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Applied Mathematical Sciences, Vol. 6, 2012, no. 73, 3623 - 3659

An Innovative Method to Modeling

Realistic Flexible Robots

Laura Celentano

Università degli Studi di Napoli Federico II

Dipartimento di Informatica e Sistemistica

Via Claudio 21, I-80125, Napoli, Italy

[email protected]

Abstract

In this paper a modular, computationally efficient and numerically stable method

is presented, which allows to obtain the dynamic model of a robot constituted by

flexible links having varying cross-section and subjected to generic ending forces

and torques and to the gravity actions. This method is based on the use of

admissible deformation functions of wavelet type, obtained by using the

Instantaneity Principle of the deflection of an element, and on the Euler-Bernoulli

beam theory if the link is slender or, otherwise, on the Timoshenko one.

Moreover, it is easy to extend the presented methodology to deal also with the

case of large link deformations.

The kinematic model of the generic link is obtained by using absolute motion and

relative deformation coordinates; the dynamic model, derived with the Lagrangian

approach, is obtained by assembling the dynamic models of the links by using a

recursive algorithm based on the congruence technique.

The proposed modeling methodology guarantees no static error independently of

the number of wavelet functions per link, both in the presence of generic forces

and torques at both ends, for generic cross-section profiles, and in the presence of

gravity actions, for several cross-section ones; moreover, it guarantees good

dynamic performance in a frequency range which increases when the number of

wavelet functions increases. It is shown that the presented methodology is also

more efficient and numerically stable than other modeling methods known in

literature.

This methodology can be used for the dynamic simulation of flexible robots

and/or for the design of the control system and for the analysis of its

performances.

Moreover significant examples, which illustrate the properties of the proposed

3624 L. Celentano

methodology, are presented; they also show that the proposed modeling

methodology is an advisable choice when it is necessary to obtain high precisions,

in particular at low frequencies, and/or not prohibitive calculus time, and/or when

other modeling methods are inapplicable because of numerical divergence

problems.

Keywords: Modeling, Simulation, Wavelet Functions, Flexible Robots,

Computational Efficiency

1. Introduction

The flexible robots are necessary and/or needful in: very cramped workspaces

(spaces of work) and/or winding ones (where less invasive robots, and hence with

slender links, are required); very wide workspaces or anyway with end-effector

very far from the base of the robot because of the presence of obstacles (rivers,

buildings, ...) or when it is impossible or not convenient to use robots with mobile

base or more cooperative robots (to build and/or to maintain mega-structures,

electric lines, ...); in this case the robots need long links and hence, to avoid their

possible breaking caused by the gravity action and to make them easily

transportable, very slender links; dangerous and/or harmful areas of work both for

the human operators and for the actuators with the electronic control (rescue and

security robots).

For these reasons the modeling of robots with flexible links and, more

generally, the modeling of controlled or not flexible structures with degree of

freedom, is a historic topic of robotic research [1]-[12] and it remains very

interesting for the scientific community [13]-[17], [20]-[26]. Nowadays, in fact,

many transport and manufacturing systems employed in the modern mass

production plants require higher and higher specifications in terms of operating

speeds and/or amplitude of areas of work; the only way to satisfy the previous

specifications is to reduce the mass and to make the structures slender, i.e. to

employ robots having flexibility properties. Obviously, in order to take full

advantage of the benefits offered by the above lightweight flexible robots, it is

necessary to develop advanced control systems based on reliable and efficient

models. In order to describe the behavior of a flexible robot it is possible to use

infinite dimensional models, exact but scarcely operative, or finite dimensional

models, which are approximate but more operative. With regard to this, the most

known approximate methods in literature are based on the assumed modes (e.g.

[1], [9], [14]), the Ritz-Kantorovich expansions with polynomials (e.g. [7]) and on

finite elements and finite differences (e.g. [4], [14]). The modeling of flexible

structures with degree of freedom has several problems to deal with, such as:

• the derivation of a model which is precise at low frequencies, in particular at

zero frequency, since the flexible structures must be almost always operated

and/or solicited at low frequencies, in order to avoid their breaking and/or

Modeling realistic flexible robots 3625

annoying noises;

• a suitable choice of the mode shape functions and/or of admissible basis

functions to describe the flexible behavior;

• the integration with the models of sensors and actuators;

• the numerical stability of the models with small errors, in order to easily

design robust closed-loop control systems which do not exhibit the spillover

phenomenon;

• the computational load, since the models of flexible structures, because of

their lability, are strongly nonlinear, even for small deformations, and very

complex;

• a method easy to apply also in the case of flexible links having varying

cross-sections.

In this paper a methodology which gives significant answers to the above

problems is provided. This method is based on an approach different from the one

presented in [26] and more general and complete than the one proposed in [25].

More in details, in Section 2 the hypotheses about the typology of the robots

considered in this paper are presented, and the Instantaneity Principle of the

deflection of an element, which is the key of the proposed modeling technique, is

stated. In Section 3, first the Lagrangian function of a generic flexible link having

varying cross-section, supposed in free-free boundary conditions, is derived in a

compact and closed form; then a very simple iterative algorithm used to calculate

the Lagrangian function of a robot, constituted by several interconnected flexible

links, is given, and finally the dynamic analytical model of the whole robot is

obtained. In Section 4 the properties of the proposed modeling methodology are

presented. In Section 5 very significant examples and comparisons with the

literature are reported. In Section 6 an experimental validation of the proposed

method is briefly illustrated. Finally, in Section 7 conclusions and future

developments are reported.

2. Hypotheses, notations and preliminaries

In this paper it is considered, for brevity, the case of a planar robot with fixed

base, constituted by ν flexible links having varying cross-section. For simplicity,

each link has a straight line as unstressed configuration, both rigid ends of

negligible dimensions with respect to its length, and rotation axes orthogonal to

the vertical plane. In Fig. 1 a detailed representation of the i-th link in the stressed

and unstressed configuration is shown.

3626 L. Celentano

Fig. 1. Schematic representation of the generic flexible link.

For the sake of clarity, the following preliminary notations are introduced for the

generic i-th link: iL is the length of the link;

iA is the cross-section area of the

link;iE is the Young’s modulus of the link’s material;

iG is the shear modulus

of the link’s material; iχ is the shear factor of the link;

iI is the area moment of

inertia for the cross-section of the link; iρ is the mass density of the flexible part

of the link; i i im Aρ= is the mass per unit length of the flexible part of the link;

,i i

M M− + are the masses of the rigid ends of the link; ,

i iJ J

− + are the inertia

moments of the rigid ends of the link with respect to rotation axes; iq is the

distributed transversal load; iτ are the distributed torques; , ,

o i o i ix y α are the

absolute motion coordinates of the link supposed rigid; ( , )id z t and ( , )i z tγ are

respectively the deflection and rotation of the generic cross-section of the link at

abscissa z and time t , due to the distributed flexibility. Moreover, suppose that

the following result holds.

Instantaneity Principle of the deflection of an element. If the inertia of an element

iL∆ of “sufficiently small” length of the i-th link is neglected, the vertical

deflection of this element, due to “slowly variable” control actions and/or

disturbances acting on the links and on the end-effector and due to the consequent

inertial actions, remains practically unchanged.

Remark 1. It is worth noting that, extending the rules, well-known by the experts

of building science, to plot the cutting diagram and the diagram of the bending

moment of a beam (sectioning principle of a structure) (see e.g. [18]), the

deflection of the element is due to (

Fig. 2):

• the torque C− and the force T − acting on the left end of the element,

corresponding to the resultant of all external torques and forces with changed

sign, including gravity actions, constraints actions and inertial forces acting

on the left-hand side of the robot;


• the torque C+ and the force T + acting on the right end of the element,

corresponding to the resultant of all external torques and forces, including

gravity actions, constraints actions due to interaction with the environment

and inertial forces acting on the right-hand side of the robot with its possible

payload;

• the generalized forces acting on the element itself including gravitational

actions, disturbances and inertial actions.

Therefore, for the calculus of the deflection of the element, it is possible to neglect

the inertial actions acting on the element itself. This approximation is as true as

the element is small, as the flexible parts of the links are lightweight with respect

to the whole robot, also with the actuators, and as the motion is slow. The above

remarks justify the Instantaneity Principle.

The Instantaneity Principle and the Euler-Lagrange theory, or the Timoshenko

one, represent the key on which the presented modeling methodology is based.

Fig. 2. Forces and torques acting on an element iL∆ of the link.

3. Flexible robot model

3.1 Model of a flexible link

In this subsection the Lagrangian function of a flexible link, supposed in free-free

boundary conditions, is derived, in a compact and closed form, by using the

Euler-Bernoulli beam theory, if the link is slender, or the Timoshenko one

otherwise, and the Instantaneity Principle of the deflection of an element.

According to the preliminary notations (see also Fig. 1), the configuration of the

generic cross-section of the i-th link can be expressed as:

3628 L. Celentano

cos( ) sin( )

sin( ) cos( )

.

i o i i i i

i o i i i i

i i i

x x z d

y y z d

α α

α α

ψ α γ

= + −

= + +

= +

(1)

Several methods to approximate the deflection ( , )id z t of a flexible link have been

proposed in literature. These methods consist in choosing a complete set of

functions { ( )}km z through which the deflection can be approximated as

1

( , ) ( ) ( )n

i k fk

k

d z t m z q t=

≈∑ , where ( )fk

q t are the Lagrangian deformation variables;

moreover, almost always it is supposed that i id zγ = ∂ ∂ . Different choices of the

set { ( )}km z have been proposed in literature. The assumed modes method uses the

modes deriving from the solution of the following Euler-Bernoulli beam dynamic

equation

2 2 2

2 2 2

( , ) ( , )( ) ( ) ( ) ( , ).

d z t d z tE z I z m z q z t

z z t

∂ ∂ ∂+ =

∂ ∂ ∂ (2)

with distributed inertial load and with ( , ) 0q z t = .

Equation (2) is solved by imposing four boundary conditions which describe the

configuration of the flexible link. In some works ([9], [10]), the configuration of

the link with clamped-free boundary conditions has been proposed, in other works

([1], [9]), instead, the one with clamped-mass boundary conditions has been

considered, typically under the hypothesis that EI const= . In other works ([7]), the

polynomial set ( ) k

km z z= has been chosen as complete set of functions { ( )}km z .

According to the Instantaneity Principle and to Remark 1 the deflection of the

generic element of the link is computed with the proposed method by

approximating the dynamic beam equation of Euler-Bernoulli (2) with the

following static equation

2 2

2 2

( , )( ) ( ) ( )

d z tE z I z q z

z z

∂ ∂=

∂ ∂ , (3)

which is obtained by neglecting the inertial term and by considering

time-independent loads.

If the shear effect is not negligible, the deflection of the generic element of the

link can be computed more accurately by approximating the dynamic beam

equations of Timoshenko

2

2

2

2

GA d dm q

z z t

GA dEI I

z z z t

γχ

γ γγ ρ τ

χ

∂ ∂ ∂ − − = −

∂ ∂ ∂

∂ ∂ ∂ ∂ + − − =

∂ ∂ ∂ ∂

(4)


with the static ones. Finally, if the link is subjected to large deformations (see e.g.

[12]), it is still possible to apply the proposed methodology, by fictitiously

subdividing the link into sublinks as shown in Fig. 3.

Fig. 3. Fictitious subdivision of the generic link into sublinks.

The above static equations, with suitable boundary and “middle” conditions, with

concentrated loads in the middle and/or at the ends, and/or with distributed loads

according to the mass density or according to its first order moment, are

numerically, or analytically if it is possible, solved on any interval kZ of an

appropriate partition of the monodimensional domain of the i-th link. For

example, by assuming that the generic domain [ ]0,i

L is uniformly partitioned into

in parts, the intervals kZ are defined as follows

0, 2 , , 3 , , ( 1) ,i i i i

i i

i i i i

L L L Ln L

n n n n

−

K . (5)

In Fig. 4 the constraint and the static loading physical schemes of a partition

element, considered in the following, are reported.

Fig. 4. The considered constraint and loading schemes.

The deflections wδk and wγk corresponding to the considered schemes, derived for

the interval ,k k kZ Z Z− + = (k=1,2,…, ni) of the partition, represent the wavelet

spatial functions used by the proposed methodology in order to approximate the

3630 L. Celentano

vertical displacement and the rotation due to the deformation.

In the light of the above considerations it is important to explicitly note that the

calculation of the wavelet functions is very easy also in the case of flexible links

having varying cross-section. Indeed it is easy to see that the static equation of

Euler-Bernoulli (or the Timoshenko ones) is linear with variable coefficients;

hence it can be easily solved with a Matlab program with four independent initial

conditions. By linearly combining the obtained four solutions, it is easy to

determine the linear combination coefficients such that the boundary conditions of

the wavelet functions are satisfied.

An example of these wavelet functions, under the hypothesis that [ ]1 0 2iZ L= ,

[ ]2 4 3 4 ,i i

Z L L= [ ]3 2i i

Z L L= and [ ]4 3 4i i

Z L L= , is reported in Fig. 5.

0 0.5 1 1.5 2 2.5 3 3.5 4-0.2

-0.1

0

0.1

0.2

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

z

wγ3

wγ2

wγ4

wδ1

wδ2

wδ3

wδ4

wγ1

Fig. 5. An example of wavelet functions ( 4, 8).in n= =

Once the wavelet functions have been calculated, the deflection of the i-th link is

approximated as

( )

1

( , ) ( ) ( ) ( ) ( )in

i k k k k

k

d z t w z t w z tδ γδ γ=

= +∑ . (6)

Remark 2. It is worth noting that the Lagrangian deformation variables i kδ , i k

γ

corresponding to these wavelet functions respectively represent the vertical

displacements and the rotations of the cross-sections of the i-th link in

correspondence to the endpoints of the partition intervals kZ .

After these preliminary considerations, by neglecting the rotational inertia, the

kinetic energy of the i-th link can be derived as follows


( )

2 2 2 2

0

22 2 2

1 1( ) ( ) ( )

2 2

1 1 1( ) .

2 2 2

i

i i i

L

i i i i i o i o i

i i i i z L i z L i i in

T x y m z dz M x y

J M x y Jα α γ

−

− + +

= =

= + + + +

+ + + + +

∫ & & & &

& & && &

(7)

After some tedious manipulations and by omitting, for the simplicity of notations,

the subscript i, it is

( ){ ( )( )

( )( ) ( )

( )( )}

2 2 2 2 2 2 21

2

2

2 cos sin

2 cos sin ,

T

o o n n f f f n

T T

f f f f n n

T

n f o o

T

f n o o

T M x y M J J q B q M L

q B q k q J M L

M h q x y

N h q M y x

δ γ δ α

γ δ α

δ α α α

α δ α α

+ + +

+ +

+

+

= + + + + + + + +

+ + + + +

− + + +

+ + + −

& & && &

&& && & &

& & &

&& & & &

(8)

where, by omitting the dependency on z, it is:

2

0 0

, , ;

L L L

o

M M mdz M J J z m dz J N M L z mdz− + − + += + + = + + = +∫ ∫ ∫

the symmetric matrix 2 2n n

fB R ×∈ is derived by using the relation

2

1

0

2

1 1 1

0 0

2

1 1

0 0 0

L

L L

f

L L L

n n n

w m dz

w w m dz w m dzB

w w m dz w w m dz w m dz

δ

δ γ γ

δ γ γ γ γ

− − −

− −

=

−

∫

∫ ∫

∫ ∫ ∫

M M O

L

; (9)

the vector 2nh R∈ is derived by using the relationship

1 1

0 0 0

L L L

T

nh w mdz w mdz w mdzδ γ γ

= ∫ ∫ ∫L ; (10)

the vector 2nk R∈ is derived by using the relation

1 1

0 0 0

L L L

T

nk w m z dz w m z dz w m z dzδ γ γ

= ∫ ∫ ∫L ; (11)

3632 L. Celentano

2n

fq R∈ is the vector of the Lagrangian deformation coordinates,

[ ]1

T

f k k n nq δ γ δ γ δ γ1= L L . (12)

Once the kinetic energy has been derived, it is necessary to calculate the elastic

potential energy eiU and the gravitational potential one giU of the i-th link. The

elastic potential energy due to the deformation of the i-th link, by neglecting the

contribute due to the shear and by omitting the subscript i for simplicity, turns out

to be

22

2

0

1 1( ) ( ) ,

2 2

L

T

e f f

dU E z I z dz q Kq

z

∂= =

∂ ∫ (13)

where the symmetric matrix 2 2n nK R ×∈ is derived by using the relationship

2

1

0

2

1 1 1

0 0

2

1 1

0 0 0

L

L L

L L L

n n n

w E I dz

w w E I dz w E I dzK

w w E I dz w w E I dz w E I dz

δ

δ γ γ

δ γ γ γ γ

′′ − − −

′′ ′′ ′′ − −

=

− ′′ ′′ ′′ ′′ ′′

∫

∫ ∫

∫ ∫ ∫

M M O

L

, (14)

in which it is supposed that ( ) ( )EI E z I z= .

Remark 3. It is worth noting that if the rotational inertia is taken into account, it is

necessary to add the term 2

0

1

2

iL

i i iI dzψ ρ∫ & to (7); moreover, in the case of the

Timoshenko model, it is necessary to add the elastic energy due to the shear

deformation 2

0

1

2

iL

i i i

i

i

G A ddz

zγ

χ

∂ − ∂

∫ to (14).

The gravitational potential energy turns out to be

0

( ) ,i

i

L

gi i i o i i i z LU g y m z dz M gy M gy

− +

== + +∫ (15)

where g is the gravitational acceleration. By substituting the second of (1) into

(15), it is


( )sin cos ,T

g o n fU Mgy Ng M h q gα δ α+= + + + (16)

in which the subscript i has been omitted.

Remark 4. It is interesting to note that the matrices Bfi and Ki and the vectors hi

and ki can be easily calculated one-off in a numerical way (or also analytically in

the case of homogeneous links having constant cross-section). Moreover, it is

important to note that the matrices fi

B and iK are very sparse (Fig. 6) and also

well-conditioned; this fact is in accordance with (9), (14) and with the wavelet

nature of the basis functions (see Fig. 5). Instead, when other basis functions, such

as the polynomial ones, are used, the above matrices are full and ill-conditioned

(see the following examples).

Fig. 6. Composition scheme of the matrices Bfi and Ki of the i-th link and

their structure.

Remark 5. It is worth noting that the compact and closed-form expression of the

kinetic energy (8) can be further simplified if the higher order terms T

f f fq B q and

2

nM δ+ are neglected.

Remark 6. It is important to note that if

2

100GAL

EI

χβ = > (17)

the Timoshenko beam theory practically coincides with the Euler-Bernoulli one.

For example, for a link made of steel having square hollow constant cross-section,

with side 40mmil = , thickness 2mmis = and length 5miL = , it is 42.88 10β = ⋅ . It

follows that, for the majority of flexible robots, it is possible to apply the

Euler-Bernoulli beam theory.

3634 L. Celentano

3.2 Interconnection algorithm

In this subsection an algorithm for the calculation of the Lagrangian function of a

robot constituted by interconnected flexible links is presented, starting from the

results, valid for a single link, stated above. The kinetic energy of the i-th link (8)

can be rewritten in a compact matricial form as follows

1

2

T

i i i iT q B q= & &%% % , (18)

where:

T T T

i ti iq q q = % , T

ti o i o iq x y = , T T

i i fiq qα = ;

11 12

12 22

i i

i T

i i

B BB

B B

=

% %%

% %, 11

0

0

i

i

i

MB

M

=

% ;

( )

( )12

cos sin sin

sin cos cos

i

i

T T

i in i fi i i i i i

iT T

i in i fi i i i i i

M h q N hB

M h q N h

δ α α α

δ α α α

+

+

− + − − = − + +

%

%

%;

( )2 2

22

i

T T

i fi fi fi i i in i

i

i fi

J q B q M L kB

k B

δ+ + + + =

%%

% %;

T

ih% is obtained by adding

iM

+ to the (2ni-1)-th element of vector T

ih ;

T

ik% is obtained by adding

i iM L

+ to the (2ni-1)-th element and i

J+ to the 2ni-th

element of vector T

ik ;

fiB% is obtained by adding i

M+ to the (2ni-1, 2ni-1)-th element and

iJ

+ to the (2ni,

2ni)-th element of the matrix fiB .

Now observe that, being the 1-st link hinged to the base, the variables 1ox and

1oy are null, hence

1 1 1 1

1

2

TT q B q= & & , (19)

where 1 221 .B B= % Moreover, the rigid translation variables 1o i

x + and

1, 1 1,

o iy i ν+ ≤ ≤ − are redundant, since they depend on , , 1, ,k kn k iα δ = K ; indeed

from (1) the following recursive relationship can be derived

1

1

( , )i

o i o i i

i i in

o i o i fi

x xA

y y q

αα δ

+

+

= +

&& &

& & &, (20)

where ( )2 2 1in

iA R

× +∈ is given by


sin cos 0 .. 0 sin 0

cos sin 0 .. 0 cos 0

i i in i i

i

i i in i i

LA

L

α δ α α

α δ α α

− − − = −

. (21)

Therefore, equation (18), for 2i ≥ , can be rewritten as function of the only

Lagrangian variables as follows

1 1 1 1 1 1 1 1

1 1

2 2

T T T

i i i i i i i i iT q A B A q q B q− −= =K K K K K K

%& & & & , (22)

where:

1 1

1 1 1 1

2 1

, ,i

iT T T

i i i

n

A A Oq q q A

O I

−

−+

= =

K K

KL (23)

in which pI denotes the identity matrix of order p and O is a zero matrix of

suitable dimensions.

Finally, the kinetic energy of the robot constituted by ν flexible links turns out to

be

1

2

TT q Bq= & & , (24)

where 1q q ν=K

and the inertia matrix B is obtained “by adding” the matrices Bi

according to the recursive scheme reported in Fig. 7.

Remark 7. It is useful to note that the matrices iA are very sparse,

well-conditioned and they always depend only on the deformation variable ii nδ ,

whatever the number of Lagrangian deformation variables (i.e. of wavelet

functions) is, and they do not depend on all the Lagrangian deformation variables,

as it happens when other basis functions are used.

Fig. 7. Composition scheme of the matrix B.

3636 L. Celentano

Concerning the elastic potential energy of the whole robot, it is easy to verify that

it turns out to be

1

2

T

eU q Kq= , (25)

where the matrix K is the following block diagonal matrix

1 2(0, ,0, , ,0, )K diag K K Kν= K . (26)

Finally, the gravitational potential energy of the whole robot is obtained as the

sum of

( ) ( )1

1

sin cos sin cos ,k i

iT

gi i k k kn k i i in i fi i

k

U M g L Ng M h q gα δ α α δ α−

+

=

+ + += +∑ (27)

where giU is the gravitational potential energy of the i-th link.

3.3 Dynamic model of the robot

In this subsection the dynamic model of the whole robot is derived in the more

suitable form presented in [19] by using the Euler-Lagrange method. It is easy to

show that this model, under the assumptions that the control actions uc and the

disturbances ud are the ones reported in Fig. 8, turns out to be

( )

1( ) ( )

2

T

g c c d d

dB q q q B q q Kq U H u H u

dt q q

∂ ∂− + + = +

∂ ∂& & & , (28)

where:

[ ] [ ]1 2 ,T T

c d d du C C C u C Fν= =L , (29)


11 12

21 22

1 2

1 1 0

0 1 0

0 1 1

,0 0 1

0 0 1

0 0 0

d d

d d

c d

d d

O O O

h h

h hO O OH H

h h

O O O

ν ν

− −

− = = −

L

L

L

L

L

M ML

L

L

L

M M M O

, (30)

in which:

[ ] [ ]1 10 0 0 , 1, , 1, 1 0 1 ,T T

di dh O i h Oνν= = − =K

(31)

( ) ( ) ( )2 sin cos sin 0 ,

i

T

di i i d in i d i dh L Oα α δ α α α α= − − − − − − (32)

being O zero vectors of suitable dimension.

Fig. 8. Control actions and disturbances acting on the robot.

4. Some advantages of the proposed method

To more quickly understand the proposed modeling methodology of flexible

structures based on the wavelet functions and its peculiarities by making a suitable

comparison with the most recent and reliable methods in literature, in the

following it will mainly be considered a robot modeled by using 8 wavelet

functions per link.

3638 L. Celentano

The model obtained by using the proposed method exhibits the following

distinctive features:

1. It is very efficient from a computational point of view. Moreover it is

numerically very stable as compared with those obtained by using the assumed

modes method and the Ritz-Kantorovich expansion.

Indeed the inertia matrix obtained with the above methods is very ill-conditioned

and/or very sensitive to parametric uncertainties, even if a single link with a

relatively small number of modes or with a relatively low maximum degree of

polynomials is considered, as the following example shows.

In Fig. 5, under the hypothesis that 4,in =

8 wavelet functions are reported. Note

that, since the wavelet functions are null out of the intervals

[ ] [ ] [ ] [ ]1 2 3 40 2 , 4 3 4 , 2 , 3 4 ,Z L Z L L Z L L Z L L= = = = (33)

the inertia matrix fi

B of the i-th link is sparse (see Fig. 6) and also

well-conditioned. Moreover the terminal deflection of each link ( ) ( , )t L tδ δ=

turns out to be the only Lagrangian variable 4 ( )tδ

and analogously the tip

displacement ( , )L tγ γ= turns out to be the only Lagrangian variable 4 ( )tγ (see

(6), Remark 2 and Fig. 5). This fact makes the calculation of the inertia matrix B

of the whole robot and of the vector c of the centrifugal and Coriolis forces little

onerous.

Supposing that the Euler-Bernoulli model is valid (if the model of Timoshenko is

used, analogous considerations, even if more complex, are true) and that the link

is homogeneous and with constant cross-section, it is well-known that the spatial

mode shapes of a clamped-free link are solutions of the equation

42

4( ) ( ) 0

dw z w z

dzω− = (34)

with the boundary conditions:

2 3

2 30 01) ( ) 0, 2) ( ) 0, 3) ( ) 0, 4) ( ) 0

z z z L z L

d d dw z w z w z w z

dz dz dz= = = == = = = . (35)

Under the hypothesis that constm

EI= , the analytic expressions of the mode shapes

turn out to be (see Fig. 9):

( ) cosh( ) cos( ) (sinh( ) sin( )), ,i i i i i i i iw z z z s z z l Lα α α α α= − − − = (36)

where:


[ [ ] ]

( )

1.87510407, 4.69409113, 7.85475744, 10.99554073 ,14.13716839, 2* 6 : 20 1 * / 2]

s 0.734095514, 1 .018467319, 0.999224497, 1 .000033553, 0.999998550, ones 1, 20 6 1 .

i

i

l pi= −

= − +

(37)

Fig. 9. The first 8 spatial mode shapes

It is easy to see that the above mode shapes are very sensitive with respect to

parameters and i il s ; indeed, despite and i il s have been provided with nine digits,

it is easy to verify that the boundary conditions and the orthogonal conditions of

the mode shapes are “perfectly” satisfied only for the first modes.

Moreover ( ) 0, ( ) 0, 1, 2,...i iz L

z L

dw z w z i

dz==

≠ ≠ ∀ = (see Fig. 9). Hence both the

terminal deflection ( , )L tδ δ= and the tip displacement ( , )L tγ γ= of each link

turn out to be linear combination of all the deformation Lagrangian variables. This

fact makes the calculation of the inertia matrix B of the whole robot and of the

vector c of the centrifugal and Coriolis forces very onerous. Moreover the total

inertia matrix B of the robot is also ill-conditioned (in the case in which the

Ritz-Kantorovich expansion is used the ill-conditioning is glaring, as it is shown

in Table IV reported in Section 5).

To become aware of this fact, consider the model of a single hinged link, with

small values of 1α , with 1,L = 1m = , 1EI = and payload with

1 11 and M 1J

+ += = .

In this case the inertia matrix can be easily enough obtained by adding the “mode

shape” 0w z= .

One of our tools (it is verified more times also by comparing the modes

frequencies in the case 1 1

0 and M 0J+ += = with the theoretic ones of a single

hinged link), by using the modes of the clamped free-link, returns

cond(B)=1.4674e+008, which turns out to be 594 times larger of the cond(B)

obtained with the wavelet based method.

This explains the simulation times of Tables VII, VIII, IX reported in Section 5.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0. 0.9 1-2

-1

0

1

2

3

1

2

3

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

-1

0

1

2

z

5 6 7 8

3640 L. Celentano

2. It allows to easily and systematically obtain (see (24), (25) and (27)) the

closed-form dynamic model of a planar robot with any number of flexible links and

with any number of elements for each link, starting from the closed-form

expression of the kinetic energy (8), of the potential elastic one (13) and of the

potential gravitational one (16), by using a symbolic calculation language.

3. The frequency response for a generic equilibrium configuration, by virtue of the

instantaneity principle, becomes more and more accurate, starting from the zero

frequency, as the number of elements in which the link has been divided increases.

4. The static accuracy is always strictly guaranteed, both in the presence of

gravitational actions due to the links and to the payload, and in the presence of

lumped control actions and disturbances in forces and torques, as it turns out to be

from the following theorem.

Theorem 1. For any static equilibrium configuration, the position and the

orientation, obtained with the proposed model (28), of all the nodes into which the

flexible links of the robot are divided, both in the presence of gravitational actions

due to links and payload, and in the presence of lumped control actions and/or

disturbances in forces and torques, strictly coincide with those achieved using the

Euler-Bernoulli beam theory.

Proof. Note that, in static conditions, the Euler-Bernoulli equation written for the

i-th link, supposed homogeneous and with constant cross-section, subjected to a

distributed load q turns out to be

4

4

( , )d z tEI q

z

∂=

∂, (38)

where, for the simplicity of notations, the subscript i is omitted.

In static conditions, the deflection of the generic element of any link with respect

to the tangent frame (Fig. 10) can be obtained by solving (38) with cosq mg α= −

and with the following boundary conditions

( ) ( )0 0, 0 0, ( ) , ( )

C Td d d l d l

EI EI

+ +

′ ′′ ′′′= = = = . (39)


Fig. 10. Deflection of the generic element.

It is easy to verify that, for z l= , it is

( ) ( )

3 2 4 2 3

cos , cos .3 2 8 3 6

l l l l l ld l T M mg d l T M mg

EI EI EI EI EI EIα α+ + + +′= + − = + − (40)

Instead, by using the approximated model (27) under static conditions, obtained

by introducing a reference frame rigidly connected to the element, having z axis

tangent into the left node, and recalling (14), it easily follows

23 2

12 6 cos / 2

6 4 cos /12

l T lmgEI

l ll C l mg

δ α

α α

+ +

+ +

− − = − −

, (41)

from which ( ) , ( )d l d lδ γ+ + ′= = and hence the proof follows.

The theoretical static deflection and the approximated one of a link constituted by

a single element, under the action of the gravitational force, are reported in Fig.

11; these deflections are practically coincident, if the link is subdivided into two

elements.

Fig. 11. Comparison between the theoretical deflection (- - -) and the

approximated one (−) of a generic link.

3642 L. Celentano

Moreover, it is worth noting that the main causes of deformation of a robot link

are: the gravitational load, the ending forces and torques, the inertial translational

and rotational forces. The mentioned deformations can be better approximated

with the wavelet functions instead of the mode shape functions. This fact explains

the values of α and δ reported in Tables II and III of the Example 2 (see

Section 5).

With regard to this, consider again a clamped homogeneous single link with

constant cross-section. It is well-known that, under the hypothesis of little

deformations, the curvature of the link 2

2

( )d d z

dz

=

in static conditions, when a

constant torque C is applied at the free end of the link, is constant C

EI

=

. But the

curvature 2

2

id w

dz

=

of all the mode shapes of the free end is always null (see

boundary condition 3) and Fig. 12).

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-200

-150

-100

-50

0

50

100

150

200

250

z

1

2

3

4

Fig. 12. Curvatures of the first 4 mode shape functions

5. It allows to easily and systematically obtain (see (24), (25) and (27)) the

closed-form dynamic model of robot also if the links have varying cross-sections

and/or if they are not homogeneous.

Indeed, if the generic link has varying cross-section and/or if it is not

homogeneous, as it is well-known, it is very difficult to compute the mode shape

functions. They can be numerically determined by using tools expensive and

difficult to use.

On the contrary the proposed wavelet based method is very easy to apply. Indeed

it is easy to see that the static equation of Euler-Bernoulli (or the Timoshenko

ones) is linear with variable coefficients. Therefore it can be easily solved with a

Matlab program with four independent initial conditions. By linearly combining

the obtained four solutions, it is easy to determine the linear combination


coefficients such that the boundary conditions of the wavelet functions are

satisfied.

It is worth noting that the wavelet based method can be used to find the mode

shape functions but not vice versa.

5. Examples and comparisons with literature

In the following three examples are reported in order to concretely show the

superiority, in terms of precision, computational efficiency and numerical

stability, of the proposed method with respect to very reliable ones available in

literature.

The models presented in the following examples, in a closed and simplified form,

have been implemented with the aid of the Matlab Symbolic Math Toolbox.

Example 1. Let be considered the case of a single flexible link robot with clamped

joint, as shown in Fig. 13.

Fig. 13. Single flexible link robot with clamped joint.

Under the assumption of absence of gravity its dynamic model turns out to be:

[ ]1 1

0 0, 0 .

0

TI F

x x H xB K B H C

δ α− −

= + = −

& (42)

By supposing that 25m, 1kg / m, 1NmL m EI= = = and by using the proposed method

with 5 elements per link, it is easy to note that:

3644 L. Celentano

312 0 54 13 0 0 0 0 0 0

0 8 13 3 0 0 0 0 0 0

54 13 312 0 54 13 0 0 0 0

13 3 0 8 13 3 0 0 0 0

0 0 54 13 312 0 54 13 0 01,

0 0 13 3 0 8 13 3 0 0420

0 0 0 0 54 13 312 0 54 13

0 0 0 0 13 3 0 8 13 3

0 0 0 0 0 0 54 13 156 22

0 0 0 0 0 0 13 3 22 4

B

− − − − − − − =

− − − −

− − −

− − − −

24 0 12 6 0 0 0 0 0 0

0 8 6 2 0 0 0 0 0 0

12 6 24 0 12 6 0 0 0 0

6 2 0 8 6 2 0 0 0 0

0 0 12 6 24 0 12 6 0 0,

0 0 6 2 0 8 6 2 0 0

0 0 0 0 12 6 24 0 12 6

0 0 0 0 6 2 0 8 6 2

0 0 0 0 0 0 12 6 12 6

0 0 0 0 0 0 6 2 6 4

K

− − − − −

− − − − =

− − − −

−

− − − −

(43)

0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 1

T

H

=

; (44)

instead, by using the Ritz-Kantorovich expansion with 10 deformation

polynomials, it is easy to verify that:

2

, 2,3,...,11; 2,3,...,11 ,1

i jLB i j

i j

+ + = = =

+ + (45)

from which

2 4 4 7 8

4 4 4 8 9

4 4 5 9 9

7 8 9 13 14

9 9 9

6.25 10 2.60 10 1.12 10 9.39 10 4.36 10

2.60 10 1.12 10 4.88 10 4.36 10 2.03 10

1.12 10 4.88 10 2.17 10 2.03 10 9.54 10

9.39 10 4.36 10 2.03 10 2.27 10 1.08 10

4.36 10 2.03 10 9.54 10 1

B

⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅=

⋅ ⋅ ⋅ ⋅ ⋅

⋅ ⋅ ⋅

L

L

L

M M M O M M

L

L14 14.08 10 5.18 10

⋅ ⋅

; (46)


{ } 3

0, 4 0

, , 2,3,...,11; 2,3,...,11 ,( 1)( 1), 4 0

3

i jij ij

i j

K k k i jij i j Li j

i j

+ −

+ − <

= = = =− −+ − ≥ + −

(47)

2 3 4 10 11

2 3 9 10

25 125 625 9765625 48828125.

20 75 500 19531250 1074218752 3 4 10 11

T T

L L L L LH

L L L L L

= =

LL

LL(48)

The condition number of the inertia matrix B, given by (46), obtained with the

proposed method is 27.85 10⋅ , whereas the one of the inertia matrix computed by

using the Ritz-Kantorovich expansion is 211.78 10⋅ !

Moreover, in the following the exact natural angular frequencies tω , the ones

calculated by using the proposed method with 5 and 10 elements, respectively 5sω

and 10sω , and finally, due to ill-conditioning problems, the eigenvalues p

λ of the

dynamic matrix computed by using the Ritz-Kantorovich expansion are reported

5

10

0.1406 0.8814 2.4679 4.8361 7.9944 11.9422 16.6796 22.2066 28.5232 35.6293

0.1406 0.8818 2.4768 4.8928 8.1208 13.4909 19.7305 28.6136 40.6478 59.7951

0.1406 0.8814 2.4685 4.8407 8.0145 12.0067 16.8460 22.5689 29.1814

t

s

s

ω

ω

ω

=

=

= 36.2766

0.1340 0.7439 1.5938 3.377 5.6917 10.99 28.46 36.99 98.28 205.57.p

i i i i i i i iλ = ± ± ± ± ± ± ± ± ± ±

(49)

It is worth noting that, because of the ill-conditioning of the inertia matrix (46)

obtained by using the Ritz-Kantorovich expansion, some eigenvalues of the

dynamic matrix are even positive real!

In Fig. 14 the frequency responses ending vertical force – vertical tip

displacement, obtained by using the proposed method with 5 elements, with 50

elements (considered true) and with 10 polynomials, are compared. From this

figure it more clearly emerges the precision and the numerical stability of the

proposed method with respect to the Ritz-Kantorovich expansion. Similar results

are valid also if the proposed method is compared with the other ones.

3646 L. Celentano

10-2

10-1

100

101

-3

-2

-1

0

1

2

3

4

rad/s

proposed method

polynomials

true

Fig. 14. Frequency responses ending vertical force – vertical tip displacement.

Furthermore, it is worth noting that if a planar robot with two flexible links is

considered, if it is modeled by using the proposed method by subdividing the first

link into 5 elements, the transformation which allows to eliminate the reduntant

variables 2 0 2 0,x y of the second link is:

11 1 5 1 1 1 1 51 1

11

11

2 1

2 1

02 01

31

02 01

3 1

41

4 1

511 1

5 1

sin cos cos sin

0 0

0 0

0 0

0 0

0 0

0 0

0 0

0 0

sin cos

0 0

TL L

x x

y y

αα δ α α δ α

δ

γ

δ

γ

δ

γ

δ

γ

δα α

γ

− − −

= +

−

&

&

&

&

&& &

&

& &&

&

&

&

&

. (50)

From (50) it is important to note that:

• by virtue of the choice made of Lagrangian deformation variables, the

transformation matrix 1A depends only on two Lagrangian variables, is sparse

and ill-conditioned;

• the above choice made of Lagrangian deformation variables allows also to

minimize the dependence of 0 0,i ix y and of the position and the orientation of

the end-effector , ,T T Tx y α on those variables.


Instead, if the Ritz-Kantorovich expansion with 10 deformation polynomials for

the first link is used, the transformation which allows to eliminate the redundant

variables 2 0 2 0,x y of the second link is:

10 101 1

1 1 1 1 1 1 1 1 1 1

1 1

2 2

1 1 1 1

3 3

1 1 1 1

4 4

1 1 1 1

5 5

1 1 1 102 01

6 6

1 1 1 102 01

7 7

1 1 1 1

8 8

1 1 1

sin cos cos sin

sin cos

sin cos

sin cos

sin cos

sin cos

sin cos

sin c

i i

i i

i i

L L q L L q

L L

L L

L L

L Lx x

L Ly y

L L

L L

α α α α

α α

α α

α α

α α

α α

α α

α

+ +

= =

− − −

−

−

−

− = + −

−

−

∑ ∑

& &

& &

1

11

2 1

31

4 1

51

6 1

7 1

1

8 19 9

1 1 1 1

9 110 10

1 1 1 1

10 111 11

1 1 1 1

os

sin cos

sin cos

sin cos

T

q

q

q

q

q

q

q

qL L

qL L

qL L

α

α

α α

α α

α α

−

− −

&

&

&

&

&

&

&

&

&

&

&

. (51)

It is worth noting that, in this case, the transformation matrix 1A depends on all

the Lagrangian deformation variables, is full and ill-conditioned.

Finally, by using the assumed mode method with the clamped-free configuration

modes, the transformation which allows to eliminate the redundant variables is

even more complex and also the matrix 1A is full and ill-conditioned.

Moreover, the inertia sub-matrices fiB (9) calculated with the proposed method

are sparse and well-conditioned, while the ones obtained with the other methods

are full and ill-conditioned; hence it follows that the expressions of the inertia

matrix of the flexible robot and of the gradient of the kinetic energy obtained by

using the other methods, besides being ill-conditioned, contain a really high

number of terms that, by the way, have to be calculated with the maximum

precision in order not to further worsen the numerical stability of the model.

In conclusion, this example shows that, already in the very simple case of a robot

constituted by only one flexible link, the Ritz-Kantorovich expansion is

inapplicable when the number of Lagrangian deformation variables is greater than

10, because the dynamic matrix of the linearized model of the robot around an

equilibrium configuration has even positive real eigenvalues, due to

ill-conditioning of the inertia matrix. Moreover, it is explicitly highlighted that the

Ritz-Kantorovich expansion, as well as the assumed mode method, are much

more burdensome, from a computational point of view, than the proposed method.

Therefore, this very simple example may be considered a significant one to show,

in an outstanding manner, the weakness of the Ritz-Kantorovich expansion and of

the assumed mode methods.

3648 L. Celentano

Example 2. Let be considered a robot constituted by two flexible links having

square hollow constant cross-section with the following parameters 10 2 3 3

1 2 1 2 1 2 1 2 1 22.5m, 30mm, 2mm, 21·10 N / m , 7.8·10 kg / m ,L l l s s EL E ρ ρ= = = = = = = = = =

where iρ denotes the density of the material which the i-th link is made of.

Let be considered the above flexible robot in the horizontal configuration with

clamped joints. In this configuration the robot can be considered as a single link

with clamped joint of length 1 2L L+ .

In

Table I are reported: the first two theoretical frequencies Tf , the ones Sf

obtained with the proposed method with one element per link, the frequencies If

obtained with the assumed mode method by using, for each link, the first two

modes of the clamped-free configuration, the ones Cf obtained with the assumed

mode method by using, for each link, the first two modes of the hinged-free

configuration and, finally, the frequencies Pf obtained by using the

Ritz-Kantorovich expansion with two deformation polynomials per link.

( )Tf Hz ( )Sf Hz ( )If Hz ( )Cf Hz ( )Pf Hz

1.33 1.33 1.37 1.52 1.33

8.34 8.41 9.34 10.07 8.41

Table I. First two frequencies of the robot with clamped joints.

In Table II and in Table III the values of the rotation α and of the vertical

displacement δ of the end-effector, produced by an ending torque 50C Nm= , by

an ending vertical force 10F N= and by the gravitational payload are respectively

reported. The above values are calculated by using the theoretical formulae of the

beam theory and the models of the robot obtained with the various methods, as

previously illustrated.

Tα (deg)

Sα (deg) Iα (deg)

Cα (deg) Pα (deg)

C 2.32 2.32 1.86 1.73 2.32

F 1.16 1.16 1.02 0.90 1.16

gq 3.31 3.31 3.02 2.53 3.31

Table II. Tip rotation for ending torque and force and for the gravitational

payload.


Tδ (mm)

Sδ (mm) Iδ (mm)

Cδ (mm) Pδ (mm)

C 101 101 90 78 101

F 67 67 62 52 67

gq 217 217 203 165 217

Table III. Tip vertical displacement for ending torque and force and for the

gravitational payload.

From Tables I, II, III it emerges that the proposed method and the

Ritz-Kantorovich expansion, which are equivalent in the considered case of links

with constant cross-section and for the chosen number of deformation freedom

degrees, result better than the ones based on the assumed mode method. However,

when the number of deformation freedom degrees increases, the proposed method

is numerically very stable with respect to the Ritz-Kantorovich expansion, as it

emerges from

Table IV, in which the condition numbers of the inertia matrices calculated

with the proposed method and with the Ritz-Kantorovich expansion are reported,

for several degrees of the polynomials Pn and numbers of the elements

Sn , such

to always consider the same number of deformation freedom degrees Fn per link.

Fn Sn

Pn Scond

Pcond

2 1 2 7.99·103 8.99·10

4

6 3 6 2.90·105 1.35·10

12

10 5 10 1.95·106 2.81·10

19

Table IV. Condition numbers of the inertia matrices when the number of

deformation freedom degrees increases.

In Fig. 14 the frequency responses ending torque – tip rotation of the robot in the

considered configuration are reported. It is worth noting that the frequency

response, obtained by using the proposed method with only one element per link,

in the considered frequency interval, practically coincide with the true one, which

is calculated still by using the proposed method, but with 10 elements per link.

3650 L. Celentano

Fig. 14. Frequency responses ending torque – tip rotation with 2 Lagrangian

deformation variables per link.

Finally, in Fig. 15 the frequency responses ending torque – tip rotation of the

robot in the considered configuration are reported, which are obtained by using

the proposed method with 5 and 10 elements per link (10 and 20 Lagrangian

deformation variables per link) and the Ritz-Kantorovich expansion with 10

deformation polynomials per link (10 Lagrangian deformation variables per link).

It is worth noting that the model obtained with the Ritz-Kantorovich expansion is

strongly impaired by the numerical instability, according to the results reported in

Table IV.

100

101

102

103

-120

-110

-100

-90

-80

-70

-60

-50

-40

-30

-20

rad/s

polynomials

proposed method ne=5

proposed method ne=10

Fig. 15. Frequency responses ending torque – tip rotation with 10 Lagrangian

deformation variables per link.

Let be considered the above robot in the same configuration, but under the

assumption that (see Fig. 1) 2

1 2 1 2 1 21 , 1 ,M M M M kg J J kg m

+ + − − − −= = = = = = ⋅


2

1 2 2J J kg m+ += = ⋅ and by supposing that each joint is controlled by a PD controller

with the following gains 410 , 2 100p d

K K= = ⋅ .

In Fig. 16 the frequency responses ending torque – tip rotation of the robot in the

considered configuration are reported, which are obtained by using the proposed

method with 2 and 10 elements per link (4 and 20 Lagrangian deformation

variables per link) and by using the assumed mode method, considering the modes

of the clamped-free configuration of the link, with 4 deformation modes per link.

It is worth noting that the frequency response obtained with the proposed method,

in the considered frequency range, practically coincide with the true response,

obtained still by using the proposed method, but with 10 elements per link.

Instead, the frequency response obtained with the assumed mode method presents

considerable errors in the whole frequency range. This result, as already observed,

is a consequence of the fact that the static precision, the modes and the modal

frequencies relating to an equilibrium configuration of the whole robot, strongly

depend on the control law and on the payload which the robot is subjected to. This

dependence, by virtue of the instantaneity principle, is intrinsic to the proposed

method, but not to the other methods which use the modes of the links in

well-defined, but scarcely recurring, configurations.

100

101

102

103

-130

-120

-110

-100

-90

-80

-70

-60

-50

-40

-30

rad/s

clamped-free

proposed-method

true

Fig. 16. Frequency responses ending torque – tip rotation of the robot controlled

by a PD controller.

Example 3. Let be considered a robot constituted by three flexible links having

square hollow constant cross-section with the following parameters

31 2 3 1 2 3 1 22.5m, 50mm, 2mm,L L l l lL ss s= = = = = = === 3

10 2

1 221·10 N / m ,E EE == =

3 3

1 2 37.8· ,10 kg / mρ ρ ρ= = = where

iρ denotes the density of the material which the

i-th link is made of.

Let be further supposed that (see Fig. 1) 1 2 3 1 2 3

1kg ,M M M M M M+ + + − − −= = = = = =

2

1 2 3 1 2 31kg m .J J J J J J

+ + + − − −= = = = = = ⋅

3652 L. Celentano

In Tables V and VI the number of multiplications required to evaluate the inertia

matrix B and the gradient of the kinetic energy c are reported, which are computed

by using the various methods, under the assumption that the number of

Lagrangian deformation variables per link is 4 (4 polynomials, 4 modes, 2

elements) and 8 (8 polynomials, 8 modes, 4 elements) respectively.

B c B and c

Proposed

method

344 878 1222

Polynomials 1147 2770 3917

Clamped-free

modes

1179 2830 4009

Hinged-free

modes

1001 2526 3527

Table V. Number of multiplications with 4 deformation variables per link.

B c B and c

Proposed

method

687 1712 2399

Polynomials 4232 10915 15147

Clamped-free

modes

4101 10997 15098

Hinged-free

modes

3665 10050 13615

Table VI. Number of multiplications with 8 deformation variables per link.

Remark 8. It is worth noting that when the number of Lagrangian deformation

variables has doubled also the number of multiplications required by the proposed

method has about doubled, instead, the one required by the other methods has

about quadrupled.

In Table VII the time costs required by the dynamic simulation of the above robot

are reported, in the hypotheses that: 4 Lagrangian deformation variables per link

are used; each joint of the robot is controlled by using a PD controller with 315 10

pK = ⋅ and 3

10d

K = ; the torque disturbance [ ]3( ) 3 10 1( 3) 1( 4)d t t t= ⋅ − − − acts on

the first joint; the initial configuration of the link is undeformed with angles

1 2 3, 0,

10 10

π πα α α= − = = − ; the numerical solvers Matlab ode45 and ode15s are

used, with variable step size and relative tolerance of 510− .


Ode45 Ode15s

Proposed

method

25.9 27.1

Polynomials 67.0 91.7

Clamped-free

modes

77.7 103.3

Hinged-free

modes

553.9 147.7

Table VII. Time costs in seconds of the dynamic simulation with the initial

conditions 1 2 3, 0,

10 10

π πα α α= − = = − .

In Table VIII the time costs required by the dynamic simulation of the above

robot are reported, under the same assumptions as above, but supposing that the

initial configuration of the link is undeformed with angles 1 2 3, 0,4 4

π πα α α= − = = − .

Ode45 Ode15s

Proposed

method

25.4 29.7

Polynomials divergence divergence

Clamped-free

modes

68.8 107.8

Hinged-free

modes

556.8 163.6

Table VIII. Time costs in seconds of the dynamic simulation with the initial

conditions 1 2 3, 0,4 4

π πα α α= − = = − .

It is interesting to note that the simulation is impossible when the polynomials of

the Ritz-Kantorovich expansion are used.

In Table IX the time costs required by the dynamic simulation of the above robot

are reported, under the same assumptions as above, but supposing that the initial

configuration of the link is undeformed with angles1 2 30.7 , 0, 0.7

2 2

π πα α α= − ⋅ = = − ⋅ .

3654 L. Celentano

Ode45 Ode15s

Proposed

method

25.1 29.7

Polynomials divergence divergence

Clamped-free

modes

divergence divergence

Hinged-free

modes

557.1 165.9

Table IX. Time costs in seconds of the dynamic simulation with the initial

condition 1 2 30.7 , 0, 0.72 2

π πα α α= − ⋅ = = − ⋅

It is worth noting that, in this case, the simulation is impossible not only with the

polynomials of the Ritz Kantorovich expansion, as in the previous case, but also

with the assumed mode method with the modes of the clamped-free configuration

of the link.

Finally, it is important to note that, in all the three analyzed cases, the proposed

method is strongly advantageous with respect to the others, when it has been

possible to make a comparison. This is a very substantial advantage also by taking

into account the high complexity of the models, as it emerges from the number of

multiplications reported in Table V and in Table VI. With reference to the last

simulation, the plots of the ending deflections iδ and of the ending rotations

iγ

of each link are reported in Figs. 17, 18 and 19.

0 1 2 3 4 5 6 7-50

-40

-30

-20

-10

0

10

20

30

40

50

time [s]

[cm

]

δ1

δ2

δ3

Fig. 17. Time histories of the terminal deflections iδ of the three links.


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-50

-40

-30

-20

-10

0

10

20

30

40

50

time [s]

[cm

]

δ1

δ2

δ3

Fig. 18. A detail of Fig. 17.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-25

-20

-15

-10

-5

0

5

10

15

20

time [s]

[de

g]

γ1

γ2

γ3

Fig. 19. A detail of the time histories of the terminal rotations iγ of the three

links.

From the above comparisons it emerges that the simulation times required by the

other models are markedly longer than those required by the proposed method,

both as shown in the above tables and because, due to numerical stability

problems, a very small integration step is sometimes required. Finally, for some

initial conditions not very far from the equilibrium configuration of the robot, the

numerical integration algorithms (implemented by the numerical solvers Matlab

ode45 and ode15s) used for the simulation of the models obtained with the

Ritz-Kantorovich expansion and with the assumed mode method (clamped-free

configuration of the link), always diverge, even if a high relative tolerance is

imposed. This situation never occurs when the model obtained with the proposed

method is used, also if a high number of Lagrangian variables of deformation is

employed.

3656 L. Celentano

6. Experimental validation

The presented modeling methodology has been experimentally validated in

several different static and almost static conditions by considering a planar robot

with two very flexible links (see Figs. 20 and 21).

Fig. 20. The considered robot in two static equilibrium configurations.

I II

III IV

Fig. 21. Four configurations of the considered robot during an almost static

motion to track the vertical line.


Remark 9. From these examples and experiments, the accurate validation of the

Authors’ tools also by comparing the results obtained with several parameters and

in different operative conditions with the ones available in literature, according to

us, justify the significance and utility of the proposed method.

The proposed modeling method is an unescapable choice when it is necessary to

obtain high precisions, in particular at high frequencies (necessary to validate a

closed-loop control law) and/or not prohibitive calculus times and/or when the

other modeling methods result inapplicable because of numerical divergence

problems.

7. Conclusions

In this paper a method to obtain a closed-form dynamic model of a planar flexible

robot has been proposed. The model obtained with this method is very simple to

implement, very efficient, from a computational point of view, and numerically

very stable. Moreover, it has been proved that this model guarantees no static

error both in the presence of the gravity actions of the links and of the payload,

and in the presence of concentrated disturbances and control actions in forces and

torques. For bounded bandwidth actions, the proposed method presents an error

strongly decreasing when the number of elements of the links increases. Finally

three significant examples have been reported in order to concretely show the

superiority, in terms of precision, computational efficiency and numerical

stability, of the proposed method with respect to very reliable ones available in

literature.

Author is successfully working in order to extend the proposed method to the case

of non planar flexible robots, having not necessarily constant cross-section links

and in order to introduce the internal friction into the model and in order to design

efficient control systems of flexible robots.

References

[1] W.J. Book, O. Maizza-Neto, D.E. Whitney, Feedback control of two-beam

two-joint systems with distributed flexibility, ASME J. Dyn. Syst., Meas.

Control (1975) 424-431.

[2] M.J. Balas, Feedback control of flexible systems, IEEE Trans. Autom. Control

23 (1978) 673-679.

[3] A. Truckenbrodt, Control of elastic mechanical systems, Regielungstechnick

30 (1982) 277-285.

[4] P. B. Usoro, R. Nadira, and S. Mahil, A Finite Element/Lagrangian Approach

to Modeling Lightweight Flexible Manipulators, ASME J. Dyn. Syst., Meas.

Control, 108, (1986) 198-205.

3658 L. Celentano

[5] P.C. Hughes, Space structure vibration modes: How many exist? Which are

important, IEEE Control Syst. Mag., 7 (1987) 22–28.

[6] G.G. Hasting, W.J. Book, A linear dynamic model for flexible robot

manipulators, IEEE Control Syst. Mag., 7 (1987) 61–64.

[7] P. Tomei, and A. Tornambe, Approximate Modeling of Robots Having Elastic

Links, IEEE Trans. Syst., Man, Cybern., 18 (5) (1988) 831–840.

[8] Y. Chait, M. Miklavcic, C. R. Maccluer, C. J. Radcliffe, A natural modal

expansion for the flexible robot arm problem via a self-adjoint formulation,

IEEE Trans. Robot. Autom., 6 (5) (1990) 601–603.

[9] A. De Luca, B. Siciliano, Closed form dynamical model of planar multilink

lightweight robots, IEEE Trans. Syst., Man, Cybern. 21 (4) (1991) 826–839.

[10] D. Wang, M. Vidyasagar, Modeling a class of multilink manipulators with

the last link flexible, IEEE Trans. Robot. Autom., 8 (1) (1992) 33–41.

[11] C. J. Li, and T. S. Sankar, Systematic methods for efficient modeling and

dynamics computation of flexible robot manipulators, IEEE Trans. Syst., Man,

Cybern., 23 (1) (1993).

[12] F. Boyer, and W. Khalil. Kinematic model of a multi-beam structure

undergoing large elastic displacement and rotations. Part one: model of an

isolated beam. J. Mech. Mach. Theory, 34 (1999) 205-222.

[13] G. Wang, Y. F. Li, W. L. Xu, Regularization-based recovery scheme for

inverse dynamics of high-speed flexible beams, Appl. Math. Comput., 115

(2-3) (2000) 161-175.

[14] R.D. Robinett, C. Dohrmann, G.R. Eisler, J. Feddema, G.G. Parker, D.G.

Wilson, D. Stokes, Flexible Robot Dynamics and Controls, Kluver

Academic/Plenum Publishers, New York, 2002.

[15] S. E. Khadem, A. A. Pirmohammadi, Analytical development of dynamic

equations of motion for a three-dimensional flexible link manipulator with

revolute and prismatic joints, IEEE Trans. Syst., Man, Cybern., 33 (2) (2003)

237–249.

[16] X. Hou, S.-K. Tsui, A feedback control and a simulation of a torsional elastic

robot arm, Appl. Math. Comput., 142 (2-3) (2003) 389-407.

[17] X. Hou, S.-K. Tsui, Analysis and control of a two-link and three-joint elastic

robot arm, Appl. Math. Comput., 152 (3) (2004) 759-777.

[18] T.H. Megson. Structural and stress analysis. Butterworth-Heinemann,

Oxford, (2005).

[19] L. Celentano, R. Iervolino, A New Method for Robot Modeling and

Simulation, ASME J. Dyn. Syst., Meas., Control, 128 (2006) 811-819.

[20] A. Macchelli, C. Melchiorri, S. Stramigioli, Port-based Modeling of a

flexible link, IEEE Trans. Robot., 23 (2007) 650-660.

[21] S. Liu, L. Wu, Z. Lu, Impact dynamics and control of a flexible dual-arm

space robot capturing an object, Appl. Math. Comput., 185 (2) (2007)

1149-1159.


[22] L. Celentano, An innovative method for robots modeling and simulation, in:

Harald Aschemann, New Approaches in Automation and Robotics, Publisher:

InTech, Vienna (2008), 173-196. Available at www.intechopen.com/ .

[23] B. Siciliano, and O. Khatib. Springer handbook of robotics. Springer, 2008.

[24] S. Moberg, Modeling and Control of Flexible Manipulators. PhD Thesis.

Department of Electrical Engineering, Linköping University (2010).

[25] L. Celentano, A. Coppola, A wavelet based method to modeling realistic

flexible robots, presented at 18th

IFAC World Congress, Milano, Italy, (28

Aug.-2 Sep. 2011), 929-937.

[26] L. Celentano, A. Coppola, A computationally efficient method for modeling

flexible robots based on the assumed modes method, Appl. Math. Comput.,

218 (8) (2011), 4483-4493.

Received: March, 2012

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