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Controller design of a one-link flexible robot arm
Comprrers & Smctwes Vol. 49, No. I. pp. 117-126, 1993 Printed in Great Britain.
CONTROLLER
004s7949/93 56.00 + 0.00 0 1993 Pergamon Preds Ltd
DESIGN OF A ONE-LINK ROBOT ARM
FLEXIBLE
Z. ABDULJABBAR, M. M. ELMADANYand H. D. AL-DOKHIEL Faculty of Engineering, Mechanical Engineering Department, King Saud University, P.O. Box 11421, Saudi Arabia (Received 27 May 1992) Ahstrati-In this paper, a one link manipulator with an actuator located at the base is considered. The model of a flexible manipulator is developed using a truncated modal expansion solution of a Bernoulli-Euler beam. A truncated model retains the rigid body mode and the first three flexible modes. Measurements are made on the flexible arm yielding time varying quantities which are linear combinations of the system states. Several controllers to control the flexible modes of the flexible arm are designed and discussed.
INTRODUCllON One of the major limitations of current industrial robots is its low payload/weight ratio. The excessive weight of the arm limits the speeds at which robots can displace from one point to another and increases the energy requirements and the size of actuators. In order to improve the payload/weight ratio, several researchers have suggested the use of light-weight robots whose links might flex and deform during the operation. However, the control problem for flexible arms is more complicated than in the case of rigid arms, due to the flexibility along the light weight mechanical structure. Several methods for modelling light weight arms (flexible beams) have been proposed in the literature. The Lagrangian-assumed modes method has been widely used [l-3]. The result is an extended number of generalized coordinates, and then state variables, to handle for control purposes. Other methods such as finite elements [4, 51and polynomial expansions [6] have also been used. Several control techniques for the design of active feedback control of the flexible vibrations of the arm have been studied. These techniques include optimal linear control theory [3,7,8], frequency domain techniques [9, lo], transfer function approaches [ 111, and adaptive control [ 121. In this paper, a one link flexible manipulator with an actuator located at the base is considered. Measurements are made on the flexible arm yielding time varying quantities which are linear combinations of the system states. Several controllers to control the flexible modes of the flexible arm are designed and discussed. Specifically, the following tasks are considered in this work. 1. A model of a flexible manipulator is presented using a truncated modal expansion solution of a Bernoulli-Euler beam. The truncated model retains
the first four modes (i.e., rigid body mode, the first, second and third flexible modes). 2. A full state optimal feedback control for the flexible arm is derived. All states are assumed perfectly measured and available for feedback. Methods for designing optimum controllers assuming zero regulation set points are presented. 3. As the desired value of the state vector is not the origin, the arm control problem is treated as a tracking problem. 4. Reconstruction and estimation of the flexible variables from multiple measurements for use in state feedback control of the flexible manipulator are discussed. A reduced order observer is briefly reviewed, and then application to the flexible manipulator is presented. MODELLING The flexible arm is modelled as a continuous, pinned-tip mass Bernoulli-Euler beam of length L, whose moment of inertia about the root is IB, (Fig. 1). Here I,, is the lumped moment of inertia of the hub, El(x) = EI = constant, is the beam’s flexural rigidity, p(x) = p = constant, is the beam’s linear density (mass per unit length), and m, is the tip mass. Gravity effects are neglected and motion is confined to a horizontal plane. The deflections are assumed to be small and any extensions are neglected. The control input is the torque T(t) at the manipulator base. The X-Y coordinate frame is stationary and the x-y coordinate is fixed to the root of the arm. The variables w(x, t) and 19(t) represent deflection at point x along the arm and hub angle, respectively. The total deflection y(x, 1) of any point on the arm is defined as y(x, 1) = xc(t) + w(x, 1). 117
(1)
118
2. ABDULJABBARet al. Y
Fig. 1. Geometry of the arm model.
The beam deflection is determined fourth-order partial differential equation
by
EI$,,$O together with the four boundary EI
!? [ a2 1
a
(2)
With additional dissipative energy and torque terms, Lagrange’s method [2,3] or the method of modal expansions gives an infinite number of decoupled ordinary second order differential equations
conditions,
g=;
(8) T
+T-I,#=0
(3)
h(O) T
xso
4,+2c,wh+o:q,=-g-~
(9)
T
Y(O, t)=O
(4)
=o
42+2C2024*+&2=~~
(5)
C+21.0.4.+oaq"=,-~. The general solution of y(x, t) is assumed to be
YCx9 ?)= f
(7)
qi(z)cPi(x)9
drp,(O)T T
dv,(‘N T 2
(10)
(11)
Retaining the first (n + 1) modes of interest, it is possible to write the final state space model
i-0
i=Ax+Bu, where pi(x) is a mode function (i.e. i = 0 represents the rigid body mode; cpo(x) =x) and qi(t) are time dependent generalized coordinates (qO(t) = O(t)). The mode shapes q,(x) and the corresponding natural frequencies for the pinned-tip mass beam are derived and used in this study.
A=
where u = 7’9 x = ho4oh
01
0
0
...
0
0
00
0
0
... ...
0
0
0
0
... ...
0 .
0
...
.
00
0
0 .
0 .
-0: .
.
.
.
00
100
1 -2&w,
0
0
0
0
(12)
... ...
. . . w&IT
(13)
(15)
0
1
_& It
-xJ%
Controller design of a one-link flexible robot arm Table 2. Eigenvalues of the undamped flexible arm
Table 1. Flexible arm physical properties Beam Length
Height Width Mass density
Parameters
Real part
l.Om 2.5 x 10e3m 5.0 x 10m3m 9,812 x lo-* kg/m
0
+0
0
f 15.88 f 55.30 f 119.99
OFTIMAL
and r,=z,q+z,+z,,. The tip displacement is given by Y, =
cx,
Imaginary part
0 0
64 x 108N/m2 2.453 x 1O-2kg 3.2063 x lo-‘m’kg
Modulus of elasticity Tip mass Hub inertia
(16)
0 . . . q?“(L) 017
DESIGN
s m
J= 0 q,(L)
REGULATOR
Numerous representations of the performance functional are available in the literature [12]. The most commonly used form is the quadratic performance functional. In this representation, ‘weights’ are assigned to the components of the output and control vectors. The performance index is defined by the integrated values of the squares of these weighted components. The performance measured minimized by the optimal controller is
where
c = [L
119
0
(17)
(y: + up:+pu’) dr.
The performance index can be. put in the following form
For the system in hand, the physical parameters are
given in Table 1. In the following analysis, the first four modes (i.e. the rigid body mode and the first, second and third flexible modes) will be retained. The undamped system is an eight-order system with all its poles on thejo-axis with two at the origin, Table 2. In reality, the poles of the beam are not exactly on the jw-axis but slightly to the left of it, since there is always some damping within the beam. However, this is so small that one should not count on the inherent damping of the beam to reduce oscillations but the -additional feedback control should be employed.
m
I
J=
(x
0
rQx + u*Ru) dt,
0.3
(19)
where Q=CTC+aG=G G=
10
c =IL
L
0 0
cp,(L) .
4,(L)
0 0
rp,(L) ._.
&WI
0 0
R = p.
0.10
0.0
(18)
0.6
0.9
1.2
Time (set) Fig. 2. Tip position response for an uncontrolled system.
1.5
P,(L)
43WIlT
OIZ
Z. ABDULJABBAR et al.
120
1.5 1.0 ‘;; 2
0.5
3 .r 0.0 0” % -0.5 P .r t-1.0 -1 .5
0.3
0.0
0.6
1.2
0.9
1.5
Time (set) Fig. 3. Tip velocity response for an uncontrolled system.
The entails .I, can matrix
solution of this optimal control problem which minimization of the performance functional be obtained in closed form using the algebraic Riccati equation, as follows: PA+A’P-PBR-‘B*P+Q=O,
(20)
where P is the Riccati matrix. The optimal control vector variation
u(t) = - Gx = - R-‘B*Px
where c = R-‘B*P
is given by
= - f B*Px,
(21)
Thus, solving the algebraic matrix Riccati equation (20) leads to a closed-loop control law given by eqn (21). Whereas, the eigenvalues of the uncontrolled system are governed by A, the eigenvalues of the new optimally controlled system are governed by matrix (A - BR -‘B*P). Solution of the algebraic Riccati equation can be solution schemes numerous achieved by (e.g. [13, 14]), all of which have their relative merits and demerits. The uncontrolled system response is first simulated. The vibration of the first mode is simulated by taking initial conditions as x(O)=[O
is the gain matrix.
0
0.15
0
0
0
150 L
w
d
A
-Y-
Uncontrolled System
-c-
p=O.OOl
4-
p=O.l
-
100
-
50
K
t0
# --50
d
fi K
a -50
K
I
I
I
I
-40
-30
-20
-10
REAL Fig. 4. Root loci for a = 0.01.
- -100 -150 0
0 01*, (22)
121
Controller design of a one-link flexible robot arm
Table 3. Typical gain vectors and eigenvalues for p = 0.1 and D = 0.001. a = 0.01
where the value 0.15 corresponds to the initial deflection for the first flexible mode. The damping ratio ((C, = C2= * * * = (.) is taken to be 0.05. The simulation was formed using a Runge+Kutta algorithm. The tip position response and the tip velocity response are shown in Figs 2 and 3, respectively. It is clear that the response is oscillatory due to the presence of small damping. For the controlled system, typical gain vectors for a = 0.01 and two values for p(p = 0.1 and p = 0.001) are given in Table 3, together with the corresponding system eigenvalues. Figure 4 shows the root loci for a = 0.01. The response of the system to the initial conditions (22) using the optimal regulator designed above and
Gain matrix
0
0.1
0.001
Eigenvalues
3.1622 1.0430 2.4996 0.1515 1.7637 0.0862 1.8238 0.0450
-4.596 f 3.05
31.6227 8.67129 35.1832 0.64805 56.9744 0.74254 107.283 0.16368
-9.729 f 2.32
-4.598 + 16.31 -6.059 f 55.19 -8.512 f 119.98
- 18.947 f 14.26 - 30.432 f 53.04 -46.598 + _ 127.96
0.08, 0.06
----
p&l
-
p-o.001
0.04 0.02 0.00 -0.02 -0.04 -0.06 -0.08 -0.10 0.0
0.3
0.6
0.9
1.2
1.5
Time (MC) Fig. 5. Tip position response for a controlled system-p
0.0
0.3
0.6
= 0.01 and p = 0.001, a = 0.01.
0.9
1.2
1.5
Time (set) Fig. 6. Tip velocity response for a controlled system-p
= 0.1 and p = 0.001, a = 0.01.
122
Z. ABDULJABBARet al. 0.8 0.6
____
p=o*
-
p=.OOl
1
0.4 -z z -
0.2
!x F c"
0.0
'=, -0.2 E 0 -0.4 -0.6 0.0
0.3
0.9
0.6
1.5
1.2
Time (set) Fig. 7. Control torque for a controlled system-p = 0.01and JJ = 0.001, Q = 0.01.
given in Table 3, is shown in Fig. 5 through Fig. 7. Using p = 0.001 results in a well damped tip position
The costate equations are
b(t)=
-E=
-Qx(t)-Ar~+Qr(t)
(26)
and the algebraic relations that must be satisfied are given by
DESIGN OF THE FLEXIBLE ARM AS A LINEAR TRACKING PROBLEM
In this section, the problem of the flexible arm is treated as a tracking problem. The state equations are
0 = 2 = Ru(t) + C&r),
(27)
therefore i(t) = Ax(t) + &4(t)
(23) u(t) = -R_‘Cp(t).
(28)
and the performance measure to be minimized is
J = iW,-- r(tfllrfWtf)
+i
s‘SW-
-
Substituting (28) in the state equations yields the state and costate equations
r(q)1
rtOlUW
- r(t)1
-BR-‘Br][;]+[Qr;rj]. -AT
(2%
10
+ ur(r)R(r)u(r)}dr,
(24)
where r(r) is the desired or reference value of the state vector. The tinal time r, is fixed, and the states and controls are not bounded. H and Q are real symmetric positive semi-definite matrices, and R is real symmetric and positive definite. The Hamiltonian is given by
4-M uW,ptt), 1) = $4 + $‘(r)Ru(r)
-
WrQbtr)
+p’Ax(r)
+p%u(r).
-WI (25)
Notice that the term Qr(r) is a forcing function; these differential equations are linear and time-varying, but not homogeneous. The boundary conditions are
PO&= fW,) - W+).
(30)
Assume a solution for the costate vector as p(r) = K(r)x(r)
+ s(r).
(31)
Differentiate both sides with respect to r, we obtain t;(r) = &)x(r)
+ K(r)2(r)
+ b(r).
(32)
Controller design of a one-link flexible robot arm
123
aXlTRoLLER r(t) PLANT
““““““““““““““‘I
I /
B(t)
A(t) L____.
_____
I
w
~___________________,______,,
i I I I I I I
““,
f Fig. 8. Plant and optimal feedback controller for the linear tracking problems.
Substituting from (29) for i(r) (31) to eliminate p(t) yields
and i(t),
u(r) = -R-‘BrK(r)x(t)
and using
- R-‘Brs(r)
= F(r)x(r) + v(r). [k(t) + K(t)A + A%(t)
- K(r)BR-‘B%(t)
A diagram of the plant and controller is shown in Fig. 8.
xx(r) + [i(t) + Ars(r) - K(r)BR-*Ps(r)
- Qr(r)] = 0.
(38)
+ Q]
(33) DESIGN OF THE REDUCED ORDER OBSERVER
Therefore, the necessary conditions for a finite time optimal control solution, for arbitrary x(r) and r(r), are
In the state feedback control treated in the previous sections, all the states are assumed to be available for measurements. But practically, some of the state variables may not be measurable, or measuring them may not be economically feasible. Therefore, in these cases, an estimate for the state vector can be used (15, 161. Joint angles, and joint rotational speeds can be measured directly as for rigid manipulators, however, for state feedback control of manipulator flexibility, it is desirable to make direct measurements of the modal variables. Three types of measurements are receiving attention for controlling the flexible structure of the manipulators, optical measurements of end point position [17,18], optical measurement of deflection, and measurement of strain on the flexible link [ 11. The measuring output is given by
k(r) = - K(r)A - A ‘K(r) + K(r)BR -IPK(r) - Q (34) and
b(r) = - A ‘s(r) + K(r)BR-lBrs(r) + Qr(t) (35) with boundary
conditions
from (30) and (31)
K(r,) = H
(36)
s(fi) = - Hr(r,).
(37)
and
Since K is symmetric and s is an n x 1 vector, eqns (34) and (35) are a set of (n(n + 1)/2) + n first-order differential equations. Equations (34) and (35) are to be integrated from 1/ to to using the boundary conditions (36) and (37), and the values for K(r) and s(r) are to be stored. Then, the optimal control law can be determined from
y = cx,
where C is determined from the measuring quantities using three sensors: the tip position sensor, the collected rate sensor and the strain sensor at distance a from the hub.
I-
I LO C=
cpl(L)
01 0
0 0
e&r)
0 -dq,(o) dx 0
(39)
R(L) 0 c%(c)
0
R(L)
0
o
dv, (0)
-dM0) ti 0
dx
c&l
Z. Amuurlenm
124
Define the state transformation s=
[I; ;
matrix S as
(41)
s-‘=e=ra,al
et al.
This system is completely observable if the original system A and C is completely observable. Therefore (n - q) state variables may be estimated by d=(&-L&)~*++LW+t7.
L is completely arbitrary as long as S is nonsingular. Transformation using Z=SPX
(42)
Insert the expressions for w and u’ d = (A,, -LA,*)&
+ L(j - A,,y - B,u)
gives
+ G&Y + +J). f = SAS-‘2 y = CS-‘f
which can be partitioned
+ SBu
(43)
= [z,o]a
(44)
as
(54)
To eliminate )‘, redefine observer state as z =&-
Ly.
Then, the state equation
Error for transformed
(55)
for the reduced order
+ (A,, - LA,,)~Y + (4 - LB,)u. For the design of the reduced order observer, only Z2 need to be estimated since f, are available as output. Using y instead of P,
(53)
(56)
state will be
e=f,-{z+Ly}
(57) (58)
j=A,,y+A,,&+BB,u
(47)
e = (AZ2- LA,*)e.
k2 = A22Z2+ A,, y + Bzu
(48)
From eqn (58) choose the eigenvalues such that the error dies out fast. The complete (estimated and measured) state vector is given by the following. For the transformed system
or k2 = A&,
+ ii
(49)
w = A&,
(50)
8={J={zcyLy}.
(59)
where (51)
17= A,,y + b2u w =j
- A,,y - Blu.
For the original system 2 = S-Ii.
(52)
This system is similar to the original system but of lower order.
The block diagram for the reduced order observer is shown in Fig. 9.
Y
A21 - ‘-A11
(60)
L
q,p Fig. 9. Block diagram for reduced order observer.
QI
Controller design of a one-link flexible robot arm
---
I
-0.05 0.0
0.3
I
I
0.6
0.9
125
tip position
I 1.5
1.2
Time (set) Fig. 10. Tip position for the flexible arm as a linear tracking problem. SIMULATION
and the initial conditions
In this problem the objective is to maintain the state q1and consequently the tip position close to the function r,(t) without excessive expenditure of control effort. In this simulation, rl(t) is given by 0.8t2 r,(t) = 0.4 - 0.8(1 - t)’ 0.4
t co.5 0.5 < t < 1.0
(61)
t > 1.0
for i=2,3,...,6
ri(t)=O
R = p = 0.01
are
XT(O)= [O 0 0 0 0 01. The simulation results are shown in Figs 10-12. A well damped tip position response is shown in Fig. 10. There is an initial transient period that is over at approximately 0.2 sec. Therefore, the arm is perfectly tracking the input. The magnitude of the control torque required for tracking oscillates and approaches zero at the final time, Fig. 12.
0.8 , 0.7
0.6
.
0.0
1 0.6
I 0.9
Time (set) Fig. 1I. Tip velocity for the flexible arm as a linear tracking problem.
z. tiBDULJABBAn et
126
al.
0.3 0.2 0.1
0.0
-0.1
0.0
0.3
0.6
0.9
1.2
1.5
Time (set) Fig. 12. Control torque for the flexible arm as a linear tracking problem. SUMMARY
AND CONCLUDING
RRMARRS
This study has presented some preliminary results on controlling the end-point of a single link flexible manipulator using different control strategies. The model used in the analysis is a single Bernoulli-Euler
beam moving in a horizontal plane, considering only a rigid body plus three flexible modes. The state space model of a flexible beam is formulated. The control schemes are based on linear quadratic regulator techniques. Control effort penalty is varied until satisfactory gains and performance in position control is achieved. The particular problem on which attention is focused is that of designing a controller that stabilizes the flexible arm governing equations, and at the same time minimizes the mean-squared tracking error in response to a particular input. From the preceding analysis it may be concluded that the optimal linear regulator is effective in reducing oscillations and settling time of the flexible arm. Formulation of the flexible arm as a tracking problem, in response to a particular input, produces a system whose response is quite rapid with minimal overshoot. The use of a reduced-order observer for controlling the single flexible arm has been successfully demonstrated. The results indicate that the observer poles must be placed in the left half plane at least five times the modes being estimated. REFERENCES
1. L. Meirovitch, Analytical Methods in Vibrations. MacMillan, New York (1967). 2. W. J. Book, Recursive Lagrangian dynamics of flexible manipulator arms. ht. J. Robotics Res. 3(3), 87-101 (1984). 3. R. H. Cannon, Jr and E. Schmitz, Initial experiments on the end-point control of a flexible one link robot. ht. J. Robotics Res. 3(3), (1984). 4. W. Sunada and S. Dubowsky, On the dynamic analysis and behavior of industrial robotic manipulators with
5.
6. 7. 8.
9.
10.
11.
2.
3. 14. 15. 16. 17. 18.
elastic members. ASME J. Mechanisms, Transmissions and Automation in Design 105, 42-51 (1983). P. B. Usoro, R. Nadria and S. S. Mahil, A finite element/Lagrange approach to modeling lightweight flexible manipulators. ASME J. Dynamic Systems, Measurement, and Control 108, 198-205 (1986). P. Tomei and A. Tomambe, Approximate modeling of robots having elastic links. IEEE Trans. Systems, Man, and Cybernetics SMC-18, 831-840 (1988). T. Fukuda, Flexibility control of elastic robotic arms. J. Robotic Systems 2, 73-88 (1985). N. G. Chalhoub and A. G. Ulsoy, Control of a flexible robot arm-experimental and theoretical results. ASME J. Dynamic Systems, Measurements and Control 109, 299-309 (1987). W. J. Book and M. Majetta, Controller design for flexible distributed parameter mechanical arms via combined state spaced and frequency domain techniques. ASME J. Dynamic Systems, Measurements and Control 105, 245-254 (1983). S. Yurkovich. F. E. Pacheco and A. P. Tzes. On line frequent; domain information for control ‘of a flexible-link robot with varying payload. IEEE Trans. on Automatic Control AC-34, 1300-1304 (1989). D. Wang and M. Vidyasagar, Transfer function for a single flexible link. Proc. 1989 IEEE Int. Conf. on Robotics and Automation, Scottsdale, AZ, pp. 1042-1047 (1989). B. S. Yuan, W. J. Book and B. Siciliano, Direct adaptive control of a one-link flexible arm with tracking. J. Robotics Systems 6, 663480 (1989). A. F. Fath, ComputaGonal aspects of the linear optimal regulator problem. IEEE Trans. on Automatic Control, Oct. 1969, pp. 547-549 (1969). D. Wikerg, State Space and Linear Systems. McGrawHill Schaums Outlines (1971). D. Luenberger, Observing the state of a linear system. IEEE Trans. on Mil. Electronics, April (1964). B. Gopinath, On the control of linear multiple input-output systems. Bell System Tech. Jnl S(l), March (1971). G. G. Hastings and W. J. Book, Experiments in the control of a flexible robot arm. Proc. SME Robots Nine, Detroit, MI, June (1985). G. G. Hastings and W. J. Book, Reconstruction and robust reduced-order observation of flexible variables. DSC-Vol.3, pp. 11-16 (1986).
CONTROLLER
004s7949/93 56.00 + 0.00 0 1993 Pergamon Preds Ltd
DESIGN OF A ONE-LINK ROBOT ARM
FLEXIBLE
Z. ABDULJABBAR, M. M. ELMADANYand H. D. AL-DOKHIEL Faculty of Engineering, Mechanical Engineering Department, King Saud University, P.O. Box 11421, Saudi Arabia (Received 27 May 1992) Ahstrati-In this paper, a one link manipulator with an actuator located at the base is considered. The model of a flexible manipulator is developed using a truncated modal expansion solution of a Bernoulli-Euler beam. A truncated model retains the rigid body mode and the first three flexible modes. Measurements are made on the flexible arm yielding time varying quantities which are linear combinations of the system states. Several controllers to control the flexible modes of the flexible arm are designed and discussed.
INTRODUCllON One of the major limitations of current industrial robots is its low payload/weight ratio. The excessive weight of the arm limits the speeds at which robots can displace from one point to another and increases the energy requirements and the size of actuators. In order to improve the payload/weight ratio, several researchers have suggested the use of light-weight robots whose links might flex and deform during the operation. However, the control problem for flexible arms is more complicated than in the case of rigid arms, due to the flexibility along the light weight mechanical structure. Several methods for modelling light weight arms (flexible beams) have been proposed in the literature. The Lagrangian-assumed modes method has been widely used [l-3]. The result is an extended number of generalized coordinates, and then state variables, to handle for control purposes. Other methods such as finite elements [4, 51and polynomial expansions [6] have also been used. Several control techniques for the design of active feedback control of the flexible vibrations of the arm have been studied. These techniques include optimal linear control theory [3,7,8], frequency domain techniques [9, lo], transfer function approaches [ 111, and adaptive control [ 121. In this paper, a one link flexible manipulator with an actuator located at the base is considered. Measurements are made on the flexible arm yielding time varying quantities which are linear combinations of the system states. Several controllers to control the flexible modes of the flexible arm are designed and discussed. Specifically, the following tasks are considered in this work. 1. A model of a flexible manipulator is presented using a truncated modal expansion solution of a Bernoulli-Euler beam. The truncated model retains
the first four modes (i.e., rigid body mode, the first, second and third flexible modes). 2. A full state optimal feedback control for the flexible arm is derived. All states are assumed perfectly measured and available for feedback. Methods for designing optimum controllers assuming zero regulation set points are presented. 3. As the desired value of the state vector is not the origin, the arm control problem is treated as a tracking problem. 4. Reconstruction and estimation of the flexible variables from multiple measurements for use in state feedback control of the flexible manipulator are discussed. A reduced order observer is briefly reviewed, and then application to the flexible manipulator is presented. MODELLING The flexible arm is modelled as a continuous, pinned-tip mass Bernoulli-Euler beam of length L, whose moment of inertia about the root is IB, (Fig. 1). Here I,, is the lumped moment of inertia of the hub, El(x) = EI = constant, is the beam’s flexural rigidity, p(x) = p = constant, is the beam’s linear density (mass per unit length), and m, is the tip mass. Gravity effects are neglected and motion is confined to a horizontal plane. The deflections are assumed to be small and any extensions are neglected. The control input is the torque T(t) at the manipulator base. The X-Y coordinate frame is stationary and the x-y coordinate is fixed to the root of the arm. The variables w(x, t) and 19(t) represent deflection at point x along the arm and hub angle, respectively. The total deflection y(x, 1) of any point on the arm is defined as y(x, 1) = xc(t) + w(x, 1). 117
(1)
118
2. ABDULJABBARet al. Y
Fig. 1. Geometry of the arm model.
The beam deflection is determined fourth-order partial differential equation
by
EI$,,$O together with the four boundary EI
!? [ a2 1
a
(2)
With additional dissipative energy and torque terms, Lagrange’s method [2,3] or the method of modal expansions gives an infinite number of decoupled ordinary second order differential equations
conditions,
g=;
(8) T
+T-I,#=0
(3)
h(O) T
xso
4,+2c,wh+o:q,=-g-~
(9)
T
Y(O, t)=O
(4)
=o
42+2C2024*+&2=~~
(5)
C+21.0.4.+oaq"=,-~. The general solution of y(x, t) is assumed to be
YCx9 ?)= f
(7)
qi(z)cPi(x)9
drp,(O)T T
dv,(‘N T 2
(10)
(11)
Retaining the first (n + 1) modes of interest, it is possible to write the final state space model
i-0
i=Ax+Bu, where pi(x) is a mode function (i.e. i = 0 represents the rigid body mode; cpo(x) =x) and qi(t) are time dependent generalized coordinates (qO(t) = O(t)). The mode shapes q,(x) and the corresponding natural frequencies for the pinned-tip mass beam are derived and used in this study.
A=
where u = 7’9 x = ho4oh
01
0
0
...
0
0
00
0
0
... ...
0
0
0
0
... ...
0 .
0
...
.
00
0
0 .
0 .
-0: .
.
.
.
00
100
1 -2&w,
0
0
0
0
(12)
... ...
. . . w&IT
(13)
(15)
0
1
_& It
-xJ%
Controller design of a one-link flexible robot arm Table 2. Eigenvalues of the undamped flexible arm
Table 1. Flexible arm physical properties Beam Length
Height Width Mass density
Parameters
Real part
l.Om 2.5 x 10e3m 5.0 x 10m3m 9,812 x lo-* kg/m
0
+0
0
f 15.88 f 55.30 f 119.99
OFTIMAL
and r,=z,q+z,+z,,. The tip displacement is given by Y, =
cx,
Imaginary part
0 0
64 x 108N/m2 2.453 x 1O-2kg 3.2063 x lo-‘m’kg
Modulus of elasticity Tip mass Hub inertia
(16)
0 . . . q?“(L) 017
DESIGN
s m
J= 0 q,(L)
REGULATOR
Numerous representations of the performance functional are available in the literature [12]. The most commonly used form is the quadratic performance functional. In this representation, ‘weights’ are assigned to the components of the output and control vectors. The performance index is defined by the integrated values of the squares of these weighted components. The performance measured minimized by the optimal controller is
where
c = [L
119
0
(17)
(y: + up:+pu’) dr.
The performance index can be. put in the following form
For the system in hand, the physical parameters are
given in Table 1. In the following analysis, the first four modes (i.e. the rigid body mode and the first, second and third flexible modes) will be retained. The undamped system is an eight-order system with all its poles on thejo-axis with two at the origin, Table 2. In reality, the poles of the beam are not exactly on the jw-axis but slightly to the left of it, since there is always some damping within the beam. However, this is so small that one should not count on the inherent damping of the beam to reduce oscillations but the -additional feedback control should be employed.
m
I
J=
(x
0
rQx + u*Ru) dt,
0.3
(19)
where Q=CTC+aG=G G=
10
c =IL
L
0 0
cp,(L) .
4,(L)
0 0
rp,(L) ._.
&WI
0 0
R = p.
0.10
0.0
(18)
0.6
0.9
1.2
Time (set) Fig. 2. Tip position response for an uncontrolled system.
1.5
P,(L)
43WIlT
OIZ
Z. ABDULJABBAR et al.
120
1.5 1.0 ‘;; 2
0.5
3 .r 0.0 0” % -0.5 P .r t-1.0 -1 .5
0.3
0.0
0.6
1.2
0.9
1.5
Time (set) Fig. 3. Tip velocity response for an uncontrolled system.
The entails .I, can matrix
solution of this optimal control problem which minimization of the performance functional be obtained in closed form using the algebraic Riccati equation, as follows: PA+A’P-PBR-‘B*P+Q=O,
(20)
where P is the Riccati matrix. The optimal control vector variation
u(t) = - Gx = - R-‘B*Px
where c = R-‘B*P
is given by
= - f B*Px,
(21)
Thus, solving the algebraic matrix Riccati equation (20) leads to a closed-loop control law given by eqn (21). Whereas, the eigenvalues of the uncontrolled system are governed by A, the eigenvalues of the new optimally controlled system are governed by matrix (A - BR -‘B*P). Solution of the algebraic Riccati equation can be solution schemes numerous achieved by (e.g. [13, 14]), all of which have their relative merits and demerits. The uncontrolled system response is first simulated. The vibration of the first mode is simulated by taking initial conditions as x(O)=[O
is the gain matrix.
0
0.15
0
0
0
150 L
w
d
A
-Y-
Uncontrolled System
-c-
p=O.OOl
4-
p=O.l
-
100
-
50
K
t0
# --50
d
fi K
a -50
K
I
I
I
I
-40
-30
-20
-10
REAL Fig. 4. Root loci for a = 0.01.
- -100 -150 0
0 01*, (22)
121
Controller design of a one-link flexible robot arm
Table 3. Typical gain vectors and eigenvalues for p = 0.1 and D = 0.001. a = 0.01
where the value 0.15 corresponds to the initial deflection for the first flexible mode. The damping ratio ((C, = C2= * * * = (.) is taken to be 0.05. The simulation was formed using a Runge+Kutta algorithm. The tip position response and the tip velocity response are shown in Figs 2 and 3, respectively. It is clear that the response is oscillatory due to the presence of small damping. For the controlled system, typical gain vectors for a = 0.01 and two values for p(p = 0.1 and p = 0.001) are given in Table 3, together with the corresponding system eigenvalues. Figure 4 shows the root loci for a = 0.01. The response of the system to the initial conditions (22) using the optimal regulator designed above and
Gain matrix
0
0.1
0.001
Eigenvalues
3.1622 1.0430 2.4996 0.1515 1.7637 0.0862 1.8238 0.0450
-4.596 f 3.05
31.6227 8.67129 35.1832 0.64805 56.9744 0.74254 107.283 0.16368
-9.729 f 2.32
-4.598 + 16.31 -6.059 f 55.19 -8.512 f 119.98
- 18.947 f 14.26 - 30.432 f 53.04 -46.598 + _ 127.96
0.08, 0.06
----
p&l
-
p-o.001
0.04 0.02 0.00 -0.02 -0.04 -0.06 -0.08 -0.10 0.0
0.3
0.6
0.9
1.2
1.5
Time (MC) Fig. 5. Tip position response for a controlled system-p
0.0
0.3
0.6
= 0.01 and p = 0.001, a = 0.01.
0.9
1.2
1.5
Time (set) Fig. 6. Tip velocity response for a controlled system-p
= 0.1 and p = 0.001, a = 0.01.
122
Z. ABDULJABBARet al. 0.8 0.6
____
p=o*
-
p=.OOl
1
0.4 -z z -
0.2
!x F c"
0.0
'=, -0.2 E 0 -0.4 -0.6 0.0
0.3
0.9
0.6
1.5
1.2
Time (set) Fig. 7. Control torque for a controlled system-p = 0.01and JJ = 0.001, Q = 0.01.
given in Table 3, is shown in Fig. 5 through Fig. 7. Using p = 0.001 results in a well damped tip position
The costate equations are
b(t)=
-E=
-Qx(t)-Ar~+Qr(t)
(26)
and the algebraic relations that must be satisfied are given by
DESIGN OF THE FLEXIBLE ARM AS A LINEAR TRACKING PROBLEM
In this section, the problem of the flexible arm is treated as a tracking problem. The state equations are
0 = 2 = Ru(t) + C&r),
(27)
therefore i(t) = Ax(t) + &4(t)
(23) u(t) = -R_‘Cp(t).
(28)
and the performance measure to be minimized is
J = iW,-- r(tfllrfWtf)
+i
s‘SW-
-
Substituting (28) in the state equations yields the state and costate equations
r(q)1
rtOlUW
- r(t)1
-BR-‘Br][;]+[Qr;rj]. -AT
(2%
10
+ ur(r)R(r)u(r)}dr,
(24)
where r(r) is the desired or reference value of the state vector. The tinal time r, is fixed, and the states and controls are not bounded. H and Q are real symmetric positive semi-definite matrices, and R is real symmetric and positive definite. The Hamiltonian is given by
4-M uW,ptt), 1) = $4 + $‘(r)Ru(r)
-
WrQbtr)
+p’Ax(r)
+p%u(r).
-WI (25)
Notice that the term Qr(r) is a forcing function; these differential equations are linear and time-varying, but not homogeneous. The boundary conditions are
PO&= fW,) - W+).
(30)
Assume a solution for the costate vector as p(r) = K(r)x(r)
+ s(r).
(31)
Differentiate both sides with respect to r, we obtain t;(r) = &)x(r)
+ K(r)2(r)
+ b(r).
(32)
Controller design of a one-link flexible robot arm
123
aXlTRoLLER r(t) PLANT
““““““““““““““‘I
I /
B(t)
A(t) L____.
_____
I
w
~___________________,______,,
i I I I I I I
““,
f Fig. 8. Plant and optimal feedback controller for the linear tracking problems.
Substituting from (29) for i(r) (31) to eliminate p(t) yields
and i(t),
u(r) = -R-‘BrK(r)x(t)
and using
- R-‘Brs(r)
= F(r)x(r) + v(r). [k(t) + K(t)A + A%(t)
- K(r)BR-‘B%(t)
A diagram of the plant and controller is shown in Fig. 8.
xx(r) + [i(t) + Ars(r) - K(r)BR-*Ps(r)
- Qr(r)] = 0.
(38)
+ Q]
(33) DESIGN OF THE REDUCED ORDER OBSERVER
Therefore, the necessary conditions for a finite time optimal control solution, for arbitrary x(r) and r(r), are
In the state feedback control treated in the previous sections, all the states are assumed to be available for measurements. But practically, some of the state variables may not be measurable, or measuring them may not be economically feasible. Therefore, in these cases, an estimate for the state vector can be used (15, 161. Joint angles, and joint rotational speeds can be measured directly as for rigid manipulators, however, for state feedback control of manipulator flexibility, it is desirable to make direct measurements of the modal variables. Three types of measurements are receiving attention for controlling the flexible structure of the manipulators, optical measurements of end point position [17,18], optical measurement of deflection, and measurement of strain on the flexible link [ 11. The measuring output is given by
k(r) = - K(r)A - A ‘K(r) + K(r)BR -IPK(r) - Q (34) and
b(r) = - A ‘s(r) + K(r)BR-lBrs(r) + Qr(t) (35) with boundary
conditions
from (30) and (31)
K(r,) = H
(36)
s(fi) = - Hr(r,).
(37)
and
Since K is symmetric and s is an n x 1 vector, eqns (34) and (35) are a set of (n(n + 1)/2) + n first-order differential equations. Equations (34) and (35) are to be integrated from 1/ to to using the boundary conditions (36) and (37), and the values for K(r) and s(r) are to be stored. Then, the optimal control law can be determined from
y = cx,
where C is determined from the measuring quantities using three sensors: the tip position sensor, the collected rate sensor and the strain sensor at distance a from the hub.
I-
I LO C=
cpl(L)
01 0
0 0
e&r)
0 -dq,(o) dx 0
(39)
R(L) 0 c%(c)
0
R(L)
0
o
dv, (0)
-dM0) ti 0
dx
c&l
Z. Amuurlenm
124
Define the state transformation s=
[I; ;
matrix S as
(41)
s-‘=e=ra,al
et al.
This system is completely observable if the original system A and C is completely observable. Therefore (n - q) state variables may be estimated by d=(&-L&)~*++LW+t7.
L is completely arbitrary as long as S is nonsingular. Transformation using Z=SPX
(42)
Insert the expressions for w and u’ d = (A,, -LA,*)&
+ L(j - A,,y - B,u)
gives
+ G&Y + +J). f = SAS-‘2 y = CS-‘f
which can be partitioned
+ SBu
(43)
= [z,o]a
(44)
as
(54)
To eliminate )‘, redefine observer state as z =&-
Ly.
Then, the state equation
Error for transformed
(55)
for the reduced order
+ (A,, - LA,,)~Y + (4 - LB,)u. For the design of the reduced order observer, only Z2 need to be estimated since f, are available as output. Using y instead of P,
(53)
(56)
state will be
e=f,-{z+Ly}
(57) (58)
j=A,,y+A,,&+BB,u
(47)
e = (AZ2- LA,*)e.
k2 = A22Z2+ A,, y + Bzu
(48)
From eqn (58) choose the eigenvalues such that the error dies out fast. The complete (estimated and measured) state vector is given by the following. For the transformed system
or k2 = A&,
+ ii
(49)
w = A&,
(50)
8={J={zcyLy}.
(59)
where (51)
17= A,,y + b2u w =j
- A,,y - Blu.
For the original system 2 = S-Ii.
(52)
This system is similar to the original system but of lower order.
The block diagram for the reduced order observer is shown in Fig. 9.
Y
A21 - ‘-A11
(60)
L
q,p Fig. 9. Block diagram for reduced order observer.
QI
Controller design of a one-link flexible robot arm
---
I
-0.05 0.0
0.3
I
I
0.6
0.9
125
tip position
I 1.5
1.2
Time (set) Fig. 10. Tip position for the flexible arm as a linear tracking problem. SIMULATION
and the initial conditions
In this problem the objective is to maintain the state q1and consequently the tip position close to the function r,(t) without excessive expenditure of control effort. In this simulation, rl(t) is given by 0.8t2 r,(t) = 0.4 - 0.8(1 - t)’ 0.4
t co.5 0.5 < t < 1.0
(61)
t > 1.0
for i=2,3,...,6
ri(t)=O
R = p = 0.01
are
XT(O)= [O 0 0 0 0 01. The simulation results are shown in Figs 10-12. A well damped tip position response is shown in Fig. 10. There is an initial transient period that is over at approximately 0.2 sec. Therefore, the arm is perfectly tracking the input. The magnitude of the control torque required for tracking oscillates and approaches zero at the final time, Fig. 12.
0.8 , 0.7
0.6
.
0.0
1 0.6
I 0.9
Time (set) Fig. 1I. Tip velocity for the flexible arm as a linear tracking problem.
z. tiBDULJABBAn et
126
al.
0.3 0.2 0.1
0.0
-0.1
0.0
0.3
0.6
0.9
1.2
1.5
Time (set) Fig. 12. Control torque for the flexible arm as a linear tracking problem. SUMMARY
AND CONCLUDING
RRMARRS
This study has presented some preliminary results on controlling the end-point of a single link flexible manipulator using different control strategies. The model used in the analysis is a single Bernoulli-Euler
beam moving in a horizontal plane, considering only a rigid body plus three flexible modes. The state space model of a flexible beam is formulated. The control schemes are based on linear quadratic regulator techniques. Control effort penalty is varied until satisfactory gains and performance in position control is achieved. The particular problem on which attention is focused is that of designing a controller that stabilizes the flexible arm governing equations, and at the same time minimizes the mean-squared tracking error in response to a particular input. From the preceding analysis it may be concluded that the optimal linear regulator is effective in reducing oscillations and settling time of the flexible arm. Formulation of the flexible arm as a tracking problem, in response to a particular input, produces a system whose response is quite rapid with minimal overshoot. The use of a reduced-order observer for controlling the single flexible arm has been successfully demonstrated. The results indicate that the observer poles must be placed in the left half plane at least five times the modes being estimated. REFERENCES
1. L. Meirovitch, Analytical Methods in Vibrations. MacMillan, New York (1967). 2. W. J. Book, Recursive Lagrangian dynamics of flexible manipulator arms. ht. J. Robotics Res. 3(3), 87-101 (1984). 3. R. H. Cannon, Jr and E. Schmitz, Initial experiments on the end-point control of a flexible one link robot. ht. J. Robotics Res. 3(3), (1984). 4. W. Sunada and S. Dubowsky, On the dynamic analysis and behavior of industrial robotic manipulators with
5.
6. 7. 8.
9.
10.
11.
2.
3. 14. 15. 16. 17. 18.
elastic members. ASME J. Mechanisms, Transmissions and Automation in Design 105, 42-51 (1983). P. B. Usoro, R. Nadria and S. S. Mahil, A finite element/Lagrange approach to modeling lightweight flexible manipulators. ASME J. Dynamic Systems, Measurement, and Control 108, 198-205 (1986). P. Tomei and A. Tomambe, Approximate modeling of robots having elastic links. IEEE Trans. Systems, Man, and Cybernetics SMC-18, 831-840 (1988). T. Fukuda, Flexibility control of elastic robotic arms. J. Robotic Systems 2, 73-88 (1985). N. G. Chalhoub and A. G. Ulsoy, Control of a flexible robot arm-experimental and theoretical results. ASME J. Dynamic Systems, Measurements and Control 109, 299-309 (1987). W. J. Book and M. Majetta, Controller design for flexible distributed parameter mechanical arms via combined state spaced and frequency domain techniques. ASME J. Dynamic Systems, Measurements and Control 105, 245-254 (1983). S. Yurkovich. F. E. Pacheco and A. P. Tzes. On line frequent; domain information for control ‘of a flexible-link robot with varying payload. IEEE Trans. on Automatic Control AC-34, 1300-1304 (1989). D. Wang and M. Vidyasagar, Transfer function for a single flexible link. Proc. 1989 IEEE Int. Conf. on Robotics and Automation, Scottsdale, AZ, pp. 1042-1047 (1989). B. S. Yuan, W. J. Book and B. Siciliano, Direct adaptive control of a one-link flexible arm with tracking. J. Robotics Systems 6, 663480 (1989). A. F. Fath, ComputaGonal aspects of the linear optimal regulator problem. IEEE Trans. on Automatic Control, Oct. 1969, pp. 547-549 (1969). D. Wikerg, State Space and Linear Systems. McGrawHill Schaums Outlines (1971). D. Luenberger, Observing the state of a linear system. IEEE Trans. on Mil. Electronics, April (1964). B. Gopinath, On the control of linear multiple input-output systems. Bell System Tech. Jnl S(l), March (1971). G. G. Hastings and W. J. Book, Experiments in the control of a flexible robot arm. Proc. SME Robots Nine, Detroit, MI, June (1985). G. G. Hastings and W. J. Book, Reconstruction and robust reduced-order observation of flexible variables. DSC-Vol.3, pp. 11-16 (1986).