07074833

Modeling and Validation of a Planar Flexible Manipulator

L. Bossi, L. Magni, C. Rottenbacher and G. Mimmi

Abstract The modeling and validation problem of a labo-ratory planar flexible manipulator with a link in compositematerial has been considered. The goal of the paper is toobtain a model useful for the synthesis of feedback controllaws. A mathematical linear model, based on the Hamiltonprinciple, is complemented with a nonlinear friction model.Suitable experiments are designed in order to identify the pa-rameters of the model that cannot be measured or analyticallycomputed. Once the single components of the model have beenidentified, the whole model has been validated with open- andclosed-loop experiments. The closed loop validation has beenperformed using two control strategies. The first one is a singleinput (position error) single output (motor torque) controllersynthesized in the frequency domain and the second one is amulti input (position error and link deflection) single output(motor torque) controller based on a Linear Quadratic (LQ)optimal control law complemented with an observer and anintegral action.

I. INTRODUCTION

Lightweight exible manipulators have been a widely

investigated topic in the eld of mechatronic systems. They

represent an attractive alternative to heavy and bulky robots

in a wide spectrum of applications because of their high

payload-weight ratio and lower energy consumption. Ex-

amples of this go from simple pick and place tasks of an

industrial robot to micro-surgery and substitution of human

operations in inhospitable environments like nuclear plant

or space [3]. Many papers on modeling and control issue of

such systems have appeared during the last decades exploring

a huge variety of different strategies some of which has

proved rather effective in real applications [1]. Very complex

and accurate models have been developed to describe the

nonlinear dynamical behavior of multilink lightweight robots

[5]. However, more experimental investigations should be

carried on to better understand the actual effectiveness of

the various techniques developed in this eld. Moreover

only very few researchers considered the possibility of

modeling manipulators in composite materials so that a

lot of experimental work is still required to control the

exibility effects [6]. This paper is intended to ll the gap

of literature in the eld of theoretical modeling validation

and control of exible manipulators with composite material

links. In particular the goal of this paper is to obtain a model

useful for the synthesis of feedback control laws. A linear

mathematical model, derived by using the Hamilton principle

L. Bossi and L. Magni are with the Dipartimento di Informatica eSistemistica, Universita degli Studi di Pavia, via Ferrata 1, 27100 Pavia,Italy. {luca.bossi, lalo.magni}@unipv.it

C. Rottenbacher and G. Mimmi are with the Dipartimento di MeccanicaStrutturale, Universita degli Studi di Pavia, via Ferrata 1, 27100 Pavia, Italy.{rottenbacher, giovanni.mimmi}@unipv.it

approach and complemented with a nonlinear friction model

is considered. Then a parameters identication procedure is

described. In particular suitable experiments are designed in

order to identify the parameters of the model that cannot

be measured or analytically computed. The closed loop

validation has been done using two control strategies. The

rst one is a single input (position error) single output (motor

torque) controller synthesized in the frequency domain and

the second one is a multi input (position error and deection)

single output (motor torque) controller based on an LQ

control law complemented with an observer and an integral

action. Conclusions are drawn in the last section.

II. EXPERIMENTAL APPARATUS

The experimental device is a part of the TEMSRAD

(Testbed for Microgravity Simulation in Robotic Arm Dy-

namics) [9] and consists of a exible robotic arm driven by

a brushless servomotor, operating in a working space com-

patible with the volume of a standard Express Pallet Adapter

(EPA) for on board experiments on the International Space

Station (ISS). It is made of an aluminium fork and composite

laminate material beam. The robotic arm is suspended on a

special air-pad oating on a planar friction-free glass surface

in order to simulate the dynamic behavior in a micro-gravity

environment. In this way the torsional vibration components

are reduced even with a payload mounted at the end point

of the manipulator. The link in kevlar bre tissue is very

exible in the operating plane; on the contrary it can be

considered rigid in the other directions. The robotic arm is

actuated with a Kollmorgen 713RBH brushless motor with

a maximum torque of 0.3 Nm driven by a sinusoidal digital

servoamplier Danhaer Motion Servostar S606, capable to

produce a maximum current of 6A. An internal current loop,

realized by the servo amplier, with parameters tuned by

the manufacturer, supplies the control torque of the motor.

A resolver is mounted on motor axis to furnish the rotor

position, needed by the amplier in order to provide the

right signal for motor driving. The system is equipped with

strain gauges in a full-bridge conguration mounted both

at the base and in the middle of the link, to measure

deections and with a potentiometer mounted on the motor

hub, to measure the angular position of the motor shaft.

The Real Time Application used is the MATLAB Real

Time Target interfaced with a PCMCIA DAQ Card 6036E

by National Instruments. This interface permits a real time

signal monitoring of potentiometer, strain gauges, actual

in-phase current (active component Iq), and enables to aneasy implementation of innovative control strategies without

wasting time on programming issues.

Proceedings of the European Control Conference 2009 Budapest, Hungary, August 2326, 2009 TuB14.5

2809 Copyright EUCA 2009978-3-9524173-9-3

III. MODEL DESCRIPTION

A. Linear Model Equations

The following boundary value problem is obtained apply-

ing the Hamilton Principle [8] using the reference frame Xp,Yp (see Fig. 1) passing through the center of mass:{

EIwp (x, t) + (wp (x, t) + x(t)) = 0(t) J(t) = 0

(1)

where x and t are respectively spatial and time coordinate,wp(x, t) is the link deection, is the linear mass density,(t) is the applied torque, (t) is the angle of the center ofmass, E is the Young modulus, I is the cross area inertia andJ is the total inertia of the system considering the rotatinginertia of the joint, the inertia of the payload at the tip and

the inertia of the beam.

For the expansion theorem the solution of (1) can be

represented by an absolutely and uniformly convergent series

in the eigenfunctions in the form

wp(x, t) =i=1

pi(x)pi(t)

where pi(x) is the exact eigenfunction and pi(t) is the timedependent term. Truncating the series at the n consideredmodes of vibration, the following system is derived [4]{

J(t) = (t)

pi (t) + 2ipi (t) + 2i pi (t) =

pi(0)(t)

i = 1, . . . , n where 2i are the eigenvalues of the systemwhich correspond to the squares of angular frequencies

i = 2i

EI/ , being i the innite solutions of the

characteristic equation of the eigenvalue problem [2]. Note

that the damping ratio includes the dissipation effects dueto the internal frictions of the link, the friction between

the air-pad and the table and the air resistance. However,

it does not represent the damping of the overall structure,

in fact, the effect of other friction sources such as the

friction produced by the motor will be described separately

in the next subsection. As it will be claried in the model

identication section, the decision to consider some friction

phenomena in the damping ratio and some explicitly inthe friction model is driven by the possibility to design

experiments able to identify them.

In order to obtain the system description with respect to

the clamped reference frame XcYc (see Fig. 1) the followingchange of coordinates is done:

ci(x) = pi(x) x

pi(0)

ci(t) = pi(t) = i(t)

where ci(x) is the exact eigenfunction and ci(t) is the timedependent term in the new reference frame.

In this way the hub angle hub(t), the tip angle tip(t)and the deection at the end point of the end-effector in the

clamped reference frame wc(l, t) can be derived as follows:

hub(t) = (t) +n

i=1

pi(0)i(t)wc(l, t) =

ni=1 ci(l)i(t)

tip(t) = hub(t) +wc(l,t)

l= (t) +

wp(l,t)l

Fig. 1. Single Link Manipulator

where

wp(l, t) =n

i=1

pi(l)i(t).

Finally dening the state and the output vectors

=[ 1 n 1 n

]y =

[hub wc(l, )

]

the system equations can be rewritten in the following state

space representation{(t) = A(t) +B(t)Y (t) = C(t)

(2)

where

A =

0 0 0 1 0 0

0 0 0... 1 0

......

. . ....

......

. . ....

0 0 0 0 0 10 0 0 0 0 00 21 0 0 211 0...

.... . .

......

.... . .

...

0 0 2n 0 0 2nn

B =[0 0 0 1

Jp1(0)

pn(0)]

C =

[1 p1(0)

pn(0) 0 0 00 c1(l) cn(l) 0 0 0

]

B. Friction model

In the considered plant the dissipative effects that are not

taken into account in the damping factor are the internal

friction of the motor and the friction due to wires torsion.

We will take into account these complex effects introducing

the following friction model, that is a simplication of the

classical Karnopp friction model described in [10]:

f =

csgn(hub) if |hub| > d

if | | < s, |hub| < d

ssgn(hub) otherwise

where f is the overall friction torque, s the static friction,c the Coulomb friction, d the velocity dead-zone.

L. Bossi et al.: Modeling and Validation of a Planar Flexible Manipulator TuB14.5

2810

C. Actuator limitation

Another non linear effect is due to the saturation on the

control variable (i.e. min (t) max) due to the limiton the torque supplied by the motor.

IV. MODEL IDENTIFICATION

In this section the values of the parameters of the plant

reported in Table II are derived. In particular the mass of

the payload mp, the linear mass density of the beam , thebeam length l, the cross area inertia I are easily measurablefrom the plant; the mass moment of inertia of the payload

Jp and the hub inertia J0 are computed starting from thegeometrical and physical characteristics of the system, while

the Young modulus E, the damping ratio and the frictionare identied on the base of experimental data obtained with

suitable experiments performed on single parts of the plant

or on the overall plant.

A. Young modulus characterization

The identication of the Young modulus E requires abeam test. In fact the theoretical estimation of this parameter

may be quite different from the real one because the compos-

ite material link has peculiar characteristics strongly depen-

dent on the manufacturing process. The composite materials

are also subject to ageing that produces link stiffness loss.

The beam test is performed as follows: the link is clamped

at one end and increasing mass payloads are hanged up at

the free end of the link.

Fig. 2. Experimental test bed

Starting from the relationships between the applied force

F and the displacement we can nd the Young modulusby the following equation derived by structural mechanic [7]

E =Fl3

3I

where F = mg is the force acting on the end point of thebeam; m and g are respectively the total applied mass andthe gravity acceleration, with m = m1 + m2 where m1 isthe weight of the applied mass reported in Table I, while

m2 = 0.019[kg] is the weight of a thin plate, inserted in thecavity of the link, in correspondence of the area where the

TABLE I

YOUNG MODULUS CHARACTERIZATION

Applied Mass [Kg] Displacement [m] Young Modulus E [GPa]

0.005 0.023 6.6867

0.010 0.029 6.4081

0.015 0.034 6.4081

0.020 0.039 6.4081

0.025 0.044 6.4081

aluminium stirrup is screwed, to avoid a beam break in that

point.

The obtained values, summarized in Table I, highlight a

quite linear elasticity model of the beam in the range of

displacements where the tests have been performed. Nev-

ertheless different experiments made on the overall system

show the presence of a nonlinear behavior of the system for

very small and very large values of the beam deections. Dif-

ferent sweep signals with increasing amplitude are applied

to the motor as torque references. As can be seen in Fig.

3 the wider is the produced displacement, the lower is the

frequency at which the resonance peak occurs. In conclusion

the most suitable choice seemed to be E = 6.408 [GPa] sinceit better describes the vibrations dynamic behavior in a wider

range.

Fig. 3. Resonance frequency shift

B. Relevant modes identification

To develop a good model for control it is very important

to understand which modes of vibration are actually excited.

The values of the resonant frequencies computed with the

analytical model are f1 = 7.27 Hz the rst one and f2 =18.21 Hz the second one. The test performed to nd outthe relevant modes, consists in the application of a torque

sweep to the motor, sweeping frequencies from 0 to 50 Hz.


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The spectral analysis of the strain gauges measurements is

shown in Fig. 4. We can recognize the rst mode, with a

frequency that is very similar to the analytical one, while

we see a relevant peak at about 28 Hz that is very differentfrom the second analytical mode. To better understand this

phenomena a second experiment has been performed with an

air-pad at the end of the arm. From the spectral analysis of

0 5 10 15 20 25 30 350

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Frequency (Hz)

Ga

in

Spectrum estimate

Airpad, Central

Airpad, Base

No Airpad, Central

No Airpad, Base

Fig. 4. Modal analysis

both strain gauges signals reported in Fig. 4 it is clear that

the amplitude of the second peak is signicantly reduced.

This is due to the fact that this is not a exural mode but a

torsional one that is not described by the model. In view of

these experimental results we decided to consider only one

exural mode of vibration.

C. Damping and friction Identification

1) Damping Identification: As it is described in the pre-

vious section the damping ratio describes also the effect ofsome friction terms. The ones due to the motor rotation are

not involved in it. Then in order to identify we clamp thelink at the joint end. Then, given an initial displacement to

the end effector, the resulting free vibrations are analyzed.

Applying the method of logarithmic decrement [11] we

identify the damping ratio value = 0.034.

2) Friction identification: To obtain the friction model of

the plant, three parameters should be identied: the static

friction s, the velocity threshold d and the coulomb frictionc. The rst parameter is identied applying increasingtorque till a joint motion is produced. The needed torque

value to move the motor is the value assigned to s. The valued is usually very small and difcult to estimate. We havearbitrarily chosen it equal to 0.001 degrees/s. The procedureto identify c follows the sequent steps. First an experimentsimilar to the one made for the identication of the damping

factor but without clamping the link is made. Based on itan estimation of an equivalent damping factor of the whole

plant is obtained. Then we perform several simulationsfor different increasing values of c. The damping value foreach simulation is computed by means of the logarithmic

decrement applied to the deection responses obtained in

TABLE II

MODEL PARAMETERS

Parameter Value Parameter Value

0.034 E 8.46E9[Pa]I 2.95E11[m4] J0 0.0016[kgm2]Jp 1.73E5[kgm2] l 0.42[m]s 0.065[Nm] 0.09[kg/m]c 0.0055[Nm] mp 0.155[kg]

max 0.214[Nm] min 0.214[Nm]d 0.002[m/s]

TABLE III

REPEATABILITY EXPERIMENTS ANALYSIS

Designed Model Actual Hub Rotation Actual HubHub Rotation Hub rotation Mean Value Rotation SD

20 18.84 10.49 0.2545 42.70 29.15 0.4560 58.25 51.60 0.4490 85.00 69.00 0.46120 112.00 67.62 2.01

simulation. Then the damping values are plotted on the y-axis vs the correspondent c values on the x-axis. Finally,starting from the knowledge of the real global damping ratio

, we can obtain an estimation of c.

V. MODEL VALIDATION FOR CONTROL

Once the single components of the model have been

identied it is necessary to validate the whole model. The

goal of this paper is to obtain a model useful for the synthesis

of feedback control laws. It is well known that this requires

a good model in a particular range of frequencies while it is

not necessary to have a very precise model at low or high

frequencies. However the rst validation experiments have

been done in open loop in order to verify the repeatability

of the experiments. Then, a closed loop validation has been

done using two control strategies. The rst one is a single

input (position error) single output (motor torque) controller

synthesized in the frequency domain and the second one is

a multi input (position error and deection) single output

(motor torque) controller based on an LQ control law com-

plemented with an observer and an integral action.

A. Open loop validation

Well suited bang-bang torque proles, designed on the

base of the system model without friction, are applied several

times to the motor in order to obtain an hub rotation of 20,

45, 60, 90 and 120 respectively. Table III summarizes

the obtained results. In particular the asymptotic value of

the rotation obtained in simulation with the full model (also

with the friction model), the mean value and the standard

deviation (SD) of the asymptotic value obtained on the

experimental set-up are reported. It is important to notice

that the standard deviations obtained are very small. On

the contrary, the mean value is rather different from the

simulated one and also with a nonlinear behavior with

respect to the amplitude of the hub rotation. This information


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suggests to synthesize a regulator with a high gain in order

to obtain a small regulation asymptotic error.

B. Single Input Single Output closed-loop validation

A standard position control scheme is implemented using

only the hub position measurement. The controller, synthe-

sized in the frequency domain, is given by

R(s) =5(s+ 1)

(s+ 5)(s+ 300)

The validation has been performed applying sinusoidal po-

sition references with different frequencies and amplitudes.

In particular Fig. 5 and 6 show the system response to a

sinusoidal position reference signal, having 40 amplitude at

0.2 Hz and 0.4 Hz frequency respectively.

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 150

50

100

150

hub [

Degre

es]

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 1550

0

50

w(l,t)

[mm

]

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 150.1

0

0.1

(t)

[N

m]

Time [s]

Simulation

Real# 1

Real# 2

Fig. 5. Sinusoidal position reference, 0.2 Hz

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 150

50

100

150

hub [D

egre

es]

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15100

0

100

w(l,t)

[mm

]

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 150.5

0

0.5

(t)

[N

m]

Time [s]

Simulation

Real#1

Real#2

Fig. 6. Sinusoidal position reference, 0.4 Hz

The blue line is obtained via simulation while the other

ones are experimental results. Finally Fig. 7 shows the

system response to the 120 amplitude position reference

signal reported with the black dashed line in the rst sub-

plot. Looking at the gures, we can observe that the model is

capable to capture with good approximation the dynamics of

Fig. 7. Filtered Step 120 Positioning

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5100

0

100

200

hub [

Degre

es]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5100

0

100

w(l,t)

[mm

]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.5

0

0.5

(t)

[N

m]

Simulation

Real#1

Real#2

the rigid mode as well as the exible one, while the torque

required by the regulator in the real plant and in the simulated

one are comparable, with only very small deviations likely

due to the approximation of friction model, such as the

chosen threshold d.

C. Multi Input Single Output closed-loop validation

A second closed-loop validation has been performed using

also the base strain gauge measure. To this aim, a regulator

based on a Linear Quadratic (LQ) optimal control law

has been adopted. In Fig. 8 the overall control scheme is

Fig. 8. LQ control scheme with state observer and integral action

reported. In view of the relevant model gain uncertainties

stressed by the open loop validation, an integral action on

the position error has been introduced in order to guarantee

an asymptotic zero-error regulation. Moreover an observer

is required. The LQ controller is synthesized, assuming that

the state is available, based only on the linear model (2),

without considering the friction model, and the model of the

integrator. For this reason the following enlarged system has

been derived:

.(t) = A(t) + B(t) + B

0hub(t), t 0, (0) = 0

where

=[ 1 1 v

]


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A =

[A 0

C1 0

], B =

[B0

], B =

[01

]

v is the state of the integrator, C1 is the rst line of thematrix C and 0hub is the reference signal for hub.The cost function to be minimized is

J (0, ()) =

0

( ()

Q () + ()R())d

where

Q =

50 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 0.05

, R = 1000

so that the control law is

(t) = K(t)

where

K = R1BP

and P is the only positive denite solution of steady Riccatiequation

0 = PA+ AP +Q PBR1BP.

The cost matrices Q and R are chosen in order to penalizemore the position error and less the integral action. Moreover

since the plant is equipped only with two transducers that

furnish a measure proportional to the motor rotation and

a measure proportional to the end-effector displacement

from the neutral axis, the state system is not completely

measurable. Then the following state observer is used:.

(t) = A(t) +B(t) L (y(t) C(t))

where (t) is an estimation of the state system (t) and Lis chosen such that the eigenvalues of the matrix A+LC areless faster than the slowest eigenvalues of the matrix ABK.In particular the vector of the eigenvalues of A+LC is equalto r =

[5 5 10 10

]. Then the overall regulator

is given by.

(t) = A(t) +B(t) L (y(t) C(t))

v(t) = C1(t) 0hub(t)

(t) = K

[(t)v(t)

]

In Fig. 9 a step response of 120 of the hub is reported.

The blue line is the simulated evolution while the other ones

are the experimental responses. The mean value and the

standard deviation of the hub position after 5 seconds for

several experiments are 121.1 and 0.3 respectively. Fromthe mean value it is possible to notice that in spite of the

presence of the integral action a small error remains. This

is due to the small value of the element (5,5) of the matrix

Q that penalizes the integral state. However, this small erroris negligible while an increase of the penalty on the integral

action increases the oscillations on the arm. Moreover, the

SD is reduced with respect to the same experiment made in

open-loop (see Table III).

!

"

!

Fig. 9. LQ control validation

VI. CONCLUSION

A mathematical model for the synthesis of a regulator for

a laboratory planar exible manipulator has been proposed

and successfully validated. Remarkably, even if the nonlinear

friction model and the actuator limitation are not directly

considered in the synthesis of the control law, they are

required to choose the values of the controller parameters

in simulation before applying them to the experimental set-

up.

REFERENCES

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[3] W. J. Book. Structural exibility of motion systems in the spaceenvironement. IEEE Trans on Robotics and Automation, 9:524530,1993.

[4] C. Canudas de Wit and B. Siciliano. Theory of Robot Control.Springer, London, 1996.

[5] A. De Luca and B. Siciliano. Closed-form dynamic model of planarmultilink lightweight robots. IEEE Trans. On Systems, Man, andCybernetics, 21:826839, 1991.

[6] S. K. Dwivedy and P. Eberhard. Dynamic analysis of exiblemanipulators, a literature review. Mechanism and Machine Theory,41:749777, 2006.

[7] M. E. Gurtin. An Introduction to Continuum Mechanics. AcademicPress, New York, 1981.

[8] L. Meirovitch. Analytical Methods in Vibrations. Mac MillanPublishing, 1967.

[9] G. Mimmi, C. Rottenbacher, and G. Bonandrini. Theoretical andexperimental sensitivity analysis of extra insensitive input shapersapplied to open loop control of exible arm. International Journalof Mechanics and Materials in Design, 2008.

[10] H. Olsson, K. AAstrom, C. Canudas-de Wit, M. Gafvert, andP. Lischinsky. Friction models and friction compensation. EuropeanJournal of Control, 4:176195, 1998.

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07074833