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Optimal control of a flexible parallel link robotic manipulator
compn?ers & strucrures Vol. 48. No. 3, pp. 375-385, 1993 Printedin Great Britain.
OPTIMAL
004s7949/93 s6.00 + 0.00 FWgUllOllRarLtd
CONTROL OF A FLEXIBLE PARALLEL ROBOTIC MANIPULATOR
LINK
J. D. LEE Department of Civil, Mechanical, and Environmental Engineering, The George Washington University, Washington, DC 20052, U.S.A. (Received 12 August 1992) Aftstrati-The parallel link robotic manipulator consists of a base platform fixed in space or attached to the end-effecter of another robot, a mobile platform, and six linear actuators (pistons) that link the two platforms together. The load carrying parts of the actuators are considered to be flexible (deformable) and modeled as mass-springdamper elements. The nonlinear dynamic equations of the manipulator have been rigorously derived. No attempt has been made to linearize the plant or the measurements. The observer and the control law are derived based on the optimal control theory. The control gain matrix and the estimate gain matrix are obtained by solving the algebraic Riccati equations. The state excitation white noises and the observation white noises are included in the plant and the measurements, respectively. The dynamic equations of the observer are solved analytically and exactly. The feasibility of real-time control is demonstrated by the computer simulation results which are presented hem.
INTRODUCTION
Most of today’s industrial robots can be classified into two types according to their configurations. The robotic manipulator of the first type consists of successive links, usually hinged at rotary joints which can be actuated in a coordinated fashion to position the end-e&&or. It may be called a serial link manip ulator. The robotic manipulator of the second type, called a parallel link manipulator, consists of two platforms, the base platform is fixed in space or attached to the end-effecter of another robot and the mobile platform which is movable with respect to the base platform, and six linear actuators which link the two platforms together. The idea of the parallel link manipulator was first used for the design of flight simulators by Stewart [l], therefore, it may be named as Stewart platform. In order to position the endeffector of the parallel link robotic manipulator to the commanded pose (location and orientation), the length of each actuator is computed and then the linear actuators are driven simultaneously to assume those values. Generally speaking, serial link manipulators have the advantage of access to large workspaces. On the other hand, parallel link manipulators are known for the simplicity of their mechanical design, and their high strength/weight and stiffness/weight ratios, because the actuators act in parallel and bear no moment loads. The parallel link manipulator is of interest in its own right, from the point of view of its design, kinematic analysis, dynamics, control, applications, and performance. Fichter [2] considered the issues of kinematics and practical construction of a Stewart platform. Landsberger and Sheridan [3] formulated a procedure to determine the pose of the
end-effector given the lengths of the six actuators. Gosselin [4] determined the workspace of the parallel link manipulator. Recently, Geng et al. [5] conducted a kinematics analysis of a class of Stewart platforms and obtained a sixteenth-order polynomial equation corresponding to the forward kinematic solution. Do and Yang[6] performed an inverse dynamic analysis of the Stewart platform using Newton-Euler equations of motion derived for the actuators. Lee et al. [7j derived the dynamic equations of the parallel link manipulator and, through computer simulation, showed that the manipulator can be used to perform tasks such as position control, force control, and path tracing. Nguyen and Pooran [8] conducted a dynamic analysis of the parallel link robotic manipulator which was built to study the telerobotic service and maintenance of NASA hardwares in space. Geng et al. [9] applied a learning control technique to a simplified parallel link manipulator to perform the task of trajectory tracking control. Nowadays, most of the work done in robotics is based on the assumption that the manipulator is just a collection of rigid bodies. In other words, after the joint angles, for a serial link manipulator, or the actuator lengths, for a parallel link manipulator, assume the computed values, all the links are presumed to be stiff enough to ensure that the end-effector will thus be in the intended pose. Even the static deflection of the manipulator due to gravity has not been taken into consideration, which is why most of today’s industrial robots are built to be massive and unwieldy. However, it is desirable to construct lightweight robotic manipulators which are capable of carrying a heavy payload and to move swiftly. To meet these requirements, the links cannot be too large and hence they have to be considered as
375
J. D.
316
bl
lhaewm
Fig. 1. A general structure of the Stewart platform.
LEE
dently controlled by a hydraulic or electrical system and are not necessarily identical in their mechanical designs and electrical characteristics. A typical (clth) piston, whose schematic drawing is shown in Fig. 2, consists of two parts: (1) the stationary part which is rigid and has mass A, and length A,, and (2) the moving part which is assumed to be deformable, i.e., a viscoelastic rod, and has mass m, and length (in the undeformed state) rlI. In this work, the standard tensor summation convention is adopted for repeated Latin indices; a superposed dot indicates the time derivative; a variable with a Greek index a as the subscript refers to the quantity associated with the ath piston, and the Greek index may be omitted if no ambiguity arises; a bold-faced variable means either a vector or a matrix; a vector or a matrix with a superscript prime means its transpose. Also, for the sake of simplicity, the following notations are employed (ALAx..
flexible (deformable) members, otherwise the accuracy and stability of the robot will be compromised. In general, for the flexible serial link manipulator, each link should be considered as an elastic or viscoelastic member which is subjected to bending (in both directions perpendicular to the axis of the link) and torsion (about the axis of the link). The finite element method will be employed to formulate the dynamic equations for the manipulator [lo-121. For the parallel link manipulator, in this work, the load carrying part, later referred to as the moving part, of each linear actuator is modeled as a generalized one-dimensional mass-spring-damper link. It is seen that the resulting dynamic equations of the manipulators, later referred to as the plant, are nonlinear and complex. More importantly, the number of state variables of the plant is larger than that of its counterpart-the manipulator with rigid links. One has to consider another factor: the number of measurements, limited by cost and feasibility, certainly cannot match the number of state variables. With these factors in mind, it is the purpose of this work to demonstrate the feasibility of applying optimal control technique to the flexible parallel link robotic manipulator.
PROBLEM
.A,), = A,.Axa. * .A,,
~(A,AI. ..A,),=
t
A,mA2m...A,,.
Let the rectangular coordinate system (X, , X2, X3) be fixed in space, later referred to as space coordinate, and the rectangular coordinate system (X, , if2, _%?3) be embedded in the mobile platform, later referred as the body coordinate. The six pairs of attaching points are located at b, (X, = Tim) and a, (XiU= X,*,), where _?i and XT are constants once the mechanical design of the Stewart platform is finalized. The pose of the mobile platform expressed in a space coordinate can be determined by three translations and three successive rotations performed in a specific sequence (Eulerian angles). There are 12 distinct sequences, each corresponding to a specific rotation matrix [13]. In this work, the sequence starts with a rotation by an angle Bs about the X3-axis, followed by an angle 0, about the $,-axis and, finally, by an angle f$ about the X,-axis; and the three translations are represented by three displacements, U,, U,, and U,, along Xi, X,, and X, axes,
DESCRIPTION
The parallel link manipulator being considered in this work is shown in Fig. 1. The base platform is assumed to be rigid and fixed in space. However, it does not have to be a flat plate, in other words, the attaching points of the six linear actuators (pistons) at the base platform, b, (a = 1,2,. . . ,6), are not necessarily coplanar. The mobile platform is also assumed to be rigid but may have an irregular shape, therefore the attaching points of the six pistons at the mobile platform, a, (a = 1,2, . . ,6), are not necessarily coplanar either. The six pistons can be indepen-
b Fig. 2. Schematic drawing of the structure of a piston.
Control of a parallel link robotic manipulator
respectively. Then the space coordinates of a generic point in the mobile platform with body coordinates S are obtained as Xi= R,.fj+
Vi,
(1)
377
and Yi can be obtained, from eqn (l), as pi= R&~.+
[
-ClS3
c2c3 -
s,s2s3
c2s3 +
sI s2 c3
ClC3
S2C3 +SlC2C3 s2s3 -slc2c3
31
-C1S2
CIC2
(9)
where
where the rotation matrix R can be written as
R=
oiii,
1(2)
Substituting
RUk2$
(10)
k eqn (9) into (8), it is obtained that
T, =;{MCji~i+2MRhj~j~i~,,,+
J,,,nRi,“jR,t#jt$},
where ci = cos 0, and s, = sin 0,. For the a th piston, the vector from b. to a, is obtained as q,=RQXj++Ui-R(i.
(11)
where (3)
(12)
The length of the piston can be calculated as 1 = (q . qy’2.
Actually, eqns (3) and (4) are the inverse kinematics formula for the Stewart platform. The piston force,f, shown in Fig. 2 acting on the moving part in the direction of q is considered as the control input. The length of the moving part of the piston, q, is simply calculated as tj=I-6,
(13)
(4) J,,,,, =
(14)
are the mass, the body coordinates of the mass center, and the J matrix of the mobile platform, respectively. It is noticed that the J matrix relates to the mass moment of inertia I of the mobile platform as
(5) Iu=Jkkksii-Jlj
where 6 (see Fig. 2) is considered as one of the state variable because the moving part is the load carrying element and, consequently, it is deformable, in other words ‘I-‘Io’I-~-$
p&C& dv
s
(6)
is the elongation. However there is a physical constraint, i.e. 0 G 6 6 A, and, to model this constraint, a reaction force,f, acting on the moving part of the piston in the direction of q is included in the formulation as follows:
or
Jii=fItiBij-Ii/,
where S, is the Kronecker delta. Now it is clear that the mass center and the principal axes of the mobile platform do not necessarily coincide with the origin and the orientation of the body coordinate axes, respectively. For the piston, let s be the one-dimensional coordinate of a generic point on the stationary part or moving part, and s in the range of [0, A] or [a, I], respectively. For the moving part, the coordinate of a generic point is xi=sqiI-‘.
Define a dimensionless
KINETIC ENERGIES Let p be
the mass density (per unit volume) of the mobile platform, then the kinetic energy of the mobile platform is
(16)
variable e as
er(s-a)/(/-6) where L is a small positive number and ICis a very large positive number. The physical constraint actually determines the workspace of the Stewart platform.
(15)
or
s=~5++(l-6).
(17)
Let p* be the mass density (per unit length) of the moving part and assume the mass is uniformly distributed along the length even when the moving part is elongated (or shortened), i.e. p*=m/(l-6).
(18)
From eqns (16) and (17) one obtains that 2i = (6 + e(i - 8)]qil-’
p.$.$ dv,
T, = ; s
(8)
+ (6 + e(l - 6)}(Qiir-* -
qil-2i).(19)
J. D. LEE
378
The kinetic energy of the moving part of the piston can be expressed as T, = f
’p*ii$
& =T
s6
’ ,$i2i de. s0
(20)
where k is the stiffness (spring constant) of the moving part of the piston. Rayleigh’s dissipation function of the moving part of the piston due to the existence of damping can be expressed as d
=
&Z/2,
c(i-
(36)
Substituting eqn (19) into eqn (20), T, is obtained as T ,=~{~Z+i2+Bi+(62+[2+61)0},
(21)
where c is the damping coefficient. The general Lagrange’s equation can be written as [14] d aL aL ao ,,,--+-=z, ap a$ 0
where
(22) b,
E
126,-
qiqj .
(23)
Similarly, the kinetic energy of the stationary part of the piston is obtained as Ts = Ci A2@/6.
(37)
where L is the Lagrangian (the kinetic energy minus the potential energy) of the system; p is the vector of the generalized coordinates; r is the vector of the generalized forces. In this work, there are twelve generalized coordinates (three displacements, three Eulerian angles, and six 6,), i.e.
(24)
It is worthwhile to note that (25)
and L = T, - Y, + c{ T, + T, - V, - V, - I/*},
(26)
(39)
D =ccfDI (28)
t
=(f’~~F~~f’dfdwf
39
(40)
r r r r r Ir
21
39
49
5,
r)‘, 6 (41)
(29) where (30)
r=f+T
(31) LAGRANGE’S
EQUATIONS
v, = -giM{Ruq
+ Uijp
(32)
where the vector g is the gravitational constant g multiplied by the unit vector in the direction of the gravity. Similarly, the potential energies of the moving part and the stationary part of the piston due to gravity are, respectively v2 = -g,m(l
+ @I-‘qJ2
V3 = -g,fiAl-‘q,/Z.
M, = M, cos 0, + My sin OS
M2 = (-M,
The potential energy of the mobile platform due to gravity is simply
(33) (34)
(42) (43)
sin O3+ My cos B,)cos 8, + M, sin 8,
(44 M,=M,
(45)
and [F,, F,, F,] are the externally applied forces acting on the origin of the mobile platform along the corresponding space coordinate axes; [M, , My, M,] are the externally applied moments acting on the mobile platform about the corresponding space coordinate axes. After lengthy but straightforward manipulation, a set of 12 Lagrange’s equations are obtained as Mii, + Matie;. f M$jkdjbk - Mgi+ ~{i-i,},
= Fi (4)
The elastic energy (another form of potential energy)
due to deformation
M@, tij + Jm Rhi Rkaj4 + J,, R/njRhjk dj dk
is
v* = k(l - 6 - ?#/2,
(35)
-M6jigj+~{@;Qj},=
M,
(47)
Control of a parallel link robotic manipulator (m8/3 -I- w{li,/6I - m&B/3 - mq&/21 -
(k(l - 6 - tt”) + c(i - 8))). -Jb’,
(48)
379
For the regulator problem, i.e., the position control problem in robotics, let (x0, u”, x”) be the operating point, in other words
where in eqns (46) and (47) i = 1,2,3; in eqn (48) a=1,2,...,6; and
N,(ti,n0)=0
(59)
z” = N2(x0, u”).
W)
f2, = Z{b,(l’tj, - 2@jkqk)}/16+ m(2f+ 8)q,/6l . . . . + m(211+ 266 + 61+ Sl)b,tjj/314
For
- m(21+ S)@q,/61 - (m& + (ma + tSiA)b,j13}gj/2
(49) 8# = Rkj&
if the desired
pose is set to be
be employed to solve for (6:) (a = 1,2, . . . ,6) and, afterwards, the control inputs (piston forces) (u:} (a=1,2,..., 6) at the operating point can be obtained from eqn (48) and then z” can be calculated from eqn (60). Perform Taylor series expansions for N, and N, about the operating point as follows:
(50)
(51) Z = {m(f’ + 6’ + IS) + tiiA*}/3.
+B(x”,uo){u-u”]+
OPTIMALCONTROL THEmtY
(53) of the plant,
.a.
N2(~,u)=~+H,(~o,~o){~-xo) + H2(xo, u”){u - u”} + -. -,
eqns
A _ dNr
ax ’
B=z,
H,+,
H2$
u= {f,LLLLLf,)‘.
(55)
in general, can be expressed as 2
=
N,(x,
(56)
u),
where N2 are also nonlinear functions of x and u and z is the vector of observations. For example, if sensors are employed to measure the lengths of the pistons, then z =
(I,,
12, I,,
14, l,,
161’
(57)
and if the pose of the mobile platform are observed, then
z=
(63)
(64)
(54)
where N, are 24 nonlinear functions of the state variables,’ x, and control inputs, u, which in this work are identified as the six piston forces, i.e.
The measurements,
(62)
and it is understood that A, B, H, , H2 are evaluated at the operating point. Define new sets of state variables, control, inputs and observations as t - x - x0, 6 = u - uo, g = jr - 20
t = N, (x, u),
(61)
where
Let the 12 generalized coordinates, p, and their first time derivatives, p, be identified as the state variables, i.e.
x = {Pt PI’
NI (x, II) = A(x”,u”) {x - x0}
(52)
Some formulae, which have been used for the derivation of eqns (46~48), are listed in Appendix 1.
then the dynamic equations (46)-(48) can be written as
example,
{Uy, Ui, U$, @, Oi, Oi}, then eqns (46) and (47) will
w,,u2, u,,4, e,,w.
(58)
then eqns (54) (56) can be rewritten as (65) g=H,I+H,g+m,(f,ti),
(66)
where f$ and m2 are the nonlinear functions which contain all the second and higher order terms of 2 and ii. It is noticed that, by keeping these nonlinear terms in eqns (65) and (66), neither the plant nor the measurements are approximated. Further, for actual plant and measurements, state excitation noises and observation noises, respectively, are introduced as %=Al+Bii+l$+#,w
(67)
Z=H,%+H,ii+m,++,w,
(68)
where w(f) is a vector of n, independent white noises; 4, and #2 are constant n x n, and n, x n, matrices,
J. D. LEE
380
respectively; )I is number of state variables; and n, is number of observations. Since w(t) are independent white noises, the expectation values of n(r) are zeros and the variance, W, is a constant n, x n, square matrix with only nonva~shing diagonal elements, i.e. E{w(t)j = 0
(69)
E{w(t, )w’(tl)} = W&t* - r*),
(70)
and the diagonal elements of matrix W are denoted by #,a:,..., d&f. Now let the ohserver and the control law be represented, respectively, by jr=Ay+Bii+G(i-H,y-H,ii)
It is assumed that W, is nonsingular. It is noticed that if the state excitation noises and the measurement noises are correlated, then W,* is not vanishing. Also, it can be proved that, if the weighting matrices, Q, and Qr , are multiplied by a scalar factor, the control gain matrix K does not change. Similarly, the estimate gain matrix G will not change even when the variance matrix, W, is multiplied by a scalar factor. This means that if the plant and the measurements have no white noises, W can be artificially specified by the control engineer in order to adjust the estimate gain matrix and afterwards, in computer sim~ation, the vector of the standard deviations of the white noises may be modified as
(71)
&I -KY,
(72)
where y is the vector of n estimated state variables; G is the n x n, estimate gain matrix; K is the n, x n control gain matrix (n, is the number of control inputs). The optimal regulator-observer problem means to determine K and G such that the following two performance indices
ei = ratio(i) x c?~, (i = 1,2, . . . , n,).
(83)
COMPUTER SIMU~~ON
Recall the equations for the plant, the measurements, the observer, and the control law, respectively, as follows: ~=N,(x,u)~~,w
(84)
z=N,(x,u)-t&w
(85)
ot I, = E
(Z’Q, x + 1’Qrii) dr 15 1
A = WW)
(73)
1 - W’W
- yWH
(74)
are minimized. The control engineer is free to choose the state weighting matrix Ql and the control weighting matrix Q2 in order for the system to achieve the preferred performance as long as Q, is semipositive definite and Qz is nonsingular and positive definite. Then, according to the optimal control theory [ l&16], K and G are dete~n~ as K=Q;‘B’#l
(75)
G = (tizH; + W,,)W;‘,
(76)
where #I and +2 are the solutions of the following algebraic Riccati equations
S,A+A'~,-Jr,BQ;'B'S,+Q,=O, (77) @~+$~~-~~zH;W;‘HISZ + W, - W,,W;‘W;,
= 0,
(78)
where A--A-W,,W;‘H,
(79)
W, = ~,W#~
(80)
W2 = cpZW&
(81)
WI2
=
cb,wrb;.
(82)
y = Ay + B(u - u”) + G{z - z” - H, y - H&i - 0”)) (86) u = u” - Ky.
(87)
In reality, once the equations of the plant, eqn (84) and the measurements, eqn (85), are given or derived, the control engineer should first determine the matrices, A and B, H, and H, which come from the Taylor series expansion of N, and N,, respectively, about the operating point (x0, u*, z*). At this point, one has to investigate whether the system is controllable and observable by finding out whether the rank of the two matrices, defined as (B, AB, A2B , . . . ,A”-‘B) and (H;, A’H;,(A’)* Hi, . . . , (A’)“-*Hi}, are both equal to n, the number of state variables. If so, the control engineer should proceed to (1) choose the two weighting matrices, Q, and Q2, based on which the control gain matrix, K, is determined, and (2) estimate the level of the white noises, i.e., the vector of the standard deviations of white noises d = {a,, &, . . . ,6”, 1’. based on which the estimate gain matrix, G, is determined. Afterwards, a digital computer, which is interconnected to the actuators and the sensors, will be used to control the system, i.e., to solve eqn (86) for the estimated state variables, y, based on the measurements from the sensors, z, as inputs and to issue commands, u, according to eqn (87) to the actuators as outputs. The measurement data should be converted from analog form to digital form before
381
Control of a parallel link robotic manipulator
P=BY+YV
(88)
/?=A-BK-GH,+GH,K
(89)
Because 1 is a constant matrix and y is a constant vector between t = t, and t = t2 = t, + At, i.e., the measurements enter the digital computer once per sampling period, the solution of the estimated state variables, y, for t E [t, , t2] can be readily obtained as
075
’
“‘I 0
0.5
I 0
I 5
’
’
’
’
’
2.0
2.5
3.0
35
4.0
(91)
YCt)= i vi(t)Yi3 i=l
where u,(t) = Zj{(y(t,) + y/oi)eoi(‘-‘l) - y/q};
(92)
and {wi,Yi} and {oi, Zi} are the eigenvalues and eigenvectors associated with the following eigenvalue problems [ 171
with the orthonormal
flYi = WiYi
(93)
/.?‘Z, = wizi
(94)
conditions Z,Y,=6,
(95)
provided that all the eigenvalues are distinct. In this work, the distinctness of the eigenvalues and the orthonormality of the eigenvectors have been verified. At t = t,, the estimated state variables are obtained as
where
070
WY
y =G(z-#).
entering the digital computer. After the digital computer has processed the inputs, it provides outputs in digital form which then goes through a digital-toanalog converter before entering the actuators. In this work, it is further assumed that the numerical data enter and leave the computer at the same fixed sampling period, At. On the other hand, according to the linear optimal control theory, if the plant and the measurements have no nonlinear terms, fi, and &, in eqns (65) and (66), respectively, then the properly obtained gain matrices, K and G, will guarantee the stability of the system provided that the sampling period, At, approaches zero. It is evident that a significant discrepancy exists between the reality and the theory when the optimal control theory is applied to the control of flexible parallel link robotic manipulator. Moreover, in this work, a digital computer is used to stimulate the plant and the measurements with not only the nonlinear terms but also the white noises as indicated in eqns (84) and (85). The vector of independent white noises, w(t), are generated by invoking a random number generator when the vector of the standard deviations of white noises, u, is specified. It is noticed that eqn (86), with eqn (87), can be expressed as
’ 4 5
0
5.0
Time beconds)
0.5
1.0
1.5
2.0
Time
7
2.5
3.0
3 5
40
45
50
(seconds) FI
z g.s
F4
$5 Y Pzc4
AY%s 3 a
F2
9
F5
’
2
F3
$1 I
po
0
05
IO
15
20 Time
25
30
(seconds)
35
40
45
5.0
-
h
.&Z-, 8 -
-2.
0
0.5
I.0
15
2.0
25
3.0
35
Time (seconds)
Fig. 3. Observed pose and control inputs as functions of time (Case 1). CAS 48
40
45
F6 50
382
J. D. LEE I 15
4.7
1.10
4.6
c JS
I05
0 ;
g ;
loo
5 44
2 095 zE 0.90
UY
28 085
45
S g 4.3
8
B 080 b_
ux
UZ
42
84,
0.75
4.0
0.70
3.9 0
05
I 0
1.5
2.0 Time
2 5
30
3.5
4.0
4.5
5.0
0
05
IO
15
(seconds)
20 Time
25
30
35
4.0
45
50
(seconds)
7 6 2g, g
4 2
$
0
AZ AY
g
-2
6
-4
5 3
-6
z P6
FI
15
F4
2 94 33
F2
.S! F5
?2 * ‘; @ ‘- 0 AX “0 $-I
I
-8 -10 0
05
I.0
I5
F3
F6
Q-2 2.0 Time
2.5
3.0
3.5
4.0
4.5
50
0
05
(seconds)
I.0
15
20 Time
2.5
3.0
3.5
40
45
50
(seconds)
Fig. 4. Observed pose and control inputs as functions of time (Case 2).
Y(G) = i {zl{(~(tl) i=l
+ Y/~iYiA’
- ~h)}Yi
(96)
and the control commands, according to eqn (87) u(t,) = u” - Ky(t,) are sent to the actuators. It is seen that (1) equations for the observer are solved analytically and exactly; (2) the eigenvalues, w,, and the eigenvectors, Y, and Zi, can be calculated once for all before the controller activates; and (3) the digital computer only needs to evaluate eqn (96) once per sampling period-this makes the real-time control of flexible robotic manipulator feasible. In this work, the sampling period is set at 1Omsec and the equations for the plant, eqn (84), are solved by employing the Runge-Kutta method. In this work, the convergency of the numerical solutions has been verified. For illustrative purpose, the numerical results of three cases are presented and discussed as follows.
from which it is seen that the elongations of the six moving parts (?I is set at 4.0) are { -0.031, -0.015, -0.003, -0.024, -0.007,0.007}. The stiffness and the damping coefficient are specified as k, = 200 and c, = 2. In this work, the units for time, mass, length, and force are respectively second, pound, inch, and pound-weight (lbf). This is a case in 1-g environment with the level of white noises being reduced to zero. The pose as measured and the control inputs (the six piston forces) are presented in Fig. 3 as functions of time. It is clearly seen that the pose and the control inputs converge to their final values associated with the operating point. Case 2
This case is the same as Case 1 except that the white noises are included. The results are presented in Fig.. 4 from which randomness is clearly seen in the obCase 1 served pose and the control inputs; however, the corresponding expectation values, especially at later In this case, the initial pose is set at time, remain the same as those in Case 1. This {1.0,0.8,4.0, -8’, 2G,0’) and the final pose, at the operating point, is set at (0.8, 1.0,4.5, -6O, lo, 3O}. characteristic has also been noticed in the control of flexible serial link manipulator [12]. If the white At operating point it is calculated that, without noises are proportionally increased to a high enough externally applied forces and moments level, although the estimate gain matrix remains the same, the system will become unstable. u” = (6.627, 3.282,0.925,5.354, 1.875, - 1.041}’ Case 3
So = {0.764,0.915, 1.068,0.757, 1.332, 1.903)’ I0 = {4.733,4.900, 5.065,4.733, 5.325, 5.910)’
This case is the same as Case 1 except that it is in the O-g environment. At operating point it is calculated that
383
Control of a parallel link robotic manipulator
4.7
I IO I.05
i=
I 00
s z
0.95
UY
_
46
E ‘-
4.5
E
44
z E OS3 g 085 u) 5 0.80
ou 2
4.3
5iYt
42
ux
0.75.
L.
41
/-
4
0.70. 0
0.5
I 0
I .5
2.5
20
3.0
3.5
4 0
4 5
0
5.0
05
I 0
I 5
.I? c
2 0
2 5
35
3.0
40
4.5
5.0
Time (seconds)
Time (seconds)
-4
$ G-6
._
!
Ax
’
’
-8 0
0.5
IO
I .5
2.0
25
30
3.5
40
4 5
5.0
Time kecondsl
0
05
I.0
I.5
2.0
2.5
30
35
4.0
4.5
50
Time (seconds)
Fig. 5. Observed pose and control inputs as functions of time (Case 3). IlO= 0 Jo= {0.733,0.900, 1.065,0.733, 1.325, 1.910)’ lo = {4.733,4.900, 5.065,4.733, 5.325,5.910}’ from which it is seen that there is no elongation, i.e., I - 6 = q” = 4.0, and no piston forces are needed to sustain the commanded pose simply because there is no gravity. The numerical results are presented in Fig. 5. The numerical values of those input parameters used for these illustrative cases are listed in Appendix 2. DISCUSSION
One of the characteristics of the control of flexible robotic manipulators is that the number of state variables is usually very large while the number of observations, limited by cost and feasibility, is much less than that. Therefore, it is impossible to have an acceptable feedback control based only on the observations, and it is then necessary to construct an observer whose state variables are estimates of the state variables of the plant. Based on the estimated state variables, the control inputs can be calculated by using the control law. Common to both serial and parallel link manipulators, the dynamic equations of the plant and the measurements are nonlinear and complex, However, in this work, no attempt has been made to approximate those equations. On the other hand, the ob-
server and the control law are derived based on the linear optimal control theory. Hence, strictly speaking, the control system only works in a small neighborhood around the operating point. This restriction could be relieved by adopting the two-stage control strategy as indicated by Lee [lo, 121. However, it is worthwhile to point out that finding a set of pre-determined control inputs to be employed in the first stage (coarse) control to bring the system close to the operating point is not a trivial task.
REFERENCES
1. D. Stewart, A platform with six degrees of freedom. Prac. Inst. Mech. Engrs 180, 371-386 (1965-1966). 2. E. F. Fichter, A Stewart platform-based manipulator: general theory and practical construction. Int. 1. Robofics Res. 5, 157-182 (1986). 3. S. E. Landsberger and T. B. Sheridan, A new design for parallel link manipulator. Proc. Systems Man and Cybernetics Con$, pp. 812-814, Tuscan, Arizona (1985). 4. C. Gosselin, Determination of the workspace of 6-DOF parallel manipulators. J. Mech. Des. 112, 331-336 (1990). 5. Z. Geng, L. S. Haynes, J. D. Lee and R. L. Carroll, On the dynamic model and kinematic analysis of a class of Stewart platforms. Robotics and Autonomous Systems 9, 237-254 (1992). 6. W. Q. D. Do and D. C. H. Yang, Inverse dynamic
analysis and simulation of a platform type of robot. J. Robotic Systems 5, 209-227 (1988). 7. J. D. Lee, J. S. Albus, N. G. Dagalakis and T. Tsai,
Computer simulation of a parallel link manipulator. Mnfg. 5, 333-342 (1989). 8. C. C. Nguyen and F. J. Pooran, Dynamic analysis of a 6 DOF CKCM robot end-effector for dual-arm telerobot Robotics Computer-Integr.
384
J. D. LEE systems. Robotics and Autonomous Systems 5, 377-394,
(1989). 9. Z. Geng, J. D. Lee, R. L. Carroll and L. S. Haynes,
Learning control system design based on 2-D theoryAn application to parallel link manipulator. IEEE Znr. Co& Robotics and Automation, pp. 1510-1515 (1990). 10. J. 6. Lee and B. L. Wang, Optimal control of a gexibie robot arm. Cornour. Sfruct. 29, 459-467 (1988). 11. J. D. Lee and B. L. Wang, Dynamic equations for a two-link flexible robot arm. Cornput. Strucf. 29, 469-477 (1988). 12. J. D. Lee, Applications of optimal control theory to flexible robotic manipulators. Robotics ComputerZntegr. Mnfg. 7, 327-335 (1990). 13. J. J. Craig, Introduction to Robotics: Mechanics and Conrrol. Addison-Wesley, Reading, MA (1989). 14. H. Goldstein, Classical Mechanics. Addison-Wesley, Reading, MA (1950). 15. H. Kwakemaak and R. Sivan, Linear Optimal Control Systems. John Wiley, New York (1972). 16. B. Friedland, Control System Design: An Introduction to State-Space Methods. McGraw-Hill, New York (1986). 17. F. B. Hildebrand, Methods of Applied Mathematics. Prentice-Hall, Englewood Cliffs, NJ (1965). APPENDIX
qj= R#X:+
1
Zr[m(Z2+S2+fS)+tiA2)/3 I = I@/2
= ZAJ2 - m(21 + 6)q#/61 + m(2Zi + 268 +
d al Z 0i@
is + 18)
-z=Z0;Aj/2-m(21+6)0$q,@/61
ae,
i-m(2li+268 -I-l&0;
d al -z= 522 0
+is
bfi,j,/31’ = 0,: ZZ,
-m(l+26)@/6
p = m (P + & + ii)/6
U,-fj
!!Ls..
aui p a9j
as=@‘i+ I
d ap zz 0
- f$ = m6/3 + m(q,@,/I + la)/6
V, = -gjM(R$,
+ U,)
-w = --MQ,g, 80,
V = -g,{m(l
I = (qjqj)“* $,=4J
;=
O;qj,,
g
+ S) + fiA}q,/2~
= -g,{mb,
+ (rnd + @iA)b,/Z’}/2
dV as=
-Q;gk{mS,+(mS
av as =
-mg,q,/21
I
i= qj4j/l
v* = k(/ - d - $)2/2
av+
w=k(‘-S
-$=4tlI $= O;qjp 6 = b&+,/P
av* ==
-k(l-6
-q”)Q;qj/l -rj”)
d = c(i - d)2/2
+fiA)b,j/l’}/2
Control of a parallel link robotic manipulator
g = c(i - d)Q;qj/l
385
$ = (-2,
-3.4641,O)
,
X6 = (0, -4,O) ad
z=
-c(i-b)
g=
(0, 0, -g)
g = 32.2 x 12 = 384.4 F = (0, 0,O)' M = (0, 0,O)' E = 1.0 x 1o-3 K =
1.0 x lo4
n = 24 APPENDIX 2
M=
10
n, = 6 n, = 6
X=(0,0,0)
I?,=30
%- -1 s: =4 km=200 c, = 2 ri& = 1 A,=4 x:
= (3.4641, -2,O)
xy = (2,3.4641, x:
0)
= (0,4,0)
Q2=&xJ CTj= 0.5
(i = 1,2, . . . , 12)
d,=5.0
(i=13,14
oi=o.15
x,* = (-4,0,0)
,...,
(i =25,26,.
24)
. .,30)
ratio(i)
= 0.00
(i = 1,2,. . . , 12)
ratio(i)
= 0.04
(i = 13, 14, 15)
2, = (4,0,0)
ratio(i)
= 0.01
(i = 16, 17, 18)
x* = (3.4641,2,0)
ratio(i)
= 0.04
(i = 19,20, . . . ,24)
x, = (-2,
ratio(i)
= 0.04
(i = 25,26,27)
ratio(i)
= 0.01
(i = 28, 29,301.
x:
= (-3.4641,
-2,O)
x,+ = (2, - 3.4641,O)
3.4641,O)
x,=(-3.4641,2,0)
OPTIMAL
004s7949/93 s6.00 + 0.00 FWgUllOllRarLtd
CONTROL OF A FLEXIBLE PARALLEL ROBOTIC MANIPULATOR
LINK
J. D. LEE Department of Civil, Mechanical, and Environmental Engineering, The George Washington University, Washington, DC 20052, U.S.A. (Received 12 August 1992) Aftstrati-The parallel link robotic manipulator consists of a base platform fixed in space or attached to the end-effecter of another robot, a mobile platform, and six linear actuators (pistons) that link the two platforms together. The load carrying parts of the actuators are considered to be flexible (deformable) and modeled as mass-springdamper elements. The nonlinear dynamic equations of the manipulator have been rigorously derived. No attempt has been made to linearize the plant or the measurements. The observer and the control law are derived based on the optimal control theory. The control gain matrix and the estimate gain matrix are obtained by solving the algebraic Riccati equations. The state excitation white noises and the observation white noises are included in the plant and the measurements, respectively. The dynamic equations of the observer are solved analytically and exactly. The feasibility of real-time control is demonstrated by the computer simulation results which are presented hem.
INTRODUCTION
Most of today’s industrial robots can be classified into two types according to their configurations. The robotic manipulator of the first type consists of successive links, usually hinged at rotary joints which can be actuated in a coordinated fashion to position the end-e&&or. It may be called a serial link manip ulator. The robotic manipulator of the second type, called a parallel link manipulator, consists of two platforms, the base platform is fixed in space or attached to the end-effecter of another robot and the mobile platform which is movable with respect to the base platform, and six linear actuators which link the two platforms together. The idea of the parallel link manipulator was first used for the design of flight simulators by Stewart [l], therefore, it may be named as Stewart platform. In order to position the endeffector of the parallel link robotic manipulator to the commanded pose (location and orientation), the length of each actuator is computed and then the linear actuators are driven simultaneously to assume those values. Generally speaking, serial link manipulators have the advantage of access to large workspaces. On the other hand, parallel link manipulators are known for the simplicity of their mechanical design, and their high strength/weight and stiffness/weight ratios, because the actuators act in parallel and bear no moment loads. The parallel link manipulator is of interest in its own right, from the point of view of its design, kinematic analysis, dynamics, control, applications, and performance. Fichter [2] considered the issues of kinematics and practical construction of a Stewart platform. Landsberger and Sheridan [3] formulated a procedure to determine the pose of the
end-effector given the lengths of the six actuators. Gosselin [4] determined the workspace of the parallel link manipulator. Recently, Geng et al. [5] conducted a kinematics analysis of a class of Stewart platforms and obtained a sixteenth-order polynomial equation corresponding to the forward kinematic solution. Do and Yang[6] performed an inverse dynamic analysis of the Stewart platform using Newton-Euler equations of motion derived for the actuators. Lee et al. [7j derived the dynamic equations of the parallel link manipulator and, through computer simulation, showed that the manipulator can be used to perform tasks such as position control, force control, and path tracing. Nguyen and Pooran [8] conducted a dynamic analysis of the parallel link robotic manipulator which was built to study the telerobotic service and maintenance of NASA hardwares in space. Geng et al. [9] applied a learning control technique to a simplified parallel link manipulator to perform the task of trajectory tracking control. Nowadays, most of the work done in robotics is based on the assumption that the manipulator is just a collection of rigid bodies. In other words, after the joint angles, for a serial link manipulator, or the actuator lengths, for a parallel link manipulator, assume the computed values, all the links are presumed to be stiff enough to ensure that the end-effector will thus be in the intended pose. Even the static deflection of the manipulator due to gravity has not been taken into consideration, which is why most of today’s industrial robots are built to be massive and unwieldy. However, it is desirable to construct lightweight robotic manipulators which are capable of carrying a heavy payload and to move swiftly. To meet these requirements, the links cannot be too large and hence they have to be considered as
375
J. D.
316
bl
lhaewm
Fig. 1. A general structure of the Stewart platform.
LEE
dently controlled by a hydraulic or electrical system and are not necessarily identical in their mechanical designs and electrical characteristics. A typical (clth) piston, whose schematic drawing is shown in Fig. 2, consists of two parts: (1) the stationary part which is rigid and has mass A, and length A,, and (2) the moving part which is assumed to be deformable, i.e., a viscoelastic rod, and has mass m, and length (in the undeformed state) rlI. In this work, the standard tensor summation convention is adopted for repeated Latin indices; a superposed dot indicates the time derivative; a variable with a Greek index a as the subscript refers to the quantity associated with the ath piston, and the Greek index may be omitted if no ambiguity arises; a bold-faced variable means either a vector or a matrix; a vector or a matrix with a superscript prime means its transpose. Also, for the sake of simplicity, the following notations are employed (ALAx..
flexible (deformable) members, otherwise the accuracy and stability of the robot will be compromised. In general, for the flexible serial link manipulator, each link should be considered as an elastic or viscoelastic member which is subjected to bending (in both directions perpendicular to the axis of the link) and torsion (about the axis of the link). The finite element method will be employed to formulate the dynamic equations for the manipulator [lo-121. For the parallel link manipulator, in this work, the load carrying part, later referred to as the moving part, of each linear actuator is modeled as a generalized one-dimensional mass-spring-damper link. It is seen that the resulting dynamic equations of the manipulators, later referred to as the plant, are nonlinear and complex. More importantly, the number of state variables of the plant is larger than that of its counterpart-the manipulator with rigid links. One has to consider another factor: the number of measurements, limited by cost and feasibility, certainly cannot match the number of state variables. With these factors in mind, it is the purpose of this work to demonstrate the feasibility of applying optimal control technique to the flexible parallel link robotic manipulator.
PROBLEM
.A,), = A,.Axa. * .A,,
~(A,AI. ..A,),=
t
A,mA2m...A,,.
Let the rectangular coordinate system (X, , X2, X3) be fixed in space, later referred to as space coordinate, and the rectangular coordinate system (X, , if2, _%?3) be embedded in the mobile platform, later referred as the body coordinate. The six pairs of attaching points are located at b, (X, = Tim) and a, (XiU= X,*,), where _?i and XT are constants once the mechanical design of the Stewart platform is finalized. The pose of the mobile platform expressed in a space coordinate can be determined by three translations and three successive rotations performed in a specific sequence (Eulerian angles). There are 12 distinct sequences, each corresponding to a specific rotation matrix [13]. In this work, the sequence starts with a rotation by an angle Bs about the X3-axis, followed by an angle 0, about the $,-axis and, finally, by an angle f$ about the X,-axis; and the three translations are represented by three displacements, U,, U,, and U,, along Xi, X,, and X, axes,
DESCRIPTION
The parallel link manipulator being considered in this work is shown in Fig. 1. The base platform is assumed to be rigid and fixed in space. However, it does not have to be a flat plate, in other words, the attaching points of the six linear actuators (pistons) at the base platform, b, (a = 1,2,. . . ,6), are not necessarily coplanar. The mobile platform is also assumed to be rigid but may have an irregular shape, therefore the attaching points of the six pistons at the mobile platform, a, (a = 1,2, . . ,6), are not necessarily coplanar either. The six pistons can be indepen-
b Fig. 2. Schematic drawing of the structure of a piston.
Control of a parallel link robotic manipulator
respectively. Then the space coordinates of a generic point in the mobile platform with body coordinates S are obtained as Xi= R,.fj+
Vi,
(1)
377
and Yi can be obtained, from eqn (l), as pi= R&~.+
[
-ClS3
c2c3 -
s,s2s3
c2s3 +
sI s2 c3
ClC3
S2C3 +SlC2C3 s2s3 -slc2c3
31
-C1S2
CIC2
(9)
where
where the rotation matrix R can be written as
R=
oiii,
1(2)
Substituting
RUk2$
(10)
k eqn (9) into (8), it is obtained that
T, =;{MCji~i+2MRhj~j~i~,,,+
J,,,nRi,“jR,t#jt$},
where ci = cos 0, and s, = sin 0,. For the a th piston, the vector from b. to a, is obtained as q,=RQXj++Ui-R(i.
(11)
where (3)
(12)
The length of the piston can be calculated as 1 = (q . qy’2.
Actually, eqns (3) and (4) are the inverse kinematics formula for the Stewart platform. The piston force,f, shown in Fig. 2 acting on the moving part in the direction of q is considered as the control input. The length of the moving part of the piston, q, is simply calculated as tj=I-6,
(13)
(4) J,,,,, =
(14)
are the mass, the body coordinates of the mass center, and the J matrix of the mobile platform, respectively. It is noticed that the J matrix relates to the mass moment of inertia I of the mobile platform as
(5) Iu=Jkkksii-Jlj
where 6 (see Fig. 2) is considered as one of the state variable because the moving part is the load carrying element and, consequently, it is deformable, in other words ‘I-‘Io’I-~-$
p&C& dv
s
(6)
is the elongation. However there is a physical constraint, i.e. 0 G 6 6 A, and, to model this constraint, a reaction force,f, acting on the moving part of the piston in the direction of q is included in the formulation as follows:
or
Jii=fItiBij-Ii/,
where S, is the Kronecker delta. Now it is clear that the mass center and the principal axes of the mobile platform do not necessarily coincide with the origin and the orientation of the body coordinate axes, respectively. For the piston, let s be the one-dimensional coordinate of a generic point on the stationary part or moving part, and s in the range of [0, A] or [a, I], respectively. For the moving part, the coordinate of a generic point is xi=sqiI-‘.
Define a dimensionless
KINETIC ENERGIES Let p be
the mass density (per unit volume) of the mobile platform, then the kinetic energy of the mobile platform is
(16)
variable e as
er(s-a)/(/-6) where L is a small positive number and ICis a very large positive number. The physical constraint actually determines the workspace of the Stewart platform.
(15)
or
s=~5++(l-6).
(17)
Let p* be the mass density (per unit length) of the moving part and assume the mass is uniformly distributed along the length even when the moving part is elongated (or shortened), i.e. p*=m/(l-6).
(18)
From eqns (16) and (17) one obtains that 2i = (6 + e(i - 8)]qil-’
p.$.$ dv,
T, = ; s
(8)
+ (6 + e(l - 6)}(Qiir-* -
qil-2i).(19)
J. D. LEE
378
The kinetic energy of the moving part of the piston can be expressed as T, = f
’p*ii$
& =T
s6
’ ,$i2i de. s0
(20)
where k is the stiffness (spring constant) of the moving part of the piston. Rayleigh’s dissipation function of the moving part of the piston due to the existence of damping can be expressed as d
=
&Z/2,
c(i-
(36)
Substituting eqn (19) into eqn (20), T, is obtained as T ,=~{~Z+i2+Bi+(62+[2+61)0},
(21)
where c is the damping coefficient. The general Lagrange’s equation can be written as [14] d aL aL ao ,,,--+-=z, ap a$ 0
where
(22) b,
E
126,-
qiqj .
(23)
Similarly, the kinetic energy of the stationary part of the piston is obtained as Ts = Ci A2@/6.
(37)
where L is the Lagrangian (the kinetic energy minus the potential energy) of the system; p is the vector of the generalized coordinates; r is the vector of the generalized forces. In this work, there are twelve generalized coordinates (three displacements, three Eulerian angles, and six 6,), i.e.
(24)
It is worthwhile to note that (25)
and L = T, - Y, + c{ T, + T, - V, - V, - I/*},
(26)
(39)
D =ccfDI (28)
t
=(f’~~F~~f’dfdwf
39
(40)
r r r r r Ir
21
39
49
5,
r)‘, 6 (41)
(29) where (30)
r=f+T
(31) LAGRANGE’S
EQUATIONS
v, = -giM{Ruq
+ Uijp
(32)
where the vector g is the gravitational constant g multiplied by the unit vector in the direction of the gravity. Similarly, the potential energies of the moving part and the stationary part of the piston due to gravity are, respectively v2 = -g,m(l
+ @I-‘qJ2
V3 = -g,fiAl-‘q,/Z.
M, = M, cos 0, + My sin OS
M2 = (-M,
The potential energy of the mobile platform due to gravity is simply
(33) (34)
(42) (43)
sin O3+ My cos B,)cos 8, + M, sin 8,
(44 M,=M,
(45)
and [F,, F,, F,] are the externally applied forces acting on the origin of the mobile platform along the corresponding space coordinate axes; [M, , My, M,] are the externally applied moments acting on the mobile platform about the corresponding space coordinate axes. After lengthy but straightforward manipulation, a set of 12 Lagrange’s equations are obtained as Mii, + Matie;. f M$jkdjbk - Mgi+ ~{i-i,},
= Fi (4)
The elastic energy (another form of potential energy)
due to deformation
M@, tij + Jm Rhi Rkaj4 + J,, R/njRhjk dj dk
is
v* = k(l - 6 - ?#/2,
(35)
-M6jigj+~{@;Qj},=
M,
(47)
Control of a parallel link robotic manipulator (m8/3 -I- w{li,/6I - m&B/3 - mq&/21 -
(k(l - 6 - tt”) + c(i - 8))). -Jb’,
(48)
379
For the regulator problem, i.e., the position control problem in robotics, let (x0, u”, x”) be the operating point, in other words
where in eqns (46) and (47) i = 1,2,3; in eqn (48) a=1,2,...,6; and
N,(ti,n0)=0
(59)
z” = N2(x0, u”).
W)
f2, = Z{b,(l’tj, - 2@jkqk)}/16+ m(2f+ 8)q,/6l . . . . + m(211+ 266 + 61+ Sl)b,tjj/314
For
- m(21+ S)@q,/61 - (m& + (ma + tSiA)b,j13}gj/2
(49) 8# = Rkj&
if the desired
pose is set to be
be employed to solve for (6:) (a = 1,2, . . . ,6) and, afterwards, the control inputs (piston forces) (u:} (a=1,2,..., 6) at the operating point can be obtained from eqn (48) and then z” can be calculated from eqn (60). Perform Taylor series expansions for N, and N, about the operating point as follows:
(50)
(51) Z = {m(f’ + 6’ + IS) + tiiA*}/3.
+B(x”,uo){u-u”]+
OPTIMALCONTROL THEmtY
(53) of the plant,
.a.
N2(~,u)=~+H,(~o,~o){~-xo) + H2(xo, u”){u - u”} + -. -,
eqns
A _ dNr
ax ’
B=z,
H,+,
H2$
u= {f,LLLLLf,)‘.
(55)
in general, can be expressed as 2
=
N,(x,
(56)
u),
where N2 are also nonlinear functions of x and u and z is the vector of observations. For example, if sensors are employed to measure the lengths of the pistons, then z =
(I,,
12, I,,
14, l,,
161’
(57)
and if the pose of the mobile platform are observed, then
z=
(63)
(64)
(54)
where N, are 24 nonlinear functions of the state variables,’ x, and control inputs, u, which in this work are identified as the six piston forces, i.e.
The measurements,
(62)
and it is understood that A, B, H, , H2 are evaluated at the operating point. Define new sets of state variables, control, inputs and observations as t - x - x0, 6 = u - uo, g = jr - 20
t = N, (x, u),
(61)
where
Let the 12 generalized coordinates, p, and their first time derivatives, p, be identified as the state variables, i.e.
x = {Pt PI’
NI (x, II) = A(x”,u”) {x - x0}
(52)
Some formulae, which have been used for the derivation of eqns (46~48), are listed in Appendix 1.
then the dynamic equations (46)-(48) can be written as
example,
{Uy, Ui, U$, @, Oi, Oi}, then eqns (46) and (47) will
w,,u2, u,,4, e,,w.
(58)
then eqns (54) (56) can be rewritten as (65) g=H,I+H,g+m,(f,ti),
(66)
where f$ and m2 are the nonlinear functions which contain all the second and higher order terms of 2 and ii. It is noticed that, by keeping these nonlinear terms in eqns (65) and (66), neither the plant nor the measurements are approximated. Further, for actual plant and measurements, state excitation noises and observation noises, respectively, are introduced as %=Al+Bii+l$+#,w
(67)
Z=H,%+H,ii+m,++,w,
(68)
where w(f) is a vector of n, independent white noises; 4, and #2 are constant n x n, and n, x n, matrices,
J. D. LEE
380
respectively; )I is number of state variables; and n, is number of observations. Since w(t) are independent white noises, the expectation values of n(r) are zeros and the variance, W, is a constant n, x n, square matrix with only nonva~shing diagonal elements, i.e. E{w(t)j = 0
(69)
E{w(t, )w’(tl)} = W&t* - r*),
(70)
and the diagonal elements of matrix W are denoted by #,a:,..., d&f. Now let the ohserver and the control law be represented, respectively, by jr=Ay+Bii+G(i-H,y-H,ii)
It is assumed that W, is nonsingular. It is noticed that if the state excitation noises and the measurement noises are correlated, then W,* is not vanishing. Also, it can be proved that, if the weighting matrices, Q, and Qr , are multiplied by a scalar factor, the control gain matrix K does not change. Similarly, the estimate gain matrix G will not change even when the variance matrix, W, is multiplied by a scalar factor. This means that if the plant and the measurements have no white noises, W can be artificially specified by the control engineer in order to adjust the estimate gain matrix and afterwards, in computer sim~ation, the vector of the standard deviations of the white noises may be modified as
(71)
&I -KY,
(72)
where y is the vector of n estimated state variables; G is the n x n, estimate gain matrix; K is the n, x n control gain matrix (n, is the number of control inputs). The optimal regulator-observer problem means to determine K and G such that the following two performance indices
ei = ratio(i) x c?~, (i = 1,2, . . . , n,).
(83)
COMPUTER SIMU~~ON
Recall the equations for the plant, the measurements, the observer, and the control law, respectively, as follows: ~=N,(x,u)~~,w
(84)
z=N,(x,u)-t&w
(85)
ot I, = E
(Z’Q, x + 1’Qrii) dr 15 1
A = WW)
(73)
1 - W’W
- yWH
(74)
are minimized. The control engineer is free to choose the state weighting matrix Ql and the control weighting matrix Q2 in order for the system to achieve the preferred performance as long as Q, is semipositive definite and Qz is nonsingular and positive definite. Then, according to the optimal control theory [ l&16], K and G are dete~n~ as K=Q;‘B’#l
(75)
G = (tizH; + W,,)W;‘,
(76)
where #I and +2 are the solutions of the following algebraic Riccati equations
S,A+A'~,-Jr,BQ;'B'S,+Q,=O, (77) @~+$~~-~~zH;W;‘HISZ + W, - W,,W;‘W;,
= 0,
(78)
where A--A-W,,W;‘H,
(79)
W, = ~,W#~
(80)
W2 = cpZW&
(81)
WI2
=
cb,wrb;.
(82)
y = Ay + B(u - u”) + G{z - z” - H, y - H&i - 0”)) (86) u = u” - Ky.
(87)
In reality, once the equations of the plant, eqn (84) and the measurements, eqn (85), are given or derived, the control engineer should first determine the matrices, A and B, H, and H, which come from the Taylor series expansion of N, and N,, respectively, about the operating point (x0, u*, z*). At this point, one has to investigate whether the system is controllable and observable by finding out whether the rank of the two matrices, defined as (B, AB, A2B , . . . ,A”-‘B) and (H;, A’H;,(A’)* Hi, . . . , (A’)“-*Hi}, are both equal to n, the number of state variables. If so, the control engineer should proceed to (1) choose the two weighting matrices, Q, and Q2, based on which the control gain matrix, K, is determined, and (2) estimate the level of the white noises, i.e., the vector of the standard deviations of white noises d = {a,, &, . . . ,6”, 1’. based on which the estimate gain matrix, G, is determined. Afterwards, a digital computer, which is interconnected to the actuators and the sensors, will be used to control the system, i.e., to solve eqn (86) for the estimated state variables, y, based on the measurements from the sensors, z, as inputs and to issue commands, u, according to eqn (87) to the actuators as outputs. The measurement data should be converted from analog form to digital form before
381
Control of a parallel link robotic manipulator
P=BY+YV
(88)
/?=A-BK-GH,+GH,K
(89)
Because 1 is a constant matrix and y is a constant vector between t = t, and t = t2 = t, + At, i.e., the measurements enter the digital computer once per sampling period, the solution of the estimated state variables, y, for t E [t, , t2] can be readily obtained as
075
’
“‘I 0
0.5
I 0
I 5
’
’
’
’
’
2.0
2.5
3.0
35
4.0
(91)
YCt)= i vi(t)Yi3 i=l
where u,(t) = Zj{(y(t,) + y/oi)eoi(‘-‘l) - y/q};
(92)
and {wi,Yi} and {oi, Zi} are the eigenvalues and eigenvectors associated with the following eigenvalue problems [ 171
with the orthonormal
flYi = WiYi
(93)
/.?‘Z, = wizi
(94)
conditions Z,Y,=6,
(95)
provided that all the eigenvalues are distinct. In this work, the distinctness of the eigenvalues and the orthonormality of the eigenvectors have been verified. At t = t,, the estimated state variables are obtained as
where
070
WY
y =G(z-#).
entering the digital computer. After the digital computer has processed the inputs, it provides outputs in digital form which then goes through a digital-toanalog converter before entering the actuators. In this work, it is further assumed that the numerical data enter and leave the computer at the same fixed sampling period, At. On the other hand, according to the linear optimal control theory, if the plant and the measurements have no nonlinear terms, fi, and &, in eqns (65) and (66), respectively, then the properly obtained gain matrices, K and G, will guarantee the stability of the system provided that the sampling period, At, approaches zero. It is evident that a significant discrepancy exists between the reality and the theory when the optimal control theory is applied to the control of flexible parallel link robotic manipulator. Moreover, in this work, a digital computer is used to stimulate the plant and the measurements with not only the nonlinear terms but also the white noises as indicated in eqns (84) and (85). The vector of independent white noises, w(t), are generated by invoking a random number generator when the vector of the standard deviations of white noises, u, is specified. It is noticed that eqn (86), with eqn (87), can be expressed as
’ 4 5
0
5.0
Time beconds)
0.5
1.0
1.5
2.0
Time
7
2.5
3.0
3 5
40
45
50
(seconds) FI
z g.s
F4
$5 Y Pzc4
AY%s 3 a
F2
9
F5
’
2
F3
$1 I
po
0
05
IO
15
20 Time
25
30
(seconds)
35
40
45
5.0
-
h
.&Z-, 8 -
-2.
0
0.5
I.0
15
2.0
25
3.0
35
Time (seconds)
Fig. 3. Observed pose and control inputs as functions of time (Case 1). CAS 48
40
45
F6 50
382
J. D. LEE I 15
4.7
1.10
4.6
c JS
I05
0 ;
g ;
loo
5 44
2 095 zE 0.90
UY
28 085
45
S g 4.3
8
B 080 b_
ux
UZ
42
84,
0.75
4.0
0.70
3.9 0
05
I 0
1.5
2.0 Time
2 5
30
3.5
4.0
4.5
5.0
0
05
IO
15
(seconds)
20 Time
25
30
35
4.0
45
50
(seconds)
7 6 2g, g
4 2
$
0
AZ AY
g
-2
6
-4
5 3
-6
z P6
FI
15
F4
2 94 33
F2
.S! F5
?2 * ‘; @ ‘- 0 AX “0 $-I
I
-8 -10 0
05
I.0
I5
F3
F6
Q-2 2.0 Time
2.5
3.0
3.5
4.0
4.5
50
0
05
(seconds)
I.0
15
20 Time
2.5
3.0
3.5
40
45
50
(seconds)
Fig. 4. Observed pose and control inputs as functions of time (Case 2).
Y(G) = i {zl{(~(tl) i=l
+ Y/~iYiA’
- ~h)}Yi
(96)
and the control commands, according to eqn (87) u(t,) = u” - Ky(t,) are sent to the actuators. It is seen that (1) equations for the observer are solved analytically and exactly; (2) the eigenvalues, w,, and the eigenvectors, Y, and Zi, can be calculated once for all before the controller activates; and (3) the digital computer only needs to evaluate eqn (96) once per sampling period-this makes the real-time control of flexible robotic manipulator feasible. In this work, the sampling period is set at 1Omsec and the equations for the plant, eqn (84), are solved by employing the Runge-Kutta method. In this work, the convergency of the numerical solutions has been verified. For illustrative purpose, the numerical results of three cases are presented and discussed as follows.
from which it is seen that the elongations of the six moving parts (?I is set at 4.0) are { -0.031, -0.015, -0.003, -0.024, -0.007,0.007}. The stiffness and the damping coefficient are specified as k, = 200 and c, = 2. In this work, the units for time, mass, length, and force are respectively second, pound, inch, and pound-weight (lbf). This is a case in 1-g environment with the level of white noises being reduced to zero. The pose as measured and the control inputs (the six piston forces) are presented in Fig. 3 as functions of time. It is clearly seen that the pose and the control inputs converge to their final values associated with the operating point. Case 2
This case is the same as Case 1 except that the white noises are included. The results are presented in Fig.. 4 from which randomness is clearly seen in the obCase 1 served pose and the control inputs; however, the corresponding expectation values, especially at later In this case, the initial pose is set at time, remain the same as those in Case 1. This {1.0,0.8,4.0, -8’, 2G,0’) and the final pose, at the operating point, is set at (0.8, 1.0,4.5, -6O, lo, 3O}. characteristic has also been noticed in the control of flexible serial link manipulator [12]. If the white At operating point it is calculated that, without noises are proportionally increased to a high enough externally applied forces and moments level, although the estimate gain matrix remains the same, the system will become unstable. u” = (6.627, 3.282,0.925,5.354, 1.875, - 1.041}’ Case 3
So = {0.764,0.915, 1.068,0.757, 1.332, 1.903)’ I0 = {4.733,4.900, 5.065,4.733, 5.325, 5.910)’
This case is the same as Case 1 except that it is in the O-g environment. At operating point it is calculated that
383
Control of a parallel link robotic manipulator
4.7
I IO I.05
i=
I 00
s z
0.95
UY
_
46
E ‘-
4.5
E
44
z E OS3 g 085 u) 5 0.80
ou 2
4.3
5iYt
42
ux
0.75.
L.
41
/-
4
0.70. 0
0.5
I 0
I .5
2.5
20
3.0
3.5
4 0
4 5
0
5.0
05
I 0
I 5
.I? c
2 0
2 5
35
3.0
40
4.5
5.0
Time (seconds)
Time (seconds)
-4
$ G-6
._
!
Ax
’
’
-8 0
0.5
IO
I .5
2.0
25
30
3.5
40
4 5
5.0
Time kecondsl
0
05
I.0
I.5
2.0
2.5
30
35
4.0
4.5
50
Time (seconds)
Fig. 5. Observed pose and control inputs as functions of time (Case 3). IlO= 0 Jo= {0.733,0.900, 1.065,0.733, 1.325, 1.910)’ lo = {4.733,4.900, 5.065,4.733, 5.325,5.910}’ from which it is seen that there is no elongation, i.e., I - 6 = q” = 4.0, and no piston forces are needed to sustain the commanded pose simply because there is no gravity. The numerical results are presented in Fig. 5. The numerical values of those input parameters used for these illustrative cases are listed in Appendix 2. DISCUSSION
One of the characteristics of the control of flexible robotic manipulators is that the number of state variables is usually very large while the number of observations, limited by cost and feasibility, is much less than that. Therefore, it is impossible to have an acceptable feedback control based only on the observations, and it is then necessary to construct an observer whose state variables are estimates of the state variables of the plant. Based on the estimated state variables, the control inputs can be calculated by using the control law. Common to both serial and parallel link manipulators, the dynamic equations of the plant and the measurements are nonlinear and complex, However, in this work, no attempt has been made to approximate those equations. On the other hand, the ob-
server and the control law are derived based on the linear optimal control theory. Hence, strictly speaking, the control system only works in a small neighborhood around the operating point. This restriction could be relieved by adopting the two-stage control strategy as indicated by Lee [lo, 121. However, it is worthwhile to point out that finding a set of pre-determined control inputs to be employed in the first stage (coarse) control to bring the system close to the operating point is not a trivial task.
REFERENCES
1. D. Stewart, A platform with six degrees of freedom. Prac. Inst. Mech. Engrs 180, 371-386 (1965-1966). 2. E. F. Fichter, A Stewart platform-based manipulator: general theory and practical construction. Int. 1. Robofics Res. 5, 157-182 (1986). 3. S. E. Landsberger and T. B. Sheridan, A new design for parallel link manipulator. Proc. Systems Man and Cybernetics Con$, pp. 812-814, Tuscan, Arizona (1985). 4. C. Gosselin, Determination of the workspace of 6-DOF parallel manipulators. J. Mech. Des. 112, 331-336 (1990). 5. Z. Geng, L. S. Haynes, J. D. Lee and R. L. Carroll, On the dynamic model and kinematic analysis of a class of Stewart platforms. Robotics and Autonomous Systems 9, 237-254 (1992). 6. W. Q. D. Do and D. C. H. Yang, Inverse dynamic
analysis and simulation of a platform type of robot. J. Robotic Systems 5, 209-227 (1988). 7. J. D. Lee, J. S. Albus, N. G. Dagalakis and T. Tsai,
Computer simulation of a parallel link manipulator. Mnfg. 5, 333-342 (1989). 8. C. C. Nguyen and F. J. Pooran, Dynamic analysis of a 6 DOF CKCM robot end-effector for dual-arm telerobot Robotics Computer-Integr.
384
J. D. LEE systems. Robotics and Autonomous Systems 5, 377-394,
(1989). 9. Z. Geng, J. D. Lee, R. L. Carroll and L. S. Haynes,
Learning control system design based on 2-D theoryAn application to parallel link manipulator. IEEE Znr. Co& Robotics and Automation, pp. 1510-1515 (1990). 10. J. 6. Lee and B. L. Wang, Optimal control of a gexibie robot arm. Cornour. Sfruct. 29, 459-467 (1988). 11. J. D. Lee and B. L. Wang, Dynamic equations for a two-link flexible robot arm. Cornput. Strucf. 29, 469-477 (1988). 12. J. D. Lee, Applications of optimal control theory to flexible robotic manipulators. Robotics ComputerZntegr. Mnfg. 7, 327-335 (1990). 13. J. J. Craig, Introduction to Robotics: Mechanics and Conrrol. Addison-Wesley, Reading, MA (1989). 14. H. Goldstein, Classical Mechanics. Addison-Wesley, Reading, MA (1950). 15. H. Kwakemaak and R. Sivan, Linear Optimal Control Systems. John Wiley, New York (1972). 16. B. Friedland, Control System Design: An Introduction to State-Space Methods. McGraw-Hill, New York (1986). 17. F. B. Hildebrand, Methods of Applied Mathematics. Prentice-Hall, Englewood Cliffs, NJ (1965). APPENDIX
qj= R#X:+
1
Zr[m(Z2+S2+fS)+tiA2)/3 I = I@/2
= ZAJ2 - m(21 + 6)q#/61 + m(2Zi + 268 +
d al Z 0i@
is + 18)
-z=Z0;Aj/2-m(21+6)0$q,@/61
ae,
i-m(2li+268 -I-l&0;
d al -z= 522 0
+is
bfi,j,/31’ = 0,: ZZ,
-m(l+26)@/6
p = m (P + & + ii)/6
U,-fj
!!Ls..
aui p a9j
as=@‘i+ I
d ap zz 0
- f$ = m6/3 + m(q,@,/I + la)/6
V, = -gjM(R$,
+ U,)
-w = --MQ,g, 80,
V = -g,{m(l
I = (qjqj)“* $,=4J
;=
O;qj,,
g
+ S) + fiA}q,/2~
= -g,{mb,
+ (rnd + @iA)b,/Z’}/2
dV as=
-Q;gk{mS,+(mS
av as =
-mg,q,/21
I
i= qj4j/l
v* = k(/ - d - $)2/2
av+
w=k(‘-S
-$=4tlI $= O;qjp 6 = b&+,/P
av* ==
-k(l-6
-q”)Q;qj/l -rj”)
d = c(i - d)2/2
+fiA)b,j/l’}/2
Control of a parallel link robotic manipulator
g = c(i - d)Q;qj/l
385
$ = (-2,
-3.4641,O)
,
X6 = (0, -4,O) ad
z=
-c(i-b)
g=
(0, 0, -g)
g = 32.2 x 12 = 384.4 F = (0, 0,O)' M = (0, 0,O)' E = 1.0 x 1o-3 K =
1.0 x lo4
n = 24 APPENDIX 2
M=
10
n, = 6 n, = 6
X=(0,0,0)
I?,=30
%- -1 s: =4 km=200 c, = 2 ri& = 1 A,=4 x:
= (3.4641, -2,O)
xy = (2,3.4641, x:
0)
= (0,4,0)
Q2=&xJ CTj= 0.5
(i = 1,2, . . . , 12)
d,=5.0
(i=13,14
oi=o.15
x,* = (-4,0,0)
,...,
(i =25,26,.
24)
. .,30)
ratio(i)
= 0.00
(i = 1,2,. . . , 12)
ratio(i)
= 0.04
(i = 13, 14, 15)
2, = (4,0,0)
ratio(i)
= 0.01
(i = 16, 17, 18)
x* = (3.4641,2,0)
ratio(i)
= 0.04
(i = 19,20, . . . ,24)
x, = (-2,
ratio(i)
= 0.04
(i = 25,26,27)
ratio(i)
= 0.01
(i = 28, 29,301.
x:
= (-3.4641,
-2,O)
x,+ = (2, - 3.4641,O)
3.4641,O)
x,=(-3.4641,2,0)