A Guaranteed Approach For Kinematic Analysis of Continuum RobotBased Catheter

Sohail Iqbal, Samer Mohammed and Yacine Amirat

Abstract— Controlling a continuum robot in a constrainedenvironment constitutes a real challenge due to the nature ofthis robot and its bending properties. The navigation in un-structured environments, such as surgical operations, presentsa potential application of the use of such robots. In this paper,a new formulation of a continuum robot forward kinematicsis presented firstly. In a second step, an analytical inversekinematics of the continuum robot is developed. It computesthe continuum robot bellows length as a function of the robotend-tool position and orientation in the task space. Then aresolution of the inverse kinematics is shown based on theuse of interval analysis. This technique has the advantageof dealing with uncertainties that may occur in the systemmodeling, parameter identification and could be used as wellfor robot control. System redundancies when computing theinverse kinematics are solved by a branch and bound basedalgorithm that minimizes a given quadratic criterion.

Index Terms— continuum robots, inverse kinematics, intervalanalysis, optimization, rehabilitation robotics

I. INTRODUCTION

Unlike classical robotic structures, continuum robots havethe originality of not containing any rigid links or anyrotational joints [1]. These structures provide, like a hyper-redundant robot with a high number of Degrees Of Freedom,a continuous, precise, and smooth bending curve motions inthe operational space [2], [3], [4]. Elephant trunk, octopustentacles, and human tongues are good examples of contin-uum structures. In fact, elephant uses his trunk not only fordrinking and respiration but also for lifting heavy tree trunksand to catch flying insects. It can also be used as a gripper tohold object of different shapes and sizes. Similarly octopusarm is used for navigation in cluttered environments [5].Another interesting examples of continuum structure is thehuman tongue, which manipulates the food in a very smallworkspace for chewing and swallowing. Tongue tip can reachto almost every place in the mouth to detect any anomalies.Various types of continuum robots have been developed [6],[7], [8], [9], [10], [11]. In robotics, continuum robots havemany potential application areas such as robot navigationin unstructured and unsafe environments. Special care wasgiven lately for applications dealing with robotized surgicaloperations where top priority is granted to the control of theposition and the orientation of the robot end-tool. In fact, thecurrent therapeutic trend is the development of minimallyinvasive techniques [12]-[19], [20], [21], [22]. This trendis justified by the clinical needs which have the following

Sohail Iqbal, Samer Mohammed and Yacine Amirat are withLISSI, University of Paris XII, 122 rue Paul Armangot, 94400,Vitry Sur Seine, France. {iqbal, samer.mohammed, amirat}@univ-paris12.fr

objectives: increasing the comfort of the patient (less painand traumatisms, reducing the period of convalescence), in-creasing the safety of both the patient and the surgical team,enhancing the surgical gestures and diagnoses while reducingtheir costs (shortening the stay period in hospital). It is nowgenerally admitted that in the perspective of a medium-term project, intelligent surgical tools should be designedand developed instead of complex autonomous robots [6].These tools should be simple, small and inexpensive, whileproviding a clinical advantage. Regarding the computationtechniques used for the determination of the forward and in-verse kinematics, classical ones based on standard arithmeticfloating point have show rapidly their limitations in termsof taking into account the bounding errors, the modelingerrors etc. Using interval analysis instead has shown a betterperformance and a guaranteed solution. This technique isable to take into account the errors that could result fromimprecisions in modeling and parameters identification [23].Interval analysis constitutes a potential technique that couldbe used to control medical robots. In fact, medical robotsused in surgical operations require particular characteristics,such as high security and error-tolerance. Consequentlymotion control based on classical techniques do not satisfythose constraints, mainly, in terms of bounding errors, precisetrajectory tracking, high-performance control, etc. [24], [25]The use of interval analysis technique has shown a relativelysatisfied results when computing the inverse kinematics bymaintaining the controllable variables within certain knownbounds. In this context and in order to improve therapeuticgestures in endovascular surgical operations, we propose inthis study a new mathematical formulation of a continuumrobot forward and inverse kinematics. The approach consists,as detailed in section II, of equipping the distal extremity ofa catheter with an active compliant microrobot to have a con-trollable bending. The paper is organized as follows, firstlya description of the microrobot that has been developedis presented. In the next section mathematical formulationsof the three bellows continuum robot forward and inversekinematics were developed. They give the robot pose in termsof bellow’s lengths and orientations. In the fourth section, theinverse kinematics was computed based on interval analysistechnique. Conclusion and perspectives are presented in thefifth section.

II. DESCRIPTION OF THE MICROROBOT

In this section, a description of the continuum microrobotthat has been developed in previous studies [6], [26], [27] ispresented. The resulting constraints of this description were

978-1-4244-4775-6/09/$25.00 © 2009 IEEE. 1573

Proceedings of the 2009 IEEEInternational Conference on Robotics and Biomimetics

December 19 -23, 2009, Guilin, China

Fig. 1. 3D Continuum robot

taken into account when computing, in the next sections,the forward and the inverse kinematics. The environment inwhich the microrobot has to operate, because of blood flow,is strongly constrained. For the medical certification of themicrorobot, stability and rigidity are important factors to beconsidered. Our active catheter, as an assisting tool, will notchange the existing techniques for endovascular surgery [22]but it will improve their performances. Thus, the microrobotshould be integrated in the standard catheters, used for thistype of operations. For the design of the microrobot, we haveadopted a parallel structure, recognized for its high perfor-mances from the point of view of compactness, precision ofpositioning and control of compliance [6], [26]. The choiceof the actuators is of primary importance to ensure an optimalcontrol of the catheter during the operation. For this purpose,we exclude Shape Memory Alloy (SMA) actuators whichhave some disadvantages, such as cooling, leaking electriccurrent, non-linear thermomechanical behavior and responsedelay [15], [28], [29]. On the other hand, fluid actuatorsare more suitable for satisfying the constraints of safety inendovascular surgery. They provide a good compliance, arewell adapted for the design of biocompatible systems anddo not use any electrical power on any part to be insertedinto the patient body. The scale of the application excludesthe complex structures whose miniaturization is difficult orimpossible. Therefore we chose a simple parallel structurecomposed of two cylindrical platforms connected by threehydraulic bellows actuators placed at 120◦ of each other. Thevariation of hydraulic pressure inside the bellows modifiestheir lengths and this combination of different lengths ofthe bellows changes the bending of the microrobot (figure2). Moreover, in order to increase the bending angle severalmicrorobots are serially connected to make the navigationeasier and safer through the sinuous vascular system. Thismodular hybrid serial-parallel structure allows complex posesof the catheter.

III. FORWARD AND INVERSE KINEMATICS OF THE

CONTINUUM ROBOT

As shown in figure 3, one module of the continuumrobot is considered when computing forward and inversekinematics. The forward kinematics consists of representingthe robot end-tool in the task space X as a function ofthe bellows length Q, while inverse kinematics consists of

Fig. 2. 3-D bending principle of the microrobot

representing the below lengths Q as a function of the end-tool position/orientation X.X = [Xp;Xo] = [x,y,z,θ ,φ ]T and Q = [l1, l2, l3]T ; where

• Xp = [x,y,z]T represent the cartesian position coordi-nates

• Xo = [θ ,φ ]T represent the orientation coordinates• Q = [l1, l2, l3]T represent joint coordinates or bellow

lengths

In the context of orientation control, computing the in-verse kinematics based only on orientation parameters, i.e.computing Q as a function of only Xo will generate aninfinit number of solutions. Two cases are then proposedand detailed, the first one consists of fixing the height of thecontinuum robot (z parameter) and the second one consistsof adding a given criterion to optimize. This criterion is ofa main importance for the assistance of the surgeon duringendovascular treatment. It gives the robot the capability ofbending, according to the instantaneous radius of curvatureof the aorta while maintaining the desired orientation.In the following, mathematical formulations of forward andinverse kinematics will be derived under some assumptions:

• bellows of the continuum robot, while bending, areassumed to make an arc of circle.

• bellows are bending in the same orientation in order tokeep constant distance between them

The first hypothesis is justified by the fact that robot bellowsare homogenous due to fluid actuations. The geometricconstraints imposed by manufacture conditions justify thesecond hypothesis.

A. Forward kinematics

The standard method to compute the forward kinematicsof a robot, consists of computing the end-tool position andorientation in the task space as a function of the jointcoordinates. Hannan et al. [5] have proposed a continuumrobot model that incorporates robot bending using a con-stant curvatures into a modified Denavit-Hartenberg notation.Jones and Walker [7] have presented a new approach forsolving the forward kinematics of the continuum robots. Thisapproach is an extension of the work done by Hannan [5].Lets define S = [s,κ ]T where s represents the arc length ofthe continuum robot and κ represents its curvature. S could

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Fig. 3. Continuum robot and its model

be expressed as function of the bellows length parameters Q

as follows:

s =l1 + l2 + l3

3(1)

κ =

√l21 + l2

2 + l23− l1l2− l2l3− l2l3

d(l1 + l2 + l3)(2)

Fig. 4. Continuum robot in bending plane

Approximating the continuum robot by a rigid body leadto the following formula θ = s

R = sκ where R is equal to therobot curvature as shown in Fig. 3.Let r = |OP′| be in XY plane (figure 4), z = |PP′| is the heightof the microrobot and θ = OAP is the angle between X-axisand OP′. From figure 4, r can be expressed as a function ofS as follows:

r = |OP′|= R−Rcos(θ ) =1κ

(1− cos(θ )) =2κ

sin2(sκ2

)(3)

Consequently, the end-tool position in the task space Xp canbe expressed as a function of S as follows:

x =2κ

sin2(sκ2

)cos(sκ) (4)

y =2κ

sin2(sκ2

)sin(sκ) (5)

z =1κ

sin(sκ) (6)

Fig. 5. Top view of robot with 3 bellows 120◦ apart from each other

Taking into account equations 1 and 2, the robot end-toolposition and orientation vector X could be expressed as afunction of the bellow length vector Q as follows:

x =d(l1 + l2 + l3)

Dsin2(

D3d

)cos

(tan−1

(l3 + l2−2l1√

3(l2− l3)

))

y =d(l1 + l2 + l3)

Dsin2(

D3d

)sin

(tan−1

(l3 + l2−2l1√

3(l2− l3)

))

z =d(l1 + l2 + l3)

2Dsin(

2D3d

)

θ = sκ =2√

l21 + l2

2 + l23− l1l2− l2l3− l2l3

3d

φ = tan−1(

l2 + l3−2l1√3(l2− l3)

)

(7)

Where d represents the robot radius and D is equal to:

D =√

l21 + l2

2 + l23− l1l2− l2l3− l2l3 (8)

B. Inverse kinematics

The goal here is to compute the bellows length Q as afunction of the end-tool position vector parameters Xp andorientation vector parameters Xo. Note that in this lattercase, the system is redundant since the rotation around thez axis is mechanically infeasible. Consequently, the problemshould be solved either by adding an additional constraint orby optimizing a given criterion. This latter case is dealt insection IV using interval analysis.Regarding the first case and in order to avoid an infinite

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number of solutions, the robot height is supposed to be fixedat a given z value. The radius of the robot curvature is linkedto the bellows length by the following formula:

l1 = s(

1−κ cos{π

2−φ

})(9)

l2 = s

(1−κ cos

{−5π6−φ

})(10)

l3 = s

(1−κ cos

{−π6−φ

})(11)

In figure 4, geometric constraints give the following re-lation: s = z θ

sin(θ) and κ = 1R = sin(θ)

z . By substituting theparameteres in equations (9), (10), (11), a new formulationof the inverse kinematics is obtained.

l1 = zθ

sin(θ )−θd sin(φ) (12)

l2 = zθ

sin(θ )−θd cos(φ − π

6) (13)

l3 = zθ

sin(θ )−θd cos(φ +

π6

) (14)

Figure 6 shows the simulations of the continuum robotorientation for different radius of curvature.

−5 0 5 10 15 200

5

10

15

20

Bending Axis (mm)

Z−ax

is (m

m)

Bendings of the continuum robot

l3 = 23 mm

l3 = 22 mm

l3 = 21 mm

Fig. 6. Matlab simulations of the continuum robot while fixing l1 and l2to 20 mm and different values of l3

Q could be expressed as well as a function of Xp asfollows:

l1 = sin−1(ϒ)[

1ϒ−d sin

{tan−1

(yx

)}](15)

l2 = sin−1(ϒ)[

1ϒ

+ d sin{π

3+ tan−1

(yx

)}](16)

l3 = sin−1(ϒ)[

1ϒ−d cos

{π6

+ tan−1(y

x

)}](17)

Where ϒ =(

2z√

x2+y2

x2+y2+z2

)

IV. INTERVAL ANALYSIS & INVERSE KINEMATICS

A. Interval Analysis

Computers, due to their finite space capacity, cannot man-age the numbers with infinite binary string representations.The standard solution is to approximate these infinite binarystrings with their finite counterparts. However, error propa-gation could lead in some cases, where system precision iscrucial, to intolerable errors. Another solution to overcomethis problem consists in enclosing every real number in aninterval with lower and upper binary strings. Resulting inter-val bounding the real number is the representative of the realnumber. Manipulating real numbers, operating over them,and consequent properties are called real analysis; similarlyproperties over interval representations of real numbers arecalled interval analysis.

Interval analysis is not only useful in dealing with theproblem of computers finite precision but also takes care ofmany factors such as uncertainties in the system modelingarising due to inaccuracies in the physical measurements,model simplifications, hypotheses and approximations. Thesefactors are vital when dealing with the medical environment,particularly, when the application concerns the developmentof minimally invasive techniques. This latter requires a safeand smooth control of the end-tool robot to increase the com-fort of the patient and to reduce the treatment period whileenhancing the surgical gestures. Interval analysis has beenalso used to find solutions for constrained and unconstrainedoptimization problems [30].

B. Inverse kinematics using interval analysis

In this section, an implementation of an interval constraintsolver interval to find an optimized inverse kinematics ofthe continuum robot has been formulated. As we have aninfinit number of solutions, the problem is solved throughan optimization formulation. The cost function that has beenadopted is:

f (l1, l2, l3) =12

(l1 + l2 + l3

3− ld

)2

(18)

Constraints to a given orientation are derived directlyfrom the robots physical properties, and adapted for theirimplementation as a Constraint Satisfaction Problem (CSP).These constraints could be expressed in a suitable form forthe CSP formulation:

Cθ = l21 + l2

2 + l23 + l1l2 + l2l3 + l2l3−αθ 2 (19)

Cφ = 2l1− [1−√

3tan(φ)]l2− [1 +√

3tan(φ)]l3 (20)

C. Interval COnstraint Solver

ICOS (Interval Constraint Solver) is a software basedon interval Branch and Bound algorithm for globaloptimization [31]. It allows to find guaranteed globalsolutions for constrained optimization problems. It consistsof minimizing an objective function taking into account

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equality and inequality constraints.The solver computes enclosures for minimizers and safebounds of the global minimum value within an initial box X .The algorithm maintains two lists: a list L of boxes to beprocessed and a list S of proved feasible boxes. It providesa rigorous enclosure [L,U] of the global optimum withrespect to a given tolerance ε [31]. For further informationson the ICOS algorithm, the reader is invited to refer to [31].

Input: Error tolerance = ε , Domain=XOutput: S, [L,U]% S: set of proved feasible points% L : list of boxes to be processed% fX : denotes the set of possible values for f in X% L denotes the Lower bound and U denotes the Upper% bound of the feasible points and ω denotes the% interval width% Xp: proved feasible pointL ← X ; S ← ∅;(L,U) ← (-∞,+∞)foreach ω([L,U]) > ε do

X ′ ← X ′′ such that fX ′′ = min{ f

X ′′ :X ′′ ∈L };f X ′ ← min( f X ′ ,U)X ′ ← Reduce(X ′);f

X ′ ← LowerBound(X ′);( f X ′ , Xp, Proved) ← UpperBound(X ′);if Proved then

S ← S ∪ {Xp};endif X ′ �= ∅ then

(X ′1,X′2)← Split(X ′); L ← L ∪ (X ′1,X

′2)

endif L = ∅ then

(L,U) ← (-∞,+∞)else

(L,U) ← ( min{ fX ′′ :X ′′ ∈L }, min{ f X ′′

:X ′′ ∈ S} )end

endend

Algorithm 1: Branch and bound algorithm [31]

Numerical application

In figure 7, the mid interval of continuum robot bellowslength is shown while tracking a circular trajectory, wheremid(X) = (Lower(X)+U pper(X))/2.θ and ld were fixed respectively to π/6 and 23mm. Now byincrementing the panning angle φ , we compute the bellowslengths intervals using ICOS solver [31]. We can noticethat both constraints (19), (20) were respected. In figure 8,execution time of the optimization process is shown. Averagetime is equal to 0.072 sec. Simulations were done on Linuxplateform using a 2.8 Ghz Pentium processor. Notice thatthe highest time execution is observed for φ equal to π /2+nπ(n=1,2,. . . ). This is due to the fact that the constraint containsa tan(φ).

Figure 9 shows the convergence of the quadratic costfunction 18. We can notice that starting from the 3rd iteration

0 50 100 150 200 250 300 35021.5

22

22.5

23

23.5

24

24.5

φ (degrees)

Be

llow

1 le

ng

th (

mm

)

Continuum robot following a circulr trajectory

0 50 100 150 200 250 300 3500

0.02

0.04

φ (degrees)

Inte

rva

l w

idth

(m

m)

Fig. 7. Continuum robot following a circular trajectory with θ = π/6; anda bellow desired length ld equal to 23 mm

5 10 15 20 25 30 350

50

100

150

200

250

Samples of the φ circular trajectory

Tim

e (

mse

c)

Fig. 8. Execution time in msec of the optimization algorithm

onwards, cost function goes below 10−3. In this simulation,the tolerance error is fixed to 10−9 while the time executionis equal to 0.228 sec.

0 2 4 6 8 10 12 14

10−8

10−6

10−4

10−2

100

No. of Iterations

Cos

t fun

ctio

n m

m2 (l

ogar

ithm

ic s

cale

)

Cost function

Fig. 9. Convergence of the cost function for given desired length ld , φand θ

V. CONCLUSIONS AND FUTURE WORKS

A. Conclusions

In this paper, mathematical formulations of a continuumrobot forward kinematics and inverse kinematics have beendevelopped. These formulation ensure computation of con-tinuum robot bellow lengths from a given point in the taskspace and vice-versa. Computation of inverse kinematicshave been computed also using interval analysis, an efficient

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technique based on Branch and Bound algorithm. Simula-tions of circular trajectory show relatively low tracking error.Execution time of the used algorithm is relatively low andmotivating for real time implementation.

B. Future Works

Future works will concern the kinematic analysis takinginto account other relevant optimization criterion for the con-trol of the continuum robot. In order to improve therapeuticgestures in endovascular surgery, the dexterity of the robot isa crucial performance characteristic. It defines the ability ofthe robot to perform arbitrary end-effector motions around anoperating configuration. Different metrics for dexterity canbe considered : manipulability and reciprocal conditioning.We are currently working on the evaluation of these differentoptimization criteria.

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