slack and excessive loading avoidance in n-tendon...

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Slack and Excessive Loading Avoidancein n-tendon Continuum RobotsMOHSEN MORADI DALVAND1,2,SAEID NAHAVANDI1(Fellow, IEEE) and ROBERT D. HOWE3 (Fellow, IEEE).1Deakin University, Geelong, Australia, Institute for Intelligent Systems Research and Innovation (IISRI) and Harvard Paulson School of Engineering andApplied Sciences, Cambridge, MA 02138 USA (e-mail: [email protected])2Deakin University, Geelong, Australia, Institute for Intelligent Systems Research and Innovation (IISRI) (e-mail: [email protected])3Harvard Paulson School of Engineering and Applied Sciences, Cambridge, MA 02138 USA (e-mail: [email protected])

Corresponding author: Mohsen Moradi Dalvand (e-mail: [email protected]).

ABSTRACT In this work, two tendon tension loading distributions (fixed and moving) are designed toprevent undesired slack and excessive loading in tendons of continuum manipulators. For any given beamconfiguration, the algorithms utilize the proposed loading distributions to find a new beam configurationas well as base displacement to control the orientation or position of the n-tendon continuum robot. Thealgorithms account for the bending and axial compliance of the manipulator as well as tendon compliance.Numerical results are provided to demonstrate the orientation and position control algorithms under fixed ormoving loading distributions. A 6-tendon continuum robot system is employed to experimentally evaluatethe effectiveness and accuracy of the proposed control algorithms. Multiple experiments are carried out andresults are reported. The results verify the performance of the proposed control algorithms in avoiding slackand excessive loading in tendons and controlling the orientation and position of continuum structures.

INDEX TERMS Continuum Robot, Catheter, Slack Avoidance, Kinematics, Load Analysis, Tendon-driven

I. INTRODUCTIONContinuum robots are inspired by natural continuumstructures like elephant trunks [1], octopus arms [2],squid tentacles [3], and snakes [4], [5]. Their con-tinuous structure and inherent compliance enablethem to exhibit elastic deformation along their entirelength and navigate safely [6]–[11]. In the medi-cal field, catheters and catheter-like instruments arewell-known examples of continuum structures thathave gained attention in minimally invasive treat-ments [12], [13].

A direct approach to manipulating a continuumstructure is the use of remotely actuated tendons[14], [15]. Tendons can only support tension (nega-tive load) and in compression they go slack becauseof their low bending stiffness [14], [16]. Actuationof a slacked tendon will first recover the slackbefore producing tension in the tendon [17]. Thislatency results in actuator backlash that is one ofthe key causes of inefficiency and inaccuracy in thecontrollers of robotic catheters [18]. Slack directly

affects the stiffness of the articulating beam andthe load distribution among tendons [13]. Althoughslack may be mitigated by increasing tension loadsin all tendons, high tendon load in continuum robotsgenerates less compliance and more friction. There-fore, slack and excessive loading are both undesir-able characteristics especially in catheters in whichtendon size and materials are constrained by theenvironment [13], [19].

Methods were introduced to prevent slack basedon a redundancy of control actions in redundantrigid-link tendon-driven mechanisms [20]–[24]. Asimilar approach was employed to prevent slack inredundant tendon-driven continuum robots based onthe numerical optimization techniques in a closed-loop control architecture [17], [25], [26]. Thesestudies do not promise reliable control algorithmscapable of performing the real-time computationsinvolved in numerical optimization process, espe-cially for higher numbers of tendons. This maycause a drift in Jacobian estimation from the true

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Author et al.: Slack and Excessive Loading Avoidance in n-tendon Continuum Robots

estimate and take longer to converge, leading therobot to diverge from the desired path [25]. Theyalso require load cells for online measurement andfeedback of the tension loading in tendons [25].Pretensioning tendons was also utilized in multiplestudies [27]–[31]. Excessive pretension causes morefriction which leads to faster wear and tear andshorter lifetime [32], [33]. Not enough pretensioncauses slack and deficiency in control [27].

The main contribution of this study is algorithmsto control orientation or tip position of n-tendoncontinuum robots guaranteeing no slack or exces-sive loading in tendons. Two loading distributions(fixed and moving) are proposed to minimize thetension loadings to a certain level that causes tensionin all tendons (no slack). For any given beam con-figuration within workspace, the algorithms utilizethe proposed loading distributions to find a newbeam configuration as well as base displacement tocontrol the orientation or position of the continuumrobot.

In the following section, the tension loadingmodel in n-tendon continuum robots used through-out this paper is explained. Slack and excessiveloading problems are described and their effectsare demonstrated using numerical analysis in Sec-tion III. Fixed and moving loading distributions andhow they are utilized in orientation and positioncontrol algorithms are explained and numerical re-sults are provided in Sections IV and V. Section VIpresents details about experimental setup, procedureand results followed by discussion and concludingremarks in Section VIII.

II. LOADING MODELFigure 1 shows the schematic description of thearticulating beam of a single segment cable-drivencontinuum robot with n equidistant tendons (n = 6in this case) in an arbitrary beam configurationdefined with bending angle (θ), pending plane angle(φ), and beam length along its centerline (L). Aspreviously developed [34], the tension load modelfor i-th tendon in n-tendon continuum robots canbe expressed as a function of beam configurationparameters as

Fi(θ, φ, L) =EI(k − 1)

nL0Raθ cos(φ− αi)−

(L0 − L)EA

nL0i = 1, . . . , n

(1)

where i is tendon number (i = 1, . . . , n), E, Iand A are the Young’s modulus, area moment of

θ

ϕRc

L

Ra

Base Plane

Reference Plane

Bending PlaneL2

L1

Li

Ro

x

y

z

o

VirtualTendon

ϕRa

o

x

y

Virtual Tendon

αi

Cross Sec�on View

FIGURE 1: Schematic description of the articulat-ing beam of a tendon-driven catheter with n equidis-tant tendons (n = 6 in this description) at a beamconfiguration ([θ, φ, L])

ϕ1

2

3

c) Fixed Loading Distribution

d) Moving Loading Distribution

Fixed Reference

Moving Reference

F1 F3 F2

F1 F3

1

2

3

a) Slack

b) Excessive Loading

Zero-load AxisLocation

F1 F3 F2

F1F3 F2

Zero-load AxisLocation

F0

F0

Dzl

Dzl

ϕ

Virtual Tendon

FIGURE 2: Schematic presentation of differentloading distributions with (a) slack (b) and excessiveloading as well as (c) fixed and (d) moving loadingdistribution in a n-tendon continuum robot (n = 3in this description)

2 VOLUME 4, 2016




inertia, and cross section area of the beam, Ra andαi are radial and angular location of tendons, L0 isthe initial length of the continuum structure, and kis representative of number of degrees of freedom(DOF) of the continuum robot and is defined as

k =

n if n = 1, 2

3 if n ≥ 3(2)

The angular location of i-th tendon (αi) of nequidistant tendons can be represented by 2π/n(i −1). In Figure 1, a virtual tendon is defined thatis located at the back of the continuum structureon the bending plane at the distance Ra from thecenterline.

Figure 2 schematically presents different loadingdistributions in a n-tendon continuum robot (n = 3in this example). As demonstrated in this figure,applying actuator displacements obtained from theinverse kinematics solution for a given configura-tion results in unique set of tendons tensions in acontinuum robot that are linearly distributed amongthe tendons [34]. The conceptual axis on the crosssection of the continuum structure, perpendicularto the bending plane, with zero tension in tendonslying on it is called zero-load axis. The locationof this axis from the centerline of the continuumstructure is denoted by Dzl and can be derivedas [34]

Dzl =AR2

a

2Iθ(L0 − L) (3)

This axis is different from the neutral axiswith zero displacement in tendons lying on it.The location of the neutral axis from the center-line of the continuum structure can be derived asDzd = (L0−L)/θ.

In order to illustrate the behavior of the ten-don loading for different beam configurations anddemonstrate the effects of slack and excessive load-ings in tendons, numerical simulations are per-formed and the results are provided in the followingsection. Similar simulations are also utilized to eval-uate the performance of the developed control al-gorithms for slack and excessive loading avoidanceand the results are also described in the followingsections.

III. SLACK AND EXCESSIVE LOADINGFigure 3 presents results for the numerical simu-lations for the estimated tendon tension loads ina 6-tendon continuum robot at different sets ofbeam configurations. The mechanical specification

Tend

onte

nsio

nlo

ads

(N)

0 30 60 90 120 150 180 210 240 270 300 330 360-12

-10

-8

-6

-4

-2

0

2

4

Tendon 1 2 3 4 5 6

Slack - Peaks: +2.63 and -4.66 (N)

a©

Bending plane angle φ (deg)

Tend

onte

nsio

nlo

ads

(N)

0 30 60 90 120 150 180 210 240 270 300 330 360-12

-10

-8

-6

-4

-2

0

2

4Tendon 1 2 3 4 5 6

Excesasive Loading - Peaks: -2.45 and -9.77 (N)

b©


FIGURE 3: Numerical simulation results for thetendon tension loads in a 6-tendon continuum robotat the beam configurations (a) [θ, φ, LSS ] and (b)[θ, φ, LE ]. Parameters are listed in Tables 1 and 2.

TABLE 1MECHANICAL SPECIFICATION

OF THE CONTINUUM ROBOT

Mechanical ParameterValue

Number of tendons (n) 6

Beam Young’s modulus (E) 5.9 MPa

Radial tendon location (Ra) 3.5 mm

Beam radius (Ro) 6 mm

Beam initial length (L0) 160 mm

Tendon Young’s modulus (Et) 48.9 GPa

Tendon radius (Rt) 0.23 mm

Tendon initial length (Lt0) 435.54 mm

of the continuum structure used in the simulationare chosen identical to those from the experimentalsetup (Table 1). The beam configurations [θ, φ, LSS ]and [θ, φ, LE ] (Table 2) served as inputs to esti-mate tendon tension loads using Equation (1) forthe first and second simulations, respectively. Theseconfigurations are selected to respectively demon-strate slack (in some of the tendons) and excessiveloading in tendons as presented in Figures 2a and 2b.Table 3 includes parametric results of the numericalanalysis.

As shown in Figure 3a, in every configuration,there are tendons with positive tension values (peak

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TABLE 2BEAM CONFIGURATIONS USED IN

NUMERICAL ANALYSIS AND EXPERIMENTS

Beam configuration parameterValue

Bending angle (θ) 40 deg

Bending plane angle (φ) [0, 10, 20,..., 360] deg

Length - Some slack (LSS ) 159 mm

Length - Excessive loading (LE ) 154 mm

Length - All slack (LAS ) 175 mm

values of +2.63 and -4.66 N). This means that inorder to reach these configurations, some of thetendons must act like a solid rod and push (positiveloading as shown in Figure 2a). However, tendonscan only support tension (negative load) and incompression they tend to buckle (go slack) becauseof their low bending stiffness. Actuation of a slackedtendon results in inefficiency and inaccuracy in thecontrol of robotic catheters [13], [34].

It should be noted that slack may be produced intendons even when the beam structure is under com-pression. For example, there is 1 mm compressionacross all beam configurations in the first simula-tion. This is because of the zero-load axis locatedinside the beam structure; Dzl < Ra (Figure 2aand Table 3, #1). This generates different loadingdirections in tendons located on opposite sides ofthis axis [34] and results in the summation of thetension loads to be as low as -6.10 N (Table 3, #1).

The second simulation is an example of excessiveloading in tendons (Figures 2b and 3b). Although θand φ used in both simulations have similar values,the beam structure in the second simulation is under5 mm additional compression which causes moreloading in tendons (peak values of -2.45 and -9.77 Nas listed in Table 3, #2). In this simulation, the zero-load axis is located outside of the beam structure;Dzl > Ra (Figure 2b and Table 3, #2). This gen-erates similar loading directions in all tendons. Asshown in Figure 3b, there is no slack in any of thetendons across all configurations; however, tendonsare always under tension even at their minimumcontribution (Figure 2b). This results in the largevalue of -36.60 N for the summation of tensionloads. High tendon tension in continuum robotsgenerates less compliance and more friction [29],[33].

Therefore, slack and excessive loading are bothundesirable characteristics especially in catheters inwhich tendon size and materials are constrained bythe environment [17]. Given the above examples,the question is how can slack in continuum robots

be avoided without producing excessive loading intendons? The proposed solution is to determine,based on fixed and moving loading distributions,new beam configuration as well as base displace-ment to satisfy no-slack and excessive loading con-ditions for orientation and position controls.

IV. FIXED LOADING DISTRIBUTIONTo prevent slack in a continuum robot, target beamconfigurations requiring tendon tensions larger thanzero should be prevented. As shown in Figure 2c,this can be achieved by pushing the location ofthe zero-load axis at or behind the virtual tendon(Dzl ≥ Ra). The bigger |Dzl − Ra|, the moretension loading in tendons. To minimize the loadingdistribution and still have no slack in any of thetendons, the zero-load axis can be located right atthe location of the virtual tendon (Dzl = Ra). Inorder to also control the stiffness of the continuummanipulator, as shown in Figure 2c, the tensionloading F0 (instead of zero) is considered at thelocation of the virtual tendon (see Discussion). Inthis approach, the location of the zero-load axis isindependent of beam configuration and same min-imum F0 loading condition can be satisfied for allconfigurations. This results in a loading distributionwith fixed zero-load axis location that is referredto as the fixed loading distribution in this paper(Figure 2c).

The location of the bending plane at which thepeak tension loads in the i-th tendon happen may bedetermined by differentiating Fi from Equation (1)with respect to φ (dFi/dφ = 0) as

φif = αi +mπ

i = 1, . . . , n

m ∈ N(4)

As implied from this equation, the peak tensionloads in each tendon happen when it lies on thebending plane. Even and odd values of parameterm result in the angular location of the maximumand minimum magnitude of the tension loadings inthe i-th tendon. Substituting odd values of m inEquation 4 determines the bending plane at whichthe i-th tendon is on the virtual tendon. To satisfythe minimum tension loading F0 in the i-th tendonat the bending plane angle of φif , Equation (4) canbe substituted in Equation (1) as

Fi(θ, φif , L) = F0

i = 1, . . . , n

m = 1(5)

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TABLE 3RESULTS OF THE NUMERICAL SIMULATIONS

Simulation Given config. Target config. Tension peaks (N) Dzl (mm) Dzd (mm) ∆Zf(m)p mm∑n

i=1 Fi (N)

#1 - Some slack [θ, φ, LSS ] [θ, φ, LSS ] [−4.66,+2.63] +0.97 +1.43 0.00 -6.10

#2 - Excessive loading [θ, φ, LE ] [θ, φ, LE ] [−9.77,−2.45] +5.84 +8.59 0.00 -36.60

#3 - Fixed - Orientation [θ, φ, LSS ] [θ, φ, Lfo] [−7.30, 0.00] +3.50 +5.14 0.00 -21.90

#4 - Fixed - Position [θ, φ, LSS ] [θfp, φ, Lfp] [−7.43, 0.00] +3.50 +5.14 +2.89 -22.30

#5 - Fixed - Position [θ, φ, LE ] [θfp, φ, Lfp] [−7.17, 0.00] +3.50 +5.14 -2.68 -21.52

#6 - Moving - Orientation [θ, φ, LSS ] [θ, φ, Lmo] [−7.30, 0.00] [+3.03 to +3.50] [+4.45 to +5.14] 0.00 [-21.90 to -18.97]

#7 - Moving - Position [θ, φ, LSS ] [θmp, φ, Lmp] [−7.43, 0.00] [+3.03 to +3.50] [+4.45 to +5.14] [+2.35 to +2.89] [-22.30 to -19.20]

#8 - Moving - Position [θ, φ, LE ] [θmp, φ, Lmp] [−7.17, 0.00] [+3.03 to +3.50] [+4.45 to +5.14] [-3.21 to -2.68] [-21.52 to -18.57]

By solving this equation with respect to L, thenew length of the continuum structure (Lf ) guaran-teeing minimum tension loading F0 is derived as afunction of θ as

Lf (θ) = C1 − C2θ k ≥ 2 (6)

where constant parameters C1 and C2 are

C1 =L0(nF0

AE+ 1)

C2 =I(k − 1)

ARa

The constraint k ≥ 2 in this equation is be-cause slack is meaningless in case of single tendonrobot. It is worth noting that Lf is independent ofparameters i, L, and φ and is a function of onlyθ. In the following sections, control algorithms aredeveloped based on the fixed loading distributionin order to enable orientation and position controlof continuum robot while satisfying minimum F0

tension constraint (no slack).

A. ORIENTATION CONTROLFigure 4 illustrates the beam structure at differentconfigurations and under various control scenarios.The beam structure illustrated with solid line repre-sents a target beam configuration in which tendonsare under excessive loading or slack. In some appli-cations, the orientation of the tip of the continuumrobot or catheter is more important than its position,e.g. pointing an ultrasound imaging catheter [35].In order to manipulate a continuum structure tothe given bending angle θ under the fixed loadingdistribution, Equation (6) may be used to find thelength of the continuum structure Lfo satisfyingminimum F0 loading in tendons as

Lfo = Lf (θ) k ≥ 2 (7)

The dashed line in Figure 4 illustrates the beamstructure under orientation control with similar θ

θ θ θfp

L(slack)

Target Configuration

Orientation Control

Position Control

pfo or pmo

pfp or pmp

Lfo or Lmo

Lfp or Lmp

ΔZfp orΔZmp

L(excessive)

p(slack orexcessive)

o Dxy

Axial Base Actuation DOF

θmpor

FIGURE 4: Schematic description the beam struc-ture at different configurations and under variouscontrol scenarios (solid lines: target configurationcausing slack or excessive loading in tendons,dashed line: orientation control under fixed or mov-ing loading distribution, dash-dot lines: positioncontrol under fixed or moving loading distributionfor slack and excessive loading cases)

as of the solid line. Based on the Equation (7), fororientation control under fixed loading distribution,only parameters θ and φ are required and the al-gorithm determines the maximum possible lengthof the continuum structure (Lfo) at which thereis minimum F0 loading in tendons (no slack orexcessive loading). Substituting Equation (7) intothe Equation (1), the tendon tensions for orientationcontrol of a n-tendon continuum robot under fixedloading distribution is derived as

Fifo(θ, φ) = F0 −EI(k − 1)

nL0Raθ×

(cos(φ− αi) + 1

) i = 1, . . . , n

k ≥ 2

(8)

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A characteristic feature of fixed loading distri-bution is the minimum F0 loading in the virtualtendon (Figure 2c). This can be demonstrated usingEquation (8) by calculating the tension load in the 4-th tendon (F4fo

) of a 6-tendon robot at φ = 0 (φ −α4 = −π) which makes it lay on the virtual tendon(F4fo

= F0). Using Equation (8), the summationof the tension loads of all tendons for orientationcontrol of a continuum robot under fixed loadingdistribution for any given configuration [θ, φ, L] canbe derived as a function of θ as

n∑i=1

Fifo = nF0 −EI(k − 1)

L0Raθ k ≥ 2 (9)

Based on the beam configuration and tendon ten-sion, tendon displacements ∆Li, tendon stretchesδi, and actuator displacements (∆Lai ) can be de-termined using

∆Li = (L0 − L) +Raθ cos (αi − φ) i = 1, ..., n(10)

δi =FiLt0EtAt

i = 1, . . . , n (11)

∆Lai = ∆Li + δi i = 1, . . . , n (12)

To control orientation of a continuum robot underfixed loading distribution, target θ and φ are usedin Equation (7) to determine the desired length ofthe continuum structure (Lfo). The new beam con-figuration [θ, φ, Lfo] can then be used to determine∆Lai to reach orientation θ and satisfy no-slack andminimum F0 loading conditions (Algorithm 1).

To demonstrate the effectiveness of the orienta-tion control algorithm under fixed loading distribu-tion in satisfying F0 loading constraint (no slack),the numerical simulation process for the beam con-figurations [θ, φ, LSS ] and [θ, φ, LE ] are repeatedbased on this control algorithm. The minimum load-ing is set to zero (F0 = 0). The results are presentedin Figure 5a. The orientation control algorithm isindependent of the given L as it determines therequired beam length (Lfo) to satisfy the desiredconditions. Therefore, the results for both configu-rations [θ, φ, LSS ] and [θ, φ, LE ] are similar.

As shown in Figure 5a compared to Figure 3,the peak values are changed to 0 and -7.3 N (Ta-ble 3, #3) which means that tendons are never underpositive tension for any configuration. Tendon ten-sion summation (Equation (9)) is a function of onlyθ that results in a constant value of -21.90 N in this

Tend

onte

nsio

nlo

ads

(N)

0 30 60 90 120 150 180 210 240 270 300 330 360-10

-8

-6

-4

-2

0

Tendon 1 2 3 4 5 6

Fixed Distribution - Orientation Control - Peaks: 0 and -7.30 (N)

a©


Tend

onte

nsio

nlo

ads

(N)

0 30 60 90 120 150 180 210 240 270 300 330 360-10

-8

-6

-4

-2

0

Tendon 1 2 3 4 5 6

Moving Distribution - Orientation Control - Peaks: 0 and -7.30 (N)

b©


FIGURE 5: Simulation examples for the estimatedtendon tension loads in a 6-tendon continuum robotat the beam configuration [θ, φ, LSS ] for orientationcontrol under (a) fixed and (b) moving loading dis-tributions. Parameters are defined in Table 2.

simulation. As expected, this value is decreased sig-nificantly compared to the summation of tensions inthe simulation for the excessive loading in tendons(Figure 3b and Table 3, #3). On the other hand, thereis no excessive loading in tendons as their tensionsreach the minimum absolute values of zero (F0 = 0in this case) when they meet the virtual tendon(Figure 2c). In all other configurations, the absolutevalues of the tensions in all tendons are alwaysbigger than |F0|, e.g. at φ = 90◦ in Figure 5a. Inother words, in case of fixed loading distribution,the magnitude of the tensions in all tendons arealways greater than F0 except that of the tendonlaid on the virtual tendon with tension equal to F0.This is another characteristic feature of the fixedloading distribution that exists for both orientationand position control algorithms.

B. POSITION CONTROL

To control the tip position of a continuum robot andsatisfy no-slack and minimum F0 loading condi-tions, the fixed loading distribution can be utilized incombination with an axial actuated DOF of the baseof the continuum structure, i.e. insertion/retraction

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Algorithm 1: Orientation control algorithm under fixed (or moving) loading distributioninput : Bending angle (θ), bending plane angle (φ), minimum loading (F0), and geometry and

mechanical properties of beam and tendons ([p])output: Tendon actuator displacements [∆Lai ]

-21 Algorithm ControlOrientation(θ, φ, F0, [p])-22-23 Determine Lfo using Equation (7) (or Lmo using Equation (21));-24 Determine tendon displacements [∆Li] using Equation (10) ;-25 Determine tendon tensions [Fifo ] using Equation (8) (or [Fimo

] using Equation (22));-26 Determine tendon stretches [δi] based on the tendon tensions [Fifo ] (or [Fimo

]) using Equation (11);-27 Determine tendon actuator displacements [∆Lai ] using Equation (12);-28 return [∆Lai ];

of the catheter shaft. The dash-dot line in Figure 4illustrates the beam structure under position con-trol. As shown in this figure, a continuum robotcommanded to [x, y, z] coordinates (point p inthe figure) can instead be manipulated to a newcoordinates [x, y, zfp] (point pfp in the figure) whilethe axial actuated DOF of the base compensates forz and zfp differences (∆Zfp). To meet minimumF0 loading condition as well as x and y coordi-nates condition, two constraints are defined. Thefirst constraint is to guarantee minimum F0 loadingin tendons (no slack) which is defined based on theEquation (6) as

Lfp = Rfpθfp = C1 − C2θfp k ≥ 2 (13)

where Lfp in the new length of the continuumstructure that can be represented by multiplying theradius of the beam structure (Rfp) by the new beambending angle (θfp). Lfp and θfp are the unknownbeam parameters satisfying the desired conditions.As shown in Figure 4, the tip position of the dash-dot line (point pfp) have similar x and y coordinatesas of those of the solid lines (point p). Therefore,both tip points p and pfp (Figure 4) projected uponxy plane should be equidistant from the origin. Thesecond constraint is defined as

Rfp =1− cos(θ)

1− cos(θfp)× L

θ(14)

To find the two unknown variables Rfp and θfp,Equation (14) is substituted into Equation (13):

(C2 +

1− cos(θ)

1− cos(θfp)× L

θ

)θfp = C1 (15)

By finding θfp from this equation and substitutingit back into Equation (14) and then in Equation (13),

both configuration variables Lfp and θfp are deter-mined. The required base displacement to compen-sate for the changes in z coordinates of points p andpfp (Figure 4) can be derived as

∆Zfp =L

θ

(sin(θ)− 1− cos(θ)

1− cos(θfp)× sin(θfp)

)(16)

As described in Algorithm 2, to manipulate acontinuum robot to a given tip position [x, y, z](corresponding to the configuration [θ, φ, L]) andalso satisfy no-slack condition under fixed loadingdistribution, the robot can be commanded to thebeam configuration [θfp, φ, Lfp] while the displace-ment ∆Zfp is simultaneously commanded to the ax-ial base DOF. Numerical simulations are performedfor the configurations [θ, φ, LSS ] and [θ, φ, LE ] andF0 = 0 based on the position control algorithmunder fixed loading distribution. The results havesimilar patterns to those presented in Figure 5abut different peak values (Table 3, #4 and #5).As implied from numerical results, position controlalgorithm satisfies no-slack condition. However, itrequires extra DOF of the base to compensate for∆Zfp.

V. MOVING LOADING DISTRIBUTIONTo prevent slack in a continuum robot, zero-loadaxis can be pushed at or behind the location of thevirtual tendon at the back of the continuum structure(Dzl ≥ Ra). In case of fixed loading distribution,the minimum loading in tendons guaranteeing noslack in tendons happens when the zero-load axisis located right at the location of the virtual ten-don (Dzl = Ra). The magnitude of this loadingdistribution can be further decreased whenever thebending plane does not coincide with a tendon loca-tion (Figure 2). To minimize the loading distribution

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Algorithm 2: Position control algorithm under fixed (or moving) loading distributioninput : Tip position ([x, y, z]), minimum loading (F0), and geometry and mechanical properties of

beam and tendons ([p])output: Tendon actuator displacements [∆Lai ], base actuator displacement ∆Zfp (or ∆Zmp)

-21 Algorithm ControlPosition([x, y, z], F0, [p])-22-23 Determine beam configuration parameters [θ, φ, L] of the tip position [x, y, z] [36];-24 Determine θfp using Equation (15) (or θmp using Equation (28));-25 Determine Lfp using Equation (13) (or Lmp using Equation (26));-26 Determine tendon displacements [∆Li] using Equation (10) ;-27 Determine tendon tensions [Fi] using Equation (1);-28 Determine tendon stretches [δi] Equation (11);-29 Determine tendon actuator displacements [∆Lai ] using Equation (12);

-210 Determine base actuator displacement ∆Zfp using Equation (16) (or ∆Zmp);-211 return [∆Lai ], ∆Zfp (or ∆Zmp);

magnitude to the lowest possible value and stillsatisfy the no-slack condition, the zero-load axis canbe located at the location of the farthest tendon at theback of the continuum structure.

To avoid slack and unnecessary loading in ten-dons under moving loading distribution and alsobe able to potentially adjust stiffness in continuumstructure, the location of the zero-load axis can bemoved behind the farthest tendon at the back of thecontinuum structure such that there is a minimumtension loading of F0 in the farthest tendon at theback. An illustration of this loading distribution ispresented in Figure 2d. As φ changes, the location ofthe farthest tendon and consequently the zero-loadaxis location and the loading distribution magnitudechanges. This results in a loading distribution that isreferred to as the moving loading distribution in thispaper. Equation (1) can be used to equate tensionin the farthest tendon at the back of the continuumstructure (tendon im) to F0 as

Fi(θ, φ, Lc) = F0 i = im (17)

where im is determined as

im = arg min{i:i=1,...,n}

(Ra cos

(φ− αi

))Using Equation (17), the new length of contin-

uum structure satisfying minimum F0 loading con-dition is derived as

Lm(θ) = C1 + C2θ cos(φ− αim

)(18)

The differences between Lm and Lf (Equation 6)is

Lm − Lf

{= 0, if αim − φ = π

> 0, otherwise(19)

As implied, the length of the beam structure undermoving loading distribution is always bigger thanits length under fixed loading distribution, unless theim-th tendon is laid on the bending plane (αim−φ =π). This is because (αim − φ) is always larger thanπ/2 which means that Lf and Lm only equate whenthe later has its minimum value

Lf = min(Lm) (20)

This produces less compression in the continuumstructure and consequently less tension in tendons.Both orientation and tip position of a continuumrobot can be controlled under moving loading distri-bution that are explained in the following sections.

A. ORIENTATION CONTROLTo control the pointing angle of a continuum robotunder moving loading distribution, Equation (18)may be used to find the length of the continuumstructure (Lmo in Figure 4) satisfying the minimumF0 loading condition as

Lmo = Lm(θ) k ≥ 2 (21)

Similar to orientation control under fixed load-ing distribution, to control orientation angle undermoving loading distribution, parameters θ and φ areneeded for the algorithm to obtain the maximumlength of the continuum structure (Lmo) satisfying

8 VOLUME 4, 2016




Tens

ion

load

s(N

)

0 30 60 90 120 150 180 210 240 270 300 330 360-23-22-21-20-19-18

F-O M-OSummation of all Tendon Tensions


FIGURE 6: Summation of the tension loads in alltendons calculated in the simulation for orientationcontrol under both fixed and moving loading distri-butions with respect to the bending plane angle φ

F0 loading condition (Algorithm 1). SubstitutingEquation (21) into the Equation (1), the requiredtendon tensions for orientation control of a n-tendoncontinuum robot under moving loading distributionis

Fimo(θ, φ) = F0 −EI(k − 1)

nL0Raθ×(

cos(φ− αi)− cos(φ− αim))

i = 1, . . . , n

k ≥ 2

(22)

The behavior of the orientation control algorithmunder moving loading distribution is studied usingthe numerical simulation for similar beam config-urations and F0 as in previous simulations. Theresults are presented in Figure 5b. This orientationcontrol algorithm is also independent of the givenL and, therefore, for both given LSS and LE , theresults are similar. The peak values of the tendontensions in this simulation (Table 3, #6) are identicalto those of the orientation control under fixed load-ing distribution (Table 3, #3 and Figure 5b) whichdemonstrates the effectiveness of this algorithm inavoiding slack and excessive loading the tendons.

The summation of the tension loadings in alltendons for orientation control of a continuum robotunder moving loading distribution can be derivedfrom (Equation 22) as

n∑i=1

Fimo = nF0 +EI(k − 1)

L0Raθ×

cos(φ− αim) k ≥ 2(23)

Comparing this equation with Equation (9) resultsin

∣∣∣ n∑i=1

Fimo

∣∣∣− ∣∣∣ n∑i=1

Fifo

∣∣∣{= 0, if αim − φ = π

< 0, otherwise(24)

which means that manipulating a continuum robotunder moving loading distribution always producesless tendon tensions than those produced under fixedloading distribution unless the im-th tendon is laidon the bending plane (αim − φ = π). This can beexpressed as∣∣∣ n∑

i=1

Fifo

∣∣∣ = max(∣∣∣ n∑i=1

Fimo

∣∣∣) (25)

Figure 6 shows the simulation results for sum-mation of the tension loads for orientation controlunder fixed and moving average distributions withrespect to φ. As shown in this figure and alsoimplied from (Equation 23), unlike the summationof tensions in orientation control under fixed load-ing distribution (Equation (9)) that is a function ofonly θ, this summation under moving distributionis a function of both θ and φ. Therefore, it hasvariable values ranging from -18.97 N to -21.9 N(Table 3, #6). The results verify Equation (25). Themoving loading distribution also causes movementsin both locations of zero-load and neutral axes (Ta-ble 3, #6).

There is no excessive loading in tendons as theirtensions reach the minimum absolute values of zero(F0 = 0 in this case) when they meet the virtualtendon (Figure 2d). A characteristic feature of themoving loading distribution, as shown in Figure 5b,is that in every beam configurations one of thetendons has the minimum F0 tension loading. Thisis a key difference between the fixed and movingloading distributions as in fixed loading distributionthe tension in one of the tendons is F0 only whenthe tendon is laid on the bending plane. Similarto fixed loading distribution, this feature exists inboth orientation and position control under movingloading distribution.

B. POSITION CONTROLIt is also possible to control the tip position of acontinuum robot under moving loading distributionto avoid slack and excessive loading in tendons.However, similar to position control under fixedloading distribution, an axial displacement of thebase of the continuum structure is needed to reachthe target tip position. Two constraints are defined

VOLUME 4, 2016 9




to satisfy no slack and excessive loading as well asx and y coordinates conditions. The first constraintis to have minimum F0 loading on the im-th tendonwhich is, based on the Equation (18), defined as

Lmp = Rmpθmp = C1 + C2θmp cos(φ− αim

)(26)

whereRmp, θmp, and Lmp are the new radius, bend-ing angle, and length of the continuum structuresatisfying the desired conditions represented by thedash-dot line in Figure 4. The second constraintguaranteeing similar x and y coordinates for bothtip points p and pmp is defined as

Rmp =1− cos(θ)

1− cos(θmp)× L

θ(27)

Substituting Equation (27) into Equation (26)leads to a single-variable equation in θmp

C1

θmp+C2 cos(φ−αim) =

1− cos(θ)

1− cos(θmp)×Lθ

(28)

Opposed to θfp determined from Equation (15)which is a function of only θ, θmp derived fromEquation (28) is a function of θ, φ, and αim . Thisintroduces dependency to φ and αim in parametersLmp and consequently ∆Zmp and Fimp

. By findingθmp from Equation (28) and substituting it back intoEquation (27) and then in Equation (26), configura-tion variables Lmp and θmp are determined. Similarto position control under moving loading distribu-tion, the required base displacement to compensatefor ∆Zmp can be derived using Equation (16) basedon θmp.

As described in Algorithm 2, to manipulate acontinuum robot to the beam configuration [θ, φ,L] corresponding to the given tip position [x, y,z] under moving loading distribution, the robot andbase DOF should be commanded to the beam con-figuration [θmp, φ, Lmp] and ∆Zmp, respectively.This moves the tip of the continuum structure to [x,y, z] coordinates and satisfy no-slack and minimumF0 loading conditions.

The numerical simulations for similar beam con-figurations and F0 = 0 for position control of a6-tendon continuum robot under moving loadingdistribution results in similar tendon tension patternas shown in Figure 5b, but with different peak values(Table 3, #7 and #8). As expected, these peak valuesare still identical with those of the position controlunder fixed loading distribution. The results verifythe expected dependency of ∆Zmp and Fimp

to φand αim parameters as well as their relations with∆Zfp and Fifp .

Actua�on and Force Measurement Unit

Image Processing Background

Camera 1

Camera 2

Actuator

Flexure Mechanism

Catheter

FIGURE 7: The developed modular continuumrobotic system capable of manipulating robots withup to 6 tendons integrated with a real-time vision-based 3D reconstruction system [34]

VI. EXPERIMENTSA. EXPERIMENTAL SETUP

The 6-tendon robotic catheter system, shown inFigure 7, is employed to experimentally evaluate thecontrol algorithms and validate simulation results.The system is integrated with a real-time vision-based shape sensing system that enables the 3Dreconstruction of the catheter tube at the rate of 200Hz with accuracy of ±0.6 mm and ±0.5 deg for thelinear and angular parameters, respectively [37]. DCgeared motors (Model EC-max 22, Maxon MotorsInc., 6072 Sachseln, Switzerland) are utilized for theactuation of the tendons. The base of this roboticsystem is fixed and not actuated. Load-cells (ModelFC22, Phidgets Inc., Calgary, Canada) are incor-porated into the actuation modules using a flexuremechanism to measure the tension loads in tendons.

Mechanical parameters of the continuum struc-ture and tendons are listed in Table 1 which are sim-ilar to those used in the numerical simulations. Al-though these parameters are readily available for theprototype in the experimental setup, they need to beeither measured or obtained from manufacturer foroff-the-shelf continuum robots and catheters. Ten-dons used in this setup are Spectra microfilamentbraided lines (Spectra, PowerPro Inc., Irvine, CA,United States). To decrease the friction between thetendon and pulleys, low-friction pulleys with ballbearing are employed in the system. The 6-lumencatheter was molded of urethane rubber (PMC 780,Smooth-On Inc., Macungie, PA, United States). TheYoung’s modulus of the tubes and tendons are mea-sured using an Instron 5566 universal testing ma-chine. Tendons are passed through the lumens and

10 VOLUME 4, 2016




knotted to the actuator spools from one end and toa cap part from the other end at termination point atdistal end of the catheter. The system was operatedby a PC with Intel Core i7 processors running at3.00 GHz with 16 Gb of memory.

B. EXPERIMENT PROCEDURE AND RESULTSTo evaluate the effectiveness of the proposed meth-ods, the cases presented in Figure 3, namely beamconfigurations [θ, φ, LSS ] and [θ, φ, LE ] respec-tively causing slack and excessive loading usinginverse kinematics solution, as well as the beamconfigurations modified by the proposed algorithmsto avoid slack and excessive loadings are exper-imentally examined and the results are presentedhere. F0 is considered zero in experiments. Be-fore commanding the robot to a configuration, thecatheter was commanded to its home configuration(at rest in vertical position with zero tension intendons) as shown in Figure 7.

Figure 8 presents the experimental results for thebeam configurations of [θ, φ, LSS] and [θ, φ, LE]causing slack and excessive loading in tendons, re-spectively. As implied from Figure 8a (peak valuesof 0 and -5.34 N), at most of the configurations,tendons have zero tension loads which is an indi-cation of slacked tendon. Slacked tendons do notproduce any load, therefore, slacked tension loadsmeasured by load-cells at these configurations arezero. Results shown in Figure 8b presents excessiveloadings in tendons ranging from -2.56 to -10.96N. Experimental results are in good confirmationwith simulation results (Figure 3) in terms of trendand magnitude for both cases of slack and excessiveloading (Table 3). It is worth noting that the tensionloads obtained from all experiments have highermagnitudes than corresponding loads obtained fromnumerical simulations. This correlation that existsacross all experiments may be due to friction, non-linearity, and viscoelasticity in the materials [32],[33].

Experimental results for orientation control algo-rithm under fixed loading distribution for beam con-figurations [θ, φ, LSS ] are presented in Figure 9a.Figures 9b and 9c display the tendon tensions mea-sured in experiments for the position control al-gorithm under fixed loading distribution for beamconfigurations [θ, φ, LSS ] and [θ, φ, LE ], respec-tively. Figure 10 presents the experimental resultsfor orientation and position control algorithms un-der moving loading distribution based on the beamconfigurations [θ, φ, LSS ] and [θ, φ, LE ]. Both ori-entation and position control algorithms under fixed

Tend

onte

nsio

nlo

ads

(N)

0 30 60 90 120 150 180 210 240 270 300 330 360-12

-10

-8

-6

-4

-2

0

Tendon 1 2 3 4 5 6

Slack - Mean Bending Angle Error: 19.77 deg

a©


Tend

onte

nsio

nlo

ads

(N)

0 30 60 90 120 150 180 210 240 270 300 330 360-12

-10

-8

-6

-4

-2

0Tendon 1 2 3 4 5 6

Excessive Loading - Mean Bending Angle Error: 1.63 deg

b©


FIGURE 8: Experimental results for the tendontension loads in a 6-tendon continuum robot for thebeam configurations (a) [θ, φ, LSS ] causing meanbending error of 19.77 deg and [θ, φ, LE ] (b) caus-ing mean bending angle error of 1.63 deg

and moving loading distributions satisfy no-slackand minimum F0 (F0 = 0 in experiments) loadingconditions. They also match the simulation resultslisted in Figure 5 in terms of peak values and loadingdistributions.

Figures 11a and 11b present errors in the bendingangles measured in experiments with respect to thedesired bending angle (Table 2) for the orientationand position control algorithms under fixed andmoving loading distributions for the beam config-urations [θ, φ, LSS ] and [θ, φ, LE ], respectively. Asshown in Figure 11a, slacked tendons negatively af-fect the control accuracy of continuum robot; meanerror of 19.77 deg in θ. On the other hand, despiteits disadvantages, excessive loading produced moreaccurate manipulation; mean error of 1.63 deg in θin Figure 11b. Orientation and position control algo-rithms under fixed and moving loading distributionsdecreased the errors down to the range of 3.51 to4.96 deg. Given the open-loop nature of the testingand evaluation process, the results demonstrate goodaccuracy and effectiveness for the proposed controlalgorithms.

For both cases of [θ, φ, LSS ] and [θ, φ, LE ], theerrors for orientation control algorithm are similar

VOLUME 4, 2016 11




Tend

onte

nsio

nlo

ads

(N)

0 30 60 90 120 150 180 210 240 270 300 330 360-10

-8

-6

-4

-2

0

Tendon 1 2 3 4 5 6

Fixed Distribution - Orientation Control - Peaks: 0 and -8.45 (N)

a©


Tend

onte

nsio

nlo

ads

(N)

0 30 60 90 120 150 180 210 240 270 300 330 360-10

-8

-6

-4

-2

0

Tendon 1 2 3 4 5 6

Fixed Distribution - Position Control - Slack - Peaks: 0 and -8.81 (N)

b©


Tend

onte

nsio

nlo

ads

(N)

0 30 60 90 120 150 180 210 240 270 300 330 360-10

-8

-6

-4

-2

0

Tendon 1 2 3 4 5 6

Fixed Distribution - Position Control - Excess. - Peaks: 0 and -7.76 (N)

c©


FIGURE 9: Experimental results for the ten-don tension loads in a 6-tendon continuum robotfor (a) orientation control of beam configurations[θ, φ, LSS ], (b) position control of beam configura-tions [θ, φ, LSS ], and (c) position control of beamconfigurations [θ, φ, LE ] under fixed loading distri-bution

as these algorithms are independent of length. Thereare no significant differences between the errors fororientation and position control algorithms for anyof the cases of slack or excessive loadings. This isbecause of small differences between the desiredθ for both cases of slack and excessive loading(40 deg) and θfp (40.74 and 39.31 deg) and θmp(40.6 to 40.74 and 39.18 to 39.31 deg) determinedrespectively from Equations (15) and (28) for thecases of slack and excessive loading.

To better demonstrate the capability of the algo-rithms in controlling orientation and position of the

Tend

onte

nsio

nlo

ads

(N)

0 30 60 90 120 150 180 210 240 270 300 330 360-10

-8

-6

-4

-2

0

Tendon 1 2 3 4 5 6

Moving Distribution - Orientation Control - Peaks: 0 and -8.28 (N)

a©


Tend

onte

nsio

nlo

ads

(N)

0 30 60 90 120 150 180 210 240 270 300 330 360-10

-8

-6

-4

-2

0

Tendon 1 2 3 4 5 6

Moving Distribution - Position Control - Slack - Peaks: 0 and -8.78 (N)

b©


Tend

onte

nsio

nlo

ads

(N)

0 30 60 90 120 150 180 210 240 270 300 330 360-10

-8

-6

-4

-2

0

Tendon 1 2 3 4 5 6

Moving Distrib. - Position Control - Excess. - Peaks: 0 and -7.68 (N)

c©


FIGURE 10: Experimental results for the ten-don tension loads in a 6-tendon continuum robotfor (a) orientation control of beam configurations[θ, φ, LSS ], (b) position control of beam configura-tions [θ, φ, LSS ], and (c) position control of beamconfigurations [θ, φ, LE ] under moving loading dis-tribution

continuum structure, a new set of beam configura-tion is chosen ([θ, φ, LAS ] - Table 2). This configu-ration is chosen to produce considerable differencesbetween the given θ and L and those determinedfrom the orientation and position control algorithms.Given the initial length of the continuum structure,commanding the robot to this configuration causesslack in all tendons which results in no movementin the continuum structure. To reach this exact beamconfiguration, base displacement is required. How-ever, as explained in previous sections, for positioncontrol requiring base displacements, a new coor-

12 VOLUME 4, 2016




θer

ror (

deg)

19.77

4.52 3.51 4.13 3.98

Excess. F-O F-P M-O M-P0

5

10

15

20

25

a©

Control Algorithms

θer

ror (

deg)

1.634.52 4.63 4.13 4.96

Excess. F-O F-P M-O M-P0

5

10

15

20

25

b©

Control Algorithms

θan

dD

xy

erro

rs

40.00

4.351.13 4.18 1.32

58.64

10.123.17

10.813.21

Slack F-O F-P M-O M-P0

10

20

30

40

50

60 Dxy

c©

Control Algorithms

FIGURE 11: Mean and standard deviation of orien-tation errors for (a) beam configurations [θ, φ, LSS ],(b) beam configuration [θ, φ, LE ] as well as meanorientation and position errors for (c) configurations[θ, φ, LAS ] for the orientation and position controlalgorithms under fixed (F-O and F-P) and moving(M-O and M-P) loading distributions in a 6-tendoncontinuum robot.

dinates is derived to satisfy x and y coordinates ofthe original target point while the axial displacementof the base compensates for the target z coordinate.Since the experimental setup does not allow basemovements, the new point satisfying x and y co-ordinates is utilized. This is not expected to haveany effects on accuracy because the required basemovement is a pure straight displacement and doesnot involve any manipulations in the continuumstructure.

The errors in experimental results for bending an-gle and tip position for the case of traditional inversekinematics as well as four control algorithms arelisted in Figure 11c. Although each algorithm hasdifferent target orientation and tip position, the er-rors are calculated with respect to the original targetbeam configurations. For the orientation angles, the

given θ (40 deg) is used as the reference. The vectorsum of x and y coordinates of the tip position (Dxy

in Figure 4) corresponding to [θ, φ, LAS ] (58.64mm) is chosen as the reference for calculating errorsin tip positions for different control algorithms. Thefirst column of this figure shows errors as big as thetarget θ and Dxy calculated from the kinematics ofthe continuum structure for the target configuration.The reason for such big errors is that command-ing the actuators displacements calculated usingthe inverse kinematics for the target configuration([θ, φ, LAS ]) simply does not cause any movementin the robot at all and only makes all tendons goslack.

As implied from Figures 11c, orientation controlalgorithms under both fixed and moving loadingdistributions produced errors as low as 4.35 and 4.18deg in the bending angle, respectively. Since the ob-jective of these control algorithms is not tip positioncontrol, the tip position differences with respect tothe given configurations is relatively large; 10.12and 10.81 mm for fixed and moving loading dis-tributions, respectively. On the other hand, positioncontrol algorithms produce small position errors of3.17 and 3.21 mm under two fixed and moving load-ing distributions, respectively. The orientation errorsof position control algorithms under the two loadingdistributions (1.13 and 1.32 mm, respectively) listedin this figure are lower than those of the orientationcontrol algorithms. This is because the listed errorsare calculated based on the given θ which means theorientation errors for the case of the position controlalgorithm do not represent the actual orientationerror. Computing these errors with respect to thenew bending angles calculated by the algorithms(θfp and θmp from Equations (15) and (28), re-spectively produces similar orientation errors as ofthose in orientation control algorithms for the twoloading distributions. These results demonstrates thecapability of the proposed control algorithms in con-trolling orientation and tip position of a continuumrobot.

Figure 12 presents the experimental results forthe summation of the tension loads in all tendonsfor each bending plane angle φ for orientation con-trol (Figure 12a), position control of beam con-figuration [θ, φ, LSS ] (Figure 12b), and [θ, φ, LE ](Figure 12c) under both fixed and moving loadingdistributions. The experimental results match thesimulation results presented in Figure 6 in terms ofthe magnitude and overall pattern. As implied fromthese figures, summation of the tension loads fororientation and position control under fixed loading

VOLUME 4, 2016 13




Tens

ion

Sum

mat

ion

(N)

0 30 60 90 120 150 180 210 240 270 300 330 360-24-22-20-18-16-14-12

F-O M-O

Fixed Distribution - Orientation Control

Mo

re T

ensi

on

a©


Tens

ion

Sum

mat

ion

(N)

0 30 60 90 120 150 180 210 240 270 300 330 360-24-22-20-18-16-14-12

F-P M-P

Moving Distribution - Position Control - Slack

Mo

re T

ensi

on

b©


Tens

ion

Sum

mat

ion

(N)

0 30 60 90 120 150 180 210 240 270 300 330 360-24-22-20-18-16-14-12

F-P M-P

Moving Distribution - Position Control - Excess.M

ore

Ten

sio

n

c©


FIGURE 12: Summation of the tension loads in alltendons measured in experiments for (a) orientationcontrol, (b) position control of beam configuration[θ, φ, LSS ] and (c) [θ, φ, LE ] under both fixed andmoving loading distributions with respect to thebending plane angle φ

distribution are mostly around the maximum abso-lute values of those of the similar control algorithmsunder moving loading distribution. In other words,moving loading distribution always produces lesstension in tendons than the fixed loading distributionunless the im-th tendon is laid on the bending plane(αim − φ = π) in which case the tensions are equalfor both loading distributions.

VII. DISCUSSIONAs shown in Figure 12, in a 6-tendon robot, the fixedand moving loading distributions lead to similartendon tension summations at the bending planeangles of [0, 60, 120, . . . , 360] deg. Interestingly, thenumber of maximum peak values (Nmp) of thetension summation is directly related to the numberof tendons n in the continuum structure. In contin-uum robots with even number of tendons, the peakvalues are repeated at the bending plane angles of0 and 360 deg. However, there is no peak value atthe bending plane angle 0 deg in continuum robotswith odd number of tendons. This results in norepetition of the peak values in continuum robotswith odd number of tendons. It is worth noting that

this difference is dependent of the location of thetendons. This relationship is represented as

Nmp =

{n when n is oddn+ 1 when n is even

(29)

Experimental results validated the numerical simu-lation and verified the performance of the proposedcontrol algorithms under fixed and moving loadingdistributions in controlling orientation and positionof a 6-tendon continuum robot. The algorithms areproved to be capable of avoiding slack and excessiveloading in multi-tendon continuum robots even inopen-loop control architecture for a polymer-basedcontinuum manipulator. Based on the applicationcontrol requirements, e.g. orientation or positioncontrol, computational capabilities of the controller,e.g. analytical or numerical iterative solutions, hard-ware specification, e.g. DOFs and robot workspace,and manipulation features, e.g. motion smoothness,the proposed control algorithms may be chosen tomanipulate a continuum robot with any number oftendons without concern about slack deficiencies orexcessive loading damage.

Stiffness of a continuum structure may be con-trolled by manipulating the tension loads in tendons.Reduction of the stiffness of the continuum robotcan enable safe navigation inside delicate confinedspaces, e.g. to avoid wall puncture. In contrast,a high stiffness continuum structure may be ad-vantageous during tissue manipulation [14], [38],[39]. In order to avoid both slack and unnecessaryloading in tendons and also to adjust stiffness inthe continuum structure, the location of the zero-load axis can be moved behind virtual tendon sothat there is a minimum tension loading of F0 in thevirtual tendon. The minimum tension loading F0 intendons may be used to trade off between tensionloading magnitudes and beam manipulator stiffnessrequired for any applications. Changing F0 directlyaffects the stiffness of the manipulator. The largerF0 values, the stiffer manipulator. The minimumallowable value of F0 is zero and negative valuescause slack in some or all of the tendons.

VIII. CONCLUSIONIn this work, orientation and position control al-gorithms developed based on two fixed and mov-ing loading distributions in n-tendon continuumrobots. These loading distributions were designedto prevent slack and excessive loading in tendons.The algorithms were developed to account for thebending and axial compliance of the continuumstructure as well as tendon compliance. Numerical

14 VOLUME 4, 2016




simulations were performed to help describe thecontrol algorithms and their key characteristic fea-tures under fixed and moving loading distributions.The effectiveness of the proposed control algorithmsin avoiding slack and excessive loadings in tendonsand controlling the orientation and tip position ofthe continuum structure for different beam con-figurations was experimentally verified using a 6-tendon continuum robotic system in open-loop con-trol architecture. Future works may take friction andexternal loading and disturbance into considerationand address experimental characterization of off-the-shelf catheters and continuum robots.

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