Kinematics and the Implementation of an Elephant's Trunk ...robot kinematics. This provides a solid...

Kinematics and theImplementation of an

Elephant’s Trunk Manipulatorand Other Continuum Style

Robots

Michael W. Hannan and Ian D. WalkerDepartment of Electrical and ComputerEngineeringClemson UniversityClemson, South Carolina 29634e-mail: [email protected];[email protected]

Received 7 April 2002; accepted 23 September 2002

Traditionally, robot manipulators have been a simple arrangement of a small number ofserially connected links and actuated joints. Though these manipulators prove to be veryeffective for many tasks, they are not without their limitations, due mainly to their lackof maneuverability or total degrees of freedom. Continuum style (i.e., continuous ‘‘back-bone’’) robots, on the other hand, exhibit a wide range of maneuverability, and can havea large number of degrees of freedom. The motion of continuum style robots is generatedthrough the bending of the robot over a given section; unlike traditional robots where themotion occurs in discrete locations, i.e., joints. The motion of continuum manipulators isoften compared to that of biological manipulators such as trunks and tentacles. Thesecontinuum style robots can achieve motions that could only be obtainable by a conven-tionally designed robot with many more degrees of freedom. In this paper we present adetailed formulation and explanation of a novel kinematic model for continuum stylerobots. The design, construction, and implementation of our continuum style robot calledthe elephant trunk manipulator is presented. Experimental results are then provided toverify the legitimacy of our model when applied to our physical manipulator. We alsoprovide a set of obstacle avoidance experiments that help to exhibit the practical imple-mentation of both our manipulator and our kinematic model. © 2003 Wiley Periodicals, Inc.

1. INTRODUCTION

Over the last several decades the research area of ro-botic manipulators has focused mainly on designsthat resemble the human arm. These conventional ro-bots can be best described as discrete manipulators,1

where their design is based on a small number of ac-tuatable joints that are serially connected by discreterigid links (see Figure 1). Discrete manipulators have-been proven to be very useful and effective for manydifferent tasks, but they are not without their limita-tions. These robots most often have five to seven de-

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Journal of Robotic Systems 20(2), 45–63 (2003) © 2003 Wiley Periodicals, Inc.Published online in Wiley InterScience (www.interscience.wiley.com). • DOI: 10.002/rob.10070

grees of freedom (DOF), thus in a spatial environment they will most often need all of their DOF justto position the end-effector. This design is very effi-cient for open environments, but as constraints areadded to the environment it is possible that the ma-nipulator can not reach its desired end-effector posi-tion. This failure is due to the lack of DOF in the robotto meet both the environmental constraint conditionsand the desired end-effector position requirements.Another drawback of discrete manipulators is thatthey require some specialized device for manipula-tion of an object. Most often this manipulation is doneby attaching a gripper/hand at the end of the robot.Once again this design works well in many cases, butit is possible that a strategy where the sections of amanipulator do the grasping, i.e., whole arm grasp-ing, might be a better solution.

The addition of more DOF for these conventionalmanipulators can further enhance their maneuver-ability and flexibility. We can relate this situation tothe diversity of biological manipulators. As notedearlier, discrete manipulators resemble the humanarm in structure, but if we look at other biological ma-nipulators we see that this is not the only design. Ani-mals such as snakes, elephants, and octopuses canproduce motions from their appendages or body thatallow the effective manipulation of objects, eventhough they are very different in structure comparedto the human arm. Even among these appendages,

i.e., trunks, tentacles, etc., their physical structure canvary, but they all share the trait of having a relativelylarge number of DOF.

Research into manipulators with a large numberof DOF or a high degree of maneuverability hasspawned many different designs. The most populardesign uses some type of ‘‘backbone’’ structure that isactuated by sets of cables.2–6 Another design usesfluid filled tubes for actuation.7,8 There are also otherdesigns that are based on more conventional robotconstruction techniques.9 These types of manipula-tors are commonly referred to as a hyper-redundantmanipulators due to their number of actuatable DOFbeing much larger than the DOF of their intendedworkspace. Due to the vast design possibilities Rob-inson and Davies1 developed three classifications forthe different possible designs. As introduced earlier,the first class which describes conventional manipu-lators is termed discrete robots. As the redundancy/maneuverability of the manipulator increases overthat of discrete manipulators by increasing the num-ber of discrete joints, it moves into the second clas-sification known as serpentine robots. This classifica-tion would include hyper-redundant manipulators.The third classification of robots, known as continuumrobots, does not contain discrete joints and rigid linksas in the previous two classifications. Instead the ma-nipulator bends continuously along its length similarto that of biological trunks and tentacles.

The increased DOF that allows hyper-redundantand continuum robots to become more maneuverablehas the adverse effect of complicating the kinematicsfor these manipulators. Chirikjian and Burdick10–13

proposed a great deal of theory that has laid a foun-dation for the kinematics of hyper-redundant robots.Mochiyama et al.14–16 presented research in the area ofkinematics and the shape correspondence between ahyper-redundant robot and a desired spatial curve.More recently, Gravagne17–19 has presented work inthe kinematics for continuum robots.

This paper focuses on continuum style robots;this includes both continuum robots and the spec-trum of serpentine robots that behave very similarlyto continuum robots. We have developed a con-tinuum style robot which resembles an elephant’strunk, see Figure 2, which is somewhat similar instructure to those developed in refs. 2–5. More impor-tantly, we present here a kinematic model that notonly applies to our robot, but also to most other typesof continuum style robots. Our kinematic approach isunlike the other models presented by researcherssuch as Chirikjian, Mochiyama, and Gravagne, in thatour model utilizes the concept of constant curvature

Figure 1. Discrete manipulator.

46 • Journal of Robotic Systems—2003

sections, and incorporates them through the use ofdifferential geometry into a modified Denavit-Hartenberg procedure to determine the kinematics.The Denavit-Hartenberg procedure is the most com-monly used approach for determining conventionalrobot kinematics. This provides a solid base for un-derstanding and implementing our approach. The re-sulting form of our kinematic model allows the use ofconventionally designed motion planning and re-dundancy resolution techniques. Our kinematicmodel can also be easily implemented on both planarand spatial continuum robots, and we present here

experimental results with our continuum style robotthat verify the viability of our kinematic model andits real time implementation. In essence, our ap-proach allows the complex kinematics of continuumstyle robots to be expressed in such a way that can beeasily understood and implemented by anyone famil-iar with conventional robotic manipulators.

2. CONTINUUM STYLE ROBOTS

First we would like to point out why we choose todistinguish between continuum style and hyper-redundant designs. Hyper-redundant robots are usu-ally defined as having many more kinematically ac-tuatable DOF than the number of DOF of the robot’sworkspace. However, continuum style robots do nothave to fit this definition. Continuum style manipu-lators can exhibit a large number of kinematic DOF,but not all of these DOF are directly actuated. There-fore it is possible to have a continuum robot that isnot technically a hyper-redundant robot. The designdifference between conventional manipulators andcontinuum style manipulators is that the conven-tional manipulator is a serial connection of actuatedjoints. That is, every joint on the robot is actuated, andit is then rigidly connected to the next joint. On theother hand, continuum style manipulators are not de-signed this way. Consider a three degree of freedomsection. The traditional approach would be to actuateall three joints, thus obtaining three actuated DOF.The continuum style robot would provide a torque atthe tip of the robot, and then some type of coupling,i.e., springs, would be used to transmit the torque tothe three joints. Thus, the available DOF are coupledin such a way that there is only one actuated DOF.Though this seems at first to be counter-productive,the kinematics generated by this configuration proveto be very beneficial in some cases.

A simple example is if the robot needs to reacharound an obstacle (see Figure 3). The discrete ma-nipulator must provide actuation to all three of thejoints to reach around the obstacle, but the continuumrobot can achieve the same motion with only one ac-tuator instead of three. Expanding this concept to ro-bots with hyper-DOF, then one can see the benefit ofcontinuum robots. The complexity of the mechanicaldesign, and the control for all of the actuators for aconventionally designed robot would be extremelydifficult to implement. Using a continuum robot de-sign we can achieve many of the same configurations,but the design can be greatly simplified.

In refs. 20 and 21 we outlined the most common

Figure 2. Elephant’s trunk manipulator.

Hannan and Walker: An Elephant’s Trunk Manipulator and Other Continuum Style Robots • 47

actuation and structural design strategies for con-tinuum robots. The prevalent designs for actuationare cable/tendon systems and pressurized fluid sys-tems. Both of these styles are remote actuation strat-egies where the bulk of the mechanisms used for ac-tuation are not located directly on the robot. This isbecause the use of actuators located directly on therobot, as in conventional designs, proves to be ex-tremely complicated and inefficient to implement. Inthe area of structural design there are also two maindesigns. In both cases the robot is based upon sometype of ‘‘backbone’’ structure, where the backboneserves to determine both the manipulator’s maneu-verability and overall shape. The two backbone de-signs can be described as either being segmented orcontinuous in structure, where in either case thestructure must contain a sufficient amount of ‘‘bend-ing stiffness’’ for proper support andmaneuverability.21

3. ELEPHANT’S TRUNK ROBOT

Though the classifications by Robinson and Davies1

help to better describe the different styles of roboticmanipulators, there are some possible manipulator

configurations that do not readily fit into one of thoseclasses. One such design is that of the elephant’strunk manipulator (see Figure 4). This design is moreof a hybrid of all three classes. As described in detailin refs. 21 and 22, the fundamental structure of therobot is composed of four sections, where each sec-tion has two actuatable DOF which are actuated by acable servo system. This yields a manipulator witheight actuatable DOF. Though this manipulator is re-dundant, it would more appropriately fall in to thediscrete classification of manipulators. However, it isthe construction of these sections that allows the ro-bot to fit into more than one classification. The struc-ture of each section is based on four very small linksserial connected by four, two-DOF joints (see Figure

Figure 3. Obstacle example: (a) discrete manipulator and(b) continuum manipulator.

Figure 4. Elephant’s trunk manipulator.


5). There are elastic connections between each joint,i.e., springs, thus coupling all of the joints. This yieldsa total of 32 kinematic DOF for the manipulator, andit would seem to fall in to the serpentine classifica-tion. However, due to this combination of 32 coupledjoints and eight cable actuatable DOF, the manipula-tor takes on the appearance of a continuum robot inthe sense that each section of the robot will take on acurved appearance when actuated. When all the sec-tions are viewed together the robot takes on a smoothcontinuous shape as a continuum robot would. Thiscan be seen easily from Figure 4, and is thus the rea-son that we model the elephant’s trunk robot as a con-tinuum style robot.

4. KINEMATICS

In this section, we introduce a new technique formodeling the kinematics of continuum style robots.First we need to understand how the motions of con-tinuum designs can be quantified. Unlike conven-tional manipulators the use of joint angles and linklengths does not provide an easy and physically re-alizable description of the manipulator’s shape dueto the continuous nature of the design. Instead a ki-nematic model that uses curvatures to describe theshape of the manipulator can provide a more physicaland intuitive description of the manipulator. Beforewe apply this concept to the manipulator, we mustfirst introduce some preliminary mathematical steps.

4.1. Fundamental Mathematics

Classical differential geometry23 parametrizes a spa-tial curve C by its arc length s . The position vector xis defined in the spatial case to be x��x ,y ,z�T. Theunit tangent vector to C at x is t�dx/ds , the principalnormal, n, is placed such that t·n�0, and the binor-mal, b, is defined as b�t�n. The collection of thesethree vectors at a point can be viewed as a new frameof reference. The formulas of Serret-Frenet describe

the motion of this frame as it changes along C . Thethree vector formulas can be written as

dtds

�� n,dnds

��t�� b, anddbds

��n. (1)

The scalar parameters � and � are called respectivelythe curvature and torsion which may be either posi-tive or negative. An important note is that Eq. (1) de-fines � uniquely, but only �2 not � can be defineduniquely.

In the above, we have assumed that the curva-tures and torsions have been a general function of arclength within a curve. If each section of a continuumrobot is optimally constructed,24 then each sectionwill bend such that the curvature and torsion withinthat section is approximately constant. Given in Fig-ure 6 is a picture of a planar continuum robot with anoverlay of an arc with constant curvature. From in-spection it can been seen that the curvature of the ro-bot is approximately constant. Our work focuses onthe description of robots that contain constant curva-ture and zero torsion sections. We proceed in thisfashion due to the fact that the constant-curvature-only model best represents the experimental robotswe are developing, and those designs which havebeen successfully implemented previously.2–5

4.2. Planar Curve Kinematics

In the following two sections we describe how the ki-nematics of a planar curve can be generated, in a formconvenient for modeling continuum manipulators.

4.2.1. Differential Geometry

Under the assumption of constant curvature and thatthe torsion is zero for x�R2, Eq. (1) can be written inthe following linear differential equation,

Figure 5. Tip section of the elephant’s trunk manipulator.

Figure 6. Planar continuum robot.


t��2t�0, (2)

where the prime indicates differentiation with respectto s , i.e., t��dt/ds . Upon solving Eq. (2) with the ini-tial conditions: t0�t(0) and n0�n(0), the equationfor the tangent vector as a function of s is

t�s��t0 cos �� s ��n0 sin �� s �. (3)

Integrating Eq. (3) from 0 to s yields the position vec-tor

x�s��t0�

sin�� s ��n0�

�1�cos�� s ��. (4)

Using Eq. (1) and that the property that the Serret-Frenet frame is a right-hand frame, n can be formedby a rotation of /2 about b. If t0 is defined as t0��t0x , t0y�

T, then n0 can be written as n0��t0y , t0x�

T. Substituting the previous two defini-tions into Eq. (4) we have

�x ,y�T�1�

�t0x ,t0y�T sin�� s �

�1�

��t0y ,t0x�T�1�cos�� s ��. (5)

The magnitude of x is given by

�x��x2�y2, (6)

where the undefined terms are

x2�1�2

t0x2 sin2�� s �

�2�2

t0xt0y sin�� s ��1�cos�� s ��

�1�2

t0y2 �1�2 cos�� s ��cos2�� s ��,

(7)

y2�1�2

t0y2 sin2�� s �

�2�2

t0xt0y sin�� s ��1�cos�� s ��

�1�2

t0x2 �1�2 cos�� s ��cos2�� s ��.

Substituting Eq. (7) into Eq. (6), and using the fact that�t0��1 we obtain

�x��&

��1�cos�� s �. (8)

If is defined to be the angle between x and t0 , as isshown in Figure 7, then

sin��x�t0�

�x� �t0��&

2�1�cos�� s �, (9)

cos��x·t0

�x� �t0��& sin�� s �

2�1�cos�� s �. (10)

Using Eq. (9), Eq. (10), and the trigonometric identityof sin(2)�2 sin()cos() we obtain the relation

�� s

2. (11)

Substituting Eq. (11) into Eq. (8) yields

�x��s

sin��. (12)

The position vector x can now be described by a mag-nitude, Eq. (12), and its angle, Eq. (11).25 Note that Eq.(12) is a function of the angle which provides a cou-pling between the magnitude and angle of x. Intu-itively, as the sections bend, the net effect is that ofbending a link of variable length.

4.2.2. Simple Geometry

Though the use of differential geometry gives use avery good understanding of the mechanics behindthe curves motion, it is not the only way to arrive at

Figure 7. Magnitude and angle.


Eqs. (11) and (12). We can use some simple geometryto obtain the same results. Using the angles labeled inFigure 8 we have

��sR

��s , (13)

��

2�

�s2

, (14)

�� , (15)

��s2

. (16)

Thus we have determined Eq. (11). Again referring toFigure 8, we then have

�x�2

�1�

sin��, (17)

�x��2�

sin��s

sin��, (18)

thus giving Eq. (12).

4.3. Planar Robot Kinematics

As discussed in refs. 25 and 20, using the above analy-sis the movement of a planar curve with constant cur-vature can be described by three coupled movements:

1. rotation by an angle ,2. translation by an amount of �x� , and3. rotation by the angle again,

where x is defined to be the position vector of the end-point of the curve. Now we have a description of themotion of the curve in terms of discrete movements.We can now apply a modified Denavit-Hartenberg(D-H) procedure for the forward kinematic analysisas would be done for conventional manipulators.26

Thus, our approach allows one to model continuumrobots using a procedure that is almost universallyaccepted and used. We term the D-H procedure asmodified because the basis of the D-H procedure isthat all of the joints are independent, but in our casethe three motions are coupled through the curvature.In effect, we are modeling a complex one DOF motionby three simple, coupled motions. If the D-H framesfor one section are set up as in Figure 9, then the D-Htable is

link d a

1 * 0 0 �90°2 0 * 0 90°3 * 0 0 0°

Using the above D-H table the homogeneous trans-formation matrix for the curve can be written in termsof the curvature � and the total arc length l as

A03��

cos(�l) �sin(�l) 01�

�cos(�l)�1�

sin(�l) cos(�l) 01�

sin(�l)

0 0 1 0

0 0 0 1

� .(19)

4.4. Spatial Robot Kinematics

As shown in Ref. 27, the forward kinematics for a spa-tial robot can be easily generated from the planarcase. After analyzing a spatial curve with only con-stant curvature and no torsion, the curve can beviewed as a planar curve rotated out of the plane. Us-ing this concept, we can apply the planar kinematictechnique with only the addition of an angle of ro-tation about the initial tangent. One possible way to

Figure 8. Standard geometry.

Figure 9. D-H frames for a planar section.


set up the frames for the spatial curve is given in Fig-ure 10. By necessity, the frames were chosen so thatframes 0 and 4 will have a direct relationship to the

orientation of the tangents. This is done so the out ofplane rotation will be about the tangent vector. TheD-H table corresponding to Figure 10 is

link d a

1 * 0 0 90°2 * 0 0 �90°3 0 * 0 90°4 * 0 0 �90°

Using the table, where 1�� , 4�2 , Eq. (11) for 2 ,and Eq. (12) for d3 we obtain the transformation ma-trix in terms of the curvature �, the total arc length lof the curve, and the rotation angle � about the tan-gent as

A04��

cos��cos��l � �sin�� cos��sin��l � �1�

cos��1�

cos��cos��l �

sin��sin��l � cos�� sin��sin��l � �1�

sin��1�

sin��cos��l �

sin��l � 0 cos��l �1�

sin��l �

0 0 0 1

� . (20)This homogeneous transformation matrix gives theforward kinematics for one section of a spatial con-tinuum robot. Note that since � can take on both posi-tive and negative values, � needs to only take on thevalues in a range of to uniquely describe the curvein space.

The forward kinematics for an n section manipu-lator can then be generated by the product of n ma-trices of the form given in Eq. (20). The forward ki-nematics for our elephant trunk robot with its foursections can be calculated as

T15�T1

2•T2

3•T3

4•T4

5 , (21)

where Tii�1�A0

4 for section i of the manipulator. Notethat the total arc length for section i��1,...,n�1� mustbe used so that section n is properly oriented, but anyarc length can be used for the final section dependingon where the point of interest lies in the section. Us-ing the total arc length for the final section gives thekinematics in terms of the end point of the section.

4.5. Prismatic Joint

Due to the fact that our kinematic model is based onthe Denavit-Hartenberg procedure modeling, the ad-dition of extra DOF is straightforward. To expand theusefulness and ‘‘flexibility’’ of the elephant’s trunkrobot we added a prismatic joint at its base.32 To in-corporate this joint into the kinematics only requiresthe multiplication of one extra matrix which has thefollowing form,

T01�� 1 0 0 00 1 0 00 0 1 d0

0 0 0 1� , (22)

where d is the prismatic joint’s amount of extension/retraction. Therefore, using Eq. (21), the forward ki-nematics for the robot are

T05�T0

1•T1

2•T2

3•T3

4•T4

5 . (23)

Figure 10. D-H frames for a spatial curve.


4.6. Velocity Kinematics

Analogous to conventional kinematic analysis,26 thevelocity kinematics can be written as

ẋ�Jq̇, (24)

where x�Rmx1 is the task space vector, i.e., positionand/or orientation, and the dot implies differentia-tion with respect to time. For our four section spatialrobot

q��d0 , �1 , �1 , �2 , �2 , �3 , �3 , �4 , �4�T.(25)

The matrix J is the Jacobian, and is a function of the‘‘joint’’ variables q.

Conceptually, Eq. (23) is found explicitly in termsof sines, cosines, q, etc., and then J is determined ex-plicitly by taking the time derivative of the neededelements. For conventional robots the dimension of qis small, and the overall complexity of the forward ki-nematics is not overwhelming. However, for con-tinuum robots the elements in the homogeneoustransformation matrix can be extremely long. Thoughrelatively simple and repetitive in nature the sheernumber of terms in each element makes explicit de-

termination of the elements for both the forward andinverse kinematics very inefficient. However, the nu-meric computation of these matrix elements is rela-tively simple. The forward kinematics can be com-puted in a straightforward manner by the matrixmultiplication of separate matrices determined bynumerically evaluating the matrix given by Eqs. (20)and (22).27 The generation of the Jacobian matrix canthen be found numerically through the use of thechain rule in differentiation. Differentiating Eq. (23)with respect to time yields

Ṫ05�� Ṫ0

1•T1

5��T01•Ṫ1

2•T2

5��T02•Ṫ2

3•T3

5�

��T03•Ṫ3

4•T4

5��T04•Ṫ4

5�. (26)

The time derivative of the prismatic joint matrixgiven by Eq. (22) is obviously

Ṫ01�� 0 0 0 00 0 0 00 0 0 ḋ0

0 0 0 0� . (27)

For each section i��1,2,3,4�, the time derivative ofEq. (20) is

Ṫ ii�1�� sin�� i�cos�� il i��̇ i�cos�� i�sin�� il i��̇ il i �cos�� i��̇ i a13 a14cos�� i�cos�� il i��̇ i�sin�� i�sin�� il i��̇ il i �sin�� i��̇ i a23 a24cos�� il i��̇ il i 0 a33 a34

0 0 0 0

� , (28)where the undefined elements are

a13�sin�� i�sin�� il i��̇ i�cos�� i�cos�� il i��̇ il i ,

a23��cos�� i�sin�� il i��̇ i�sin�� i�cos�� il i��̇ il i ,

a33��sin�� il i��̇ il i ,

a14��̇ i� i

2 �cos�� i��cos�� i�cos�� il i��̇ i� i

�sin�� i�

�sin�� i�cos�� il i��̇ il i� i

cos�� i�sin�� il i�,

a24��̇ i� i

2 �sin�� i��sin�� i�cos�� il i��

��̇ i� i

�cos�� i�cos�� il i��cos�� i��

��̇ il i� i

sin�� i�sin�� il i�,

a34��̇ i� i

2 sin�� il i��̇ il i� i

cos�� il i�.

Now, using Eqs. (22), (27), (20), and (28), the matrix inEq. (26) can be computed numerically. Once Eq. (26)


is computed, the Jacobian matrix can be generated byfactoring the appropriate elements from the matrix oftime derivatives. If

Ṫ05�� 11 12 13 1421 22 23 2431 32 33 34

41 42 43 44

� (29)and x��x ,y ,z�T, then Jq̇ can be constructed from Eq.(29) as

Jq̇��14 , 24 , 34�T. (30)

The Jacobian matrix J can then be factored out ofEq. (30).

If it is desired to have x��x ,y ,z ,x ,y�T, where

��x ,y�T are the orientation angles of the tangent

vector t��tx ,ty ,tz� with respect to the x and y axes,then it is possible to determine J as follows. If the ori-entation angles are defined as

�� arctan� tytz �arctan� txtz � � , (31)

where the tangent vector t can be determined fromthe third column of the forward kinematic’s matrix,and the time derivative of Eq. (31) can be written as

�̇�� ṫ ytz�tyṫztz2�ty2ṫ xtz�txṫztz

2�tx2

� , (32)the vector ṫ is formed as

ṫ�Jtq̇��1323

33

� , (33)where Jt��Jt1 ,Jt2 ,Jt3�

T is the associated Jacobian ma-trix for the tangent vector. Using Eqs. (32) and (33),the Jacobian matrix for the orientation angles can beconstructed as

J�� J1J2�� 1tz2�ty2 �tzJt2�tyJt3�1tz

2�tx2 �tzJt1�txJt3�

� . (34)Thus, given x��x ,y ,z ,x ,y�

T, the velocity kinemat-ics are

Jq̇�� 142434J1q̇J2q̇

� . (35)As before, the Jacobian matrix J can be factored out ofEq. (35).

4.7. Motion Planning

Now that the forward and velocity kinematics havebeen established we can use them for the motionplanning of the robot. If we are given the vector ofdesired task space velocities, then the generalsolution29 to Eq. (24) is

q̇�J�q��ẋ��I�J�q��J�q�� . (36)

The term J(q)� in Eq. (36) is the pseudo inverse ofJ(q) and is given in its most commonly used form as

J��W�1JT�JW�1JT��1, (37)

where W is a positive definite weighting matrix. If Wis equal to the identity matrix, then the term J�ẋ is theleast squares solution of Eq. (24).29 The term (I�J�J)� is the null space projection of the solution ofEq. (24). This solution only generates motion in the‘‘joint’’ space of the manipulator, and not the taskspace of the robot.29 This joint space motion is alsoknown as the self motion of the robot. The manipu-lator can now be controlled using any of the standardredundancy techniques, i.e., refs. 29–31, to generatethe desired robot motion.

5. EXPERIMENTS

Now that a theoretical kinematic model has been pre-sented for continuum style robots the next logicalstep is to provide experimental verification of themodel, and also the viability and ‘‘flexibility’’ of both


the elephant’s trunk manipulator and our kinematicmodel. A video of a real time implementation of simi-lar experiments can be viewed in ref. 32.

5.1. Implementation

Before various experiments are presented, some im-portant information is presented to help understandhow the kinematic model was implemented. Wepresent how the curvature values were determined,how the curvatures obtained from the robot relate tothe kinematic variables, and the control used to servothe robot.

5.1.1. Real Time

Real time implementation of the elephant trunk ro-botic manipulator was conducted using the QMotor3.0 program by Quality Real-Time Systems, and isrun under the QNX 4.0 real time operating system.The basis of the control program was implementedusing a C++ program, and real time data acquisitionfor the main sections of the robot was performed bya Quanser MultiQ III I/O board, and a ServoToGoI/O board was used for the prismatic joint. To in-crease the ‘‘flexibility’’ of the system the desired ve-locity trajectories could be given by a pair of three de-gree of freedom joysticks. The servo output from theI/O boards was connected to four dual channelTechron linear amplifiers which powered the el-ephant’s trunk dc motors. All of the calculations andI/O operations were conducted at a frequency of 1500Hz on a AMD 1300 MHz CPU.

5.1.2. Determination of Curvature

As with most continuum style robots, due to theelephant’s trunk construction, in particular the largenumber of joints and the inability to mount measure-ment devices for the joint angles, the determination ofthe manipulators shape is a problem. There are sev-eral different technologies that could help solve thisproblem, but they are very difficult and costly toimplement on a three dimensional robot. The solutionadapted for the elephant’s trunk incorporates the ge-ometry of the manipulator, the cable lengths, and thebasic assumption that each of the joints in a sectionrotate by the same amount. This assumption is a di-rect result of the assumption that the sections shouldbend with constant curvature. From this informationan approximate curvature can be fitted to each sec-tion of the manipulator as follows.

Figure 11 shows a simplified diagram of one side

of one of the four segments in each section, where wis the radius of the segment, h is the distance from thecenter of the segment’s joint to the end of the seg-ment, r��w2�h2, l1 is the length of cable over thesegment before it has been actuated, and l2 is thecable length after actuation. The important value thatneeds to be determined is the angle � that each jointin a section moves through given a change in cablelength. We define �l as

�l�l1�l2 . (38)

From Figure 11 we have the relation

��2�� , (39)

where the angle is

�arctan� wh � (40)and the angle � is

��arctan��4r4��2r2�l22�22r2�l22 � . (41)Combining Eqs. (39), (40), and (41) the total angle ofrotation of one joint in a section after actuation is

��2 arctan� wh ��arctan��4r4��2r2�l22�22r2�l22 � .(42)

Figure 11. Diagram for the determination of curvature.


The cable length l2 can be determined by solving Eq.(38), where �l is the amount of cable retracted for agiven section. Using Eq. (11) and noting that each sec-tion contains n segments we have

�n��s2

, (43)

and solving for the curvature yields

��2n�

s. (44)

For the elephant’s trunk manipulator we have n�4and s�8 in., and thus for section i��1,2,3,4� we have

� i�12 � i , (45)

where the units of the curvature are 1/in. The calcu-lated curvature is an approximation to the exact cur-vature for each section due to the fact that s does notremain exactly a constant value, but through experi-mentation proves to be very close to constant.

We would like to make the note that the design ofthe cable drive system has the cable pairs of eachplane running parallel to each other off the center axisof the manipulator by some distance. This has the ef-fect of coupling the cable actuation of a section withall of the previous sections. When one of the previoussections moves, then, since the cables are not on thecenter line of the robot, the following sections’ cur-vatures will also change. Therefore by Eqs. (42) and(45) all the cables that run through that section mustbe altered by the amount of cable length in the actu-ated section to maintain the proper curvature of thesections.

5.1.3. Curvature to Rotation Angle Relationship

Up to this point the kinematics of a continuum stylerobot’s sections have been described by a curvatureand a rotation angle. However, the elephant’s trunkmanipulator physically does not move in this fashion.The information that is retrieved from each of the ro-bot’s sections is two orthogonal curvature measure-ments via Eqs. (42) and (45), instead of one curvature

and one rotation angle. However, there is a con-venient transformation between these two sets ofvariables.

To transform from the two orthogonal curvaturesto one curvature and one rotation angle only requiresthe following transformation:

��x2��y2,(46)

��arctan� �y�x� .If � /2 �� /2, then � is positive, else � is nega-tive and � is altered by radians so that � /2�� /2.

The inverse transformation from the kinematicvariables to the robot variables involves a slightly dif-ferent approach. The problem here lies in that the ro-tation angle of one section is a function of the previ-ous section’s rotation angle. If one can imaginelooking down the ‘‘backbone’’ of a continuum robotthat is in a straight configuration [see Figure 12(A)],one can see that as each section in the kinematicmodel rotates all the sections after that section rotatearound the backbone by the same amount [see Fig-ures 12(B) and (C)]. The calculation of the two or-thogonal curvatures in a section with respect to thatsection’s corresponding rotation angle is

�x�� cos��,(47)

�y�� sin��.

These values now must be transformed to theirproper values by accounting for the rotationangles of all the previous sections. The transforma-tion is simply

�� i ,x0� i ,y0 �� cos�� i�1�� i�2�¯ � �sin�� i�1�� i�2�¯ �sin�� i�1�� i�2�¯ � cos�� i�1�� i�2�¯ � � �� i ,x� i ,y� , (48)where i��1,2,3,4�.

Figure 12. Effects of rotation on kinematic frames.


5.1.4. Shape Control

The real time low level controller for the elephant’strunk is composed of two main steps. During the firststep the controller converts the kinematic variables tothe robot’s curvature variables via Eq. (48). Then forthe second step a PD controller servos between thedesired curvature �d and the present curvature �found through Eq. (45) for each axis in a section. Thecontrol algorithm for one axis in a section is as fol-lows:

��Gp��d��Gd�̇ , (49)

where � is the voltage to the servo amplifiers, Gp�2000, Gd�1.0, and �̇ is the velocity of the curvature.Since �̇ is not directly measurable, it was obtainedthrough a two step approach. First �� was calculatedusing the following equation,

��

�t, (50)

where �� is the change in the curvature over thesample time of the control. For a control frequency of1500 Hz, �t�667 �s. A simple digital filter was thenused to smooth ��. The equation for the digital filterwas

�̇�� 12��c ��c��c�0��c�2 ��̇0�, (51)where �c�9.42�10

3 for a control frequency of 1500Hz, �0� is the value of �� during the previous controlcycle, and �̇0 is the value of �̇ during the previouscontrol cycle.

The input to the controller for the robot is ob-tained in two ways. The first approach is through theuse of a trajectory simulator that uses an iterativescheme based on Eq. (36) to compute the trajectory.The complete trajectory of kinematic variables is thenfed into the robot’s low level controller. The secondapproach follows the real time acquisition of desiredendpoint velocities, ẋd , that are provided by the joy-sticks, i.e., tele-operation. This approach uses the fol-lowing form of Eq. (36),

q̇d�J�qd��ẋd , (52)

where the kinematic variables are calculated as

qd� q̇d dt. (53)These kinematic variables are then used in the lowlevel joint controller in the same way as in the firstapproach.

5.2. Configuration Tracking

Given a predefined set of kinematic variables the con-figuration tracking approach compares the differ-ences in the configuration of a simulated robot basedon the theoretical kinematics to that of a physical con-tinuum style robot. The idea is that any inaccuraciesin the kinematic model will show up as deviations inthe configuration of the physical robot from that ofthe configuration for the simulated robot. If the kine-matics are valid, then there should be no discernibledifference between the configurations.

In both of the following experiments the initialconfiguration was q��0,0,0,0,0,0,0,0,0�T, yielding aninitial end-point location of x��0,0,32� in., and thedesired end-point location was xd��10,0,10� in. Ineach of the experiments a figure shows a comparisonbetween the simulated configurations at severalpoints in the linear trajectory and the correspondingphotographic snapshots of the physical robot at thesame points. Snapshots of the manipulator are showndue to the lack of ability to precisely measure the ro-bot’s curvatures and end-point location, therefore theerrors xd�x and qd�q cannot be verified precisely. Itis important to note that manipulators of this designare not well suited to high precision applications dueto their actuation and physical construction, but areinstead best suited for applications where their flex-ibility and compliance can be exploited.

We would like to point out that a great deal of therobot’s inaccuracy is inherited from the method usedfor the calculation of curvature and the coupling be-tween the cables of the sections. It is clear that a moreaccurate determination of a section’s curvature, i.e.,direct measurement, and a better cable actuation de-sign would yield better results. Even with these limi-tations in measurement and actuation, the robot’sconfigurations are very close to that of the simulation.We would also like to point out that even though ourkinematic model has no problem dealing with a 3Dworkspace, the following experiments were con-ducted in 2D for ease of presentation.

5.2.1. Prismatic Joint Excluded (Figure 13)

In this experiment the weighting matrix in Eq. (37) isset such that W�1��0,1,1,1,1,1,1,1,1� . This has the ef-


fect of nullifying any motions of the prismatic joint,thus allowing only the four sections of the manipu-lator to move in order to obtain the proper motion ofthe end point of the manipulator. The results of theexperiment are shown in Figure 13. The manipulatordoes not exactly follow the desired trajectory, but theamount of error is very small given the physical limi-tations of our manipulator.

5.2.2. Prismatic Joint Included (Figure 14)

In this experiment the weight matrix is set to W�1

��1000,1,1,1,1,1,1,1,1� . The reason that the weightingfactor for the prismatic joint is set to a much greatervalue than the joint variables is to help equalize itscontribution to the kinematics. The result of the ex-periment is shown in Figure 14. Even with the addi-tion of the prismatic joint, our kinematic model re-mains a very close approximation of themanipulator’s kinematics.

5.3. End-point Tracking

The end-point velocity tracking approach comparesthe actual end-point trajectory of the robot to that ofthe desired trajectory. The desired trajectory is ob-tained by specifying velocities for the end-point.Thus, the desired kinematic variables can be calcu-lated with Eqs. (52) and (53), where ẋd was the desired

velocity provided via one of the elephant’s trunk set-up’s joystick. In this situation there is no feedbackcontrol on the end-point’s position. The idea is thatany error in the kinematic model will show up as de-viations in the robot’s end-point trajectory from thatof the desired end-point trajectory.

5.3.1. Prismatic Joint Excluded (Figure 15)

Shown in Figure 15 are six sequential photographsthat show the robot’s movement when following thedesired end-point velocity. In this experiment theprismatic joint was disabled by again using W�1

��0,1,1,1,1,1,1,1,1� . From the figure it can be seenthat our kinematic model still provides a very goodapproximation to the robot’s actual kinematics.

5.3.2. Prismatic Joint Included (Figure 16)

In this experiment the weight matrix is set to W�1

��1000,1,1,1,1,1,1,1,1� . The results of the experiment

Figure 13. Configuration tracking without prismaticjoint.

Figure 14. Configuration tracking with the prismaticjoint.


are shown in Figure 16, and again it can be seen thatour kinematic model very closely approximates thephysical setup of the manipulator.

5.4. Obstacle Avoidance

As was demonstrated in Figure 3, continuum stylemanipulators are well suited for obstacle avoidance.Avoiding obstacles inherently requires a manipulatorto exhibit a large number of degrees of freedom, andthough continuum style robots may not have a largenumber of actuated degrees of freedom, in generaltheir physical structure does exhibit a large numberof degrees of freedom. This helps them to achievecomplex motions over large ranges with minimumactuation, and thus makes them very efficient for ob-stacle avoidance. Conventional style manipulators,on the other hand, would require many more actua-tors to achieve the same motions. Many different ob-stacle avoidance strategies have been developed forconventional manipulators, but none of them havebeen applied to continuum style robots. In the follow-

ing section two different strategies are presented tohelp demonstrate the viability of continuum robots inobstacle avoidance.

5.4.1. Reference Configuration

As presented in ref. 31, a simple approach to obstacleavoidance is through the concept of subtasks. Inmany cases obstacle avoidance is a secondary objec-tive. The manipulator’s main task is that of followinga given end-point trajectory, and whatever ability formotion is left over after satisfying the end-point taskcan be used to avoid the obstacle. The idea is to definea static reference configuration for the robot that suf-ficiently avoids the obstacle, and then use the nullspace projection, which does not affect the end-pointtrajectory, to cause the manipulator to adopt postureswhich resemble this configuration. Thus, the robotwill try and maintain a configuration that will exactlysatisfy the end-point constraint as well as trying tomaintain the reference configuration that avoids theobstacle. This approach was implemented by defin-ing � in Eq. (36) to be

��k�qr�q�, (54)

where k�0.001 is positive control gain and qr��0,0,/2,0.35,0,�0.25,0,�0.15�T is the desired ref-erence configuration that avoids the obstacle. Note,this experiment was performed before the prismaticjoint was added, and therefore qr has only eight vari-ables. The experimental results are shown in Figure17. The first picture shows the spatial relationship be-tween the robot and the obstacle. The second pictureshows the configuration of the robot before Eq. (54) isimplemented. The position was obtained after mov-ing from the initial position, first picture, to that po-sition with ��0. This configuration of the robot is notconducive for avoiding the obstacle. If only the leastsquares solution was implemented from this configu-ration, then the robot would collide with the obstacle.In the third picture, as the robot begins to approachthe obstacle, Eqs. (36) and (54) are used to calculate �.In this and the following pictures one can see the con-figurations of the robot evolving in such a way thatthey try to obtain the reference position, and thus suf-ficiently avoid the obstacle.

5.4.2. Reference Curve

The previous experiment is useful in showing howthe robot can avoid an obstacle given that we alreadyknow the exact manipulator configuration which

Figure 15. End-point tracking without prismatic joint.

Figure 16. End-point tracking with the prismatic joint.


avoids the obstacle. This approach can be used inmany different situations, but it does have its limita-tions. In this section a new alternative strategy forplanar obstacle avoidance is presented. Based on re-search presented in refs. 14–16, another possible wayto design an obstacle avoidance strategy would be todefine a curve that would sufficiently clear the ob-stacle. The robot would then be ‘‘servoed’’ in such away that it would take on the basic shape of thecurve. The idea is that the defined curve is simple,and, with the exception of a few constraints, requireslittle knowledge of the robots kinematics. With theaid of some basic information about the environment,the operator defines a curve that avoids the obstacle,and once the curve is generated, the kinematics of therobot are used to servo the robot to the curve.

As shown in Figure 18, to implement such a strat-egy the predetermined obstacle avoidance curve isdefined as

xc�f�sc�, (55)

and from the forward kinematics the position of onesection of the robot can be generally defined as

xr�g�sr�, (56)

where s is the arc length in each case. The basis of thisapproach is that the ends of each of the robot’s sec-tions do not initially line up with the curve. The ap-

proach is to servo the robot in such a way that the endof each section will lie on the curve. The beginning ofeach section is the end of the previous section so thereis no need to servo this point for each section. Also,the constraint that the first section’s origin must be onthe curve is imposed. This condition is easily satisfiedby impressing the constraint that the curve must bedesigned such that it and the robot originate from thesame location. The problem in servoing the robot tothe curve is that given xc , Eq. (11), and Eq. (12) cannotbe solved directly for �. Therefore, some type of it-erative scheme can be used.

If it is given that the origin of the manipulator’ssection, xr(0), is on the curve, then some way isneeded to determine the proper � that places the endof the section on xc . The error between the curve andthe manipulator is defined as

e�xc�xr . (57)

The problem is: given xr(l), (l is total length of the sec-tion), how can xc be determined? This is done by de-fining xc in terms of its magnitude and angle such thatit is solvable for sc , then equation �xc��xr� is solvedfor sc . Once sc has been obtained, xc and thus e canbe calculated. Now that the error vector has been de-termined, it can be used to determine � by iteratingthe following equation,

Figure 17. Obstacle avoidance using a reference configu-ration. Figure 18. The desired shape verses the manipulator’s

shape.


d�dt

��kpe, (58)

where kp is a positive constant. What is important tonote about this strategy is that the desired result isthat the end of each section lies on the curve. Thisdoes not guarantee that the manipulator will alwayshave the exact shape of the desired curve, but it willhave the same general shape.

To show the viability of this strategy, a simple ex-periment that incorporates not only the different sec-tions of the robot, but also the prismatic joint is pre-sented. As shown in the right hand column of Figure19, a curve was generated by using several key pointsthat avoid the obstacle. A cubic spline was then fittedthrough those points to determine xc . The robot isthen moved along the curve with the prismatic joint.It can be seen from the different configurations in Fig-ure 19 that even though the end of each section is onthe curve, the robot does not have the exact shape ofthe curve. However, the robot does maintain the samebasic shape, and thus this strategy still provides avery useful and easily implementable strategy forcomplicated obstacle avoidance.

5.5. Grasping

Not only are continuum style robots well suited forobstacle avoidance, but their design is also very con-ducive for grasping. Unlike conventional manipula-tors which require some type of end-effector or grasp-ing mechanism, continuum style manipulators canperform what is commonly known as whole armgrasping. In whole arm grasping the manipulatoruses itself to grasp an obstacle without the aid of anyadditional mechanisms. The manipulator uses its‘‘whole arm’’ to grasp the object, very much like an

elephant can do with its trunk. This allows the ma-nipulator to grasp a large range of objects indepen-dent of the objects size or shape.

To demonstrate the ability of continuum robots toperform whole arm grasping we presented a simplestrategy in ref. 21. The basic approach is to use onejoystick to provide the information for Eq. (36), with��0 and W�1��0,1,1,1,1,1,1,1,1� , to maneuver themanipulator around. As a second control input, an-other joystick was used to control the last sectionsshape by providing the desired ‘‘joint’’ velocities. Thelast section was manipulated in such a way that itcould grasp the desired object. The weighting matrixwas then set to W�1��0,1,1,1,1,1,1,1,0� so the last sec-tion would not change its grasping position. Settingthe weighting matrix in this fashion has the effect of‘‘locking’’ the last section of the robot in such a waythat it will not change its shape, but the inverse ki-nematics will still function properly. Figure 20 showsone of the images from ref. 21 that demonstrates thephysical robot grasping a small cylinder. Figure 21shows a different image of the physical robot grasp-ing a large diameter ball.

6. CONCLUSIONS

In this paper the elephant’s trunk robotic manipula-tor is presented. The elephant’s trunk is a new andnovel design for a continuum style robot. The robot’sbasic structure is very similar to that of a backbone.It is composed of a serial connection of 16 two degreeof freedom joints, and is actuated by a cable servo sys-tem. To facilitate intelligent motion for continuumstyle manipulators, such as the elephant’s trunk,a new kinematic model was developed. Unlike

Figure 19. Obstacle avoidance using a static curve (top row is simulation, bottom column is experiment).


conventional manipulators which use link lengthsand joints as a basis for their kinematic model, themodel presented uses the concept of arc length andcurvature to describe the kinematics. The kinematicmodel uses the idea that the bending motions dem-onstrated by continuum style robots can be best de-scribed by using the concept of constant curvature.These motions exhibited by curves that can only bendwith sections of constant curvature is examined, andthe resulting analysis demonstrates that the complex

motion exhibited by these curves can be dissectedinto a series of simple, but coupled motions. This ideais then incorporated into the widely used Denavit-Hartenberg procedure for kinematic analysis. Thisability to rewrite the kinematics of continuum robotsin a form that can be incorporated into the Denavit-Hartenberg procedure exposes continuum robots tomuch of the kinematic analysis and motion planningresearch that is used for conventional robots. Themodel has the ability to account for any number andarrangement of robot sections. Also the model caneasily account for not only continuum style robot ac-tuators, but also the incorporation any type of con-ventional joints, i.e., rotational and prismatic joints.

Experimental results are presented to validateboth the accuracy and viability of the designed kine-matic model using the elephant’s trunk continuumstyle robot. With the kinematic model verified, sev-eral different strategies were implemented on the el-ephant’s trunk to demonstrate the usefulness of thesynthesized kinematic model. The first set of strate-gies dealt with obstacle avoidance. First a simpleavoidance strategy is used to demonstrate the viabil-ity of continuum robots for obstacle avoidance. Theadopted strategy uses the idea of subtasks and a pre-defined reference configuration for the robot to avoidan obstacle. This strategy uses the idea of subtasksand a predefined reference configuration for the robotto avoid an obstacle. A second new obstacle avoid-ance strategy is also provided. This strategy uses theidea of a predefined curve that avoids the obstacle. Asimple obstacle avoidance experiment based on

Figure 20. Whole arm grasping.

Figure 21. Grasping of a ball.


inserting the elephant’s trunk robot into a clutteredenvironment is performed, and thus demonstratessome of the more powerful uses for continuum stylemanipulators. Also, work is presented that helps todemonstrates the usefulness of continuum style ro-bots for whole arm grasping.

The research presented in this paper provides asolid foundation for both the mechanical design andthe kinematics of continuum style robotic manipula-tors. Both the elephant’s trunk and the kinematicmodel prove to work very well.

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