsurfaces of corresponding disks the distance of the tworotation axes is always equal to h regardless of the robotconfigurations In addition a pair of antagonistic cableinputs only control the 1-DOF rotational output and therotational output of each DOF of the joint is only deter-mined by a single pair of antagonistic cable inputs With thisdesign the mapping between actuator space and joint spacecan be eventually simplified Besides the driving mechanismdesign of the robot is illustrated in Section 4

3 Kinematics

e kinematics of the cable-driven hyper-redundant robotrequires establishing the mapping between actuator space

joint space and task space [35] e following assumptionsare made in this study In this proposed robot there is no gapbetween the cables and the cable holes e cablesrsquo shearstrains and elongation are negligible e cable tensionexerting on each joint is the same

Based on these assumptions [36] the joint kinematics isfirst established to analyze the decoupled effect in theproposed 2-DOF joint e relationship between the sum ofthe cable length change and bending angles theoreticallyvalidates that the cables in the robot will not become slackBesides the robot kinematics is established and the robotworkspace is analyzed

31 Joint Kinematics Since the proximal segment and distalsegment have the same bending motion we consider a singlejoint in the proximal segment as an example to establish thejoint kinematics as shown in Figure 6

e coordinate systems Oi O1i and Oi+1 areestablished respectively on the center of the upper surface

Driving mechanism

linkage

disk

tw1

w2

h

r

Joint structure parameters

Bending joint

Proximal segment Distal segment

linkage linkagedisk

disk

Distal segment

Proximal segment

Figure 4 Cable-driven hyper-redundant robot prototype

Table 1 Parameters of the joint structure

Symbol Description ValueH e distance between two axes 8mmt Disk thickness 25mmr Cable distribution circle radius 425mmd Robot diameter 10mm(θ φ) Joint variables in the proximal segment (minusπ18 π18)(α β) Joint variables in the distal segment (minusπ18 π18)


of the lower disk the lower surface of the upper disk and theupper surface of the upper disk Axis xi coincides with theaxis w1 axis y1i coincides with the axis w2 and axis yi+1 isparallel to the axis w2 e transformation from the coor-dinate system Oi to Oi+1 is as follows First the coor-dinate system Oi rotates angle θ around xi axis to obtain thecoordinate system O0i Second the coordinate system O0imoves h along the z0i axis and then rotates angle φ aroundthe y0i axis to obtain the coordinate system O1i ird thecoordinate system O1i moves t along the z1i axis to obtainthe coordinate system Oi+1 Hence the homogeneoustransformation matrix from the coordinate system Oi toOi+1 can be obtained as

ii+1T rot xi θ( 1113857trans z0i h( 1113857rot y0iφ( 1113857trans z1i t( 1113857 (1)

To establish the relationship between the cable lengthand bending angles it is assumed that the position vector ofany point p1 in Oi on the upper surface of the lower disk isrepresented by ipp1 while the position vector of any point p2in O1i on the lower surface of the upper disk is representedby 1ipp2 According to the coordinate transformation theposition vector of any point p2 in Oi on the lower surface ofthe upper disk is represented by ipp2e transformation canbe described as

ipp2 i1iR

1ipp2 +ip1i (2)

B2BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB222222222222222222222222222

E

GE2

G2

w2

E1w1

D2

D1

D

B1

B

F2

F1 H1

H2

HF

A2

A1

C2

AC

C1

O2

G1

O1

(a)

E4

EH

E3

G3

G

G4F4

F

F3H4

H4O4

O3

(b)

Figure 5 e 2-DOF joint of the proximal and distal segment

zi+1

xi+1

x1i

w2

w1 xi(x0i)

A2

C2B2

D2

A1

C1B1

D1

z1i

z0i

y0i

zi

yi+1

y1i

Figure 6 Kinematics coordinate system of the single joint


where i1iR is the rotation matrix from Oi to O1i and ip1i is

the position vector of O1i relative to Oi erefore therelationship between cable length and angles θ and φ can beexpressed as

l ipp2 minus

ipp1

11138681113868111386811138681113868

11138681113868111386811138681113868 (3)

where pp1 and pp2 respectively represent the intersectionpoints between the cables the upper surface of the lowerdisk and the lower surface of the upper disk

Taking the cable length in a single joint of the proximalsegment as an example the coordinates of the points A1 B1C1 and D1 in Oi are represented by ipA1 (r 0 0)ipB1 (0 minus r 0) ipC1 (minusr 0 0) and ipD1 (0 r 0)while the coordinates of the pointsA2 B2 C2 andD2 in O1iare represented by 1ipA2 (r 0 0) 1ipB2 (0 minus r 0)1ipC2 (minusr 0 0) and 1ipD2 (0 r 0) erefore the rela-tionship between the cable length and angles θ and φ can beexpressed as

lA ipA2 minus

ipA1

11138681113868111386811138681113868

11138681113868111386811138681113868

2r2

minus 2r2cφ + h

2minus 2rh middot sφ

1113969

lB ipB2 minus

ipB1

11138681113868111386811138681113868

11138681113868111386811138681113868

(minusr middot cθ minus h middot sθ + r)2

+(r middot sθ + h middot cθ)2

1113969

lC ipC2 minus

ipC1

11138681113868111386811138681113868

11138681113868111386811138681113868

2r2

minus 2r2cφ + h

2+ 2rh middot sφ

1113969

lD ipD2 minus

ipD1

11138681113868111386811138681113868

11138681113868111386811138681113868

(r middot cθ minus h middot sθ minus r)2

+(minusr middot sθ + h middot cθ)2

1113969

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)

where lA lB lC and lD represent the cable length in the jointof the proximal segment cθ cosθ sθ sinθ cφ cosφ andsφ sinφ

When angles θ and φ are equal to zero each cable lengthin the joint is equal to h and the proximal segment keeps astraight configuration According to equation (4) the lengthof cable A and cable C only depends on the angle φ while thelength of cable B and cable D only depends on the angle θConsequently during the 2-DOF bending motion of theproximal segment a pair of antagonistic cable inputs onlycontrols the 1-DOF rotation output and the rotation outputof each DOF of the joint is only determined by a single pairof antagonistic cable inputs e 2-DOF bending motion inthe proximal segment is completely decoupled Similarly the2-DOF decoupled effect of the distal segment is the same asthe decoupled effect of the proximal segment

According to the literature [37 38] if the sum of thecables length changes in each one pair of antagonistic cablesis positive the cables will not become slack When theproposed joint bends from a straight configuration to anarbitrary bending configuration around axes w1 and w2 therelationship between the sum of the cable length changes ineach one pair of antagonistic cables and bending angles isshown in Figure 7 e sum of length changes in the an-tagonistic cables is positive regardless of the bending anglesis indicates that the cables will not become slack when theproximal segment and the distal segment achieve respec-tively 2-DOF bending motion

32 Robot Kinematics Based on the joint kinematics thekinematics of the cable-driven hyper-redundant robot isestablished e mapping between the actuator space jointspace and task space is also obtained [39] e base coor-dinate system O0 is established at the center of the lowersurface of the base disk as shown in Figure 8 Axis x0 isparallel to the axis w1 and axis z0 is perpendicular to thelower surface of the base disk According to the joint ki-nematics the establishment principle of the coordinatesystems O1minus On in the proximal segment is the same asthe coordinate system Om1minus Omn in the distal segmenterefore the mapping between actuator space and jointspace is first establishedWe assume that each segment of therobot contains n identical joints In the straight configu-ration each cable length in the proximal and distal segmentcan be obtained as

Lp nh +(n + 1)t

Ld 2nh + 2(n + 1)t1113896 (5)

where Lp represents the initial length of each cable drivingthe proximal segment and Ld represents the initial length ofeach cable driving the distal segment

According to equation (2) and equation (3) the rela-tionship between each cable length and bending angles in thearbitrary configurations can be derived as

LPprime (n + 1)t + n

ipp2 minusipp1

11138681113868111386811138681113868

11138681113868111386811138681113868

Ldprime LPprime +(n + 1)t + n

jpp4 minusjpp3

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎨

⎪⎩(6)

where Lpprime represents the cablesrsquo length in the proximal

segment and Ldprime represents the cablesrsquo length in the distal

segment In the proximal segment ipp1 and ipp2 are theintersection description between the cable on any side andthe upper surface of the lower disk and the lower surface ofthe upper disk in the coordinate system Oi respectivelyIn the distal segment and jpp3 and jpp4 are the intersectiondescription between the cable on any side and the uppersurface of the lower disk and the lower surface of the upperdisk in the coordinate system Oj respectively ereforethe mapping between the actuator space and joint space canbe established by equation (6) According to the trans-formation shown in Figure 8 the mapping between jointspace and task space can be established Hence the ho-mogeneous transformation matrix from the coordinatesystem O0 to the coordinate system Omn can bewritten as

0mnT

01T times(

ii+1T)

n times nm1T times(

j

j+1T)n

(7)

In the proposed cable-driven hyper-redundant robotthe adjacent two axes in each joint are perpendicular todifferent bending planes which causes that the inverse ki-nematics is difficult to solve through the analytical method[40ndash42]e NewtonndashRaphson iterative method can be usedto solve the inverse kinematics but it is not the researchfocus in this paper


33 Workspace Analysis e workspace of the cable-drivenhyper-redundant robot is determined by the joint geometrybending angles and the joint number [43] Based on therobot kinematics the robot workspace is obtainedFigure 9(a) shows the workspace of the proximal segmentwhile Figure 9(b) shows nine bending configurations of theproximal segment when the joint angles (θ φ) are re-spectively (0 0) (0 π36) (0 minusπ36) (π36 0) (minusπ36 0)(π36 minusπ36) (minusπ36 π36) (minusπ36 minusπ36) and (π36π36) Based on this Figure 9(c) shows the workspace of therobot while Figure 9(d) shows multiple bending configu-rations of the robot when joint angles (θ φ α β) are re-spectively (0 0 0 0) (0 0 0 π36) (0 0 0 minusπ36) (0 0π36 0) (0 0 minusπ36 0) (π36 0 0 0) (minusπ36 0 0 0) (0minusπ36 0 0) and (0 π36 0 0) According to the aboveanalysis the more the segment number is the larger theworkspace of the robot becomes

4 Kinematics Linear Fitting

Based on equation (4) and equation (6) the relationshipbetween each cable length and bending angles is the

nonlinear function However the following kinematicslinear fitting and error analysis will show that the rela-tionship can be well fitted to a linear function in a certainrange of joint variables and the tightened amount of thecable on one side is almost equal to the released amount ofthe antagonistic cable when the robot configuration changesHence any pair of antagonistic cables in the robot can bedriven by a motor and a circle cable pulley as shown inFigure 3(c) which not only simplifies the driving mecha-nism but also reduces the control complexity e followingcontents are the kinematic linear fitting and error analysis intwo segments Based on the results the driving mechanismsof the two segments are designed

41 Kinematics Linear Fitting in the Proximal SegmentSince the 2-DOF bending motion in the proximal segment isdecoupled cable A and cable C are considered as an exampleto perform the kinematic linear fitting using the polynomialfitting methode error values between the original and thefitting function are analyzed by the percentage error modele percentage error el (φ) is defined as

el(φ) 100 middotl(φ) minus lprime(φ)

l(φ)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (8)

where l (φ) represents the original function and lrsquo (φ)represents the fitting function e fitting curve and errorvalues between the original function and the fitting functionof cable A and cable C are solved by the MATLAB curvefitting tool as shown in Figure 10 e fitting functions ofcable A and cable C are represented by lAprime(φ) minus4237φ + 8and lCprime(φ) 4237φ + 8 respectively as shown inFigure 10(a)

Within the range of bending angles shown in Table 1 themaximum fitting error between the original function andfitting function is 0025 and the maximum angle error of

0

1

2

3

4∆l

A+∆l

C (m

m)

5

6times10-4

-π18 π18-π36 π360

Bending angle φ (rad)

(a)

0

1

2

3

4

∆lB+

∆lD

(mm

)

5times10-3

-π18 π18-π36 π360

Bending angle θ (rad)

(b)

Figure 7 e sum of the antagonistic cablesrsquo length changes in any pair of cables (a) Bending angle φ (rad) (b) Bending angle θ (rad)

x1 x2xi

xnyn-1

y2y1 yixm1

xm2

xj xmn

zmnymn

yj

ym2

ym1

x0

yn

xn-1

Figure 8 Coordinate system of the cable-driven hyper-redundantrobot


the end disk in the proximal segment is less than 015 asshown in Figure 10(b) Based on the same kinematics fittingmethod the fitting functions of cable B and cable D are

represented by lBprime(θ) minus4237θ + 8 and lDprime(θ) 4237θ + 8respectively erefore the relationship between each cablelength and bending angles is linear through the special

65

60

55Z

(mm

)

50

45

40

4020

-20-40

-30 -20 -1010 20 30

Y (mm)0

0X (mm)

(a)

Z (m

m)

6070

5040302010

0-20 -10

10 20 -20 -10 0

Y (mm)10 20

0X (mm)

(b)

Z (m

m)

60

80

100

120

40

20-100 -50

0X (mm)50 100

10050-50

-1000

Y (mm)

(c)

Z (m

m)

6080

100120140

4020

0-50 -25

-60 -30 0 30 60

Y (mm)0

X (mm)25 50

(d)

Figure 9 e workspace and bending configurations of the robot (a) e workspace of the proximal section (b) Nine bending con-figurations of the proximal section (c) e robotic workspace (d) Multiple bending configurations of the robot

88

86

84Cable A Cable C

82

8

e l

engt

h of

the c

able

Aan

d ca

ble C

(mm

)

78

76

74

72-π18 π18-π36 π360


lcrsquo(φ)=4237φ+8 lArsquo(φ)=-4237φ+8

Original function

Fitting function

(a)

0025

002Cable A Cable C

0015

Erro

rs e l

001

0005

0-π18 π18-π36 π360


(b)

Figure 10 Cable A and cable C error between the original and fitted function (a) Bending angle φ (rad) (b) Bending angle θ (rad)


distribution of two rotation axes It means the mappingbetween actuator space and joint space is simplifiedMoreover the 2-DOF bending motion of the proximalsegment can be driven by two motors and two circularpulleys Based on the literature [29] the robot motion ac-curacy is satisfied

42 Kinematics Linear Fitting in the Distal SegmentAccording to equation (2) and equation (3) when theproximal segment undergoes 2-DOF bending motion thelength of all eight cables will change is means that thebending motion between the proximal segment and distalsegment is coupled erefore the kinematics linear fittingof cable E cable G cable F and cable H in proximal anddistal segments should be considered to design the drivingmechanism of the distal segment

When the proximal segment keeps the straight config-uration and the distal segment keeps an arbitrary bendingconfiguration the relationship between the length of cable Ecable G cable F and cableH and bending angles α and β canbe expressed as

lE1

2r2

minus 2r2cα + h

2minus 2rh middot sα

1113969

lF1

(minusr middot cβ minus h middot sβ + r)2

+(r middot sβ + h middot cβ)2

1113969

lG1

2r2

minus 2r2cα + h

2+ 2rh middot sα

1113969

lH1

(r middot cβ minus h middot sβ minus r)2

+(minusr middot sβ + h middot cβ)2

1113969

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)

where lE1 lF1 lG1 and lH1 represent the length of a singlejoint in the distal segment cα cosα sα sinα cβ cosβand sβ sinβ According to equation (9) the 2-DOF bendingmotions of the distal segment are decoupled when theproximal segment does not achieve the 2-DOF bendingmotion In addition the relationship between the cablelength of the distal segment and the bending angles α and β isalso approximately linear

If the proximal segment achieves 2-DOF bending mo-tion the lengths of cable E cable G cable F and cable H willalso change Hence the cable length change relationship inthe proximal segment should be considered to achieve thekinematics linear fitting of the distal segment According toequation (2) and equation (3) the relationship between thelength of cable E cable G cable F and cable H and bendingangles θ and φ can be calculated as

lE2 ipE2 minus

ipE1

11138681113868111386811138681113868

11138681113868111386811138681113868

lF2 ipF2 minus

ipF1

11138681113868111386811138681113868

11138681113868111386811138681113868

lG2 ipG2 minus

ipG1

11138681113868111386811138681113868

11138681113868111386811138681113868

lH2 ipH2 minus

ipH1

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)

where lE2 lF2 lG2 and lH2 represent the cable length of thesingle joint in the proximal segment Since the 2-DOFbending motion in the proximal segment has similar ki-nematics cable E and cable G are considered as an exampleto perform the kinematic linear fitting using the polynomial

fitting method Within the range of bending angles shown inTable 1 the kinematics linear fitting results of cable E andcable G in each joint of the proximal segment are shown inFigure 11

Similarly the MATLAB curve fitting tool is also used tosolve the fitting functione fitting functions of cable E andcable G are represented by lEprime(θ φ) minus2996θ minus 2996φ + 8lGprime(θφ) 2996θ + 2996φ + 8 e percentage error el isredefined as

el(θφ) 100 middotl(θ φ) minus lprime(θφ)

l(θ φ)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (11)

where l (θ φ) represents the original function and lrsquo(θ φ)represents the fitting function

According to equation (11) the maximum fitting errorsbetween the original function and fitting function of cable Eand cable G are less than 004 while the maximum fittingerrors of the cable F and cable H are less than 025 asshown in Figure 12 erefore the relationship between thelength of cable E cable G cable F and cable H and bendingangles can be approximately linear is indicates that themapping between actuator space and joint space in the distalsegment is also simplified erefore the 2-DOF bendingmotion of the distal segment can be achieved by two motorsand two circle pulleys Based on the literature [29] the robotmotion accuracy is satisfied

5 Experiment Validation

In this section a 4-DOF cable-driven hyper-redundant robotprototype is established to validate the robot design eproposed robot includes the proximal segment the distalsegment and the driving mechanism as shown inFigure 13(a)e total length of the proximal segment and thedistal segment is 131mme cablesrsquo diameter is 04mmedriving mechanism of the robot prototype includes a guidingdevice a motor driving device and a cable tension adjustingdevice as shown in Figure 13(b)e rated speed of themotoris 10 rmin and the rated torque is 70 kg cme range of theforce sensor is 0ndash10 kg with an accuracy of 003

According to the kinematic linear fitting relationship inSection 4 the driving mechanisms of the proximal and distalsegments are the same For the proximal segment cable Aand cable C are the two ends of one cable that is driven bymotor 1 to control the proximal segment bending in the x0z0plane Cable B and cableD are also the two ends of one cablethat is driven by motor 2 to control the proximal segmentbending in the y0z0 plane

For the distal segment cable E and cable G are the twoends of one cable that is driven bymotor 3 to control the distalsegment bending in the xm1zm1 plane Cable F and cableH arethe two ends of one cable that is driven by motor 4 to controlthe distal segment bending in the ym1zm1 plane e two endsof each cable are fixedly connected to the end disks of theproximal segment and distal segment through knotting emiddle of each cable passes through each joint disk and windsaround the guide device driving device and tension adjustingdevice as shown in Figure 13(b) Each cable tension is


adjusted by changing the position of the sliding block ecable tension values are tested by the tension sensors efollowing experiments include the free bending motion testthe cable tension test and payload experiments

51 Free Bending Motion In this section the multi-DOFbending motions of the proximal segmentand distal segment have experimented as shown inFigure 14

002

0015

001

0005

e e

rror

the c

able

E (e

l)

0π18 π36 0 -π36 -π18 -π18 -π36

π36 π180

Bending angle θ (rad)Bending angle φ (rad)

(a)

025

020

015

01

005

0

e e

rror

the c

able

F (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(b)

003500300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(c)

00300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π360

-π36 -π18 -π18 -π36π36 π18

0



(d)

Figure 12 Kinematics linear fitting error analysis of cables E G F and H

95

Cable GCable E

Cable GCable E

e len

gth

of th

e cab

le E

and

cabl

e G (m

m)

85

75

65

7

8

9

-π18

π18 π18 π36-π36

0-π18

-π36 0 π36


Original function

Fitting function

Figure 11 Kinematics linear fitting analysis of cable E and cable G


When the distal segment keeps a straight configurationthe bending motion of the proximal segment in the x0z0plane requires the coordinated work of motor 1 motor 3and motor 4 Motor 2 does not work to ensure that the cablelengths of cable B and cable D are unchanged e bendingconfiguration outputs of the proximal segment in the x0z0plane are only determined by the inputs of motor 1 Motor 3and motor 4 are driven to keep the straight configuration ofthe distal segment e bending configurations are shown inFigures 14(a)ndash14(c)

When the distal segment keeps a straight configurationthe bending motion of the proximal segment in the y0z0plane requires the coordinated work of motor 2 motor 3and motor 4 Motor 1 does not work to ensure that thelengths of cable A and cable B are unchanged e bendingconfiguration outputs of the proximal segment in the x0z0plane are only determined by the inputs of motor 2 Motor 3and motor 4 are driven to keep the straight configuration ofthe distal segment e bending configurations are shown inFigures 14(d)ndash14(g)

When the proximal segment keeps a straight config-uration the bending motion of the distal segment in thexm1zm1 plane only requires motor 3 working to change thelength of cable E and cable G e other motors do notwork to ensure that the cable lengths of cable A cable Ccable E cable G cable F and cable H are unchanged ebending configuration outputs of the proximal segment inthe xm1zm1 plane are only determined by the inputs ofmotor 3 Motor 1 motor 2 and motor 4 are not driven tokeep the straight configuration of the proximal segmente bending configurations are shown in Figures 14(h)ndash14(j)

Similarly the bending motion of the distal segment inthe ym1zm1 plane only requires motor 4 working to changethe length of cable F and cable H Besides to verify themulti-DOF bending motion of the robot we consider thebending configuration of the proximal segment in the x0z0plane and the bending configuration of the distal segmentin the xm1zm1 plane as an example as shown inFigures 14(k)ndash14(n)

Motor 3Motor 4

e distal segment

e proximal segment

Tension adjustingdevice Driving device

Driving device

Guiding device

Guiding device

e driving mechanism of the proximal segment

e driving mechanism of the distal segment

Tension adjustingdevice

Tension pulley

Tension pulley



52 Cable Tension Test During the multiple bending mo-tions the cable average tension curves are used to illustratethe design rationalization of the driving mechanism for theproximal segment and distal segment [44] For the proximalsegment the bending configuration in the x0z0 plane isdetermined by the angle θ while the bending configurationin the y0z0 plane is determined by the angle φ When thedistal segment keeps a straight configuration and theproximal segment keeps a bending configuration in the x0z0plane and y0z0 plane the average cable tension of each cablevaries with the bending angles θ and φ as shown inFigure 15

Within the joint angle ranges of [minusπ18 π18] whenonly the proximal segment bends in the x0z0 plane theaverage tension of cable A and cable C in the proximalsegment varies in the range of 13Nndash15N as shown inFigure 15(a) When only the proximal segment bends in they0z0 plane the average tension of cable B and cable D in theproximal segment changes within the range of 10Nndash13N asshown in Figure 15(b) For the cables of the distal segmentthe average tension of the cable E cable G cable F and cableH varies in the range of 8Nndash10N and 6Nndash10N Since the 2-DOF bending motion of the proximal segment will change

the length of each cable driving the distal segment theaverage tension of each cable driving the distal segment willincrease

When the proximal segment keeps a straight configu-ration and the distal segment keeps a bending configurationin the xm1zm1 plane and ym1zm1 plane the average cabletension of each cable varies with the joint angles α and β asshown in Figure 16 For the proximal segment the bendingconfiguration in the xm1zm1 plane is determined by the angleβ while the bending configuration in the ym1zm1 plane isdetermined by the angle α

Within the range of the bending angles of [minusπ18 π18]when only the distal segment bends in the xm1zm1 plane theaverage tension of the cable F and cable H in the distalsegment varies in the range of 10Nndash12N as shown inFigure 16(a) When only the distal segment bends in theym1zm1 plane the average tension of cable E and cable G inthe distal segment varies in the range of 11Nndash13N as shownin Figure 16(b) For the cables of the proximal segment theaverage tension of cable A cable B cable C and cable Dvaries in the range of 12Nndash15N erefore the phenom-enon of the cables slack does not appear during the multi-DOF bending motion e results indicate that the driving

0deg

(a)

+30deg

(b)

+60deg

(c) (d)

(e) (f) (g)

0deg

(h)

-30deg

(i)

-60deg

(j) (k)

(l) (m) (n)

Figure 14 Robot bending motion experiments


e a

vera

ge ca

ble t

ensio

n (N

)

14

16

12

10

8

ndashπ18 π18ndashπ36 π360


Cables of the proximal segmentCables of the distal segment

(a)

e a

vera

ge ca

ble t

ensio

n (N

) 14

16

12

10

8

6




(b)

Figure 15 Cable tension during the bending motion of the proximal segment (a) Bending angle φ (rad) (b) Bending angle θ (rad)

14

13

12

11

e a

vera

ge ca

ble t

ensio

n (N

)

10

9


Bending angle β (rad)


(a)

Figure 16 Continued


mechanism design of the proximal and distal segments isreasonable

53 Payload Experiments When most of the prior cable-driven hyper-redundant robots are subjected to small ex-ternal disturbance the robots easily appear in the S con-figuration and even other uneven configurations [45] In thissection a 1N weight is loaded at different positions ofdifferent bending configurations in the proposed robot asshown in Figure 17 According to the observation theproposed robot does not appear in the S configurationwhich means the proposed robot can resist small external

disturbances Since there are eight cables in the proximalsegment and four cables in the distal segment the loadcapacity of the proximal segment is stronger than the loadcapacity of the distal segment Besides when the load po-sition keeps moving away from the driving mechanism thedeformation of the terminal position of the robot becomeslarger During the payload experiments it can be easilyknown that the inevitable clearance between the cables andthe cable holes and the assembly errors of the initial con-figuration of each joint will aggravate the deformation of therobot under the external disturbance In addition the robotload capacity is also determined by the robot prototypestructure parameters

e a

vera

ge ca

ble t

ensio

n (N

)

14

15

13

12

11

10ndashπ18 π18ndashπ36 π360

Bending angle α (rad)


(b)

Figure 16 Cable tension during the bending motion of the distal segment (a) Bending angle β (rad) (b) Bending angle α (rad)

Figure 17 e 1N payload experiments in the different positions of the robot


6 Conclusions and Future Works

In this paper a novel 2-DOF decoupled joint is first pro-posed A 4-DOF cable-driven hyper-redundant robot pro-totype is developed based on the 2-DOF decoupled jointekinematics model is established and the workspace is ana-lyzed e kinematic fitting approach is presented for bothproximal and distal segments e mapping between actu-ator space and joint space is simplified through the kine-matics linear fitting and error analysis It means that thedriving mechanism design of the robot is also simplifiedExperiment results indicated that the 2-DOF bending mo-tion of each segment is decoupled According to the averagecable tension and the payload experiments the proposedrobot design is feasible and the hyper-redundant robot doesnot appear in the S configuration when 1N weight is loadedat different positions of different bending configurations inthe proposed robot In future research the tip position andshape under the external forces and the position accuracyanalysis of the robot will be analyzed and evaluated in detailBesides the method avoiding the kinematic coupling be-tween the proximal segment and distal segment is also aresearch focus in the future

Data Availability

e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

e authors declare no conflicts of interest

Acknowledgments

is work was supported by the National Natural ScienceFoundation of China (project nos 51805047 and 52175003)Natural Science Foundation of Hunan Province (project no2021JJ40259) Outstanding Youth Program of Hunan Ed-ucation Department (project nos 20B307 and 20B017)Open Research Project of the State Key Laboratory of In-dustrial Control Technology and Zhejiang UniversityChina (project no ICT2021B02)

References

[1] O M Omisore S Han J Xiong H Li Z Li and L Wang ldquoAreview on flexible robotic systems for minimally invasivesurgeryrdquo IEEE Transactions on Systems Man and Cyber-netics Systems pp 1ndash14 2020

[2] C Yang S Geng I Walker et al ldquoGeometric constraint-based modeling and analysis of a novel continuum robot withshape memory alloy initiated variable stiffnessrdquo e Inter-national Journal of Robotics Research vol 39 no 14pp 1620ndash1634 2020

[3] H Yuan L Zhou andW Xu ldquoA comprehensive static modelof cable-driven multi-section continuum robots consideringfriction effectrdquo Mechanism and Machine eory vol 135pp 130ndash149 2019

[4] J Wang S Wang J Li X Ren and R M Briggs ldquoDevel-opment of a novel robotic platform with controllable stiffness

manipulation arms for laparoendoscopic single-site surgery(LESS)rdquo International Journal of Medical Robotics andComputer Assisted Surgery vol 14 no 1 Article ID e18382018

[5] K Xu J Zhao and M Fu ldquoDevelopment of the SJTUunfoldable robotic system (SURS) for single port laparos-copyrdquo IEEE vol 20 no 5 pp 2133ndash2145 2014

[6] L Huang L Yin B Liu and Y Yang ldquoDesign and errorevaluation of planar 2DOF remote center of motion mech-anisms with cable transmissionsrdquo Journal of MechanicalDesign vol 143 no 1 2021

[7] S Kolachalama and S Lakshmanan ldquoContinuum robots formanipulation applications a surveyrdquo Journal of Roboticsvol 2020 Article ID 4187048 19 pages 2020

[8] M H Korayem A Zehfroosh H Tourajizadeh andS Manteghi ldquoOptimal motion planning of non-linear dy-namic systems in the presence of obstacles and movingboundaries using SDRE application on cable-suspendedrobotrdquo Nonlinear Dynamics vol 76 no 2 pp 1423ndash14412014

[9] M H Korayem and M Bamdad ldquoDynamic load-carryingcapacity of cable-suspended parallel manipulatorsrdquo Interna-tional Journal of Advanced Manufacturing Technology vol 44no 7-8 pp 829ndash840 2009

[10] M H Korayem M Bamdad H TourajizadehA H Korayem and S Bayat ldquoAnalytical design of optimaltrajectory with dynamic load-carrying capacity for cable-suspended manipulatorrdquo International Journal of AdvancedManufacturing Technology vol 60 no 1 pp 317ndash327 2012

[11] L Huang Y Yang J Xiao and P Su ldquoType synthesis of 1R1Tremote center of motion mechanisms based on pantographmechanismsrdquo Journal of Mechanical Design Transactions ofASME vol 138 no 1 Article ID 014501 2016

[12] F Jelınek E A Arkenbout PW Henselmans R Pessers andP Breedveld ldquoClassification of joints used in steerable in-struments for minimally invasive surgery-a review of the stateof the artrdquo Journal of Medical Devices vol 9 no 1 2015

[13] M C Lei and R Du ldquoGeometry modeling and simulation ofthe wire-driven bending section of a flexible ureteroscoperdquo inProceedings of the World Congress on Engineering andComputer Science Year vol 2 San Francisco USA October2017

[14] Z Wang T Wang B Zhao et al ldquoHybrid adaptive controlstrategy for continuum surgical robot under external loadrdquoIEEE Robotics and Automation Letters vol 6 no 2pp 1407ndash1414 2021

[15] A Gao J Li Y Zhou Z Wang and H Liu ldquoModeling andtask-oriented optimization of contact-aided continuum ro-botsrdquo IEEE vol 25 no 3 pp 1444ndash1455 2020

[16] W S Rone and P Ben-Tzvi ldquoMechanics modeling of mul-tisegment rod-driven continuum robotsrdquo Journal of Mecha-nisms and Robotics vol 6 no 4 2014

[17] Y J Kim S Cheng S Kim and K Iagnemma ldquoA stiffness-adjustable hyperredundant manipulator using a variableneutral-line mechanism for minimally invasive surgeryrdquo IEEETransactions on Robotics vol 30 no 2 pp 382ndash395 2013

[18] J Kim S I Kwon Y Moon and K Kim ldquoCable-movablerolling joint to expand workspace under high external load ina hyper-redundant manipulatorrdquo IEEE 2021

[19] W Xu T Liu and Y Li ldquoKinematics dynamics and controlof a cable-driven hyper-redundant manipulatorrdquo IEEEvol 23 no 4 pp 1693ndash1704 2018


[20] A Kanada and T Mashimo ldquoSwitching between continuumand discrete states in a continuum robot with dislocatablejointsrdquo IEEE Access vol 9 pp 34859ndash34867 2021

[21] W Shen G Yang T Zheng Y Wang K Yang and Z FangldquoAn accuracy enhancement method for a cable-driven con-tinuum robot with a flexible backbonerdquo IEEE Access vol 8pp 37474ndash37481 2020

[22] M Dehghani and S A A Moosavian ldquoDynamics modeling ofa continuum robotic arm with a contact point in planargrasprdquo Journal of Robotics vol 2014 Article ID 30828313 pages 2014

[23] Z Li M Zin Oo V Nalam et al ldquoDesign of a novel flexibleendoscope-cardioscoperdquo Journal ofMechanisms and Roboticsvol 8 no 5 2016

[24] X Dong D Axinte D Palmer et al ldquoDevelopment of a slendercontinum robotic system for on-wing inspectionrepair of gasturbine enginesrdquo Robotics and Computer-IntegratedManufacturing vol 44 pp 218ndash229 2017

[25] X Jin J Zhao M Feng L Hao and Q Li ldquoSnake-like surgicalforceps for robot-assisted minimally invasive surgeryrdquo In-ternational Journal of Medical Robotics and Computer AssistedSurgery vol 14 no 4 Article ID e1908 2018

[26] J Kim S-i Kwon and K Kim ldquoNovel block mechanism forrolling joints in minimally invasive surgeryrdquo Mechanism andMachine eory vol 147 Article ID 103774 2020

[27] K Kim H Woo and J Suh ldquoDesign and evaluation of acontinuum robot with discreted link joints for cardiovascularinterventionsrdquo in Proceedings of the 2018 7th IEEE Interna-tional Conference on Biomedical Robotics and Biomechatronics(Biorob) IEEE pp 627ndash633 Enschede Netherlands Auguest2018

[28] R J Webster III and B A Jones ldquoDesign and kinematicmodeling of constant curvature continuum robots a reviewrdquoe International Journal of Robotics Research vol 29 no 13pp 1661ndash1683 2010

[29] J Barrientos-Diez X Dong D Axinte and J Kell ldquoReal-timekinematics of continuum robots modelling and validationrdquoRobotics and Computer-Integrated Manufacturing vol 67Article ID 102019 2021

[30] X Dong M Raffles S C Guzman D Axinte and J KellldquoDesign and analysis of a family of snake arm robots con-nected by compliant jointsrdquoMechanism and Machine eoryvol 77 pp 73ndash91 2014

[31] L Tang J Wang Y Zheng G Gu L Zhu and X ZhuldquoDesign of a cable-driven hyper-redundant robot with ex-perimental validationrdquo International Journal of AdvancedRobotic Systems vol 14 no 5 Article ID 17298814177344582017

[32] A Yeshmukhametov K Koganezawa and Y YamamotoldquoDesign and kinematics of cable-driven continuum robot armwith universal joint backbonerdquo in Proceedings of the 2018IEEE International Conference on Robotics and Biomimetics(ROBIO) IEEE pp 2444ndash2449 Kuala Lumpur MalaysiaDecember 2018

[33] R Xue B Ren Z Yan and Z Du ldquoA cable-pulley systemmodeling based position compensation control for a lapa-roscope surgical robotrdquo Mechanism and Machine eoryvol 118 pp 283ndash299 2017

[34] J W Suh and K Y Kim ldquoHarmonious cable actuationmechanism for soft robot joints using a pair of noncircularpulleysrdquo Journal of Mechanisms and Robotics vol 10 no 62018

[35] Z Li and R Du ldquoDesign and analysis of a bio-inspired wire-driven multi-section flexible robotrdquo International Journal ofAdvanced Robotic Systems vol 10 no 4 Article ID 209 2013

[36] Y Liu and F Alambeigi ldquoEffect of external and internal loadson tension loss of tendon-driven continuum manipulatorsrdquoIEEE Robotics and Automation Letters vol 6 no 2pp 1606ndash1613 2021

[37] S M Segreti M D Kutzer R J Murphy and M ArmandldquoCable length estimation for a compliant surgical manipu-latorrdquo in Proceedings of the 2012 IEEE International Con-ference on Robotics and Automation IEEE pp 701ndash708 SaintPaul MN USA May 2012

[38] H In H Lee U Jeong B B Kang and K J Cho ldquoFeasibilitystudy of a slack enabling actuator for actuating tendon-drivensoft wearable robot without pretensionrdquo in Proceedings of the2015 IEEE International Conference on Robotics and Auto-mation (ICRA) IEEE pp 1229ndash1234 Seattle WA USA May2015

[39] B A Jones and I D Walker ldquoKinematics for multisectioncontinuum robotsrdquo IEEE Transactions on Robotics vol 22no 1 pp 43ndash55 2006

[40] Y Y Zhou J H Li M Q Guo Z D Wang and H LiuldquoModeling and optimization analysis of a continuum robotfor single-port surgeryrdquo Robot vol 42 no 3 pp 316ndash3242020

[41] G Palmieri and C Scoccia ldquoMotion planning and control ofredundant manipulators for dynamical obstacle avoidancerdquoMachines vol 9 no 6 Article ID 121 2021

[42] S Jin S K Lee J Lee and S Han ldquoKinematic model and real-time path generator for a wire-driven surgical robot arm witharticulated joint structurerdquo Applied Sciences vol 9 no 19Article ID 4114 2019

[43] T Kato I Okumura H Kose K Takagi and N HataldquoTendon-driven continuum robot for neuroendoscopy vali-dation of extended kinematic mapping for hysteresis opera-tionrdquo International Journal of Computer Assisted Radiologyand Surgery vol 11 no 4 pp 589ndash602 2016

[44] M Hwang and D-S Kwon ldquoStrong continuum manipulatorfor flexible endoscopic surgeryrdquo IEEE vol 24 no 5pp 2193ndash2203 2019

[45] H Yuan P W Y Chiu and Z Li ldquoShape-reconstruction-based force sensing method for continuum surgical robotswith large deformationrdquo IEEE Robotics and AutomationLetters vol 2 no 4 pp 1972ndash1979 2017


Based on multiple cylindrical rolling joints Kim et aldesigned a cable-driven hyper-redundant robot which di-minishes the bending coupling effect by enlarging the spacefor the passage of the center cable [17]

In this paper a novel 2-DOF decoupled joint is proposedby adjusting the distribution of two rotation axes For thedecoupled joint a pair of antagonistic cable inputs onlycontrols a 1-DOF rotational output and the rotationaloutput of each DOF is only determined by a single pair ofantagonistic cable inputs By connecting two 2-DOF seg-ments in serial a cable-driven hyper-redundant robot ispresented Each segment consists of multiple identical 2-DOF decoupled joints in serial A kinematics linear fittingapproach is presented to simplify the mapping betweenactuator space and joint space Based on the linear fittingerror analysis each pair of antagonistic cables is driven by amotor through a circular pulley which can simplify thedesign of the driving mechanism e proposed robot isverified by the bending motion experiments the cabletension test and the load experiments

e rest of this paper is organized as follows Section 2introduces the challenges of the existing robots Moreover anovel cable-driven hyper-redundant robot is also introducedin Section 2 e kinematics and the robot workspace areanalyzed in Section 3e presented kinematics linear fittingand error analysis of the robot are discussed in Section 4erobot prototype through some experiments is verified inSection 5 Section 6 presents the conclusion

2 Robot Design

21 Challenges of the Existing Robots

211 Coupling Effect For most of the prior designs of cable-driven hyper-redundant robots the 2-DOF bending motionin each segment of the robot is coupled e rotationaloutput of each DOF of the joint is controlled by multiplepairs of antagonistic cable inputs Consequently the cou-pling effect will increase the complexity of the robotrsquos drivingmechanism and the risk of slack in the driving cables

e relationship between the cable length and the ro-tation angles is determined by the joint types and theirstructural parameters For instance the coupling effect of therevolute joints is determined by the cable distribution circleradius the number of cables and the distribution of therotation axes Since the cable distribution circle radius andthe number of cables are confined by the practical factorsthe distribution of joint rotation axes is a significant factor toavoid the coupling effect Figure 1 shows a conventionaluniversal joint with two intersecting rotation axes located atthe middle of two disks [31] When the upper disk rotatesaround rotation axesw1 through the releasing of cable A andtightening of cable C cable B and cable Dmust be tightenedsimultaneously to avoid slack as shown in Figure 1(b)Similarly the rotation around axis w2 requires the control ofall four cables Otherwise cable A and cable B will becomeslack It can be proved that the rotational output of eachDOF requires the control of multiple pairs of antagonisticcables regardless of the location of the two intersecting axes

in the joint For the universal joint with two nonintersectingrotation axes located between the two disks [32] it can bealso proved that the coupling effect exists in the joint when atleast one rotation axis is located at the middle place of twodisks which is similar to the universal joint as shown inFigure 1

Based on the above analysis we tried to change thedistribution of the joint rotation axes to avoid the couplingeffect erefore this paper proposes a novel joint withoutcoupling effects as shown in Figure 2 e two axes in thejoint are respectively coincident with the upper surface ofthe lower disk and the lower surface of the upper disk Sincepoints A2 and B2 are located at the axis w1 points A2 and B2achieve circular motion around points A1 and B1 when theupper disk rotates around the axis w1 erefore the lengthof cable A and cable B will not change while the length ofcable C and cable D will also change Since the points C2 andD2 are located at the axis w2 the points C2 and D2 can beconsidered as the fixed points when the upper disk rotatesaround the axisw2 In consequence the length of cableC andcableDwill not change while the length of cable A and cableB will changeis indicates that the coupling effect does notexist in the proposed joint with the special distribution oftwo axes positions

212 Challenge of the Driving Mechanism Design e re-lationship between cable length and bending angles shouldbe considered to design the driving mechanism of the cable-driven hyper-redundant robot [33 34] For most cable-driven hyper-redundant robots the relationship between thelength inputs of each cable and bending angles is a nonlinearfunction It means the tightened amount on one cable is notequal to the released amount on the antagonistic cable whenthe robot bends to an arbitrary configuration It is difficultfor cable-driven hyper-redundant robots to design a com-pact and simple driving mechanism

e following driving mechanism design approaches areadopted e first approach is that each cable is driven by aseparate motor and a circular cable pulley It is convenient todesign the driving mechanism However this approachincreases the complexity of the control system as shown inFigure 3(a) e second approach is that noncircular cablepulleys are designed to realize a motor driving a pair ofcables based on the nonlinear function as shown inFigure 3(b)is method significantly reduces the number ofthe motor but the fabrication and assembly of noncircularcable pulleys require high accuracy e third approach isthat a pair of cables is driven by a separate motor and acircular cable pulley as shown in Figure 3(c) is methodrequires the releasing amount of one cable is equal to thetightening amount of the antagonistic cable which cansimplify the driving mechanism

22 A Novel Robot Design with the Decoupled Joints issection proposes a cable-driven hyper-redundant robotbased on the multiple decoupled joints as shown in Figure 4e robot is composed of a proximal segment a distalsegment and a drivingmechanism Each segment consists of


A2 A2

A1 A1

A AB B

DDC C

B2

B2

w2w2

r

h

w1

w1

B1 B1

D1 D1

D2

D2

C1 C1

C2

C2Disk

Disk


Driving mechanism

Tighten

TightenRelease

Joint


A2 A2

A1 A1

A AB B

D

DCC

B2

B2

B1 B1

D2D2

D1 D1

C2

C2

C1 C1

w2

w2

w1 w1

Upper disk

Lower disk


Driving mechanism

Tighten


Joint


Tighten Release

(a)

Tighten Release

(b)

Tighten Release

(c)





3 Kinematics






Driving mechanism

linkage

disk

tw1

w2

h

r


Bending joint


linkage linkagedisk

disk

Distal segment

Proximal segment








ipp2 i1iR

1ipp2 +ip1i (2)


E

GE2

G2

w2

E1w1

D2

D1

D

B1

B

F2

F1 H1

H2

HF

A2

A1

C2

AC

C1

O2

G1

O1

(a)

E4

EH

E3

G3

G

G4F4

F

F3H4

H4O4

O3

(b)


zi+1

xi+1

x1i

w2

w1 xi(x0i)

A2

C2B2

D2

A1

C1B1

D1

z1i

z0i

y0i

zi

yi+1

y1i





l ipp2 minus

ipp1

11138681113868111386811138681113868

11138681113868111386811138681113868 (3)



lA ipA2 minus

ipA1

11138681113868111386811138681113868

11138681113868111386811138681113868

2r2

minus 2r2cφ + h


1113969

lB ipB2 minus

ipB1

11138681113868111386811138681113868

11138681113868111386811138681113868



1113969

lC ipC2 minus

ipC1

11138681113868111386811138681113868

11138681113868111386811138681113868

2r2

minus 2r2cφ + h

2+ 2rh middot sφ

1113969

lD ipD2 minus

ipD1

11138681113868111386811138681113868

11138681113868111386811138681113868



1113969

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)





Lp nh +(n + 1)t

Ld 2nh + 2(n + 1)t1113896 (5)




ipp2 minusipp1

11138681113868111386811138681113868

11138681113868111386811138681113868


jpp4 minusjpp3

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎨

⎪⎩(6)




0mnT

01T times(

ii+1T)

n times nm1T times(

j

j+1T)n

(7)









l(φ)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (8)



0

1

2

3

4∆l

A+∆l

C (m

m)

5

6times10-4

-π18 π18-π36 π360


(a)

0

1

2

3

4

∆lB+

∆lD

(mm

)

5times10-3

-π18 π18-π36 π360


(b)


x1 x2xi

xnyn-1

y2y1 yixm1

xm2

xj xmn

zmnymn

yj

ym2

ym1

x0

yn

xn-1





65

60

55Z

(mm

)

50

45

40

4020

-20-40

-30 -20 -1010 20 30

Y (mm)0

0X (mm)

(a)

Z (m

m)

6070

5040302010

0-20 -10

10 20 -20 -10 0

Y (mm)10 20

0X (mm)

(b)

Z (m

m)

60

80

100

120

40

20-100 -50

0X (mm)50 100

10050-50

-1000

Y (mm)

(c)

Z (m

m)

6080

100120140

4020

0-50 -25

-60 -30 0 30 60

Y (mm)0

X (mm)25 50

(d)


88

86

84Cable A Cable C

82

8

e l

engt

h of

the c

able

Aan

d ca

ble C

(mm

)

78

76

74

72-π18 π18-π36 π360



Original function

Fitting function

(a)

0025

002Cable A Cable C

0015

Erro

rs e l

001

0005

0-π18 π18-π36 π360


(b)






lE1

2r2

minus 2r2cα + h


1113969

lF1



1113969

lG1

2r2

minus 2r2cα + h

2+ 2rh middot sα

1113969

lH1



1113969

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)



lE2 ipE2 minus

ipE1

11138681113868111386811138681113868

11138681113868111386811138681113868

lF2 ipF2 minus

ipF1

11138681113868111386811138681113868

11138681113868111386811138681113868

lG2 ipG2 minus

ipG1

11138681113868111386811138681113868

11138681113868111386811138681113868

lH2 ipH2 minus

ipH1

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)





l(θ φ)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (11)










002

0015

001

0005

e e

rror

the c

able

E (e

l)

0π18 π36 0 -π36 -π18 -π18 -π36

π36 π180


(a)

025

020

015

01

005

0

e e

rror

the c

able

F (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(b)

003500300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(c)

00300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π360

-π36 -π18 -π18 -π36π36 π18

0



(d)


95

Cable GCable E

Cable GCable E

e len

gth

of th

e cab

le E

and

cabl

e G (m

m)

85

75

65

7

8

9

-π18

π18 π18 π36-π36

0-π18

-π36 0 π36


Original function

Fitting function







Motor 3Motor 4

e distal segment

e proximal segment


Driving device

Guiding device

Guiding device




Tension pulley

Tension pulley








0deg

(a)

+30deg

(b)

+60deg

(c) (d)

(e) (f) (g)

0deg

(h)

-30deg

(i)

-60deg

(j) (k)

(l) (m) (n)



e a

vera

ge ca

ble t

ensio

n (N

)

14

16

12

10

8




(a)

e a

vera

ge ca

ble t

ensio

n (N

) 14

16

12

10

8

6




(b)


14

13

12

11

e a

vera

ge ca

ble t

ensio

n (N

)

10

9




(a)

Figure 16 Continued





e a

vera

ge ca

ble t

ensio

n (N

)

14

15

13

12

11




(b)






Data Availability




Acknowledgments


References

















































A2 A2

A1 A1

A AB B

DDC C

B2

B2

w2w2

r

h

w1

w1

B1 B1

D1 D1

D2

D2

C1 C1

C2

C2Disk

Disk


Driving mechanism

Tighten

TightenRelease

Joint


A2 A2

A1 A1

A AB B

D

DCC

B2

B2

B1 B1

D2D2

D1 D1

C2

C2

C1 C1

w2

w2

w1 w1

Upper disk

Lower disk


Driving mechanism

Tighten


Joint


Tighten Release

(a)

Tighten Release

(b)

Tighten Release

(c)





3 Kinematics






Driving mechanism

linkage

disk

tw1

w2

h

r


Bending joint


linkage linkagedisk

disk

Distal segment

Proximal segment








ipp2 i1iR

1ipp2 +ip1i (2)


E

GE2

G2

w2

E1w1

D2

D1

D

B1

B

F2

F1 H1

H2

HF

A2

A1

C2

AC

C1

O2

G1

O1

(a)

E4

EH

E3

G3

G

G4F4

F

F3H4

H4O4

O3

(b)


zi+1

xi+1

x1i

w2

w1 xi(x0i)

A2

C2B2

D2

A1

C1B1

D1

z1i

z0i

y0i

zi

yi+1

y1i





l ipp2 minus

ipp1

11138681113868111386811138681113868

11138681113868111386811138681113868 (3)



lA ipA2 minus

ipA1

11138681113868111386811138681113868

11138681113868111386811138681113868

2r2

minus 2r2cφ + h


1113969

lB ipB2 minus

ipB1

11138681113868111386811138681113868

11138681113868111386811138681113868



1113969

lC ipC2 minus

ipC1

11138681113868111386811138681113868

11138681113868111386811138681113868

2r2

minus 2r2cφ + h

2+ 2rh middot sφ

1113969

lD ipD2 minus

ipD1

11138681113868111386811138681113868

11138681113868111386811138681113868



1113969

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)





Lp nh +(n + 1)t

Ld 2nh + 2(n + 1)t1113896 (5)




ipp2 minusipp1

11138681113868111386811138681113868

11138681113868111386811138681113868


jpp4 minusjpp3

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎨

⎪⎩(6)




0mnT

01T times(

ii+1T)

n times nm1T times(

j

j+1T)n

(7)









l(φ)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (8)



0

1

2

3

4∆l

A+∆l

C (m

m)

5

6times10-4

-π18 π18-π36 π360


(a)

0

1

2

3

4

∆lB+

∆lD

(mm

)

5times10-3

-π18 π18-π36 π360


(b)


x1 x2xi

xnyn-1

y2y1 yixm1

xm2

xj xmn

zmnymn

yj

ym2

ym1

x0

yn

xn-1





65

60

55Z

(mm

)

50

45

40

4020

-20-40

-30 -20 -1010 20 30

Y (mm)0

0X (mm)

(a)

Z (m

m)

6070

5040302010

0-20 -10

10 20 -20 -10 0

Y (mm)10 20

0X (mm)

(b)

Z (m

m)

60

80

100

120

40

20-100 -50

0X (mm)50 100

10050-50

-1000

Y (mm)

(c)

Z (m

m)

6080

100120140

4020

0-50 -25

-60 -30 0 30 60

Y (mm)0

X (mm)25 50

(d)


88

86

84Cable A Cable C

82

8

e l

engt

h of

the c

able

Aan

d ca

ble C

(mm

)

78

76

74

72-π18 π18-π36 π360



Original function

Fitting function

(a)

0025

002Cable A Cable C

0015

Erro

rs e l

001

0005

0-π18 π18-π36 π360


(b)






lE1

2r2

minus 2r2cα + h


1113969

lF1



1113969

lG1

2r2

minus 2r2cα + h

2+ 2rh middot sα

1113969

lH1



1113969

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)



lE2 ipE2 minus

ipE1

11138681113868111386811138681113868

11138681113868111386811138681113868

lF2 ipF2 minus

ipF1

11138681113868111386811138681113868

11138681113868111386811138681113868

lG2 ipG2 minus

ipG1

11138681113868111386811138681113868

11138681113868111386811138681113868

lH2 ipH2 minus

ipH1

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)





l(θ φ)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (11)










002

0015

001

0005

e e

rror

the c

able

E (e

l)

0π18 π36 0 -π36 -π18 -π18 -π36

π36 π180


(a)

025

020

015

01

005

0

e e

rror

the c

able

F (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(b)

003500300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(c)

00300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π360

-π36 -π18 -π18 -π36π36 π18

0



(d)


95

Cable GCable E

Cable GCable E

e len

gth

of th

e cab

le E

and

cabl

e G (m

m)

85

75

65

7

8

9

-π18

π18 π18 π36-π36

0-π18

-π36 0 π36


Original function

Fitting function







Motor 3Motor 4

e distal segment

e proximal segment


Driving device

Guiding device

Guiding device




Tension pulley

Tension pulley








0deg

(a)

+30deg

(b)

+60deg

(c) (d)

(e) (f) (g)

0deg

(h)

-30deg

(i)

-60deg

(j) (k)

(l) (m) (n)



e a

vera

ge ca

ble t

ensio

n (N

)

14

16

12

10

8




(a)

e a

vera

ge ca

ble t

ensio

n (N

) 14

16

12

10

8

6




(b)


14

13

12

11

e a

vera

ge ca

ble t

ensio

n (N

)

10

9




(a)

Figure 16 Continued





e a

vera

ge ca

ble t

ensio

n (N

)

14

15

13

12

11




(b)






Data Availability




Acknowledgments


References



















































3 Kinematics






Driving mechanism

linkage

disk

tw1

w2

h

r


Bending joint


linkage linkagedisk

disk

Distal segment

Proximal segment








ipp2 i1iR

1ipp2 +ip1i (2)


E

GE2

G2

w2

E1w1

D2

D1

D

B1

B

F2

F1 H1

H2

HF

A2

A1

C2

AC

C1

O2

G1

O1

(a)

E4

EH

E3

G3

G

G4F4

F

F3H4

H4O4

O3

(b)


zi+1

xi+1

x1i

w2

w1 xi(x0i)

A2

C2B2

D2

A1

C1B1

D1

z1i

z0i

y0i

zi

yi+1

y1i





l ipp2 minus

ipp1

11138681113868111386811138681113868

11138681113868111386811138681113868 (3)



lA ipA2 minus

ipA1

11138681113868111386811138681113868

11138681113868111386811138681113868

2r2

minus 2r2cφ + h


1113969

lB ipB2 minus

ipB1

11138681113868111386811138681113868

11138681113868111386811138681113868



1113969

lC ipC2 minus

ipC1

11138681113868111386811138681113868

11138681113868111386811138681113868

2r2

minus 2r2cφ + h

2+ 2rh middot sφ

1113969

lD ipD2 minus

ipD1

11138681113868111386811138681113868

11138681113868111386811138681113868



1113969

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)





Lp nh +(n + 1)t

Ld 2nh + 2(n + 1)t1113896 (5)




ipp2 minusipp1

11138681113868111386811138681113868

11138681113868111386811138681113868


jpp4 minusjpp3

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎨

⎪⎩(6)




0mnT

01T times(

ii+1T)

n times nm1T times(

j

j+1T)n

(7)









l(φ)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (8)



0

1

2

3

4∆l

A+∆l

C (m

m)

5

6times10-4

-π18 π18-π36 π360


(a)

0

1

2

3

4

∆lB+

∆lD

(mm

)

5times10-3

-π18 π18-π36 π360


(b)


x1 x2xi

xnyn-1

y2y1 yixm1

xm2

xj xmn

zmnymn

yj

ym2

ym1

x0

yn

xn-1





65

60

55Z

(mm

)

50

45

40

4020

-20-40

-30 -20 -1010 20 30

Y (mm)0

0X (mm)

(a)

Z (m

m)

6070

5040302010

0-20 -10

10 20 -20 -10 0

Y (mm)10 20

0X (mm)

(b)

Z (m

m)

60

80

100

120

40

20-100 -50

0X (mm)50 100

10050-50

-1000

Y (mm)

(c)

Z (m

m)

6080

100120140

4020

0-50 -25

-60 -30 0 30 60

Y (mm)0

X (mm)25 50

(d)


88

86

84Cable A Cable C

82

8

e l

engt

h of

the c

able

Aan

d ca

ble C

(mm

)

78

76

74

72-π18 π18-π36 π360



Original function

Fitting function

(a)

0025

002Cable A Cable C

0015

Erro

rs e l

001

0005

0-π18 π18-π36 π360


(b)






lE1

2r2

minus 2r2cα + h


1113969

lF1



1113969

lG1

2r2

minus 2r2cα + h

2+ 2rh middot sα

1113969

lH1



1113969

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)



lE2 ipE2 minus

ipE1

11138681113868111386811138681113868

11138681113868111386811138681113868

lF2 ipF2 minus

ipF1

11138681113868111386811138681113868

11138681113868111386811138681113868

lG2 ipG2 minus

ipG1

11138681113868111386811138681113868

11138681113868111386811138681113868

lH2 ipH2 minus

ipH1

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)





l(θ φ)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (11)










002

0015

001

0005

e e

rror

the c

able

E (e

l)

0π18 π36 0 -π36 -π18 -π18 -π36

π36 π180


(a)

025

020

015

01

005

0

e e

rror

the c

able

F (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(b)

003500300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(c)

00300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π360

-π36 -π18 -π18 -π36π36 π18

0



(d)


95

Cable GCable E

Cable GCable E

e len

gth

of th

e cab

le E

and

cabl

e G (m

m)

85

75

65

7

8

9

-π18

π18 π18 π36-π36

0-π18

-π36 0 π36


Original function

Fitting function







Motor 3Motor 4

e distal segment

e proximal segment


Driving device

Guiding device

Guiding device




Tension pulley

Tension pulley








0deg

(a)

+30deg

(b)

+60deg

(c) (d)

(e) (f) (g)

0deg

(h)

-30deg

(i)

-60deg

(j) (k)

(l) (m) (n)



e a

vera

ge ca

ble t

ensio

n (N

)

14

16

12

10

8




(a)

e a

vera

ge ca

ble t

ensio

n (N

) 14

16

12

10

8

6




(b)


14

13

12

11

e a

vera

ge ca

ble t

ensio

n (N

)

10

9




(a)

Figure 16 Continued





e a

vera

ge ca

ble t

ensio

n (N

)

14

15

13

12

11




(b)






Data Availability




Acknowledgments


References




















































ipp2 i1iR

1ipp2 +ip1i (2)


E

GE2

G2

w2

E1w1

D2

D1

D

B1

B

F2

F1 H1

H2

HF

A2

A1

C2

AC

C1

O2

G1

O1

(a)

E4

EH

E3

G3

G

G4F4

F

F3H4

H4O4

O3

(b)


zi+1

xi+1

x1i

w2

w1 xi(x0i)

A2

C2B2

D2

A1

C1B1

D1

z1i

z0i

y0i

zi

yi+1

y1i





l ipp2 minus

ipp1

11138681113868111386811138681113868

11138681113868111386811138681113868 (3)



lA ipA2 minus

ipA1

11138681113868111386811138681113868

11138681113868111386811138681113868

2r2

minus 2r2cφ + h


1113969

lB ipB2 minus

ipB1

11138681113868111386811138681113868

11138681113868111386811138681113868



1113969

lC ipC2 minus

ipC1

11138681113868111386811138681113868

11138681113868111386811138681113868

2r2

minus 2r2cφ + h

2+ 2rh middot sφ

1113969

lD ipD2 minus

ipD1

11138681113868111386811138681113868

11138681113868111386811138681113868



1113969

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)





Lp nh +(n + 1)t

Ld 2nh + 2(n + 1)t1113896 (5)




ipp2 minusipp1

11138681113868111386811138681113868

11138681113868111386811138681113868


jpp4 minusjpp3

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎨

⎪⎩(6)




0mnT

01T times(

ii+1T)

n times nm1T times(

j

j+1T)n

(7)









l(φ)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (8)



0

1

2

3

4∆l

A+∆l

C (m

m)

5

6times10-4

-π18 π18-π36 π360


(a)

0

1

2

3

4

∆lB+

∆lD

(mm

)

5times10-3

-π18 π18-π36 π360


(b)


x1 x2xi

xnyn-1

y2y1 yixm1

xm2

xj xmn

zmnymn

yj

ym2

ym1

x0

yn

xn-1





65

60

55Z

(mm

)

50

45

40

4020

-20-40

-30 -20 -1010 20 30

Y (mm)0

0X (mm)

(a)

Z (m

m)

6070

5040302010

0-20 -10

10 20 -20 -10 0

Y (mm)10 20

0X (mm)

(b)

Z (m

m)

60

80

100

120

40

20-100 -50

0X (mm)50 100

10050-50

-1000

Y (mm)

(c)

Z (m

m)

6080

100120140

4020

0-50 -25

-60 -30 0 30 60

Y (mm)0

X (mm)25 50

(d)


88

86

84Cable A Cable C

82

8

e l

engt

h of

the c

able

Aan

d ca

ble C

(mm

)

78

76

74

72-π18 π18-π36 π360



Original function

Fitting function

(a)

0025

002Cable A Cable C

0015

Erro

rs e l

001

0005

0-π18 π18-π36 π360


(b)






lE1

2r2

minus 2r2cα + h


1113969

lF1



1113969

lG1

2r2

minus 2r2cα + h

2+ 2rh middot sα

1113969

lH1



1113969

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)



lE2 ipE2 minus

ipE1

11138681113868111386811138681113868

11138681113868111386811138681113868

lF2 ipF2 minus

ipF1

11138681113868111386811138681113868

11138681113868111386811138681113868

lG2 ipG2 minus

ipG1

11138681113868111386811138681113868

11138681113868111386811138681113868

lH2 ipH2 minus

ipH1

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)





l(θ φ)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (11)










002

0015

001

0005

e e

rror

the c

able

E (e

l)

0π18 π36 0 -π36 -π18 -π18 -π36

π36 π180


(a)

025

020

015

01

005

0

e e

rror

the c

able

F (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(b)

003500300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(c)

00300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π360

-π36 -π18 -π18 -π36π36 π18

0



(d)


95

Cable GCable E

Cable GCable E

e len

gth

of th

e cab

le E

and

cabl

e G (m

m)

85

75

65

7

8

9

-π18

π18 π18 π36-π36

0-π18

-π36 0 π36


Original function

Fitting function







Motor 3Motor 4

e distal segment

e proximal segment


Driving device

Guiding device

Guiding device




Tension pulley

Tension pulley








0deg

(a)

+30deg

(b)

+60deg

(c) (d)

(e) (f) (g)

0deg

(h)

-30deg

(i)

-60deg

(j) (k)

(l) (m) (n)



e a

vera

ge ca

ble t

ensio

n (N

)

14

16

12

10

8




(a)

e a

vera

ge ca

ble t

ensio

n (N

) 14

16

12

10

8

6




(b)


14

13

12

11

e a

vera

ge ca

ble t

ensio

n (N

)

10

9




(a)

Figure 16 Continued





e a

vera

ge ca

ble t

ensio

n (N

)

14

15

13

12

11




(b)






Data Availability




Acknowledgments


References



















































l ipp2 minus

ipp1

11138681113868111386811138681113868

11138681113868111386811138681113868 (3)



lA ipA2 minus

ipA1

11138681113868111386811138681113868

11138681113868111386811138681113868

2r2

minus 2r2cφ + h


1113969

lB ipB2 minus

ipB1

11138681113868111386811138681113868

11138681113868111386811138681113868



1113969

lC ipC2 minus

ipC1

11138681113868111386811138681113868

11138681113868111386811138681113868

2r2

minus 2r2cφ + h

2+ 2rh middot sφ

1113969

lD ipD2 minus

ipD1

11138681113868111386811138681113868

11138681113868111386811138681113868



1113969

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)





Lp nh +(n + 1)t

Ld 2nh + 2(n + 1)t1113896 (5)




ipp2 minusipp1

11138681113868111386811138681113868

11138681113868111386811138681113868


jpp4 minusjpp3

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎨

⎪⎩(6)




0mnT

01T times(

ii+1T)

n times nm1T times(

j

j+1T)n

(7)









l(φ)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (8)



0

1

2

3

4∆l

A+∆l

C (m

m)

5

6times10-4

-π18 π18-π36 π360


(a)

0

1

2

3

4

∆lB+

∆lD

(mm

)

5times10-3

-π18 π18-π36 π360


(b)


x1 x2xi

xnyn-1

y2y1 yixm1

xm2

xj xmn

zmnymn

yj

ym2

ym1

x0

yn

xn-1





65

60

55Z

(mm

)

50

45

40

4020

-20-40

-30 -20 -1010 20 30

Y (mm)0

0X (mm)

(a)

Z (m

m)

6070

5040302010

0-20 -10

10 20 -20 -10 0

Y (mm)10 20

0X (mm)

(b)

Z (m

m)

60

80

100

120

40

20-100 -50

0X (mm)50 100

10050-50

-1000

Y (mm)

(c)

Z (m

m)

6080

100120140

4020

0-50 -25

-60 -30 0 30 60

Y (mm)0

X (mm)25 50

(d)


88

86

84Cable A Cable C

82

8

e l

engt

h of

the c

able

Aan

d ca

ble C

(mm

)

78

76

74

72-π18 π18-π36 π360



Original function

Fitting function

(a)

0025

002Cable A Cable C

0015

Erro

rs e l

001

0005

0-π18 π18-π36 π360


(b)






lE1

2r2

minus 2r2cα + h


1113969

lF1



1113969

lG1

2r2

minus 2r2cα + h

2+ 2rh middot sα

1113969

lH1



1113969

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)



lE2 ipE2 minus

ipE1

11138681113868111386811138681113868

11138681113868111386811138681113868

lF2 ipF2 minus

ipF1

11138681113868111386811138681113868

11138681113868111386811138681113868

lG2 ipG2 minus

ipG1

11138681113868111386811138681113868

11138681113868111386811138681113868

lH2 ipH2 minus

ipH1

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)





l(θ φ)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (11)










002

0015

001

0005

e e

rror

the c

able

E (e

l)

0π18 π36 0 -π36 -π18 -π18 -π36

π36 π180


(a)

025

020

015

01

005

0

e e

rror

the c

able

F (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(b)

003500300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(c)

00300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π360

-π36 -π18 -π18 -π36π36 π18

0



(d)


95

Cable GCable E

Cable GCable E

e len

gth

of th

e cab

le E

and

cabl

e G (m

m)

85

75

65

7

8

9

-π18

π18 π18 π36-π36

0-π18

-π36 0 π36


Original function

Fitting function







Motor 3Motor 4

e distal segment

e proximal segment


Driving device

Guiding device

Guiding device




Tension pulley

Tension pulley








0deg

(a)

+30deg

(b)

+60deg

(c) (d)

(e) (f) (g)

0deg

(h)

-30deg

(i)

-60deg

(j) (k)

(l) (m) (n)



e a

vera

ge ca

ble t

ensio

n (N

)

14

16

12

10

8




(a)

e a

vera

ge ca

ble t

ensio

n (N

) 14

16

12

10

8

6




(b)


14

13

12

11

e a

vera

ge ca

ble t

ensio

n (N

)

10

9




(a)

Figure 16 Continued





e a

vera

ge ca

ble t

ensio

n (N

)

14

15

13

12

11




(b)






Data Availability




Acknowledgments


References























































l(φ)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (8)



0

1

2

3

4∆l

A+∆l

C (m

m)

5

6times10-4

-π18 π18-π36 π360


(a)

0

1

2

3

4

∆lB+

∆lD

(mm

)

5times10-3

-π18 π18-π36 π360


(b)


x1 x2xi

xnyn-1

y2y1 yixm1

xm2

xj xmn

zmnymn

yj

ym2

ym1

x0

yn

xn-1





65

60

55Z

(mm

)

50

45

40

4020

-20-40

-30 -20 -1010 20 30

Y (mm)0

0X (mm)

(a)

Z (m

m)

6070

5040302010

0-20 -10

10 20 -20 -10 0

Y (mm)10 20

0X (mm)

(b)

Z (m

m)

60

80

100

120

40

20-100 -50

0X (mm)50 100

10050-50

-1000

Y (mm)

(c)

Z (m

m)

6080

100120140

4020

0-50 -25

-60 -30 0 30 60

Y (mm)0

X (mm)25 50

(d)


88

86

84Cable A Cable C

82

8

e l

engt

h of

the c

able

Aan

d ca

ble C

(mm

)

78

76

74

72-π18 π18-π36 π360



Original function

Fitting function

(a)

0025

002Cable A Cable C

0015

Erro

rs e l

001

0005

0-π18 π18-π36 π360


(b)






lE1

2r2

minus 2r2cα + h


1113969

lF1



1113969

lG1

2r2

minus 2r2cα + h

2+ 2rh middot sα

1113969

lH1



1113969

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)



lE2 ipE2 minus

ipE1

11138681113868111386811138681113868

11138681113868111386811138681113868

lF2 ipF2 minus

ipF1

11138681113868111386811138681113868

11138681113868111386811138681113868

lG2 ipG2 minus

ipG1

11138681113868111386811138681113868

11138681113868111386811138681113868

lH2 ipH2 minus

ipH1

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)





l(θ φ)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (11)










002

0015

001

0005

e e

rror

the c

able

E (e

l)

0π18 π36 0 -π36 -π18 -π18 -π36

π36 π180


(a)

025

020

015

01

005

0

e e

rror

the c

able

F (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(b)

003500300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(c)

00300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π360

-π36 -π18 -π18 -π36π36 π18

0



(d)


95

Cable GCable E

Cable GCable E

e len

gth

of th

e cab

le E

and

cabl

e G (m

m)

85

75

65

7

8

9

-π18

π18 π18 π36-π36

0-π18

-π36 0 π36


Original function

Fitting function







Motor 3Motor 4

e distal segment

e proximal segment


Driving device

Guiding device

Guiding device




Tension pulley

Tension pulley








0deg

(a)

+30deg

(b)

+60deg

(c) (d)

(e) (f) (g)

0deg

(h)

-30deg

(i)

-60deg

(j) (k)

(l) (m) (n)



e a

vera

ge ca

ble t

ensio

n (N

)

14

16

12

10

8




(a)

e a

vera

ge ca

ble t

ensio

n (N

) 14

16

12

10

8

6




(b)


14

13

12

11

e a

vera

ge ca

ble t

ensio

n (N

)

10

9




(a)

Figure 16 Continued





e a

vera

ge ca

ble t

ensio

n (N

)

14

15

13

12

11




(b)






Data Availability




Acknowledgments


References



















































65

60

55Z

(mm

)

50

45

40

4020

-20-40

-30 -20 -1010 20 30

Y (mm)0

0X (mm)

(a)

Z (m

m)

6070

5040302010

0-20 -10

10 20 -20 -10 0

Y (mm)10 20

0X (mm)

(b)

Z (m

m)

60

80

100

120

40

20-100 -50

0X (mm)50 100

10050-50

-1000

Y (mm)

(c)

Z (m

m)

6080

100120140

4020

0-50 -25

-60 -30 0 30 60

Y (mm)0

X (mm)25 50

(d)


88

86

84Cable A Cable C

82

8

e l

engt

h of

the c

able

Aan

d ca

ble C

(mm

)

78

76

74

72-π18 π18-π36 π360



Original function

Fitting function

(a)

0025

002Cable A Cable C

0015

Erro

rs e l

001

0005

0-π18 π18-π36 π360


(b)






lE1

2r2

minus 2r2cα + h


1113969

lF1



1113969

lG1

2r2

minus 2r2cα + h

2+ 2rh middot sα

1113969

lH1



1113969

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)



lE2 ipE2 minus

ipE1

11138681113868111386811138681113868

11138681113868111386811138681113868

lF2 ipF2 minus

ipF1

11138681113868111386811138681113868

11138681113868111386811138681113868

lG2 ipG2 minus

ipG1

11138681113868111386811138681113868

11138681113868111386811138681113868

lH2 ipH2 minus

ipH1

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)





l(θ φ)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (11)










002

0015

001

0005

e e

rror

the c

able

E (e

l)

0π18 π36 0 -π36 -π18 -π18 -π36

π36 π180


(a)

025

020

015

01

005

0

e e

rror

the c

able

F (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(b)

003500300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(c)

00300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π360

-π36 -π18 -π18 -π36π36 π18

0



(d)


95

Cable GCable E

Cable GCable E

e len

gth

of th

e cab

le E

and

cabl

e G (m

m)

85

75

65

7

8

9

-π18

π18 π18 π36-π36

0-π18

-π36 0 π36


Original function

Fitting function







Motor 3Motor 4

e distal segment

e proximal segment


Driving device

Guiding device

Guiding device




Tension pulley

Tension pulley








0deg

(a)

+30deg

(b)

+60deg

(c) (d)

(e) (f) (g)

0deg

(h)

-30deg

(i)

-60deg

(j) (k)

(l) (m) (n)



e a

vera

ge ca

ble t

ensio

n (N

)

14

16

12

10

8




(a)

e a

vera

ge ca

ble t

ensio

n (N

) 14

16

12

10

8

6




(b)


14

13

12

11

e a

vera

ge ca

ble t

ensio

n (N

)

10

9




(a)

Figure 16 Continued





e a

vera

ge ca

ble t

ensio

n (N

)

14

15

13

12

11




(b)






Data Availability




Acknowledgments


References




















































lE1

2r2

minus 2r2cα + h


1113969

lF1



1113969

lG1

2r2

minus 2r2cα + h

2+ 2rh middot sα

1113969

lH1



1113969

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(9)



lE2 ipE2 minus

ipE1

11138681113868111386811138681113868

11138681113868111386811138681113868

lF2 ipF2 minus

ipF1

11138681113868111386811138681113868

11138681113868111386811138681113868

lG2 ipG2 minus

ipG1

11138681113868111386811138681113868

11138681113868111386811138681113868

lH2 ipH2 minus

ipH1

11138681113868111386811138681113868

11138681113868111386811138681113868

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(10)





l(θ φ)

11138681113868111386811138681113868111386811138681113868

11138681113868111386811138681113868111386811138681113868 (11)










002

0015

001

0005

e e

rror

the c

able

E (e

l)

0π18 π36 0 -π36 -π18 -π18 -π36

π36 π180


(a)

025

020

015

01

005

0

e e

rror

the c

able

F (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(b)

003500300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(c)

00300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π360

-π36 -π18 -π18 -π36π36 π18

0



(d)


95

Cable GCable E

Cable GCable E

e len

gth

of th

e cab

le E

and

cabl

e G (m

m)

85

75

65

7

8

9

-π18

π18 π18 π36-π36

0-π18

-π36 0 π36


Original function

Fitting function







Motor 3Motor 4

e distal segment

e proximal segment


Driving device

Guiding device

Guiding device




Tension pulley

Tension pulley








0deg

(a)

+30deg

(b)

+60deg

(c) (d)

(e) (f) (g)

0deg

(h)

-30deg

(i)

-60deg

(j) (k)

(l) (m) (n)



e a

vera

ge ca

ble t

ensio

n (N

)

14

16

12

10

8




(a)

e a

vera

ge ca

ble t

ensio

n (N

) 14

16

12

10

8

6




(b)


14

13

12

11

e a

vera

ge ca

ble t

ensio

n (N

)

10

9




(a)

Figure 16 Continued





e a

vera

ge ca

ble t

ensio

n (N

)

14

15

13

12

11




(b)






Data Availability




Acknowledgments


References



















































002

0015

001

0005

e e

rror

the c

able

E (e

l)

0π18 π36 0 -π36 -π18 -π18 -π36

π36 π180


(a)

025

020

015

01

005

0

e e

rror

the c

able

F (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(b)

003500300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π36 0 -π36 -π18 -π18 -π36π36 π18

0


(c)

00300025

0020015

00050

001

e e

rror

the c

able

G (e

l)

π18 π360

-π36 -π18 -π18 -π36π36 π18

0



(d)


95

Cable GCable E

Cable GCable E

e len

gth

of th

e cab

le E

and

cabl

e G (m

m)

85

75

65

7

8

9

-π18

π18 π18 π36-π36

0-π18

-π36 0 π36


Original function

Fitting function







Motor 3Motor 4

e distal segment

e proximal segment


Driving device

Guiding device

Guiding device




Tension pulley

Tension pulley








0deg

(a)

+30deg

(b)

+60deg

(c) (d)

(e) (f) (g)

0deg

(h)

-30deg

(i)

-60deg

(j) (k)

(l) (m) (n)



e a

vera

ge ca

ble t

ensio

n (N

)

14

16

12

10

8




(a)

e a

vera

ge ca

ble t

ensio

n (N

) 14

16

12

10

8

6




(b)


14

13

12

11

e a

vera

ge ca

ble t

ensio

n (N

)

10

9




(a)

Figure 16 Continued





e a

vera

ge ca

ble t

ensio

n (N

)

14

15

13

12

11




(b)






Data Availability




Acknowledgments


References





















































Motor 3Motor 4

e distal segment

e proximal segment


Driving device

Guiding device

Guiding device




Tension pulley

Tension pulley








0deg

(a)

+30deg

(b)

+60deg

(c) (d)

(e) (f) (g)

0deg

(h)

-30deg

(i)

-60deg

(j) (k)

(l) (m) (n)



e a

vera

ge ca

ble t

ensio

n (N

)

14

16

12

10

8




(a)

e a

vera

ge ca

ble t

ensio

n (N

) 14

16

12

10

8

6




(b)


14

13

12

11

e a

vera

ge ca

ble t

ensio

n (N

)

10

9




(a)

Figure 16 Continued





e a

vera

ge ca

ble t

ensio

n (N

)

14

15

13

12

11




(b)






Data Availability




Acknowledgments


References






















































0deg

(a)

+30deg

(b)

+60deg

(c) (d)

(e) (f) (g)

0deg

(h)

-30deg

(i)

-60deg

(j) (k)

(l) (m) (n)



e a

vera

ge ca

ble t

ensio

n (N

)

14

16

12

10

8




(a)

e a

vera

ge ca

ble t

ensio

n (N

) 14

16

12

10

8

6




(b)


14

13

12

11

e a

vera

ge ca

ble t

ensio

n (N

)

10

9




(a)

Figure 16 Continued





e a

vera

ge ca

ble t

ensio

n (N

)

14

15

13

12

11




(b)






Data Availability




Acknowledgments


References

















































e a

vera

ge ca

ble t

ensio

n (N

)

14

16

12

10

8




(a)

e a

vera

ge ca

ble t

ensio

n (N

) 14

16

12

10

8

6




(b)


14

13

12

11

e a

vera

ge ca

ble t

ensio

n (N

)

10

9




(a)

Figure 16 Continued





e a

vera

ge ca

ble t

ensio

n (N

)

14

15

13

12

11




(b)






Data Availability




Acknowledgments


References




















































e a

vera

ge ca

ble t

ensio

n (N

)

14

15

13

12

11




(b)






Data Availability




Acknowledgments


References



















































Data Availability




Acknowledgments


References

















































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