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Structural dynamics analysis of spatial robots with finite element approach
Computers & S~rucrures Vol. 46, No. 4, pp. 703-716, 1993 Printed in Great Britain.
0
0045-7949193 $6.00 + 0.00 1993 Pergamon Press Ltd
STRUCTURAL DYNAMICS ANALYSIS OF SPATIAL ROBOTS WITH FINITE ELEMENT APPROACH SHIUH-JER HUANG and TZYY-YI WANG Department of Mechanical Engineering, National Taiwan Institute of Technology, 43 Keelung Road, Sec. 4, Taipei, Taiwan 10772, R.O.C. (Received 25 November 1991)
Abstract-This paper develops the generalized equations of motion for a robotic mechanism with elastic structures by using the finite element theory. The derivation and final form of the equations of motion can mode1 two- or three-dimensional complex elastic mechanisms and include the nonlinear coupling terms of rigid body and elastic motions in a general representation. Here, the dynamic model of an R-R-R spatial robotic manipulator with the finite element approach is established. A robot mechanism consists of links and joint transmission systems. The equations of motion of elastic links are derived based on the displacement finite element method. The dynamic model of the flexible joint transmission systems are modeled as a general spherical joint with multi-degree of freedoms. The complete dynamic model of an elastic robotic structure can then be developed by combining the equations of motion of elastic links and joints. According to this dynamic model, we can investigate the structural dynamic properties of this robotic structure from the system dynamic responses. These data are useful for the controller design and optimal dynamic operation planning of robotic manipulator.
INTRODUmION The structure of a conventional industrial robot has been rigidly designed and built without detailed structural analysis. Each component of a manipulator was made strong enough to eliminate link deflection and to overcome the deterioration of system performance
due to payload vibration. With the requirements of higher operational speed and greater positioning accuracy, the robotic mechanism needs to be constructed as lightweight as possible in order to reduce the inertial forces and the driving torque requirements. However, lighter members are more likely to vibrate due to the inertial or external forces. The performance of the manipulator may become unacceptable due to the positioning inaccuracies of the deformation variation of the end-effector. Furthermore, the unpredicted resonant condition without the vibration analysis may influence the system’s operational performance. Structural analysis is a useful tool to provide the necessary information for robotic mechanism design, optimal performance controller design and operation planning in order to overcome the above problems. The purpose of this paper is to develop a general model for spatial robotic structures with flexible links and elastic joints. It can be used to analyze the dynamic response of not only open chain, but also closed chain robotic mechanisms. The optimal lightweight manipulator and optimal dynamic operation can then be designed based on the dynamic response properties of this dynamic model. PREVIOUS WORK The earlier research of robotic structural dynamic analysis can be classified into two categories. The 703
end-effector position deviation due to the effect of flexible links [l-9] and that of elastic joints [lO-151 were investigated. Recently, the combination of flexible links and elastic joints effects were studied [16-201. There are two approaches for modeling the deformation of flexible links. The one is the finite mode shapes method whose vibration modes or eigenfunction are directly derived from the partial differential equation with appropriate boundary conditions. This is suitable for simple boundary conditions and simple geometry configuration of manipulator. The other one is the finite element method for more complicated structures with the expense of additional computer time. According to its different reference configuration, this approach can be divided into two categories [21] whose reference coordinates of links are the Cartesian coordinate of mechanism system or the rigid body motion coordinate attached to the link. Book [3] and Judd [22] used Lagrange’s equation and the latter coordinate system accompanied by a 4 x 4 homogeneous coordinate transformation matrix to derive the recursive Lagrange dynamics of spatial flexible manipulator arms equipped with revolute joints. The scheme referring to the system’s reference Cartesian coordinate is more suitable for the finite element method. Although in this scheme the homogeneous transformation matrix is a function of joint variables only, the transformation matrix of this scheme refers to the rigid link’s reference coordinate is a function of joint variables and elastic deflection. Winfrey [23] and Iman et al. [24] first used the finite element method for the dynamic analysis of an elastic planar four-bar linkage mechanism with beam element consideration. Sunada and Dubowsky [ 181
704
SHKJH-JERHUANGand
employed the perturbation coordinate and coordinate elimination scheme to derive the dynamic equations of the whole mechanism. Turcic and Midha [25-271 derived the generalized equations of motion of elastic mechanism systems using one displacement finite element method. They then used the modal analysis technique accompanied by a numerical recursive scheme to solve the equations of motion which included the nonlinear coupling terms. They also provided experimental results for comparison. According to the Newton-Euler equation, Naganathan and Soni [28] employed the finite element method together with Timoshenko beam theory in order to derive the nonlinear dynamic equations of robotic structure. Experimental results are provided to demonstrate the accuracy of the numerical results from Newmark integration solution. The joint compliance including the deflection of motor shaft, gears and belts comes from the transmission system of each driving joint. The small angular deviation due to joint compliance will significantly influence the end-effector position accuracy by the amplification effect of link length. The dynamic model of the elastic joint has been modeled as a torsion spring in parallel with a viscous damper in previous research. Spong [13, 141 investigated the modeling and control problem for flexible joints manipulators. Good et al. [29], De Luca [ 121and Soni and Dado [30] also studied the dynamic response and control properties of manipulators with elastic joints. Dubowsky and Freudenstein [31] investigated the dynamic model and response of a mechanical system with clearances. Recently, Yang and Donath [19,20] investigated the combination effect of link flexibility and joint
14 = M (u’} = {Id;
preach can be used not only for open chain, but also for closed chain robotic mechanisms. The structural dynamics properties can then be identified from the simulation analysis. The information is useful for mechanism design, controller design and operational planning. MATHEMATICAL
MODEL
The robotic structure flexibility consists of the flexible link deflection and the elastic joint compliance. Here the dynamic model of the robotic mechanism’s flexible link is modeled as a flexible Timoshenko beam which can be derived from Lagrange’s equation using the finite element approach. The dynamic model of each joint transmission system with multidegree of freedoms can be established based on the Newton’s second law. Then the nonlinear dynamic model of the whole elastic manipulator mechanism, including the coupling effects of rigid body motion and elastic vibration, can be derived using the substructure concepts. The basic assumption of the displacement finite element method is that the local displacement vector of any point in an element can be represented as a linear combination of the nodal displacements
where d is the local displacement vector, ae is the element nodal displacement vector and N is the matrix of the shape functions. The terms ue and dare with respect to coordinate X, Y,&. In order to simplify the derivation of the dynamic mode1 of this spatial mechanism with the Timoshenko beam, twonode beam elements are discussed. Hence
dy 4 0, ‘$ RI
uf uf e:
-N, 0
lN1= : 0
-0
TZYY-YIWANG
e;
e:
u:
u;
U;
eg e;
et}'
0
0
N2
0
0
0
0
0
0
0
0
0
N2
0
0
0
0
:x,0
0
0
0
0
0
0
0
0
0
0
N, 0
N2
0
0
0
0
0
0
0
N,
0
0
0
N,
0
0
0
0
N,
0
0
0
0
compliance by combining a simple assumed mode shapes model of beams with spring-damper models. In this paper, we combine the finite element model of flexible links using the Timoshenko beam theory with multi-degree of freedom models of elastic joints to obtain a reasonable model for elastic robotic structure. The dynamic model established by this ap-
N,
0
0
0
0
0
-
’
0 N2_
where N /-x-, I
I
N2=f
and 1 is the length of the element. X is defined as the length along the element between 0 and 1. In order to
105
Structural dynamics analysis of spatial robots
{ir:} is the rigid body motion of the element node with respect to the X, Y, Z, coordinate and [r:] is the constant transformation matrix of the orientation between the Xi Y, Z, and XYZ coordinates. Then the translation velocity of any point in the element can be described as
The translation
kinetic energy of this element is
KE,=;p{d}‘{h}
dl’.
(5)
s
Substituting eqn (4) into (5) after some rearrangement, gives
Fig. 1. Configuration of spatial beam deformation. guarantee the solution convergence, the shape functions must satisfy the continuity, compatibility and completeness conditions. The kinetic energy of this three-dimensional element consists of translational and rotational kinetic energy. In order to simplify and reduce the computing time, they are derived individually. The configurations of the single element before and after deformation are shown in Fig. 1. XYZ is the manipulator Cartesian base reference coordinate, X, Y,Z, is the moving coordinate attached to the undeformed element equipped with rigid body motion and the X, Y,Z, coordinate has a common origin point with the XYZ coordinate while its direction is parallel to that of the X, Y,Z, coordinate. The moving coordinate Xi Y;Z; is attached to the deflection element which has a relative displacement d, with respect to the X, YrZ, coordinate. Then the positional vector of any point on the element can be written as
where [T,,,] is the transformation matrix between X, Y,Z, and XYZ coordinate systems. It is a function of the rigid body motion only. R, is the relative positional vector of the moving coordinate X, Y,Z, with respect to the reference coordinate XYZ. Then the velocity of this point can be derived by differentiating eqn (2) with respect to time
(4 = MO> + Knlh41+ KJVJ
+ 2(~:}1~,l’[~~~[~,l[~,lI~:})dt’. Similarly, the rotational described as KE,=;
where [T,,,] is a function of the joint angular and [iV,] is function of the shape function only. Additionally,
~{~)V,l{~jdx,
(7)
The rotation velocity is
@I = h31 +Kl@rI~
(9)
where (w} is the rotation speed of the X; Y;Z; coordinate with respect to the base reference coordinate XYZ and {w,} is the element rigid body rotation speed which can be described as {%j =
(3)
kinetic energy can be
where p is the mass density and [Z,] is the cross-sectional moment of inertia matrix. If the coordinate axes coincide with the element principal axes, then
bt>= W,lW,I IM = K1[~~1Iir:L
s
(6)
The rotation element is
EJW,l~W.
(10)
velocity vector of any point in the Id:) =
W,l{6L
(11)
706
SHIUH-JERHUANGand TZYY-YI WANG
where {ti;} = {et
s;,
e:
eg, s_; e;}’
transverse deformation and notational deformation can be represented as the function of the nodal displacement w = [N,
N,
0
0
N,
0
0
Wrl = i 0
N,
0
0
N,
0
0
0
N,
0
0
N2
Substituting
eqns (10) and (11) into (9) we obtain
(01 = KlW,l{~:~
N,]{uf
u;}’
v=[N,
N*]{uJ
Id;}’
n= W,
%I{4
41
8, = [N,
Nz]{e;
ep.
+[~mIP’,li$J
and
KE, =;
I
~({ri:)‘[N,l’[r~l’[~,l[~l[N,l{Li:}
+ ~ri:~‘~~,1’~~61’~~,1~T,1~~,l(li:j
After some rearrangement, described as
(16)
the shear strain can be
+ ~~:~‘~~,I’~~~l’~~,I~~:I~~~l~ir:~ (13) Y”=x + ~~:~‘[N,1’~~,1’[I,l~T,1[N,l(li:j)dx.
dv
- ez
The total kinetic energy of this element is
dN’
_,$I
’
KE = KE,+ KE,. The potential energy of this elastic element comes from the link’s deformation strain energy which can be represented as
dx
y = dw + 0 zx dx ’
PE,=f{u'}'[K'l{u') and
WI =
s
PWI PI dK
where [K’] is the stiffness matrix, [B] is the transformation matrix between the strain and nodal displacement and [E] is the material properties matrix. According to the Timoshenko beam theory and the configuration of the beam’s rotational deformation, the relationship between rotational deformation and shear strain can be described as
x{e: e;y.
The axial strain due to the bending and axial deformation is
dBz de, G=dxY -dxr+x
dN, dx where &/ax and awlax are the slope of the rotation central line of Y and Z axes, respectively, while v and w are transverse shear deformation, 0, and 0, are the rotational deformation of Y and Z axes, respectively. yXP,Y,,~and y, are the shear strain. The
(17)
dp
dN, -“x
x {u.: e:
dNI dN, ydxdx
ei. g
ej
dN, -rdx
e: j-1,
dN, ydx
1
(18)
where y and z are the distances from the central line.
Structural dynamics
analysis of spatial robots
Combining the results of eqns (17) and (18), we can obtain
dN, dx
0
0
Q
0
dNI dx
0
0
=
OOdxO
0
Use the following equation as
dN, -Zdxdn5 ydx
dN,
0
symbol
i
0
0
0
0
0
-N2
0
N2
0
0
0
0
-N,
0
dN2 d*
N,
0
N,
0
0
0
0
(lo)
The stress and strain relationship
= [;
0
2
0
rdN2 ’ dx
-
dN2 dN, Zdx ydx
to simplify the above
k) = VW).
{ ;}
dN, dx
;
; j
The total potential energy of this element due to bending and axial deformation is
can be written as
[?#
(20)
The above equation can be expressed as
(22) (21) where the element stiffness matrix [K’] is
d"
708
SHIIJH-JER HUNG and
TZYY-YIWANG
The element dynamic equations can be obtained by substituting the element kinetic energy and potential energy into Lagrange’s equation
aKE apE, -m+a{Ue)=lQ}.
row and the column numbers of the matrix, respectively. Similarly, the other matrices or vectors of equation of motion can be obtained, except the generalized forces due to the adjacent elements {Q:,}. {QE,} becomes the internal forces to the link system (23) and these forces will cancel themselves. The dynamic equations of the link system are now
After some tedious mathematical operation and appropriate matrix assembly arrangement, the element dynamic equation is obtained as Wl{~‘) + Kl{a’> =
Wl{~‘l + WI{4 = {Q;) + {Qkl -WI{@) - [Gil{
IQ:,>+ {Q;}+ {Q:x>
- [rrfl{ii’} - [C:]{ ri’} - [C:]{ti’} - [K;]{u'}, (24) where
WI = ~[~,l’[~J’[~mlI~,ldx I + P~~,I’~~~I’~I,I~~~1~~,1 dx s [Cl = P[~~l’[~~l’[~,l[~:I[~‘l d-x i [Cl = 2 ~W,1’[~ml’~~mlPJ,1 dx s + P(~~,l’~~~l’~~,l[~,l~~~l s + [N,1’[T,l’[Z,l[i’,l[N11) dx
ri’} - [Cl(d) - K;l(u’I, (26)
where vector {u’} contains all the degrees of freedom of the link system and {CJ? includes the absolute rigid body motion of the degrees of freedom of the link. The joint transmission system is the other source of robotic structure flexibility. In order to match the three degrees of freedom finite element model of a link, the spring in parallel with a damper model of a compliance joint in previous research may not be accurate enough to represent the dynamic properties of the transmission mechanism. Here, a spherical joint model is employed with six degrees of freedom including three pairs of translation springs accompanied with a damper and three pairs of torsional springs in parallel with a damper in the X, Y and Z directions, respectively. The forces acting on the joint can be derived from the force equilibrium condition of Fig. 2
(Q,)= [M,]({;4’]+ {ii’) - {ii’- ‘1) + [C,]
[&I = ~[~,I’[~ml’[~mI[~,1 dx I
x ({h’} - {&I}) + &]({a’} - {u’- I},,
(27)
and {Q;>and {Q:,l are the forces acting on this element from the neighboring link and element, respectively, while {Q&} is the extra external force acting on this element, i.e. the gravity force. The fourth term of the right-hand side of eqn (24) is the force due to rigid body acceleration. The fifth term is the coriolis acceleration force due to rigid body motion. The sixth term is the coupling coriolis acceleration force due to rigid body motion and elastic vibration. The last term contains forces resulted from normal and tangential type accelerations. The equations of motion for all elements of one link are derived by appropriately assembling the elemental matrices of eqn (24) to form the corresponding link matrices as follows: (m%j=
2
(m'hj,
(25)
e-l
where n, is the total element number, superscript I denotes an element of link matrix, i andj and are the
Fig. 2. Multi-degree of freedoms of spherical joint model.
709
Structural dynamics analysis of spatial robots where [M,], [C,] and [Kj] are the mass, damping and stiffness coefficient matrices
I"jl
=
_
m,
0
0
0
0
0
0
my
0
0
0
0
:
:
0
0
0
0
z,
0
0
0
0
0
0
z, _
0
k, 0
0
0
0
0
k,
[C3 = [~!lWW!l
“d zyx
0
0
0
0
0
0
0
{Q:I = V:l’{Qf)= - {Qf) + IQ;+I1
0 1
{QhI = P’:l’{Qh}.
1
By combining the results of eqns (30) and (27) and making some mathematical rearrangements, the dynamic equations of a single link combined with elastic joints can be represented as
0
0
0
0
k,. 0
k,z 0
C,
0
0
0
0
o-
0
cu
0
0
0
0
:
i
‘d Oxi
z y
0
0
0
0
cry
0
_ 0
0
0
0
0
c,
Icjl =
([m']+[M,']+[M,'+,]){ii'} -[ikfj-"]{ii'-I'} -[M$;']{ti'+"} + ([Ci'l[c;+,])(d)
_
+ (1’) is the rigid body acceleration of the Ith link. While {u’-‘) and {a’) are the displacements of the (I - 1)th and Ith links at the jth joint common coordinate system. During the system dynamic equation derivation, the nodal degrees of freedom {u’} of each link at the common joint should be transformed into the common coordinate in order to guarantee the displacement, velocity and acceleration compatibility at the junction of two or more adjacent links
WI + bql + Dq+,l)W
- [Kj- “]{u’-“}- [fq; ;‘]{u’+ 1’) ={Qh) - Ctmrl{~r)- P%~‘f - [MI]{CF}- [Mj: !‘I){U’+“}- (2[mi]) +
[C9{li’)- &&I + [Gl + [K:l)WJ. (31)
According to the finite element sub-structure theory, the dynamic equation of the whole robotic (24’)= [T,l’{u’), (733) system can be formed by appropriately assembling all the elements of mass, damping and stiffness matrices with respect to each nodal degrees of freedom based where [T,] is the orthogonalized matrix. [7’,] is equal on the equations of motion of the single link and joint to [T,]-1 combination (4 = [T,1{4. (29) [Msl{ii’} + [Cs]{ti’} + [K”l{u”}= {Q:x} - [M”l{ii’} By substituting eqn (29) into (26) and taking some mathematical operations we obtain - [c;]{ ri”} - [Mj]{ ti”} - (2[M”,] + [c:l){ri”}
[m’l@ff+ [K’I{u~I= IQ:1+ IQ:> -[mr](Ifr} -@&1+ where
-[Cl](V) Gf1+
-
- (2[mi] + [Cj)(ti’}
bq>b4
(30)
(Mfdl + cfl + rq){u%
(32)
where {us} includes all the nodal degrees of freedom of the robotic structure, while (MS] and [K”] are the mass inertia and stiffness matrices of the system, respectively, which contain the assembling effect of link flexibility and joint elasticity. Figure 3 shows the difference of system matrices assembled from substructure matrices between link flexibility only and the combination of link flexibility and joint compliance. The damping coefficient matrix [Cl consists of
710
SIWH-JERHUANGand TZYY-YIWANG
[K31 =
(a)
[K?
With link flexibility
= Fig. 5.
(b) With tiik
flexibiUty
and joint ccmpUwm
Fig. 3. The assembly of system stiffness matrix. joint damping matrix [Cj] and the assumed material damping coefficient matrix [C,] of the manipulator structure [32] WI = [Gl + Gl.
(33)
NUMERICALSIMULATION In order to demonstrate the performance of the equations of motion of elastic robotic systems derived in the last section, two manipulator models, an open chain and a closed chain spatial mechanism, are employed for the structural dynamic simulations. The open chain manipulator used in the simulation is simplifi~ R-R-R spatial m~hanism (Fig. 4) of the first three degrees of freedom of an ITRI-U type closed chain robot. The numerical data of this mech-
e 6 0
Link
Link
2
I
4
83
0
Fig. 4. An R-R-R spatial manipulator mechanism.
The configuration of an ITRI-U type robot.
anism are: density 2.71 x 10m6kg/mm3, Young’s modulus 69 x lo9 Pa and shear modulus 26 x lo9 Pa of aluminum alloy. The lengths of links 1 and 2 are 600 and 720 mm, respectively. The cross-sections of links 1 and 2 are 4.48 x lo3 and 3.4 x 103mm2, respectively. The principal axes cross-sectional moments of inertia of links 1 and 2 are 2.9367 x 10’ and 1.4034 x 10’ mm4 for I,,, 2.108 x 10’ and 7.058 x 106mm4 for Zyy and 8.287 x lo6 and 6.976 x 1O6mm4 for Z,, respectively. The translation stiffness and damping coefficients of the three compliance joints are 3 x 109, 2 x lo9 and lo9 NT/mm and 6, 4 and 2 NT set/mm, respectively. Those are assumed the same for the X, Y and 2 directions. The rotational stiffnesses of joint 1 are 3.0 x 10E4, 3.0 x lOI4 and 3.0 x lOI NT mmjrad for the X, Y and 2 directions, respectively. The rotation stiffness of joint 2 are 2.0 x 10i4, 2.0 x 1014 and 2.0 x lOI NTmm/rad for X, Y and Z directions, respectively. The rotational stiffnesses of joint 3 are 1.0 x 1014, 1.0 x lOI and 1.0 x 10” NT mm/rad for the X, Y and 2 directions, respectively. The torsional damping of the three compliance joints are 6, 4 and 2 NT mm sec/rad, respectively. The finite element approach can be used to model and analyze the complicated structure. The method developed in this paper is used to analyze the structural dynamics properties of a closed chain robot of ITRI-U type as shown in Fig. 5, with a parallel four-bar linkage. The material of this structure is aluminum alloy 60617, which has physical properties of density 2.71 x 10e6 kg/mm3, Young’s modulus 69 x IO9Pa and shear modulus 26 x 10’ Pa. The numerical data of this manipulator are shown in Table 1. According to the driving situation and force actuating direction at each compliance joint, the dominant elasticity of each joint is considered only in
Structural dynamics analysis of spatial robots
711
Table 1. Numerical data of ITRI-U type robot
Link length (mm) Area of cross section (mm2) Area moment of inertia I,, IYY (mm4) I ZI Estimated compliance values Translation stiffness (NT/mm) Translation damping (NT secjmm) Rotation stiffness K,, (NT mm/rad) KrY KZ Rotation damping (NT mm sec/rad)
Link 1
Link 2
Link 3
Link 4
600
720 3.6 x 10’ 1.4 x 10’ 7.1 x 106 6.98 x IO6
200 800 1.66 x 105 1.27 x lo5 2.94 x 104
600
590
800 1.66 x lo5 1.27 x lo5 3.94 x 104
6300 5.23 x 10’ 1.27 x 10’ 3.17 x 10’
Joint 1 3.0 x lo4
Joint 2 2.0 x 109
Joint 3 1.0 x 109
Joint 4 1.0 x lo9
Joint 5 1.0 x lo9
3.0 ,” lOi 3.0 x lOI 3.0 x 10’) 6
2.0: 10’4 2.0 x lOI4 2.0 x 10’3 4
1.o ,” 10’4 1.0 x 10’4 1.0 x 10’3 2
4.48 x 2.94 x 2.11 x 8.29 x
lo3 10’ 10’ lo6
system modeling in order to reduce the total degrees of freedom of the whole system. The accuracy of the numerical solution and the cost of computer time depends on the element numbers of each link and the nodal number of each element in the finite element modeling. The compromise between them should be investigated before Fwst
mode
notural
frequency
vs joint
1.0 ,’ lOI 1.0 x lOI 1.0 x lOI 2
dynamic simulation. Figure 6 shows the natural frequencies of the first three modes which are the function of rotation angle and element numbers of each link. The system natural frequency function of rotation angle and nodal number of each element are shown in Fig. 7. According to these results, three elements for each link and four nodes for each
angle
First
1300
Link 5
mode
natural
frequency
vs joint
angle
1300
1
](a) I loo Three
element
900
xx)
-1
,600
i ’
second
mode
natuml
frequency
vs Joint
angle
,600
Secwid
mode
natural
frequency
vs joint
angle
mode
natuml
frequency
vs
angle
_ (b) 1400-
E 12Oo% & &
lcco-
2
3 P
aoo600
,
,
Third
,
I
mode
,
,
notuml
,
,
frequerq
,
, vs
,
,
joint
6001
I angle
Third
-I joint
(c) 3400 J
5cO
8,,‘ 0
I
2
,,,,,,,,, 3
4
5
Rotation
angle
(rad)
6
i
7
Fig. 6. The relationship between natural vibration frequencies and the number of elements of each link.
Rotation
angle
(rad)
Fig. 7. The relationship between natural vibration frequencies and the number of nodes of each element.
712
SHIUH-JERHUANG and TZYY-YI WANG
-oooo2] 0
I 05
1 I.0
I 15
I 20
25
3'0
-".Cd20do
Time tsec)
Time
(a)
(see)
Cb)
Fig. 8. The tip-end position deviation due to deformation in the X, Y and Z axes: (a) the first link, (b) the second link.
RESULTS AND DISCUSSlON
element are a reasonable
choice. The system dynamic equations (32) can then be established by the previous derivation. The system dynamic response is found from the numerical integration by using the mode superposition approach. In this simulation, the robot accelerates from the starting point, then moves with constant speed and finally decelerates to stop at another position. This causes each degree of freedom to move with the same type. Hence, the three joints of this spatial mechanism move simultaneously. During the simulation, the effects of gravity, joint compliance and link flexibility can be investigated indi~dually or simultaneously. From Figs 6 and 7, the first mode of vibration frequency of this spatial mechanism without joint elasticity and rigid body motion effects can be found with respect to the chosen ~nfi~ration. The frequency is about 100 Hz for a small rotation angle of each joint. The natural frequency will be reduced to 50-50 Hz or even less due to the effects of the joint’s rigid body motion dynamics and the introduction of the joint elasticity and the gravity force; this can be determined from the computer simulation given below.
Here four case studies are used to describe the structural dynamic response of the finite element model developed in this paper. In the figures the solid line shows the response curve of numerical simulation and the dashed line shows the results of quasi-static analysis. Cafe 1 The open chain spatial robot moves first with constant acceleration, then with constant speed and finally with a constant deceleration to stop at another point. Due to the acceleration discontinuity, the inertia force acts on the mechanism on the instant of the motion situation changing. Figure 8 shows the position deviation of the tip end of the manipulator in the X, Y and Z directions, respectively. The inertia effect of rotation speed results in the displacement in the X direction which increases from zero toward negative direction due to the deflection of link 1 in the Y direction and reaches its maximum value when the constant speed begins. Then the impulse force due to discontinuing acceleration makes it deviate from the
713
Structural dynamics analysis of spatial robots
(POJ) a@o
m-m---r
z
SHIUH-JERHUANG
714
and
TZYY-YI WANG
robotic mechanism configuration. The vibration frequency of the tip-end in the Z direction is increased due to the inertia effect increasing. Case 3
The motion situation is similar to case 1 except that the link flexibility and multi-degrees of freedom joint elasticity are considered simultaneously. Figure 10 plots the joint compliance deformation of a six degrees of freedom spherical joing 2. The end-effector displacements in the X, Y and Z directions are shown in Fig. 11. From these results, we find that the oscillation amplitude dampens more quickly due to the additional damping included in the joint. Case 4
0 0030
-3
-o.oolo+
0
05
I
I
I
I 0
15 Tim
1
20
I
2.5
I
The first three degrees of freedom of a closed chain ITRI-U type robot mechanism, which has a parallel four bar linkage, are used as the simulation model. The motion planning is the same as in case 1. The equations of motion of this robotic mechanism include the finite element model of link flexibility and the spherical joint model of tr~smission elasticity. Additionally, the effect of gravity is considered. From the simulation results, the dominant elasticity of each joint is shown in Fig. 12. The displacement response curves of the end-effector of this spatial robot are shown in Fig. 13. The oscillation converges more quickly because of the additional joint damping.
30
isec)
Fig. I I. The end-effector position deviation in the X, Y and
2 directions. maximum value and oscillate with a high frequency. Owing to structural damping, this vibration amplitude will dampen quickly. The response of the Y and 2 axes is very similar except that the 2 axis has a small vibration frequency because the dimension of link in the 2 direction is narrower than that of the Y axis. The Y and Z axes have impulse oscillation due to the inertia forces at the instant the motion situation changes. Their vibration amplitude will dampen due to structural damping. Case 2 The motion situation is the same as in case 1 except that the effect of gravity is included during the numerical simulation. The response curves of the tip-end of the manipulator are shown in Fig. 9. The magnitude of the position deviation in the X and Y directions is enlarged due to the gravity force being included. The vibration frequencies of the tip-end position in the X and Y directions are reduced. As the 2 axis is always perpendicular to the gravity field direction, the displacement of the end-point in the 2 direction is not affected. The displacement direction fully depends on the inertia force direction and the
CONCLUSION
In this paper the derivation of a generalized equation of motion for a spatial robotic mechanism with link flexibility and joint elasticity is presented. Here the dynamic equations of the flexible link are derived from Lagrange’s equation based on the displacement finite element scheme and the Timoshenko beam theory. The dynamic model of a multi-degrees of freedom spherical joint is established from Newton’s second law. They are assembled by substructure combination concepts. The dynamic model presented in this study can be used to analyze the dynamic response properties of spatial mechanisms and determine the nonlinear effects of link flexibility, joint elasticity and gravity forces. Additionally, the element numbers of each link and the nodal numbers of each element are investigated based on the natural frequency convergence in order to provide an accurate and low-cost computer model for application. According to the computer simulation results of some case studies, this approach can be used to model not only open chain, but also closed chain spatial robotic mechanisms. The availability of this scheme including the link flexibility, socket joint elasticity and gravity effect simultaneously or individually is very powerful. This is useful in structural dynamic analysis for manipulator operation planning and controller design.
Structural dynamics analysis of spatial robots
(ww)
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SHIUH-JERHUANGand TZYY-YI WANG 0007
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I ,
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Fig. 13. The tip-end position deviation of an ITRI-U type robot in the X, Y and 2 axes. REFERENCES 1. E. Bayo, A finite element approach to control the end-point motion of a single-link flexible robot. J. Robotic Systems 63-75 (1987). 2. W. J. Book, Analysis of massless elastic chains with servo controlled joints. ASME J. Dynamic Systems, Measurement and Control 101, 187-192 (1979). 3. W. J. Book, Recursive Lagrangian dynamics of flexible manipulator arms. Int. J. Robotics Research 3, 87-101 (1984). 4. G. G. Hastings and W. J. Book, Verification of a linear dynamicmodel for flexible roboticmanipulators. Proc. IEEE Conf. Robotics and Automation, pp. 1024-1029 (1986). 5. N. 0. Maizza, Model analysis and control of flexible manipulators. Ph.D. thesis, Department of Mechanical Engineering, MIT (1974). 6. S. Nicosia, P. Tomei and A. Tomambe, Dynamic modeling of flexible robot manipulators. Proc. Int. IEEE Conf. Robotics and Automation, pp. 365-372 (1986). 7. D. M. Rovner and R. H. Canon, Experiments towards on-line identification and control of a very flexible onelink manipulator. Int. J. Robotics Reserach 6,3-19 (1987). 8. R. Johanni, On the automatic generation of the equations of motion for robots with elastically deformable arms. Proc. IFAC Symposium on Theory of Robots, Vienna, pp. 195-199 (1986). 9. S. S. Mahil, On the application of Lagrange methods to the description of dynamic systems. IEEE Trans. System Man and Cybernetics SMC-12, No. 6 (1982). 10. M. H. F. Dado and A. H. Soni, Dynamic response analysis of a 2-R robot with flexible joints. IEEE Inr. Conf on Robotics and Automation, pp. 479483 (1987).
11. C. C. Dewit and 0. Lys, Robust control and parameter estimation of robots with flexible joints. IEEE Inr. Conf. on Robotics and Automation, pp. 324-331 (1988). 12. A. De Luca, Dynamic control of robots with joint elasticity, IEEE Int. Conf: on Robotics and Automation, pp. 152-158 (1988). 13. M. W. Spong, An integral manifold approach to feedback control of flexible joint robots. IEEE J. Robotics and Automation, Vol. 3 (1987). 14. M. W. Spong, Modeling and control of elastic joint robots. ASME J. Dynamic System, Measurement and Control 109, 310-319 (1987). 15. S. Liang, A. A. Frank and B. Ravani, Dynamic simulation of legged machines using a compliant joint model. Int. J. Robotic Research 6, 3346 (1987). 16. F. Toshio, Flexibility control of elastic robotic arms. J. Robotic Systems 2, 73-78 (1985). 17. B. Gebler, Feed-forward control strategy for an industrial robot with elastic links and joints. IEEE Inr. Conf. Robotics and Automation, pp. 923-928 (1987). 18. W. H. Sunada and S. Dubowsky, On the dynamic analysis and behavior of industrial robotic manipulators with elastic members. Trans ASME Mech. Trans. Automat. Design 105, 422451 (1983). 19. G. B. Yang and M. Donath, Dynamic model of a one-link robot manipulator with both structural and joint flexibility. IEEE Int. Conf: Robotics and Automation, pp. 476-481 (1988). 20. G. B. Yang and M. Donath, Dynamic model of a two-link robot manipulator with both structural and joint flexibility. ASME 1988 Winter Annual Meeting, pp. 374 (1988). 21. M. Geradin, G. Robert and C. Bemardin, Dynamic modeling of manipulators with flexible members. Advanced Software in Robotics, pp. 27-39 (1984). 22. P. Judd, Dynamics of nonrigid articulated robot linkages. IEEE ConJ of Robotics and Automation (1983). 23. R. C. Winfrey, Elastic link and mechanism dynamics. ASME J. Engng Industry 93, 268-272 (1971). 24. I. Iman, G. N. Sandor and S. N. Kramer, Deflection and stress analysis in high-speed planar mechanisms with elastic link. ASME J. Engng Industry 95,541-548 (1973). 25. D. A. Turic and A. Midha, Generalized equations of motion for the dynamic analysis of elastic mechanism systems. ASME J. Dynamic Systems, Measurement and Control 106, 243-248 (1984). 26. D. A. Turic and A. Midha, Dynamic analysis of elastic mechanism systems, part I: applications. ASME J. Dynamic Systems, Measurement and Control 106, 249-254 (1984). 27. D. A. Turic, A. Midha and J. R. Bosnik, Dynamic analysis of elastic mechanism systems, part II: experimental results. ASME J. Dynamic Systems, Measurement and Control 106, 255-260 (1984). 28. G. Naganathan and A. H. Soni, Nonlinear modeling of kinematic and flexibility effects in manipulator design. ASME J. Mechanisms, Transmission and Automation in Design 110, 243-254 (1988). 29. M. C. Good, L. W. Sweet and K. L. Strobel, Dynamic models for control system design of integrated robot and drive systems. ASME J. Dynamic Syslems, Measurement and Control 107, 53-59 (1985). 30. A. H. Soni and M. H. F. Dado, Dynamic response analysis of 2-R robot with flexible joints. IEEE Conf on Robotic and Automation, pp. 419-483 (1987). 31. S. Dubowsky and F. Freudenstein, Dynamic analysis of mechanical systems with clearances part 1: formulation of dynamic model. ASME J. Engng for Industry 93, 305-309 (1971). 32. A. Midha, A. G. Erdman and D. A. Frohrib, Finite element approach to mathematical modeling of highspeed elastic linkages. J. Mech. Machine Theory 13, 603-618 (1975).
0
0045-7949193 $6.00 + 0.00 1993 Pergamon Press Ltd
STRUCTURAL DYNAMICS ANALYSIS OF SPATIAL ROBOTS WITH FINITE ELEMENT APPROACH SHIUH-JER HUANG and TZYY-YI WANG Department of Mechanical Engineering, National Taiwan Institute of Technology, 43 Keelung Road, Sec. 4, Taipei, Taiwan 10772, R.O.C. (Received 25 November 1991)
Abstract-This paper develops the generalized equations of motion for a robotic mechanism with elastic structures by using the finite element theory. The derivation and final form of the equations of motion can mode1 two- or three-dimensional complex elastic mechanisms and include the nonlinear coupling terms of rigid body and elastic motions in a general representation. Here, the dynamic model of an R-R-R spatial robotic manipulator with the finite element approach is established. A robot mechanism consists of links and joint transmission systems. The equations of motion of elastic links are derived based on the displacement finite element method. The dynamic model of the flexible joint transmission systems are modeled as a general spherical joint with multi-degree of freedoms. The complete dynamic model of an elastic robotic structure can then be developed by combining the equations of motion of elastic links and joints. According to this dynamic model, we can investigate the structural dynamic properties of this robotic structure from the system dynamic responses. These data are useful for the controller design and optimal dynamic operation planning of robotic manipulator.
INTRODUmION The structure of a conventional industrial robot has been rigidly designed and built without detailed structural analysis. Each component of a manipulator was made strong enough to eliminate link deflection and to overcome the deterioration of system performance
due to payload vibration. With the requirements of higher operational speed and greater positioning accuracy, the robotic mechanism needs to be constructed as lightweight as possible in order to reduce the inertial forces and the driving torque requirements. However, lighter members are more likely to vibrate due to the inertial or external forces. The performance of the manipulator may become unacceptable due to the positioning inaccuracies of the deformation variation of the end-effector. Furthermore, the unpredicted resonant condition without the vibration analysis may influence the system’s operational performance. Structural analysis is a useful tool to provide the necessary information for robotic mechanism design, optimal performance controller design and operation planning in order to overcome the above problems. The purpose of this paper is to develop a general model for spatial robotic structures with flexible links and elastic joints. It can be used to analyze the dynamic response of not only open chain, but also closed chain robotic mechanisms. The optimal lightweight manipulator and optimal dynamic operation can then be designed based on the dynamic response properties of this dynamic model. PREVIOUS WORK The earlier research of robotic structural dynamic analysis can be classified into two categories. The 703
end-effector position deviation due to the effect of flexible links [l-9] and that of elastic joints [lO-151 were investigated. Recently, the combination of flexible links and elastic joints effects were studied [16-201. There are two approaches for modeling the deformation of flexible links. The one is the finite mode shapes method whose vibration modes or eigenfunction are directly derived from the partial differential equation with appropriate boundary conditions. This is suitable for simple boundary conditions and simple geometry configuration of manipulator. The other one is the finite element method for more complicated structures with the expense of additional computer time. According to its different reference configuration, this approach can be divided into two categories [21] whose reference coordinates of links are the Cartesian coordinate of mechanism system or the rigid body motion coordinate attached to the link. Book [3] and Judd [22] used Lagrange’s equation and the latter coordinate system accompanied by a 4 x 4 homogeneous coordinate transformation matrix to derive the recursive Lagrange dynamics of spatial flexible manipulator arms equipped with revolute joints. The scheme referring to the system’s reference Cartesian coordinate is more suitable for the finite element method. Although in this scheme the homogeneous transformation matrix is a function of joint variables only, the transformation matrix of this scheme refers to the rigid link’s reference coordinate is a function of joint variables and elastic deflection. Winfrey [23] and Iman et al. [24] first used the finite element method for the dynamic analysis of an elastic planar four-bar linkage mechanism with beam element consideration. Sunada and Dubowsky [ 181
704
SHKJH-JERHUANGand
employed the perturbation coordinate and coordinate elimination scheme to derive the dynamic equations of the whole mechanism. Turcic and Midha [25-271 derived the generalized equations of motion of elastic mechanism systems using one displacement finite element method. They then used the modal analysis technique accompanied by a numerical recursive scheme to solve the equations of motion which included the nonlinear coupling terms. They also provided experimental results for comparison. According to the Newton-Euler equation, Naganathan and Soni [28] employed the finite element method together with Timoshenko beam theory in order to derive the nonlinear dynamic equations of robotic structure. Experimental results are provided to demonstrate the accuracy of the numerical results from Newmark integration solution. The joint compliance including the deflection of motor shaft, gears and belts comes from the transmission system of each driving joint. The small angular deviation due to joint compliance will significantly influence the end-effector position accuracy by the amplification effect of link length. The dynamic model of the elastic joint has been modeled as a torsion spring in parallel with a viscous damper in previous research. Spong [13, 141 investigated the modeling and control problem for flexible joints manipulators. Good et al. [29], De Luca [ 121and Soni and Dado [30] also studied the dynamic response and control properties of manipulators with elastic joints. Dubowsky and Freudenstein [31] investigated the dynamic model and response of a mechanical system with clearances. Recently, Yang and Donath [19,20] investigated the combination effect of link flexibility and joint
14 = M (u’} = {Id;
preach can be used not only for open chain, but also for closed chain robotic mechanisms. The structural dynamics properties can then be identified from the simulation analysis. The information is useful for mechanism design, controller design and operational planning. MATHEMATICAL
MODEL
The robotic structure flexibility consists of the flexible link deflection and the elastic joint compliance. Here the dynamic model of the robotic mechanism’s flexible link is modeled as a flexible Timoshenko beam which can be derived from Lagrange’s equation using the finite element approach. The dynamic model of each joint transmission system with multidegree of freedoms can be established based on the Newton’s second law. Then the nonlinear dynamic model of the whole elastic manipulator mechanism, including the coupling effects of rigid body motion and elastic vibration, can be derived using the substructure concepts. The basic assumption of the displacement finite element method is that the local displacement vector of any point in an element can be represented as a linear combination of the nodal displacements
where d is the local displacement vector, ae is the element nodal displacement vector and N is the matrix of the shape functions. The terms ue and dare with respect to coordinate X, Y,&. In order to simplify the derivation of the dynamic mode1 of this spatial mechanism with the Timoshenko beam, twonode beam elements are discussed. Hence
dy 4 0, ‘$ RI
uf uf e:
-N, 0
lN1= : 0
-0
TZYY-YIWANG
e;
e:
u:
u;
U;
eg e;
et}'
0
0
N2
0
0
0
0
0
0
0
0
0
N2
0
0
0
0
:x,0
0
0
0
0
0
0
0
0
0
0
N, 0
N2
0
0
0
0
0
0
0
N,
0
0
0
N,
0
0
0
0
N,
0
0
0
0
compliance by combining a simple assumed mode shapes model of beams with spring-damper models. In this paper, we combine the finite element model of flexible links using the Timoshenko beam theory with multi-degree of freedom models of elastic joints to obtain a reasonable model for elastic robotic structure. The dynamic model established by this ap-
N,
0
0
0
0
0
-
’
0 N2_
where N /-x-, I
I
N2=f
and 1 is the length of the element. X is defined as the length along the element between 0 and 1. In order to
105
Structural dynamics analysis of spatial robots
{ir:} is the rigid body motion of the element node with respect to the X, Y, Z, coordinate and [r:] is the constant transformation matrix of the orientation between the Xi Y, Z, and XYZ coordinates. Then the translation velocity of any point in the element can be described as
The translation
kinetic energy of this element is
KE,=;p{d}‘{h}
dl’.
(5)
s
Substituting eqn (4) into (5) after some rearrangement, gives
Fig. 1. Configuration of spatial beam deformation. guarantee the solution convergence, the shape functions must satisfy the continuity, compatibility and completeness conditions. The kinetic energy of this three-dimensional element consists of translational and rotational kinetic energy. In order to simplify and reduce the computing time, they are derived individually. The configurations of the single element before and after deformation are shown in Fig. 1. XYZ is the manipulator Cartesian base reference coordinate, X, Y,Z, is the moving coordinate attached to the undeformed element equipped with rigid body motion and the X, Y,Z, coordinate has a common origin point with the XYZ coordinate while its direction is parallel to that of the X, Y,Z, coordinate. The moving coordinate Xi Y;Z; is attached to the deflection element which has a relative displacement d, with respect to the X, YrZ, coordinate. Then the positional vector of any point on the element can be written as
where [T,,,] is the transformation matrix between X, Y,Z, and XYZ coordinate systems. It is a function of the rigid body motion only. R, is the relative positional vector of the moving coordinate X, Y,Z, with respect to the reference coordinate XYZ. Then the velocity of this point can be derived by differentiating eqn (2) with respect to time
(4 = MO> + Knlh41+ KJVJ
+ 2(~:}1~,l’[~~~[~,l[~,lI~:})dt’. Similarly, the rotational described as KE,=;
where [T,,,] is a function of the joint angular and [iV,] is function of the shape function only. Additionally,
~{~)V,l{~jdx,
(7)
The rotation velocity is
@I = h31 +Kl@rI~
(9)
where (w} is the rotation speed of the X; Y;Z; coordinate with respect to the base reference coordinate XYZ and {w,} is the element rigid body rotation speed which can be described as {%j =
(3)
kinetic energy can be
where p is the mass density and [Z,] is the cross-sectional moment of inertia matrix. If the coordinate axes coincide with the element principal axes, then
bt>= W,lW,I IM = K1[~~1Iir:L
s
(6)
The rotation element is
EJW,l~W.
(10)
velocity vector of any point in the Id:) =
W,l{6L
(11)
706
SHIUH-JERHUANGand TZYY-YI WANG
where {ti;} = {et
s;,
e:
eg, s_; e;}’
transverse deformation and notational deformation can be represented as the function of the nodal displacement w = [N,
N,
0
0
N,
0
0
Wrl = i 0
N,
0
0
N,
0
0
0
N,
0
0
N2
Substituting
eqns (10) and (11) into (9) we obtain
(01 = KlW,l{~:~
N,]{uf
u;}’
v=[N,
N*]{uJ
Id;}’
n= W,
%I{4
41
8, = [N,
Nz]{e;
ep.
+[~mIP’,li$J
and
KE, =;
I
~({ri:)‘[N,l’[r~l’[~,l[~l[N,l{Li:}
+ ~ri:~‘~~,1’~~61’~~,1~T,1~~,l(li:j
After some rearrangement, described as
(16)
the shear strain can be
+ ~~:~‘~~,I’~~~l’~~,I~~:I~~~l~ir:~ (13) Y”=x + ~~:~‘[N,1’~~,1’[I,l~T,1[N,l(li:j)dx.
dv
- ez
The total kinetic energy of this element is
dN’
_,$I
’
KE = KE,+ KE,. The potential energy of this elastic element comes from the link’s deformation strain energy which can be represented as
dx
y = dw + 0 zx dx ’
PE,=f{u'}'[K'l{u') and
WI =
s
PWI PI dK
where [K’] is the stiffness matrix, [B] is the transformation matrix between the strain and nodal displacement and [E] is the material properties matrix. According to the Timoshenko beam theory and the configuration of the beam’s rotational deformation, the relationship between rotational deformation and shear strain can be described as
x{e: e;y.
The axial strain due to the bending and axial deformation is
dBz de, G=dxY -dxr+x
dN, dx where &/ax and awlax are the slope of the rotation central line of Y and Z axes, respectively, while v and w are transverse shear deformation, 0, and 0, are the rotational deformation of Y and Z axes, respectively. yXP,Y,,~and y, are the shear strain. The
(17)
dp
dN, -“x
x {u.: e:
dNI dN, ydxdx
ei. g
ej
dN, -rdx
e: j-1,
dN, ydx
1
(18)
where y and z are the distances from the central line.
Structural dynamics
analysis of spatial robots
Combining the results of eqns (17) and (18), we can obtain
dN, dx
0
0
Q
0
dNI dx
0
0
=
OOdxO
0
Use the following equation as
dN, -Zdxdn5 ydx
dN,
0
symbol
i
0
0
0
0
0
-N2
0
N2
0
0
0
0
-N,
0
dN2 d*
N,
0
N,
0
0
0
0
(lo)
The stress and strain relationship
= [;
0
2
0
rdN2 ’ dx
-
dN2 dN, Zdx ydx
to simplify the above
k) = VW).
{ ;}
dN, dx
;
; j
The total potential energy of this element due to bending and axial deformation is
can be written as
[?#
(20)
The above equation can be expressed as
(22) (21) where the element stiffness matrix [K’] is
d"
708
SHIIJH-JER HUNG and
TZYY-YIWANG
The element dynamic equations can be obtained by substituting the element kinetic energy and potential energy into Lagrange’s equation
aKE apE, -m+a{Ue)=lQ}.
row and the column numbers of the matrix, respectively. Similarly, the other matrices or vectors of equation of motion can be obtained, except the generalized forces due to the adjacent elements {Q:,}. {QE,} becomes the internal forces to the link system (23) and these forces will cancel themselves. The dynamic equations of the link system are now
After some tedious mathematical operation and appropriate matrix assembly arrangement, the element dynamic equation is obtained as Wl{~‘) + Kl{a’> =
Wl{~‘l + WI{4 = {Q;) + {Qkl -WI{@) - [Gil{
IQ:,>+ {Q;}+ {Q:x>
- [rrfl{ii’} - [C:]{ ri’} - [C:]{ti’} - [K;]{u'}, (24) where
WI = ~[~,l’[~J’[~mlI~,ldx I + P~~,I’~~~I’~I,I~~~1~~,1 dx s [Cl = P[~~l’[~~l’[~,l[~:I[~‘l d-x i [Cl = 2 ~W,1’[~ml’~~mlPJ,1 dx s + P(~~,l’~~~l’~~,l[~,l~~~l s + [N,1’[T,l’[Z,l[i’,l[N11) dx
ri’} - [Cl(d) - K;l(u’I, (26)
where vector {u’} contains all the degrees of freedom of the link system and {CJ? includes the absolute rigid body motion of the degrees of freedom of the link. The joint transmission system is the other source of robotic structure flexibility. In order to match the three degrees of freedom finite element model of a link, the spring in parallel with a damper model of a compliance joint in previous research may not be accurate enough to represent the dynamic properties of the transmission mechanism. Here, a spherical joint model is employed with six degrees of freedom including three pairs of translation springs accompanied with a damper and three pairs of torsional springs in parallel with a damper in the X, Y and Z directions, respectively. The forces acting on the joint can be derived from the force equilibrium condition of Fig. 2
(Q,)= [M,]({;4’]+ {ii’) - {ii’- ‘1) + [C,]
[&I = ~[~,I’[~ml’[~mI[~,1 dx I
x ({h’} - {&I}) + &]({a’} - {u’- I},,
(27)
and {Q;>and {Q:,l are the forces acting on this element from the neighboring link and element, respectively, while {Q&} is the extra external force acting on this element, i.e. the gravity force. The fourth term of the right-hand side of eqn (24) is the force due to rigid body acceleration. The fifth term is the coriolis acceleration force due to rigid body motion. The sixth term is the coupling coriolis acceleration force due to rigid body motion and elastic vibration. The last term contains forces resulted from normal and tangential type accelerations. The equations of motion for all elements of one link are derived by appropriately assembling the elemental matrices of eqn (24) to form the corresponding link matrices as follows: (m%j=
2
(m'hj,
(25)
e-l
where n, is the total element number, superscript I denotes an element of link matrix, i andj and are the
Fig. 2. Multi-degree of freedoms of spherical joint model.
709
Structural dynamics analysis of spatial robots where [M,], [C,] and [Kj] are the mass, damping and stiffness coefficient matrices
I"jl
=
_
m,
0
0
0
0
0
0
my
0
0
0
0
:
:
0
0
0
0
z,
0
0
0
0
0
0
z, _
0
k, 0
0
0
0
0
k,
[C3 = [~!lWW!l
“d zyx
0
0
0
0
0
0
0
{Q:I = V:l’{Qf)= - {Qf) + IQ;+I1
0 1
{QhI = P’:l’{Qh}.
1
By combining the results of eqns (30) and (27) and making some mathematical rearrangements, the dynamic equations of a single link combined with elastic joints can be represented as
0
0
0
0
k,. 0
k,z 0
C,
0
0
0
0
o-
0
cu
0
0
0
0
:
i
‘d Oxi
z y
0
0
0
0
cry
0
_ 0
0
0
0
0
c,
Icjl =
([m']+[M,']+[M,'+,]){ii'} -[ikfj-"]{ii'-I'} -[M$;']{ti'+"} + ([Ci'l[c;+,])(d)
_
+ (1’) is the rigid body acceleration of the Ith link. While {u’-‘) and {a’) are the displacements of the (I - 1)th and Ith links at the jth joint common coordinate system. During the system dynamic equation derivation, the nodal degrees of freedom {u’} of each link at the common joint should be transformed into the common coordinate in order to guarantee the displacement, velocity and acceleration compatibility at the junction of two or more adjacent links
WI + bql + Dq+,l)W
- [Kj- “]{u’-“}- [fq; ;‘]{u’+ 1’) ={Qh) - Ctmrl{~r)- P%~‘f - [MI]{CF}- [Mj: !‘I){U’+“}- (2[mi]) +
[C9{li’)- &&I + [Gl + [K:l)WJ. (31)
According to the finite element sub-structure theory, the dynamic equation of the whole robotic (24’)= [T,l’{u’), (733) system can be formed by appropriately assembling all the elements of mass, damping and stiffness matrices with respect to each nodal degrees of freedom based where [T,] is the orthogonalized matrix. [7’,] is equal on the equations of motion of the single link and joint to [T,]-1 combination (4 = [T,1{4. (29) [Msl{ii’} + [Cs]{ti’} + [K”l{u”}= {Q:x} - [M”l{ii’} By substituting eqn (29) into (26) and taking some mathematical operations we obtain - [c;]{ ri”} - [Mj]{ ti”} - (2[M”,] + [c:l){ri”}
[m’l@ff+ [K’I{u~I= IQ:1+ IQ:> -[mr](Ifr} -@&1+ where
-[Cl](V) Gf1+
-
- (2[mi] + [Cj)(ti’}
bq>b4
(30)
(Mfdl + cfl + rq){u%
(32)
where {us} includes all the nodal degrees of freedom of the robotic structure, while (MS] and [K”] are the mass inertia and stiffness matrices of the system, respectively, which contain the assembling effect of link flexibility and joint elasticity. Figure 3 shows the difference of system matrices assembled from substructure matrices between link flexibility only and the combination of link flexibility and joint compliance. The damping coefficient matrix [Cl consists of
710
SIWH-JERHUANGand TZYY-YIWANG
[K31 =
(a)
[K?
With link flexibility
= Fig. 5.
(b) With tiik
flexibiUty
and joint ccmpUwm
Fig. 3. The assembly of system stiffness matrix. joint damping matrix [Cj] and the assumed material damping coefficient matrix [C,] of the manipulator structure [32] WI = [Gl + Gl.
(33)
NUMERICALSIMULATION In order to demonstrate the performance of the equations of motion of elastic robotic systems derived in the last section, two manipulator models, an open chain and a closed chain spatial mechanism, are employed for the structural dynamic simulations. The open chain manipulator used in the simulation is simplifi~ R-R-R spatial m~hanism (Fig. 4) of the first three degrees of freedom of an ITRI-U type closed chain robot. The numerical data of this mech-
e 6 0
Link
Link
2
I
4
83
0
Fig. 4. An R-R-R spatial manipulator mechanism.
The configuration of an ITRI-U type robot.
anism are: density 2.71 x 10m6kg/mm3, Young’s modulus 69 x lo9 Pa and shear modulus 26 x lo9 Pa of aluminum alloy. The lengths of links 1 and 2 are 600 and 720 mm, respectively. The cross-sections of links 1 and 2 are 4.48 x lo3 and 3.4 x 103mm2, respectively. The principal axes cross-sectional moments of inertia of links 1 and 2 are 2.9367 x 10’ and 1.4034 x 10’ mm4 for I,,, 2.108 x 10’ and 7.058 x 106mm4 for Zyy and 8.287 x lo6 and 6.976 x 1O6mm4 for Z,, respectively. The translation stiffness and damping coefficients of the three compliance joints are 3 x 109, 2 x lo9 and lo9 NT/mm and 6, 4 and 2 NT set/mm, respectively. Those are assumed the same for the X, Y and 2 directions. The rotational stiffnesses of joint 1 are 3.0 x 10E4, 3.0 x lOI4 and 3.0 x lOI NT mmjrad for the X, Y and 2 directions, respectively. The rotation stiffness of joint 2 are 2.0 x 10i4, 2.0 x 1014 and 2.0 x lOI NTmm/rad for X, Y and Z directions, respectively. The rotational stiffnesses of joint 3 are 1.0 x 1014, 1.0 x lOI and 1.0 x 10” NT mm/rad for the X, Y and 2 directions, respectively. The torsional damping of the three compliance joints are 6, 4 and 2 NT mm sec/rad, respectively. The finite element approach can be used to model and analyze the complicated structure. The method developed in this paper is used to analyze the structural dynamics properties of a closed chain robot of ITRI-U type as shown in Fig. 5, with a parallel four-bar linkage. The material of this structure is aluminum alloy 60617, which has physical properties of density 2.71 x 10e6 kg/mm3, Young’s modulus 69 x IO9Pa and shear modulus 26 x 10’ Pa. The numerical data of this manipulator are shown in Table 1. According to the driving situation and force actuating direction at each compliance joint, the dominant elasticity of each joint is considered only in
Structural dynamics analysis of spatial robots
711
Table 1. Numerical data of ITRI-U type robot
Link length (mm) Area of cross section (mm2) Area moment of inertia I,, IYY (mm4) I ZI Estimated compliance values Translation stiffness (NT/mm) Translation damping (NT secjmm) Rotation stiffness K,, (NT mm/rad) KrY KZ Rotation damping (NT mm sec/rad)
Link 1
Link 2
Link 3
Link 4
600
720 3.6 x 10’ 1.4 x 10’ 7.1 x 106 6.98 x IO6
200 800 1.66 x 105 1.27 x lo5 2.94 x 104
600
590
800 1.66 x lo5 1.27 x lo5 3.94 x 104
6300 5.23 x 10’ 1.27 x 10’ 3.17 x 10’
Joint 1 3.0 x lo4
Joint 2 2.0 x 109
Joint 3 1.0 x 109
Joint 4 1.0 x lo9
Joint 5 1.0 x lo9
3.0 ,” lOi 3.0 x lOI 3.0 x 10’) 6
2.0: 10’4 2.0 x lOI4 2.0 x 10’3 4
1.o ,” 10’4 1.0 x 10’4 1.0 x 10’3 2
4.48 x 2.94 x 2.11 x 8.29 x
lo3 10’ 10’ lo6
system modeling in order to reduce the total degrees of freedom of the whole system. The accuracy of the numerical solution and the cost of computer time depends on the element numbers of each link and the nodal number of each element in the finite element modeling. The compromise between them should be investigated before Fwst
mode
notural
frequency
vs joint
1.0 ,’ lOI 1.0 x lOI 1.0 x lOI 2
dynamic simulation. Figure 6 shows the natural frequencies of the first three modes which are the function of rotation angle and element numbers of each link. The system natural frequency function of rotation angle and nodal number of each element are shown in Fig. 7. According to these results, three elements for each link and four nodes for each
angle
First
1300
Link 5
mode
natural
frequency
vs joint
angle
1300
1
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element
900
xx)
-1
,600
i ’
second
mode
natuml
frequency
vs Joint
angle
,600
Secwid
mode
natural
frequency
vs joint
angle
mode
natuml
frequency
vs
angle
_ (b) 1400-
E 12Oo% & &
lcco-
2
3 P
aoo600
,
,
Third
,
I
mode
,
,
notuml
,
,
frequerq
,
, vs
,
,
joint
6001
I angle
Third
-I joint
(c) 3400 J
5cO
8,,‘ 0
I
2
,,,,,,,,, 3
4
5
Rotation
angle
(rad)
6
i
7
Fig. 6. The relationship between natural vibration frequencies and the number of elements of each link.
Rotation
angle
(rad)
Fig. 7. The relationship between natural vibration frequencies and the number of nodes of each element.
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I 05
1 I.0
I 15
I 20
25
3'0
-".Cd20do
Time tsec)
Time
(a)
(see)
Cb)
Fig. 8. The tip-end position deviation due to deformation in the X, Y and Z axes: (a) the first link, (b) the second link.
RESULTS AND DISCUSSlON
element are a reasonable
choice. The system dynamic equations (32) can then be established by the previous derivation. The system dynamic response is found from the numerical integration by using the mode superposition approach. In this simulation, the robot accelerates from the starting point, then moves with constant speed and finally decelerates to stop at another position. This causes each degree of freedom to move with the same type. Hence, the three joints of this spatial mechanism move simultaneously. During the simulation, the effects of gravity, joint compliance and link flexibility can be investigated indi~dually or simultaneously. From Figs 6 and 7, the first mode of vibration frequency of this spatial mechanism without joint elasticity and rigid body motion effects can be found with respect to the chosen ~nfi~ration. The frequency is about 100 Hz for a small rotation angle of each joint. The natural frequency will be reduced to 50-50 Hz or even less due to the effects of the joint’s rigid body motion dynamics and the introduction of the joint elasticity and the gravity force; this can be determined from the computer simulation given below.
Here four case studies are used to describe the structural dynamic response of the finite element model developed in this paper. In the figures the solid line shows the response curve of numerical simulation and the dashed line shows the results of quasi-static analysis. Cafe 1 The open chain spatial robot moves first with constant acceleration, then with constant speed and finally with a constant deceleration to stop at another point. Due to the acceleration discontinuity, the inertia force acts on the mechanism on the instant of the motion situation changing. Figure 8 shows the position deviation of the tip end of the manipulator in the X, Y and Z directions, respectively. The inertia effect of rotation speed results in the displacement in the X direction which increases from zero toward negative direction due to the deflection of link 1 in the Y direction and reaches its maximum value when the constant speed begins. Then the impulse force due to discontinuing acceleration makes it deviate from the
713
Structural dynamics analysis of spatial robots
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TZYY-YI WANG
robotic mechanism configuration. The vibration frequency of the tip-end in the Z direction is increased due to the inertia effect increasing. Case 3
The motion situation is similar to case 1 except that the link flexibility and multi-degrees of freedom joint elasticity are considered simultaneously. Figure 10 plots the joint compliance deformation of a six degrees of freedom spherical joing 2. The end-effector displacements in the X, Y and Z directions are shown in Fig. 11. From these results, we find that the oscillation amplitude dampens more quickly due to the additional damping included in the joint. Case 4
0 0030
-3
-o.oolo+
0
05
I
I
I
I 0
15 Tim
1
20
I
2.5
I
The first three degrees of freedom of a closed chain ITRI-U type robot mechanism, which has a parallel four bar linkage, are used as the simulation model. The motion planning is the same as in case 1. The equations of motion of this robotic mechanism include the finite element model of link flexibility and the spherical joint model of tr~smission elasticity. Additionally, the effect of gravity is considered. From the simulation results, the dominant elasticity of each joint is shown in Fig. 12. The displacement response curves of the end-effector of this spatial robot are shown in Fig. 13. The oscillation converges more quickly because of the additional joint damping.
30
isec)
Fig. I I. The end-effector position deviation in the X, Y and
2 directions. maximum value and oscillate with a high frequency. Owing to structural damping, this vibration amplitude will dampen quickly. The response of the Y and 2 axes is very similar except that the 2 axis has a small vibration frequency because the dimension of link in the 2 direction is narrower than that of the Y axis. The Y and Z axes have impulse oscillation due to the inertia forces at the instant the motion situation changes. Their vibration amplitude will dampen due to structural damping. Case 2 The motion situation is the same as in case 1 except that the effect of gravity is included during the numerical simulation. The response curves of the tip-end of the manipulator are shown in Fig. 9. The magnitude of the position deviation in the X and Y directions is enlarged due to the gravity force being included. The vibration frequencies of the tip-end position in the X and Y directions are reduced. As the 2 axis is always perpendicular to the gravity field direction, the displacement of the end-point in the 2 direction is not affected. The displacement direction fully depends on the inertia force direction and the
CONCLUSION
In this paper the derivation of a generalized equation of motion for a spatial robotic mechanism with link flexibility and joint elasticity is presented. Here the dynamic equations of the flexible link are derived from Lagrange’s equation based on the displacement finite element scheme and the Timoshenko beam theory. The dynamic model of a multi-degrees of freedom spherical joint is established from Newton’s second law. They are assembled by substructure combination concepts. The dynamic model presented in this study can be used to analyze the dynamic response properties of spatial mechanisms and determine the nonlinear effects of link flexibility, joint elasticity and gravity forces. Additionally, the element numbers of each link and the nodal numbers of each element are investigated based on the natural frequency convergence in order to provide an accurate and low-cost computer model for application. According to the computer simulation results of some case studies, this approach can be used to model not only open chain, but also closed chain spatial robotic mechanisms. The availability of this scheme including the link flexibility, socket joint elasticity and gravity effect simultaneously or individually is very powerful. This is useful in structural dynamic analysis for manipulator operation planning and controller design.
Structural dynamics analysis of spatial robots
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Fig. 13. The tip-end position deviation of an ITRI-U type robot in the X, Y and 2 axes. REFERENCES 1. E. Bayo, A finite element approach to control the end-point motion of a single-link flexible robot. J. Robotic Systems 63-75 (1987). 2. W. J. Book, Analysis of massless elastic chains with servo controlled joints. ASME J. Dynamic Systems, Measurement and Control 101, 187-192 (1979). 3. W. J. Book, Recursive Lagrangian dynamics of flexible manipulator arms. Int. J. Robotics Research 3, 87-101 (1984). 4. G. G. Hastings and W. J. Book, Verification of a linear dynamicmodel for flexible roboticmanipulators. Proc. IEEE Conf. Robotics and Automation, pp. 1024-1029 (1986). 5. N. 0. Maizza, Model analysis and control of flexible manipulators. Ph.D. thesis, Department of Mechanical Engineering, MIT (1974). 6. S. Nicosia, P. Tomei and A. Tomambe, Dynamic modeling of flexible robot manipulators. Proc. Int. IEEE Conf. Robotics and Automation, pp. 365-372 (1986). 7. D. M. Rovner and R. H. Canon, Experiments towards on-line identification and control of a very flexible onelink manipulator. Int. J. Robotics Reserach 6,3-19 (1987). 8. R. Johanni, On the automatic generation of the equations of motion for robots with elastically deformable arms. Proc. IFAC Symposium on Theory of Robots, Vienna, pp. 195-199 (1986). 9. S. S. Mahil, On the application of Lagrange methods to the description of dynamic systems. IEEE Trans. System Man and Cybernetics SMC-12, No. 6 (1982). 10. M. H. F. Dado and A. H. Soni, Dynamic response analysis of a 2-R robot with flexible joints. IEEE Inr. Conf on Robotics and Automation, pp. 479483 (1987).
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