complete dynamic modeling and approximate state … · complete dynamic modeling and approximate...

Journal of Mechanical Science and Technology 26 (9) (2012) 2845~2856

www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-012-0731-x

Complete dynamic modeling and approximate state space

equations of the flexible link manipulator†

Habib Esfandiar1,* and Saeed Daneshmand2 1Department of Mechanical Engineering, Firuzkooh Branch, Islamic Azad University, Firuzkooh, Iran

2Department of Mechanical Engineering, Majlesi Branch, Islamic Azad University, Isfahan, Iran

(Manuscript Received November 4, 2011; Revised March 17, 2012; Accepted May 3, 2012)

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Abstract

This work treats the problem of dynamic modeling and state space approximation for robotic manipulators with flexibility. Case stud-

ies are planar manipulators with a single flexible link together with clamped-free ends and tip mass conditions. In this paper, complete

dynamic modeling of the flexible beam without premature linearization in the formulation of the dynamics equations is developed,

whereby this model is capable of reproducing nonlinear dynamic effects, such as the beam stiffening due to the centrifugal and the Corio-

lis forces induced by rotation of the joints, giving it the capability to predict reliable dynamic behavior. On the other hand, in order to

show the joint flexibility effects on the model dynamic behaviors, manipulator with structural and joint flexibility is considered. Thus, a

reliable model for flexible beam is then presented. The model is founded on two basic assumptions: inextensibility of the neutral fiber

and moderate rotations of the cross sections in order to account for the foreshortening of the beam due to bending. To achieve flexible

manipulator control, the standard form of state space equations for a flexible manipulator system (flexible link and actuator) is very im-

portant. In this study, finite difference method for discretization of the dynamic equations is used and the state space equations of the

flexible link with tip mass considering complete dynamic of the system are obtained. Simulation results indicated substantial improve-

ments on dynamic behavior and it is shown that the joint flexibility has a considerable effect on the dynamic behavior of rotating flexible

arm that should not be simply neglected. The effects of tip mass is proved to be increasing the elastic deformations' amplitudes and in-

creasing stability.

Keywords: Flexible link; Manipulator; Inextensibility; Tip mass; State space; Finite difference

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1. Introduction

The robots used in industry are built mechanically rigid.

This feature simplifies their kinematics and dynamics. It also

simplifies their sensor and control systems because accurate

estimations of the tip position are obtained simply by measur-

ing the angles of motors. Unfortunately, this rigidity require-

ment which facilitates the control strongly determines the

mechanical design leading to robots whose links are unneces-

sarily heavy (considering the load to be carried) and thick

Furthermore, the number of robotic applications where the

arms have to be lightweight (relative to their size) is growing:

aerospace applications, robots mounted on vehicles, large

manipulators for construction, etc. These arms must have links

of small cross-section (compared to their length), but then they

exhibit an undesirable flexibility, leading to mechanical vibra-

tions, that make the control of their tip position extremely

difficult. Therefore, there has been a major research effort

over the last few years to model and control flexible structures

and, in particular, flexible arms. In spite of the flexibility of

robot arm, many modeling methodologies have been proposed

for the flexible link manipulators as analytical (exact), lumped

mass and spring, eigenfunctions expansion, assumed mode,

finite element and finite difference methods. Book initiated

research on the dynamic modeling and control of the flexible-

link manipulators in the 1970s [1]. Since then, many works on

the subject have been published. Finding the exact solution of

partial differential equations, as equations of motion of the

flexible link is very difficult. Widely used methods, to model

the flexible manipulators are based on the assumed mode of

the method used [2, 3], the finite element method [4] or the

Ritz-expansion method [5], making use of either the Lagran-

gian formulation or the Newton–Euler recursive formulation.

The Hamilton principle [6, 7] has also been frequently used to

derive the dynamics of flexible manipulators. Theodore et al.

[8], Kopacek et al. [9] and Ge et al. [10] made some compari-

sons between the assumed mode method and the finite ele-

ment method for modeling the flexible multi-link manipula-

tors. In order to solve a large set of differential equations de-

*Corresponding author. Tel.: +98 9124207065, Fax.: +98 53115223229

E-mail address: [email protected], [email protected] † Recommended by Associate Editor Eung-Soo Shin

© KSME & Springer 2012

2846 H. Esfandiar and S. Daneshmand / Journal of Mechanical Science and Technology 26 (9) (2012) 2845~2856

rived by the finite element method, a lot of boundary condi-

tions should be considered, which are, in most situations, un-

certain for the flexible manipulator. Using the assumed mode

method to derive the motion equations of the flexible manipu-

lators, only the first several vibration modes are usually re-

tained by truncation and the higher modes are neglected. Re-

ducing the system state variables in the assumed mode method

makes it easier to utilize real-time control. In general, the first

several vibration modes play the dominant role in the dynamic

characteristics of the flexible manipulator. Each modeling

approach has its own characteristics from the complexity,

adaptability and accuracy point of view. Tso et al. [11] used a

measurement based method incorporating the assumed mode

in conjunction with Lagrangian formulation to derive the mo-

tion equations for the flexible-link manipulator. A comparison

between different modeling techniques like the simple con-

centrated (lumped mass and spring) model and the finite ele-

ment method was reported in Ref. [12]. Zhou et al. [13] em-

ployed finite difference method in descretizing the partial

differential equations of a single flexible link. If accurate

modeling of the dynamics of the flexible beam is not required

one may model it as a rigid beam with a mass-spring combi-

nation at the free end [14]. Somaraju [15] extracted equations

of motion for a single flexible beam with the Hamilton princi-

ple and used eigenfunction expansion method for descretizing.

Yokohama [16] investigated the effect of shear deformations

on the free vibrations characteristics of rotating beams.

Velthius et al. [17] reported a dynamic model for a rotating

flexible arm impacted on its tip. They utilized the finite ele-

ment method in descretizing the beam bending deformations

and their model resulted in a coupled dynamic model for the

beam reference motion and the beam elastic deformations.

The effect of payload on the vibrations of a rotating beam was

reported by Low [18]. Al-Bedoor and Almusallam [19] pre-

sented a dynamic model of a rotating flexible arm driven

through a joint which is flexible in torsion. The small defor-

mation theory is adopted for both the joint torsional deforma-

tions and the arm bending deformations. The finite element

method related to the Euler-Bernoulli beam theory is adopted

to discretize the arm bending deformations. The Lagrangian

dynamics is utilized in developing a completely coupled and

nonlinear dynamic model. Featherstone and Orin examined

some of the accomplishments in the field of robot dynamics

research, from the development of the recursive Newton-Euler

algorithm to the present day. Equations and algorithms are

given for the most important dynamics computations, ex-

pressed in a common notation to facilitate their presentation

and comparison [20]. The design, dynamic modeling and ex-

perimental validation of a new three degree of freedom flexi-

ble arm were described by Somolinos et al. The arm was de-

signed on the assumption that all its mass was concentrated at

the tip [21]. Moberg [22] treated three different aspects of

modeling and control of flexible, i.e. elastic, manipulators. An

improved model where the elasticity is described by a number

of localized multidimensional spring-damper pairs is therefore

proposed. This model is called "the extended flexible joint

model". The main contribution of this work aims at the devel-

opment of efficient models and state space representation for a

single flexible link robot. A payload is considered to be car-

ried at the tip of the link and this subject is motivated by mod-

eling of flexible link together with tip mass. Therefore, in this

work, dynamic model of a flexible beam in two cases as,

clamped-free ends beam considering transverse and longitudi-

nal vibrations and flexible beam with tip mass are extracted.

In spite of the advantages of finite element method for model-

ing of the flexible link, among the several approximate meth-

ods, the finite difference method is employed in descretizing

the flexibility of link and the Hamilton principle is utilized in

deriving the equations of motion. Because, this method has

simple programming, compatible to solution domain, it can be

used in forward and backward form of derivatives for bounda-

ries and central form of derivatives for other places and the

algebraic equations that result from approximating differential

equations almost always have simple structure (e.g. they may

be tridiagonal), so computing a solution is relatively inexpen-

sive. The paper is organized in the following sections: in Sec-

tion 2 a model for the deformation of beams and necessary

kinematic assumptions is presented, in Section 3 dynamic

models are presented for an Euler–Bernoulli link, in Section 4

the discretization procedure for the previous models is de-

scribed, in Section 5 simulation results are presented, in Sec-

tion 6, the overall system’s dynamics is presented, in section 7

state space equations for the overall system (flexible arm and

actuator) are derived and the general control problem is dis-

cussed and in the last section the final conclusions are per-

ceived.

2. Mechanical issues

2.1 Kinematic assumptions

The most basic assumption that is usually considered for the

analysis of slender beams is that plane beam cross sections

remain plane during deformation. This assumption is moti-

vated by the Euler–Bernoulli hypothesis, which states that

during bending, plane cross sections which are normal to the

beam axis before deformation remain plane after deformation

and normal to the deformed axis. Our kinematic assumptions,

which consists in relaxing the Euler–Bernoulli hypothesis, can

be synthesized as follows:

(1) Plane beam cross sections before deformation, remain

plane after deformation.

(2) The deformation due to shear is considered.

(3) Beam neutral fiber does not suffer extension, and fore-

shortening is considered.

(4) The beam axis before deformation is a straight line.

The coordinate systems used in developing the model are

shown in Fig. 1. Wherein, 0 0 0 0,O X Y Z is the inertial refer-

ence frame, 1 1 1 1,O X Y Z is a body coordinate system at-

tached to the motor shaft and 2 2 2 2,O X Y Z is another body

coordinate system attached to the hub such that its 2X axis is

H. Esfandiar and S. Daneshmand / Journal of Mechanical Science and Technology 26 (9) (2012) 2845~2856 2847

directed along the initial situation of the arm. The rotation of

frame 1 1 1 1,O X Y Z relative to frame 0 0 0 0,O X Y Z is de-

scribed by the angle ( )tθ and α denotes rotation of frame

2 2 2 2,O X Y Z relative to frame 1 1 1 1,O X Y Z . Consider a

flexible beam with length l fixed on the hub in the horizon-

tal plane. The joint is taken to have small torsional deforma-

tions. The hub is assumed to be rigid and the flexible link is

radially attached to the hub. The motion of the manipulator

system is described by the rigid body rotation ( )θ , the tor-

sional deformation angle ( )α which is measured relative to

the motor coordinate system and flexible displacement

( , )y x t of the beam. The rotatory inertia is also considered.

The beam is divided into n elements with the i th element

having two nodes i and 1i + and the transverse component

of the displacement vector of an infinitesimal link element is

described by ( , )y x t .

Having introduced the kinematics assumptions for the beam

deformation, the floating frame approach is now adopted in

order to develop the model equations. A coordinate system

2 2 2 2,O X Y Z is oriented such that the 2 2O X axis is coinci-

dent with the beam neutral axis, while axes 2 2O Y and

2 2O Z

are in the directions of the principal moments of the cross

section (Fig. 2).

3. Dynamic modeling of a flexible beam as the robot’s

arm

In this section we will consider flexible beam without joint

flexibility modeling in two different cases as follows:

(1) Flexible beam with both transverse and longitudinal vi-

brations and clamped (fixed)-free ends condition.

(2) Flexible beam with transverse vibration and tip mass.

3.1 Flexible beam with both transverse and longitudinal

vibrations and clamped-free ends conditions

In this section, we will consider dynamic modeling of a sin-

gle flexible link without joint flexibility with clamped-free

condition together with both transverse and longitudinal vibra-

tions. The position of an infinitesimal element of the link lo-

cated at a distance x from the frame origin in the 2 2O X di-

rection, relative to the 0 0 0 0,O X Y Z reference frame is given

by

( )( )

1

2

( , ) ( , ) cos ( , )sin

( , ) sin ( , )cos

d x t x u x t y x t e

x u x t y x t e

θ θ

θ θ

= + − +

+ +

(1)

where 1 2,e e ,3e are the unit vectors along the

0 0,X Y and

0Z axis, respectively.

According to this expression, the kinetic energy is involved

kinetic energy of the hub and kinetic energy of flexible link as:

( )

( ) ( ) ( ) ( )

1 12

2 2

1 2

2

1 22 2 2 2 22 22

r lKE I d d dx

H r

KE IH

r lx u x u y y x u y x u dx

r

θ ρ

θ

ρ θ θ θ θ

+= + ⋅∫

= +

++ + + + + + + − +∫

ɺ ɺɺ

ɺ

ɺ ɺ ɺ ɺɺ ɺ ɺ ɺ ɺ ɺ

(2)

where

⋅− denotes time derivative of ( )− and ( ) ( )− ⋅ −

denotes the dot product of vectors ( )− . The potential energy is

composed of the internal energy due to the elastic deformation

of the link due to bending and tension.

( ) ( )2 2

2

2

, ,1 1

2 2

r l r l

r r

y x t u x tPE EI dx EA dx

x x

+ +

∂ ∂= +∫ ∫

∂ ∂

(3)

where

( )2

,1

2

r l

r

u x tEA dx

x

+ ∂ ∂

∫

is the potential energy due to longitudinal vibration and the

potential energy due to gravity is not accounted for since only

the motion in the plane perpendicular to the gravitational field

is considered. For extracting equations of equations of motion,

the extended Hamilton principle is used as following:

( )2

1

0t

t

KE PE W dtδ δ δ− + =∫ (4)

Fig. 1. Rotary arm system.

Fig. 2. Mechanical model of a single flexible link.


where

( )LW T tδ δθ= (5)

is the virtual work done by the torque applied at the hub,

( )T tL

. After a long process of computations, the following

dynamic equations can be obtained in terms of the two flexi-

ble displacements and rotation of the hub,

( ) ( )2 22 2 2

2 2 2 2

,y x td y dy EI x u

dt x x t dt

θ θρ ρ ρ

∂∂ ∂− + + + + +

∂ ∂ ∂

( , )2 0

u x t dfor r x r l

t dt

θρ

∂= ≤ ≤ +

∂ (6)

( )2 2

2

2

2

( , )

( , )2 0

d u u x tx u EA

dt x x t

d y x t dy

dt t dt

θρ ρ

θ θρ ρ

∂ ∂ ∂+ + − +

∂ ∂ ∂

∂+ =

∂

(7)

( ) ( ) ( ) ( )22

2

2 2

2 22

2 2

,2

( , ) ( , )2

r l

r

y x tu d dx u x u x u

t dt dt t dxy x t d d u x t

y y yt dt dt t

θ θρ ρ ρ

θ θρ ρ ρ

+

∂∂+ + + + + +

∂ ∂ +∫∂ ∂

+ −∂ ∂

( )2

2H L

dI T t

dt

θ= (8)

( )

( , ) 0

, 0

( , ) 0

y x t

yx r x t

x

u x t

=

∂= =∂

=

(9)

2

2

3

3

0

0

0

yEI

x

yx r l EIx

uEA

x

∂=

∂∂= + =∂∂

=∂

(10)

where Eqs. (6) and (7) are the system equations of motion, Eq.

(8) is the constraint equation expressing the total balance of

angular momentum of the system, Eq. (9) expresses the three

geometric boundary conditions at the clamped end of link and

Eq. (10) are the natural boundary conditions at the free end of

link. Moreover, initial conditions for ( ) ( ), ,t y x tθ and

( ),u x t require for solution set of the above equations such that

( ) ( )0 00 ,

d tin t t

dt

θθ θ θ= = = ɺ (11)

( ) ( )0 0

,, ,

dy x ty x t y y for r x r l

dt= = ≤ ≤ +ɺ

( ) ( )0 0

,, ,

du x tu x t u u

dt= = ɺ

where ( )0

− denotes value of ( )− , when 0t = . Thus, a

fourth order partial differential equation respected to ( , )y x t

and second order partial differential equation relative to

( , )u x t , six boundary conditions, six initial conditions and a

constraint equation expressing the overall balance of angular

momentum are obtained. In the set of mentioned equations,

terms are defined as Table 1.

3.2 Flexible beam with transverse vibration and tip mass

A payload is considered to be carried at the tip of the link

and this subject is motivated by modeling the flexible link

carrying a tip mass. In this section we consider a single flexi-

ble robot link without flexibility joint fixed in the hub along

with a lumped mass on the other end, therefore, the total ki-

netic energy of the flexible beam with a tip mass is the sum-

mation of translational and rotational kinetic energy of the

flexible beam, kinetic energy of the hub, translational and

rotational kinetic energy of the point mass, whose kinetic en-

ergy is as follows:

b b Tip Tip

t rot hub t rKE KE KE KE KE KE= + + + + (12)

( )( ) ( ) ( )( )2 2 2 2 21 1. , 2 ,

2 2

r l

Hr

KE I x y x t y x t x y x t dxθ ρ θ θ+

= + + + + +∫ɺ ɺ ɺɺ ɺ

( ) ( ) ( )( )2 21, 2 ,

2

r l

r

x y x t y x t dxφ θ θ+

′ ′+ + +∫ ɺ ɺɺ ɺ

( ) ( )( ) ( ) ( ) ( )2 2 2 21. , 2 ,

2t

M r l y r l t y r l t r l y r l tθ θ

+ + + + + + + + +ɺ ɺɺ ɺ

( ) ( )2 2 .1

, 2 ,2

J y r l t y r l tt θ θ

′ ′+ + + +ɺ ɺɺ ɺ (13)

To simplify

( )

( ) ( )( )

2 2 2

2 2

,

1.

2

r l r l

br r

t t

x y dx x dx I

M r l y r l t I

ρ ρ+ +

+ ≈ =∫ ∫

+ + + ≈ (14)

that tI is the inertia moment of the tip mass around the hub

and bI is the beam rotation inertia about the origin

0O , as if

it would be rigid. Using the same potential energy and virtual

work in the extended Hamilton principle, the following equa-

Table 1. Interpretation of the mathematical expression of Eqs. (6)-(8).

Description Transverse component Longitudinal

component

Centrifugal force

2d

ydt

θρ

( )2

dx u

dt

θρ

+

Force due to tan-

gential accelera-

tion

( )2

2

dx u

dt

θρ +

2

2

dy

dt

θρ

Coriolis force ( , )

2u x t d

t dt

θρ

∂

∂

( , )2

y x t d

t dt

θρ

∂

∂

Torque due to the

Coriolis force ( ) ( , )

2u x t d

x ut dt

θρ

∂+

∂

( , )2

y x t dy

t dt

θρ

∂

∂

Torque due to the

force due to tan-

gential accelera-

tion

( )2

2

2

dx u

dt

θρ +

22

2

dy

dt

θρ


tions are derived,

( ) ( )

2 2 2 2

2 2 2 2

2

2

( , )0

y y dEI x

t x x dt

y x t dx x for r x r l

x t x t dt

θρ ρ

θφ φ

∂ ∂ ∂+ + −

∂ ∂ ∂

∂ ∂ ∂ ∂+ = ≤ ≤ +

∂ ∂ ∂ ∂

(15)

( )

( ) ( )

( ) ( ) ( )

2 2

2 2

2

2

2

2

( , )

,

r l

H b tr

r l

r

t L

d yI I I x dx

dt t

y x t dx x dx

t x t dt

d y r l tM r l T t

dt

θρ

θφ φ

+

+

∂+ + + +∫

∂

∂ ∂ ∂+ +∫

∂ ∂ ∂

++ =

(16)

( )

( , ) 0

, 0

y x t

for x r yx t

x

== ∂

=∂

(17)

( )

2 2 2

2 2 2

2 2 3

2 2 3

( , )0

0 .

d y x t yJ J EIt tdt t x x

for x r ly d y

M M r l EIt tt dt x

θ

θ

∂ ∂ ∂+ + =

∂ ∂ ∂= +

∂ ∂+ + − =

∂ ∂

(18)

As mentioned before, the initial conditions for ( )tθ and

( ),y x t are necessary for solving the set of above equations

illustrated in Eq. (11).

4. Finite dimensional equations (discretization)

The continuous models consist of partial differential equa-

tions difficult to deal with both analytically and computation-

ally. In order to obtain a set of ordinary differential equations,

a discretization of the equations developed previously must be

performed. The finite difference method is used to approxi-

mate partial differential equations described in the previous

sections with a set of ordinary differential equations. In this

method, the interval r x r l≤ ≤ + is divided into n uniform

segments with

lx h

n∆ = =

and , 0,1, ,ix r ih i n= + = … . Space derivatives are substituted

by finite differences such that the second and fourth spatial de-

rivatives are approximated with the second-order central differ-

ence approximation and the first spatial derivative is approxi-

mated with the first-order backward difference formula.

4

1 1 2 2

4

2

1 1

2 2

0 0 1

( ) 4 6 41,2, ,

4

( ) 2

( )

i i i i i i

n n n n

d y t y y y y yi n

dx h

d y t y y y

dx h

dy t y y

dx h

− + − +

− +

−

− + − + += =

− +=

−=

…

(19)

And third spatial derivative is approximated by

3

1 1 2

3 3

( ) 3 3n n n n nd y t y y y y

dx h− + +− + − +

= (20)

where the abbreviated notation ( )iy t and 0 ( )y t are used for

( , )iy x t and ( , )y r t , respectively. The point requiring atten-

tion is, that fourth order derivative in every point depends on

the information of five points, thus, meshing must have at

least four elements or five nodes. The elements consisting the

integral are discretized; the integral is replaced with summa-

tion ( )∑ and the beam is meshed from 0 to n intervals

such that first node in the mesh is on the hub and equals 0

and the final node is in x r l= + and equals n .

In this section, equations of the tip mass are considered and

terms relevant to rotational kinetic energy of the flexible beam

is neglected and the set of ordinary differential equations are

presented. Using finite difference, ordinary differential equa-

tions are,

1 1 2 2

4

4 6 40 1,2, ,

i i

i i i i i

y x

y y y y yEI for i n

h

ρ ρ θ

− + − +

+ +

− + − + += =

ɺɺɺɺ

…

(21)

( ) ( ) ( )2

20

ni

H b t i t n Li

yI I I x M r l y T t

tθ ρ

=

∂+ + + + + =∑

∂ɺɺ ɺɺ (22)

( )0

0 1

0

00

y

for x r i y y

h−

== = −

= (23)

( )

( )

1 1 1

2

1 1 2

20

3 30 .

3

n n n n nt t

n n n nt n t

or x r l i n

y y y y yJ J EI

h h

y y y yM y M r l EI

h

θ

θ

− + −

− + +

= + =

− − ++ + =

− + − ++ + − =

ɺɺ ɺɺɺɺ

ɺɺɺɺ

(24)

Since the boundary value of the solution is given for

0i = therefore, this approximation is applied for 1,2, ,i n= … ,

where:

For 1i =

1 2 31 1 4

6 40

y y yx y EI

hρ θ ρ

− ++ + =ɺɺ ɺɺ (25)

For 2,3, , 2i n= −…

1 1 2 2

4

4 6 40

i i

i i i i i

x y

y y y y yEI

h

ρ θ ρ

− + − +

+ +

− + − + +=

ɺɺ ɺɺ

(26)

For 1i n= −


1 12 2 3

2 1 3

4

4 5 20

t t tn n n

n n n n

J J Jx y y

h h h

y y y yEI

h

θ ρ ρ− −

− − −

+ + + + − +

− + − +=

ɺɺ ɺɺ ɺɺ

(27)

For i n=

( )2 3

1 21 3 4

20 .

t t t tn n

t n n nn

M r l J M Jx y

h h h h

J y y yy EI

h h

θ ρ ρ

− −−

++ + + + +

− + ++ − + =

ɺɺ ɺɺ

ɺɺ

(28)

Discretization of the partial differential equation with re-

spect to the spatial variable replaces it by a set of ordinary

differential equations that can be written in matrix form as

1 1MZ KZ τ+ =ɺɺ (29)

where

0 0 0 0 0 0 0 0 0 0

0 6 4 1 0 0 0 0 0 0

0 4 6 4 1 0 0 0 0 0

0 1 4 6 4 1 0 0 0 0

4

0 0 0 0 0 0 4 6 4 1

0 0 0 0 0 0 1 4 5 2

0 0 0 0 0 0 0 1 2 1 1 1

EIK

h

n n

−

− −

− −=

− −

− −

− + × +

…

…

…

…

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

…

…

…

( )

( )

51 2 3 4 3 2 1

0 0 0 0 0 0 0 01

0 0 0 0 0 0 0 02

0 0 0 0 0 0 0 03

0 0 0 0 0 0 0 02

0 0 0 0 0 0 01 2 2 3

0 0 0 0 0 0 02 3 3

I I I x x x x x x x x x M r lnt tH n n nb

x

x

x

M

xn

J J Jt t txn h h h

M r l J J M Jt t t t txn h hh h h

ρ ρ ρ ρ ρ ρ ρ ρ ρ

ρ ρ

ρ ρ

ρ ρ

ρ ρ

ρ ρ

ρ ρ

+ + + +− − −

=

−

+ + −−

++ + − + +

…

…

…

…

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

…

…

…

( ) ( )1 1n n

+ × +

(30)

where 1Z is variables vector with 1n + elements defined

as:

1 1 2 ., , , , nZ y y yθ = … (31)

5. Simulation results

In order to illustrate the behavior of the previous models

under excitation pulse, the following section is presented.

5.1 Single flexible link under excitation pulse

Time and amplitude of the pulse are 2 seconds and

0.55 .N m , respectively.

As Fig. 3 depicts, it is predicted that hub angle graph must

be composed of two parts such that, at time interval 1 sect = ,

the graph has its global maximum value and the angle value

doesn’t vary from 2 sect = to the end of the time interval,

where this situation consistent on physic of problem. Angular

velocity of the hub is also composed of two sections. Velocity

of rigid motion of the link and velocity relevant to elastic mo-

tion of the link around the situation. The produced graph must

explain the physic of the problem such that, until 1sect =

velocity increases, from 1 sect = to 2 sect = velocity de-

creases, and after 2 sect = , velocity is constant.

As Fig. 4 shows the hub angle and the angular velocity

graphs have their maximum values after 1 sect = and de-

crease until 2.3 sect = and afterwards, the trends in both

graphs remain constant. The source of this time delay is here-

by explained.

This difference is because of the flexibility of the joint such

that a percent of the input is dissipated in the actuator system

to maintain flexibility; thus, the system response is different

from the ideal case as depicted in Fig. 4. Produced graphs

almost resemble the presented results in Refs. [7, 15, 23 and

19]. The delicate and important point almost neglected is, the

sudden variation of the angular displacement and velocity of

the hub at the beginning of the motion. With a little attention it

can be founded that, this event is also relevant to the flexibility

in the actuator system, and the rigidity assumption in the ac-

tuator isn't logical. In other words, the angular displacement of

the hub is equal to the rigid and flexible motions of the gear’s

system.

θ αΘ = + (32)

Table 2. Key parameters of the flexible robot system.

Parameter Value Parameter Value

( )l m 1 2( )

tJ Kg m 0.002

( )w m 0.01 ( )tM Kg 0.5

( )b m 0.02 ( )r m 0.075

2( )EI N m 2000 ( )2

HI Kg m 0.0174

( )kg

mρ 1 ( )h m 0.02

Fig. 3. Excitation pulse as applied torque.


where α and θ are flexible and rigid components of the

joint, respectively. Therefore, for this claim to be valid, equa-

tions of motion of a single flexible link were derived consider-

ing flexibility of the joint and by drawing flexible displace-

ment of the actuator is seen to existed decreasing similar to

Fig. 4, where is decreasing reason of the hub angle in Fig. 4.

Therefore, total angular displacement of the actuator is the

summation of rigid and flexible components of the joint as

depicted in Fig. 5(b). The following figures are results of the

flexible link with tip mass of 0.5tM Kg= . As Fig. 7(a) de-

picts by increasing of the tip mass, displacement of the end

effector of the beam raises in the beginning of motion and is

caused of stability in the short time than flexible link with

small tip mass. By increasing of point mass, the flexibility

joint effect decreases, so that after start of motion figure

doesn’t decay and the stability is produced at an earlier time

(Fig. 7(b)).

6. Overall system dynamic model

The actuator dynamics is incorporated with the link state

space equation into the overall system. A DC motor with con-

stant field is assumed to be the actuator for this study that is

driven by application of a voltage to its armature terminals.

Through a set of gears, the motor drives a load with moment

of inertia J , subject to an external torque. Its dynamics is

described by

(a) Joint flexibility

(b) Total angular displacement of the hub

Fig. 5. Joint flexibility effect on total displacement of the hub.

Fig. 6. Angular velocity of the flexible link with tip mass with zero

initial conditions.

(a) Angular displacement of the hub

(b) Angular velocity of the hub

(c) Tip displacement of a single flexible link

Fig. 4. Transverse and longitudinal vibrations case with zero initial

conditions.


m m m eJ T Tω = −ɺ (33)

where Te is the torque exerted on the motor shaft by the load,

transmitted through the gears. Thus,

e m m mT T J ω= − ɺ (34)

So the torque at the motor shaft is seen to be the torque gen-

erated by the motor, minus the torque required to accelerate

the rotor. The torque exerted by the motor on the load shaft,

transmitted through the gears, is eNT . Newton's second law

applied to the load is

.e L m L m m

J NT T NT T NJω ω= − = − −ɺ ɺ (35)

Since ,m Nω ω= this becomes

( )2

m m LJ N J NT Tω+ = −ɺ (36)

or

e m LJ NT Tω = −ɺ (37)

where 2

e mJ J N J= + is effective inertia seen at the load

shaft. We need the current i by Kirchhoff voltage law

m m

diL Ri V K

dtω+ = − (38)

where mK ω is the back emf and equations in matrix form is:

.

0 1 0 0 0

10 0 0

100

m

e L

m

Vd NK

dt J J Tei i

NK R

LL L

θ θω ω

= + −

− −

(39)

By selection of θ and ω as the outputs, the output equa-

tion is:

.1 0 0

0 1 0i

θθ

ωω

= (40)

The simulation conditions are as follows: with zero

conditions for the state variables and ( ) 0,LT t =

( ) 3 0 2V t volt t= ≤ ≤ and ( ) 3 2 4V t volt t= − < ≤ , and

0.05 / , 1.2 , 0.05mK Nm A R L H= = Ω = 4 2 28 10 . , 0.02 .

mJ kg m J kg m−= × = and 12N = . Time

response of DC motor has been shown in Fig. 8.

7. Approximate state space equations

For the purpose of the flexible manipulator control (future

work), we need a standard form of the state space equations

for a flexible manipulator system (flexible link and actuator).

The finite difference method is used to approximate partial

differential equations with a set of ordinary differential equa-

tions as,

1 1.MZ KZ τ+ =ɺɺ (41)

And from latter section, we derived the state space equation of

actuator. In this section, we want to derive state space equa-

tions of total system. Therefore, for writing the closed form of

Fig. 8. Time response for the DC motor.

(a) Tip displacement

(b) Angular displacement of the hub

Fig. 7. The effect of increasing tip mass on tip displacement and angu-

lar velocity of the hub.


the state space equations of system, we should transform state

space representation of DC motor (Eq. (39)) to same order of

equation motion of flexible beam.

With introduction of variables 2 1 2, , , , T

nZ v v vω = … , i

and *Z , where 2 1

Z Z= ɺ with 1n + elements, i is current

of motor and *

1 2, ,T

Z i Z Z = , expansion of the state space

representation of driver motor is:

( )

1

22 3 1n

id

Zdt

Z+ ×

=

2

0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

m

m

m n

R NK

L L

NK

Je

A

− −

… …

… …

… …

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

… …

… …

… …

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

… …

( ) ( )

1

2

3 2 3n

i

Z

Z

+ × +

+

( )

1

2

2 3 2

10

0 0

0 0

0 0

10

0 0

0 0

L

e

m n

L

iV

ZT

Z

J

B + ×

+

−

⋮ ⋮

⋮ ⋮

11 12 11 12

21 22 21 22

,m m m m

m m

m m m m

A A B BA B

A A B B

= = (42)

where mA and

mB are system and input matrices of DC

motor and

( )

( ) ( ) ( ) ( ) ( )

1 1

11

1 1 1 12 2

0

0 0

n

m

n n nn n

R

LA and× +

+ × + × ++ × +

−= (43)

( ) ( )( ) ( )

1

12

1 12 1

0n

m

n nn n

NKmLAI

×

+ × ++ × +

−=

( ) ( )

21

1 2

0 0

0 0 0

0 0 0

m

m

n n

NK

Je

A and

+ × +

=

…

…

⋮ ⋮ ⋮ ⋮

…

( ) ( )

22

1 1

0 0 0 0

0 0 0 0

0 0 0 0

m

n n

A

+ × +

=

…

…

⋮ ⋮ ⋮ ⋮ ⋮

…

( ) ( )21 121 1 2 10 0

m mn nB and B

+ × + × = =

( ) ( )

11 22

2 1 1 1 .

11

0 0

0 0

e

m m

n n

JL

B and B

+ × + ×

−

= =⋮ ⋮

Thus, with this separation, let:

11 12 1 11 12

1

2

2 21 22 1 21 22

2

.

m m m mL

m m m m

L

iVid

A A Z B BTdt Z

Z

iVd

Z A A Z B BTdt

Z

= +

= +

(44)

Now, the state space representation equations of the motion

of the flexible beam along tip mass with total form as Eq. (41)

where

( )1 1,0,0, ,0

T

L LnT T qτ

+ × = =…

and ( )1 1

1,0,0, ,0T

nq

+ × = … , must be extracted:

( )

( )

2 1

1 1 1

2 1 1

1 1

2 1

2

0 | 0 0

L

L L

L

MZ KZ T q

Z M T q KZ M T q M KZ

iVd

Z M K Z M qTdt

Z

− − −

− −

+ =

= − = −

= − +

ɺ

ɺ

(45)

where 10 | M K− is the 1M K− matrix with a zero column

added to it until its dimensions turn into ( ) ( )1 2n n+ × + .

Using Eqs. (44) and (45), the total description of the state

space equations of the overall system is

( )1

2 21 22 1

2

1

21 22

2 0 |m m

m m

L

id

Z A M K A Zdt

Z

VB B M q

T

−

−

= − +

+


( )1

21 22 1

22

1

21 22

0 |1

2

m m

m mL

i

A M K A Zd ZZdt

VB B M q

T

−

−

− +

=

+

(46)

and the closed form is

( )( ) ( )

11 12

11 1

21 22

2 2

1 10 |

2 2

m m

m m

i iA Ad

Z ZA M K Adt

Z Z

A

−

= +−

( ) ( )11 12

1

21 22

1 1

2 2

m m

Lm m

B BV

TB B M q

B

−

+

.X AX BU= +ɺ (47)

Selecting angular position of the flexible beam and tip de-

flection of the beam as output variables, as the output equation,

let

( ),Y

y r l t

θ

=+

( )

1

2

2 2 3

0 1 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0

n

i

Y Z

ZC × +

= +

… …

… …

( )2 2

0 0

0 0 L

V

T

D

Y CX DU

×

= +

(48)

where only ( ) ( )1,2 2, 2 1Y Y n= + = .

7.1 General control problem

In this section, we consider a general control problem. Fig.

9 shows the general control configuration, where ( )G s is the

generalized plant model and it will include the interconnection

structure between the plant and the controller. ( )G s is the

generalized controller.

For control purposes, Eqs. (47) and (48) can be rewritten to

the augmented state space model as,

1 2

1 11 12

2 21 22

( )

A B B

G s C D D

C D D

= (49)

where

( )( ) ( )( ) ( )

11 12

1

21 22

2 3 2 3

1 10 |

2 2

m m

m m

n n

A A

AA M K A−

+ × +

=−

(50)

( )( )

( )( )

11 12

1 2 1

21 22

2 3 1 2 3 1

,1 1

2 2

m m

m m

n n

B B

B BB B M q−

+ × + ×

= =+

( )1 1 2 30 1 0 0 0 0 0 0 0

nC × + = … …

( )2 1 2 30 0 0 0 1 0 0 0 0

nC × + = … …

11 12 21 22 0 .D D D D= = = =

Therefore, the augmented state space model has been ob-

tained and it is ready for implementing control methods.

Clearly, state equation (Eq. (47)) has incorporated manipulator

motion (hub rotation and link deflection), actuator dynamics,

as well as sensor specification. The system matrix A is a

function of link construction (mass and stiffness distributions)

and actuator design (motor parameters), while the output ma-

trix C is a function of beam position and tip deflection.

8. Conclusions

This paper treated the problem of dynamic modeling and

state space representation for robotic manipulators with flexi-

bility. A systematic top-down approach to the modeling of

flexible robotic links was undertaken. In this paper, complete

dynamic modeling of a flexible beam without the premature

linearization in the formulation of the dynamics’ equations

was developed, whereby the model is capable of reproducing

nonlinear dynamic effects, such as the beam stiffening due to

the centrifugal and the Coriolis forces induced by rotation of

the joints. On the other hand, joint flexibility effects on the

model dynamic behaviors were shown. The model is founded

on two basic assumptions: inextensibility of the neutral fiber

and moderate rotations of the cross sections in order to ac-

count for the foreshortening of the beam due to bending. It is

essential to notice that the sudden variation of the angular

displacement and velocity of the hub in the beginning of the

motion, is an event relevant to the flexibility in the actuator

system. In other words, angular displacement of the hub is

equal to rigid and flexible motions of the actuator system. For

Fig. 9. Most general control system.


the flexible arm with tip mass, when the value of the lumped

mass increases thus the end- effector displacement of the arm

also increases in the beginning of motion and is caused of the

stability in the short time than flexible link with small tip mass

and the flexibility joint effect decreases, so that after start of

the motion, angular displacement and velocity don’t decay

and the stability is produced at an earlier time. Simulation

results and resulting model have indicated considerable im-

provements on dynamic behavior and have shown that the

joint flexibility has a substantial effect on the dynamic behav-

ior of rotating flexible arm that should not be simply neglected.

To achieve enhanced control, the state space equations were

obtained and work is ready for implementing control methods.

Nomenclature------------------------------------------------------------------------

l : Length of link

w : Width of link

E : Height of link

I : Cross section moment of inertia

A : Cross section of link

E : Young Modulus

EI : Stiffness of the link

ρ : Specific mass per unit of length of beam

x : The position of an infinitesimal element of the link

from the absolute frame origin

HI : Moment of inertia of the hub

r : Radius of the hub

( )tθ : Reference angular motion

α : Torsional deformation angle

LT : Torque applied at the hub

( , )y x t : Transverse vibration component of an infinitesimal

element of beam

( , )u x t : Longitudinal vibration component of an infinitesimal

element of beam

ψ : Rotation of the flexible beam

tM : Tip mass

tJ : Moment of inertia of tip mass

( )xφ : Moment of inertia per unit length of beam b

tKE : Translational kinetic energy of flexible link

b

rotKE : Rotational kinetic energy of flexible link

hubKE : Rotational kinetic energy of rigid hub Tip

tKE : Translational kinetic energy of tip mass

Tip

rKE : Rotational kinetic energy of tip mass

KEδ : Virtual differential of kinetic energy

PEδ : Virtual differential of potential energy

Wδ : Virtual work

M : System mass matrix

K : System stiffness matrix

τ : Force column vector

eJ : Effective inertia of load shaft of a DC motor

N : Gear ratio of motor

V : Armature voltage

i : Armature current

R : Resistance of armature circuit

L : Inductance of armature circuit

mT : Produced torque by a DC motor

J : Moment of inertia of load

n : Number of elements

h : Step size

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H. Esfandiyar is a PhD student of Me-

chanical Engineering at Department of

Mechanical and Aerospace Engineering,

Science and Research Branch, IAU,

Tehran, Iran. His current research inter-

ests include robotic, composite and vi-

bration.

S. Daneshmand is a PhD student of

Mechanical Engineering at Department

of Mechanical and Aerospace Engineer-

ing, Science and Research Branch, IAU,

Tehran, Iran. His current research inter-

ests include robotic, rapid prototyping

and rapid tooling.

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