is the product of elemental mass and elemental volume of the manipulator. Note that the system ..... manipulators" ,IEEE Control Systems Magazine, Vol.87, 61-64. Lin, J. and ... International Conference on Robotics and Automation, Cincinnati.
Modelling of a Flexible Robot Manipulator Using Finite Element Methods: A Symbolic Approach M. O. Tokhi l , Z. Mohamed- and A. W. I. Hashim! 'Department of Automatic Control and Systems Engineering, The University of Sheffield, UK. 2FacultJ' of Electrical Engineering, University Teknologi Malaysia, Malaysia, received 4 luly 1998
ABSTRACT This paper presents the development of a symbolic approach in characterising the dynamic behaviour offlexible robot manipulators using finite element (FE) methods. A constrained planar single linkflexible manipulator is considered. A simulation algorithm of the system is developed using a symbolic language which enables system characterisation with varying parameters. Analyses and investigations in terms system transfer function, stabiIitv, response and endpoint vibration to (11/ input command arc presented. An assessment of the performance the approach is presented and discussed through numerical simula tions.
or
or
1. INTRODUCTION Flexible robot manipulators offer several advantages over their traditional counterparts (rigid robot manipulators). These include speed of response, efficiency and cost. However, the dynamics of the system are highly non-linear involving complex processes due to flexural effects. This, in turn, makes the control of the system complicated. If the advantages associated with lightweight manipulators are not to be sacrificed, accurate model and efficient control have to be developed. Several approaches have previously been considered for modelling of flexible robot manipulators (Azad, 1995). These can be divided into two categories. The first approach looks at obtaining approximate modes by solving the partial differential equation (PDE) characterising the dynamic behaviour of a flexible robot manipulator. Previous investigations at utilisation of a linear state-space model for a single-link flexible manipulator have shown that the model eigen values agree well with experimentally determined frequencies of the vibratory model (Cannon and Schmitz, 1984; Hastings and Book, 1987). The second approach uses numerical analysis methods to solve the PDE. Investigations at utilising the finite difference (FD) method have shown that acceptable results in comparison to analytical values in characterising the system are obtained (Tokhi and Azad, 1995). The FE method has widely been used in modelling flexible robot manipulators (Meng and Chen, 1988; Tokhi et al., 1997; Usoro et al., 1986). It has been demonstrated that with the FE method a good model is obtained with large number of elements. It has also been reported that the FE method has certain advantages over the FD method in characterising flexible manipulators (Tokhi et al., 1997). However, most of these investigations are numerical-based. System characteristics such as
Journal ofLow Frequency Noise, I ibration and Aui\'c Control liil. 18 No.:: IYYY
63
MODELLING OF A FLEXIBLE ROBOT MANIPULATOR
stability, characteristic frequencies and modes of vibration are interpreted on the basis of certain system parameters. Moreover, numerical systems must operate using numeric approximations with limited precision due to processor wordlength. This limitation could be overcome by carrying out computations in a symbolic form. As a result, exact quantities can be obtained using this approach. Symbolic approaches for modelling and simulation of flexible manipulators have previously been investigated (Cetinkunt and Ittoop, 1992; Lin and Lewis, 1994; Tzes et al., 1991). These have used Euler-Langrange equation and Newton-Euler inverse dynamic model, to model the system. These have demonstrated that the approach has some advantages, such as allowing independent variation of flexure parameters. The aim of the work presented in this paper is to explore and investigate the symbolic approach to modelling a flexible robot manipulator using FE methods. Not much work has been done on such a study. Using such an approach, an accurate model of the flexible robot manipulator with different number of elements can be obtained and investigated. The rest of the paper is structured as follows: Section 2 provides a brief description of the flexible manipulator system considered in this study, Section 3 describes the FE approach to modelling the system. The symbolic simulation algorithm is presented in Section 4. An assessment of the results of the algorithm is given in Section 5. Numerical simulation results are presented in Section 6 and the paper is concluded in Section 7.
I
a
p Figure I: The flexible robot manipulator system. 2. THE FLEXIBLE ROBOT MANIPULATOR The single-link flexible manipulator considered in this paper is described in figure 1, where, I h represents the hub inertia of the manipulator. A payload mass Mp with its associated inertia Ip is attached to the end-point. A control torque ttt] is applied at the hub by an actuator motor. The angular displacement of the manipulator, in moving in the POQ - plane, is denoted by 6(t). The manipulator is assumed to be stiff in vertical bending and torsion, thus, allowing it to vibrate (be flexible) dominantly in the horizontal direction (POQ - plane). Moreover, the shear deformation and rotary inertia effects are ignored. For an angular displacement 6 and an elastic deflection u the total (net) displacement y(x,t) of a point along the manipulator at a distance x from the hub can be described as a function of both the rigid body motion ()(t) and elastic
64
MODELLING OF A FLEXIBLE ROBOT MANIPULATOR
deflection utx.t) measured from the line OX;
y(x, r) = xe(t) + u(x. r)
(I)
This describes the displacement of the manipulator from the reference line OP.
3. THE INFINITE ELEMENT METHOD Since its introduction in the 1950s, the FE method has been continually developed and improved (Fagan, 1992; Rao, 1989). The FE method involves decomposing a structure into several simple pieces or elements. The elements are assumed to be interconnected at certain points, known as nodes. For each element, an equation describing the behaviour of the element is obtained through an approximation technique. The elemental equations are then assembled together to form the system equation. It is found that by reducing the element size of the structure, that is, increasing the number of elements, the overall solution of the system equation can be made to converge to the exact solution. The main steps in an FE analysis include (I) discretisation of the structure into elements; (2) selection of an approximating function to interpolate the result: (3) derivation of the basic element equation; (4) calculation of the system equation; (5) incorporation of the boundary conditions and (6) solving the system equation with the inclusion of the boundary conditions. In this manner, the flexible manipulator is treated as an assemblage of n elements and the development of the algorithm can be divided into three main parts: the FE analysis, state-space representation and obtaining and analysing the system transfer function. Using the FE method to solve dynamic problems leads to the well-known equation (Fagan, 1992)
u(x, r) = N(x)Q(t)
(2)
where Q(t) and N(x) represent the nodal displacement and shape function respectively. Thus, in this work,
(3)
where I is the elemental length. For the flexible manipulator under consideration, utxi) in equation (2) represents the residual motion. Substituting for utx.t) from equation (I) into equation (2) and simplifying yields
y(x, r) = N(xfQ(tf
(4)
where
Q(tf=[e(t) Q(t)y
and
N(xf =[x N(x)]
The new shape function N* and nodal displacement vector Q* in equation (4) incorporate local and global variables. Among these, the angle e(t) and the
65
MODELLING OF A FLEXIBLE ROBOT MANIPULATOR
distance x are global variables while N(x) and Q(t) are local variables. Defining n-\
W
=X
-
2);, where I; is the length of the ith element, as a local variable of ;=\
the nth element, the new shape function can be obtained as
Using the above, the elemental mass matrix Me and stiffness matrix K" can be obtained as [7] I
=pA J(N')T (N')dw
M'
o
(6)
I
K'
=El f (B' /
B' dw
o
where A, p, I and E are the cross-sectional area, linear mass density, area moment of inertia of the manipulator and the Young Modulus respectively and
Hence the elemental mass and stiffness matrices can be obtained for the n elements as
140/(3n 2 -3n+ I) 21(lOn-7) 7/(5n-3) 21(lOn- 3) -7/(5n-2) -131 21(lOn-7) 54 221 156 2 M n = pAL -3/ 2 7/(5n - 3) 131 221 41 420
21(lOn-3)
54
-7/(5n - 2)
-131
131 -312
156
-221
-221
4/ 2 (7)
K"
= El3 1
0
0
61
-12
61
2
-61
21 2
0 -12 -61
12
-61
2
-61
41 2
0
0
0
0
12
61
0 0
61
41
21
Note in the above that, the elemental mass matrix depends on the element number, whereas the elemental stiffness matrix has the same value regardless of the element number. The matrices from above are assembled to obtain system mass and stiffness matrices, M and K, and used in the Lagrange equation to obtain the dynamic equation of the flexible manipulator as
MQ(t) + KQ(t) =F(t) where F(t) is the vector of applied forces and torques and
66
(8)
MODELLING Of A flEXIBLE ROBOT MANIPULATOR
The M and K matrices in equation (8) are of size m X m, Q(t) and Fit] each are of size m X I, where m = 2n + I. For the manipulator, considered as a pinned-free arm, with the applied torque T at the hub, the flexural and rotational displacement, velocity and acceleration are all zero at the hub at f = 0 and the external force is Fit) = [T 0 ... 0 f. Moreover, in this work, it is assumed that Q(O) =O.
4. SYMBOLIC ALGORITHM In this section, the development of the symbolic algorithm is presented. To investigate the performance of the symbolic approach in simulating the dynamic characteristic of the flexible manipulator system using the FE method, the algorithm is implemented with one and two elements. In each use, the algorithm is tested and verified. In this work, for simplicity reasons, the effects of hub inertia and payload are ignored in implementing the simulation algorithm. The matrix differential equation in equation (8) can be represented in a statespace form as
v=Av+Bu
(9)
y=Cv+Du where
Om is an m X m null matrix, 1m is an m X m identity matrix, 0mx, is an mX I null vector, 0, Xm is a I X m null vector and L is the manipulator length. Solving the state-space representation gives the system transfer function and vector of states v, that is, the angular nodal flexural and rotational displacements and velocities. 4.1. Simulation of one element Using one element, that is n = I, the dynamic equation of the flexible manipulator can be obtained as in equation (8) where the system mass matrix,
1401 2 63l M = pAL 141 2 420 147l
-21l
2
156
141 2 22l
22l
41 2
63l
54
13l -13l -3l 2
147l -21l 2 54 -13l 13l -3l 2 , 156 -22l -22l 41 2
the system stiffness matrix,
UI
0 -12
0 -12 -6l 0 6l 21 2
12
12
K= EI 0
6l
l3
e» =[8
0 6l 41 2
0 0
0
8 1 U2 8 2
t. 67
-6l -6l
and F(t)
0 6l 21 2 -6l 41 2
=rt
0 0 0
Or·
MODELLING OF A FLEXIBLE ROBOT MANIPULATOR
Incorporating the initial conditions, with flexural and rotational displacements at the hub as zero, the second and third rows and columns in M, K, Q and F are thus ignored. This yields 2
-21l [0 0 -221' K = EI 0 12 2
1471 156
140/ M = pAL 1471 420 [ -2112
]
Q(t)
13
'
4/ 2
-221
0 -61
[;}d =m
=
F(t)
Thus, the state-space matrices can be obtained as
A=~
0
0
0
1 0 0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0 0
I
0
0
0
3780
19801
0 0 0
0 -31501 15901
0 0 0
pAIs 0 0
3150
,
1 pAI
, B=3
-20701 0 0 0 C=[L
I
0
0
0
-2701 90 -2701 2551 2 -151 -151 495 90 300
(10)
0] and D = O.
Using equation (9) the system transfer function can be expressed as G(s)
= Y(s) = C(sI -
U(s)
Ar B + D l
( II )
where s is the Laplace variable and I is the identity matrix. Substituting for the matrices A, B, C and D from equation (10) into equation (11) and simplifying yields
This is the system transfer function with the FE simulation incorporating a single element.
4.2. Simulation of two elements Following the same procedure as in case of one element, the dynamic equation of the flexible robot manipulator using two elements can be obtained as in equation (8) with
120/ 2 4201
pAL M=- 281 2 420
3571 -56/ 2
4201
281 2
312
0 2
81 54 131 -131 -30/ 2 0
3571 -56/ 2 -131 54
0
0
0
0
0
0 24 0 -12 61 . K= EI 0 0 81 2 61 2/ 2 131 -3/ ' 13 0 -12 -61 12 -61 156 -221 2 2 0 61 2/ 2 61 4/ -221 4/ 2
68
MODELLING OF A FLEXIBLE ROBOT MANIPULATOR
Q(r)
8
't
U2
0
= 82
and F(r)
=0
U3
0
83
0
Similarly, the state-space matrices can be obtained as
A=
0
0
0
0
0
I
0
0 0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0 0
I
0
0
0
0
0
0
0
0
0
0
I
0
0 EI 24pA[s 0
0
0
0
0
0
0
0
0
I
19872
-367201
27783
-13788[
0
0
0
0
0
0
-272161
36240[2
0 0
0
0
0
0
-28512
-64801
0
-483841
0
-100224
576001
2
-374401
-251371 12132z2
0
0
0
0
-362881
-117721 0 161281 2 0
0
0
0
0
92232
-707041 0
0
0 0
0
20727
0 0 0 0 0 0 0 0 0 0 B=_I_ 3 56pA/ 10947 -11253/ -11253/ 11667/2 -5517 5883/ -21504/ 22176f -5724 67561 C=
[L
0 0
I
0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5517 -21504/ -5724 5883/ 22176/2 6756/ 2041 10404 7587 2 11424/ 43008/ 16128/ 10404 16128/ 44208 0
0
0] and D = O.
Substituting for the matrices, A , B , C and D from above into equation (II) and simplifying yields 195(pAI4)4 S8 _ 522720(pA/4)3 EIs6 + 308188800(pAI4)2 (E/)2 S4 G s _ -37884672000(pAI 4)(E/)3 S2 + 274337280000(EI)4 ( ) - S2 (28 p s A S[18 S8 + 146736p 4 A 4[14 EIs6 + I54483200p 3 A 3['0 (EI)2 S4 (13)
+26606361600p 2A 2[6 (E1)3 S2 + 365783040000pA[2 (EI)4) This is the system transfer function with the FE simulation incorporating two elements.
5. SYSTEM CHARACTERISTICS In this section, the transfer functions obtained in the previous section are assessed in the dynamic characterisation of the flexible manipulator system.
69
MODELLING OF A FLEXIBLE ROBOT MANIPULATOR
This involves obtaining and investigating system characteristics such as poles, zeros, stability and system response. Note that in this work, the effect of damping is ignored. Therefore, the system is expected to be marginally stable, exhibiting a response of oscillatory nature. It is noted in equations (12) and (13) that the system transfer function has two poles at the origin of the s - plane. Thus, the flexible manipulator is a type two system. This implies that zero steady-state error can only be achieved with step and ramp command inputs to the manipulator. Factoring the numerator and denominator polynomials of equation (12) yields system zeros as = + 38.9944
I
S
ri
ri
± 9.97181
"If'
(14)
"If
I
and system poles as S
=±J' 17.5444 I
ri
"If'
+ . 70.087 I
-J
ri
0
"If' ,
0
(15)
with one element, where a = EJ represents the flexural rigidity and (3 = (p/)(AI) is the product of elemental mass and elemental volume of the manipulator. Note that the system has six poles, two of which are at the origin and the remaining four are on the jw - axis in the s - plane. Thus, the system, as expected, is marginally stable. Using numerical computation methods, such level of accuracy is not achieved due to computational errors, as reported previously (Tokhi et al., 1997). This shows an advantage of using the symbolic approach. 70
- - . - - - . - . - - - - - - . - - - - - - - - - - - - - - .. - - -
- - -
60
Mode 2
.., ..
50
'
g >-
g
40
! c
.-
o 30
.~ .Q
;> 20
Mode I 10
-1"---+---+---+---+---+---+---+---+---+----; 0.5
'5
35
2.5
4.5
5
l
Flexural rigidify(Nm )
Figure 2:
Effect of flexural rigidity on vibration frequency «(3 kgm').
= 0.1668
Since the system is marginally stable, the poles on the jw - axis give the system natural frequencies. These, in turn, determine vibration modes of the system. Evalu7;in equation (15) yields vibration frequencies at modes I and 2 17.5444 a 70.087 as - Hz and - Hz respectively, This shows that the 27f1 (3 27f1 (3
fi
. .
frequencies depend on a, (3 and I. Figure 2 shows the effect of a on the vibration frequency of modes I and 2 with a constant (3. Practically, changing the value of a implies changing the manipulator material. In this case, (3 = 0.1668 kgrn ' is used. Similarly, since I is determined by geometry of material
70
MODELLING OF A FLEXIBLE ROBOT MANIPULATOR
and is independent of A, the inter-relation between 13 and the vibration frequencies of the system with a constant a can be investigated. The result with a = 3.6918 Nm 2 is shown in Figure 3. It is also important to investigate, for the same material, the effect of changing the manipulator length on vibration frequency. Figure 4 shows the result with two different materials. It follows from these results that the system vibration frequencies can be increased by either (i) increasing the flexural rigidity of the manipulator i.e. using a more rigid material, or (ii) reducing the volume and/or mass of the manipulator i.e. using a stiffer structure, or (iii) reducing the length of the manipulator. These are further evidenced in Figures 5 and 6 which show the relation of a and 13 with vibration frequencies at modes I and 2 respectively. These show that high system vibration frequency is obtained with small 13 and large a of a flexible manipulator. 70
, 60
50 N
e ~
40
f o
.2 30
]
:> 20
. ~
Mode 2
Mode I
----_---.!~----
0-!---+----+----+---4----...,1----+---;----+--+----I
o
0.5
1.5
2.5
3.5
4.5
Massx volume
Figure 3.
Effect of product of elemental mass and elemental volume on vibration frequency (a = 3.6918 Nm 2) .
70
60
50
g e 40
I o
~
30
.0
:> 20
. ~
Mode2
........ -
.
~-----~---Mode I
0+----+---1----...,---+---1----+----+---+--+----1
o
0.5
2.5
1.5
3.5
4.5
Massx voJume
Figure 4.
Effect of length on vibration frequency with two types of material.
71
MODELLING OF A FLEXIBLE ROBOT MANIPULATOR
Another important result that can be observed in equation (14) is that the system is non-minimum phase since two zeros of the system transfer function are on the right half s - plane (rhp). This result agrees with those reported previously (Meng and Chen, 1988; Park and Asada, 1990). Since control of a non-minimum phase system is rather involved, this aspect is further analysed. It is important to study whether the zeros can be relocated to the left half s plane (1hp) using different system parameters. If so, certain parameter values can be used to make the system minimum phase. The analysis is done using the RouthHurwitz (R-H) method (Nise, 1995). Accordingly, if there is no sign change in the first column of the R-H table, then all roots will be on the Ihp. The first column of the R-H table for numerator using one element is shown in Table I. It is noted that there are two sign changes, thus two zeros exist on the rhp. These agree with the results shown above. Furthermore, the table shows that the zeros cannot be relocated by changing any parameter value since all terms are single. Thus, the flexible robot manipulator characterises a non-minimum phase system. TABLE 1 First column of R-H table for numerator using one element 30p 2A2/H I20p2A2/H -24300pA/4£/ 22302.8pA/4£/ 4.54 106£2[2
It is noted in equation (13) that the numerator and denominator polynomials of the system transfer function with two elements are of orders 8 and 10 respectively. As these can not easily be explicitly factored into first-order factors, the R-H method can be used to assess the stability of the system and determine whether the system is non-minimum-phase. The first column of R-H table for denominator and numerator polynomials of the system transfer function are shown in Tables 2 and 3 respectively. TABLE 2. First column of R-H table for denominator using two elements 28 p5A5/IH 280p5A5/IH 2.93 X lO'lp4A4/14£[ 5.84 X J05p4A4/14£[ 2.29 X J07p 3A3/I()£2[2 4.99 X JOH p3A3/IO£2[2 6.32 X J09p2A2t6£3J3 7.94 X JOI()p2A 2t6E'[3 7.94 X IOI(J p2A2t6£3J3
Since the denominator of the system transfer function is an even polynomial, a row of zero exists at s9. Thus, if no sign change occurs, the system has all poles on the jw - axis. It is clearly noted in Table 2 that no sign change occurs. Thus, all system poles lie on the jw - axis and the system is marginally stable. These agree with the results above using one element. Zeros of the system are assessed from Table 3. It is noted that there are four sign changes in the first column of the R-H table for the numerator polynomial of equation (13). This implies that four zeros of the system transfer function are on the rhp and, thus, the system is non-minimum phase. Moreover, as noted, the zeros cannot be
72
MODELLING OF A FLEXIBLE ROBOT MANIPULATOR
relocated to the Ihp by altering any parameter values.
TABLE 3. First column of R-H table for numerator using two elements 195p4A 4 { 16 1560p4A4{16 -130680p 3A3{12£/ -1.29 X 106p3A 3{12£/ 6.41 X IOx p2A2{X£2/2 4.66 X IOxp2A 2{x£2/2 -1.19 X lO"pA{4£3[3 5.62 X lO"pA{4£3[3 2.74 X 10"£4/4
20
¥
."
>:15 o
.
c::
"
... .- r ..
, .
"
"
, ,
s 10
,
,
"
"
,
I
.,
.
0"
~ c:: o
~ 5
.D
s
o
6
6
Figure 5.
Mass x Volume 0 0 Flexural rigidity Effect of flexural rigidity and product of elemental mass and elemental volume on vibration frequency (mode I).
80
" , , "
.:>
,.
•
1,
..
'.
,
I .........
, "
.
"
"
'.
.'
'.
,
'
, ,
.. ' o
6
6
Mass x Volume
Figure 6.
0
0
Flexural rigidity
Effect of flexural rigidity and product of elemental mass and elemental volume on vibration frequency (mode 2).
73
MODELLING OF A FLEXIBLE ROBOT MANIPULATOR
6. SIMULATIONS To demonstrate the performance of the developed symbolic algorithm in characterising the dynamic behaviour of the system simulated examples are given in this section. To allow this, an aluminium type flexible manipulator of length L =0.96 m, mass density p = 2710 kg/rn'', Young Modulus E = 7.11 X 1010 N/m 2 , cross-sectional area A = 6.1544 X 10-5 m 2 and area moment of inertia I = 5.1924 X 1O- 11m4 was considered. In these exercises a bang-bang input of amplitude 0.1 Nm, shown in Figure 7, is used as input torque and applied at the hub of the manipulator. With this torque the manipulator initially accelerates when the input torque is positive and decelerates when the torque is negative, thus causing it to stop at a certain position. The system response at the end-point is obtained and analysed over a period of 1.4 sec.
Q15
-
-
Ql
--
Q05
.
-,
.
-
--
,-
-
.
-
-
-
'-
---
-,
-
-,
.
-'
Q6
(
,-
.
-
,-
-
.
-
-,
-
-,
'.
E ~
0
Q)
::s
B' o
Q2
Q4
8
E-
-oos
-
.
.
. -.
:
-
-
-D.I
-D.15
-
1
-.
-
-
-
:
-
.
-
.
-
.
-
. .
-
:
-
-
---
1: 4
1:2
---. .
.
TDre(Sl:)
Figure 7. 0.2
Input torque applied at the hub of the manipulator. • - • - - , - • • • • -, - • - • • • , - - • • • • ,- - •• - -
~
••••••••• - - ,
0.18 0.16
I _
j
0.14
. , . - • ; • • • • - -, - • - - .
0.12
- - .. -
!
-
- - - • -
~
- • - - - p' - -
.
-
.
-
.' - - -
- -
• •••• - "
•• - • • ,- •• - • • ; - - • - • "
P' •• - • - .'. - - - - - '. - - - - - !. • - • • • .'
,
,
0.1 0.08
I
~
• • _' - - • • - - '_ _ _ _ _ _ '. _ • _ • _
I
,
•
,
• • • • _ :
I
,
0?6
... - . -t- - - ... ,. - - ....- .. - - - ; - - - . - .,
0.04 0.02
• - - - -
.!
.0.02
. ----
0:2 ~
-
-
•
-
-
p' - • - • - p' •• - - - - '. - - - - - '. - - - • - !. - - - - •
0:4
-----
.'
0'6
----
- .' - - .
--
0:8 -'
-. .
. --
-
-----
1:2 ~
Time (sec)
Figure 8.
System response at the end-point using one element.
74
~
1:4
--. . . :
MODELLING OF A FLEXIBLE ROBOT MANIPULATOR
0.2 O.lS
~
- ---
.-
~
~
,
.. .. - -, - - - - - - , . - - ,
•
....... I
..
- .. I
- .. .. .. - .. .. •
..
.. .. ..
0.16
.,
0.14
......
-
IC
., .,e
t , '
'C
0.12 0.1 O.OS
...... ....
J
Q. 0.06
..........
i
0.04
..........
J
0.02
..........
"i
?1
'" Q
-
,
,
_'
..
_
-
'
, - . ,
"
'_
,
","
'
I
-,
..
. "
I
","
..
_
-
"
-
.
..•
,
I
I
/
L.
.J
,-
i
i
,
'
I.
J
'
I·
;
"i
,
_
'
0
-0.02
l1.2
l1.4
0~6
l1.S
1
1~2
1!4
......................................................................................
TirreIsec)
Figure 9.
System response at the end-point using two elements.
The algorithm was implemented using one and two elements. Figures 8 and 9 show the simulated time-domain system response at the end-point using one and two elements respectively. It is noted that the response of the system at the end-point reached a steady state level of 0.18 m within 0.8 sec with persistent oscillations. The response shows that the system is non-minimum phase, as the response slightly undershoots at start up. The natural frequencies of the system can be obtained by calculating system poles. These, for the first two vibration modes are found as 13.67 Hz and 54.65 Hz.
7. CONCLUSION A symbolic algorithm for simulation of a flexible robot manipulator has been developed and investigated. It has been demonstrated that the algorithm can be used in characterising the dynamic behaviour of the manipulator, allowing to assess the stability, response and vibration characteristics of the system. The inter-relation between physical parameters and vibration characteristics of the system has been obtained and discussed. Simulated examples have been presented demonstrating a further advantage of the symbolic approach in numerical simulation of the system.
8. REFERENCES Azad, A. K. M. (1995). Analysis and design of control mechanisms for flexible manipulator systems. PhD thesis, Department of Automatic Control and Systems Engineering, The University of Sheffield, UK. Cannon, R. H. and Schmitz, E. (1984). "1nitial experiments on the control of ajlexible manipulator", The International Journal ofRobotics Research, Vol.3, 62-75. Cetinkunt, S. and Ittoop, B. (1992). "Computer-automated symbolic modelling of robotics manipulators with flexible links", 1EEE Transactions on Robotics and Automation, Vol.8, 94-105. Fagan, M. J. (1992). Finite element analysis, theory and practice, Longman. Harlow.
75
MODELLING OF A FLEXIBLE ROBOT MANIPULATOR
Hastings, G. G. and Brook, W. J. (1987). "A linear dynamic modelfor flexible manipulators" ,IEEE Control Systems Magazine, Vol.87, 61-64. Lin, J. and Lewis, F. L. (1994). "A symbolicformulation ofdynamic equations for a manipulator with rigid and flexible links", International Journal of Robotics Research, Vol.13. 454-466. Meng, C. -H. and Chen, J. -S. (1988). "Dynamic modelling and payload adaptive control of a flexible manipulator". Proceedings of IEEE Conference on Robotics and Automation, Philadelphia, April 1988, 448-453. Nise, N. S. (1995). Control systems engineering, Benjamin-Cummings Publishing, California. Park, J. -H. and Asada, H. (1990). "Design and control of minimum-phase flexible arms with torque transmission mechanism", Proceedings of IEEE International Conference on Robotics and Automation, Cincinnati. April 1990, 1790-1795. Rao, S. S. (1989). The finite element method in engineering, Pergamon Press, Oxford. Tokhi, M. O. and Azad, A. K. M. (1995). "Real-time finite difference simulation of a single-link flexible manipulator system incorporating hub inertia and payload", Proceedings of IMechE - I: Journal of" Systems and Control Engineering, Vol.209, 21-33. Tokhi, M. 0., Mohamed, Z. and Azad, A. K. M. (1997). "Finite difference and finite element approaches to dynamic modelling ofa flexible manipulator", Proceedings of IMechE - I: Journal of Systems and Control Engineering, Vol.2l1 , 145-156. Tzes, A. P., Yurkovich, S. and Langer, F. D. (1991). "A symbolic manipulation package for modelling of rigid or flexible manipulators", Proceedings of the 1991 IEEE International Conference on Robotics and Automation, Sacremento, April 1991, 2356-2362. Usoro, P. B., Nadira, R. and Mahil, S. S. (1986). "A finite element/Lagrange approach to modelling lightweight flexible manipulators", Transactions of ASME Journal of Dynamic Systems, Measurement and Control, Vol.108, 198205.
76
