Reconfigurable Parallel Continuum Robots for Incisionless Surgery

Arthur W. Mahoney, Member, IEEE, Patrick L. Anderson, Student Member, IEEE,Philip J. Swaney Student Member, IEEE, Fabien Maldonado, and Robert J. Webster III Senior Member, IEEE

Abstract— We propose a new class of robotic device forminimally-invasive surgery that lies at the intersection of con-tinuum, parallel, and reconfigurable robotics. This ContinuumReconfigurable Incisionless Surgical Parallel (CRISP) paradigminvolves the use of multiple needle-diameter devices insertedthrough the skin and assembled into parallel structures insidethe body. The parallel structure can be reconfigured inside thepatient’s body to satisfy changing task requirements such asreaching initially inaccessible locations or modifying mechan-ical stiffness for manipulation or palpation. Another potentialadvantage of the CRISP concept is that many small (needle-sized) entry points into the patient may be preferable in termsof both patient healing and cosmesis to the single (or multiple)larger ports needed to admit current surgical robots. Thispaper presents a mechanics-based model for CRISP forwardand inverse kinematics, along with experimental validation.

I. INTRODUCTION

Continuum robots are tentacle-like devices with the abilityto perform manipulation in confined workspaces reachablethrough narrow, tortuous pathways [1], [2]. Motivated byapplications in exploration and minimally-invasive medicine,researchers have developed a variety of continuum devicesactuated by tendons [3], backbones [4], concentric tubes [5],[6], and pneumatics [7]. These robots typically consist ofserially connected curved sections [2].

When designing continuum robots for manipulation tasksthrough narrow pathways, a tradeoff is often made betweenthe task’s geometric and mechanical requirements. For exam-ple, the task’s geometry may require a long, thin robot to passthrough a small opening, but the mechanics of such a robottypically precludes the application of large tip forces. Thismotivated the development of continuum manipulators withelastic elements arranged in a parallel architecture. Simaanet al. originally suggested the use of parallel combinationsof elastic members to form sections of a continuum robot[4]. Xu and Simaan showed how robots of this type can beused for intrinsic force sensing [8]. Recently this concept hasbeen generalized to use backbones that are not constrainedbetween the disks at the end of each section, and insteadfollow general paths [9]. This creates a flexible parallel robot

This material is based upon work supported by the National Institutes ofHealth under awards R21 EB017952 and R01 EB017467, and the NationalScience Foundation under award IIS-1054331. Any opinions, findings, andconclusions or recommendations expressed in this material are those of theauthors and do not necessarily reflect the views of the NIH or the NSF.

A. W. Mahoney, P. L. Anderson, P. J. Swaney, and R. J. Web-ster III are with the Department of Mechanical Engineering at Van-derbilt University, Nashville, TN 37235, USA (e-mail: {art.mahoney,robert.webster}@vanderbilt.edu). F. Maldonado is with the Vanderbilt Uni-versity Medical Center, Nashville, TN 37232

ParallelContinuum(a)

snarewire

snareneedle

tools

CRISP

Robot

(b)

Fig. 1: (a) We propose a new surgical robot concept called aContinuum Reconfigurable Incisionless Surgical Parallel (CRISP)robot. (b) The working end of a CRISP robot for minimally-invasive surgery that can be assembled with two flexible forcepsfor manipulation, a thin flexible endoscope for visualization, andthree snare needles that can grasp and assist in manipulating theforceps and scope with the ability to reconfigure.

that is in some ways analogous to the rigid-link Stewart-Gough platform [10]. Parallel continuum manipulators canapply greater forces than their serial continuum counterpartswith the ability to maneuver around obstacles [9].

For most tasks, a continuum robot will only be requiredto intermittently apply large tip forces. The required forcesalso typically vary in direction and magnitude. Any fixed (i.e.non-reconfigurable) design must compromise with respectto device diameter, stiffness, and workspace. Indeed, task-based design problems in general involve finding the optimaltradeoff of these properties, but all cannot be optimizedsimultaneously. This has motivated the development of self-reconfigurable robotic systems that alter their physical mor-phology to adapt to new circumstances or changing taskrequirements [11], [12]. Reconfigurability removes the needto compromise on performance during the design process–one robot can assemble itself into multiple designs.

This paper describes a new class of robot system con-sisting of elastic elements that form closed kinematic chainswith reconfigurable morphology. These robots lie at the in-tersection of continuum, parallel, and reconfigurable robotics(Fig. 1(a)). We refer to this as the Continuum ReconfigurableIncisionless Surgical Parallel (CRISP) concept. In minimally-invasive medicine, a CRISP robot can be assembled intoparallel structures inside the human body. The robot consistsof several needle-diameter flexible tools, as well as a set ofhollow needles through which wire snare loops are deployed(Fig. 1(b)). The system is assembled inside the patientby snaring the flexible tools with the wire loops, forming

2016 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)Daejeon Convention CenterOctober 9-14, 2016, Daejeon, Korea

978-1-5090-3761-2/16/$31.00 ©2016 IEEE 4330

(b)

parallel structuresnare needle

snare

(a)

snare needle

manipulator

Fig. 2: (a) CRISP robots assemble parallel structures inside apatient’s body that can be (b) controlled outside the body usingrobot manipulators.

(a) (b) (c)

Fig. 3: CRISP robots are assembled by (a) percutaneously insertinga needle and deploying a snare, (b) navigating the flexible toolthrough the open snare loop, (c) and tensioning the snare. Parallelstructures that consist of multiple tools and snare needles can beassembled inside the body this way.

a parallel structure (Fig. 3). The parallel structure can beactuated by manipulating the snare needles and tool outsidethe patient’s body using robot manipulators. Fig. 1 shows theworking end of a CRISP system consisting of three snareneedles that control two flexible forceps for manipulationand a flexible endoscope (in this case a 2.5mm diameterureteroscope) for visualization.

Key aspects of this concept are that the flexible tooland snare needles are inserted percutaneously, which re-duces invasiveness (no sutures are required for diameters< 3mm [13]). Additionally, parallel structures are stifferfor manipulation, and additional needles increase stiffnesswithout increasing invasiveness. The parallel structure canbe reconfigured inside the patient to satisfy changing taskrequirements by regrasping the flexible tool in differentlocations and morphological arrangements.

II. MOTIVATING CLINICAL EXAMPLES

There are a variety of clinical scenarios where the recon-figurability, stiffness modulation, and small diameter entry

points of CRISP systems are desirable. One promising ap-plication is to enable physicians in closed-chest proceduresto access the entirety of the lung cavity while maintainingdexterity and adapting stiffness as needed for palpationand tissue manipulation. New approaches to lung cancerdiagnosis and treatment are urgently needed because of bothits prevalence and high mortality rate (150,000 deaths in theUnited States each year [14]), as well as how challenginglung cancer is to diagnose and treat with current instrumen-tation. For example, it is extremely challenging to localizesubsurface tumors [15]. While it is possible to use continuummanipulators with intrinsic force sensing to palpate tissue tolocalize tumors (e.g., [8], [16], [17]), an inherent tradeoffexists. To maximize signal to noise in intrinsic force sensing,low stiffness is desirable. However, high stiffness is desirableto apply forces to tissue during interventional procedures. Inclosed-chest lung procedures the reconfigurability of CRISPmanipulators enables the robot to be stiff when necessaryand compliant when advantageous.

CRISP robots may also be useful in the abdominal pro-cedures. In the past several years, surgeons have developeda number of “micro-laparoscopic” or “needlescopic” proce-dures [18]. The objective of these is to use small diameter(< 3mm) tools to reduce postoperative pain, recovery time,and scarring. Ports for tools of this size do not require sutureclosure after the procedure. However, current needlescopictechniques are challenging for surgeons to perform due tothe inherent flexibility of tools of this diameter [19]. CRISProbots offer the potential to enhance stiffness in needlescopictechniques without requiring larger port diameters.

Other clinical applications where the CRISP approachmay be beneficial include fetal surgery, where small toolsare needed due to the size of the anatomy and to reducethe (currently significant) risk of complications like fetalmembrane rupture [20]. Similarly in neonatal surgery, smalldiameter instruments are preferable to adult-sized tools [21].Both applications would benefit from robotic dexterity [21].

III. KINEMATICS OF A CRISP SYSTEM

We model the flexible tool and the snare needles using theCosserat rod model as described in the appendix. The stateof the system x is constructed by packing the rod states ofeach element of the parallel structure into a single vector

x =[xt x1 . . . xn

](1)

where xt is the Cosserat-rod states of the flexible tool (i.e.,backbone position and orientation, and internal force andmoment), x1 . . .xn are the Cosserat-rod states of the snareneedles. (Note that x is a column-vector but we express itin the form of (1) for compactness.)

The state vector x(s) is a function of the arc-lengthparameter s, which is defined so that s = 0 is at the proximalend of the flexible tool and s = `t at the distal end of theflexible tool (`t and `i are the lengths of tool and ith snare-needle, respectively). The tool and snare needle Cosserat-rodstates are packed into x so that the corresponding physical

4331

x1 x2xt

l1 l2

s1s2

rt r1 r2

lt

d1 d2dt

=RCM points

(b) Free grasp DOFs

snare

(a)

freeDOFs

Fig. 4: (a) A mechanics-based model predicts the arc-length statesof the flexible tool xt and snare needles, x1 and x2. The snaresgrasp the tool at the tool’s arc-length s1 and s2. Remote centers ofmotion (RCM) of the tool and each snare is enforced at the RCMpoints rt, r1, and r2. (b) The unconstrained degrees of freedompermitted by the snare-grasp constraints expressed in (7).

location on the tool and snares at arc-length s is located ata distance of `t − s from the flexible tool’s distal end.

The propagation of the state vector in arc-length s isgoverned by the arc-length derivative vector

x′ =[x′t x′1 . . . x′n

]= f(x, s) (2)

The arc-length derivatives of the flexible tool and the ith snareneedle rod states are defined piecewise in arc-length as

x′t(s) =

{[p′t q

′t m

′t +α n′t + β] , if 0 ≤ s ≤ `t

0, otherwise(3)

x′i(s) =

{[p′i q

′i m

′i n

′i] , if si − `i ≤ s ≤ si

0, otherwise, (4)

and are given by the Cosserat-rod equations (13), where siis the grasp location of the ith snare (Fig. 4(a)).

The terms α(s) ∈ R3 and β(s) ∈ R3 propagate the pointloads applied to the tool by the snares toward the tool’s distalend and are defined in arc-length as

α(s) =n∑

i=1

(I −AiA†i )miδ(si − s) (5)

β(s) =n∑

i=1

niδ(si − s) (6)

where Ai(s) =[p′t(s) p′i(s)

], δ(s) is the Dirac delta

function, and † denotes the Moore-Penrose pseudoinverse.We model the snare grasps as unable to support momentsabout the tool and needle shaft directions (p′t and p′i, i.e. the“Free DOFs” illustrated in Fig. 4). Equation (5) ensures thatonly the moment applied by each needle about the directionsperpendicular to the tool and needle shafts is propagatedtoward the tool’s distal end. Equation (6) propagates the pointforces applied to the tool by the snares toward the tool’s distalend. It is worth noting that these are assumptions. The freemotion about the needle shaft is a good assumption, sincethe snare wire is free to move within the needle shaft. Thefree motion about the tool shaft is a good assumption if thecoefficient of friction between the snare and the tool is low,but it is certainly possible to use materials that have higherfrictional interaction for these components.

0 0.05 0.1 0.15 0.2 0.250

0.05

0.1

0.15

0.2

0.3x (m)

y (m

)

4

n, x (N)

n, y

(N

)

(a) circle trajectory (b) proximal x and y load, n(s0)

1

2

34

1

23

4

23

1

-0.1 -0.05 0 0.05 0.1

0

-0.1

-0.05

0.05

0.1

1

2

3

4

1

2

3

4

�exibletool

snareneedle

RCMRCM

snareneedle

�exibletool

Fig. 5: (a) The end-effector of a simulated single-snare CRISP robotfollows a 2-cm circle trajectory in the x-y plane while performingremote-center motion. A curve showing the proximal position ofthe flexible tool and snare needle as the end-effector follows thecircular trajectory is shown, with four configurations shown. (b)The proximal tension cycle of the flexible tool and snare needle,which only lies in the x-y plane, required for the tool’s end-effectorto follow the circle trajectory.

A. Geometric and Wrench Constraints

Grasping the flexible tool with the ith snare at arc-lengthlocation si creates a state constraint that relates the com-ponents of the flexible tool state xt(si) to the componentsof the snare needle’s state xi(si). We approximate thegeometric grasp interaction as a position constraint thatenforces the tip of the needle to be coincident with the tooland a constraint that enforces the needle and tool shafts tobe orthogonal. We assume the snare grasp cannot supportmoments in the p′t and p′i directions, which is enforced bytwo constraints on the snare needle’s moment mi.

The geometric and wrench constraints can be representedfor each of the n needles at arc-length si as

ci(si) =

pt − pi tip coincidencep′t · p′i shaft orthogonalityp′i ·mi needle shaft momentp′t ·mi tool shaft moment

= 0 (7)

We assume that the system is quasistatic which leads toconstraints that enforce the force and moment at the tool’sdistal end to be balanced with any tip applied force F ormoment T , represented by the tip constraint

ct(`t) =

[mt − Tnt − F

]= 0. (8)

The n grasp constraints ci and the tip constraint ct canbe packed into the combined constraint vector

c =[ct c1 . . . cn

]= 0. (9)

B. Body Wall Fulcrum Constraints

Surgical robots that penetrate the skin perform remote-center motion (RCM) around a fulcrum, located at the bodywall, that prevents the robot from pulling the patient’s skinby minimizing the entry point’s spatial motion [22].

An element of the CRISP system can perform remote-center motion around a virtual center. A member of the

4332

8765

4321

Fig. 6: Eight possible morphologies of a CRISP robot consisting of one flexible tool and one to three snare needles. The tool and snaresare highlighted in white for visibility.

parallel structure passes through an RCM point r if thereexists an arc-length d on its body such that the constraint

crcm = p(d)− r = 0 (10)

is satisfied as the CRISP robot moves.We denote the RCM points of the tool and snares to be

rt and r1, . . . , rn, respectively, with the arc-length positionwhere the tool and snares intersect their RCM points denotedas dt and d1, . . . , dn. The RCM constraint (10) can beincorporated into the kinematics framework for the flexibletool and each of the snare needles by augmenting the systemstate x with the scalar arc-lengths dt and d1, . . . , dn, andaugmenting (9) with a constraint of the form (10) for thetool and each of the snare needles.

C. Forward and Inverse Kinematics

The forward kinematics of the parallel structure can befound by piecewise integrating (2) from arc-length s0 =min{0, s1−`1, . . . , sn−`n} to arc-length `t with initial con-ditions x(s0) = x0. The initial condition vector x0 is packedwith a vector of inputs u0, which are the initial positions andorientations of the tool and snare needles’ proximal ends. Theinitial conditions are also packed with a vector v0 containingthe initial internal moments and forces of the tool and snareneedles at their proximal ends, which are unknown a prioriand must be solved for in order to satisfy the constraints(9). Given the known inputs u0, we solved the forwardkinematics using a numerical optimization routine that findsthe unknowns v0 that minimize ‖c‖.

The inverse kinematics can be computed by augmentingthe constraint vector (9) with a constraint on the tool’s distalpose at arc-length `t, of the form

cinv(`t) =

[pt − P

log(qtQ−1)

]= 0 (11)

where P and Q denote the desired position and orientation,respectively. In this formulation of the inverse kinematics, thevector v0 of unknowns includes the proximal pose of the tooland snare needles along with the proximal internal momentsand forces. The same numerical optimization method can beused to solve the inverse and forward kinematics.

Fig. 5(a) shows the distal end of a hypothetical single-snare robot, with the tool and snare constructed out ofNitinol tubing with properties identical to the tool reportedin Table I, following a circular trajectory in the x-y planewith its heading pointing in the x direction and with RCMconstraints illustrated. The necessary proximal tool and snareneedle poses are computed by solving the inverse kinematicsusing the constraint (11). The tool and snare needle work inconcert to position and orient the end-effector as shown. Inall configurations, the tool and snare needles are antagonisticin that their proximal loads, which lie in the x-y plane,balance one another (Fig. 5(b)).

As noted in prior literature, the forward and inverse kine-matics of elastic parallel and serial continuum manipulatorsmay have more than one solution [9], where multiple vectorsv0 satisfy the constraints (9). This can occur in “buckled”configurations where multiple static equilibrium solutionsexist that locally minimize the system’s elastic potentialenergy. In this case, each of the buckled configurations canbe found from the kinematic equations by appropriatelyselecting v0. Elastic instability has been observed in othercontinuum devices, including concentric tube robots [23],[24] and has been explored for a robotic manipulation systemthat holds Kirchhoff elastic rods on both ends [25].

IV. RECONFIGURING A CRISP ROBOT

Reconfiguration can be used to change the properties ofa CRISP robot to satisfy changing application requirements;this distinguishes the system from other types of parallelcontinuum robot devices. Here our system shares several keychallenges with reconfigurable and parallel robot systems,notably the challenge of determining the optimal configura-tion/design for a given task.

The system can be configured into a variety of morpholo-gies in which the complexity increases with the numberof flexible tools and snares. Fig. 6 shows some possiblearrangements with one tool and from one to three snareneedles. There are many possible morphologies whose utilityvaries depending on the task. A morphology like “5” couldbe used to control the flexible tool’s body along with itstip, while a morphology like that of “6”, where one snare

4333

TABLE I: Parameters of the CRISP device.

Tool Snare 1 Snare 2Outer Diameter (mm) 1.02 3.00 3.00Inner Diameter (mm) 0.84 2.30 2.30

Length (mm) 475 157 153Young’s Modulus (GPa) 50 180 180

Poisson’s Ratio 0.33 0.305 0.305Grasp-Point Location (mm) – 207 434

grasps another, could be used to decrease a snare needle’scompliance or exploit mechanical advantage.

It may be tempting to reduce the compliance by regraspingthe flexible tool with needles that are much stiffer than thetool. However, using snare needles that are much stiffer thanthe flexible tool they manipulate reduces the system’s abilityto control tip orientation independently of the tip position.This is because the tool and snare needles work in concertto position and orient the tip, i.e., the tool bends the snareneedles and vice versa as shown in the example of Fig. 5(a).When the snare needles are much stiffer than the tool, thenthe pose of the distal-most snare dominates the pose of thetip. This effect can be observed in the Jacobian’s conditionnumber. We anticipate that the compliance of a snared-toolsystem can be decreased without affecting the ability tocontrol the tip pose by increasing the number (but not thestiffness) of snare needles.

V. EXPERIMENTAL RESULTS

We performed experiments to verify the kinematic modeldescribed in Sec. III by creating a CRISP device with twostainless steel snare needles and a superelastic Nitinol tubefor the tool. Table I lists the system’s parameters. The snareswere constructed out of 1.52mm wide, 0.28mm thick su-perelastic Nitinol strip, and were tightened by hand.

A Northern Digital Inc. Aurora tabletop electromagnetictracking system with a hand-held measurement probe wasused to localize the base positions and orientations of theflexible tool and the snare needles. We measured the grasp-point location of each snare on the tool by hand. Ground-truth measurements of the tool’s backbone position weretaken by manually sliding a 0.3mm diameter electromagneticsensor through the tool with the device placed in five config-urations and comparing it to the backbone position predictedby the model of Sec. III. The ground-truth measurements areplotted on top of the predicted backbone position for all fiveconfigurations in Fig. 7.

Fig. 7(a) shows an example of a snare needle and toolproximal configuration where the system’s kinematic equa-tions have more than one solution as described in Sec. III-C.In this example, there are two solutions which are shown.The mechanics-based model can predict the snare-systemstates, even if there are multiple solutions, by appropriatelyselecting the initial moment and load (which are unique)at the proximal ends of the tool and snare needles. Ournumerical method was able to find both solutions by startingfrom two different initial values of the vector v0. The two

solutions are plotted alongside their corresponding ground-truth measurements in the front and side views of Fig. 7(a).

Fig. 8 shows the ground-truth error between the predictedtool backbone curve and the raw data obtained with the elec-tromagnetic tracker for each of the configurations presentedin Fig. 7. The error e(s) was computed at arc-length s as

e(s) = mink‖pt(s)− p∗k‖ (12)

where pt is the predicted backbone curve and p∗k is a rawdata point indexed by integer k = 1, . . . , N and N is thenumber of gathered data points.

The average error for each configuration is shown in thelegend of Fig. 8. The average error for all configurationsexcept (d) was less than 4mm. In the case of configuration(d), the average error was 10.3mm, which we expect wascaused by unmodeled static friction at the interface of thesnare needles and flexible tool. Static friction was dominatedby the flexible tool’s internal moments and forces in the otherconfigurations. When scaled by the shortest snare’s length,the normalized error of configuration (d) was 6.7%.

Fig. 8 shows that the flexible tool’s backbone error de-creases near arc-lengths where the snare needles grasp thetool. This is a result of the snare needle’s stiffness preventingit from deflecting under the loads applied by the flexibletool. The model predicts the deflection, but using stifferneedles reduces uncertainty. This can be exploited by placingmultiple grasp points along the flexible tool’s body, as Fig. 8illustrates, to reduce the uncertainty of the predicted toolbackbone curve by preventing errors at the proximal endfrom propagating down the tool’s body. CRISP robots sharethis property with continuum [9] and rigid-link parallelrobots, whose structure prevents error in the individual jointsfrom producing an amplified error at its end-effector [10].

VI. DISCUSSION AND OPEN RESEARCH CHALLENGES

The CRISP robot concept is a novel approach tominimally-invasive medicine that reduces invasiveness byincreasing the number of instruments that enter the body (incontrast to single-port approaches [26]) while decreasing thediameter of each instrument down to that of a needle, requir-ing no incisions. These systems are excellent candidates fora robotic approach since the necessary motion of the snareneedles and flexible tool outside the body is nonintuitive,particularly in the presence of body-wall fulcrum (i.e., RCM)constraints. This is demonstrated by the example trajectoryof Fig. 5. Here, we discuss some of the many open design,planning, and control challenges relating to the system; inparticular, we focus on those at the intersection of continuum,parallel, and reconfigurable robotics.

Reconfiguration is a unique aspect of this concept thatmakes it particularly versatile as a robotic system. Anopen problem that CRISP robots share with reconfigurablerobots is that of planning when and how to reconfigureas task requirements change. Planning for reconfigurablerobots includes both planning for individual components andplanning the connectivity of the reconfigurable parts [12].In the context of CRISP robots in the operating room, the

4334

(a)front view (simulated) side view (simulated)

(c)

(b) (d)

(e)

front view (actual)

front view (actual) front view (actual)front view (simulated) front view (simulated)

1

2

1

2

Fig. 7: Five configurations compare the ground-truth magnetic tracker measurements of the tool’s backbone position to that predicted bythe model of Sec. III. Configuration (a) demonstrates the potential for multiple tool configurations for the same inputs, both found by themodel. In all configurations, a photo of the experimental setup is shown on the left with the simulation results and overlaid tracker dataon the right. The tool and the snares are highlighted in white for visibility.

Tool Arc-length (mm)0 100 200 300 400 500

Err

or (

mm

)

0

5

10

15

20

25

Config (b), Avg: 2.46 mm

Config (c), Avg: 3.15 mm Config (d), Avg: 10.3 mmConfig (e), Avg: 2.93 mm

Config (a)-1, Avg: 3.68 mmConfig (a)-2, Avg: 2.20 mm

Snare 1 grasp point

Snare 2 grasp point

Fig. 8: Backbone error (12) measured between the mechanics-basedmodel and the raw tracker data for each configuration shown inFig. 7. Error is shown as a function of the flexible tool’s backbonearc-length, with average backbone error in the legend. The arc-length locations (Table I) where the snares grasp the tool are shown.

planning problem includes both planning the motion of thesystem inside the body while incorporating anatomical andsafety constraints, and determining what configurations thesystem should form itself into based on the surgeon’s needs(e.g., what forces are needed at the tool tip). The ability ofthe parallel structure to reconfigure from stiff to compliantconfigurations could be important for surgical applicationsthat require localizing tumors via palpation (e.g., in the lungor liver). Further explorations of the effect of reconfigurationon stiffness and manipulability are necessary.

The planning problem is highly related to the problemof designing a system in that the choices made duringthe design process (e.g., how many snare needles to use

and the relative stiffnesses of the tool and snares) directlyaffects the configurations it can achieve. When designingserial continuum robots for manipulation in constrainedenvironments, the geometry of the application guides theselection of the continuum robot’s shape and mechanicaldesign [27]. We expect that similar principles can be appliedto CRISP robots to guide the selection of the number ofsnare needles, where they should be inserted into the patient’sbody, and even what their shape should be. Naturally, theproblem of planning paths that are free of both collisionswith the environment and self-collisions between snare-needle manipulators becomes increasingly difficult as snareneedles are added to the system.

As with rigid-link and continuum parallel robots, deter-mining the workspace of a CRISP robot is challenging[10]. Its workspace depends on the configurations that therobot can achieve and the elasticity of its members. Onecontributing factor to the limits of the workspace is thepossibility of buckling in extreme configurations. Recentwork in continuum robotics shows that elastic instabilitycan be understood [23], [24], actively avoided by planning[28], and can be eliminated through design [29], [30]. Weanticipate that the much of this work for continuum robotswill also be applicable to CRISP robots and can be used toinform the design and reconfiguration planning processes.

VII. CONCLUSION

We have introduced a new class of robotic system, theCRISP robot, that lies at the intersection of parallel, con-tinuum, and reconfigurable robotics. These robots have thepotential to perform minimally invasive surgery without any

4335

incisions. Using a Cosserat rod-based modeling approach,we derived the forward and inverse kinematics for CRISPdevices. We explored the reconfigurability of these systemsand we conducted experiments verifying our mechanics-based models. The CRISP concept opens up a wealth ofdesign, planning, and control problems for future study.

APPENDIXKINEMATICS OF COSSERAT RODS

We model each member of the parallel structure as anunshearable and inextensible Cosserat rod with a state vectorthat contain states defining its material position p(s) ∈ R3,material orientation represented as a unit quaternion q(s),internal force n(s) ∈ R3, and internal moment m(s) ∈ R3.The states vary as a function of scalar arc-length s along therod’s body, measured from a proximal reference.

The rod’s position, orientation, internal force, and internalmoment propagate in arc-length according to

p′ = qe3q−1 q′ =

1

2qu

m′ = n× p′ − l n′ = −r(13)

where u ∈ R3 is angular rate-of-change of the rod’s bodyreference frame expressed in the body frame, r ∈ R3 andl ∈ R3 are externally applied distributed forces and momentsper unit rod length, and ′ indicates derivative with respect toarc-length s. We assume that r = 0 and l = 0 in this paper.

The internal moment can be related to the angular rate ofchange by a linear constitutive law of the form

m = q [Km(u− u∗)] q−1 (14)

where Km = diag(EI,EI, JG) maps bending and torsionto internal moment, E is the Young’s modulus, G is theshear modulus, I is the second moment of area about thebody e1 and e2 axes, and J is the polar area moment aboutthe body e3 axis. The vector u∗ is the rod’s precurvature inits undeformed state as represented in the rod’s undeformedbody frame. For example, u∗ = 0 for a straight rod.

REFERENCES

[1] J. Burgner-Kahrs, D. C. Rucker, and H. Choset, “Continuum robotsfor medical applications: A survey,” IEEE Transactions on Robotics,vol. 31, no. 6, pp. 1261–1280, 2015.

[2] G. Chirikjian, “Conformational modeling of continuum structures inrobotics and structural biology: A review,” Advanced Robotics, vol. 29,no. 13, pp. 817–829, 2015.

[3] D. Camarillo, C. Milne, C. Carlson, M. Zinn, and J. Salisbury, “Me-chanics modeling of tendon-driven continuum manipulators,” IEEETrans. on Robot., vol. 24, no. 6, pp. 1262–1273, 2008.

[4] N. Simaan, R. Taylor, and P. Flint, “A dexterous system for laryngealsurgery,” in Proc. IEEE Int. Conf. on Robot. Autom., 2004, pp. 351–357.

[5] P. E. Dupont, J. Lock, B. Itkowitz, and E. Butler, “Design and controlof concentric-tube robots,” IEEE Trans. on Robot., vol. 26, no. 2, pp.209–225, 2010.

[6] D. C. Rucker, B. A. Jones, and R. J. Webster III, “A geometricallyexact model for externally loaded concentric-tube continuum robots,”IEEE Trans. on Robot., vol. 26, no. 5, pp. 769–780, 2010.

[7] W. McMahan, V. Chitrakaran, M. Csencsits, D. Dawson, I. Walker,B. Jones, M. Pritts, D. Dienno, M. Grissom, and C. Rahn, “Field trialsand testing of the octarm continuum manipulator,” in Proc. IEEE Int.Conf. on Robot. Autom., 2006, pp. 2336–2341.

[8] K. Xu and N. Simaan, “An investigation of the intrinsic force sensingcapabilities of continuum robots,” IEEE Trans. on Robot., vol. 24,no. 3, pp. 576–587, June 2008.

[9] C. E. Bryson and D. C. Rucker, “Toward parallel continuum manipu-lators,” in Proc. IEEE Int. Conf. on Robot. Autom., 2014, pp. 778–785.

[10] J.-P. P. Merlet, Parallel Robots. Norwell, MA, USA: KluwerAcademic Publishers, 2000.

[11] M. Yim, P. White, M. Park, and J. Sastra, Encyclopedia of Complexityand Systems Science. New York, NY: Springer New York, 2009, ch.Modular Self-Reconfigurable Robots, pp. 5618–5631.

[12] M. Yim, W.-M. Shen, B. Salemi, D. Rus, M. Moll, H. Lipson,E. Klavins, and G. Chirikjian, “Modular self-reconfigurable robotsystems [grand challenges of robotics],” IEEE Robotics AutomationMagazine, vol. 14, no. 1, pp. 43–52, 2007.

[13] F. Ghezzi, A. Cromi, G. Siesto, L. Boni, S. Uccella, V. Bergamini,and P. Bolis, “Needlescopic hysterectomy: incorporation of 3-mminstruments in total laparoscopic hysterectomy,” Surgical Endoscopy,vol. 22, no. 10, pp. 2153–2157, 2008.

[14] American Cancer Society, “Cancer facts & figures 2010,” AmericanCancer Society, Tech. Rep., 2010.

[15] L. Solaini, F. Prusciano, P. Bagioni, F. Francesco, and D. B. Pod-die, “Video-assisted thoracic surgery (VATS) of the lung,” SurgicalEndoscopy, vol. 22, no. 2, pp. 298–310, 2007.

[16] K. Xu and N. Simaan, “Intrinsic wrench estimation and its perfor-mance index for multisegment continuum robots,” IEEE Trans. onRobot., vol. 26, no. 3, pp. 555–561, 2010.

[17] S. Sanan, S. Tully, A. Bajo, N. Simaan, and H. Choset, “Simultaneouscompliance and registration estimation for robotic surgery,” in Proc.Robotics: Science and Systems, 2014.

[18] M. Gagner and A. Garcia-Ruiz, “Technical aspects of minimallyinvasive abdominal surgery performed with needlescopic instruments.”Surgical Laparoscopy Endoscopy & Percutaneous Techniques, vol. 8,no. 3, pp. 171–179, 1998.

[19] D. M. Krpata and T. A. Ponsky, “Needlescopic surgery: what’s in thetoolbox?” Surgical endoscopy, vol. 27, no. 3, pp. 1040–1044, 2013.

[20] J. Deprest, J. Jani, L. Lewi, N. Ochsenbein-Kolble, M. Cannie,E. Done, X. Roubliova, T. Van Mieghem, A. Debeer, F. Debuck et al.,“Fetoscopic surgery: encouraged by clinical experience and boostedby instrument innovation,” in Seminars in fetal and neonatal medicine,vol. 11, no. 6. Elsevier, 2006, pp. 398–412.

[21] A. G. Coran, A. A. Caldamone, N. S. Adzick, T. M. Krummel, J.-M.Laberge, and R. Shamberger, Pediatric Surgery, 7th ed. Philadelphia,PA: Elsevier, 2012.

[22] R. Taylor, J. Funda, B. Eldridge, S. Gomory, K. Gruben, D. LaRose,M. Talamini, L. Kavoussi, and J. Anderson, “A telerobotic assistant forlaparoscopic surgery,” Engineering in Medicine and Biology Magazine,IEEE, vol. 14, no. 3, pp. 279–288, 1995.

[23] H. B. Gilbert, R. J. Hendrick, and R. J. Webster III, “Elastic Stabilityof Concentric Tube Robots: A Stability Measure and Design Test,”IEEE Transactions on Robotics, vol. 32, no. 1, pp. 20–35, 2016.

[24] J. Ha, F. Park, and P. Dupont, “Elastic stability of concentric tuberobots subject to external loads,” IEEE Transactions on BiomedicalEngineering, vol. 63, no. 6, pp. 1116–1128, 2016.

[25] T. Bretl and Z. McCarthy, “Quasi-static manipulation of a kirchhoffelastic rod based on a geometric analysis of equilibrium configura-tions,” The International Journal of Robotics Research, vol. 33, no. 1,pp. 48–68, 2014.

[26] N. Simaan, A. Bajo, A. Reiter, L. Wang, P. Allen, and D. Fowler,“Lessons learned using the insertable robotic effector platform (irep)for single port access surgery,” Journal of Robotic Surgery, vol. 7,no. 3, pp. 235–240, 2013.

[27] C. Bergeles, A. Gosline, N. V. Vasilyev, P. J. Codd, P. J. del Nido, andP. E. Dupont, “Concentric tube robot design and optimization basedon task and anatomical constraints,” IEEE Trans. on Robot., vol. 31,no. 1, pp. 67–84, Feb 2015.

[28] C. Bergeles and P. E. Dupont, “Planning stable paths for concentrictube robots,” in Proc. IEEE/RSJ International Conference on Intelli-gent Robots and Systems, 2013, pp. 3077–3082.

[29] J. Ha, F. C. Park, and P. E. Dupont, “Achieving elastic stability ofconcentric tube robots through optimization of tube precurvature,” inProc. IEEE/RSJ International Conference on Intelligent Robots andSystems, 2014, pp. 864–870.

[30] R. J. Hendrick, H. B. Gilbert, and R. J. Webster III, “Designing snap-free concentric tube robots: A local bifurcation approach,” in Proc.IEEE Int. Conf. on Robot. Autom., 2015, pp. 2256–2263.

4336