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A composite laminated box-section beam design for obtaining optimal elastodynamic responses of a flexible robot manipulator
Int. J. Mech. Sci. Vol. 32, No. 5, pp. 391-403, 1990
0020-7403/90 $3.00 + .00 © 1990 Pergamon Press pie
Printed in Great Britain.
A COMPOSITE LAMINATED BOX-SECTION BEAM DESIGN FOR OBTAINING OPTIMAL ELASTODYNAMIC RESPONSES OF A FLEXIBLE ROBOT MANIPULATOR C. K. SUNGand S. S. SHYL Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China
(Received 31 July 1989) Abstract--A design methodology is presented for synthesizing a high-performance articulating robot manipulator fabricated with optimally tailored composite laminates. By optimally specifying the geometric configurations, the types of fiber, matrix, stacking sequence, fiber volume fraction, fiber layups, etc., the synthesized composite material may possess such superior characteristics as high damping, high stiffness, high strenigth and low mass. In accordance with the design requirements, e.g. minimum deflection during articulating motion or fast settling time after the power has been stopped, the design objectives and constraint conditions were specified. As an illustrative example, a two-link robot manipulator fabricated with the aforementioned composite laminates is employed for demonstrating the proposed design methodology.
NOTATION coordinate system oxyz: x-axis along the fiber coordinate system 0123: 1-axis along the principle longitudinal axis of the beam aij i,j = I, 2, 6; component of the in-plane compliance matrix [ao] Aj~ j, k = 1, 2, 6, component of the in-plane modulus matrix [A~] A the cross-section area of the beam b the width of the beam bl the prescribed internal width of the beam [c] the global linear damping matrix [Ccor] equivalent damping matrix associated with Coriolis acceleration d o i,j = 1, 2, 6; component of the compliance matrix [do] D~j i,j = 1, 2, 6; component of the flexural modulus matrix [D~] E the effective flexural Young's modulus El the effective in-plane longitudinal Young's modulus Ef elastic Young's modulus of the fiber Em elastic Young's modulus of the matrix {~} prescribed surface traction imposed on area St G effective in-plane shear modulus Gt shear modulus of the fiber Gm shear modulus of the matrix h the height of the beam h, the prescribed internal height of the beam I the second moment of inertia associated with the moment M~ [K] global linear stiffness matrix [gt] the inertia coupling matrix linking the deformation and rigid-body fields 1 the length of the beam [M] the global mass matrix [M,] the global mass matrix coupling the shape functions defining the discretized rigid-body kinematic motion and the elastic deformations M 1 bending moment along the 1-axis M3 bending moment along the 3-axis IN] the row vector containing the shape function Ni i = 1, 2, 6; the in-plane shear resultants [~,] the vector containing the discretized rigid-body acceleration q total number of layers {q} the global deformation displacement vector Qtil ik j, k = 1, 2, 6; component of the flexural moduli of the ith layer ti the thickness of the ith layer It,, the fiber volume fraction of the ith layer w total strain energy 391
392 AW AWi {X} yi z~ gj p pf Pm vt Vm q~ ~P~ ~P ai, e~ #~, ~
C.K. SUNGand S.S. SHYL total strain energy dissipation i = x, y, s; the strain energy dissipated per cyclealong each axis the body force vector imposed on volume V the horizontal distance from the 3-axis to the ith layer the vertical distance from the 2-axis to the ith layer the ply angle of the ith layer density density of the fiber density of the matrix longitudinal Poisson's ratio of fiber longitudinalPoisson's ratio of matrix i = x, y, s; specificdamping capacity component of each layer i = x, y, s; component of the total specificclamping capacity total specificdamping capacity i = x, y, s; stress and strain componentsof the ith layer of the upper and lowerflange'salong each axis i = x, y, s; stress and strain componentsof the ith layer of the side walls along each axis INTRODUCTION
The intense competition in the international marketplace for robots and machine systems which significantly enhance manufacturing productivity by operating at high speeds with more accurate endpoint positioning has resulted in the evolution of a new frontier in machine design. These objectives cannot be achieved with existing designs because the current articulating members possess large moments of inertia, high stiffness and also substantial weight, in an attempt to provide an acceptable degree of repeatability and endpoint accuracy with payload-weight to arm-weight ratios of only about 1:20 [-1]. These parameters significantly constrain the response of these devices. For example, the beam deflections are primarily due to the weight of the mechanical linkage itself, in addition, the large mass, and hence inertia, of the moving members of the robot naturally inhibit the ability of the device to respond quickly. Hence, if higher operating speeds are to be achieved, a lighter mechanism is desirable provided structural stiffness is not sacrificed. The design goals of high-speed operation and improved end-effector positioning have received considerable attention in the literature [2-8], with the emphasis being upon the manipulator control system, the elastodynamic characteristics of the linkage and also the interaction between these diverse design areas. A variety of techniques have been proposed. The concepts of designing mechanical components possessing high strength, high stiffness and low mass in order to improve the performance of manipulators have been employed in the design of high-speed linkages by optimally designing the geometries of the beam crosssections [9-11]. An alternative design philosophy has been proposed for the design of high-speed linkages and robot arms, which requires the members to be fabricated in modern composite materials I-12, 13]. As is well known, these materials possess much higher strength-to-weight and stiffness-to-weight ratios than the commercial metals and consequently they promise manipulator designs with much smaller deflections at the end-effector and also higher speeds of operation using existing actuators. In addition, modern fibrous composite laminates have a larger number of design variables which must be carefully selected in order to achieve the desired material properties. These variables typically include the properties of both the fibers and matrices, the stacking sequence, the fiber layups, the fiber volume fraction and the ply thicknesses. Literature regarding the optimal tailoring of composite laminates are included in Refs 1-14-16]. However these publications generally focus on strength or stiffness characteristics and do not address dynamical problems. Furthermore they do not address the optimal synthesis of members specifically tailored for robotic applications in which optimal damping and stiffness characteristics are required, subject to a variety of practical constraints. Based on the aforementioned design variables and geometries, Sung and Thompson [17] proposed a design methodology for constructing a robot manipulator which possesses superior dynamic characteristics. This paper proposes an alternative methodology for designing a robot arm which possesses not only superior dynamic properties but the characteristic of good manufactur-
Optimal design of compositerobot arms
393
ability. In the previous design methodology the upper and lower flanges and side-walls of the hollow-box-section beams are, firstly, fabricated in accordance with the design specifications, and these four pieces are, then, bonded into the desired shape. However, this proposed methodology suggests that these hollow-box-section beams may be manufactured in the process of bag molding or silicone rubber molding after each prepared layer being staked up according to the design specifications [18]. It is evident that the latter provides better strength properties and manufacturability. Because the cross-section area to length ratio of the designed beams is rather large, it is modeled as a Timoshenko-type beam. Furthermore, the mathematical models for the shear and longitudinal Young's moduli of the anisotropic materials are derived and will be employed as components of the stiffness matrix. Finally, the elastodynamic response analyses are performed to illustrate the superior dynamic characteristics which fulfil the design requirements, minimum endpoint deflection during articulating motion and fast settling time upon power stopped.
EXPRESSIONS FOR DAMPING AND FLEXURAL YOUNG'S MODULUS The elastodynamic deflections and the setting time upon completion of a prescribed maneuver of general robotic systems are primarily governed by the flexural rigidity and the damping capacity of the link material and geometrical properties [17]. The objective, herein, is to develop expressions for the damping capacity and also the flexural Young's modulus with respect to the design variables. A laminated hollow-box-section beamlike structural member of a general robotic system shown in Fig. 1 is considered, which is constructed by a number of thin, unidirectional layers of continuous fibers oriented at specified angles relative to the beam's principal longitudinal axis ol. For the considerations of manufacturability, these beams are fabricated using processes such as bag molding or silicone rubber molding, and each layer with the identical material properties is stacked up in accordance with the design specification. The model for material damping in fibrous polymeric composite laminates is developed for hollow-box-section beams subjected to a planar bending moment. Because of the favorable correlation between the theoretical predictions and the experimental results, this model, originated from Ni and Adams [ 19], was selected as a basis for a viable methodology for designing thin solid rectangular beams [16]. This model was further extended for designing hollow-box-section beams subjected to bending moments M 1 and M 3 (Fig. 1). According to this model the specific damping capacity, W, associated with the flexural response of an arbitrary beam subjected to moment M 1 alone may be defined as
V = AW/W.
FIG. 1. Modelfor a laminated hollow-box-sectionbeam.
(1)
394
C.K. SUNGand S. S. SHYL
where A W is the strain energy dissipated during a stress cycle and W is the total strain energy. The energy dissipated per cycle may be divided into three components AW = AW~ + AW~ + AW~.
(2)
These three components may be written as
AW x = 4
;
(f:
)
' 1,6") ,r(i)~(i)dy dz
AW r = 4
(1) (i~ i=1
+
i-I
; (f;
1 ~(i) (y(i)gr(0dy i-i 2vy r
) ] dz
dl
(4)
AW~=4 i=1
i-1
+ f:'-l(fyi'l~b(~i)f~i)g~i)dy)dzldl.
(5)
The first integrals of equations (3)-(5) represent the contribution from the layers of the sidewalls of the hollow box section and the second integrals are contributed by the upper and lower flanges. The total strain energy, W, of the general robot arm due to the flexural deformation as the result of the bending moment M 1 may be written as
W = .fl bZM2 d l,
(6)
where I is the second moment of area associated with the moment M1 and E denotes the effective flexural Young's modulus b
E - ld 11"
(7)
According to [20] the component, dl 1, of the compliance matrix [d~j], which is the inverse of flexural modulus matrix [D~j], may be obtained. Combining equations (1)-(7) the specific damping capacity, ~P, may be determined by = ~x + qJy + qJ~,
(8)
where
~Fx_ AWx
3bd~4 1 { ~,=1¢~)Sl')m~'[YiC~)(z3 - z3-1) + t'm~dxlz3-~]}
(9)
qlr - AWy 4 ( ,=1 ~ ¢(oS(i)m~. [y, Cr( o(z, 3 -z3_~)+ tin~dllZ3_x]} W - 3bdxx
(10)
AW~ 4 { ~ dp],)S(,)m,n.[yiC],)(z~_z3_~)+ 2t, m,n, dllz3 1] } tlJs = - W - = 3bd x-~l i =,
(11)
The expressions for the abbreviations, m~, n~, C~ ), Ctr°, C~i~ and S "~, are written as follows:
C~) = m2dll + n2d21 + minid61 C~i) =
n2dll + m2d21 - minid61
C]i) = 2mi nid 11 - 2mi nid 21 - (m2 - n2 ) d61 S(1)=
Q~i)ld11 + Q~i}2d21+ Q~}6d61
m i = cos eel;
ni = sin ctl.
(12)
Optimal design of compositerobot arms
395
EXPRESSIONS FOR LONGITUDINAL AND SHEAR MODULI Having completed the derivation of damping and effective flexural Young's modulus of a composite laminated hollow-box-section beam subjected to a bending moment, the objective herein is to develop the mathematical expressions for the effective longitudinal and shear moduli resulting from the in-plane stress resultants N 1, N 2 and N 6. These moduli will be employed later to express the elements of the stiffness matrix of the developed equations of motion. Assume that the hollow-box-section beam shown in Fig. 1 is subjected to in-plane stress resultants N:, N2 and N 6. The mathematical expressions for these stress resultants, based on laminate theory, may be written as
N1= 4
a~i) dydz + /=1
t-I
'~
1-1 •
a~)dydz +
N2 = 4
N 6 -=
a~)dydz i-I
4
l-I
tr~Jdydz + i=l
t-I
(13)
a~ ) dy dz •
dydz
,
t-I
where a~°, a~ ) and a~ ~are the resulting stress components of the ith layer along the principal axes. Substituting the stress-strain relationship of Ref. [20] into equation (13), these equations may be rewritten as q
N 1 = 4~, (Q~el + Q[°2e2 + Q~i~.e.6)[yi'(zi- zi-1) + z i - t " ( y i - y i - l ) ] i=1 q
N 2 = 4 E (O (~) ~1 + ~.22to{i}g2 "~ Q(~e.6)[_yi'(zi- zi-1) + zi-1 "(yi - yi-1)]
(14)
i=l q
N6 =
4
E i=l
(Q(6)I/31 q- "~to(i)62° 2 q- Q ~ r e 6 ) [ Y i ' ( z / - zi-1) + z i - , " ( Y i - Y~- 1)].
Since t i = z / - z/_ 1 = Y / - Yi-1, equation (14) may be rewritten in the compact form N 1 -- All,~ 1 + A12/32 nt- A16~6 N2 -- A21,~1 + A22,~2 "1- A26~ 6
(15)
N 6 = A61,~1 -.]-A62/32 nt- A66,g6, where q
Ajk = 4 ~, Qjk" ti'(Yi + zi- 1), J, k = 1, 2, 6. i--1
If the hollow-box-section beam is subjected to a pure tension, N1, the resulting strain, el, along the 1-axis may be expressed by e I =
allN1,
(16)
where a ~ is a compliance component of the inverse of the in-plane modulus matrix l-Ajk]. From the stress-strain relationship, the effective in-plane longitudinal modulus may be derived 1
E1 = aliA,
(17)
where A is the cross-section area. Similar to the derivation of the longitudinal modulus, the effective in-plane shear modulus is defined by 1
- - .
G = a66A
(18)
396
C.K. SONGand S. S. SHYL MATHEMATICAL MODELS FOR OPTIMAL DESIGNS
In this section the mathematical models for the optimal design of the composite arms will be considered and a proper mathematical programming algorithm will also be selected for systematically changing design variables in order to satisfy design objectives. The determination of the design objectives and variables are based on the situation of practical application of the composite robot arms. Two classes of design requirements will be investigated: minimum vibration during articulating motion such as in the processes of welding, deburring and grinding, etc.; and the fast settling time upon completion of a maneuver such as in the processes of drilling and assembly. Based on the above two design requirements, the design task may be structured as two optimization problems:
Design 1 EI pA
maximize - -
S u b j e c t t o kI/rnin ~ train ~. t i <~ tmax
(Vf)min .< V, i .< (Vf)ma x (Xmin ~ Oti ~ 0~max
i = 1, 2 , . . . , q ;
q=constant.
Design 2 maximize qJ
<,EI
Subject to (ff-~) rain
pA
train ~. t i <~ tma x
(V~)m, n .< Vf, .< (Vf)m. x Grain ~ 0~i ~ 0Cmax
i = 1,2 . . . . . q; q=constant, where Vf and ~t are fiber volume fraction and ply angle, respectively. The mathematical models for the two designs are obviously in the nature of nonlinear programming problems. For solving this class of constrained optimization problems, the method of feasible directions is adopted herein. This method consists of step-by-step solutions, where the direction vector and the step size are chosen successively so that a set of design variables in the feasible domain can finally be obtained and the objective function value is minimized ['21]. This class of problems in the optimally tailored design of composite robot arms is a nonconvex programming problem, hence this optimization method will only converge to a local minimum. It is therefore evident that several different initial points must be chosen in order that an approximate global solution may be determined. After the optimal material properties and the geometrical cross-sections have been selected, these characteristics are then embodied in a finite element formulation for analysing the elastodynamic response of an assemblage of articulating interconnected flexible bodies. EQUATIONS OF MOTION Similar to the work of [17], the elastodynamic responses of a robot manipulator fabricated in optimally tailored composite laminates and subjected to prescribed maneuvers will be analysed here. The hollow-box-section beams of the robot manipulator are modeled by the Timoshenko beam theory rather than the BernouUi-Euler assumption, and the isoparametric beam element is employed in the finite element formulation. This formulation is based on a variational theorem for an assemblage of interconnected articulating flexible bodies whose constitutive relationship is modeled as a simple linear Kelvin solid [22]. The
Optimal design of composite robot arms
397
equations of motion for this linear flexible robot system may be written as [M] {L~) + ([Ccor'] -Jv [C]){~} - t - ( [ g l ] Jl- [K]){q} = - [M,]{/~r} + f~, [ N ] t { 0 } d s +
foeN]'{X)dV.
(19)
ILLUSTRATIVE EXAMPLES
Having completed the design and the mathematical model for analysing the elastodynamic responses of an optimally tailored composite robot manipulator, two examples are presented, herein, for illustrating the superior dynamic characteristics which fulfil the design requirements, viz. the minimum deflection during articulating motion and the fast settling time after the power has been stopped. A robot manipulator consisting of two flexible links fabricated with optimally designed composite laminates and two perfect revolutes is shown schematically in Fig. 2. The perfect revolute joint is defined as the revolute joint with no flexibility, no clearance, and hence no damping and it may not exist in reality. For the purpose of demonstrating the advantages of this material, the assumption of a perfect joint is therefore made to avoid complexity in the analysis. Assume that these two links are identical in length, 500 mm long, and the patterns for the prescribed maneuver of the links 1 and 2 are expressed by the relationship between angular accelerations and time (Figs 3, 4). These two examples will be studied in the following: Example 1 A solution was sought in which the minimum endpoint vibration during articulating motion is required. The data sets which represent the constraint conditions for the design variables were tmi, = 0.5 m m (~f)min = 0.5 ~min
=
--
tmax = 0.6 mm (~f)max = 0.6
~/2
Ctmax = r~/2
and q= 9
hi = 20 mm
b 1 = 30 mm,
where h~ and b 1 are the prescribed internal dimensions of the hollow-box-section beams. For this design problem two robots with different arm geometries and fabricated with different composite laminate configurations were modeled and their responses simulated in order to illustrate the proposed methodology. Both robots were assumed to be fabricated in a graphite/epoxy material, and the properties of the principal constituents were obtained
~
Jon i2t
FIG. 2. A two-link robot manipulator.
NS 32:5-C
C. K. SUNG and S. S. SHYL
398 40 30 ~'
20 10
~
o
i
-10
~
-20 -30 -40 0
0.1
0.2
0.3
0.4
Time (sec)
FIG. 3. The pattern for the prescribed maneuver of the link 1.
20
15 10
g
5 0
0
-5 -10
-15
I
-2(
0.1
0.2
0.3
0.4
Time (sec)
FIG. 4. The pattern for the prescribed maneuver of the link 2.
from [20] as Ef = 230 GPa
E m = 3.45 G P a
Gf = 8.27 G P a
G m = 1.8 G P a
vf =
vm =
0.2
pf = 1.75
gm/cm a
0.35
Pm = 1.2
gm/cm 3.
The effective elastic moduli of the composite laminates were then computed using laminate theory. The damping capacities for the fiber and matrix were abstracted from [ 19], and then the damping capacities of the composite laminate were obtained. Prior to obtaining the optimal solution of the nonlinear programming problem by the method of feasible direction, the constraint conditions of the damping capacity were firstly normalized and the criteria of convergence was also prescribed by the value of 0.0001. By choosing several different initial conditions an approximate solution for this nonlinear programming problem was determined and shown in Table 1. It is noted that the thickness of each layer approaches the upper limit upon the completion of the optimization process. The reason is that the objective function is significantly influenced by the increase of stiffness rather than that of mass while the thickness of each layer was increasing.
399
O p t i m a l design of c o m p o s i t e r o b o t a r m s TABLE 1. DESIGN VARIABLESOF AN APPROXIMATE OPTIMAL SOLUTION FOR DESIGN 1
No.
(i)
S t a c k i n g sequence
Thickness
[- -I- 0]9
(ii)
[ + 03/-
(iii)
Optimal solution Fiber v o l u m e Ply-angle fraction
0 3 / + 03]9
[ + 0/- 0/+ 0/- 0/ + 0 / - 0 / + 0 / - 0 / + 0]9
EI/pA
tx= t2 = ta = t4 = ts = t6 = t7 = ts = t9 =
0.599426 0.598484 0.60000 0.59900 0.59900 0.59900 0.59908 0.59892 0.60000
~q = ~t2 = ~t3 = ~t4 = ~ts = ~ts = ~t7 = % = % =
0.258106 0.257671 0.257191 0.256663 0.256084 0.254554 0.254769 0.255027 0.254251
Vfl = 0.501095 Vf2 = 0.501301 Vt3 = 0.501531 Vf4 = 0.501785 Vf5 = 0.502065 Vf6 = 0.502371 Vt7 = 0.502705 Vfs = 0.503068 Vf9 = 0.503461
3.52363E + 11
tI = t2 = ta = t4 = t5 = t6 = t7 = ts = t9 =
0.594851 0.598299 0.60000 0.60000 0.60000 0.60000 0.59239 0.60000 0.60000
al = at2 = % = ~t4 = % = % = at7 = % = ~t9 =
0.517209 0.516668 0.516069 - 0.512419 - 0.511760 - 0.512570 0.513054 0.512131 0.511134
Vfl Vf2 Vfa Vt4 V~s Vf6 Vf7 Vf8 Vt9
= = = = = = = = =
0.539013 0.539116 0.539567 0.541386 0.541957 0.542580 0.540718 0.541182 0.542031
3.52469E + 11
t1= t2 = t3 = t4 = t5 = t6 = t~ = ts = t9 =
0.595657 0.60000 0.60000 0.60000 0.60000 0.60000 0.60000 0.60000 0.59913
~t1 = % = ~t3 = ~, = % = % = ~ = % = ~t9 =
0.517830 - 0.516876 0.516902 - 0.515682 0.515779 - 0.514251 0.51A.A.A.A. - 0.512562 0.512879
Vt~ Vf2 Vf3 Vf4 Vf5 Vt6 Vt7 Vf8 Vf9
= = = = = = = = --
544108 0.544108 0.544546 0.545113 0.545082 0.543888 0.541926 0.540886 0.544594
3.55785E + 11
TABLE 2. LAMINATE CHARACTERISTICS. ARM GEOMETRIES. AND ARM PROPERTIES. FOR INITIAL AND OPTIMAL ROBOTIC DESIGNS OF DESIGN 1 Parameter
Initial design
s. D . C . Ell ( N / m m 2) p ( N / m m 3) I (mm 4) A (ram 2) E A ( N / m m 2) G ( N / r a m 2) Ell I/pA (ram a) Vn cti (rad)
0.04953 9226.6 1.446E - 5 59264.25 531 9226.6 4834.82 7.124E + 10 0.5 i = 1 ~ 9 1.22173 i = 1~ 9 0.5 i = 1 ~ 9
ti (mm)
By substituting
the values
(18), the mechanical shown
in Table
arbitrarily assumed than In
2. T h e
selected.
order
variables
of the optimally
initial
design
In the case that
to be identical,
that
of design
properties
O p t i m a l design
the value of
in Table
0.04953 44822.14 1.469E - 5 79215.4 655.9 44797.33 25208.42 3.558E + 11 see T a b l e l see T a b l e 1 see T a b l e 1
in Table tailored
2 into equations
composite
2 assumes
the specific damping
EI/pA
of the optimal
that
arms
the design
capacities design
(7), (8), ( 1 7 ) a n d
were obtained
for both
is about
and
variables
were
designs
are
five times greater
of the initial design. to
optimal-designed
compare robot
the
elastodynamic
manipulators,
responses
each manipulator
performed
by
was assumed
the
initial-
and
to be subjected
the
to the
400
C.K.
SUNG a n d S. S. SHYL
same kinematic motion while the end-effector was carrying a 10 kg payload. The results of the finite-element simulations are presented in Fig. 5 for the dynamic responses of the endeffectors of the two robotic designs. As shown in Figs 3 and 4 the power inputs for the prescribed maneuvers were stopped at 0.3 s and the phase of settling motion starts, the optimal design clearly provides better dynamic response.
6 5 4
/ \ [ ~ I ~
3 2
! I
0
~ .....
Optimot design Initiot. design
~'-/I
0.2
0.6
0.4
Time (sec) FIG. 5. D y n a m i c r e s p o n s e s a t t h e r o b o t e n d - e f f e c t o r for t h e i n i t i a l d e s i g n a n d t h e o p t i m a l d e s i g n o f d e s i g n 1.
TABLE 3. DESIGN VARIABLES OF AN APPROXIMATE OPTIMAL SOLUTION FOR DESIGN 2 Optimal solution Fiber volume No.
Stacking sequence
(i)
[+8]9
(ii)
(iii)
t 1 = 0.509526
[ - I - 0 3 / -- 8 3 / - I - 0 3 ] 9
[ + Ol
Thickness
-
81 + 81 - 8 / + 8 /
/-8/+ 8/-0/+819
Ply-angle
fraction
S.D.C.
al = 0.368666
t2 = 0 . 5 2 0 0 0 4
~2 = 0 . 3 6 8 6 6 6
t 3 = 0.53
a3 = 0 . 3 6 8 6 6 6
t4 = 0 . 5 3
~4 = 0 . 3 6 8 6 6 6
Vfl = 0.5
t s = 0.53
~5 = 0 . 3 6 8 6 6 6
i=1~9
t6 = 0.53
~6 = 0 . 3 6 9 0 8 0
t 7 = 0.53
~7 = 0 . 3 6 9 5 4 0
t s = 0.53
~8 = 0 . 3 6 9 5 4 0
t9=0.53087
~9 = 0 . 3 6 9 5 4 0
t I = 0.568796
~1 = 0 . 6 1 0 2 6 9
t2 = 0.57
~z = 0 . 6 1 0 2 6 9
t 3 = 0.57
~3 = 0 . 6 1 0 2 6 9
t 4 = 0.57
~4 =
- 0.610269
Vn = 0.5
t 5 = 0.57
~s =
- 0.610269
i=1~9
t 6 = 0.5724
~6 =
- 0.610269
t 7 = 0.5676
~7 = 0 . 6 1 0 2 6 9
t 8 = 0.57
~8 = 0 . 6 1 0 2 6 9
t 9 = 0.57
~9 = 0 . 6 1 0 2 6 9
tl = 0.557043
~1 = 0 . 6 1 4 0 3 9
t 2 = 0.595917
~2 =
t3 = 0 . 5 4 4 0 8 0
~a = 0 . 6 1 4 0 3 9
t4 = 0.595920
~4 =
t 5 = 0.544080
as = 0 . 6 1 4 0 3 9
0.0586846
0.0625734
-0.623822 -0.623822
t6 = 0.595920
~6 =
t 7 = 0.544080
~7 = 0 . 6 1 4 0 3 9
- 0.623822
t8 = 0 . 5 9 5 9 2
~8 =
t 9 = 0.553870
~9 = 0 . 6 1 6 0 6 6
-0.623822
Vfi = 0.5 i=1~9
0.0628792
Optimal design of composite robot arms
401
Example 2 A solution was sought in which a fast settling time upon completion of prescribed maneuvers is required. The data sets for the constraint conditions of the design variables were tmin = 0.5 mm (Vf)mi
n =
0.5
0~mi n =
--
tm.~ = 0.6 mm (Vf)max =
re/2
0Cma x =
0.6
n/2
EI/pA >10.2062 E + 12 and q= 9
ha = 20 mm
bl = 30 mm.
The value of EI/pA was constrained to prevent high flexibility, and both robots were assu/ned to be fabricated with graphite/epoxy composite laminates. Similar to the previously mentioned procedures, the design variables for the design 2 problem were determined (Fig. 3). Then the laminate characteristics, arm geometries, and arm properties for the initial and optimal designs were computed and shown in Table 4. From Table 3 it is noted that the fiber volume fraction approaches the lower bound because the specific damping capacity is relatively sensitive to the matrix content Table 4 shows that
TABLE 4. LAMINATE CHARACTERISTICS, ARM GEOMETRIES, AND ARM PROPERTIES, F O R I N I T I A L A N D O P T I M A L R O B O T I C D E S I G N S O F D E S I G N 2 Parameter
Initial design
S. D . C . E n ( N / m m 2) p ( N / m m 3) I (mm 4) A (mm 2) E A ( N / m m 2) G ( N / m m 2) E n I/pA(mm s) Vn cti (rad) ti (mm)
1.0045 116725 1.446E 10328.99 404.532 116725 3702.3 2.062E 0.5, i 0.0, i 0.5, i
Optimal design
- 5
+ = = =
11 1~ 9 1~ 9 1~ 9
0.06288 25261.5 1.4455E - 5 72887.632 617.82 25276.5 27482.95 2.062E + 11 see Table 3~ see Table 3 see Table 3
3 Optimot
g
o
design
11 | 1 ~ I ! I ~.l.q il.t.. I I I 11 1! . I ! I.| ! l -i t q
O
0.2
0.4
0.6
0.8
1.0
1.2
,
1.4
Time (sec)
FIG. 6. Dynamic responses at the robot end-effector for the initial design and the optimal design of design 2.
402
c.K. SUNGand S. S. SHVL
the specific d a m p i n g c a p a c i t y of the o p t i m a l design is 14 times greater t h a n t h a t of initial design. The d y n a m i c responses c o m p u t e d by the finite-element simulations are presented in Fig. 6. The settling phase starts at 0.3 s after the a r t i c u l a t i n g m o t i o n - s t o p p e d . Because of space restrictions only p o r t i o n s of this response are presented in the figure. These simulations clearly d e m o n s t r a t e the r e d u c e d settling-time a n d s u p e r i o r response exhibited b y the r o b o t with the o p t i m a l link member. CONCLUSIONS A design m e t h o d o l o g y has been presented for synthesizing h i g h - p e r f o r m a n c e articulating r o b o t i c systems fabricated with o p t i m a l l y tailored c o m p o s i t e laminates. W i t h this m e t h o d o logy, a r t i c u l a t i n g m e m b e r s can be m a n u f a c t u r e d using the processes of b a g m o l d i n g or silicone r u b b e r m o l d i n g which m a y e n h a n c e the failure-resistanc¢ characteristics. By o p t i m a l l y specifying the types of fiber, matrix, stacking sequence, fiber volume fraction, fiber layup, a n d the cross-section geometries, the designer is able to synthesize the structural links of the r o b o t i c linkage so as to possess high stiffness, high strength, high d a m p i n g a n d low mass. This a p p r o a c h was illustrated in a s t u d y of a two-link r o b o t m a n i p u l a t o r to d e m o n s t r a t e the s u p e r i o r characteristics u n d e r two o p e r a t i o n a l conditions: m i n i m u m deflection d u r i n g a r t i c u l a t i n g m o t i o n a n d fast settling time after the p o w e r has been stopped. Acknowledgement--This study was funded by the National Science Council of Republic of China, under Grant
Number NSC77-0611-E007-03R. This support is gratefully acknowledged. REFERENCES 1. K. SADAMATO,R&D on intelligent robots being spurted. Jap. Robot News 1(4), 1 (1982). 2. A. ZALUSand D. E. HARDT,Active control of robot structure deflections. In: Robotics Research and Advanced Applications (edited by W. J. BOOK). ASME, H00236 (1982). 3. W. J. BOOK and M. MAJETT, Controller design for flexible, distributed parameter mechanical arms via combined state space and frequency domain techniqueg. In: Robotics Research and Advanced Applications (edited by W. J. BOOK). ASME, H00236 (1982). 4. W.J. BOOK,Characterization of strength and stiffness constraints on manipulator control. In: Proc. I F T o M M Conf. Theory and Practice of Robots and Manipulators (edited by A. MORECKIand D. DEDZIOR),Warsaw, Poland (1976). 5. W. J. BOOK, O. MAIZZA-NETOand D. D. WHITNEY,Feedback control of two beam, two joint system with distributed flexibility. ASME J. Dynamic Syst. Control 97, 424 (1975). 6. F, A. KELLYand R. L. HUSTON,Modeling of flexibility effects in robot arms. In: Proc. 1981 Joint Automotive Control Conference, Charlottesville, VA (1981). 7. W. H. SUNADAand S. DUaOWSKY,On the dynamic analysis and behavior of industrial robotic manipulators with elastic members. ASME J. Mech. Transm. Aut. Design 105, 42 (1983). 8. G. NAGANATHANand A. H. SONI, Coupling effects of kinematics and flexibility in manipulators. Int. J. Robotics Res. 6(1), 75 (1987). 9. G. G. LOWENand C. CHASSAPIS,The elastic behavior of linkages: an update. Mech. Much. Theory 21(1), 33 (1986). 10. W. L. CLEGHORN,R. C. FENTONand B. TABARROK,Optimum design of high-speed flexible mechanisms. Mech. Much. Theory 16, 339 (1981). 11. A. J. KAKATSIOSand S. J. TRICAMO,Integrated kinematic and dynamic optimal design of flexible planar mechanisms. ASME J. Mech. Transm. Aut. Design 109, 338 (1987). 12. B. S. THOMPSON,Composite laminate components for robotic and machine systems. ASME Appl. Mech. Rev. 40(11), 145 (1987). 13. C. K. SUNG,B. S. THOMPSON,P. CROWLEYand J. Cocoo, An experimental study to demonstrate the superior response characteristics mechanisms constructed with composite laminates. Mech. Much. Theory 21(2), 103 (1986). 14. S. ADALI,Multiobjective design of an antisymmetric angle-ply laminate by nonlinear programming. ASME J. Mech. Transm. Aut. Design 105, 214 (1983). 15. T. R. TAUCHERTand S. ADIBHATLA,Design of laminated plates for maximum stiffness.J. Compos. Mater. 18, 58 (1984). 16. D. X. LIAO,C. K. SUNGand B. S. THOMPSON,The optimal design of symmetric laminated beams considering damping. J. Compos. Mater. 21(5), 485 (1986). 17. C. K. SUNGand B. S. THOMPSON,A methodology for synthesizing high-performance robots fabricated with optimally-tailored composite laminates. ASME J. Mech. Transm. Aut. Design 1090), 74 (1987). 18. M. M. SCHWARTZ,Composite Materials Handbook. McGraw-Hill, New York (1984).
Optimal design of composite robot arms
403
19. R.G. NI and R. D. ADAMS,A rational method for obtaining the dynamic mechanical properties of laminate for predicting the stiffness and damping of laminated plates and beams. Composites 15(3), 193 (1984). 20. S. W. TSAI and H. T. HAHN, Introduction to Composite Materials. Technomic Publishing Co., Westport, CT (1980). 21. U. KmSCH, Optimum Structure Design. McGraw-Hill, New York (1981). 22. B. S. THOMPSONand C. K. SUNG,A variational formulation for the nonlinear finite-element analysis of flexible linkages: theory, implementation and experiment~il results. ASME J. Mech. Transm. Aut. Design 1116(4), 482 (1984).
0020-7403/90 $3.00 + .00 © 1990 Pergamon Press pie
Printed in Great Britain.
A COMPOSITE LAMINATED BOX-SECTION BEAM DESIGN FOR OBTAINING OPTIMAL ELASTODYNAMIC RESPONSES OF A FLEXIBLE ROBOT MANIPULATOR C. K. SUNGand S. S. SHYL Department of Power Mechanical Engineering, National Tsing Hua University, Hsinchu, Taiwan 30043, Republic of China
(Received 31 July 1989) Abstract--A design methodology is presented for synthesizing a high-performance articulating robot manipulator fabricated with optimally tailored composite laminates. By optimally specifying the geometric configurations, the types of fiber, matrix, stacking sequence, fiber volume fraction, fiber layups, etc., the synthesized composite material may possess such superior characteristics as high damping, high stiffness, high strenigth and low mass. In accordance with the design requirements, e.g. minimum deflection during articulating motion or fast settling time after the power has been stopped, the design objectives and constraint conditions were specified. As an illustrative example, a two-link robot manipulator fabricated with the aforementioned composite laminates is employed for demonstrating the proposed design methodology.
NOTATION coordinate system oxyz: x-axis along the fiber coordinate system 0123: 1-axis along the principle longitudinal axis of the beam aij i,j = I, 2, 6; component of the in-plane compliance matrix [ao] Aj~ j, k = 1, 2, 6, component of the in-plane modulus matrix [A~] A the cross-section area of the beam b the width of the beam bl the prescribed internal width of the beam [c] the global linear damping matrix [Ccor] equivalent damping matrix associated with Coriolis acceleration d o i,j = 1, 2, 6; component of the compliance matrix [do] D~j i,j = 1, 2, 6; component of the flexural modulus matrix [D~] E the effective flexural Young's modulus El the effective in-plane longitudinal Young's modulus Ef elastic Young's modulus of the fiber Em elastic Young's modulus of the matrix {~} prescribed surface traction imposed on area St G effective in-plane shear modulus Gt shear modulus of the fiber Gm shear modulus of the matrix h the height of the beam h, the prescribed internal height of the beam I the second moment of inertia associated with the moment M~ [K] global linear stiffness matrix [gt] the inertia coupling matrix linking the deformation and rigid-body fields 1 the length of the beam [M] the global mass matrix [M,] the global mass matrix coupling the shape functions defining the discretized rigid-body kinematic motion and the elastic deformations M 1 bending moment along the 1-axis M3 bending moment along the 3-axis IN] the row vector containing the shape function Ni i = 1, 2, 6; the in-plane shear resultants [~,] the vector containing the discretized rigid-body acceleration q total number of layers {q} the global deformation displacement vector Qtil ik j, k = 1, 2, 6; component of the flexural moduli of the ith layer ti the thickness of the ith layer It,, the fiber volume fraction of the ith layer w total strain energy 391
392 AW AWi {X} yi z~ gj p pf Pm vt Vm q~ ~P~ ~P ai, e~ #~, ~
C.K. SUNGand S.S. SHYL total strain energy dissipation i = x, y, s; the strain energy dissipated per cyclealong each axis the body force vector imposed on volume V the horizontal distance from the 3-axis to the ith layer the vertical distance from the 2-axis to the ith layer the ply angle of the ith layer density density of the fiber density of the matrix longitudinal Poisson's ratio of fiber longitudinalPoisson's ratio of matrix i = x, y, s; specificdamping capacity component of each layer i = x, y, s; component of the total specificclamping capacity total specificdamping capacity i = x, y, s; stress and strain componentsof the ith layer of the upper and lowerflange'salong each axis i = x, y, s; stress and strain componentsof the ith layer of the side walls along each axis INTRODUCTION
The intense competition in the international marketplace for robots and machine systems which significantly enhance manufacturing productivity by operating at high speeds with more accurate endpoint positioning has resulted in the evolution of a new frontier in machine design. These objectives cannot be achieved with existing designs because the current articulating members possess large moments of inertia, high stiffness and also substantial weight, in an attempt to provide an acceptable degree of repeatability and endpoint accuracy with payload-weight to arm-weight ratios of only about 1:20 [-1]. These parameters significantly constrain the response of these devices. For example, the beam deflections are primarily due to the weight of the mechanical linkage itself, in addition, the large mass, and hence inertia, of the moving members of the robot naturally inhibit the ability of the device to respond quickly. Hence, if higher operating speeds are to be achieved, a lighter mechanism is desirable provided structural stiffness is not sacrificed. The design goals of high-speed operation and improved end-effector positioning have received considerable attention in the literature [2-8], with the emphasis being upon the manipulator control system, the elastodynamic characteristics of the linkage and also the interaction between these diverse design areas. A variety of techniques have been proposed. The concepts of designing mechanical components possessing high strength, high stiffness and low mass in order to improve the performance of manipulators have been employed in the design of high-speed linkages by optimally designing the geometries of the beam crosssections [9-11]. An alternative design philosophy has been proposed for the design of high-speed linkages and robot arms, which requires the members to be fabricated in modern composite materials I-12, 13]. As is well known, these materials possess much higher strength-to-weight and stiffness-to-weight ratios than the commercial metals and consequently they promise manipulator designs with much smaller deflections at the end-effector and also higher speeds of operation using existing actuators. In addition, modern fibrous composite laminates have a larger number of design variables which must be carefully selected in order to achieve the desired material properties. These variables typically include the properties of both the fibers and matrices, the stacking sequence, the fiber layups, the fiber volume fraction and the ply thicknesses. Literature regarding the optimal tailoring of composite laminates are included in Refs 1-14-16]. However these publications generally focus on strength or stiffness characteristics and do not address dynamical problems. Furthermore they do not address the optimal synthesis of members specifically tailored for robotic applications in which optimal damping and stiffness characteristics are required, subject to a variety of practical constraints. Based on the aforementioned design variables and geometries, Sung and Thompson [17] proposed a design methodology for constructing a robot manipulator which possesses superior dynamic characteristics. This paper proposes an alternative methodology for designing a robot arm which possesses not only superior dynamic properties but the characteristic of good manufactur-
Optimal design of compositerobot arms
393
ability. In the previous design methodology the upper and lower flanges and side-walls of the hollow-box-section beams are, firstly, fabricated in accordance with the design specifications, and these four pieces are, then, bonded into the desired shape. However, this proposed methodology suggests that these hollow-box-section beams may be manufactured in the process of bag molding or silicone rubber molding after each prepared layer being staked up according to the design specifications [18]. It is evident that the latter provides better strength properties and manufacturability. Because the cross-section area to length ratio of the designed beams is rather large, it is modeled as a Timoshenko-type beam. Furthermore, the mathematical models for the shear and longitudinal Young's moduli of the anisotropic materials are derived and will be employed as components of the stiffness matrix. Finally, the elastodynamic response analyses are performed to illustrate the superior dynamic characteristics which fulfil the design requirements, minimum endpoint deflection during articulating motion and fast settling time upon power stopped.
EXPRESSIONS FOR DAMPING AND FLEXURAL YOUNG'S MODULUS The elastodynamic deflections and the setting time upon completion of a prescribed maneuver of general robotic systems are primarily governed by the flexural rigidity and the damping capacity of the link material and geometrical properties [17]. The objective, herein, is to develop expressions for the damping capacity and also the flexural Young's modulus with respect to the design variables. A laminated hollow-box-section beamlike structural member of a general robotic system shown in Fig. 1 is considered, which is constructed by a number of thin, unidirectional layers of continuous fibers oriented at specified angles relative to the beam's principal longitudinal axis ol. For the considerations of manufacturability, these beams are fabricated using processes such as bag molding or silicone rubber molding, and each layer with the identical material properties is stacked up in accordance with the design specification. The model for material damping in fibrous polymeric composite laminates is developed for hollow-box-section beams subjected to a planar bending moment. Because of the favorable correlation between the theoretical predictions and the experimental results, this model, originated from Ni and Adams [ 19], was selected as a basis for a viable methodology for designing thin solid rectangular beams [16]. This model was further extended for designing hollow-box-section beams subjected to bending moments M 1 and M 3 (Fig. 1). According to this model the specific damping capacity, W, associated with the flexural response of an arbitrary beam subjected to moment M 1 alone may be defined as
V = AW/W.
FIG. 1. Modelfor a laminated hollow-box-sectionbeam.
(1)
394
C.K. SUNGand S. S. SHYL
where A W is the strain energy dissipated during a stress cycle and W is the total strain energy. The energy dissipated per cycle may be divided into three components AW = AW~ + AW~ + AW~.
(2)
These three components may be written as
AW x = 4
;
(f:
)
' 1,6") ,r(i)~(i)dy dz
AW r = 4
(1) (i~ i=1
+
i-I
; (f;
1 ~(i) (y(i)gr(0dy i-i 2vy r
) ] dz
dl
(4)
AW~=4 i=1
i-1
+ f:'-l(fyi'l~b(~i)f~i)g~i)dy)dzldl.
(5)
The first integrals of equations (3)-(5) represent the contribution from the layers of the sidewalls of the hollow box section and the second integrals are contributed by the upper and lower flanges. The total strain energy, W, of the general robot arm due to the flexural deformation as the result of the bending moment M 1 may be written as
W = .fl bZM2 d l,
(6)
where I is the second moment of area associated with the moment M1 and E denotes the effective flexural Young's modulus b
E - ld 11"
(7)
According to [20] the component, dl 1, of the compliance matrix [d~j], which is the inverse of flexural modulus matrix [D~j], may be obtained. Combining equations (1)-(7) the specific damping capacity, ~P, may be determined by = ~x + qJy + qJ~,
(8)
where
~Fx_ AWx
3bd~4 1 { ~,=1¢~)Sl')m~'[YiC~)(z3 - z3-1) + t'm~dxlz3-~]}
(9)
qlr - AWy 4 ( ,=1 ~ ¢(oS(i)m~. [y, Cr( o(z, 3 -z3_~)+ tin~dllZ3_x]} W - 3bdxx
(10)
AW~ 4 { ~ dp],)S(,)m,n.[yiC],)(z~_z3_~)+ 2t, m,n, dllz3 1] } tlJs = - W - = 3bd x-~l i =,
(11)
The expressions for the abbreviations, m~, n~, C~ ), Ctr°, C~i~ and S "~, are written as follows:
C~) = m2dll + n2d21 + minid61 C~i) =
n2dll + m2d21 - minid61
C]i) = 2mi nid 11 - 2mi nid 21 - (m2 - n2 ) d61 S(1)=
Q~i)ld11 + Q~i}2d21+ Q~}6d61
m i = cos eel;
ni = sin ctl.
(12)
Optimal design of compositerobot arms
395
EXPRESSIONS FOR LONGITUDINAL AND SHEAR MODULI Having completed the derivation of damping and effective flexural Young's modulus of a composite laminated hollow-box-section beam subjected to a bending moment, the objective herein is to develop the mathematical expressions for the effective longitudinal and shear moduli resulting from the in-plane stress resultants N 1, N 2 and N 6. These moduli will be employed later to express the elements of the stiffness matrix of the developed equations of motion. Assume that the hollow-box-section beam shown in Fig. 1 is subjected to in-plane stress resultants N:, N2 and N 6. The mathematical expressions for these stress resultants, based on laminate theory, may be written as
N1= 4
a~i) dydz + /=1
t-I
'~
1-1 •
a~)dydz +
N2 = 4
N 6 -=
a~)dydz i-I
4
l-I
tr~Jdydz + i=l
t-I
(13)
a~ ) dy dz •
dydz
,
t-I
where a~°, a~ ) and a~ ~are the resulting stress components of the ith layer along the principal axes. Substituting the stress-strain relationship of Ref. [20] into equation (13), these equations may be rewritten as q
N 1 = 4~, (Q~el + Q[°2e2 + Q~i~.e.6)[yi'(zi- zi-1) + z i - t " ( y i - y i - l ) ] i=1 q
N 2 = 4 E (O (~) ~1 + ~.22to{i}g2 "~ Q(~e.6)[_yi'(zi- zi-1) + zi-1 "(yi - yi-1)]
(14)
i=l q
N6 =
4
E i=l
(Q(6)I/31 q- "~to(i)62° 2 q- Q ~ r e 6 ) [ Y i ' ( z / - zi-1) + z i - , " ( Y i - Y~- 1)].
Since t i = z / - z/_ 1 = Y / - Yi-1, equation (14) may be rewritten in the compact form N 1 -- All,~ 1 + A12/32 nt- A16~6 N2 -- A21,~1 + A22,~2 "1- A26~ 6
(15)
N 6 = A61,~1 -.]-A62/32 nt- A66,g6, where q
Ajk = 4 ~, Qjk" ti'(Yi + zi- 1), J, k = 1, 2, 6. i--1
If the hollow-box-section beam is subjected to a pure tension, N1, the resulting strain, el, along the 1-axis may be expressed by e I =
allN1,
(16)
where a ~ is a compliance component of the inverse of the in-plane modulus matrix l-Ajk]. From the stress-strain relationship, the effective in-plane longitudinal modulus may be derived 1
E1 = aliA,
(17)
where A is the cross-section area. Similar to the derivation of the longitudinal modulus, the effective in-plane shear modulus is defined by 1
- - .
G = a66A
(18)
396
C.K. SONGand S. S. SHYL MATHEMATICAL MODELS FOR OPTIMAL DESIGNS
In this section the mathematical models for the optimal design of the composite arms will be considered and a proper mathematical programming algorithm will also be selected for systematically changing design variables in order to satisfy design objectives. The determination of the design objectives and variables are based on the situation of practical application of the composite robot arms. Two classes of design requirements will be investigated: minimum vibration during articulating motion such as in the processes of welding, deburring and grinding, etc.; and the fast settling time upon completion of a maneuver such as in the processes of drilling and assembly. Based on the above two design requirements, the design task may be structured as two optimization problems:
Design 1 EI pA
maximize - -
S u b j e c t t o kI/rnin ~ train ~. t i <~ tmax
(Vf)min .< V, i .< (Vf)ma x (Xmin ~ Oti ~ 0~max
i = 1, 2 , . . . , q ;
q=constant.
Design 2 maximize qJ
<,EI
Subject to (ff-~) rain
pA
train ~. t i <~ tma x
(V~)m, n .< Vf, .< (Vf)m. x Grain ~ 0~i ~ 0Cmax
i = 1,2 . . . . . q; q=constant, where Vf and ~t are fiber volume fraction and ply angle, respectively. The mathematical models for the two designs are obviously in the nature of nonlinear programming problems. For solving this class of constrained optimization problems, the method of feasible directions is adopted herein. This method consists of step-by-step solutions, where the direction vector and the step size are chosen successively so that a set of design variables in the feasible domain can finally be obtained and the objective function value is minimized ['21]. This class of problems in the optimally tailored design of composite robot arms is a nonconvex programming problem, hence this optimization method will only converge to a local minimum. It is therefore evident that several different initial points must be chosen in order that an approximate global solution may be determined. After the optimal material properties and the geometrical cross-sections have been selected, these characteristics are then embodied in a finite element formulation for analysing the elastodynamic response of an assemblage of articulating interconnected flexible bodies. EQUATIONS OF MOTION Similar to the work of [17], the elastodynamic responses of a robot manipulator fabricated in optimally tailored composite laminates and subjected to prescribed maneuvers will be analysed here. The hollow-box-section beams of the robot manipulator are modeled by the Timoshenko beam theory rather than the BernouUi-Euler assumption, and the isoparametric beam element is employed in the finite element formulation. This formulation is based on a variational theorem for an assemblage of interconnected articulating flexible bodies whose constitutive relationship is modeled as a simple linear Kelvin solid [22]. The
Optimal design of composite robot arms
397
equations of motion for this linear flexible robot system may be written as [M] {L~) + ([Ccor'] -Jv [C]){~} - t - ( [ g l ] Jl- [K]){q} = - [M,]{/~r} + f~, [ N ] t { 0 } d s +
foeN]'{X)dV.
(19)
ILLUSTRATIVE EXAMPLES
Having completed the design and the mathematical model for analysing the elastodynamic responses of an optimally tailored composite robot manipulator, two examples are presented, herein, for illustrating the superior dynamic characteristics which fulfil the design requirements, viz. the minimum deflection during articulating motion and the fast settling time after the power has been stopped. A robot manipulator consisting of two flexible links fabricated with optimally designed composite laminates and two perfect revolutes is shown schematically in Fig. 2. The perfect revolute joint is defined as the revolute joint with no flexibility, no clearance, and hence no damping and it may not exist in reality. For the purpose of demonstrating the advantages of this material, the assumption of a perfect joint is therefore made to avoid complexity in the analysis. Assume that these two links are identical in length, 500 mm long, and the patterns for the prescribed maneuver of the links 1 and 2 are expressed by the relationship between angular accelerations and time (Figs 3, 4). These two examples will be studied in the following: Example 1 A solution was sought in which the minimum endpoint vibration during articulating motion is required. The data sets which represent the constraint conditions for the design variables were tmi, = 0.5 m m (~f)min = 0.5 ~min
=
--
tmax = 0.6 mm (~f)max = 0.6
~/2
Ctmax = r~/2
and q= 9
hi = 20 mm
b 1 = 30 mm,
where h~ and b 1 are the prescribed internal dimensions of the hollow-box-section beams. For this design problem two robots with different arm geometries and fabricated with different composite laminate configurations were modeled and their responses simulated in order to illustrate the proposed methodology. Both robots were assumed to be fabricated in a graphite/epoxy material, and the properties of the principal constituents were obtained
~
Jon i2t
FIG. 2. A two-link robot manipulator.
NS 32:5-C
C. K. SUNG and S. S. SHYL
398 40 30 ~'
20 10
~
o
i
-10
~
-20 -30 -40 0
0.1
0.2
0.3
0.4
Time (sec)
FIG. 3. The pattern for the prescribed maneuver of the link 1.
20
15 10
g
5 0
0
-5 -10
-15
I
-2(
0.1
0.2
0.3
0.4
Time (sec)
FIG. 4. The pattern for the prescribed maneuver of the link 2.
from [20] as Ef = 230 GPa
E m = 3.45 G P a
Gf = 8.27 G P a
G m = 1.8 G P a
vf =
vm =
0.2
pf = 1.75
gm/cm a
0.35
Pm = 1.2
gm/cm 3.
The effective elastic moduli of the composite laminates were then computed using laminate theory. The damping capacities for the fiber and matrix were abstracted from [ 19], and then the damping capacities of the composite laminate were obtained. Prior to obtaining the optimal solution of the nonlinear programming problem by the method of feasible direction, the constraint conditions of the damping capacity were firstly normalized and the criteria of convergence was also prescribed by the value of 0.0001. By choosing several different initial conditions an approximate solution for this nonlinear programming problem was determined and shown in Table 1. It is noted that the thickness of each layer approaches the upper limit upon the completion of the optimization process. The reason is that the objective function is significantly influenced by the increase of stiffness rather than that of mass while the thickness of each layer was increasing.
399
O p t i m a l design of c o m p o s i t e r o b o t a r m s TABLE 1. DESIGN VARIABLESOF AN APPROXIMATE OPTIMAL SOLUTION FOR DESIGN 1
No.
(i)
S t a c k i n g sequence
Thickness
[- -I- 0]9
(ii)
[ + 03/-
(iii)
Optimal solution Fiber v o l u m e Ply-angle fraction
0 3 / + 03]9
[ + 0/- 0/+ 0/- 0/ + 0 / - 0 / + 0 / - 0 / + 0]9
EI/pA
tx= t2 = ta = t4 = ts = t6 = t7 = ts = t9 =
0.599426 0.598484 0.60000 0.59900 0.59900 0.59900 0.59908 0.59892 0.60000
~q = ~t2 = ~t3 = ~t4 = ~ts = ~ts = ~t7 = % = % =
0.258106 0.257671 0.257191 0.256663 0.256084 0.254554 0.254769 0.255027 0.254251
Vfl = 0.501095 Vf2 = 0.501301 Vt3 = 0.501531 Vf4 = 0.501785 Vf5 = 0.502065 Vf6 = 0.502371 Vt7 = 0.502705 Vfs = 0.503068 Vf9 = 0.503461
3.52363E + 11
tI = t2 = ta = t4 = t5 = t6 = t7 = ts = t9 =
0.594851 0.598299 0.60000 0.60000 0.60000 0.60000 0.59239 0.60000 0.60000
al = at2 = % = ~t4 = % = % = at7 = % = ~t9 =
0.517209 0.516668 0.516069 - 0.512419 - 0.511760 - 0.512570 0.513054 0.512131 0.511134
Vfl Vf2 Vfa Vt4 V~s Vf6 Vf7 Vf8 Vt9
= = = = = = = = =
0.539013 0.539116 0.539567 0.541386 0.541957 0.542580 0.540718 0.541182 0.542031
3.52469E + 11
t1= t2 = t3 = t4 = t5 = t6 = t~ = ts = t9 =
0.595657 0.60000 0.60000 0.60000 0.60000 0.60000 0.60000 0.60000 0.59913
~t1 = % = ~t3 = ~, = % = % = ~ = % = ~t9 =
0.517830 - 0.516876 0.516902 - 0.515682 0.515779 - 0.514251 0.51A.A.A.A. - 0.512562 0.512879
Vt~ Vf2 Vf3 Vf4 Vf5 Vt6 Vt7 Vf8 Vf9
= = = = = = = = --
544108 0.544108 0.544546 0.545113 0.545082 0.543888 0.541926 0.540886 0.544594
3.55785E + 11
TABLE 2. LAMINATE CHARACTERISTICS. ARM GEOMETRIES. AND ARM PROPERTIES. FOR INITIAL AND OPTIMAL ROBOTIC DESIGNS OF DESIGN 1 Parameter
Initial design
s. D . C . Ell ( N / m m 2) p ( N / m m 3) I (mm 4) A (ram 2) E A ( N / m m 2) G ( N / r a m 2) Ell I/pA (ram a) Vn cti (rad)
0.04953 9226.6 1.446E - 5 59264.25 531 9226.6 4834.82 7.124E + 10 0.5 i = 1 ~ 9 1.22173 i = 1~ 9 0.5 i = 1 ~ 9
ti (mm)
By substituting
the values
(18), the mechanical shown
in Table
arbitrarily assumed than In
2. T h e
selected.
order
variables
of the optimally
initial
design
In the case that
to be identical,
that
of design
properties
O p t i m a l design
the value of
in Table
0.04953 44822.14 1.469E - 5 79215.4 655.9 44797.33 25208.42 3.558E + 11 see T a b l e l see T a b l e 1 see T a b l e 1
in Table tailored
2 into equations
composite
2 assumes
the specific damping
EI/pA
of the optimal
that
arms
the design
capacities design
(7), (8), ( 1 7 ) a n d
were obtained
for both
is about
and
variables
were
designs
are
five times greater
of the initial design. to
optimal-designed
compare robot
the
elastodynamic
manipulators,
responses
each manipulator
performed
by
was assumed
the
initial-
and
to be subjected
the
to the
400
C.K.
SUNG a n d S. S. SHYL
same kinematic motion while the end-effector was carrying a 10 kg payload. The results of the finite-element simulations are presented in Fig. 5 for the dynamic responses of the endeffectors of the two robotic designs. As shown in Figs 3 and 4 the power inputs for the prescribed maneuvers were stopped at 0.3 s and the phase of settling motion starts, the optimal design clearly provides better dynamic response.
6 5 4
/ \ [ ~ I ~
3 2
! I
0
~ .....
Optimot design Initiot. design
~'-/I
0.2
0.6
0.4
Time (sec) FIG. 5. D y n a m i c r e s p o n s e s a t t h e r o b o t e n d - e f f e c t o r for t h e i n i t i a l d e s i g n a n d t h e o p t i m a l d e s i g n o f d e s i g n 1.
TABLE 3. DESIGN VARIABLES OF AN APPROXIMATE OPTIMAL SOLUTION FOR DESIGN 2 Optimal solution Fiber volume No.
Stacking sequence
(i)
[+8]9
(ii)
(iii)
t 1 = 0.509526
[ - I - 0 3 / -- 8 3 / - I - 0 3 ] 9
[ + Ol
Thickness
-
81 + 81 - 8 / + 8 /
/-8/+ 8/-0/+819
Ply-angle
fraction
S.D.C.
al = 0.368666
t2 = 0 . 5 2 0 0 0 4
~2 = 0 . 3 6 8 6 6 6
t 3 = 0.53
a3 = 0 . 3 6 8 6 6 6
t4 = 0 . 5 3
~4 = 0 . 3 6 8 6 6 6
Vfl = 0.5
t s = 0.53
~5 = 0 . 3 6 8 6 6 6
i=1~9
t6 = 0.53
~6 = 0 . 3 6 9 0 8 0
t 7 = 0.53
~7 = 0 . 3 6 9 5 4 0
t s = 0.53
~8 = 0 . 3 6 9 5 4 0
t9=0.53087
~9 = 0 . 3 6 9 5 4 0
t I = 0.568796
~1 = 0 . 6 1 0 2 6 9
t2 = 0.57
~z = 0 . 6 1 0 2 6 9
t 3 = 0.57
~3 = 0 . 6 1 0 2 6 9
t 4 = 0.57
~4 =
- 0.610269
Vn = 0.5
t 5 = 0.57
~s =
- 0.610269
i=1~9
t 6 = 0.5724
~6 =
- 0.610269
t 7 = 0.5676
~7 = 0 . 6 1 0 2 6 9
t 8 = 0.57
~8 = 0 . 6 1 0 2 6 9
t 9 = 0.57
~9 = 0 . 6 1 0 2 6 9
tl = 0.557043
~1 = 0 . 6 1 4 0 3 9
t 2 = 0.595917
~2 =
t3 = 0 . 5 4 4 0 8 0
~a = 0 . 6 1 4 0 3 9
t4 = 0.595920
~4 =
t 5 = 0.544080
as = 0 . 6 1 4 0 3 9
0.0586846
0.0625734
-0.623822 -0.623822
t6 = 0.595920
~6 =
t 7 = 0.544080
~7 = 0 . 6 1 4 0 3 9
- 0.623822
t8 = 0 . 5 9 5 9 2
~8 =
t 9 = 0.553870
~9 = 0 . 6 1 6 0 6 6
-0.623822
Vfi = 0.5 i=1~9
0.0628792
Optimal design of composite robot arms
401
Example 2 A solution was sought in which a fast settling time upon completion of prescribed maneuvers is required. The data sets for the constraint conditions of the design variables were tmin = 0.5 mm (Vf)mi
n =
0.5
0~mi n =
--
tm.~ = 0.6 mm (Vf)max =
re/2
0Cma x =
0.6
n/2
EI/pA >10.2062 E + 12 and q= 9
ha = 20 mm
bl = 30 mm.
The value of EI/pA was constrained to prevent high flexibility, and both robots were assu/ned to be fabricated with graphite/epoxy composite laminates. Similar to the previously mentioned procedures, the design variables for the design 2 problem were determined (Fig. 3). Then the laminate characteristics, arm geometries, and arm properties for the initial and optimal designs were computed and shown in Table 4. From Table 3 it is noted that the fiber volume fraction approaches the lower bound because the specific damping capacity is relatively sensitive to the matrix content Table 4 shows that
TABLE 4. LAMINATE CHARACTERISTICS, ARM GEOMETRIES, AND ARM PROPERTIES, F O R I N I T I A L A N D O P T I M A L R O B O T I C D E S I G N S O F D E S I G N 2 Parameter
Initial design
S. D . C . E n ( N / m m 2) p ( N / m m 3) I (mm 4) A (mm 2) E A ( N / m m 2) G ( N / m m 2) E n I/pA(mm s) Vn cti (rad) ti (mm)
1.0045 116725 1.446E 10328.99 404.532 116725 3702.3 2.062E 0.5, i 0.0, i 0.5, i
Optimal design
- 5
+ = = =
11 1~ 9 1~ 9 1~ 9
0.06288 25261.5 1.4455E - 5 72887.632 617.82 25276.5 27482.95 2.062E + 11 see Table 3~ see Table 3 see Table 3
3 Optimot
g
o
design
11 | 1 ~ I ! I ~.l.q il.t.. I I I 11 1! . I ! I.| ! l -i t q
O
0.2
0.4
0.6
0.8
1.0
1.2
,
1.4
Time (sec)
FIG. 6. Dynamic responses at the robot end-effector for the initial design and the optimal design of design 2.
402
c.K. SUNGand S. S. SHVL
the specific d a m p i n g c a p a c i t y of the o p t i m a l design is 14 times greater t h a n t h a t of initial design. The d y n a m i c responses c o m p u t e d by the finite-element simulations are presented in Fig. 6. The settling phase starts at 0.3 s after the a r t i c u l a t i n g m o t i o n - s t o p p e d . Because of space restrictions only p o r t i o n s of this response are presented in the figure. These simulations clearly d e m o n s t r a t e the r e d u c e d settling-time a n d s u p e r i o r response exhibited b y the r o b o t with the o p t i m a l link member. CONCLUSIONS A design m e t h o d o l o g y has been presented for synthesizing h i g h - p e r f o r m a n c e articulating r o b o t i c systems fabricated with o p t i m a l l y tailored c o m p o s i t e laminates. W i t h this m e t h o d o logy, a r t i c u l a t i n g m e m b e r s can be m a n u f a c t u r e d using the processes of b a g m o l d i n g or silicone r u b b e r m o l d i n g which m a y e n h a n c e the failure-resistanc¢ characteristics. By o p t i m a l l y specifying the types of fiber, matrix, stacking sequence, fiber volume fraction, fiber layup, a n d the cross-section geometries, the designer is able to synthesize the structural links of the r o b o t i c linkage so as to possess high stiffness, high strength, high d a m p i n g a n d low mass. This a p p r o a c h was illustrated in a s t u d y of a two-link r o b o t m a n i p u l a t o r to d e m o n s t r a t e the s u p e r i o r characteristics u n d e r two o p e r a t i o n a l conditions: m i n i m u m deflection d u r i n g a r t i c u l a t i n g m o t i o n a n d fast settling time after the p o w e r has been stopped. Acknowledgement--This study was funded by the National Science Council of Republic of China, under Grant
Number NSC77-0611-E007-03R. This support is gratefully acknowledged. REFERENCES 1. K. SADAMATO,R&D on intelligent robots being spurted. Jap. Robot News 1(4), 1 (1982). 2. A. ZALUSand D. E. HARDT,Active control of robot structure deflections. In: Robotics Research and Advanced Applications (edited by W. J. BOOK). ASME, H00236 (1982). 3. W. J. BOOK and M. MAJETT, Controller design for flexible, distributed parameter mechanical arms via combined state space and frequency domain techniqueg. In: Robotics Research and Advanced Applications (edited by W. J. BOOK). ASME, H00236 (1982). 4. W.J. BOOK,Characterization of strength and stiffness constraints on manipulator control. In: Proc. I F T o M M Conf. Theory and Practice of Robots and Manipulators (edited by A. MORECKIand D. DEDZIOR),Warsaw, Poland (1976). 5. W. J. BOOK, O. MAIZZA-NETOand D. D. WHITNEY,Feedback control of two beam, two joint system with distributed flexibility. ASME J. Dynamic Syst. Control 97, 424 (1975). 6. F, A. KELLYand R. L. HUSTON,Modeling of flexibility effects in robot arms. In: Proc. 1981 Joint Automotive Control Conference, Charlottesville, VA (1981). 7. W. H. SUNADAand S. DUaOWSKY,On the dynamic analysis and behavior of industrial robotic manipulators with elastic members. ASME J. Mech. Transm. Aut. Design 105, 42 (1983). 8. G. NAGANATHANand A. H. SONI, Coupling effects of kinematics and flexibility in manipulators. Int. J. Robotics Res. 6(1), 75 (1987). 9. G. G. LOWENand C. CHASSAPIS,The elastic behavior of linkages: an update. Mech. Much. Theory 21(1), 33 (1986). 10. W. L. CLEGHORN,R. C. FENTONand B. TABARROK,Optimum design of high-speed flexible mechanisms. Mech. Much. Theory 16, 339 (1981). 11. A. J. KAKATSIOSand S. J. TRICAMO,Integrated kinematic and dynamic optimal design of flexible planar mechanisms. ASME J. Mech. Transm. Aut. Design 109, 338 (1987). 12. B. S. THOMPSON,Composite laminate components for robotic and machine systems. ASME Appl. Mech. Rev. 40(11), 145 (1987). 13. C. K. SUNG,B. S. THOMPSON,P. CROWLEYand J. Cocoo, An experimental study to demonstrate the superior response characteristics mechanisms constructed with composite laminates. Mech. Much. Theory 21(2), 103 (1986). 14. S. ADALI,Multiobjective design of an antisymmetric angle-ply laminate by nonlinear programming. ASME J. Mech. Transm. Aut. Design 105, 214 (1983). 15. T. R. TAUCHERTand S. ADIBHATLA,Design of laminated plates for maximum stiffness.J. Compos. Mater. 18, 58 (1984). 16. D. X. LIAO,C. K. SUNGand B. S. THOMPSON,The optimal design of symmetric laminated beams considering damping. J. Compos. Mater. 21(5), 485 (1986). 17. C. K. SUNGand B. S. THOMPSON,A methodology for synthesizing high-performance robots fabricated with optimally-tailored composite laminates. ASME J. Mech. Transm. Aut. Design 1090), 74 (1987). 18. M. M. SCHWARTZ,Composite Materials Handbook. McGraw-Hill, New York (1984).
Optimal design of composite robot arms
403
19. R.G. NI and R. D. ADAMS,A rational method for obtaining the dynamic mechanical properties of laminate for predicting the stiffness and damping of laminated plates and beams. Composites 15(3), 193 (1984). 20. S. W. TSAI and H. T. HAHN, Introduction to Composite Materials. Technomic Publishing Co., Westport, CT (1980). 21. U. KmSCH, Optimum Structure Design. McGraw-Hill, New York (1981). 22. B. S. THOMPSONand C. K. SUNG,A variational formulation for the nonlinear finite-element analysis of flexible linkages: theory, implementation and experiment~il results. ASME J. Mech. Transm. Aut. Design 1116(4), 482 (1984).