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Uniform Regulation of a Multi-Section Continuum Manipulator

Ian A. Gravagne and Ian D. Walker

Clemson University, Clemson, SC

igravag@clemson.edu, ianw@ces.clemson.edu

Abstract

Continuum manipulators are robotic manipulators built

using one continuous, elastic, and highly deformable

“backbone” instead of multiple rigid links and joints. In

previous work, we illuminated various kinematic and dy-

namic properties of continuum robots, but the question

of controller design remained open. This paper presents a

basic result for continuum robots that has long been known

for rigid-link robots: that a simple PD-plus-feedforward

controller can exponentially regulate the position of a ma-

nipulator.

1 Introduction

“Continuum robots”, a phrase referenced from a survey

paper by Robinson and Davies [1], refers to a class of ro-

botic manipulators that essentially discard the traditional

robot design paradigm that joins stiff, rigid links with

rotational or prismatic joints. That this design method-

ology has been highly successful in the past is certainly

not under debate; in fact, here and in previous work we

suggest that rigid-link designs will continue to fulÞll the

majority of automation and manipulation need for the

foreseeable future. However, rigid-link design and analy-

sis seems to have reached some practical limits. Among

these, it is difficult and often expensive to design com-

pliance into a rigid-link robot at any point except the

end-effector. We are beginning to see that compliance is

a critical ingredient in the creation of safe, comfortable

and interactive human-robot environments [2]. Compli-

ance is also a signiÞcant asset for the manipulation or ex-

ploration of unknown environments. Additionally, rigid-

link robots generally do not have the ability to use their

entire structure to manipulate things, called “whole-arm

manipulation”, and tend to be complex to build, heavy

(in order to impart stiffness) and somewhat bulky.

Some of the aforementioned issues are being addressed

by creative mechanical designs, which often accept some

ßexibility and vibration in the robot links in exchange for

lighter weight, less complexity and greater safety margin.

The question at hand is, why not abandon the joints al-

together, and make the entire mechanism out of one long

ßexible member? Inspired by trunks, tentacles and snake

backbones, continuum robot designs such as the simple

prototype seen in Þgure 1 attempt to answer this ques-

tion. However, with new properties and capabilities, con-

tinuum robots emphasize the need for new or expanded

results in some areas that long ago matured for rigid-link

robots. In fact, the purpose of this paper is to estab-

lish a basic but extremely important result in continuum

manipulator control, namely, that a PD-plus-feedforward

controller can exponentially regulate the position of a con-

tinuum robot — a result long understood for rigid-link de-

signs.

Figure 1: The Clemson Tentacle Manipulator is a two-

section, four degree-of-freedom continuum robot.

2Background

Our work in continuum manipulators grew out of the

early studies of hyper-redundant and high-degree-of-

freedom (HDOF) devices.

design attempts appear in [3], [4]. More recent designs

appear in [5] and [6]. For background into the kinematics

and path-planning of continuum and hyper-redundant ro-

bots, see [7]- [15]. Also, [10] and [11] discuss the dynamics

of continuum planar robots. Some of the fundamentals in

this work are derived from the Þeld of elastica mechanics,

and details can be found in [17]-[19] and [21].

Some of the initial HDOF

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3Dynamic Model

The fundamental difference between a rigid-link robot

and a continuum robot is that a continuum backbone

exhibits inÞnite-dimensional kinematics, described by dif-

ferential equations. Because currently practical actuation

schemes cannot continuously actuate such a backbone,

the range of actuation is limited to a Þnite-dimensional

subspace of the inÞnite-dimensional kinematics.

“left-over” region permits a desirable property termed

“inherent passive compliance”, but also permits the ex-

istence of vibrations that, depending on the system con-

Þguration and physical properties, may or may not be

controllable. (See [11] for an example of uncontrollable

axial vibration of a continuum robot.)

modes are only one of several ways that a continuum ro-

bot design can tax the theoretical limits of our ability to

model and analyze it. For this reason, the majority of

our work has so far been limited to planar robots, and

in this paper we further restrict the design parameters to

non-extensible backbones with distributed damping and

negligible shear effects.

The

Uncontrollable

Figure 2: A 3-actuator (3-section) continuum robot. Ac-

tuators apply moments to points s1, s2and s3= L.

Figure 2 illustrates a backbone of length L. Arc length

s ranges from 0 to L, so we employ s as an indepen-

dent parameter, in addition to time t. Thus, at every

point s, the backbone has centerline position x(s,t) =

£

bone has bending stiffness EI, with linear mass density

ρ and rotational inertial density Iρ. The kinetic and po-

tential energies of the robot are then

x1(s,t)

θ(s,t) counterclockwise from the horizontal. The back-

x2(s,t)

¤T, and its tangent subtends angle

KE=

1

2

ZL

ZL

0

ρkú xk2+ Iρúθ

EI¡θ0¢2ds.

2ds(1)

PE =

1

2

0

(2)

In addition, deriving the system dynamics requires the

Þrst variation of the work W,

δW=

Z

s1< ... < sn−1< L.

(−búθ)δθds + τnδθ(L,t) +

n−1

X

i=1

τiδθ(si,t), (3)

0<

The term inside the integral represents energy “extract-

ed” from the system by a linear viscoelastic damping

mechanism with damping coefficient b. The remaining

terms reßect intermediate actuator input torques τ1 to

τn−1, applied at points s1to sn−1, as well as boundary

torque τnapplied at sn= L. Note that the work does

not reßect the presence of external shear/axial inputs,

because the robot does not possess actuators capable of

applying those types of forces.

To ease the notational and calculational complexities

associated with the intermediate actuators, we rewrite

the last term above as

"n−1

i=1

n−1

X

i=1

τiδθ(si,t) =

ZL

0

X

τi(t)δ(s − si)

#

δθ(s,t)ds (4)

where δ(si) is the Dirac delta distribution, not to be con-

fused with the variational operator. (The notation should

be contextually clear.) Then, we deÞne the “distributed

moment” as

m(s,t) ,

n−1

X

i=1

τi(t)δ(s − si)(5)

so that the new virtual work expression is

Z

Using the previously deÞned potential and kinetic ener-

gies, the dynamics can be derived as [22], [21]

δW = (m − búθ)δθds + τnδθ(L,t). (6)

−Iρ¨θ − búθ + fTTTq + EIθ00+ m=0(7a)

f0− ρ¨ x

EIθ0(L,t)

=0(7b)

=τn

0

(7c)

f(L,t)= (7d)

along with the geometric conditions θ(0,t) = 0 and

x(0,t) = 0. In a non-extensible thin beam with relatively

large shear modulus, the tangent vector obeys[12]

·

and it easily follows that

x0(s,t) =

cos(θ)

sin(θ)

¸

, q(s,t)(8)

δx0= TTqδθ;T =

·

01

0−1

¸

,(9)

a relationship that we will employ repeatedly in the com-

ing derivations. Note that primes indicate partial differ-

entiation

∂s.

θ

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The force f(s,t) in (7a) identiÞes closely with the role of

a temporary variable. Because we have chosen to neglect

the effects of shear and axial deformations — a physically

realistic assumption for thin backbones — f does not re-

late directly to the primary system position coordinates,

and consequently does not have an associated governing

equation. (Note that its relation to θ is direct in the case

of more complicated models that account for shear, e.g.

[21].) Regardless of whether shear/axial forces contribute

to the deformation of the backbone, they do exist and

boundary condition (7ad) reßects the static shear/axial

end-effector loading, if there is any. For simplicity here,

we assume such loading is negligible but also remark that

this is another important difference between continuum

robots and traditional robots: end-effector loads do de-

form the robot’s shape, and must be accounted for in the

kinematics as well as the dynamics. Details of the pre-

ceding dynamic derivation appear in [22].

4Regulation Stability

A regulation controller attempts to drive the system to

some desired “set-point”. For an inÞnite-dimensional de-

vice with only a Þnite number of actuators, the desired

system solution θd(s) must be restricted to a class of func-

tions deÞned by the allowable static system solutions.

SpeciÞcally, canceling the time-derivative terms in (7)

yields

EIθ00

d(s) +

n−1

X

i=1

τd,iδ(s − si)=0 (10a)

EIθ0

d(L)=τd,n

(10b)

where τd,iis the ithdesired static backbone moment ap-

plied by the actuators.These solutions correspond to

shapes consisting of semi-circular sections end-to-end, as

explored in [11] and [12].

Now consider the control law

τi= τd,i− kd,iúθ(si,t) − kp,i˜θ(si,t)

where τd,i are the desired static holding torques and

kd,i> 0. Referring to (7c) and (10b), this gives the bound-

ary control law

(11)

EI˜θ

0(L,t) = −kd,núθ(L,t) − kp,i˜θ(L,t)

where˜θ = θ − θd.

The energy-based Lyapunov candidate

ZL

(12)

V1=1

2

0

ρkú xk2+Iρúθ

2+EI

³˜θ

0´2

ds+

n

X

i=1

kp,i

2

˜θ(si,t)2

(13)

has time derivative

ZL

úV1=

0

ρú xT¨ x+Iρúθ¨θ +EIúθ

0˜θ

0ds+

n

X

i=1

kp,i˜θ(si,t)úθ(si,t).

(14)

Substituting for the dynamics gives, after integration by

parts and some cancelation,

úV1

=

ZL

+EIúθ(L,t)˜θ

0

úθ(−búθ + m + EIθ00) − EIúθ˜θ

00ds(15)

0(L,t) +

n

X

i=1

kp,i˜θ(si,t)úθ(si,t)

=

ZL

0

úθ

"

−búθ +

n−1

X

i=1

τd,iδ(s − si) + EIθ00

d

#

ds

−

ZL

n−1

X

i=1

kd,iúθ(si,t)2− kd,núθ(L,t)2

=

0

−búθ

2ds −

n

X

i=1

kd,iúθ(si,t)2.

Note that (10a) was used in the second step. Thus, the

individual quantities in V remain bounded for any given

set of initial conditions and desired backbone moments,

and furthermore, since˜θ(s,t)2≤ LRL

in the next section.

0(˜θ

0)2ds, then˜θ is

point-wise bounded as well, a fact that will be employed

5Regulation Convergence

The new Lyapunov candidate is

V=V1+ V2

ZL

(16)

V2

=ε

0

Iρúθ˜θ + ρ

µZs

0

˜θqTTdσ

¶

ú x +1

2b˜θ

2ds (17)

where V1is given in the previous section. First we prove

that V > 0 for a sufficiently small ε > 0. With the use of

standard inequalities,

ZL

0

w(s)2ds≤L2

ZL

0

w0(s)2dsfor w(0) = 0(18)

pq≤αp2+1

αq2

for α > 0(19)

we may upper bound V2as

ZL

+1

V2

≤ε

0

2bL2³˜θ

ZL

½

Iρúθ

2+ IρL2³˜θ

ds

0´2

+ ρL2˜θ

2+ ρú xTú x(20)

0´2¾

2+ L2

≤ε

0

Iρúθ

µ

Iρ+1

2b + ρL2

¶³˜θ

0´2

+ ρú xTú xds.

Consequently, we may choose ε sufficiently small that V

remains positive deÞnite.

At this point we make the important observation that,

while V1appears to be positive deÞnite in four generalized

coordinates (two linear velocities, one angular velocity

and one curvature), in fact only two are required,úθ(s,t)

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and˜θ

that the inequality (18) gives

0(s,t). This observation can be formalized by noting

ZL

0

ρú xTú xds ≤ L2

ZL

0

ρúθ

2ds.(21)

So, if we deÞne the “minimal” functional,

V∗=1

2

ZL

0

úθ

2+ (˜θ

0)2ds +

n

X

i=1

kp,i˜θ(si,t)2

(22)

then we may arrive at new upper and lower bounds for

V ,

C1V∗

≤

=

V ≤ C2V∗

min©(1 − ε)(ρL2+ Iρ),

max©(1 + ε)(ρL2+ Iρ),

(23)

C1

EI − εL2(2ρL2+ 2Iρ+ b),1ª

EI + εL2(2ρL2+ 2Iρ+ b),1ª

C2

=

where C1and C2are constants depending only upon sys-

tem parameters. Note that C1can always be made posi-

tive by proper choice of ε.

The time derivative of V is, after some cancelation,

úV=

úV1+ ε

ZL

0

n

Iρúθ

µZs

2+˜θ¡EIθ00+ m¢

úθ˜θqTdσ

(24)

+ρú xTú x − ρ

0

¶

ú x

¾

ds.

Integrating the third term in (24) gives (25) [next page].

In second step of (25), by (10a), we are simply subtract-

ing zero but implicitly signifying that we restrict allowable

desired solutions θd(s) to those that satisfy (10a). Also

note that, in the fourth step, control law (12) was im-

plemented, and in the Þnal step, we choose γ sufficiently

small that (EI − γnL) > 0. Finally, the last term in (24)

gives

ZL

ZL

ZL

ZL

ZL

0

−ρ

µZs

0

úθ˜θqTdσ

¶

ú xds (26)

≤

0

ρú xTú x + ρ

µZs

0

úθ˜θqTdσ

¶µZs

qúθ˜θ

0

qúθ˜θdσ

¶

ds

≤

0

ρL2úθ

2+ ρL2húθ˜θqTih

2+ ρL2húθ

µ

i

ds

=

0

ρL2úθ

·

2˜θ

2i

ds

≤

0

ρL2

1 + max

s,t

˜θ

2¶¸

úθ

2ds.

For the last step here, we refer to the section on “Regu-

lation Stability”, where it was proven that, for any initial

conditions and Þnite desired backbone moments,˜θ(s,t)2

is point-wise bounded. Assembling (21), (25) and (26)

together, along with (15), yields

úV≤

ZL

−ε(EI − γnL)(˜θ

n

X

Thus, for any given trajectory, ε can always be chosen

small enough to ensure thatúV is negative deÞnite. With

(23), we have that

0

−

½

b − ερL2

µ

0)2ds

¾

2 + max

s,t

˜θ

2¶¾

úθ

2

(27)

−

i=1

½

kd,i−ε

γk2

d,i

úθ(si,t)2−

n

X

i=1

εkp,i˜θ(si,t)2

úV≤−λV,

min

(28)

¾

λ=

½

b − ερL2

µ

2 + max

s,t

˜θ

2¶

,ε(EI − γnL),ε

C1

the solution of which leads to

V∗(t) ≤C2

C1V∗(0)e−λt,

exponential convergence of V∗, and consequently point-

wise exponential convergence of˜θ(s,t). Of course, a given

choice of ε for one trajectory may not, in general, be

suitable for another trajectory because max˜θ

between the two.

The reader may note that, in the Þnal analysis, the

point dampers associated with each actuator do not

contribute to the convergence result. However, we ex-

perimentally veriÞed in [11] that additional lumped-

parameter damping does in fact signiÞcantly improve per-

formance for the PD controller without the feedforward

term; thus it is necessary to prove that the point dampers

do not disturb the stability and convergence results re-

gardless of their utility in the said results per se.

We make one Þnal remark regarding the presence of

rotational inertia, Iρ. Since linear mass density ρ over-

whelmingly eclipses Iρ for most “long and thin” elastic

members, it may be stricken from the preceding deriva-

tions (i.e. Iρ = 0). With only slight modiÞcation, the

essential results remain.

2may differ

6 Simulation and Conclusions

In [22] we derive a Þnite-element model for the dynamics

of a 2-section continuum robot. Figure 3 illustrates the

time plot for the simulated robot with parameters EI =

1, ρ = 0.7, and Iρ= 0.001. These constants are scaled

to realistic ratios for long thin backbones; note how small

the rotational density is compared to the inertial density.

The robot has unit length, and is driven with control gains

of kd,i = 0.4 and kp,i = 20. The damping coefficient is

b = 3.5. Figure 4 illustrates a time-lapse image of the

robot.

Page 5

ZL

0

˜θ¡EIθ00+ m¢ds=

ZL

ZL

0

˜θ

Ã

Ã

h

EIθ00+

n−1

X

(

i=1

h

τd,i− kd,iúθ(si,t) − kp,i˜θ(si,t)

i

δ(s − si)

!

ds(25)

=

0

˜θEIθ00−EIθ00

d(s) +

n−1

X

i=1

τd,iδ(s − si)

)

+

n−1

X

EI˜θ˜θ

i=1

τd,i− kd,iúθ(si,t) − kp,i˜θ(si,t)

i

δ(s − si)

!

ds

=

ZL

ZL

ZL

ZL

ZL

0

00ds +

n−1

X

i=1

˜θ(si,t)

h

−kd,iúθ(si,t) − kp,i˜θ(si,t)

i

=

0

−EI

³˜θ

³˜θ

³˜θ

0´2

0´2

0´2

ds +

n−1

X

n

X

n

X

0)2ds +

i=1

h

h

·1

−kd,iúθ(si,t)˜θ(si,t) − kp,i˜θ(si,t)2i

−kd,iúθ(si,t)˜θ(si,t) − kp,i˜θ(si,t)2i

+˜θ(L,t)EI˜θ

0(L,t)

=

0

−EI ds +

i=1

;sn= L

≤

0

−EI ds +

i=1

γk2

d,iúθ(si,t)2+ γ˜θ(si,t)2− kp,i˜θ(si,t)2

·1

¸

;γ > 0

≤

0

−(EI − γnL)(˜θ

n

X

i=1

γk2

d,iúθ(si,t)2− kp,i˜θ(si,t)2

¸

.

00.2 0.40.60.811.21.41.6 1.82

0

0.5

1

1.5

Time (s)

Angle (rads)

θ(1/2)

θ(1)

Figure 3: Angles θ(1

section robot.

2) and θ(1) for the simulated two-

To summarize, this work illustrates that a basic con-

trol property of rigid-link robots also holds for contin-

uum robots: a PD plus feed-forward controller can ex-

ponentially stabilize the system, under the assumption

that distributed damping exists on the continuum back-

bone. Though space does not permit closer examination

here, future work will disclose the details leading up to

the Lyapunov candidate (16), as well as arguments de-

fending the use of viscous damping on the distributed

coordinate θ(s,t). Reference [12] contains these details,

and additional derivations that closely link the structure

of continuum and “traditional” manipulators, lending fur-

ther credence to the concept of a “uniÞed” theory of ma-

nipulation that describes continuum and rigid-link robots

simply as special cases.

0 0.20.4 0.60.81

−0.1

0.1

0.3

0.5

0.7

0.9

x1

x2

t=0

t=0.1

t=0.2

t=0.3

t=2.0

Figure 4: The two-section continuum robot moves from

the horizontal straight conÞguration to the pose θ(1

π/2 and θ(1) = 0.

2) =

As the idea of continuum robots is still relatively new, a

multitude of questions remain unanswered. Among those