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Centralized, decentralized, and independent control of a flexible manipulator on a flexible base
Acta Astronautica Vol. 29, No. 3, pp. 159-168, 1993 Printed in Great Britain.All rights reserved
0094-5765/93 $6.00+ 0.00 Copyright © 1993PergamonPress Ltd
CENTRALIZED, DECENTRALIZED, A N D I N D E P E N D E N T CONTROL OF A FLEXIBLE MANIPULATOR ON A FLEXIBLE BASEt~ FEIYUELI, PETER M. BAINUMand JIANKEXU Department of Mechanical Engineering, Howard University, Washington, DC 20059, U.S.A. (Received 5 February 1992; received for publication 16 November 1992)
Abstract--The dynamics and control of a flexiblemanipulator arm with payload mass on a flexible base in space are considered. The controllers are provided by one torquer at the center of the base and one torquer at the connection joint of the robot and the base. The nonlinear dynamics of the system is modeled by applying the finite element method and Lagrangian formula. Three control strategies are considered and compared, i.e. centralized control, decentralized control, and independent control. All these control designs are based on the linear quadratic regulator theory. A mathematical decomposition is used in the decentralization process so that the coupling between the subsystems is weak, while a physical decomposition is used in the independent control design process. For both the decentralized and the independent controls, the stability of the overall linear system is checked before a numerical simulation is initiated. Two numerical examples show that the responses of the independent control system are close to those of the centralized control system, while the responses of the decentralized control system are not.
1. INTRODUCTION To meet the design requirements of the Space Station, particularly when it is used as a space transportation node, the controls/structures interaction technology and the robotics/teleoperator technology are of current interest. The robots will play an important role in the autonomous space structural assembly work in the Space Station. The effect on the Space Station or spacecraft platform, due to the movement of the robot or manipulator during the assembly process such as the reaction forces/torques on the platform, needs to be analyzed. Also, a comprehensive control law needs to be designed for such a multi-body interaction system. The structural flexibility of large, light weight, space-based robots introduces many problems. For example, as demonstrated by Longman[1], robot vibrations will almost always produce a secular term in the torque applied by the robot to the satellite, and hence, these vibrations will try to tumble the spacecraft after the robot maneuver is completed. The extra torque disturbance due to vibrations must be canceled by the satellite attitude control system. Mah et al. [2] illustrated a numerical example of the motion of a two-body system (one main body and one appendage) within the framework of their general dynamics simulation program for multibodies. It was tPaper IAF-91-357 presented at the 42nd Congress of the International Astronautical Federation, Montreal, Canada, 7-11 October 1991. ~Research supported by NASA Grant NSG-1414; Mr Stanley E. Woodard, Technical Monitor NASA Grant NAGW-2950 and NASA/Howard Center for the Study of Terrestrial and Extraterrestrial Atmospheres..
shown that the translational motion and the slewing maneuver of the appendage (manipulator) influence the librational response of the main body due to a shift in the system center of mass and the satellite inertia matrix. On the other hand, Meirovitch and France [3] considered the maneuvers of a flexible antenna (cantilever beam) on a rigid platform. The platform was stabilized relative to inertia space. A pseudo-decentralized control and nonlinear (on-off) control techniques were proposed. All these researches indicate that there is a need to deeply investigate the dynamics and control issues asbout space systems involving flexible robots and articulated members. In this paper, we analyze three control strategies for the manipulator arm slewing control and the vibration suppression control of a simplified flexible platform-based articulated manipulator system.
2. SYSTEM CONFIGURATION The system configuration essentially consists of two flexible planar beams with tip masses and inertias, connected through a joint, J2, (Fig. 1). It is assumed that the first beam represents the simplified main bus structure of a space system, for example, the Geostationary Platform, while the second beam resembles a manipulator arm operated in a space environment. A rigid body, Pl, fixed on the left end of the main bus, through a fixed joint, JI, is a simplified model of an antenna system. Also, a mass with inertia, P2, attached to the tip of the manipulator represents the payload. Two control torquers, ut and u2, are assumed to be provided, ut is located at the mass center of the bus used for the attitude and vibration control of the 159
160
FEXYUELI et Payloa~ Antenna ~ P t A
01 ul
T J1
T,
~
......
""
Arm / .
: - .
J2
.'''"
....
Bus beam Deformed shape
Fig. I. Drawing of the platform and manipulator system. bus. u2 is placed at J2 and is used for the orientation and vibration control of the arm. In this paper, we assume that the masses of the robot arm and the payload are much less than the mass of the main bus beam. Therefore, the shift of the mass center of the system due to the motion of the robot arm is neglected. The mass center of the bus beam is chosen to be the mass center of the system. It is understood that this assumption can be removed without having substantial effect on the results obtained here.
al.
The nonlinear dynamics of the system will be modeled by applying the finite element method and the Lagrangian formula [4]. Other formulations can also be used such as those reviewed in [5]. Several coordinate systems are sketched in Fig. 2(a-c). OXY is an inertial reference system. O~ Xj Y~ is a body-fixed coordinate system with O~ located at the mass center of the bus beam. 02X: Y2 is the robot arm body-fixed coordinate system with 02 located at J2 [Fig. 2(a)]. The finite elements for each beam will have two nodal displacements (transverse, u+ji and ujj 3, and rotational, uo2 and u,j4) at its two ends [Fig. 2(b)]. 01 in Fig. 1 depicts the attitude angle of the bus, i.e. the angle between OXY and OiXj Y~. 02 is the rigid rotation angle of the arm measured from the tangent line at the right end of the bus beam element connected to J2.
3. KINETIC ENERGY AND POTENTIAL ENERGY The kinetic energy and the potential energy of the system can be written as
Xp2
a)
p_ r'~l, o~'l) X2 v
YP1
Y
~ /~"2 ~"..~...4~
'
(02) ~
X
P, ~ , ' f f (01)\m...~.~l'~/"~ ' J1 ~ tn (Opl)~ ~ -" 'ment j
u
I Oijl.,~.d ~ ' 'I I I xij I uij3
Oi
si
~ Xii - Xi
(3)
where Q. is the generalized force vector associated with the control. The kinetic energy Tlj for the j t h element of the bus beam may be computed as [4]:
Tjj = 2 J0 Pli'Tr dxlj
/~.~j 4
i ij
(2)
M4+Kq=~q(oTMq)'0 I --d-[ + ] O+Q,,
Tangent
Yi
V = V, + V 2 = ½qTKq
X1
- - A l j ~-
(b)
(1)
where T~, T2, Tpl, and Tp2 are the kinetic energies of the bus beam, the robot arm, mass, PI, and mass, P2, respectively. Vl and V2 are the potential energies of the bus beam and the robot arm, respectively. K is the stiffness matrix of the system. The Lagrangian equations of the system can be written as
Y ~L~x2i/
Element 1 " ~ l Xp1 ~
T = T~ + T2 + Tpi + Tp2 = ½qTMq
(4)
where r is the generic position vector (in matrix form) of a differential element of mass dm on the j t h finite element of the bus beam, Pl is the density of the bus beam, sl is the length of the j t h finite element, x~j is the dintance along the O~jXt j axis in the local element coordinate system, O~jX~j Ylj [Fig. 2(b)]. r is related to a body-fixed vector r~ by a transformation matrix
To': (c)
FCOS01
Yi I YPi
Oi
/
Pi
dm Xpi
r = Tlorl, T°l = Lsin 01
- s i n 01] cos 01
(5)
For the elements on the positive O1Xl side, the vector rl is given by
" Xi
Fig. 2. Coordinate systems. (a) Global coordinate systems. (b) Element coordinate system. (c) Rigid mass coordinate system.
r I = [(J-- l)sl + xlj 1 , j = 1, 2 . . . . . n 1/2. Y~j
(6)
where nl (even number) is the total number of elements of the bus beam; the deformation displacement, Ylj, can be described in terms of shape functions
Flexible manipulator
(utjk,k =
[4,6], Sk(Xu), and nodal displacements 1, 2, 3, 4) as: 4
~ Ok(Xtj)Uljk(1)
yU(xu, t)=
X2
X2
~z(x)=x
3~+2s~,
-
-
X3
0~(x)=3~--2
2 - sX2
-s3,
X3 +so-v
X3
04(x)=----+~-~.s
(8)
From eqns (5)-(8), and dr
dr
dr
where T~ is the transformation matrix from 05 X2 Y2 to 01Xt Yt, i.e T~ =
[ c~ -sY], ,2=02+um, s~
X3
~(x)=l
(7)
k=l
161
~r
1.
Or
c~
Here, Lj2 is the distance from O~ to J2 along the OtXt axis, um and um are the flexible transverse and rotational displacements of the bus beam at J2, respectively. The displacement Y2j has the same form as Yu in eqn (7) and the associated ~'k(X) are defined in eqn (8). After finding the time derivative of r
Or .
i'= -E~,OutitOuu2Outy,O-~u;jztj=~zt)
c~ = cos 72 . (16,17) s~ = sin ~2
f
(9)
=
-¢3r Or Or car ~gr
Orl
aura aura 002 Ou2jl • . du2/4]Z2J
dr
where
zTj=[OI
Utj t
Ulj 2
Utj 3
Utj4].
(10)
azlj e2j
where
Equation (4) can be expressed as ,~1
z~j=[O~
Or
(18)
uj2t
um
0 2 U2j t U2j 2 U2j 3 U2j4]
(19)
one can obtain = ~t z.Tt . / M t j Z •l j
where M u is a 5 x 5 matrix. Similarly, for the elements on the negative OtXt side, vector rt can be written as rl=
r--jst + Xtj],
L
YU
j=l,2,
nl . . . . 2-"
(12)
nl
t_.T Ji ~ j--iqlMlqt _l.r " Tt= ~ Tu= ~i~lj~U
j=l
(13)
j=l
w h e r e ql = [01 fiT)T, fit is t h e u n c o n s t r a i n e d n o d a l dis-
placement vector of the bus, and Mt is the associated mass matrix. Note that the flexible displacements at the middle point of the bus (the origin of the O1X~ Yt system) are constrained to be zero and, therefore, are absent from fil. The kinetic energy T2j for the j t h element of the robot arm may be computed as
=~
TEj
p2fTrdx2j
/'2=~ jffil
n2
T 2 j = ~ I - ~z~jl,,2j ' T Ai ~2j _--l ~q2 . r M2q2 • (21) if1
where q2 = [0t um um 05 fl~]T, t2 is the unrestrained nodal displacement vector of the arm, and M s is the associated mass matrix. Note that the flexible nodal displacements of the robot arm at J2 are constrained to be zero. The generic position vector of the differential element of mass, din, on the masses, PI and P2, can be expressed as, respectively,
rpl~ T~[-m ~'r-mJluPl-Jl+l TIPI[YX]}
(22)
rm=T~{[~:]+ T2(r\mL2up2, l + _]T~2[;])}
(23,
where Lsm ?Pl
where los is the density of the robot arm, x2j is the distance along the 02jX2j axis in the local element coordinate system, s2 is the length of the j t h element. The generic position vector, r, can be written as
rcos ~ T~ = Lsin T~
To~{[~] + T2r(J--l)s2+x2' Y2j ]}
j = 1, 2 . . . . . n2.
(15)
(20)
where M2j is an 8 x 8 matrix. Then, the kinetic energy for the robot arm can be obtained by the following summation:
(14)
r=
]
I "T = "~z2jM2jz2j
n2
It is easy to see from eqns (6) and (12) that by replacing j by 1 - j in the expressions for the righthand side elements, we can obtain the expressions for the left-hand-side elements. Then, the kinetic energy for the bus beam can be obtained by the following summation: nl
•
(11)
cos ?Ptd sin ~e2l ' cos TeLl
~'e,=Ue,~
(24)
?e~ = u~22
(25)
where LjI is the distance from O t to Jr, Uetl and uet2 are the flexible transverse and rotational displacements of the bus beam at Jr, respectively. Similarly, U~l and u~2 are the flexible transverse and rotational displacements of the robot arm at the joint, J3. x and y are the coordinates in the body-fixed systems, Oel Xel Yet
162
FEI'/UEL~ et al.
and Op2 X'p2 Yp:. Since/dpl 2 and Lip22 are small, T~t and T P2 can be approximately expressed as Tm~
I' v]
,
T~" ~
UpI2
i' v]
. (26)
UP22
Note that the masses, P~ and P2, are assumed to be fixed to the bus and the robot, respectively. Therefore, there will be no additional dynamic degrees of freedom associated with these masses. By using a procedure similar to that outlined above, one can arrive at Tp I = ~qPl i . T M p I t~pl '
Tp,. = ~qp2Mp2qp2 I -T •
m40 : ~b4 4- mp2 (22 C'~ 4- )"2 S:. ),
UpII
qp2 = [01
Uj21 L/j22 02
m57=~21,
ms0 :
mj+6,k+6=P~ik,
02
j=l,2;
m99 = P 3'3+ mm,
mgo = P ~'~+mm £2
where l l = ~ 2 m b i L ~ ",
flTIT.
2, 3,4
k=j,
moo = P Z + trp2 + mp2Xp2 (27)
! 2 12=~mb2L2
6,=L+rr,,
a , = f l ~ P , fl,,
(28)
and Mp: are the associated mass matrices. Note that there are some overlaps in the coordinates between q~, q2, qm, and qp:. To obtain eqn (1), we have to assemble the system mass matrix, M, by using MI, ME, Mm, and Mp:. The system generalized coordinate vector is arranged in the following order fit
m59=:z23+mp2(L2+22)
k =6, 7, 8, 9, 10
m6.k=ms.k,
i=1,2.
P4 = Lj2c,~ + uj21s~.
Pp = Lj2P~ + uj21Py,
Mpl
q=[O,
m58=~22,
~24 4- Op2 4- H'/P2(-~.p2Sp2 4- Pl )
L/PI2]T, Up21 1,/P22] T
ms.
m56 = 62 + ap2 + mp2(Sm + uml + 2pl)
where qpt = [01
m55 = p ~ +
' Px = 5m~2L2c;. - ~%s:, 1
py = ~rob2 L2 s,. + 4) gfl2 C:.
~=[q~.
~2 ~s ~ , ] = p 2 s : [ l
0 ½ --%]
[~u ~12 ~13 ~J4]=(PlS~/20)[ - 7
(29)
[c~2, ~¢22 ~23 ~2,]=P2S~[ 1 ~s2
In the numerical simulations of the present paper, we will use only two finite elements for each beam. In that case, both fll and 132 will be 4 x 1 vectors since the flexible displacements of the middle node of the bus beam and the node of the arm at Jz are constrained to be zero. The entries of the mass matrix of the system are listed in the following: roll = ¢~1 4- d2 4- m~z(L~2 + u~z) 4- 2p,
- -
7 --Sj]
,7
i = 1,2,
c
¢rpi - Iai + mr,iy~i,
SpI = LjI 4- Xpl ,
i=1,2.
.Vi ~ YPi 4- Xpil'lpi2,
2 i ~ Xpi -- ypi~lPi2,
Sp2 = L2 4- Xp2,
--Sj
ap = - - L j l y m UpI2 4- Upllffl e l = -L2yP2up22 + Up21Y2
Px2 = (L2 + 22)c,,. - (Up21-~ ff2)s,i Px2 = (L2 + 22)s~. + (up21 + 9:)c~;
4- ~Pl + rnpl (s~,l 4- u~,ll 4- 2ap)
P2 = Lj2Px2 4- Uj21P~2
4- O'p24- mp2(L~2 + S~2 + Uj21
P3
+ Upz, + 2p, + 2p~)
=
Pa22
--
(LJ2S~'- UJ21C~')ff2
and
m~2 = ~u + mPl (LJI + 2t ) m13 = 0~12+ apl 4- n'/pl (XpI Spl "~ ap)
, plsl PI = [Pa] =420-
m~4 = ~13 + rnb2Lj2 + mp2(LJ~ + Px2)
156 22sl 0
22sl 4s~ 0
0 0 156
0 ] 0 -22sl
0
0
-22s t
312
0 54 - 13s2 8s22 13s2 -3s22 156 - 22s2
(30)
4s~
m15 : ~14 4- m16 m16 = ~ : 4-pp 4- mr,2(s~2 -F u221 4- 2pl 4- P2) m17 = 0~214- ~ I P 4 ,
I ,,
m18 = ~22 4- q~2P4
m19 = ct23+ ~b3p. + mpz(Sp: + p, - y p : Up2~)
symmetric
mlo = ~:, + dp~p4 + ap2 + mp2(Xp2Sp2 +Pl 4-P3) m2z=p;l+mpl, m33=P22+apl
m~=p~s+mb2+mp
m45 ~- p ~ + m46, m46 = p~ + mpzpx: m47 -~- ~1C7,
m48 = q~2c).,
m49 = q~3c7 4-mp2c;,
(31)
4s~
The stiffness matrix of the system in eqn (2) can be obtained as follows:
m23=P~2+mpl£1 +mpiX21,
pzsz
P2 = [Pij] = 4 - ~
~
I°°°JI; ]1 oKb~
00
K~2
(32)
Flexible manipulator
EIt
K~l = S-'T
12
6sl
0
0
6sl
4s~
0
0
0
0
12
--6si
0
0
--6S 1
4Sl2
24
0
- 12
6S2
8s~
-6s 2 12
--6S 2
I
El2 ] Kb2---- S-~- /
Mc2 = (33)
2s~
Lsymmetric
4s22
M£1
[
ml~
m~
m~
0
0
0
0
0
0
0
0
0
m,~l
t~l
t~2
~3 q- mp2
t~4"~-mp2Xp2
~21
~22
~X23At-mp2Sp2
L2 m 15
(39) mC32= g22 + (~2Lj2
m ~ = ~23 + O3Lj2 + mp2(Lj2 + sp2)
where
~12 "~ 0.Pl "4- msl Xpl
~13
(XI4
P I2 + mpl Xpt
0
0
Ph+aPl+mplx~l
0
0
Ph
Ph
P~, + mpl
ml~
m ~ = ~21 "[- ~,Lj2,
L]+r
~x,l+m,l
mC~ 0
(34)
The linearized mass matrix (constant) for the systern configuration 0~ = 0 and 02 = 0 can be expressed as follows:
mf~
163
ph
symmetric
(36) rolL( = II "4- 0.PI "~- met S2pI, msl = mpl Sp!
m ~ = 0~24+ ~4Lj2 + 0"1,2 + mp2xp2(Lj2 + Sp:).
Mass matrices for the subsystems
In the control law design that follows, we need to use two subsystem models to develop the independent control law. One subsystem consists of the bus beam and the mass, P,, only. The constant mass matrix of the subsystem is MLI in eqn (36). The associated stiffness matrix is just K~ defined in eqns (32) and (33). The other subsystem contains the remaining parts of the system, i.e. the robot arm and the payload mass, P2. This subsystem is modeled as a one arm link with the payload mass fixed to one of its ends. The other end of the arm can rotate about a fixed joint. By using the same technique for the development of the kinetic energy of the bus beam and mass, P,, one can prove that this mass matrix is ML2 in eqn (37), and the associated stiffness matrix is/(2 in eqns (32) and (34).
ML2
4. LINEAR DYNAMIC EQUATIONS
m~
0C21
tZ22
p;~
p;':
~t23+mp2sp2
L2 m15
p;~
p;',
After substituting the mass matrix, ML, and the generalized force, Qu, into eqn (3), one can obtain the linearized equations of the system
P 34+ mP2Xp2
MLt] = --Kq + B t u
P~'2
P~3
P~'3 +mp2 symmetric
/t
p24 + m n x ~2
(40)
where
(37) BL= m ~ = 12 + o n + m ~ s ~ ,
m~
0C24"[" 0.P2 "[- mp2XF2SI~
=
0 ~' [lo °0 °0 °0 °0 °1 °0 °0 °0 °]
l
(41)
U2
m cl 0
0
0
Mci
m cl
mtC~
0
0
o
o
mb2 + m ~
mf~
L symmetric
5. CONTROL LAW DESIGN
(38)
m~
m cl = I2+mb2Lj2(L2+Lj2) + op2 + rap2(Lj2 + sp2)2
In this paper, only the linear control laws based on the linear models of the system will be discussed. However, the validation of the control law will be tested numerically by using the original nonlinear model of the system, eqn (3). 5.1. Centralized control
m ° = mb2(L2/2 + Lj2) + mp2(Lj2 + sp2) m cl = 12 + (l/2)mb2Lj2L2 + 0.p2+ mp2sp2(Lj2 + Sp2) mC~ = (1/2)mb2L2 + mmsm AA29D--C
Equation (40) can be rewritten in the following state form: {~=y (wherex=q) = - M z t K x + M Z l B t u = A x + Bu.
(42)
['tlYUt [.1 el a[.
164
To minimize a given cost func6onal J=
(43)
,, (:JQz + u r R u ) d t
values are, therelbre, associated with the subsystem consisting of the robot arm and the mass, P:, onl). Let the (permutation) matrix while will rearrange the eigenvalues be denoted b\ P Thus, we havc
where : = [x*yI] j and
i< ~ ~ t,;
i i -I>,,t,/' <, _ i B = PB,,,
"i/:-- pi:
Q
[Q~
,
R=
r,:
r
(44)
the standard linear quadratic regulator technique is employed here to obtain the feedback gain matrix, K , and the control law can be expressed as u = - Ai::
(45)
5.2. Decentralized control
The decentralized control technique by using the subsystem decomposition approach discussed in [7] will be used here. The philosophy of this method is first to decompose mathematically the whole coupled system into several subsystems with relatively "weak" connections between them. By disregarding these connections, these subsystems will be decoupled from each other. Then, for each small-sized subsystem a control law can be designed independently via many advanced and effective control design methods. Finally, a stability test needs to be performed for the whole closed-loop system. Many other decentralized techniques applied to robotic systems can be found from the literature [7-9]. For example, Yuan and Book [9] proposed a decentralized adaptive control for a two-link, rigid, ground-based robotic system. The authors consider that the key in the decentralized control with the subsystem decomposition approach is how to decompose the system. There are many approaches from which we can choose, for example, the input-decentralization scheme discussed in Section 3.2 of [7] and Section 7.2 of [10]. But, by using this scheme, we end up with very large matrix entries which will make the subsequent subsystem control law design very difficult. However, in this paper, we propose the following decomposition approach which can result in a stable closed-loop system. Step 1. The matrix A in eqn (42) can be diagonalized by using a similarity transformation matrix, V
V-ly
then {~
B,,~= I / ~ I B
=f = A,.~£ + B,,,u.
(47)
Step 2. Since the ten eigenvalues of the whole system along the diagonal of matrix A are usually written in ascending order, we need to reorder them so that the first five eigenvalues are close to the eigenvalues of the subsystem composed of the bus beam and the mass, P~, only. The remaining five eigen-
(48)
then
(49) Note that P = P~, and after this reordering, the matrix A is still diagonal. Step 3. A rearranging matrix, A~, is generated to transform B into
hi,.
bl2. )
hs)
b~,
h1
0
()
h6
6
b'H)
o
bm,l bio2
=
B.
(50)
This transformation can be performed by first zeroing the last five entries of column 1, and then eliminating the upper five entries of column 2, or vice versa. For the elimination in each column, the element with maximum absolute value will be selected to perform the elimination. This process is similar to the Gaussian elimination and pivot selection in solving algebraic equations [11]. By calculating .~ =.4~: JA = A~.4A r ' I.V = A~)? ~/~ = a~/~
(51)
we can obtain
/.{ = f
(52)
= ,i.i +
where
=I
*:1
-
(53)
It can be seen that dtj and d22 have the largest nonzero elements predominantly along the principal diagonal, and that most of the entries of all2 and ,'i2~ will be zero because the number of controls is much less than the dimensionality of the system. Also, the magnitude of most of the nonzero entries of/]21 (all2) will be less than the magnitude of the entries of ,,i H (A22) because of the pivot selection in the elimination process, eqn (50). By neglecting ,4J2 and d2~, we obtain two subsystems
{
2~ = ¢ j
Y,
"t,,-fl + Blu,
22 = .f'2 !52 = e422.• 2 -~- B2u2 •
(54)
(55)
Flexible manipulator F o r each subsystem, the LQR technique is applied to derive the controls which will minimize
J,=
(#TO,~+rllU~)dt
(56)
&=
( ~ 0.2~_:+ r=u~) dt
(57)
where zl = [XlT ¢~]T, ~2 = [-/~ )~]s; and rl~ and r= are defined in eqn (44). To compare the responses of the system with different control logic, we need to use the QI and Q2 which are comparable to Q in eqn (43). From Step 1 to Step 3, we can obtain the relation between z and ~
z:[;]:[:
0
2
0 .
where [x'] ~ =
~2'
~ = ['9' 1 "
T = V P 'A:'.
(59)
LY2]'
Then by using the transformation in eqn (58), we have
zTQz =
~T[TIQuT T~QI2T]~ LTTQT2T TTQz2T z [fir
H
e.
(6o)
By letting G _[Gn
G~21, H _ ~ Hll H~21
o22j
-LHT2 H223'
I-F,, 1121 -LI2, 122]
(61)
and using eqn (59), we can obtain
[Gn 1.1
_l-G2~ ,=]
0'=LIT~ H,d' O-2-LIT= H2=J
(62)
Note that other semi-definite 01 and (~z can also be used here.
5.3. Independent control Instead of mathematically decomposing the system as in the last section, we will use a decomposing method which is based on the physical configuration of the system. This idea has been introduced by Young [12] and applied to the structural and control design problem, called controlled component synthesis. This concept has also been used in [3], called substructure decentralized control, for the maneuvering of a flexible antenna on a rigid platform. However, we will use this idea for a different structural configuration and develop a different approach. In the present paper, the system is composed of two component parts: (1) the bus beam and the fixed mass, PI and (2) the robot arm and the payload mass, P2. The control law will be designed for each part independently as if they were disconnected, or isolated.
165
Certainly, the stability of the original connected system should be guaranteed upon integrating the two subsystems together. The difference between the current separation method and the decomposition method in the decentralized control is that one subsystem derived by using the decomposition method will contain some information about another subsystem, while the subsystem defined in the independent control method here does not include any information about the other subsystem. The equations of the two subsystems can be written as MIAql = - KI ql + Ul
(63)
Mt.2g]2= --K2q2 + u2
(64)
where KI, /<2, M u , and ML= are defined in eqns (32)-(34), (36), and (37). The performance cost for each subsystem will be the same as used in the centralized control except that the weighting on the cross coupling term will not be used. Again, the poles of the original, linear closed-loop system are examined to ensure the stability of the whole system with these two independent controls. It can be seen that the current method involves less mathematics. Note that the substructure component in [12] is a part of a continuous structure (truss structure) and there is no relative rigid motion between the two substructure components. But, in the current problem, a rigid relative rotation is allowed between the interconnected subsystems. On the other hand, in [3], the equations of motion for the substructures were represented by the modal coordinates, and the number of actuators was assumed to be larger than or equal to the number of controlled modes. In the current paper, the equations of motion in the physical coordinates are used and the number of controllers is less than the number of dynamic degrees of freedom of the system. Also, the decomposition step here, in the independent control approach, is accomplished before the inverse of the mass matrix, ML, is calculated. On the other hand, in the decentralized control approach, the decomposition step is performed after this inverse is obtained. 6. NUMERICAL RESULTS
The structural parameters for the example simulated in this paper can be summarized as: bus length, LI = 30m; manipulator arm length, L: = 6 m; bus mass, mb~ = 600 kg; arm mass, rob2 = 18 kg; antenna mass, m m = 40 kg; payload mass, me2 = 36 kg; bus rigidity, E11 = 5E5 N-m2; arm rigidity, E12 = 320N-mS; bus density, p~ = 20 kg/m; arm density, P2 = 3 kg/m; the moments of inertia for the bus, the arm, mass P~, and mass P2 are, respectively (kg-m2): I I = 4.5E4, 15 = 216, I m = 562.5, Ip2 = 5; the locations of the center of mass of P~ and P2 in the local coordinate system are (m): xm = 1.5, YPt = 1, xp2 = 0, YP2 = 0.25.
I:EI~'UE LI el al.
166
The simulation responses plotted in Fig. 3 are tor the following case. It is assumed that the initial position of the robot arm is in the vertical position, i.e. 0 : = 9 0 d e g . Then, the arm is rotated to the horizontal position, 02 - 0, while the attitude of the bus is kept at 0~ = 0. For this case, the state weighting matrix, Q, is a unit matrix, and the control weighting matrix, R, is a diagonal matrix, R-=-Diag (0.001, 0.001). To compare the three methods discussed in this (a) Attitude, 01 (deg)
0
x
[ -4 [ - --8 ~ -12 ~
--~-----
D
Lin ctr. _ Centr. x Decen. , lndep,
0
i0
'
x ~ x ~ _ i
I 30
20
~ ] 40
(b) Arm rotation, 02 (deg) 90 ~- x -~ , . , ~ 60 ~ - ~
|
"~\
0
i0
30
[] --
L i n ctr. Centr.
× -
Decen. •
20
30
(C) Bus right end disp. (m) 0.04 1-[] Lin ctr. 0.02 | x - Centr.
r*
o
v ,r r v
40
× -
yVV
Decen. lndep.
V
-0.02 -0.04
|
I 20
l tO
0
_
i 30
I
40
paper, four responses of these systems are plotted. The first is the linear case for which the centralized control is used. All remaining cases are the nonlinear responses of the system by using the centralized, the decentralized, and the independent controls. Figure 3(a-f) plot the responses for 0~, 02, the transverse displacements of the right end of the bus beam and the manipulator arm tip, the bus torque, u~, and the arm joint torque, u2, respsectively. It can be seen that the responses for the linear case, centralized, and the independent controls are very close to each other. But the responses for the decentralized control have large differences from all the other responses. It is not surprising that the decentralized control shows a relatively poor response as compared with the centralized control design, because the decentralization process eliminates some coupling information. One could argue that the independent control also discards some connection terms, but it shows relatively good results. This may be explained by the fact that the decentralization neglects some information which is very significant, while the independent control design does not. In other words, the information retained in the subsystems, after the decentralization, does not represent the main stream of the whole system very well, or the decomposition process does not distribute the information very well in the decentralization method for the example considered here. The similarities and differences between the three controls can also be found by examining the pole locations of the three linear closed-loop systems, plotted in Fig. 4(a) (large negative real parts) and
(d) Arm tip disp. (m) 1 2 ~,x • Ilk.., 0.8 [J¢~
D --
~
×
04[
III
(a)
L m ctr. Centr.
Centr. + Indep. x Decen. o
Decen.
-
Indop
x
40
+
O+x
-0.4 I
I ~
0
~
l0
20
(e) Bus torque, u 1 (N-m) 60 ~x [] Lin ctr. 30 [-k Centr.
0~ -30 F
,
~
x'X. . . .
-60 ~0
~'X~x/X" 10
~
I 40 - - x.--..-o
x -
@) - ×4
x O+x
l
I
J
20
30
40
-20 K-~.~J¢,, -40 1-'~ x
L
I
i
i
-1.6
-1.3
-1.0
-0.7
xl
o
'~
• ~ I -20
+
-0.4
-40 -0.1
x~
I 20 Time (s)
o Centr. + Indep. x Decen.
x-
x
r
(b) --
[]
I 10
@ @
+
Decen. Indep.
_
20
x~
O+ 0 0 O+
rx/
(f) Arm torque, u 2 (N-m) 20 0 L . ~ x ×~., ~ x m ' ° I ' x ' ' a ' ~ x ~ ° I
-60 { - -80 / 0
I 30
_i I 30
Lin ctr. Centr. Decen. Indep. I 40
Fig. 3. Transient responses for the slew from 02 = rG2 to 02 = O.
10
-- 5
o x
-------o
[M--- x - -
0
I
I
I
I
-0.10
-0.08
-0.06
-0.04
x
-10
I -0.02
'"
~
o×
0
Real
Fig. 4. Pole locations of the closed-loop systems. (a) Large real parts; (b) small real parts.
Flexible manipulator
Fig. 4(b) (small negative real parts). Each system has 20 poles, but the pole with the largest negative real part ( - 5 3 2 . 2 for the centralized, -532.1 for the decentralized, and -559.0 for the independent) is not shown. All the poles of the three systems have negative real parts, indicating that the closed-loop system are asymptotically stable. It can be seen that (a) Attitude, 01 (deg)
I -10
"xx
I [
.,-1"
~'~x
0
Centr.
I
I
~.x/
x ;
Decen.
10
20
30
lndep. I 40
(b) Arm rotation, 02 (deg)
9o [-
×
60 ~ 30 L_ 0~
~
~ x ~ _/x~./ ~'f
x --1
0 10 20 (C) Bus right end disp. (m) 0.08~ 0.04 0
f~
~L~'~
1"~ ~
. .
0
.
10
Indep"
'
3O
40 Centr. Decen.
×
x Indep.
~ I ~ L . ~
20
, 40
30
(d) Arm tip disp. (m)
o[
-0.8
,<
0
10
20
30
40
(e) Bus torque, u I (N-m) 100 o
x
x\ ^ r' - . ~ - x - - - ~ - x ~ - , ~ - ~ < , _ . , . . . × , ~ , .
-100
-200 -300 -400 |
-500
0
[
[
10
20
-40
0
I" 10
(I, I, l, l, l, 0.5, I, I, I, I, I, I, I, l, l, l, l, I, l, I), while the stateweighting matrices for the decentralized control arc, Ql = Diag (70, I, l, l, l, l, l, I, I, I) and Q2 = Diag (0.2, l, l, l, l, l, I, I, l, l). The control weighting matrix is the same for all three control designs, R = Diag (0.001,0.001). Again, the independent control responses are close to those of the centralized control.
[
30
7. CONCLUSIONS
Three control design strategies: namely, centralized, decentralized, and independent controls, are considered. A mathematical decomposition method for the decentralized control approach and a physical decomposition method for the independent control are proposed. It is shown that the space robot control problem can be examined within the decentralized control and the independent control framework. The results for the independent control and the centralized control are close to each other. Therefore, the independent method deserves some credit because of the small dimensionality of each subsystem and the ease of obtaining the subsystem equations as compared with the development for the deccntralized method. The decentralized control shows a relatively poor response here. A more elaborate selection of the algebraic transformation may lead to a better result. For further research related to the decentralized and the independent controls, the selections of different decomposition methods and different weighting matrices in the cost functions are recommended.
REFERENCES
Centr. x__ Decen. -Indep.
(f) Arm torque, u 2 (N-m) 40 V L x 20 W ~ ' - - . / I A ---
-2o I
the majority of the poles for the independent control arc closer to the associated poles of the centralized control than those of the decentralized control. Therefore, the responses of the independent control are close to the responses of the centralized control. It is understood that the locations of the poles can also bc changed if different wcightings, Q and R, arc selected. Figure 5(a-f) show the responses for another case in which the arm is rotated from 02 = 0 to 02 = n/2. In this case, the state weighting matrix for the centralizcd control and independent control is Q = Diag
Centr.
Decen. I
167
I 40
Centr. Decen.
Indep,
[
I
I
20 Time (s)
30
40
Fig. 5. Transient responses for the slew from 02 = 0 to 02 = it/2.
I. R. W. Longrnan, Attitude tumbling due to flexibility in satellite-mounted robots. J. astronaut. Sci. 38, 487-509 (1990). 2. H. W. Mah, V. J. Modi, Y. Morita and H. Yokota, Dynamics during slewing and translational meneuvers of the space station based MRMS. J. astronaut. Sci. 38, 557-579 (1990). 3. L. Meirovitch and M. E. B. France, Discrete-time control of a spacecraft with retargetable flexible antennas. Advances in the Astronautical Sciences. Guidance and Control, pp. 119-143, Paper AAS 89-007 0989).
4. P. B. Usoro, R. Nadira and S. S. Mahil, A finite element/Lagrange approach to modeling lightweight flexible manipulators. J. Dynam. Syst. Msmt Control Trans. A S M E 108, 198-205 0986). 5. P. Th. L. M. Van Woerkom and M. Guelman, Dynamics modeling, simulation and control of a
168
6. 7. 8.
9.
J;'~IY[It El ~'[ a/.
spacecraft/manipulator system. Proceedings q! the First European In-Orbit Operations Technology Symposium, Darmstadt, Germany, ESA SP-272 (1987). R. R. Craig Jr, Structural Dynanlic.s, 4n bltroduction to Computer Methods. Wiley, New York ( 1981 ). D. D. ~iljak, Large-Scale D)'namic Systems, Stability and Structure. North-Holland, New York (1978). H. Tamura and T. Yoshikawa~ l,arge-Scale Systems Control and Decision Makin#,,. Marcel Dekker, New York (1990). B-S. Yuan and W. J. Book, Decentralized adaptive
control of robot manipulators with robust stabilization design. Proceedings of the American Control Conference, Atlanta, Ga, Vol. 1, pp. 102 107 (1988). I0. C-T. (Then, Linear ,~vstem Theo O' and Design. Holt, Rinehart & Winston, New York (1984). 1 I. J. A. Cochran, Applied Mathematics, Principles, Techniques. and Applications, Chap. 1. Wadsworth (1982). 12. K. D. Young, Distributed finite-element modeling and control approach for large flexible structures. J. Guidance Control Dynam. 13, 703 713 (1990).
0094-5765/93 $6.00+ 0.00 Copyright © 1993PergamonPress Ltd
CENTRALIZED, DECENTRALIZED, A N D I N D E P E N D E N T CONTROL OF A FLEXIBLE MANIPULATOR ON A FLEXIBLE BASEt~ FEIYUELI, PETER M. BAINUMand JIANKEXU Department of Mechanical Engineering, Howard University, Washington, DC 20059, U.S.A. (Received 5 February 1992; received for publication 16 November 1992)
Abstract--The dynamics and control of a flexiblemanipulator arm with payload mass on a flexible base in space are considered. The controllers are provided by one torquer at the center of the base and one torquer at the connection joint of the robot and the base. The nonlinear dynamics of the system is modeled by applying the finite element method and Lagrangian formula. Three control strategies are considered and compared, i.e. centralized control, decentralized control, and independent control. All these control designs are based on the linear quadratic regulator theory. A mathematical decomposition is used in the decentralization process so that the coupling between the subsystems is weak, while a physical decomposition is used in the independent control design process. For both the decentralized and the independent controls, the stability of the overall linear system is checked before a numerical simulation is initiated. Two numerical examples show that the responses of the independent control system are close to those of the centralized control system, while the responses of the decentralized control system are not.
1. INTRODUCTION To meet the design requirements of the Space Station, particularly when it is used as a space transportation node, the controls/structures interaction technology and the robotics/teleoperator technology are of current interest. The robots will play an important role in the autonomous space structural assembly work in the Space Station. The effect on the Space Station or spacecraft platform, due to the movement of the robot or manipulator during the assembly process such as the reaction forces/torques on the platform, needs to be analyzed. Also, a comprehensive control law needs to be designed for such a multi-body interaction system. The structural flexibility of large, light weight, space-based robots introduces many problems. For example, as demonstrated by Longman[1], robot vibrations will almost always produce a secular term in the torque applied by the robot to the satellite, and hence, these vibrations will try to tumble the spacecraft after the robot maneuver is completed. The extra torque disturbance due to vibrations must be canceled by the satellite attitude control system. Mah et al. [2] illustrated a numerical example of the motion of a two-body system (one main body and one appendage) within the framework of their general dynamics simulation program for multibodies. It was tPaper IAF-91-357 presented at the 42nd Congress of the International Astronautical Federation, Montreal, Canada, 7-11 October 1991. ~Research supported by NASA Grant NSG-1414; Mr Stanley E. Woodard, Technical Monitor NASA Grant NAGW-2950 and NASA/Howard Center for the Study of Terrestrial and Extraterrestrial Atmospheres..
shown that the translational motion and the slewing maneuver of the appendage (manipulator) influence the librational response of the main body due to a shift in the system center of mass and the satellite inertia matrix. On the other hand, Meirovitch and France [3] considered the maneuvers of a flexible antenna (cantilever beam) on a rigid platform. The platform was stabilized relative to inertia space. A pseudo-decentralized control and nonlinear (on-off) control techniques were proposed. All these researches indicate that there is a need to deeply investigate the dynamics and control issues asbout space systems involving flexible robots and articulated members. In this paper, we analyze three control strategies for the manipulator arm slewing control and the vibration suppression control of a simplified flexible platform-based articulated manipulator system.
2. SYSTEM CONFIGURATION The system configuration essentially consists of two flexible planar beams with tip masses and inertias, connected through a joint, J2, (Fig. 1). It is assumed that the first beam represents the simplified main bus structure of a space system, for example, the Geostationary Platform, while the second beam resembles a manipulator arm operated in a space environment. A rigid body, Pl, fixed on the left end of the main bus, through a fixed joint, JI, is a simplified model of an antenna system. Also, a mass with inertia, P2, attached to the tip of the manipulator represents the payload. Two control torquers, ut and u2, are assumed to be provided, ut is located at the mass center of the bus used for the attitude and vibration control of the 159
160
FEXYUELI et Payloa~ Antenna ~ P t A
01 ul
T J1
T,
~
......
""
Arm / .
: - .
J2
.'''"
....
Bus beam Deformed shape
Fig. I. Drawing of the platform and manipulator system. bus. u2 is placed at J2 and is used for the orientation and vibration control of the arm. In this paper, we assume that the masses of the robot arm and the payload are much less than the mass of the main bus beam. Therefore, the shift of the mass center of the system due to the motion of the robot arm is neglected. The mass center of the bus beam is chosen to be the mass center of the system. It is understood that this assumption can be removed without having substantial effect on the results obtained here.
al.
The nonlinear dynamics of the system will be modeled by applying the finite element method and the Lagrangian formula [4]. Other formulations can also be used such as those reviewed in [5]. Several coordinate systems are sketched in Fig. 2(a-c). OXY is an inertial reference system. O~ Xj Y~ is a body-fixed coordinate system with O~ located at the mass center of the bus beam. 02X: Y2 is the robot arm body-fixed coordinate system with 02 located at J2 [Fig. 2(a)]. The finite elements for each beam will have two nodal displacements (transverse, u+ji and ujj 3, and rotational, uo2 and u,j4) at its two ends [Fig. 2(b)]. 01 in Fig. 1 depicts the attitude angle of the bus, i.e. the angle between OXY and OiXj Y~. 02 is the rigid rotation angle of the arm measured from the tangent line at the right end of the bus beam element connected to J2.
3. KINETIC ENERGY AND POTENTIAL ENERGY The kinetic energy and the potential energy of the system can be written as
Xp2
a)
p_ r'~l, o~'l) X2 v
YP1
Y
~ /~"2 ~"..~...4~
'
(02) ~
X
P, ~ , ' f f (01)\m...~.~l'~/"~ ' J1 ~ tn (Opl)~ ~ -" 'ment j
u
I Oijl.,~.d ~ ' 'I I I xij I uij3
Oi
si
~ Xii - Xi
(3)
where Q. is the generalized force vector associated with the control. The kinetic energy Tlj for the j t h element of the bus beam may be computed as [4]:
Tjj = 2 J0 Pli'Tr dxlj
/~.~j 4
i ij
(2)
M4+Kq=~q(oTMq)'0 I --d-[ + ] O+Q,,
Tangent
Yi
V = V, + V 2 = ½qTKq
X1
- - A l j ~-
(b)
(1)
where T~, T2, Tpl, and Tp2 are the kinetic energies of the bus beam, the robot arm, mass, PI, and mass, P2, respectively. Vl and V2 are the potential energies of the bus beam and the robot arm, respectively. K is the stiffness matrix of the system. The Lagrangian equations of the system can be written as
Y ~L~x2i/
Element 1 " ~ l Xp1 ~
T = T~ + T2 + Tpi + Tp2 = ½qTMq
(4)
where r is the generic position vector (in matrix form) of a differential element of mass dm on the j t h finite element of the bus beam, Pl is the density of the bus beam, sl is the length of the j t h finite element, x~j is the dintance along the O~jXt j axis in the local element coordinate system, O~jX~j Ylj [Fig. 2(b)]. r is related to a body-fixed vector r~ by a transformation matrix
To': (c)
FCOS01
Yi I YPi
Oi
/
Pi
dm Xpi
r = Tlorl, T°l = Lsin 01
- s i n 01] cos 01
(5)
For the elements on the positive O1Xl side, the vector rl is given by
" Xi
Fig. 2. Coordinate systems. (a) Global coordinate systems. (b) Element coordinate system. (c) Rigid mass coordinate system.
r I = [(J-- l)sl + xlj 1 , j = 1, 2 . . . . . n 1/2. Y~j
(6)
where nl (even number) is the total number of elements of the bus beam; the deformation displacement, Ylj, can be described in terms of shape functions
Flexible manipulator
(utjk,k =
[4,6], Sk(Xu), and nodal displacements 1, 2, 3, 4) as: 4
~ Ok(Xtj)Uljk(1)
yU(xu, t)=
X2
X2
~z(x)=x
3~+2s~,
-
-
X3
0~(x)=3~--2
2 - sX2
-s3,
X3 +so-v
X3
04(x)=----+~-~.s
(8)
From eqns (5)-(8), and dr
dr
dr
where T~ is the transformation matrix from 05 X2 Y2 to 01Xt Yt, i.e T~ =
[ c~ -sY], ,2=02+um, s~
X3
~(x)=l
(7)
k=l
161
~r
1.
Or
c~
Here, Lj2 is the distance from O~ to J2 along the OtXt axis, um and um are the flexible transverse and rotational displacements of the bus beam at J2, respectively. The displacement Y2j has the same form as Yu in eqn (7) and the associated ~'k(X) are defined in eqn (8). After finding the time derivative of r
Or .
i'= -E~,OutitOuu2Outy,O-~u;jztj=~zt)
c~ = cos 72 . (16,17) s~ = sin ~2
f
(9)
=
-¢3r Or Or car ~gr
Orl
aura aura 002 Ou2jl • . du2/4]Z2J
dr
where
zTj=[OI
Utj t
Ulj 2
Utj 3
Utj4].
(10)
azlj e2j
where
Equation (4) can be expressed as ,~1
z~j=[O~
Or
(18)
uj2t
um
0 2 U2j t U2j 2 U2j 3 U2j4]
(19)
one can obtain = ~t z.Tt . / M t j Z •l j
where M u is a 5 x 5 matrix. Similarly, for the elements on the negative OtXt side, vector rt can be written as rl=
r--jst + Xtj],
L
YU
j=l,2,
nl . . . . 2-"
(12)
nl
t_.T Ji ~ j--iqlMlqt _l.r " Tt= ~ Tu= ~i~lj~U
j=l
(13)
j=l
w h e r e ql = [01 fiT)T, fit is t h e u n c o n s t r a i n e d n o d a l dis-
placement vector of the bus, and Mt is the associated mass matrix. Note that the flexible displacements at the middle point of the bus (the origin of the O1X~ Yt system) are constrained to be zero and, therefore, are absent from fil. The kinetic energy T2j for the j t h element of the robot arm may be computed as
=~
TEj
p2fTrdx2j
/'2=~ jffil
n2
T 2 j = ~ I - ~z~jl,,2j ' T Ai ~2j _--l ~q2 . r M2q2 • (21) if1
where q2 = [0t um um 05 fl~]T, t2 is the unrestrained nodal displacement vector of the arm, and M s is the associated mass matrix. Note that the flexible nodal displacements of the robot arm at J2 are constrained to be zero. The generic position vector of the differential element of mass, din, on the masses, PI and P2, can be expressed as, respectively,
rpl~ T~[-m ~'r-mJluPl-Jl+l TIPI[YX]}
(22)
rm=T~{[~:]+ T2(r\mL2up2, l + _]T~2[;])}
(23,
where Lsm ?Pl
where los is the density of the robot arm, x2j is the distance along the 02jX2j axis in the local element coordinate system, s2 is the length of the j t h element. The generic position vector, r, can be written as
rcos ~ T~ = Lsin T~
To~{[~] + T2r(J--l)s2+x2' Y2j ]}
j = 1, 2 . . . . . n2.
(15)
(20)
where M2j is an 8 x 8 matrix. Then, the kinetic energy for the robot arm can be obtained by the following summation:
(14)
r=
]
I "T = "~z2jM2jz2j
n2
It is easy to see from eqns (6) and (12) that by replacing j by 1 - j in the expressions for the righthand side elements, we can obtain the expressions for the left-hand-side elements. Then, the kinetic energy for the bus beam can be obtained by the following summation: nl
•
(11)
cos ?Ptd sin ~e2l ' cos TeLl
~'e,=Ue,~
(24)
?e~ = u~22
(25)
where LjI is the distance from O t to Jr, Uetl and uet2 are the flexible transverse and rotational displacements of the bus beam at Jr, respectively. Similarly, U~l and u~2 are the flexible transverse and rotational displacements of the robot arm at the joint, J3. x and y are the coordinates in the body-fixed systems, Oel Xel Yet
162
FEI'/UEL~ et al.
and Op2 X'p2 Yp:. Since/dpl 2 and Lip22 are small, T~t and T P2 can be approximately expressed as Tm~
I' v]
,
T~" ~
UpI2
i' v]
. (26)
UP22
Note that the masses, P~ and P2, are assumed to be fixed to the bus and the robot, respectively. Therefore, there will be no additional dynamic degrees of freedom associated with these masses. By using a procedure similar to that outlined above, one can arrive at Tp I = ~qPl i . T M p I t~pl '
Tp,. = ~qp2Mp2qp2 I -T •
m40 : ~b4 4- mp2 (22 C'~ 4- )"2 S:. ),
UpII
qp2 = [01
Uj21 L/j22 02
m57=~21,
ms0 :
mj+6,k+6=P~ik,
02
j=l,2;
m99 = P 3'3+ mm,
mgo = P ~'~+mm £2
where l l = ~ 2 m b i L ~ ",
flTIT.
2, 3,4
k=j,
moo = P Z + trp2 + mp2Xp2 (27)
! 2 12=~mb2L2
6,=L+rr,,
a , = f l ~ P , fl,,
(28)
and Mp: are the associated mass matrices. Note that there are some overlaps in the coordinates between q~, q2, qm, and qp:. To obtain eqn (1), we have to assemble the system mass matrix, M, by using MI, ME, Mm, and Mp:. The system generalized coordinate vector is arranged in the following order fit
m59=:z23+mp2(L2+22)
k =6, 7, 8, 9, 10
m6.k=ms.k,
i=1,2.
P4 = Lj2c,~ + uj21s~.
Pp = Lj2P~ + uj21Py,
Mpl
q=[O,
m58=~22,
~24 4- Op2 4- H'/P2(-~.p2Sp2 4- Pl )
L/PI2]T, Up21 1,/P22] T
ms.
m56 = 62 + ap2 + mp2(Sm + uml + 2pl)
where qpt = [01
m55 = p ~ +
' Px = 5m~2L2c;. - ~%s:, 1
py = ~rob2 L2 s,. + 4) gfl2 C:.
~=[q~.
~2 ~s ~ , ] = p 2 s : [ l
0 ½ --%]
[~u ~12 ~13 ~J4]=(PlS~/20)[ - 7
(29)
[c~2, ~¢22 ~23 ~2,]=P2S~[ 1 ~s2
In the numerical simulations of the present paper, we will use only two finite elements for each beam. In that case, both fll and 132 will be 4 x 1 vectors since the flexible displacements of the middle node of the bus beam and the node of the arm at Jz are constrained to be zero. The entries of the mass matrix of the system are listed in the following: roll = ¢~1 4- d2 4- m~z(L~2 + u~z) 4- 2p,
- -
7 --Sj]
,7
i = 1,2,
c
¢rpi - Iai + mr,iy~i,
SpI = LjI 4- Xpl ,
i=1,2.
.Vi ~ YPi 4- Xpil'lpi2,
2 i ~ Xpi -- ypi~lPi2,
Sp2 = L2 4- Xp2,
--Sj
ap = - - L j l y m UpI2 4- Upllffl e l = -L2yP2up22 + Up21Y2
Px2 = (L2 + 22)c,,. - (Up21-~ ff2)s,i Px2 = (L2 + 22)s~. + (up21 + 9:)c~;
4- ~Pl + rnpl (s~,l 4- u~,ll 4- 2ap)
P2 = Lj2Px2 4- Uj21P~2
4- O'p24- mp2(L~2 + S~2 + Uj21
P3
+ Upz, + 2p, + 2p~)
=
Pa22
--
(LJ2S~'- UJ21C~')ff2
and
m~2 = ~u + mPl (LJI + 2t ) m13 = 0~12+ apl 4- n'/pl (XpI Spl "~ ap)
, plsl PI = [Pa] =420-
m~4 = ~13 + rnb2Lj2 + mp2(LJ~ + Px2)
156 22sl 0
22sl 4s~ 0
0 0 156
0 ] 0 -22sl
0
0
-22s t
312
0 54 - 13s2 8s22 13s2 -3s22 156 - 22s2
(30)
4s~
m15 : ~14 4- m16 m16 = ~ : 4-pp 4- mr,2(s~2 -F u221 4- 2pl 4- P2) m17 = 0~214- ~ I P 4 ,
I ,,
m18 = ~22 4- q~2P4
m19 = ct23+ ~b3p. + mpz(Sp: + p, - y p : Up2~)
symmetric
mlo = ~:, + dp~p4 + ap2 + mp2(Xp2Sp2 +Pl 4-P3) m2z=p;l+mpl, m33=P22+apl
m~=p~s+mb2+mp
m45 ~- p ~ + m46, m46 = p~ + mpzpx: m47 -~- ~1C7,
m48 = q~2c).,
m49 = q~3c7 4-mp2c;,
(31)
4s~
The stiffness matrix of the system in eqn (2) can be obtained as follows:
m23=P~2+mpl£1 +mpiX21,
pzsz
P2 = [Pij] = 4 - ~
~
I°°°JI; ]1 oKb~
00
K~2
(32)
Flexible manipulator
EIt
K~l = S-'T
12
6sl
0
0
6sl
4s~
0
0
0
0
12
--6si
0
0
--6S 1
4Sl2
24
0
- 12
6S2
8s~
-6s 2 12
--6S 2
I
El2 ] Kb2---- S-~- /
Mc2 = (33)
2s~
Lsymmetric
4s22
M£1
[
ml~
m~
m~
0
0
0
0
0
0
0
0
0
m,~l
t~l
t~2
~3 q- mp2
t~4"~-mp2Xp2
~21
~22
~X23At-mp2Sp2
L2 m 15
(39) mC32= g22 + (~2Lj2
m ~ = ~23 + O3Lj2 + mp2(Lj2 + sp2)
where
~12 "~ 0.Pl "4- msl Xpl
~13
(XI4
P I2 + mpl Xpt
0
0
Ph+aPl+mplx~l
0
0
Ph
Ph
P~, + mpl
ml~
m ~ = ~21 "[- ~,Lj2,
L]+r
~x,l+m,l
mC~ 0
(34)
The linearized mass matrix (constant) for the systern configuration 0~ = 0 and 02 = 0 can be expressed as follows:
mf~
163
ph
symmetric
(36) rolL( = II "4- 0.PI "~- met S2pI, msl = mpl Sp!
m ~ = 0~24+ ~4Lj2 + 0"1,2 + mp2xp2(Lj2 + Sp:).
Mass matrices for the subsystems
In the control law design that follows, we need to use two subsystem models to develop the independent control law. One subsystem consists of the bus beam and the mass, P,, only. The constant mass matrix of the subsystem is MLI in eqn (36). The associated stiffness matrix is just K~ defined in eqns (32) and (33). The other subsystem contains the remaining parts of the system, i.e. the robot arm and the payload mass, P2. This subsystem is modeled as a one arm link with the payload mass fixed to one of its ends. The other end of the arm can rotate about a fixed joint. By using the same technique for the development of the kinetic energy of the bus beam and mass, P,, one can prove that this mass matrix is ML2 in eqn (37), and the associated stiffness matrix is/(2 in eqns (32) and (34).
ML2
4. LINEAR DYNAMIC EQUATIONS
m~
0C21
tZ22
p;~
p;':
~t23+mp2sp2
L2 m15
p;~
p;',
After substituting the mass matrix, ML, and the generalized force, Qu, into eqn (3), one can obtain the linearized equations of the system
P 34+ mP2Xp2
MLt] = --Kq + B t u
P~'2
P~3
P~'3 +mp2 symmetric
/t
p24 + m n x ~2
(40)
where
(37) BL= m ~ = 12 + o n + m ~ s ~ ,
m~
0C24"[" 0.P2 "[- mp2XF2SI~
=
0 ~' [lo °0 °0 °0 °0 °1 °0 °0 °0 °]
l
(41)
U2
m cl 0
0
0
Mci
m cl
mtC~
0
0
o
o
mb2 + m ~
mf~
L symmetric
5. CONTROL LAW DESIGN
(38)
m~
m cl = I2+mb2Lj2(L2+Lj2) + op2 + rap2(Lj2 + sp2)2
In this paper, only the linear control laws based on the linear models of the system will be discussed. However, the validation of the control law will be tested numerically by using the original nonlinear model of the system, eqn (3). 5.1. Centralized control
m ° = mb2(L2/2 + Lj2) + mp2(Lj2 + sp2) m cl = 12 + (l/2)mb2Lj2L2 + 0.p2+ mp2sp2(Lj2 + Sp2) mC~ = (1/2)mb2L2 + mmsm AA29D--C
Equation (40) can be rewritten in the following state form: {~=y (wherex=q) = - M z t K x + M Z l B t u = A x + Bu.
(42)
['tlYUt [.1 el a[.
164
To minimize a given cost func6onal J=
(43)
,, (:JQz + u r R u ) d t
values are, therelbre, associated with the subsystem consisting of the robot arm and the mass, P:, onl). Let the (permutation) matrix while will rearrange the eigenvalues be denoted b\ P Thus, we havc
where : = [x*yI] j and
i< ~ ~ t,;
i i -I>,,t,/' <, _ i B = PB,,,
"i/:-- pi:
Q
[Q~
,
R=
r,:
r
(44)
the standard linear quadratic regulator technique is employed here to obtain the feedback gain matrix, K , and the control law can be expressed as u = - Ai::
(45)
5.2. Decentralized control
The decentralized control technique by using the subsystem decomposition approach discussed in [7] will be used here. The philosophy of this method is first to decompose mathematically the whole coupled system into several subsystems with relatively "weak" connections between them. By disregarding these connections, these subsystems will be decoupled from each other. Then, for each small-sized subsystem a control law can be designed independently via many advanced and effective control design methods. Finally, a stability test needs to be performed for the whole closed-loop system. Many other decentralized techniques applied to robotic systems can be found from the literature [7-9]. For example, Yuan and Book [9] proposed a decentralized adaptive control for a two-link, rigid, ground-based robotic system. The authors consider that the key in the decentralized control with the subsystem decomposition approach is how to decompose the system. There are many approaches from which we can choose, for example, the input-decentralization scheme discussed in Section 3.2 of [7] and Section 7.2 of [10]. But, by using this scheme, we end up with very large matrix entries which will make the subsequent subsystem control law design very difficult. However, in this paper, we propose the following decomposition approach which can result in a stable closed-loop system. Step 1. The matrix A in eqn (42) can be diagonalized by using a similarity transformation matrix, V
V-ly
then {~
B,,~= I / ~ I B
=f = A,.~£ + B,,,u.
(47)
Step 2. Since the ten eigenvalues of the whole system along the diagonal of matrix A are usually written in ascending order, we need to reorder them so that the first five eigenvalues are close to the eigenvalues of the subsystem composed of the bus beam and the mass, P~, only. The remaining five eigen-
(48)
then
(49) Note that P = P~, and after this reordering, the matrix A is still diagonal. Step 3. A rearranging matrix, A~, is generated to transform B into
hi,.
bl2. )
hs)
b~,
h1
0
()
h6
6
b'H)
o
bm,l bio2
=
B.
(50)
This transformation can be performed by first zeroing the last five entries of column 1, and then eliminating the upper five entries of column 2, or vice versa. For the elimination in each column, the element with maximum absolute value will be selected to perform the elimination. This process is similar to the Gaussian elimination and pivot selection in solving algebraic equations [11]. By calculating .~ =.4~: JA = A~.4A r ' I.V = A~)? ~/~ = a~/~
(51)
we can obtain
/.{ = f
(52)
= ,i.i +
where
=I
*:1
-
(53)
It can be seen that dtj and d22 have the largest nonzero elements predominantly along the principal diagonal, and that most of the entries of all2 and ,'i2~ will be zero because the number of controls is much less than the dimensionality of the system. Also, the magnitude of most of the nonzero entries of/]21 (all2) will be less than the magnitude of the entries of ,,i H (A22) because of the pivot selection in the elimination process, eqn (50). By neglecting ,4J2 and d2~, we obtain two subsystems
{
2~ = ¢ j
Y,
"t,,-fl + Blu,
22 = .f'2 !52 = e422.• 2 -~- B2u2 •
(54)
(55)
Flexible manipulator F o r each subsystem, the LQR technique is applied to derive the controls which will minimize
J,=
(#TO,~+rllU~)dt
(56)
&=
( ~ 0.2~_:+ r=u~) dt
(57)
where zl = [XlT ¢~]T, ~2 = [-/~ )~]s; and rl~ and r= are defined in eqn (44). To compare the responses of the system with different control logic, we need to use the QI and Q2 which are comparable to Q in eqn (43). From Step 1 to Step 3, we can obtain the relation between z and ~
z:[;]:[:
0
2
0 .
where [x'] ~ =
~2'
~ = ['9' 1 "
T = V P 'A:'.
(59)
LY2]'
Then by using the transformation in eqn (58), we have
zTQz =
~T[TIQuT T~QI2T]~ LTTQT2T TTQz2T z [fir
H
e.
(6o)
By letting G _[Gn
G~21, H _ ~ Hll H~21
o22j
-LHT2 H223'
I-F,, 1121 -LI2, 122]
(61)
and using eqn (59), we can obtain
[Gn 1.1
_l-G2~ ,=]
0'=LIT~ H,d' O-2-LIT= H2=J
(62)
Note that other semi-definite 01 and (~z can also be used here.
5.3. Independent control Instead of mathematically decomposing the system as in the last section, we will use a decomposing method which is based on the physical configuration of the system. This idea has been introduced by Young [12] and applied to the structural and control design problem, called controlled component synthesis. This concept has also been used in [3], called substructure decentralized control, for the maneuvering of a flexible antenna on a rigid platform. However, we will use this idea for a different structural configuration and develop a different approach. In the present paper, the system is composed of two component parts: (1) the bus beam and the fixed mass, PI and (2) the robot arm and the payload mass, P2. The control law will be designed for each part independently as if they were disconnected, or isolated.
165
Certainly, the stability of the original connected system should be guaranteed upon integrating the two subsystems together. The difference between the current separation method and the decomposition method in the decentralized control is that one subsystem derived by using the decomposition method will contain some information about another subsystem, while the subsystem defined in the independent control method here does not include any information about the other subsystem. The equations of the two subsystems can be written as MIAql = - KI ql + Ul
(63)
Mt.2g]2= --K2q2 + u2
(64)
where KI, /<2, M u , and ML= are defined in eqns (32)-(34), (36), and (37). The performance cost for each subsystem will be the same as used in the centralized control except that the weighting on the cross coupling term will not be used. Again, the poles of the original, linear closed-loop system are examined to ensure the stability of the whole system with these two independent controls. It can be seen that the current method involves less mathematics. Note that the substructure component in [12] is a part of a continuous structure (truss structure) and there is no relative rigid motion between the two substructure components. But, in the current problem, a rigid relative rotation is allowed between the interconnected subsystems. On the other hand, in [3], the equations of motion for the substructures were represented by the modal coordinates, and the number of actuators was assumed to be larger than or equal to the number of controlled modes. In the current paper, the equations of motion in the physical coordinates are used and the number of controllers is less than the number of dynamic degrees of freedom of the system. Also, the decomposition step here, in the independent control approach, is accomplished before the inverse of the mass matrix, ML, is calculated. On the other hand, in the decentralized control approach, the decomposition step is performed after this inverse is obtained. 6. NUMERICAL RESULTS
The structural parameters for the example simulated in this paper can be summarized as: bus length, LI = 30m; manipulator arm length, L: = 6 m; bus mass, mb~ = 600 kg; arm mass, rob2 = 18 kg; antenna mass, m m = 40 kg; payload mass, me2 = 36 kg; bus rigidity, E11 = 5E5 N-m2; arm rigidity, E12 = 320N-mS; bus density, p~ = 20 kg/m; arm density, P2 = 3 kg/m; the moments of inertia for the bus, the arm, mass P~, and mass P2 are, respectively (kg-m2): I I = 4.5E4, 15 = 216, I m = 562.5, Ip2 = 5; the locations of the center of mass of P~ and P2 in the local coordinate system are (m): xm = 1.5, YPt = 1, xp2 = 0, YP2 = 0.25.
I:EI~'UE LI el al.
166
The simulation responses plotted in Fig. 3 are tor the following case. It is assumed that the initial position of the robot arm is in the vertical position, i.e. 0 : = 9 0 d e g . Then, the arm is rotated to the horizontal position, 02 - 0, while the attitude of the bus is kept at 0~ = 0. For this case, the state weighting matrix, Q, is a unit matrix, and the control weighting matrix, R, is a diagonal matrix, R-=-Diag (0.001, 0.001). To compare the three methods discussed in this (a) Attitude, 01 (deg)
0
x
[ -4 [ - --8 ~ -12 ~
--~-----
D
Lin ctr. _ Centr. x Decen. , lndep,
0
i0
'
x ~ x ~ _ i
I 30
20
~ ] 40
(b) Arm rotation, 02 (deg) 90 ~- x -~ , . , ~ 60 ~ - ~
|
"~\
0
i0
30
[] --
L i n ctr. Centr.
× -
Decen. •
20
30
(C) Bus right end disp. (m) 0.04 1-[] Lin ctr. 0.02 | x - Centr.
r*
o
v ,r r v
40
× -
yVV
Decen. lndep.
V
-0.02 -0.04
|
I 20
l tO
0
_
i 30
I
40
paper, four responses of these systems are plotted. The first is the linear case for which the centralized control is used. All remaining cases are the nonlinear responses of the system by using the centralized, the decentralized, and the independent controls. Figure 3(a-f) plot the responses for 0~, 02, the transverse displacements of the right end of the bus beam and the manipulator arm tip, the bus torque, u~, and the arm joint torque, u2, respsectively. It can be seen that the responses for the linear case, centralized, and the independent controls are very close to each other. But the responses for the decentralized control have large differences from all the other responses. It is not surprising that the decentralized control shows a relatively poor response as compared with the centralized control design, because the decentralization process eliminates some coupling information. One could argue that the independent control also discards some connection terms, but it shows relatively good results. This may be explained by the fact that the decentralization neglects some information which is very significant, while the independent control design does not. In other words, the information retained in the subsystems, after the decentralization, does not represent the main stream of the whole system very well, or the decomposition process does not distribute the information very well in the decentralization method for the example considered here. The similarities and differences between the three controls can also be found by examining the pole locations of the three linear closed-loop systems, plotted in Fig. 4(a) (large negative real parts) and
(d) Arm tip disp. (m) 1 2 ~,x • Ilk.., 0.8 [J¢~
D --
~
×
04[
III
(a)
L m ctr. Centr.
Centr. + Indep. x Decen. o
Decen.
-
Indop
x
40
+
O+x
-0.4 I
I ~
0
~
l0
20
(e) Bus torque, u 1 (N-m) 60 ~x [] Lin ctr. 30 [-k Centr.
0~ -30 F
,
~
x'X. . . .
-60 ~0
~'X~x/X" 10
~
I 40 - - x.--..-o
x -
@) - ×4
x O+x
l
I
J
20
30
40
-20 K-~.~J¢,, -40 1-'~ x
L
I
i
i
-1.6
-1.3
-1.0
-0.7
xl
o
'~
• ~ I -20
+
-0.4
-40 -0.1
x~
I 20 Time (s)
o Centr. + Indep. x Decen.
x-
x
r
(b) --
[]
I 10
@ @
+
Decen. Indep.
_
20
x~
O+ 0 0 O+
rx/
(f) Arm torque, u 2 (N-m) 20 0 L . ~ x ×~., ~ x m ' ° I ' x ' ' a ' ~ x ~ ° I
-60 { - -80 / 0
I 30
_i I 30
Lin ctr. Centr. Decen. Indep. I 40
Fig. 3. Transient responses for the slew from 02 = rG2 to 02 = O.
10
-- 5
o x
-------o
[M--- x - -
0
I
I
I
I
-0.10
-0.08
-0.06
-0.04
x
-10
I -0.02
'"
~
o×
0
Real
Fig. 4. Pole locations of the closed-loop systems. (a) Large real parts; (b) small real parts.
Flexible manipulator
Fig. 4(b) (small negative real parts). Each system has 20 poles, but the pole with the largest negative real part ( - 5 3 2 . 2 for the centralized, -532.1 for the decentralized, and -559.0 for the independent) is not shown. All the poles of the three systems have negative real parts, indicating that the closed-loop system are asymptotically stable. It can be seen that (a) Attitude, 01 (deg)
I -10
"xx
I [
.,-1"
~'~x
0
Centr.
I
I
~.x/
x ;
Decen.
10
20
30
lndep. I 40
(b) Arm rotation, 02 (deg)
9o [-
×
60 ~ 30 L_ 0~
~
~ x ~ _/x~./ ~'f
x --1
0 10 20 (C) Bus right end disp. (m) 0.08~ 0.04 0
f~
~L~'~
1"~ ~
. .
0
.
10
Indep"
'
3O
40 Centr. Decen.
×
x Indep.
~ I ~ L . ~
20
, 40
30
(d) Arm tip disp. (m)
o[
-0.8
,<
0
10
20
30
40
(e) Bus torque, u I (N-m) 100 o
x
x\ ^ r' - . ~ - x - - - ~ - x ~ - , ~ - ~ < , _ . , . . . × , ~ , .
-100
-200 -300 -400 |
-500
0
[
[
10
20
-40
0
I" 10
(I, I, l, l, l, 0.5, I, I, I, I, I, I, I, l, l, l, l, I, l, I), while the stateweighting matrices for the decentralized control arc, Ql = Diag (70, I, l, l, l, l, l, I, I, I) and Q2 = Diag (0.2, l, l, l, l, l, I, I, l, l). The control weighting matrix is the same for all three control designs, R = Diag (0.001,0.001). Again, the independent control responses are close to those of the centralized control.
[
30
7. CONCLUSIONS
Three control design strategies: namely, centralized, decentralized, and independent controls, are considered. A mathematical decomposition method for the decentralized control approach and a physical decomposition method for the independent control are proposed. It is shown that the space robot control problem can be examined within the decentralized control and the independent control framework. The results for the independent control and the centralized control are close to each other. Therefore, the independent method deserves some credit because of the small dimensionality of each subsystem and the ease of obtaining the subsystem equations as compared with the development for the deccntralized method. The decentralized control shows a relatively poor response here. A more elaborate selection of the algebraic transformation may lead to a better result. For further research related to the decentralized and the independent controls, the selections of different decomposition methods and different weighting matrices in the cost functions are recommended.
REFERENCES
Centr. x__ Decen. -Indep.
(f) Arm torque, u 2 (N-m) 40 V L x 20 W ~ ' - - . / I A ---
-2o I
the majority of the poles for the independent control arc closer to the associated poles of the centralized control than those of the decentralized control. Therefore, the responses of the independent control are close to the responses of the centralized control. It is understood that the locations of the poles can also bc changed if different wcightings, Q and R, arc selected. Figure 5(a-f) show the responses for another case in which the arm is rotated from 02 = 0 to 02 = n/2. In this case, the state weighting matrix for the centralizcd control and independent control is Q = Diag
Centr.
Decen. I
167
I 40
Centr. Decen.
Indep,
[
I
I
20 Time (s)
30
40
Fig. 5. Transient responses for the slew from 02 = 0 to 02 = it/2.
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6. 7. 8.
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