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IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 4, AUGUST 2014 935

A Variable Curvature Continuum Kinematicsfor Kinematic Control of the Bionic

Handling AssistantTobias Mahl, Alexander Hildebrandt, and Oliver Sawodny

Abstract—We present a new variable curvature continuum kine-matics for multisection continuum robots with arbitrarily shapedbackbone curves assembled from sections with three degrees offreedom (DoFs) (spatial bending and extension, no torsion). Forthese robots, the forward kinematics and the differential forwardkinematics are derived. The proposed model approach is capableof reproducing both the constant and variable backbone curvaturein a closed form. It describes the deformation of a single sectionwith a finite number of serially connected circular arcs. This yieldsa section model with piecewise constant and, thus, a variable sec-tion curvature. Model accuracy and its suitability for kinematicreal-time control applications are demonstrated with simulationsand experimental data. To solve the redundant inverse kinemat-ics problem, a local resolution of redundancy at the velocity levelthrough the use of the robot’s Jacobian matrix is presented. TheJacobian is derived analytically, including a concept for regular-ization in singular configurations. Experimental data are recordedwith Festo’s Bionic Handling Assistant. This continuum robot ischosen for experimental validation, as it consists of a variable back-bone curvature because of its conically tapering shape.

Index Terms—Biologically inspired robots, continuum kinemat-ics, motion control, redundant robots.

I. INTRODUCTION

CONTINUUM manipulators are robots that perform theirtool motion by a continuous deformation of their ma-

nipulator structure. Inspired by their biological counterparts,many continuum robot designs mimic snakes [1], [2], elephanttrunks [3], [4], or octopus tentacles [5], [6]. To achieve the de-sired manipulator motion, one or more serially connected andindividually controlled sections are deformed using actuationprinciples like tendon drives [3], [4], precurved tubes [7], [8],and fluid-driven muscles or bellows [6], [9]. This results in ahuge diversity of different robot concepts, which are utilized ina wide range of applications [10]–[13]. See [14]–[17] for a moredetailed collection.

Manuscript received October 24, 2013; revised March 11, 2014; acceptedMarch 27, 2014. Date of publication April 24, 2014; date of current versionAugust 4, 2014. This paper was recommended for publication by Editor B. J.Nelson upon evaluation of the reviewers’ comments.

T. Mahl and O. Sawodny are with the Institute for System Dynamics,University of Stuttgart, Stuttgart D-70569, Germany (e-mail: [email protected]; [email protected]).

A. Hildebrand is with the Research Department, Festo AG & Co. KG,Esslingen D-73726, Germany (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TRO.2014.2314777

Fig. 1. Festo’s BHA (source: Festo AG & Co. KG).

The state-of-the-art kinematics of continuum robots modelthe backbone curve of a single section depending on the actuatorstate and robot geometry. To do so, they either describe the shapeof the deformed backbone with a curve having a variable or con-stant curvature. However, due to the simplifications in real-timeevaluation of constant curvature (cc) kinematics, almost all ex-isting continuum robot designs prefer sectional backbone curvesthat are approximately circular. This design strategy does notmatch the biological counterparts of continuum robots. Theseare commonly conic in direction to the manipulator end(s),which yield a backbone curve with a continuously changingcurvature. Besides Festo’s award-winning Bionic Handling As-sistant (BHA) [18], which is displayed in Fig. 1, only a fewother continuum robots with a conic manipulator structure aredocumented [19], [20]. For these robots, a kinematic model witha variable section curvature is highly desirable.

Rated as the most accurate, variable curvature (vc) contin-uum kinematics that are based on equilibrium conformationsof Cosserat-rod models [21]–[25] and other elastic beam theo-ries [8], [26] are proposed in the literature. Although there existfirst attempts to evaluate these often called “geometrically ex-act” models in real time [27], [28], it is not yet clear whetherthese models can be used for the feedback control design ina wide variety of applications [15]. A computationally feasibleapproach for real-time evaluation of vc continuum kinematics isthe modal approach that was initially proposed by Chirikjian andBurdick [29] in order to resolve redundancy for hyper-redundantrigid link robots. This approach fits the robot’s backbone curveto linear combinations of shape functions such as trigono-metric functions [29], [30], clothoides [14], or wavelets [31].However, the main drawback of this approach is that most

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936 IEEE TRANSACTIONS ON ROBOTICS, VOL. 30, NO. 4, AUGUST 2014

function sets are not suited to model all physically achievablebackbone shapes, even if an infinite number of functions areused [17]. Thus, further efforts have to be made to eliminate theresulting model error.

Giving up the model property of a variable section curva-ture, the extensively studied and frequently used approach of cccontinuum kinematics (see, e.g., [3], [9], and [32]) provides alow-cost alternative to approximate the robot’s backbone shape.Different derivations of this approach are collected and unifiedin [15]. However, applied to manipulators with a variable sectioncurvature, systematic model errors obviously reduce the accu-racy of the kinematic model. In order to minimize the impact ofthese systematic errors, the authors of [33], e.g., propose a gridsearch over the segment radii to fit the cc model to the variablebackbone curvature of the BHA.

A. Contribution

The vc continuum kinematics that is proposed in this pa-per tries to combine the key advantages of the aforementionedmodeling techniques. These are high model accuracy for bothconstant and variable section curvature robots, adjustable com-putational work, and universal applicability. To reach this am-bitious goal, we model the backbone deformation of a singlesection by using a piecewise cc approximation. This closes thegap between the previously mentioned kinematics which ei-ther use section models with variable, continuously changingsection curvature or constant section curvature. Extending pre-vious work [34], a detailed derivation and discussion of the newkinematic model, its velocity kinematics, and concepts to dealwith singular configurations are presented and experimentallyvalidated for robots which satisfy the following conditions.

Condition 1: The robot consists of a finite amount of inde-pendent, serially connected sections.

Condition 2: Each section has three degrees of freedom(DoFs). Two DoFs for spatial bending and one DoF for itsextension. Sectional torsion does not exist.

To demonstrate that this new kinematics is suited for model-ing as well as for kinematic real-time control of vc continuumrobots, it is applied to Festo’s BHA. For this nine-actuator con-tinuum manipulator having three individual sections in series, a5-DoF feed-forward pose control of the manipulator tool is im-plemented. To solve the redundant inverse kinematics problem,a first-order resolved rate algorithm according to [35] and [36] isused. It generates weighted least-norm solutions of actuator ve-locities in order to track a desired end-effector trajectory, whileavoiding actuator end stops.

Note that a comparable idea of kinematic continuumrobot modeling is proposed in a recent work of Rone andBen-Tzvi [37]. The referenced work presents a dynamic modelfor single-section continuum robots with spatial bending andtorsion capability (no extension), using a section kinematicswith a piecewise cc. As this paper addresses the variable sectioncurvature due to the robot dynamics and gravitational effects,we want to focus on the variable section curvature due to thebiomimic manipulator design.

Fig. 2. Components of the BHA.

B. Outline

After a short presentation of the BHA in Section II, thevc continuum kinematics for continuum robots satisfyingConditions 1–2 is described in Section III. Section IV is fo-cused on the kinematic control. As the control algorithm is inneed of the robot’s velocity kinematics, its analytical derivationis presented in Section V. Simulation and test bench results thatvalidate the proposed kinematics and demonstrate its applica-bility in real-time tasks are discussed in Section VI. Concludingremarks are given at the end.

II. BIONIC HANDLING ASSISTANT

The BHA, a manipulator whose complex structure is com-pletely made of laser-sintered polyamide [38], belongs to theclass of biomimic continuum robots with a variable section cur-vature. It has a trunk-like manipulator arm with three in seriesarranged sections. Additionally, it may consist of an attachedtool. A labeled picture is shown in Fig. 2. The manipulator arm,which is fastened to the robot mounting on one side, executesthe spatial movement of the tool holder on the other side. Eachof its conically shaped sections has three DoFs. By definition,the section index i is a consecutive number starting with 1 atthe robot mounting and increasing value in the direction towardthe tool. A second index k labels the actuator number of eachsection ranging from 1 to 3. As the tool actuation will not beconsidered in the following, the end-effector coordinate frame isdefined to coincide with the coordinate frame of the tool holder.Its origin defines the tool center point (TCP).

To have a closer look at a single section, two cross sectionsare displayed in Fig. 3. It can be seen that each section con-sists of a base, actuator, and a head region. The base and headregions are rigid plates that provide the section’s connectivity.The actuator region is located in between these two plates. Itis composed of three pneumatic actuators that are arranged inparallel and controlled separately. These bellows-like actuators,which expand when filled with the compressed air, taper with anangle of 4.5◦. This creates the section’s conic tapering shape aswell as the variable backbone curvature in the deformed state.To be able to reconstruct the actual section deformation, eachbellows-actuator possesses its own wire-cable potentiometer

MAHL et al.: VARIABLE CURVATURE CONTINUUM KINEMATICS FOR KINEMATIC CONTROL OF THE BIONIC HANDLING ASSISTANT 937

Fig. 3. Cross sections of a single BHA section (source: Festo AG & Co. KG).

Fig. 4. Composition of the forward kinematics of a continuum robot modeledunder the assumption of vc.

measuring its individual length. The wire-cables are routed bycable guides attached to each bellows of an actuator. The mea-sured wire-cable lengths are assembled to become the robot’sactuator state. This actuator state is the model input of the for-ward kinematics which is presented next.

III. FORWARD KINEMATICS

The proposed vc continuum kinematics for continuum robotssatisfying Conditions 1–2 is strongly related to the cc theorydescribed in [15]. Instead of modeling the backbone curve ofa continuum robot by one circular arc per section, it uses afinite number of serially attached circular arcs having individualarc parameters each. This yields a section kinematics with apiecewise cc providing the ability to be fitted to backbone curvesof vc continuum robots.

Based on this model concept, the forward kinematics can bedecomposed into two parts. As presented in Fig. 4, these are arobot-specific kinematics described by the mapping f sp and ageneral continuum kinematics described by the mapping f gen .The mapping f sp computes the configuration state k dependingon the robot geometry and actuator state q. With the knownconfiguration state, the end-effector’s pose h is computed bythe general continuum kinematics. This kinematics is termed ageneral one, as it models the robot’s backbone curve with noinformation other than the configuration state k.

Beginning with the general continuum kinematics, the de-scription of the robot-specific one will follow afterward.

A. General Variable Curvature Continuum Kinematics

The general vc continuum kinematics models the backbonecurve of any continuum robot satisfying Conditions 1–2 usingthe following assumptions.

Assumption 1: The general kinematics of a continuum robotis described by an open kinematic chain of n general sectionkinematics.

Fig. 5. Structure of the general vc continuum kinematics for multisectioncontinuum robots according to Assumptions 1–3.

Assumption 2: The general kinematics of the ith section (i ∈{1 . . . n}) consists of an open kinematic chain of three kinematicmodels approximating the section’s base, actuator, and headregion. The kinematics of the actuator region is described byan open kinematic chain of mi general unit kinematics. Theoptional kinematics of the base and head regions extend thebackbone curve of the actuator region at its respective endstangentially.

Assumption 3: The general kinematics of the ijth unit (i ∈{1 . . . n}, j ∈ {1 . . . mi}) equals a circular arc.

The resulting model structure of the proposed general con-tinuum kinematics according to Assumptions 1–3 is illustratedin Fig. 5. On the left-hand side of Fig. 5, the backbone curveof a multisection continuum robot according to Assumption 1is presented. The displayed coordinate frames Oib and Oih arethe body-fixed base and head coordinate frames of each section.Their respective indices ib and ih consist of the section numberi and an additional b or h for labeling the base or head. Fur-thermore, the world coordinate frame Ow is pictured. Note thatOi−1h equals Oib , as the sections are serially attached to eachother.

The graphic in the middle of Fig. 5 illustrates the backbonecurve of section i according to Assumption 2. The base andhead regions extend the backbone curve at the respective endsby �ib and �ih . Attached between the base and head, there aremi serially connected units. The backbone curve of each unit ijis enclosed between the two coordinate frames Oij−1 and Oij ,with j labeling the unit’s index.

A further detailed illustration on the right-hand side of Fig. 5shows the circular arc-shaped backbone curve of unit ij ac-cording to Assumption 3. Its individual arc parameters are therotational deflection angle φij , the arc length �ij , and the curva-ture κij , which is the reciprocal of the arc radius. Note that 1) thez-axes of all coordinate frames are adjusted tangentially to thebackbone curve pointing toward the section’s head and 2) thecoordinate frames Oij−1 and Oij are oriented such that φij isthe angle between their respective x-axes and the unit’s deflec-tion plane displayed in gray.

In order to model a continuum robot using the structure pre-sented previously, the kinematics of Assumptions 1–3 have to bederived. Beginning with the general unit kinematics, the generalsection kinematics and the general robot kinematics are derivedsuccessively.


1) General Unit Kinematics: In order to model the circulararc-shaped backbone curve of the ijth unit, the homogenoustransformation

ij−1Uij (kij ) =

[ij−1Rij

ij−1oij

01×3 1

](1)

is used. ij−1Rij and ij−1oij describe the rotation and trans-lation of the coordinate frame Oij in relation to the coordinateframe Oij−1 , respectively. In accordance with [15], with thecoordinate frame conventions displayed in Fig. 5, ij−1Rij andij−1oij are

ij−1Rij =

⎡⎢⎣

cφ(cκ� − 1) + 1 sφcφ(cκ� − 1) cφsκ�

sφcφ(cκ� − 1) c2φ(1 − cκ�) + cκ� sφsκ�

−cφsκ� −sφsκ� cκ�

⎤⎥⎦ ,

(2)

ij−1oij =

⎧⎪⎨⎪⎩[

cφ(1 − cκ�)kij2

sφ(1 − cκ�)kij2

sκ�

kij2

]T

, kij2 �= 0

[ 0 0 kij3 ]T , kij2 = 0

(3)

with the substitutions sφ , cφ , sκ� , and cκ� being

sφ = sin (kij1) , sκ� = sin (kij2kij3) , (4)

cφ = cos (kij1) , cκ� = cos (kij2kij3) . (5)

The unit’s configuration state kij is defined as

kij = [ kij1 kij2 kij3 ]T = [φij κij �ij ]T . (6)

Note that the unit’s backbone curve fits a straight line, if theunit’s curvature kij2 = κij equals zero. This backbone shape isdescribed in the second case of (3).

2) General Section Kinematics: According to Assumption2, the kinematic model of the ith section

ibSih(ki) = ibSi0

⎛⎝mi∏

j=1

ij−1Uij (kij )

⎞⎠ im i Sih (7)

is an open kinematic chain that is composed of the kinematicsof the rigid base ibSi0 and head im i Sih regions with

ibSi0 =

⎡⎢⎣

1 0 0 00 1 0 00 0 1 �ib

0 0 0 1

⎤⎥⎦ and im i Sih =

⎡⎢⎣

1 0 0 00 1 0 00 0 1 �ih

0 0 0 1

⎤⎥⎦

(8)as well as mi kinematics of the arc-shaped units with cc ij−1Uij

according to (1). The section’s configuration state ki assemblesto

ki = [kTi1 . . . kT

imi]T , ki ∈ R3mi . (9)

Thus, the general kinematics of the ith section is described by3mi states with the arc parameters φij , κij , and �ij in additionto two optional geometric parameters �ib and �ih , which are thebase and head region lengths, respectively.

3) General Robot Kinematics: According to Assumption 1,the general robot kinematics equals an open kinematic chain ofn serially connected sections, whose kinematics are describedin (7). Choosing the task space, without limiting the generality,to be the world coordinate frame Ow , the homogenous transfor-mation of the end-effector coordinate frame yields

wHnh = wH1b

n∏i=1

ibSih(ki) (10)

with wH1b describing the relative position and orientation of therobot’s first base coordinate frame O1b referred to the world co-ordinate frame Ow . The robot’s configuration state k assemblesto

k = [kT1 . . . kT

n ]T , k ∈ R3∑ n

i = 1 mi . (11)

On the basis of the end-effector’s homogenous transformationwHnh , the mapping of the general vc continuum kinematicsf gen is computed. Defining a singularity-free vectorial poserepresentation h ∈ R12 by rearranging the 12 nonconstant ele-ments of wHnh such that

f gen(k) : h = [h1 · · · h12 ]T (12)

= [ wHnh,(1,1)wHnh,(2,1) · · · wHnh,(3,4) ]T

(13)

yields the analytic expression of f gen . The indices in bracketslabel the position of a specific matrix element with the first indexindicating the row and the second index indicating the columnnumber, respectively. Note that the resulting robot kinematicshas an (internal) singularity [35], if at least one unit ij has astraight backbone curvature (κij = 0). As this singularity is ar-tificial due to the choice of the configuration state definition,the velocity kinematics can be regularized in these robot con-figurations. In summary, the general mapping f gen computesthe end-effector pose in the task space depending on 3

∑ni=1 mi

states that are defined in the configuration space. These are thethree arc parameters φij , κij , and �ij of each unit ij of sectioni (i ∈ {1 . . . n}, j ∈ {1 . . . mi}). How these states refer to theactuator state q of a specific continuum robot is described in thefollowing section.

B. Robot-Specific Variable Curvature Continuum Kinematics

The goal of the robot-specific vc continuum kinematics f sp(refer to Fig. 4) is to map the actuator state q of the BHA to theconfiguration state k of the general continuum kinematics ac-cording to Assumptions 1–3. Due to the section’s independence(Condition 1), the ith section’s configuration state ki is solelydependent on the ith section’s actuator state qi . Thus, the robotspecific mapping f sp decomposes into n robot-specific sectionkinematics f sp,i

f sp(q) : k = [kT1 · · · kT

n ]T with (14)

ki = f sp,i(qi). (15)

To model the robot-specific section kinematics of the BHA, arobot structure similar to the structure of cable-driven continuum


Fig. 6. Exemplary structure of the robot-specific section kinematics for cable-driven continuum robots according to Assumptions 4–6

robots is used. This is due to the close constructive accordance ofthe BHA’s wire-cables with the actuator cables of cable-drivencontinuum robots. As displayed in Fig. 6, each section’s actuatorregion is described using the following assumptions.

Assumption 4: The actuator region of the ith section consistsof three wire-cables that are arranged in a rotatory distance of120◦, each. The wire-cables route linearly between two succes-sive cable guides.

Assumption 5: The backbone curve of the ith section’s actu-ator region is modeled according to Assumptions 2–3. A unit isdefined to be the region between two successive cable guides.

Assumption 6: The radial distances of the ijth unit’s cableguides referred to the backbone curve are dij−1 and dij . Thefractions of the wire-cable lengths lik that belong to the ijthunit are termed unit wire-cable lengths lijk .

Looking at the coordinate frame definitions, all coordinateframes are 1) adjusted tangentially to the backbone curve point-ing toward the section’s head and 2) orientated such that theirrespective x-axes cross the wire-cable with the actuator indexk=1. This yields coordinate frames compatible with the coor-dinate frame definitions that are proposed in the general robotkinematics.

Note that a model parameterization with nonconstant radialdistances of the cable guides within a single section yields abiomimic continuum robot kinematics. Furthermore, the amountof units used to model a single section can be chosen freely.However, an ideal model accuracy is expected for a sectionsegmentation which fits to the wire-cable routing of the robothardware.

For these model assumptions, the robot specific section kine-matics and the robot specific unit kinematics are discussed.

1) Robot-Specific Section Kinematics: According to (14),the robot-specific section kinematics f sp,i computes the sec-tion’s configuration state ki depending on the section’s actuatorstate qi . The actuator state is defined by the lengths of the threewire-cables lik

qi = [ qi1 qi2 qi3 ]T = [ li1 li2 li3 ]T . (16)

The configuration state ki is computed by the robot-specific unitkinematics f sp,ij together with the wire-cable fragmentationf frag ,ij

f sp,i(qi) : ki = [kTi1 · · · kT

imi]T with (17)

Fig. 7. Geometry of the ijth unit according to Assumptions 4–6.

kij = f sp,ij (lij ) and (18)

lij = [ lij1 lij2 lij3 ]T = f frag ,ij (qi). (19)

Note that the input variables of the unit kinematics are theunit wire-cable lengths lij . In order to match the DoF of thesection model with the section’s hardware DoF, the wire-cablefragmentation f frag ,ij has to describe the unit wire-cable lengths

lij as a function of the actuator state qi , such that the followingcondition holds.

Condition 3: For each actuator k, the sum of the fractionalunit wire-cable lengths lijk has to equal the wire-cable lengthlik

qik =mi∑j=1

lijk , k ∈ {1, 2, 3}. (20)

Without limiting the generality, the BHA kinematics satisfiesCondition 3 by the following assumption.

Assumption 7: The section model is subdivided equallyspaced into mi units. The wire-cable lengths are fragmentedhomogenously.

Thus, the respective wire-cable fragmentation yields

f frag ,ij (qi) : lij =1

miqi , j ∈ {1 . . . mi}. (21)

Note that choosing a suitable section subdivision and wire-cablelength fragmentation is fundamental to yield an accurate robotkinematics. Referred to the BHA, (21) describes a global mini-mum potential energy approximation, if gravitation is neglected.This is shown in Appendix A.

2) Robot-Specific Unit Kinematics: The geometry of a sin-gle unit is displayed in Fig. 7. On the left-hand side of Fig. 7,a top view of the deflected unit ij is presented. The right-handside of Fig. 7 illustrates a projection to its deflection plane. Theunit’s arc parameters are φij , κij , and �ij which depend on theunit wire-cable lengths lij . lij =[lij1 lij2 lij3 ]T label virtualunit wire-cable lengths which refer to the nonconic unit withdij−1 = dij pictured by the dashed circle. The angles betweenthe virtual nonconic wire-cables and the xij−1yij−1-plane indirection to the respective wire-cables are labeled with βijk .

Using the cc theory for cable-driven robots according to [15],with the coordinate frame definitions given previously,1 the

1Note that these do not match the coordinate frame definitions of [15].


wire-cable lengths for the nonconic unit map to the unit config-uration state are as follows:

f sp,ij (lij ) : kij = [φij κij �ij ]T (22)

with

φij = atan2 (√

3(lij3 − lij2), lij2 + lij3 − 2lij1) (23)

κij =2lsqrt

dij lsum(24)

�ij =

⎧⎪⎪⎨⎪⎪⎩

dij lsum

lsqrtarcsin

(lsqrt

3dij

), lsqrt �= 0

13lsum , lsqrt = 0,

(25)

and lsum and lsqrt representing the substitutions

lsum = lij1 + lij2 + lij3 (26)

lsqrt =√

l2ij1 + l

2ij2 + l

2ij3 − lij1 lij2 − lij1 lij3 − lij2 lij3 . (27)

Note that 1) atan2 is defined such that

φ = atan2(sin (φ) , cos (φ)), φ ∈ ] − π, π] (28)

and 2) lsqrt =0, iff lij1 = lij2 = lij3 . In this case, the unit’s back-bone curvature fits a straight line (κij =0). Remember that, thisrobot configuration is singular in the configuration space. Hence,in the actuator space, any modification of lij yields a κij �=0, re-sulting in a variation of the end-effector pose. This implies thatevery end-effector velocity can be realized by a finite actuatorstate velocity. Thus, neither this nor any other robot configura-tion within the robot’s workspace is singular, showing that theconfiguration space singularity is artificial due to the selectedcoordinates of the configuration state.

In order to apply this cc mapping to the conic unit, (22) hasto be expressed depending on the unit wire-cable lengths lijinstead of the virtual unit wire-cable lengths lij . The relationbetween those two wire-cable lengths is given by the law ofcosines

l2ijk = l2ijk + (dij−1 − dij )2 − 2lijk (dij−1 − dij ) cos (βijk )

(29)with cos (βijk ) being

cos (βijk ) = sin(

κij �ij

2

)cos

(23π(k − 1) − φij

). (30)

A detailed derivation of (30) is given in Appendix B. Solving(29) for the virtual unit wire-cable length yields

lijk =√

l2ijk − (dij−1 − dij )2 + (dij−1 − dij )2 cos2(βijk ) . . .

+ (dij−1 − dij ) cos (βijk ) . (31)

As can be seen, the virtual unit wire-cable lengths lijk arefunctions of the unit wire-cable lengths lijk , the unit conicity(dij−1 − dij ), and the angles βijk . The latter are again functionsof the arc parameters φij , κij , and �ij . Thus, inserting (31) into(22) does not yield an analytical solution of the robot-specific

unit kinematics

f sp,ij (lij ) : kij = [φij κij �ij ]T . (32)

As no analytical solution was found, an analytical approximatesolution is discussed instead.

To derive an analytical approximate mapping between theunit wire-cable lengths lij and the configuration state kij , theinfluence of βijk on the virtual wire-cable lengths accordingto (31) has to be negligibly small. As the unit conicity cannotbe changed, angles βijk close to π

2 are necessary.2 This canbe reached by choosing mi large, as the unit’s deflection angleκij �ij decreases with the rising amount of units per section. Inthis case, (31) simplifies to

lijk = +√

l2ijk − (dij−1 − dij )2 . (33)

Inserting (33) into (22) yields the analytical approximate so-lution of the robot-specific unit kinematics f sp,ij as requestedin (32). Note that the necessary simplification of (31) does notresult in a robot mapping with a constant section curvature, aseach unit still consists of its individual actuator distance dij . Itjust neglects the impact of conicity on a single unit.

In summary, the robot-specific vc continuum kinematics f spis composed of

∑ni=1 mi robot-specific unit kinematics f sp,ij ,

each solved using an analytical approximate solution. To matchthe surplus DoF of the configuration state k with the DoF ofthe actuator state q, a section model subdivision and wire-cablelength fragmentation has to be assumed such that Condition 3holds.

Note that for sections with constant wire-cable distances(dij−1 =dij , ∀j ∈ {1 . . . mi}): 1) the analytical approximate so-lution (33) equals the exact analytical solution and 2) Equation(17) with (22) and (33) reduces to the cc approach for cable-driven robots (see [15]). Thus, having a robot with nonconicsection geometry, the proposed vc continuum kinematics yieldsidentical results as the cc one. However, applied to a robot withthe biomimic robot structure, the approach is able to approx-imate the backbone curve more precisely, as a piecewise ccdescription within a single section becomes possible.

Using the proposed vc model, its application in a kinematiccontrol algorithm is presented next.

IV. KINEMATIC CONTROL

The kinematic control problem is defined as follows: givena trajectory of desired task variables xd(t) within the robot’sworkspace, find a trajectory of the actuator state q(t) such that

1) the end-effector tracks the trajectory of task variables;2) the actuator state must not violate actuator end stops.A basic approach to solveing such a commonly termed in-

verse kinematics problem is to use a first-order resolved ratealgorithm according to [35]. This algorithm is based on therobot’s task-specific velocity kinematics. Thus, as displayed inFig. 8, the robot’s forward kinematics has to be extended by atask-specific mapping f task in order to yield the task-specificforward kinematics f as2ts able to compute the task variables x

2Note: limβ i j k → π2

cos(βij k ) = 0


Fig. 8. Composition of the task-specific forward kinematics of a continuumrobot under the assumption of vc.

depending on the actuator state q. Then, the inverse kinematicsproblem f−1

as2ts from task variables to the actuator state has tobe solved.

A. Task-Specific Mapping

An arbitrary task is defined by the following mapping:

f task(h) = x, x ∈ Rp (34)

with x being a vector of p scalar task variables. These taskvariables may, e.g., consist of the end-effector’s position andorientation. Note that p always has to be smaller than or equal tothe robot’s kinematic DoF and task space dimension. Choosing psmaller than the robot’s kinematic DoF results in a kinematicallyredundant robot.

In the following, without loss of generality, a 5-DoF taskdefinition is presented. Its task variables, which are computeddepending on the pose representation h according to (12), arethe Cartesian end-effector position (x, y, z) and orientation in 2DoFs (θx , θy ) described by

f 5DoFs(h) : x = [x y z θx θy ]T (35)

with x = h10 , y = h11 , z = h12 , (36)

θx = atan2(−h8 , h9), θy = atan2(h7 , h9). (37)

B. Inverse Kinematics Problem

First-order resolved rate algorithms use a local resolution ofredundancy at the velocity level to compute the inverse kine-matics problem. For continuum robots modeled according toFig. 8, the velocity kinematics is derived by differentiating thetask-specific forward kinematics

x = f as2ts(q) = f task(f gen

(f sp (q)

))(38)

with respect to time. This yields

x =∂f task

∂h

∂f gen

∂k

∂f sp

∂qq(t) = J(q)q (39)

where the upper dot denotes the time derivative. J labels the(p × 3n)-dimensional Jacobian matrix of the robot, which is afunction of its actuator state q. An analytical way of computingthe Jacobian is given in Section V. In order to invert the velocitykinematics (39), a weighed least-norm solution proposed byChan and Dubey [36] is used. This yields a computation of theactuator state velocities q according to

q = W−1JT (JW−1JT )−1 (xd + K(xd − x)) (40)

where the elements of the diagonal matrix W ∈ R3n×3n arechosen in a way to avoid actuator end stops. The positive-definite and, usually, diagonal matrix K ∈ Rp×p is introduced

Fig. 9. Control structure of the described first-order resolved rate algorithm.

to reduce tracking errors between the desired xd and actual xtask variables. Furthermore, a final integration of the actuatorstate velocities q is necessary to get the actuator state trajectoryq. This yields a control algorithm as presented in Fig. 9.

Note that (40) becomes a pure feed-forward kinematic con-trol depending on the robot’s Jacobian if all elements of Kare 0. Using this feed-forward control to steer a robot, errorsin the Jacobian result in errors of the actuator state velocities.Integrated and applied to the plant as the control input yieldsmarginally stable (drifting) task variable errors Δx. Thus, thedrift of the task variable errors is an excellent measure of theJacobian matrix accuracy and will be used to validate the pro-posed velocity kinematics afterward. Summing up, the pro-posed kinematic control solves the task-specific inverse kine-matics problem by using a first-order resolved rate algorithm.To solve this algorithm efficiently, an analytical computation ofthe robot’s velocity kinematics is necessary, which is presentednext.

V. VELOCITY KINEMATICS

This section analytically derives all equations of the velocitykinematics

x = J(q)q (41)

belonging to the vc continuum kinematics proposed inSection III. According to (39), the Jacobian matrix J equals

J(q) =∂f task(h)

∂h

∂f gen(k)∂k

∂f sp(q)∂q

= Jtask(h) Jgen(k) Jsp(q). (42)

Beginning from left to right, the three components of theJacobian are discussed. Note that these matrices describe thenonsingular robot configuration (κij �=0). Afterward, a matrixregularization concept is presented that computes the Jacobianin the artificial singularity of the configuration space model.This avoids a decrease of the Jacobian’s row rank.

A. Task-Specific Jacobian Matrix

The task specific Jacobian matrix Jtask depends on the de-fined control task [see (34)]. In general, it computes to

Jtask(h) =∂f task(h)

∂h=

∂x

∂h, Jtask ∈ Rp×12 . (43)

Applied to the 5-DoF task proposed in (35), it yields

∂f 5DoFs

∂h=


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 − h9

h28 + h2

9

h8

h28 + h2

90 0 0

0 0 0 0 0 0h9

h27 + h2

90 − h7

h27 + h2

90 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

(44)

B. General Continuum Robot Jacobian Matrix

Differentiating (12) with respect to the configuration stateresults in the general continuum robot Jacobian matrix

Jgen(k) =∂f gen(k)

∂k=

∂h

∂k, Jgen ∈ R12×3

∑ ni = 1 mi . (45)

Representing Jgen in terms of the robot’s homogenous matrixwHnh yields

Jgen(k) =⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂wHnh,(1,1)

∂k111

∂wHnh,(1,1)

∂k112· · ·

∂wHnh,(1,1)

∂knmn 3

∂wHnh,(1,2)

∂k111

∂wHnh,(1,2)

∂k112· · ·

∂wHnh,(1,2)

∂knmn 3

......

. . ....

∂wHnh,(3,4)

∂k111

∂wHnh,(3,4)

∂k112· · ·

∂wHnh,(3,4)

∂knmn 3

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (46)

Using the product rule, its elements are computed by

∂wHnh

∂kijν= wH1b

(i−1∏ξ=1

ξbSξh

)∂ibSih(ki)

∂kijν

( n∏ξ=i+1

ξbSξh

)

(47)with

∂ibSih(ki)∂kijν

= ibSi0

(j−1∏χ=1

iχ−1Uiχ

)∂ij−1Uij (kij )

∂kijν

·( mi∏

χ=j+1

iχ−1Uiχ

)im i Sih (48)

as

∂ξχ−1Uξχ(kξχ)∂kijν

= 04×4 , ∀ i �= ξ, j �= χ. (49)

Thus, Jgen depends on the homogenous transformations thatare described in Section III-A and the partial derivatives of theunits’ homogenous transformations, whose explicit equationsare given in Appendix C. Note that a Jacobian matrix regular-ization is necessary for the artificial singularity of the generalcontinuum kinematics (kij2 =0).

C. Robot-Specific Jacobian Matrix

In order to yield the robot-specific Jacobian matrix Jsp , themapping f sp according to (14) has to be differentiated with

respect to the actuator state q

Jsp(q) =∂f sp(q)

∂q=

∂k

∂q, Jsp ∈ R3

∑ ni = 1 mi ×3n . (50)

As each section is independent (Condition 1), Jsp results in thefollowing block-diagonal matrix:

Jsp(q) =

⎡⎢⎣

Jsp,1(q1) · · · 0...

. . ....

0 · · · Jsp,n (qn )

⎤⎥⎦ with (51)

Jsp,i(qi) =∂f sp,i(qi)

∂qi

=∂ki

∂qi

, Jsp,i ∈ R3mi ×3 . (52)

According to (17), the robot-specific section kinematics f sp,i isassembled of the robot-specific unit kinematics f sp,ij and thewire-cable fragmentation f frag ,ij . Thus, Jsp,i computes to

Jsp,i(qi) =

⎡⎢⎢⎢⎢⎢⎢⎣

∂f sp,i1

∂ li1· · · 0

.... . .

...

0 · · ·∂f sp,im i

∂ lim i

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎢⎣

∂f frag ,i1

∂qi

...

∂f frag ,im i

∂qi

⎤⎥⎥⎥⎥⎥⎥⎦

.

(53)The elements of the block-diagonal matrix describe the partialderivatives of the unit kinematics. They can further be decom-posed in the unit kinematics depending on the virtual unit wire-cable length according to (22) and the analytic approximatesolution (33). This yields

∂f sp,ij

∂ lij=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂kij1

∂lij

∂lij

∂lij1

∂kij1

∂lij

∂lij

∂lij2

∂kij1

∂lij

∂lij

∂lij3

∂kij2

∂lij

∂lij

∂lij1

∂kij2

∂lij

∂lij

∂lij2

∂kij2

∂lij

∂lij

∂lij3

∂kij3

∂lij

∂lij

∂lij1

∂kij3

∂lij

∂lij

∂lij2

∂kij3

∂lij

∂lij

∂lij3

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(54)

with the partial derivatives of the configuration state kij beinggiven in Appendix C and the partial derivatives of the virtualunit wire-cable length being

∂lij

∂lijk

=[

∂lij1

∂lijk

∂lij2

∂lijk

∂lij3

∂lijk

]T

with (55)

∂lijξ

∂lijk

=

⎧⎨⎩

0, ξ �= k

lijk

(√l2ijk − (dij−1 − dij )2

)−1, ξ = k.

(56)

The elements of the second matrix of (53) describe the par-tial derivatives of the wire-cable fragmentation. With respect toAssumption 7, these are

∂f frag ,ij

∂qi

=1

miI3×3 (57)

with I3×3 being the identity matrix. Note that the robot spe-cific Jacobian Jsp is valid for lsqrt �=0 only. For lsqrt =0, theregularized Jacobian matrix has to be used.


D. Jacobian Matrix Regularization

A mathematically correct approach to obtaining the regular-ized Jacobian matrix J in the artificial singularity of the con-figuration space with κij =0 is to compute it by the centraldifference quotient of the forward kinematics. This yields

J(q) = [ J1 · · · J3n ] with (58)

Jξ =f as2ts(q + Δqξ ) − f as2ts(q − Δqξ )

2Δqand (59)

Δqξ = [ Δqξ1 · · · Δqξ3n ]T , Δqξχ ={

Δq, ξ = χ

0, ξ �= χ.

(60)

Note that the resulting Jacobian is regular, as the mapping fromthe actuator state to task variables f as2ts does not consist of anysingularities within the robot’s workspace. However, the maindisadvantage of this numerical computation is its high computa-tional cost of 2 · dim(q) evaluations of the forward kinematics.

A less accurate, however, much cheaper regularizationmethod is to replace the Jacobian J in a singular configura-tion with an analytically derived Jacobian J, which belongs to anonsingular robot configuration very close to the artificial sin-gularity. To do so, Jgen and Jsp are computed depending onregularized unit configuration states kij and regularized virtualunit wire-cable lengths lij , respectively. Defining the region ofthe Jacobian matrix regularization by the threshold εls q r t

, kij

and lij yield

kij =

⎧⎪⎨⎪⎩

kij , lsqrt ≥ εls q r t[kij1

2εls q r t

dij lsum

13lsum

]T

, lsqrt < εls q r t,

(61)

lij = [ lij1 lij2 lij3 ]T , with (62)

lijk = sin

(kij2 kij3

2

)(2

kij2− 2 dij cos

(2π(k − 1)

3− kij1

)).

The latter are computed by the inverse of (23)–(25). The substi-tutions lsum and lsqrt are defined according to (26) and (27).

VI. SIMULATIONS AND EXPERIMENTAL VALIDATION

In this section, simulation studies and test bench experimentsare discussed. The simulations focus on the accuracy improve-ments of the proposed vc continuum kinematics related to thecc approach and a validation of the robot’s velocity kinemat-ics including its regularization. Using test bench experiments,the simulated forward kinematics of a multisection continuumrobot is compared with measured data. Furthermore, results ofthe kinematic control are shown.

A. Simulation Studies

In the following, the robot’s forward, velocity, and inversekinematics are discussed by simulations. All simulations arebased on a continuum robot whose geometric data are pre-sented in Table I. The parameters describe a conic cable-

TABLE IGEOMETRIC ROBOT DATA USED IN ALL SIMULATIONS

driven continuum robot consisting of three serially connectedsections. Each section’s radial wire-cable distance reduces bydih−dib =0.03m in direction to its head. The section’s base andhead heights are 0.0m and its wire-cable lengths are limited be-tween 0.2 and 0.3m. This yields an average section tapering ofapproximately 7 ◦.

Using these geometric robot parameters, four different kine-matics are simulated and compared with a reference kinematicsafterward. The first kinematics is a cc model whose wire-cabledistance is parameterized by di = 1

2 (dib +dih). A correction ofthe wire-cable length similar to (33) compensates the section’sconicity of this model. Additionally, three different vc kinemat-ics having 5, 10, and 15 units per section are simulated. Theseare modeled as proposed in Section III using the analytical ap-proximate solution (33) to yield the closed-form kinematics.The simulated reference kinematics is also computed accordingto Section III. However, instead of using the analytical approx-imate solution (33), (29) is solved numerically for all of its 50units per section.

1) Simulation of the Forward Kinematics: This simulationanalyzes the modeled backbone curves of section i=3 belong-ing to the robot parameterized according to Table I. Choosingthe wire-cable lengths of actuators 2 and 3 to be lmin,3k andincreasing the wire-cable length of actuator 1 from lmin,3k tolmax,3k in four steps yields a section deflection within the xzplane. The respective simulation results of the kinematics de-scribed previously are displayed in Fig. 10. Fig. 10(a)–(d) showthe backbone curves of the cc and vc kinematics (black, dash-dotted lines) compared with the respective deflection curves ofthe reference robot (gray, solid lines). The white dots mark theboundaries of the modeled units. To display the end-effectorpose, thin red lines mark the x- and z-axes of its coordinateframe.

Looking at the backbone curve of the reference robot, an in-creasing section curvature in the direction of its head is modeled.The cc approach tries to approximate this backbone shape withone circular arc. Thus, clear discrepancies between both deflec-tion curves are visible. Obviously, these discrepancies result inunequal end-effector poses. However, choosing a vc kinematicsenables a more precise backbone approximation, especially ifthe amount of units per section increases. This yields kinematicmodels that reconstruct the end-effector pose with enhancedaccuracy.

2) Validation of the Velocity Kinematics: In order to showthe validity of the analytically derived Jacobian matrix, accord-ing to Section V, together with its regularization, according to(61) and (62), Fig. 11 displays a comparison of the infinity


(a) (b)

(c) (d)

Fig. 10. Backbone deflection simulation. Single section reference robot (gray)compared with (a) a cc kinematics and three vc kinematics using (b) 5 units, (c)10 units, and (d) 15 units.

Fig. 11. Jacobi matrix validation in regular and singular robot configurations.

norm ‖Janalytic − Jnumeric‖∞ with ‖Janalytic‖∞ for differentactuator states including the artificial singularity. Janalytic la-bels the regularized analytically derived Jacobian (42) of a vcsection model with 15 units parameterized according to the thirdsection of Table I. Jnumeric is computed numerically using thecentral difference quotient as described in (58). On the left-handside of Fig. 11, the actuator state of each matrix evaluation isdisplayed. On the right of Fig. 11, the infinity norm values ofthe analytically derived Jacobian matrix and its differences tothe numerically one are shown in a semilogarithmic plot.

Focusing on matrix evaluations close to the artificial singu-larity (near sample 2500), an increased difference between bothmatrices due to the matrix regularization is visible. However,comparing the infinity norm values of the Jacobian matrix dif-ference with those of the Jacobian itself, the latter are more thanfive powers of ten bigger. This demonstrates that the matrix dif-ference caused by the regularization is negligibly small for allrobot configurations.

3) Simulation of the Inverse Kinematics: This final simula-tion presents the results of the kinematic robot control accord-ing to Section IV. As a plant model, the multisection referencerobot parameterized according to Table I is used. The velocitykinematics of the cc and the three vc models are utilized to pa-rameterize the respective kinematic control algorithms. In order

Fig. 12. Comparison of four feed-forward control simulations applied to thereference robot using different models of the robot’s velocity kinematics.

to compare the accuracy of these inverse kinematics, the purefeed-forward part of the controller with K = 0 is simulated.The results are shown in Fig. 12. The top-left of Fig. 12 displaythe robot’s backbone deflection realized by tracking the desiredtask variable trajectory xd presented at the top right. The light-and dark-gray backbone color is used to distinguish between thethree sections of the robot. Again, thin red lines mark the end-effector’s coordinate frame. Its spatial trajectory is displayed bythe black line. As the desired task variable trajectories yd andθx,d are constant to zero, only the nonzero trajectories xd , zd ,and θy,d are shown. These describe an end-effector trajectorywithin the xz plane including a rotation around the y-axis. Thethree graphics at the bottom of Fig. 12 display the errors betweenthe desired and actual task variables

Δx = x − xd,Δz = z − zd,Δθy = θy − θy,d (63)

belonging to the kinematic feed-forward control using the ccand vc kinematics having 5, 10, and 15 units per section, re-spectively.

Comparing the task variable errors, it is apparent that thesmallest deviation is obtained by the kinematic control algo-rithm which uses the vc model with 15 units per section. Re-ducing the amount of units per section yields a successivelyrising drift of the error. Thus, choosing a high number of unitsper section increases the accuracy of the velocity kinematicsif robots with a variable section curvature are modeled. Notethat this finding also correlates with the results of the forwardkinematics simulation.


Fig. 13. BHA test bench with an optical motion capture system.

TABLE IIGEOMETRIC SECTION DATA OF THE BHA

B. Experimental Validation

The following validation experiments utilize a BHA (seeSection II and Fig. 13) as a continuum robot. To measure therobot’s backbone curve, four reflective markers are attached toeach head of its three sections. These markers are tracked byan optical 3-D motion capture system in order to reconstructthe pose of each head. To operate the robot, a dynamic trackingcontrol of its actuator state, which is the wire-cable lengths ofits pneumatic actuators, is implemented. The robot’s geometricdata are collected in Table II. The selected amount of units persection is chosen to fit the physical segmentation of the BHA,which consists of 11 wire-cable guides per section (see Fig. 3).Thus, the linear wire-cable routing from one cable guide to thenext is approximated as close as possible.

1) Forward Kinematics: Results of the experimental valida-tion of the forward kinematics are presented in Fig. 14. Theupper part of Fig. 14 illustrates the robot’s end-effector trajec-tory and backbone deflection together with its correspondingactuator state. The actuator state is partitioned in three graphics,each figuring the wire-cable lengths belonging to one of the threerobot sections. The measured end-effector pose is displayed be-low. On the left-hand side of Fig. 14, the Cartesian position isillustrated. The right-hand side of Fig. 14 shows the end-effectororientation. The angles θh and θt are the end-effector’s headingand tilt angle with

θh = atan2 ( h8 , h7 ) , and (64)

θt = atan2 (√

h27 + h2

8 , h9 ) . (65)

At the bottom of Fig. 14, the model errors of both the cc and vckinematics are illustrated. The scalar position error ep representsthe spacial distance between measured ·meas and simulated ·simCartesian end-effector position

ep =√

Δx2 + Δy2 + Δz2 , with Δ· = ·sim − ·meas .(66)

Fig. 14. Experimental validation of the forward kinematics. (Top) Backbonedeflection and actuator state. (Bottom) End-effector pose and model errors.

The scalar orientation error eO is the rotation angle that is neededto align the simulated with the measured end-effector orienta-tion. According to [35], this error can be deduced from therotation matrix R with

R = wR3h,measwRT

3h,sim (67)

by computing the angle of its respective axis-angle orientationrepresentation. wR3h labels the rotational part of the homoge-nous matrix wH3h describing the end-effector pose.

Referring to the model errors, the vc kinematics is able to sim-ulate the robot’s end-effector pose with position displacementsless than 25 mm and orientation errors smaller 6.5◦. Focusingon regions with stationary or slowly changing actuator state, alower position error of approximately 15 mm is reached. Thisenhanced model accuracy is due to the missing temporary de-formations of the backbone curve caused by inertia forces andmoments. Remaining model errors are expected to result fromgravity or torsional effects which cannot be considered explicitlyby the kinematic model. Compared with the robot’s sphericallyapproximated workspace with 0.6-m radius and end-effector tiltangles of more than 100◦, this yields relative errors ranging from2.5 to 6.5%. The cc kinematics simulates the end-effector posethroughout with lower position accuracy and an approximatelyequal orientation error. Especially, regions with low robot dy-namics maintain an additional position error of more than3 mm. Thus, even for continuum robots with low conicity (here,4.5◦), a reduction of the position error of approximately 15%


Fig. 15. Kinematic control of the BHA using an open-loop resolved ratealgorithm based on the proposed vc and cc velocity kinematics.

compared with the cc kinematics is possible if the vc kinematicsis used.

It should be noted that the model accuracy obviously cannotcompete with computationally expensive steady-state approx-imations of kinetic continuum robot models able to considerequilibrium shape deformations caused by gravitational and tor-sional effects. Furthermore, modal approaches might also yieldbetter pose accuracy if calibrated carefully to the chosen set(s)of shape functions. However, because of the easy parameteriza-tion of the kinematics, which exclusively uses geometric robotdata, the ability to adjust the computational work (by choosingthe amount of units per section mi), as well as the improvedaccuracy compared with cc section models, the proposed vccontinuum kinematics provides an excellent alternative to beused in real-time control applications. Moreover, calibrating thekinematic model to specific load cases might further increasethe accuracy of the kinematics.

2) Kinematic Control: In accordance with the simulationstudies, experimental results of the kinematic control are dis-cussed at the end. Again, the feedback part of the resolved ratealgorithm is switched off (K = 0). Thus, errors in the mod-eled velocity kinematics yield a marginally stable displacementof the end-effector pose from its reference trajectory. The driftvelocity of the displacement is reduced with rising model ac-curacy. Measurements captured with the BHA using a controlalgorithm that is based on the vc as well as the cc velocitykinematics parameterized according to Table II are presented inFig. 15. The 3-D plot in the top-left of Fig. 15 displays the de-sired and measured end-effector trajectories. Furthermore, therobot’s backbone deflection and end-effector coordinate frameare illustrated at specific time steps. The top-right graphics inFig. 15 show the actuator state of each section in separate plots

(vc measurement). In the following, the desired end-effector tra-jectory composed in its Cartesian components xd , yd , and zd , aswell as the resulting position errors of both control algorithms,is presented.

Discussing the results, it is desired that the robot’s tip followsa 0.7-m-long path consisting of three straight lines beginningat the end-effector’s initial position. At the end of this path,the end-effector displacement accumulates to 0.05 m (vc) and0.08 m (cc). This yields relative path displacements with com-parable sizes to the relative errors of the forward kinematics.Thus, open-loop control performances close to the accuraciesof the kinematic models are reached. This demonstrates the ben-eficial usage of the vc velocity kinematics in real-time controlapplications. Of course, this path displacement can be furtherreduced by closing the feedback loop with a positive-definitematrix gain K > 0.

VII. CONCLUSIONS AND FUTURE WORK

We presented a new closed-form vc continuum kinematics,including its respective velocity kinematics and possible inversekinematics, able to model hyper-redundant or continuum robotsthat consist of multiple extensible sections with spatial bending(no torsion). The kinematics focuses on the variable sectioncurvature due to the biomimic manipulator design, modeling thebackbone curve of each section by approximating its shape withseveral serially connected circular arcs. This yields a piecewisecc section kinematics, which closes the gap between existingsection models with variable continuously changing curvatureor constant section curvature.

Having a continuum robot with a variable section curvature,the model is expected to yield less accurate results comparedwith continuum kinematics that is based on equilibrium con-formations of distributed parameter kinetics and more accurateresults compared with constant section curvature models. Its keyadvantages are its low computational work, easy parameteriza-tion, and universal applicability. For these reasons, the proposedkinematics provides an excellent alternative to be used in real-time control applications.

The closed-form forward kinematics models the robot’s back-bone curve by using robot-specific and general continuum kine-matics. A first-order resolved rate algorithm is presented tosolve the inverse kinematics problem. To solve this algorithmefficiently, the robot’s velocity kinematics is given analytically,including a regularization concept to deal with internal singu-larities. For experimental validation, Festo’s BHA was used.

The presented kinematics is expected to be valuable, as bio-logically inspired mechanisms gain ongoing importance in therobotic area. Thus, future continuum robots are likely to copythe variable backbone curvature of elephant-trunks, tentacles,or snakes. The proposed kinematics constitutes a competitivealternative to be used for kinematic real-time modeling andcontrol of these robots. Furthermore, the existence of accurateand universally applicable real-time kinematics may encouragemore researchers to design and explore these biomimic robots.Finally, dynamic robot modeling and control may also be ad-vanced. Using the new vc continuum kinematics as the basis for


dynamic-lumped parameter models yields nonlinear kinetics ofcontinuum robots with ongoing research potential.

APPENDIX A

MINIMUM POTENTIAL ENERGY APPROXIMATION

We assume that 1) the potential energy of the BHA equalsthe energy stored in its spring-like bellows-actuators; 2) theactuators possess a constant stiffness characteristic distributedover the whole length of the actuator; and 3) the potential energydue to gravitation can be neglected. With these assumptions, theglobal minimum of the robot’s potential energy is reached, if thepotential energy Eik of each actuator ik with i ∈ {1 . . . n}, k ∈{1, 2, 3} is minimal. The potential energy of the ikth actuatorcomputes to

Eik =∫ li k

l0 , i k

Fik (lik ) dlik (69)

with l0,ik being the initial wire-cable lengths of the vented actu-ator and Fik (lik ) its restoring elastic force. Computing the ikthactuator’s potential energy depending on the potential energiesof its corresponding unit regions yields

Eik =mi∑j=1

(∫ l i j k

l0 , i j k

Fijk (lijk ) dlijk

)(70)

with l0,ijk being the initial unit wire-cable lengths of the ventedactuator and Fijk (lijk ) the unit’s restoring elastic force. Notethat for arbitrary combinations of section subdivisions andwire-cable length fragmentations, Eik ≤ Eik . A minimum po-tential energy approximation realizes Eik = Eik . Referring toAssumption 7 and 2), the equally spaced section subdivisionrealizes

l0,ijk =l0,ik

mi, Fijk (lijk ) = Fik (milijk ). (71)

The homogenous wire-cable fragmentation yields

lijk =likmi

. (72)

Inserting (71) and (72) into (70) results in

Eik = mi

(∫ l i km i

l 0 , i km i

Fik

(mi

likmi

)d

likmi

)= Eik . (73)

Thus, (21) describes a global minimum potential energy approx-imation for the BHA.

APPENDIX B

DERIVATION OF THE ANGULAR DEPENDENCE OF cos (βijk )

In order to verify the angular dependence of cos (βijk ) previ-ously presented in (30), the two unit vectors u and v, accordingto Fig. 16, are introduced. u is parallel to the xij−1yij−1-planepointing from the kth virtual nonconic wire-cable to the kthconic wire-cable. v is parallel to the virtual nonconic wire-cablepointing toward the section’s head. The angle βijk is enclosed

Fig. 16. Detailed illustration of the ijth unit’s geometry.

between u and v. Describing u and v referred to as the Carte-sian coordinate frame Odp , whose x- and z-axes span the unit’sdeflection plane, yields

dpu =

⎡⎢⎢⎢⎢⎢⎢⎣

cos(

23π(k − 1) − φij

)

sin(

23π(k − 1) − φij

)

0

⎤⎥⎥⎥⎥⎥⎥⎦

, dpv =

⎡⎢⎢⎢⎢⎢⎣

sin(

κij �ij

2

)

0

cos(

κij �ij

2

)

⎤⎥⎥⎥⎥⎥⎦.

(68)

Evaluating the dot product of dpu with dpv results in theangular dependence of cos (βijk ) as described in (30).

APPENDIX C

PARTIAL DERIVATIVES

Depending on ν, the partial derivatives of the ijth unit’s ho-mogenous transformation are described by

ν = 1:∂ij−1Uij (kij )

∂kij 1=

⎡⎢⎢⎢⎢⎢⎢⎣

−2sφ cφ (cκ � − 1) (1 − 2s2φ )(cκ � − 1) −sφ sκ �

sφ (cκ � − 1)kij 2

(1 − 2s2φ )(cκ � − 1) 2sφ cφ (cκ � − 1) cφ sκ � −

cφ (cκ � − 1)kij 2

sφ sκ � −cφ sκ � 0 0

0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎦

(74)


∂kij 2=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−c2φ sκ � kij 3 −sφ cφ sκ � kij 3 cφ cκ � kij 3

cφ sκ � kij 3

kij 2+

cφ (cκ � − 1)k2

ij 2

−sφ cφ sκ � kij 3 −s2φ sκ � kij 3 sφ cκ � kij 3

sφ sκ � kij 3

kij 2+

sφ (cκ � − 1)k2

ij 2

−cφ cκ � kij 3 −sφ cκ � kij 3 −sκ � kij 3cκ � kij 3

kij 2− sκ �

k2ij 2

0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(75)



∂kij 3=

⎡⎢⎢⎢⎣

−c2φ sκ � kij 2 −sφ cφ sκ � kij 2 cφ cκ � kij 2 cφ sκ �

−sφ cφ sκ � kij 2 −s2φ sκ � kij 2 sφ cκ � kij 2 sφ sκ �

−cφ cκ � kij 2 −sφ cκ � kij 2 −sκ � kij 2 cκ �

0 0 0 0

⎤⎥⎥⎥⎦ (76)

with the substitutions sφ , cφ , sκ� , and cκ� according to (4) and(5).

The partial derivatives of the configuration state kij referredto as the virtual unit wire-cable length lij are

∂kijξ

∂lij=[

∂kijξ

∂lij1

∂kijξ

∂lij2

∂kijξ

∂lij3

](77)

with

∂kij1

∂lij1=

√3(lij3 − lij2)

2l2sqrt

∂kij1

∂lij2=

√3(lij1 − lij3)

2l2sqrt

(78)

∂kij1

∂lij3=

√3(lij2 − lij1)

2l2sqrt

(79)

∂kij2

∂lij1=

3(lij1 lij2 + lij1 lij3 − l2ij2 − l

2ij3)

dij lsqrtl2sum

(80)

∂kij2

∂lij2=


2ij3)

dij lsqrtl2sum

(81)

∂kij2

∂lij3=


2ij2)

dij lsqrtl2sum

(82)

∂kij3

∂lij1=

dij arcsin(

ls q r t3di j

)lsqrt

(1 − lsum(2lij1 − lij2 − lij3)

2l2sqrt

). . .

+dij lsum(2lij1 − lij2 − lij3)

2l2sqrt

√9d2

ij − l2sqrt

(83)

∂kij3

∂lij2=

dij arcsin(

ls q r t3di j

)lsqrt


2l2sqrt

). . .


2l2sqrt

√9d2

ij − l2sqrt

(84)

∂kij3

∂lij3=

dij arcsin(

ls q r t3di j

)lsqrt


2l2sqrt

). . .


2l2sqrt

√9d2

ij − l2sqrt

(85)

using lsqrt and lsum according to (26) and (27).

REFERENCES

[1] S. Hirose and M. Mori, “Biologically inspired snake-like robots,” in Proc.IEEE Int. Conf. Robot. Biomimetics, 2004, pp. 1–7.

[2] G. Chirikjian and J. Burdick, “Kinematically optimal hyper-redundantmanipulator configurations,” IEEE Trans. Robot. Autom., vol. 11, no. 6,pp. 794–806, Dec. 1995.

[3] M. W. Hannan and I. D. Walker, “Kinematics and the implementationof an elephant’s trunk manipulator and other continuum style robots,” J.Robot. Syst., vol. 20, pp. 45–63, 2003.

[4] R. Cieslak and A. Morecki, “Elephant trunk type elastic manipulator-a toolfor bulk and liquid materials transportation,” Robotica, vol. 17, pp. 11–16,1999.

[5] B. A. Jones and I. D. Walker, “Practical kinematics for real-time im-plementation of continuum robots,” IEEE Trans. Robot., vol. 22, no. 6,pp. 1087–1099, Dec. 2006.

[6] I. S. Godage, E. Guglielmino, D. T. Branson, G. A. Medrano-Cerda, andD. G. Caldwell, “Novel modal approach for kinematics of multisectioncontinuum arms,” in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 2011,pp. 1093–1098.

[7] P. Dupont, J. Lock, B. Itkowitz, and E. Butler, “Design and control ofconcentric-tube robots,” IEEE Trans. Robot., vol. 26, no. 2, pp. 209–225,Apr. 2010.

[8] D. C. Rucker, R. J. Webster, G. S. Chirikjian, and N. J. Cowan, “Equilib-rium conformations of concentric-tube continuum robots,” Int. J. Robot.Res., vol. 29, no. 10, pp. 1263–1280, Sep. 2010.

[9] B. Jones and I. Walker, “Kinematics for multisection continuum robots,”IEEE Trans. Robot., vol. 22, no. 1, pp. 43–55, Feb. 2006.

[10] D. Lane, J. Davies, G. Casalino, G. Bartolini, G. Cannata, G. Veruggio,M. Canals, C. Smith, D. O’Brien, M. Pickett, G. Robinson, D. Jones,E. Scott, A. Ferrara, D. Angelleti, M. Coccoli, R. Bono, P. Virgili, R. Pallas,and E. Gracia, “Amadeus: Advanced manipulation for deep underwatersampling,” IEEE Robot. Autom. Mag., vol. 4, no. 4, pp. 34–45, Dec.1997.

[11] K. Suzumori, S. Endo, T. Kanda, N. Kato, and H. Suzuki, “A bendingpneumatic rubber actuator realizing soft-bodied manta swimming robot,”in Proc. IEEE Int. Conf. Robot. Autom., 2007, pp. 4975–4980.

[12] T. Mahl, A. Hildebrandt, and O. Sawodny, “Forward kinematics of acompliant pneumatically actuated redundant manipulator,” in Proc. IEEE7th Conf. Ind. Electron. Appl., 2012, pp. 1267–1273.

[13] Y. Bailly and Y. Amirat, “Modeling and control of a hybrid continuumactive catheter for aortic aneurysm treatment,” in Proc. IEEE Int. Conf.Robot. Autom., 2005, pp. 924–929.

[14] S. Hirose, Biologically Inspired Robots, Snake-Like Locomotors and Ma-nipulators. London, U.K.: Oxford Univ. Press, 1993.

[15] R. J. Webster and B. A. Jones, “Design and kinematic modeling of con-stant curvature continuum robots: A review,” Int. J. Robot. Res., vol. 29,no. 13, pp. 1661–1683, 2010.

[16] I. Walker, “Robot strings: Long, thin continuum robots,” in Proc. IEEEAerosp. Conf., 2013, pp. 1–12.

[17] I. D. Walker, “Continuous backbone continuum robot manipulators,”ISRN Robot., vol. 2013, pp. 1–40, 2013.

[18] Geschaftsstelle Deutscher Zukunftspreis. (2010). German futureaward: Laureates 2010. [Online]. Available: http://www.deutscher-zukunftspreis.de/en/preistraeger/laureates-2010

[19] M. Cianchetti, F. Renda, A. Licofonte, and C. Laschi, “Sensorization ofcontinuum soft robots for reconstructing their spatial configuration,” inProc. IEEE RAS EMBS Int. Conf. Biomed. Robot. Biomechatronics, 2012,pp. 634–639.

[20] J. Zhang and N. Simaan, “Design of underactuated steerable electrodearrays for optimal insertions,” J. Mechanisms Robot., vol. 5, pp. 011 008-1–011 008-11, 2013.

[21] F. Renda and C. Laschi, “A general mechanical model for tendon-drivencontinuum manipulators,” in Proc. IEEE Int. Conf. Robot. Autom., 2012,pp. 3813–3818.

[22] D. C. Rucker, B. A. Jones, and R. J. Webster III, “A geometrically exactmodel for externally loaded concentric-tube continuum robots,” IEEETrans. Robot., vol. 26, no. 5, pp. 769–780, Oct. 2010.

[23] D. C. Rucker and R. J. Webster III, “Statics and dynamics of continuumrobots with general tendon routing and external loading,” IEEE Trans.Robot., vol. 27, no. 6, pp. 1033–1044, Dec. 2011.

[24] D. Trivedi, A. Lotfi, and C. Rahn, “Geometrically exact models for softrobotic manipulators,” IEEE Trans. Robot., vol. 24, no. 4, pp. 773–780,Aug. 2008.

[25] I. Tunay, “Spatial continuum models of rods undergoing large deformationand inflation,” IEEE Trans. Robot., vol. 29, no. 2, pp. 297–307, Apr. 2013.

[26] K. Xu and N. Simaan, “Analytic formulation for kinematics, statics, andshape restoration of multibackbone continuum robots via elliptic inte-grals,” J. Mechanisms Robot., vol. 2, pp. 011 006-1–011 006-13, 2009.


[27] J. Burgner, D. Rucker, H. Gilbert, P. Swaney, P. Russell, K. Weaver, andR. Webster, “A telerobotic system for transnasal surgery,” IEEE/ASMETrans. Mechatronics, vol. 19, pp. 996–1006, 2014.

[28] B. Jones, R. Gray, and K. Turlapati, “Three dimensional statics for con-tinuum robotics,” in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 2009,pp. 2659–2664.

[29] G. Chirikjian and J. Burdick, “A modal approach to hyper-redundant ma-nipulator kinematics,” IEEE Trans. Robot. Autom., vol. 10, no. 3, pp. 343–354, Jun. 1994.

[30] F. Fahimi, H. Ashrafiuon, and C. Nataraj, “An improved inverse kinematicand velocity solution for spatial hyper-redundant robots,” IEEE Trans.Robot. Autom., vol. 18, no. 1, pp. 103–107, Feb. 2002.

[31] I. Gravagne and I. Walker, “Kinematics for constrained continuum robotsusing wavelet decomposition,” in Proc. Robotics, 2000, pp. 292–298.

[32] P. Sears and P. Dupont, “A steerable needle technology using curvedconcentric tubes,” in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 2006,pp. 2850–2856.

[33] M. Rolf and J. Steil, “Constant curvature continuum kinematics as fastapproximate model for the bionic handling assistant,” in Proc. IEEE/RSJInt. Conf. Intell. Robots Syst., 2012, pp. 3440–3446.

[34] T. Mahl, A. E. Mayer, A. Hildebrandt, and O. Sawodny, “A variable curva-ture modeling approach for kinematic control of continuum manipulators,”in Proc. Am. Control Conf., 2013, pp. 4952–4957.

[35] B. Siciliano, L. Sciavicco, L. Villani, and G. Oriolo, Robotics: Modelling,Planning and Control. London, U.K.: Springer-Verlag, 2009.

[36] T. F. Chan and R. Dubey, “A weighted least-norm solution based schemefor avoiding joint limits for redundant joint manipulators,” IEEE Trans.Robot. Autom., vol. 11, no. 2, pp. 286–292, Apr. 1995.

[37] W. Rone and P. Ben-Tzvi, “Continuum robot dynamics utilizing the prin-ciple of virtual power,” IEEE Trans. Robot., no. 30, no. 1, pp. 275–287,Feb. 2014.

[38] A. Grzesiak, R. Becker, and A. Verl, “The bionic handling assistant: Asuccess story of additive manufacturing,” Assembly Autom., vol. 31, no. 4,pp. 329–333, 2011.

Tobias Mahl received the Dipl.-Ing. degree in engi-neering cybernetics from the University of Stuttgart,Germany, in 2007, where he has since been workingtoward the Ph.D. degree with the Institute for SystemDynamics.

He has also been engaged as a Research Engineerwith the Institute for System Dynamics. His currentresearch interests include modeling, identification,and control of mechatronic systems and industrialapplications of system engineering.

Alexander Hildebrandt received the Dipl.-Ing. de-gree in electrical engineering from Ulm University,Germany, in 2002. He received the Ph.D. degree fromthe University of Stuttgart, Germany, in 2009.

Since 2006, he has been an Engineer of automationand control with the Research Department, Festo AG& Co. KG, Esslingen, Germany. He was a ResearchAssociate with the University of Ilmenau, Germanyand the University of Stuttgart, Germany. His keyactivities are the control and dimensioning of pneu-matic, electrical and biomechanical drives.

Oliver Sawodny received the Dipl.-Ing. degree inelectrical engineering from the University of Karl-sruhe, Germany, in 1991, and the Ph.D. degree fromthe University of Ulm, Germany, in 1996.

In 2002, he became a Full Professor at the Tech-nical University of Ilmenau, Germany. Since 2005,he has been the Director of the Institute for SystemDynamics, University of Stuttgart, Germany. His cur-rent research interests include methods of differentialgeometry, trajectory generation, and applications tomechatronic systems.

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