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Interconnection and damping assignment passivity-based experimental

control of a single-link flexible robot arm

Arantza Sanz and Victor Etxebarria

Abstract Energy-based control design methodology (theso-called IDA-PBC, interconnection and damping assignmentpassivity-based control) is a well established technique thathas shown to be very powerful to design robust controllersfor physical systems described by Euler-Lagrange equationsof motion. Their application potential is particularly importantfor under-actuated systems. This paper presents the applicationof this control design methodology to a single-link flexiblerobotic arm. It is shown that the method is well suited tohandle this kind of under-actuated devices not only from atheoretical viewpoint but also in practice. A Lyapunov analysisof the closed-loop system stability is given and the designperformance is illustrated by means of a set of simulationsand laboratory control experiments, comparing the resultswith those obtained using conventional control schemes formechanical manipulators.

I. INTRODUCTION

Flexible-link robotic manipulators have many advantages

with respect to conventional rigid robots. These mechanisms

are built using lighter, cheaper materials, which improve the

payload to arm weight ratio, thus resulting in an increase of

the speed with lower energy consumption.

However, for a flexible-link manipulator the number of

controlled variables is strictly less than the number of me-

chanical degrees of freedom, so the control aim is double in

this case. First, a flexible manipulator controller must achieve

the same motion objectives as a rigid manipulator. Second,it must also stabilize the vibrations that are naturally excited.

The existence of the slow (rigid) and fast (flexible) modes

allows the application of the singular perturbation theory to

flexible arms ([5]).Then, a composite control strategy can be

adopted, with the controller having slow and fast terms.This

combined slow-fast strategy has proved to be a promising

control method for robotic applications ([6], [7]).

In recent years, control design methods based on the

shaping of energy functions have attracted a lot of attention.

This fact added to the growing interest in the control of

under-actuated mechanical systems, has resulted in a suc-

cessful development of the so-called IDA-PBC technique,

which is a formulation based on passivity very well suited

to the problem of stabilization of under-actuated systems

([3], [11]). This energy-based control design method, initially

introduced in [8], [9], [10], adds, to the customary modifi-

cation of the potential energy, the kinetic energy shaping,

as an indispensable via to save the obstacles related to the

under-actuation. The method produces a closed-loop system

A. Sanz and V. Etxebarria are with the Dept. Electricidad yElectrnica, Facultad de Ciencia y Tecnologa, Universidad del PasVasco, Apdo. 644, Bilbao 48080, Spain, wmbsabaa@lg.ehu.es,victor@we.lc.ehu.es

with hamiltonian structure, where it is wanted the closed-

loop energy to have a minimum in the desired equilibrium for

this assures its stability. The main advantage of this technique

based on passivity is that the Lyapunov function is obtained

in a natural way by the structure of the system itself.

In this paper the application of the IDA-PBC method to

the control of a single-link flexible robotic arm is presented.

Both a theoretical stability analysis and experimental results

on a laboratory flexible arm are provided, to illustrate the

effectiveness of the proposed approach.

II. IDA-PBC METHOD

In this section a brief outline of the IDA-PBC method is

presented, (see [9], [10] for complete details). The goal is to

control the position of an under-actuated mechanical system

with total energy:

H(q, p) =1

2pTM1(q)p + U(q) (1)

where q n, p n, are the generalized positions andmomenta respectively, M(q) = MT(q) > 0 is the inertiamatrix and U(q) is the potential energy.

If it is assumed that the system has no natural damping,

then the equations of motion of such system can be written

as:

q

p

=

0 In

In 0

qHpH

+

0

G(q)

u (2)

In the IDA-PBC method two basic steps of the passivity-

based control (see [8]) are followed. The first of them is

energy shaping, where the total energy function of the system

is modified so that a predefined energy value is assigned

to the desired equilibrium. The second step is damping

injection, to achieve asymptotic stability.

The following form for the desired (closed-loop) energy

function is proposed:

Hd(q, p) =1

2pTM1d (q)p + Ud(q) (3)

where Md = MTd > 0 is the closed-loop inertia matrix and

Ud the potential energy function.

It will be required that Ud have an isolated minimum at

the equilibrium point q, that is:

q = argminUd(q) (4)

In PBC, a composite control law is defined:

Proceedings of the 2006 IEEEInternational Conference on Control ApplicationsMunich, Germany, October 4-6, 2006

FrA08.2

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u = ues(q, p) + udi(q, p) (5)

where the first term is designed to achieve the energy shaping

and the second one injects the damping.

The desired (closed-loop) dynamics can be expressed in

the following form:

q

p

=

Jd(q, p) Rd(q, p) qHd

pHd

(6)

where

Jd = JTd =

0 M1Md

MdM1 J2(q, p)

(7)

Rd = RTd =

0 00 GKvG

T

0 (8)

represent the desired interconnection and damping structures.

J2 is a skew-symmetric matrix, and can be used as free

parameter in order to achieve the kinetic energy shaping (see

[9]).The second term in (5) can be expressed as:

udi = KvGTpHd (9)

where Kv = KTv > 0.

A. Energy shaping

To obtain the energy shaping term ues of the controller, (5)

and (9) are replaced in (2) and this is equated to (6):

Gues = qH MdM1qHd + J2M

1

d p (10)

In the under-actuated case, G is not invertible, but only fullcolumn rank. Thus, multiplying (10) by the left annihilator

of G, GG = 0, it is obtained:

G{qH MdM1qHd + J2M

1

d p} = 0 (11)

If a solution (Md, Ud) exists for this differential equation,then ues will be expressed as:

ues = (GTG)1GT(qH MdM

1qHd + J2M1

d p)(12)

The PDE (11) can be separated in terms corresponding to

the kinetic and the potential energies:

G{q(pTM1p)MdM

1q(pTM1d p)+2J2M

1

d p}=0(13)

G{qU MdM1qUd} = 0 (14)

So the main difficulty of the method is in solving the

nonlinear PDE (13). Once the closed-loop inertia matrix, Md,

is known, then it is easier to obtain Ud of the linear PDE

(14).

Fig. 1. Photograph of the experimental flexible manipulator.

III. SINGLE-LINK FLEXIBLE ROBOT ARM

A. Mathematical model

The dynamics of a flexible manipulator can be described by

means of different methods. One of the more frequently em-

ployed combines the modal analysis with Lagranges method

(see [1]). The use of this technique provides a dynamical

model expressed by ordinary differential equations starting

from a representation given by partial differential equations.

The object of the study is a flexible arm with one grade offreedom that accomplishes the conditions of Euler-Bernoulli.

In this case, the elastic deformation of the arm w(x, t) isexpressed as a superposition of a infinite number of modes

where the spatial and time variables are separated:

w(x, t) =

i=1

i(x)qi(t) (15)

where i(x) is the mode shape and qi(t) is the modeamplitude.

In order to derive a finite-dimensional ODE, a finite

number of modes m is considered. Lagrange equations lead

to a dynamical system defined by m + 1 second orderdifferential equations:

It 00 I

q0(t)qi(t)

+

0 00 Kf

q0(t)qi(t)

=

1

(0)

u

(16)

where q0 and qi are the dynamic variables; q0(t) is therigid generalized coordinate, qi(t) is the vector of flexiblemodes, It is the total inertia, Kf is the stiffness matrix that

depends on the elasticity of the arm, and it is defined as

Kf = diag(2

i ), where i is the resonance frequency of eachmode; are the first spatial derivatives of i(x) evaluated at

the base of the robot. Finally u includes the applied controltorques.

Defining the incremental variables as q = q qd, whereqd is the desired trajectory for the robot such that q0d = 0and qid = 0, then the dynamical model is given by:

It 00 I

q0(t)qi(t)

+

0 00 Kf

q0(t) + q0d

qi(t)

=

1

(0)

u

(17)

The total energy of the mechanical system is obtained as

the sum of the kinetic and potential energy:

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H(q, p) =1

2pTM1p +

1

2qTKq (18)

where p =

It q0 IqiT

and

M =

It 00 I

K =

0 00 Kf

The point of interest is q = (0, 0), which corresponds tozero tracking error in the rigid variable and null deflections.

B. Controller design

The controller design will be made in two steps; first, we will

obtain a feedback of the state that produces energy shaping

in closed-loop to stabilize the position globally, then it will

be injected the necessary damping to achieve the asymptotic

stability by means of the negative feedback of the passive

output.

The inertia matrix, M, that characterizes to the system, is

independent of q, hence it follows that it can be chosen J2 =

0, (see [9]). Then, from (13) it is deduced that the matrixMd should also be constant. The resulting representationcapturing the first flexible mode is

Md =

a1 a2a2 a3

where the condition of positive definiteness of the inertia

matrix, lead to the following inequations:

a1 > 0 a1a3 > a2

2(19)

Now, considering only the first flexible mode, equation

(14) corresponding to the potential energy can be written as:

(a2

(0)a1It

)Ud

q0+ (a3

(0)a2)Ud

q1= q1

2

1(20)

which it is a trivial linear PDE whose general solution is

Ud = I2t

2

1(a3(0)a2)

2(a2+ (0)a1)2q20

It

2

1

(a2+ (0)a1)q0q1+F(z) (21)

z = q0 + q1 (22)

=It(a3 (0)a2)

(a2 + (0)a1)(23)

where F is an arbitrary differentiable function that we should

choose to satisfy condition (4) in the points q.

Some simple calculations show that the necessary restric-

tion qUd(0) = 0 is satisfied if F(z(0)) = 0, whilecondition 2qUd(0) > 0 is satisfied if:

Fz >21

(a3 (0)a2)(24)

Under these restrictions it can be proposed F(z) =1

2K1z

2, with K1 > 0, which produces the condition:

a3 > (0)a2 (25)

So now, the term corresponding to the energy shaping in

the control input (12) is given as

ues = (GTG)1GT(qUMdM

1qUd) = Kp1q0+Kp2q1(26)

where

Kp1 =It(a1a3 a22)

(a2 + (0)a1)2(K1(a3 +

(0)a2) + 2

1)

Kp2 =a1

2

1 K1(a1a3 a22)

(a2 + (0)a1)

The controller design is completed with a second term,

corresponding to the damping injection. This is achieved via

negative feedback of the passive output GTpHd, (9). AsHd = 12p

TM1d p + Ud(q), and Ud only depends on the qvariable, udi only depends on the appropriate election ofMd:

udi = Kv1 q0 + Kv2 q1 (27)

with

Kv1 = KvIt(a3 +

(0)a2)

(a1a3 a22)

Kv2 = Kv(a2 (0)a1)

(a1a3 a22)

A simple analysis on the constants Kp1 and Kv1, with the

conditions previously imposed, implies that both should be

negative to assure the stability of the system.

C. Stability Analysis

Proposition 1: Consider the robot manipulator model (17)

in closed-loop with the control law

u = Kp1q0 + Kp2q1 + Kv1 q0 + Kv2 q1 (28)

The closed-loop equation (17), (28) is given by

ddt

q0

q1q0

q1

=

q0

q1

1

It(Kp1q0 + Kp2q1 + Kv1 q0 + Kv2 q1)

Kfq1+(0)(Kp1q0+Kp2q1+Kv1 q0+Kv2 q1)

(29)

It is simple to check that there exists a unique equilibrium

point (q, q) = (0, 0) for the closed-loop system. The point(q, q) = (0, 0) is globally asymptotically stable for (17), (28).

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Proof: To analyze the stability of the closed-loop system we

consider the energy-based Lyapunov function candidate ([2],

[4])

V(q, q) =1

2pTM1d p + Ud =

12

I2t q2

0a3 2It q0 q1a2 + q2

1a1

a1a3 a22

12

I2t 21(a3 (0)a2)q20(a2 + (0)a1)2

It

21

q0q1

a2 + (0)a1+

1

2K1(q0 + q1)

2 (30)

which is globally positive definite, i.e: V(0, 0) = (0, 0) andV(q, q) > 0 for every (q, q) = (0, 0).

The time derivative of (30) along the trajectories of the

closed-loop system (29), can be written as

V(q, q)=Kv(It q0(a3+

(0)a2)+ (a2(0)a1) q1)

2

(a1a3 a2

2)2

0

(31)

where Kv > 0, so V is negative semidefinite, and (q, q) =(0, 0) is stable (not necessarily asymptotically stable).

By using LaSalles invariant set theory, it can be defined

the set R as the set of points for which V = 0

R = {

q0q1q0

q1

4 : V(q, q) = 0} =

= { q0, q1 1 & q0 + q1 = 0 } (32)

Following LaSalles principle, given R and defined N asthe largest invariant set of R, then all the solutions of the

closed loop system asymptotically converge to N when t .

Any trajectory in R should verify:

q0 + q1 = 0 (33)

and therefore it also follows that:

q0 + q1 = 0 (34)

q0 + q1 = K (35)

Considering the closed-loop system, described by (29) and

the conditions described by (34) and (35), the following

expression is obtained:

(

It+(0)Kf)K+(

21

(a2+(0)a1)+(0))q0= 0 (36)

As a result of the previous equation, it can be concluded

that q0(t) should be constant, in consequence q0(t) = 0, andreplacing it in (33), it also follows that q

1(t) = 0. Thereforeq0 = q1 = 0 and replacing these in (29) this leads to thefollowing system:

Property Value

Motor inertia, Ih 0.002kgm2

Link length, L 0.45mLink height, h 0.02mLink thickness, d 0.0008mLink mass, Mb 0.06kgLinear density, 0.1333kg/mFlexural rigidity, EI 0.1621Nm2

TABLE I

FLEXIBLE LINK PARAMETERS

1

It(Kp1q0 + Kp2q1) = 0

Kfq1 + (0)(Kp1q0 + Kp2q1) = 0

whose solutions are of the formq0(t)q1(t)

=

00

In other words, the largest invariant set N is just the

origin (q, q) = (0, 0), so we can conclude that any trajectoryconverge to the origin when t , so the equilibrium isin fact asymptotically stable.

IV. SIMULATION AND EXPERIMENTAL RESULTS

A. Simulations

To illustrate the performance of the proposed IDA-PBC

controller, in this section we present some simulations. We

use the model of a flexible robotic arm presented in [1]

and the values of Table I which correspond to the real arm

displayed in Fig. 1. The results are shown in Figs. 2 to 4. In

these examples the values a1 = 1, a2 = 0.01 and a3 = 50

have been used to complete the conditions (19) and (25). InFig. 2 the parameters are K1 = 10 and Kv = 1000. In Figs.3 and 4 the effect of modifying the damping constant Kvis demonstrated. With a smaller value of Kv, Kv = 10, therigid variable follows the desired trajectory reasonably well.

For Kv = 1000, the tip position exhibits better trackingof the desired trajectory, as it can be seen comparing Fig.

2(a) and Fig. 3(a). Even more important, it should be noted

that the oscillations of elastic modes are now attenuated

quickly (compare Fig. 2(b) and Fig. 3(b)). But If we continue

increasing the value of Kv, Kv = 100000, the oscillationsare attenuated even more quickly (compare Fig. 2(b) and

Fig. 4(b)), but the tip position exhibits worse tracking of the

desired trajectory.

B. Experimental results

The effectiveness of the proposed control schemes has been

tested by means of real time experiments on a laboratory

single flexible link. This manipulator arm, fabricated by

Quanser Consulting Inc. (Ontario, Canada), is a spring steel

bar that moves in the horizontal plane due to the action of a

DC motor. A potentiometer measures the angular position of

the system, and the arm deflections are measured by means

of a strain gauge mounted near its base (see Fig. 1).

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(a)0 2 4 6 8 10 12 14 16 18 20

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

Rigidvariableandref(rad)

(b)0 2 4 6 8 10 12 14 16 18 20

4

3

2

1

0

1

2

3

4x 10

4

Time (s)

Flexibledeflections(rad)

Fig. 2. Simulation results for IDA-PBC control withKv = 1000: (a) Timeevolution of the rigid variable q0 and reference qd ; (b) Time evolution ofthe flexible deflections.

(a)0 2 4 6 8 10 12 14 16 18 20

1

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

Time (s)

Rigidvariableandref

(rad)

(b)0 2 4 6 8 10 12 14 16 18 20

8

6

4

2

0

2

4

6

8x 10

3

Time (s)

Flexibledeflectio

ns(rad)

Fig. 3. Simulation results for IDA-PBC control with Kv = 10: (a) Timeevolution of the rigid variable q0 and reference qd ; (b) Time evolution ofthe flexible deflections.

(a)0 2 4 6 8 10 12 14 16 18 20

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

Rigidvariableandref(rad)

(b)0 2 4 6 8 10 12 14 16 18 20

2

1.5

1

0.5

0

0.5

1

1.5

2x 10

4

Time (s)

Flexibledeflections(rad)

Fig. 4. Simulation results for IDA-PBC control with Kv = 100000: (a)Time evolution of the rigid variable q0 and reference qd ; (b) Time evolutionof the flexible deflections.

The experimental results are shown on Figs. 5 and 6.

In Fig. 5 the control results using a conventional PD rigid

control design are displayed. The rigid variable tracks the

reference (with a certain error), but the naturally excited

flexible deflections are not well damped (Fig. 5(b)).

In Fig. 6 the results using the IDA-PBC design philosophy

are displayed. As seen in the graphics, the rigid variable

follows the desired trajectory, and moreover the flexible

modes are now conveniently damped, (compare Fig. 5(b)

and Fig. 6(b)). It is shown that vibrations are effectively

attenuated in the intervals when qd reaches its upper and

lower values which go from 1 to 2 seconds, 3 to 4 s., 5 to

6 s., etc.

V. CONCLUSIONS

The IDA-PBC method is a control design tool based on

energy concepts. This work has presented a theoretical andexperimental study of this method for controlling a labora-

tory single flexible link, valuing in this way the potential of

this technique in its application to under-actuated mechanical

systems, in particular to flexible manipulators. The study is

completed with the Lyapunov stability analysis of the closed-

loop system that is obtained with the proposed control law.

Then, as an illustration, a set of simulations and laboratory

control experiments have been presented. The experimented

scheme, based on an IDA-PBC philosophy, has been shown

to achieve good tracking properties on the rigid variable and

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(a)0 2 4 6 8 10 12 14 16 18 20

0.6

0.4

0.2

0

0.2

0.4

0.6

Time (s)

Rigid

variableandref(rad)

(b)0 2 4 6 8 10 12 14 16 18 20

0.03

0.02

0.01

0

0.01

0.02

0.03

Time (s)

Flexible

deflections(rad)

Fig. 5. Experimental results for a conventional PD rigid controller: (a) Timeevolution of the rigid variable q0 and reference qd ; (b) Time evolution ofthe flexible deflections.

(a)0 2 4 6 8 10 12 14 16 18 20

0.6

0.4

0.2

0

0.2

0.4

0.6

Time (s)

Rigid

variableandref

(rad)

(b)0 2 4 6 8 10 12 14 16 18 20

0.03

0.02

0.01

0

0.01

0.02

0.03

Time (s)

Flexible

deflectio

ns(rad)

Fig. 6. Experimental results for IDA-PBC control: (a) Time evolution ofthe rigid variable q0 and reference qd ; (b) Time evolution of the flexibledeflections.

definitely superior damping of the unwanted vibration of

the flexible variables compared to conventional rigid robot

control schemes.

ACKNOWLEDGMENT

The authors are grateful to UPV/EHU and to the Basque

Government for partial support of this work through project

9/UPV 00224.310-15254/2003 and grant BFI04.440, respec-

tively.

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