continuum robots

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A Hyper-Redundant Continuous Robot

*Jingzhou Yang, Potratz Jason, and Karim Abdel-Malek

Center for Computer Aided Design, The University of Iowa, Iowa City, IA 52242

*

Corresponding Author: [email protected]

Abstract - This paper presents a novel design and analysis of a hyper-redundant continuous robot (biological continuum stylemanipulator), actuation system, and control strategy. The robotincludes 8 flexible segments, though it can be extended to more

segments as necessary. In this study the gravity of the springs isneglected due to the manipulation force is much larger than thesegravity forces. This mechanism exhibits a wide range of maneuverability and has a large number of degrees of freedom.Each segment is designed using a novel flexible mechanism basedon the loading of a compression spring in both transverse andaxial directions and using cable-conduit systems. The forwardkinematics of the mechanism is also presented and lends itself

well to computer control. It is shown that the solution of thetransverse deflection of each segment is obtained in a generalform, while the stiffness coefficients are obtained in closed form.A prototype robot segment is experimentally tested, the resultsare verified. A bench-top actuation system has been developedand a control scheme used in prosthetic hand control has been

implemented to control the mechanism.

Index Terms - Continuum style robot, maneuverability, biological manipulators.

I. I NTRODUCTION

Radioactive waste handling and inspection of dangerousenvironments require dexterous, long- reach roboticmanipulators that can maneuver through constrainedenvironments and accurately position payloads. These roboticdevices can be used for inspection of nuclear reactors,maneuvering through rubble to look for survivors of anearthquake, pipe inspecting and cleaning, among other uses.

Ref. [1] broke down manipulators into threeclassifications. The first is termed discrete robots. These arethe traditional industrial manipulators with 6 DOFs or less.With increased redundancy of the robotic manipulators, themanipulators are in the second classification, known asserpentine robots. The third classification is termed continuumrobots. In the third classification, robots do not contain jointsand rigid links as do the previous two types; instead, they bendcontinuously along their length similar to biological trunks.However, the last two types, broadly categorized as hyper-redundant, or high-degree-of-freedom (HDOF) manipulators,

share many similarities, and a significant body of work hasbeen done on HDOF. Refs. [2-6] proposed the continuousbackbone curve model that minimizes a weighted combinationof bending, twisting, and local extension/contraction of backbone curve while also satisfying task constraints. Refs.[7-9] studied the shape Jacobian of manipulators with HDOFand the shape correspondence between a hyper-redundantrobot and a desired spatial curve.

Several designs of elephant trunk or snake-like robotshave been proposed. Ref. [10] developed a robotic arm calledthe Elastor. Ref. [11] pioneered the development of snake-likerobots, especially with regards to locomotion. Ref. [12] didsignificant work in flexible hydraulic micro-actuators forgrippers, which are essentially small, flexible, 3-DOFmanipulators. Ref. [13] developed an elephant trunk-typeelastic manipulator for bulk and liquid materials transportation(Fig. 1). Researchers at Clemson University [14-22] developedtwo types of manipulators: Elephant Trunk Manipulators (Fig.2) and Clemson Tentacle Manipulators (Fig. 3). Thefundamental construction of Elephant Trunk Manipulators isbased on a segmented backbone. The backbone is composedof a serial connection of 16, two-DOF joints, giving thebackbone a total of 32 DOFs. However, only eight degrees areactuated. Clemson Tentacle Manipulator consists of a highlyelastic rod as its backbone, with antagonistic cable pairsperiodically able to exert moments on the backbone to deformits shape. Although this is truly a continuous device, as a realtentacle is, it is only actuated with four DOF. Its underactuatedbackbone may adopt an infinite number of poses for a givenactuator displacement.

This paper proposes a hyper-redundant continuous robot.We will first introduce the design and prototyping of the robotin Section 2. Section 3 studies the analysis of each segmentunder different external loads. Section 4 introduces theactuation system and is followed by the paper’s conclusion.

II. Design and Prototyping of the Continuous Robot

We are developing a robot system that is analogous to anelephant trunk. Controllable spring elements replace clustersof muscles. The mechanical design is much simpler thancurrent designs for legged or wheeled robots, and thismechanical simplicity permits reliable, robust, and low-costimplementations. Consider a relatively stiff spring. Alsoconsider three cables attached to an upper plate from theinside of the spring (as shown in Fig 4). Pulling one of thecables causes the spring to flex in one direction and causes theupper plate to slightly rotate and translate. Due to theresistance induced by the coil, this action is very similar to thegroup of muscles in an elephant trunk. Similarly, exerting an

equal force on all three cables will cause the upper plate totranslate downwards.

Proceedings of the 2006 IEEE International Conference on Robotics and AutomationOrlando, Florida - May 2006

0-7803-9505-0/06/$20.00 ©2006 IEEE 1854



Fig. 1 Elastic Manipulator Fig. 2 Clemson Elephant Trunk Robot

Fig. 3 Clemson Tentacle Robot

Upper plate

Base plate

Cable

Spring

Diameter

(a) (b) (c)Fig. 4 (a) One spring and two cables are shown (this is a cross section of a

large diameter spring) (b) Pulling on one of the cables results in a rotational

motion (c) Pulling on all three cables yields a translational motion

With three cables, the control of the motion of the upperplate in 3D is established. Now consider each spring with two

plates and three cables to be similar to a cluster of muscles of an elephant trunk. We use this cluster in series to generate achain of spring elements. Each cluster is driven by threeactuators, and there are eight such stages, thus the total DOFof the robot is 48. Motion on the ground can now take placeby controlling the various stages, and effecting a caterpillar-like motion. The continuous robot comprises a number of springs, compression links, cables, and conduits. Each springacts as a segment. Affecting a tension force on three cablesthrough the conduits will yield a deformation in the spring,both in transverse and in compression. The result of controlling the 24 cables to perform manipulation of this chainis shown in Figs. 5 and 6.

III. Analysis of the Continuous RobotAnalysis of the robot includes two parts: analysis of a 2-

dimensional model and analysis of a 3-dimensional model. If only one of the three cables in one segment (Fig. 7) isactuated, the model will be 2 dimensional and will build aclosed form to simulate the segment. If two of three cables orall three cables are actuated, there will be a 3-dimensionaldeflection and a 3-dimensional model will be used to simulateit. In the following sections we will present the 2- and 3-dimensional models in detail.

A. 2-Dimensional Model (Closed form method)

The displacement of the spring element at each

connecting end is denoted by t δ and the deflection angle by

θ "

(Fig. 10). Written in terms of the compression force V and

the bending moment M , it can be shown that

t v mV M δ τ τ = + (1a)

v mV M θ λ λ = +" (1b)

wherev

τ ,m

τ ,v

λ , andm

λ are the compliance coefficients.

Fig. 5 Prototype of the continuous robot

Fig. 6 Elephant trunk robot

δ t

δ a

θ "

MV

T

δ t

δ t

δ a

δ a

θ "θ "

MV

T

Fig. 7 Nomenclature for the flexing of a spring element

θ

p

x

y

MV

T

t

uθ θ

p

x

y

MV

T

t

u

VT

M

S

N

B

t y−

t x−

ϕ

VT

M

S

N

B

t y−

t x−

ϕ

(a). Center line of the loaded spring (b). Load resultants for a segmentFig. 8 The beam model of the helical spring

In order to consider the spring behaving like an elasticrod, its rigidity in bending, shear, and compression must equalthese values:

4

/32 (1 )

2

b

B Ed L K

E d dunD

G

ϕ = − =

+

(2a)

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where b K is the bending stiffness, B is the moment, and

/d duϕ is the bending rotation angle for the element length

du , E and G are the material elastic modula, respectively, in

the normal and tangential directions, n is the number of activecoils, d is the wire diameter, D is the mean spring diameter, Lis the length of the loaded spring, ϕ is the bending rotation

angle.4

38

sS Ed L K

nDφ = = (2b)

where s

K is the shear stiffness, S is the shear load, and φ is

the shearing angle.

4

38a

V Gd K

nDδ = = (2c)

where K is the axial stiffness, V is the axial load, aδ is the

axial displacement, γ is the shearing deformation angle (Fig.

10).The expressions of (1) are written in terms of the stiffness

coefficients (inverse of compliance) as

t t V δ θ σ δ σ θ = +"

"(3a)

t t M δ θ ξ δ ξ θ = − +"

"(3b)

Because the continuous robot is composed of multiplesegments, but each segment is independently actuated, it ispossible to assume one end as relatively fixed and rigid, theother as rotating and translating. As a flexible element, it isnow possible to analyze each segment independently, where ateach section of the spring, we denote the normal force by N ,the shear force by S , and the bending moment by B, the

bending moment at 0, 0 x y= = by N M such that

cos sin N V T ϕ ϕ = − (4a)

sin cosS V T ϕ ϕ = + (4b)( ) ( )

N B M V t y T L x M Vy Tx= + − + − = − − (4c)

and the geometric relationship between the coordinates of thespring is:

cosdx

dpθ = − (5a)

sindy

dpθ = − (5b)

and θ is defined by the following relationship

θ ϕ γ = + (6)

where arctan[ /((1 / ) )] s

S N K K γ = + . The strain of the spring

is obtained by2 2(1 ) ( )

s

dp N S

du K K = + + (7)

Differentiation of (2a) with respect to p yields combined

with (5a) and (5b):

( ) sin cosb

d d K V T

dp du

ϕ ϕ ϕ = + (8)

Multiplyd

du

ϕ on both sides of (8) and integrate with respect to

p

2( ) ( sin cos )

2

b K d dpV T d C

du du

ϕ ϕ ϕ ϕ = + +∫ (9)

where C is obtained from the boundary conditionb

d M

du K

ϕ = −

at l ϕ θ = , and l θ is the angle of rotation of the end surface at

the free end of the spring.

Define ( ) ( sin cos )dp

V T d du

ϕ ϕ ϕ ϕ Φ = +∫ , then ( ) ( )l

l ϕ θ ϕ θ

=Φ = Φ ,

and (9) will be2

( )2

l

b

M C

K θ = − Φ (10)

substituting (10) into (9) yields2

2( ) ( ) ( )2 2

b

l

b

K d M

du K

ϕ ϕ θ = Φ + − Φ (11)

Simplifying (11) to obtain a representation for du yields

2/ 2

( ) ( )2

b

l

b

K du d M

K

ϕ

ϕ θ

= −

Φ + − Φ

(12)

Integrating both sides of (12) yields

0

20

/ 2

( ) ( )2

l

Lb

l

b

K L du d

M

K

θ ϕ

ϕ θ

= = −

Φ + − Φ∫ ∫ (13)

In order to calculate l θ , we can now substitute different

values for l θ into (13) until satisfied. From (5a), (7), (12), we

further simplify

2 2

2

/ 2cos( ) [(1 ) ( ) ] cos

( ) ( )2

b

sl

b

K N S dx dp d

K K M K

θ θ ϕ

ϕ θ

= − = + +

Φ + − Φ

(14)

From (4) and (14), we obtain an expression for as follows

0

sin cos( ) cos[ arctan ]

cos sin(1 )

l

s

V T x d

V T K

K

θ ϕ ϕ ϕ ϕ ϕ

ϕ ϕ

+= Ψ +

−+

∫ (15)

where

2 2

2

/ 2 cos sin sin cos( ) [(1 ) ( ) ]

( ) ( )2

b

sl

b

K V T V T

K K M

K

ϕ ϕ ϕ ϕϕ

ϕ θ

− +Ψ = + +

Φ + − Φ

Similarly, we obtain an expression for y as follows

0

sin cos

( ) sin[ arctan ]cos sin(1 )

l

s

V T

y d V T K

K

θ ϕ ϕ

ϕ ϕ ϕ ϕ ϕ

+

= Ψ + −+∫ (16)

B. 3-Dimensional Model

In our continuous robot design, there are a total of threecables for one segment. If applying loads to two or three of thecables, the spring will deflect in 3-dimensional space, and wemust develop a 3-dimensional model for this case. In thispaper we use the finite element method to simulate thedeflection of the spring in 3-dimensional space.

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Consider a curved beam element of a uniform crosssection with positive local coordinate directions, nodal forces,moments and other parameters as shown in Fig. 9. The forces

are represented by1

S ,2

S ,3

S ,7

S ,8

S , and9

S , while4

S ,5

S ,

6S ,

10S ,

11S and

12S are moments acting on this element.

Fig. 9 A curved beam element with coordinates and forces

Each curved beam element is placed on the helical curveto form a mechanical spring (Fig. 10). Consider two

coordinate systems, a fixed frame [i j k], located in the centerof the first compression link (fixed end of the spring), and asecond moving frame [t n b].

Fig. 10 Local and global coordinate systems

An arbitrary point P is located at angle φ measured

clockwise. The internal forces and moments at P may be

expressed in terms of the forces at the node I as follows:'

1 1 2cos( ) sin( )S S S φ φ = − (16a)

'2 1 2

sin( ) cos( )S S S φ φ = + (16b)'

3 3S S = (16c)'

4 4 5 3cos( ) sin( ) (1 cos( ))S S S S Rφ φ φ = − − − (16d)'

5 4 5 3sin( ) cos( ) sin( )S S S S Rφ φ φ = + + (16f)'

6 6 1 2(1 cos( )) sin( )S S S R S Rφ φ = − − − (16g)

The strain energy in the beam (Palaninathan 1985) can beexpressed as

0

' 2 ' 2 ' 2' 2 ' 2 ' 2

3 5 61 2 4

01 2 2

( ) ( ) ( )( ) ( ) ( )

2

K S S S S K S S RU d

EA GA GA GI EI EI

φ

φ ⎡ ⎤

= + + + + +⎢ ⎥⎣ ⎦

∫ (17)

Using Castigliano’s theorem, the deformation components canbe obtained from (17) as

1 2 3 4 5 6

1 2 3 4 5 6

, , , , ,U U U U U U

u u u u u uS S S S S S

∂ ∂ ∂ ∂ ∂ ∂= = = = = =

∂ ∂ ∂ ∂ ∂ ∂(18)

Then, we set up the relationship between the forces anddeformation for one element. Assembling the elementequations yields the final global equation of the relationshipbetween the external forces and the deflection. For the firstversion of the elephant trunk robot, we have used 161 nodesand 160 elements in the FEM model. Fig. 11 shows therelationship of the load and the deflection of the spring. The

red dot represents the initial position of the spring and the bluedot represents the finial position. Three loads are applied on

this spring. 10= −F1 k at the point [ ]1.7 0 6T

, 2 5= −F k

at the point [ ]0.65 1.57 6T

− and 3 20= −F k at the point

[ ]0.65 1.57 6T

− − .

F1F3 F2 F1F3 F2

Fig. 11 Load and deflection of three loads

C. Implement In this section, we implement the deformation equations.

The mechanical properties of the helical spring are shown inTable 1.

TABLE IMECHANICAL PROPERTIES

Wire diameter 0.0023d m=

Mean spring diameter 0.034 D m=

Number of active coils 10n =

Length of the spring 0.06 L m=

Pitch of the spring 0.09549h m=

Modulus of elasticity 210 E GPa=

Stiffness of the compression 4 38 K Gd nD=

Modulus of rigidity 80G GPa=

A mechanism is set up to test the flexing/load relationshipfor each mechanical spring. The loads are supplied bydifferent weights through wires in every segment. In ourexperiment we used four different weights: 1, 5, 10, and 15Newtons. The load cases are Load 1: V=1 N, T=0 N, M=0.017N.m; Load 2: V=5 N, T=0 N, M=0.085 N.m; Load 3: V=10 N,T=0 N, M=0.17 N.m; Load 4: V=15 N, T=0 N, M=0.255 N.m.The experimental results are compared with the results of the2-dimensional closed form and the 3-dimensional models inTables II and III.

From Table III, it can be seen that the FEM modeling andthe experimental results match very well. The method of loading the compression spring used in this design induces alateral deflection with a relative translation and rotation of theupper compression link. Recall that the lower compressionlink is considered fixed because the motions of the differentsegments are independent.

IV. Actuation System

A. Power Transmission

For the current state of development of the continuousrobot, a bench-top actuation system serves well forexperimentation and further development. It includes thecables that actuate each robot segment and the conduit thathouses and routes the cables. The actuation system alsocontains 24 stepper motors which each motor actuates onecable. The rest of the actuation system consists of assorted

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hardware to connect each pulley to each motor and to connectone cable to each pulley, and Fig. 12 shows a fixture to hold 6motors and pulleys of the total twenty-four.

The overall control and actuation system consists of stepper motors, stepper motor drives, a power supply, amotion controller PCI card located in a host desktop PC, andthe interconnect module. Twenty-four NEMA Size-17, bi-

polar, hybrid, 1.8 ° , DC stepper motors equipped with 3.6:1

gear reduction via an offset spur gear provide mechanicalpower. Twenty-four R208 microstepping drives power themotors. The drives receive direction and velocity commandsfrom the controller and then energize the windings of motoraccordingly. The electrical power is provided by two variablevoltages (0 to 15 V), 40 Amp DC power supplies connected inseries, providing variable voltage over the input range of thedrives (12 to 24 V). The motion controller is DMC-1850twenty-four-axis motion controller that resides in the PCI Busof a standard desktop PC. It has its own microprocessor andmemory and performs all of the motion commands internallywithout using the computers resources. The interconnectmodule is basically an extension of the motion controller andprovides terminals to handle all of the input/output for the

motion controller. This includes committed I/O for each axisalong with several digital and analogy inputs and outputs,which allows for a great deal of expandability ideal for thisstage of development (Fig. 13).

TABLE IICALCULATED AND MEASURED RESULTS FOR 2D MODEL

Closed Model Measured

Load

Axial(m)

Lateral(m)

Angle(rad)

Axial(m)

Lateral(m)

Angle(rad)

Load 1 0.00142 0.0024 0.0780 0.0013 0.0023 0.0721

Load 2 0.00695 0.0113 0.3583 0.0067 0.0108 0.3464

Load 3 0.01382 0.0259 0.8023 0.0135 0.0211 0.7266

Load 4 0.02497 0.0347 1.1093 0.0206 0.0318 1.0677

TABLE III

CALCULATED AND MEASURED RESULTS FOR 3D MODEL

FEA Model Measured

Load

Axial

(m)

Lateral

(m)

Angle

(rad)

Axial

(m)

Lateral

(m) Angle (rad)

Load 1 0.0013 0.0021 0.0725 0.0013 0.0023 0.0721

Load 2 0.0068 0.0108 0.3627 0.0067 0.0108 0.3464

Load 3 0.0136 0.0217 0.7255 0.0135 0.0211 0.7266

Load 4 0.0204 0.0326 1.0882 0.0202 0.0318 1.0677

Pulley

Assemblies

Stepper

Motors

Motor

DrivesPulley

Assemblies

Stepper

Motors

Motor

Drives

Fig. 12 Motor assembly

Host PC

Power Supply12 – 24 VDC

- +

Motion

Controllers

Interconnect

Modules

Drives

MotorsHost PC


- +


- +

Motion

Controllers

Interconnect

Modules

Interconnect

Modules

DrivesDrives

MotorsMotors

Fig. 13 Actuation and control setup

B. Motors and Control SetupThis system provides excellent performance

characteristics suitable for actuating the mechanism. Theangular resolution of the motors allows for very precisecontrol of the cable displacement and, therefore, robotconfiguration. With the gear reduction, this setup is capable

of step sizes as small as 0.5 ° without micro stepping or usingthe finest level of micro stepping the drives are capable of

rotations as small as 0.0625 ° . Depending on pulley size,these rotations correspond to cable displacements as small asapproximately 0.036 mm and 0.0045 mm, respectively. Thedrives use current limiting technology, which allows the use of voltages greater than the motors rated voltage. Each time awinding in the motor is energizing, the higher voltagedecreases the amount of time for the current in the winding tocome up to the rated current after a voltage is applied acrossthe winding, increasing the torque of the motor at higherangular velocities. The holding torque produced when themaximum current is supplied to the windings while the motoris stationary would be approximately equal to the maximumrunning torque, but the friction associated with the gearreduction increases this amount.

C. Motion ControlTo demonstrate the capabilities of the mechanism, a

control scheme [23] in Fig. 14 is implemented to demonstratethe functionality of the flexible mechanism. The signal spaceis divided into four regions, “Hold”, “Open”, “Close”, and“Change Grasp Pattern”. When the Hold command is active,which corresponds to both sets of muscles being at rest, thehand mechanism stays in its current posture. When theextensor muscles are excited, the Close command is activated,

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which causes the hand to tighten its grasp. Likewise, when theflexor muscles are excited, the Open command is activatedwidening the grasp. When both sets of muscles are excited,the Change Grasp Pattern command is activated; it can beused to toggle between different grasping postures.

Fig. 14 State activation chart

To incorporate this control scheme to operate thecontinuous robot, software was developed to use a joystick tosimulate the signals. The joystick, which contains two linearpotentiometers that are each manipulated by one DOF of the

joystick, is connected to the interconnect module. The voltageacross the two potentiometers is monitored. One DOFsimulates the extensor signal, while the other simulates theflexor signal. Proportional control has been developed bybringing into correspondence the distance traveled into eitherthe Open or Close regions with how far the mechanism opensor closes. Joint limits are also incorporated.

V. Conclusion

The design and analysis of a novel hyper-redundantcontinuous robot has been introduced, a computer-controlledbench-top actuation system has been developed and a simplecontrol scheme was implemented. It was shown that loadingof a compression spring in both transverse and axial directions

and using cable-conduit systems allows for controllablemanipulation of the continuous robot. The forward kinematicsof the mechanism is coupled with the element’s dynamiccharacteristics and is also presented. It was shown that for 2-dimensional analysis, the solution of the transverse deflectionof each segment is obtained in a general form. This closedform analytical solution is best used for computer control of the manipulator prosthesis. In the 3-dimensional model, thefinite element method has been developed and implementedinto code. The performance of each segment was testedexperimentally and compared with the numerical results.

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