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Adaptive Control of a Flexible Manipulator with Varying Payload
Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993
ADAPTIVE CONTROL OF A FLEXIBLE MANIPULATOR WITH VARYING PAYLOAD D.K. Anand*, M. Anjanappa** and K.H. Sung*** *MeciJanical Engineering. University of Maryland - UMCP. College Park. MD 20742, USA **Mechanical Engineering. University of Maryland - UMBC, MD 2/228, USA ***Satellite Business Group. Korea Telecom. 680-63 Jayand-dong, Seoul, 133-/90. Korea
ABSTRACT. This research is concerned with the control of the tip position of a flexible manipulator carrying variable payload. The tip of the manipulator considered here has three degrees of frcedom. viz. one rotational and two translational. The desired rotational motion is the result of a command whereas the undesired translational motions are due to the deflection caused by the payload. Lagrangian equations with assumed mode shape functions are used to derive the dynamic equations of the manipulator. A recursive least square algorithm was used to identify the parameters. Numerical problems of the algorithm were overcome by introducing a scale factor and a time varying forgetting factor. The self-tuning adaptive control algorithm scheme was successfully implemented to control the rotational motion of the tip of the flexible manipulator carrying varying payloads. For the remaining two translational motions. the PID control algorithms were developed and fine tuned. The overall simulation results demonstrated the ability of the developed control system to maneuver the flexible manipulator so that the tip reaches the target position accurately under varying payloads. Keywords. Adaptive control; Robots; Self-tuning regulators; Pole placement; Controllers
degrees of freedom are the rotational tip angles (11) and the two translational motions (r & z) of the manipUlator tip. The hub angle, 11, represents the orientation of its centerline. A body-fixed reference frame with i, j and k is attached to the base of the beam. The deformations measured normal and vertical to the undeflected eenterline of the beam are represented by w,(x,t) and w,(x,t) , respectively.
1. BACKGROUND Improved dynamic performance ean be achieved through the use of "flexible manipulators" which typically have less weight. consumes less power. are less expensive and have better maneuverability than rigid robots. Mechanical flexibility, however. generates a fairly severe problem in controlling the motion due to the inevitable excitation of structural vibrations which affect the accuracy of the manipulator. Hence. the suceessful implementation of flexible manipulators is contingent upon achieving acceptable performance, taking into consideration variations in load, task specification. and the ability to compensate for any environmental disturbances.
The equation of motion can be derived from the Lagrangian.
!!..(ilL) _ilL
_Q (1) dJ a', a-/ ' where, L: Lagrangian function (L = T - V), T: Kinetic energy for the system, V: Potential energy for the system, Q;: Generalized force, and r;: Generalized coordinates of the system. The kinetic energy of the system, T, is.
In addition, the control action required to suppress the structural vibration must be minimized to keep the cycle time of the manipulator short. Here, the cycle time is defined as the total time taken from the command given to pick a part and to place the part accurately at its destination. The dynamic effects due to changes in configuration. load, moments of inertia. speed and unpredictable disturbanees tend to degrade the performance of the flexible manipulators. Hence. the controller must be able to adapt to these dynamic effects. A review of literature (Simpson et al. 1982; Uicker, 1969; WaIters et al. 1982) reveals that the vertical deflections of a flexible manipUlator carrying varying payload have been neglected by making the manipulator stiff in the vertical direction. If this restriction is removed. then, the vibrations in the horizontal plane could be reduced thereby decreasing the cycle time. However. the restriction can be removed if the controller considers three degrees of motion of the flexible manipulator.
,
T-.!. 2
J pR' R dx 0
+
..!.2 MR· R, p,
(2)
where. p: the mass per unit length of the system, R: a position vector which represents a point on the link's deformed centerline relative to an inertial frame. Mp: the mass of a payload. and Rp: the position vector of the tip. The potential energy can be expressed in terms of the generalized coordinates as.
i - dx V - -1 J' El [(]lW - -X.t)]2 2 0 .. dl: 2
1 J' El [(]lw/X,t)]2 - - - dx
+ -
2
0
u
dl: 2
1 J' P
+ -
2
0
(3) where. w.(x,t) and w,(x,t) are the deflections normal and vertical to the centerline of the undeformed beam, E: Young's modulus, g: acceleration of gravity. and I,,, I,,: area moment of inertia about x-axis and z-axis.
This paper deals with the development of a control system to adaptive1y control the tip position in all of its three degrees of freedom (DOF) of a single-link flexible manipUlator under varying payload conditions to achieve increased load carrying capacity and minimized cycle time.
The generalized force of the system is given by, N
Qk
-
ah
EF"-a', i-I
r.
k - 1,2, .... n
(4)
where h"F, are the physical coordinates and the applied forces in terms of the generalized coordinates. The substitution of eq.(2-4) in (1) yields the dynamic equations.
2. MATHEMATICAL MODEL
From Fig. 1, a position vector on the link, R. is given by ,
The flexible manipulator studied here (see Fig. 1) carries a payload at its tip and moves in the horizontal plane. The active
R - [:uX(t)] i +w.(x,t)j -[w,(x,t)+z(t)] k
567
(5)
(J+ ..).
(13)
[ symmlttnc
2.1. Simulation Results From the simulation work. it was found that to obtain a satisfactory description of the motion only one mode shape need be included. This work has been previously reported by Sung et al. (1987). The three OOF dynamic equations are then simulated to demonstrate the general dynamic behavior of the flexible manipulator. An aluminum beam of dimensions 22" x 1 5/8" x 1/16" was used to model the flexible manipulator. An input torque in the form lOsin'(brt) for t<0.5 was applied to the system about the IJ axis without input forces in two linear directions. The simulation of eq.(12). was conducted. Fig. 2 shows the comparison of the tip angle for the three different DOF motions. One and two OOF cases resulted in the same response. The motions of these angles are independent and decoupled as suggested in the dynamic equations. However. three DOF motions result in different responses compared to one or two DOF because the r and theta motions are coupled as indicated in the dynamic equations.
payloMJ
/
A
tIP angle (8 r J
==~"'--=+---
1.2
-
~."ll='.I:..:'•.::dl
~
---- Iip(2 DOF)
--- lip(! DOF)
-6-
lip(3 OOF)
0.8
Fig.1. Flexible manipulator with three OOF
0.6 0.4
0.2
where. w(x,t) -w.(x.t)i - w,(x,t)k. w.(x,t) and w,(x,t). are obtained by the assumed mode shape method in the form of.
t
w.(x.t) -
4> :{x)q/(t), w,(x.t) -
~I
t
~l
4> ~(x)q/(t)
o ..............., -..--....-=-===--------------j .0.2 L _ _---'-
o
(6)
[X, +X(t)]i
+
y,i - [zp +z(t)]k
E 4> :{l)q/(t) - w.(l,t).
(7)
t
zp -
i-1
J
-'-
0.15
0.2
.L-_ _- l
0.3
0.25
lime(sec)
Also. the simulation work showed that the dynamic equations can be decoupled if motion is very small. Based on the observation made here. the translational motions will be added to the manipulator after the tip approaches the desircd angle for the control of tip position.
where. xp -x(l). Yp -
-'--0.1
Fig.2. Comparison of tip angle with one. two and three OOF
where the!(x) and ,'{x) are the I" mode shape function about x-axis and z-axis, respectively. and q!(t) and q,'{t) are called the f' generalized displacement ofx-axis and z-axis. respectively. The tip position vector can be similarly defined by Rp -
0.05
4>:(l)q/(t) - wp,t).1I1:
3. PARAMETER IDENTIFICATION
i-I
time derivative of R is also obtained as, It is extremely difficult to sct up the mathematical model for the whole system. sincc the dynamic behavior of flexible manipulators vary with changes in configurations. loads, and speeds which are complex and nonlinear. It is therefore proposed that an equivalent linear model be set up and an on-line identification scheme. based on least squares method, be used to estimate the parametcrs of an equivalent linear model for on-line adaptive control scheme.
R-[i -w.(x.t)eJ i +[(x +X)e +w.(x.t)]i - [w,(x.t) +iJk (8) Substituting eq.(8) into eq.(2). the kmetic energy for the link can be obtained in terms of the incrtial coordinates as. I
T, -
t J[[i
-w.(x.t)6j + [(X+X)6+w.(x,t)j + [w,(x.t) +ifJlh(9) o In a similar way. the kinetic energy for the payload can be obtained as. Tp -
p
t
M ,[[i-Yp6j+ [(xp +X)6+Ypj+ [ip+ifJ
The system to be identified was assumed to be a single input. single output. deterministic. time-variant discrete system.
(10)
By substituting eqs.(3.4,9 and 10) into eq.(I). the equations of motion are derivcd in terms of the generalized coordinates.
y(z) - G(z)u(z)
The generalized forces for the clamped-free condition becomes.
Qe - T . aa and Q _ T . c36
• aa
q"
(11)
iXI i
Substituting eq.(3) and eqs.(9) through (11) into eq.(l). the dynamic equation is given by.
.
-Em,'q,I i-I
[01
101
{ol
[mj lm/'l
101
M
0
y(k) - 4>T(k-I)6(k-l)
w her e ij
T.+ ..
fq~
lE.} lE.)
-, i f
(14)
where y(z) and u(z) are the z-transform of the output and the input of the system. respectively. and G (z) is the transfer function of the system. The cquivalent linear model was used to estimate the unknown parameters of the system as shown in Fig. 3. The output is assumed to have the form.
4>(k-l) T Input
~
_
(15)
6(k-l) -[a l' a 2' ... 'n' a bl' b··· b ]T 2' , 11
fy(k-l) •...•y(k-n).u(k-l) •...u(k-n)] where n is the delay. FLEXIBLE MANIPULATOR (PROCESS)
,-mg F,+ ..
sym_tric M The governing three OOF equations (12) can be simplified by decoupling the larger amplitude rotational motion from small amplitude translational motions. Therefore. eq. (1) reduces to a one OOF equation for rotational motion only as.
IDENTIFICATION ALGORITHM
OF AN EQUIVALENT LINEAR MODEL
Fig.3. Block diagram of the idcntification process
568
and
output
The parameter vector 6 is estimated from measurements of input and output. Following Gauss, the unknown parameters of a model are chosen in such a way as to minimize the loss function,
where u(k) and y(k) are the input and output respectively and z·J is the backward shift operator. The transfer function is given by"
L
h/(k)z-I
G(Z-I)- y(k) _ B(k,z-I) _---=-1-.:..1
t
V(6,k) -
.
.
.!. ~>W
(16)
2/_ 1
u(k)
where, e(1) _y(1) - Y(i) -y(i) - cjlT(i-1)6(i-1).· Here y(i) are the observation output and y(i) are the estimated output. As a result of minimization of the loss function, the estimate parameters are,
A(k,z-I)
_
(22)
l+"to/(k)z-1 i-I
where,
a{ and h{ are the estimates of ai and hi'
From Fig. 4, one can formulate the following (Sung et al. 1987). 6(k) - p(k)[i cjl(i-1)Y(i)]
,-I
Lt
]-1
(17)
(Q(z-I)A(k.z- I ) - K(k.z-I)A(k,z-I) - H(k.z·I)B(k,z·I»Y(k) - Q(z·I)BO
(18)
(23)
To make eq.(18) computationally efficient, the least squares estimate can be shown to bc given in a rccursive form by,
Let Ajk,z·I) y(k) - B(k,z-I) w(k) be the desired closed-loop system such that the polynomial AJk,Z·I) -1 +U1dZ· 1 +auZ-2 + ... +UndZ·· has the desired set of
P(k)-
[ /-1
cjl(i-1)
6(k) - 6(k-l) + P(k)cjl(k-l)[y(k) - cjlT(k-1)6(k-l)]
(19)
P(k) _ ..!.[P(k-1) - P(k-1)cjl(k-1)T(k-1)P(k-1)] J.1 J.1 + T(k-1)P(k-1)(k-1)
(20)
poles. The coefficients of A(k.z- I ) and B(k.z- I ) are not known and are obtained as estimates from the identification algorithm. If the polynomials K(k,z·J) and H(k,z·J) are chosen to satisfy,
l
K(k,Z·I)A(k,z·l) + H(k,z·I)B(k,z-l) - Q(z-I)(A(k,z-l) -Ajk,z·l) (J4)
then, the following desired relationship would be satisfied,
Here p, the forgetting factor ranging from 0 to I, can be selected based upon spced of convergcncc dcsired. For proper identification, the input should excite all the modes of the system. This is satisfied by choosing the input signal as a periodic square wave. The two main factors that are taken into consideration in speeding up the convcrgence of the identification scheme are the initial gain matrix prO) and the forgetting factor, p. Details of this is given in Sung et al. (1987).
(25)
AJk,Z'I)y(k) - B(k,z-I)w(k).
The coefficients, K(k,z'/) and H(k,z'/) can be shown to be, 1 0 000
al °2
hi °1
0
0
... b2 bl
0
'.
Although, while the equivalent linear model has to be at least a fourth order system, identitication studies were conducted on 3'd to 6'h order equivalent linear models to achieve a better understanding of the effects of P(O) , p(O), and a on the speed of convergence. Reasonably good results were obtained when the manipulator was identified by a 3'd order model, withJ.1(O)-O.7, «-0.97, P(O)-lO"[l]and kp =lO. However, for the 4'h and 5'h order models, a had to be increased to 0.98 and 0.985 for the best convergence. See (Sung et at. 1987) for more details.
a. a._ 1 0 a.
... 0 b._I
hi b2
0
... 0
b3
0
b. b._ 1 0
(26) 0 0
By solving eq.(26), the coefficients of polynomials K(k,z·J) and H(k,z·J) are obtained. Hence, this control scheme performs satisfactorily only when the model is exact and all disturbances are also modeled. However, it is desirable for the system to be robust against minor errors in the process model. One way to eliminate this steady state error is to introduce an integrator as shown in Fig. 4. This closed·loop system is of order (n+ 1). For an overall desired pole locations, it can be shown that,
4. CONTROL METHODOLOGY A self·tuning adaptive controller was chosen to control the rotational motion in one DOF. Since, the linear motions are small a linear time-invariant PlO controller was chosen. Fig. 4 shows the block diagram of the three DOF motion control.
0 -1
0
bl
1+ 1
0 ... 0
b2
P2
b3
P3
0 ...
0
-1 1 ...
0
0
0 ...
0
0
0
0
npz
...
p.
b. -1
0
(27)
c
.+1
Thus, with a desired choice of the overall closed-loop poles given by D(z·J) and the coefficients ofB(k.z-I), the coefficients ofAAk,z' J) and c(k) , would be obtained by solving eq.(27). With
coefficients of Aik,z·J) thus obtained, eq.(26) can be solved for K(k,z'/) and H(k,z·J). It may, however, be difficult to specify all the desired closed-loop pole locations. One possibility is to specify only the dominant poles and require that the remaining poles be close to the origin (in the discrete time framework). In practice, it is often satisfactory to choose D(z·J)
Fig.4. Control structure of the three DOF motion
D(z'I)-1+PIZ-I+P2Z-2
whVt l,
where, PI - -2e"Wh coo( _(2 . . , is the natural frequency, (the damping ratio, and h is the sampling period.
4.1 Self-Tuning Pole Placement Control for rotational motion While a variety of configurations can be found in the literature (Elliot et al. 1979; Goodwin et al. 1985) for pole placement, the Luenberger observer structure was choscn. From Fig. 4, the relationship between input and output is, A(Z-I)Y(k) - B(Z-I)u(k)
(28) P2 _ e- 2,wh,
4.2 Implementation of the Self-Tuning Control Scheme The equivalent linear model is assumed to be of 4th order. The experimental setup included a rectangular cross-section
(21)
569
beam driven by a computer controlled torque motor at its hub in a horizontal plane. The tip angle was measured in real time with a video camera. The mass of payload is considered to be the main cause of change in the dynamic effccts of manipulators. By implementing the self-tuning adaptive control, a satisfactory performance can be achieved even when a high load to weight ratios are encountered. ...
.
Table 1: Fundamental Frequencies (unit: rad/sec) payload
I"
·p.ylo.d: 1.JSlb(2161.)
23.540 21.778 19.184 17.316 11.428
612.183 566.297 498.833 450.268 297.150
Table 2: Cycle Time for Various Payloads (Unit: seconds)
,_~.d A I :o~:~ \ /
Rectangular crosssectional beam(z)
5% 10% 20% 30% 100%
be,la; 0.491b
I _ lU~'
Rectangular crosssectional beam(x)
payload
5%
10%
20%
30%
cycle time
2.67
2.69
2.79
3.30
plck\lpl.1Slb
relenr
O.4r'
(9Z1)
.. l _ .. L.
J .••.1....
~1IJ.01.J
ANGLEIAAD)
0.12 r----'--'---·
I
M'_ .. J .... 1
~
•...l...
·Ht
tB lin. HU I W[ (SEC)
... 1 •.•• __.. 1.. __ •
0.1
aM. m:l
Fig.5. Commanded and actual tip angles with fixed parameters
-,
"I'"
r" ..
.,
bUII\;0.491b ~110.4: I,JSlb(V61.)
l-----------;,.-='7'-==-----
0.08 .
rei ang.
0.06 .
tip ang(3radlsec)
0.04
tip ang(2radlsec) tip ang(3.5radlsec)
0.02
oL.L--'---'-"---'---~-~--~-~---'
o
d T1MEISEC)
Fig.7. Plot of tip angles for different system frequencies
plctupl.JSlb
.....
~1."ll
L . . . L .•..
!'
_.J. ...J.
~l~."~
, __ I.
•..
,L.
_.
·Ht . .,,,
f I ",(ISEC)
_.1. _•.•. ' lj~'.H"
Large oscillation of the tip during the transition period sometimes resulted in numerical instability for higher frcquencies as shown in Fig. 7. Results indicated that the performance of the PID controller deteriorates with increasing payload. It was also found that the control of the longitudinal motion can be omitted with no loss of accuracy.
~,--,-.l
.'
1• • .1
00.nn
Fig.6. Commanded and actual tip angles with resetting parameters
5. CONCLUSIONS A scheme for the tip position control of a three degree of freedom(DOF) flexible manipulator carrying variable payload was developed and simulated. The controller included a selftuning adaptive scheme for the rotation (9), and two PID controls for the translational motions (r and z). A linearized model to describe thc dynamics of the flexible manipulator was utilized in the identification and controller design. The results of the implementation for one DOF demonstrated the ability of the controller to achieve accurate positioning under varying payloads. The simulation studies for the three DOF motion showed that, by controlling the vertical motion, settling time can be minimized so that it does not dominate the cycle time.
In this experiment, a variable payload was used to change the system parameters. After the manipulator picked up the payload, it continuously released part of the payload, and it went back to the original position after it had released all of the payload. Fig. 5 shows the tip response for fixed gain scheme. The tip response with the adaptive control is shown in Fig. 6 with resetting parameters. The manipulator had the same uniform performance regardless of the weight of the payloads. These results show that the flexible manipulator can easily handle high payload to weight ratios, about 276%, with the implementation of the adaptive control. 4.3 Simulation of Three DOF Control
6. REFERENCES
In the previous experiment, it has been assumed that there is no deflection in the vertical direction of thc manipulator by imposing limits on the payload size. However, a flexible manipulator should maintain the requircd accuracy and achieve the fast settling time while carrying heavy payloads in the work place. As a result, more than one DOF is required for more practical uses of a flexible manipulator.
Elliot, H. and Wolovich W.A. , "Parameter Adaptive Identification and Control," IEEE Trans. on Auto. ConI. AC-24, Aug. 1979, pp.592-599. Goodwin G.C. and Sin KS., "Adaptive Control of Nonminimum Phase Systems," IEEE Trans. on Auto. ConI. AC-26, April 1985, pp.478-483. Simpson, J.A., Hocken, R.J., and A1bus, J.A., "The Automated Manufacturing Research Facilities of the N.B.S.," National Bureau of Standards, Washington D.C., 1982 Sung, KH. and Yang, J.C.S., "A Dynamic Study of a Flexible One Link Manipulator: Theoreticsl and Experimental Approach," Second Int. Conf. on Robotics and Factories of the Future, San Diego, July 1987. Sung, KH., Kuduva, P., and Yang, J.C.S., "Parameter Identification of a Flexible Manipulator," IEEE Systems, Man and Cybernetics Annual Conference, October 1987. Uicker, J.J., "Dynamic Behavior of Spatial Linkages," ASME J. Eng. Ind., Feb. 1969, pp.251-265. Waiters R.G. and Byoumi M.M.,"Application of a Self-Tuning Pole-Placement Regulator to an Industrial Manipulator," IEEE Proc. on CDC, 1982, pp.323-329.
The desired position for the z-axis, always set to zero here, can be reached by moving the manipulator using a linear motor installed at the origin and controlled using a PID scheme. One more controller, in the longitudinal (r) direction, is necessary to reach the desired tip positions. (see Fig. 4) Simulation studies, using rectangular cross section beam, were conducted to evaluate the proposed controller to adaptively control the tip position of a single link flexible manipulator under varying payload conditions. Different payloads were attached to the tip of the manipulator and their weights were represented by a percentage of the weight of the manipulator. Table 1 presents the fundamental frequency of the beam carrying different payloads. The cycle time attained for various payloads are listed in Table 2.
570
ADAPTIVE CONTROL OF A FLEXIBLE MANIPULATOR WITH VARYING PAYLOAD D.K. Anand*, M. Anjanappa** and K.H. Sung*** *MeciJanical Engineering. University of Maryland - UMCP. College Park. MD 20742, USA **Mechanical Engineering. University of Maryland - UMBC, MD 2/228, USA ***Satellite Business Group. Korea Telecom. 680-63 Jayand-dong, Seoul, 133-/90. Korea
ABSTRACT. This research is concerned with the control of the tip position of a flexible manipulator carrying variable payload. The tip of the manipulator considered here has three degrees of frcedom. viz. one rotational and two translational. The desired rotational motion is the result of a command whereas the undesired translational motions are due to the deflection caused by the payload. Lagrangian equations with assumed mode shape functions are used to derive the dynamic equations of the manipulator. A recursive least square algorithm was used to identify the parameters. Numerical problems of the algorithm were overcome by introducing a scale factor and a time varying forgetting factor. The self-tuning adaptive control algorithm scheme was successfully implemented to control the rotational motion of the tip of the flexible manipulator carrying varying payloads. For the remaining two translational motions. the PID control algorithms were developed and fine tuned. The overall simulation results demonstrated the ability of the developed control system to maneuver the flexible manipulator so that the tip reaches the target position accurately under varying payloads. Keywords. Adaptive control; Robots; Self-tuning regulators; Pole placement; Controllers
degrees of freedom are the rotational tip angles (11) and the two translational motions (r & z) of the manipUlator tip. The hub angle, 11, represents the orientation of its centerline. A body-fixed reference frame with i, j and k is attached to the base of the beam. The deformations measured normal and vertical to the undeflected eenterline of the beam are represented by w,(x,t) and w,(x,t) , respectively.
1. BACKGROUND Improved dynamic performance ean be achieved through the use of "flexible manipulators" which typically have less weight. consumes less power. are less expensive and have better maneuverability than rigid robots. Mechanical flexibility, however. generates a fairly severe problem in controlling the motion due to the inevitable excitation of structural vibrations which affect the accuracy of the manipulator. Hence. the suceessful implementation of flexible manipulators is contingent upon achieving acceptable performance, taking into consideration variations in load, task specification. and the ability to compensate for any environmental disturbances.
The equation of motion can be derived from the Lagrangian.
!!..(ilL) _ilL
_Q (1) dJ a', a-/ ' where, L: Lagrangian function (L = T - V), T: Kinetic energy for the system, V: Potential energy for the system, Q;: Generalized force, and r;: Generalized coordinates of the system. The kinetic energy of the system, T, is.
In addition, the control action required to suppress the structural vibration must be minimized to keep the cycle time of the manipulator short. Here, the cycle time is defined as the total time taken from the command given to pick a part and to place the part accurately at its destination. The dynamic effects due to changes in configuration. load, moments of inertia. speed and unpredictable disturbanees tend to degrade the performance of the flexible manipulators. Hence. the controller must be able to adapt to these dynamic effects. A review of literature (Simpson et al. 1982; Uicker, 1969; WaIters et al. 1982) reveals that the vertical deflections of a flexible manipUlator carrying varying payload have been neglected by making the manipulator stiff in the vertical direction. If this restriction is removed. then, the vibrations in the horizontal plane could be reduced thereby decreasing the cycle time. However. the restriction can be removed if the controller considers three degrees of motion of the flexible manipulator.
,
T-.!. 2
J pR' R dx 0
+
..!.2 MR· R, p,
(2)
where. p: the mass per unit length of the system, R: a position vector which represents a point on the link's deformed centerline relative to an inertial frame. Mp: the mass of a payload. and Rp: the position vector of the tip. The potential energy can be expressed in terms of the generalized coordinates as.
i - dx V - -1 J' El [(]lW - -X.t)]2 2 0 .. dl: 2
1 J' El [(]lw/X,t)]2 - - - dx
+ -
2
0
u
dl: 2
1 J' P
+ -
2
0
(3) where. w.(x,t) and w,(x,t) are the deflections normal and vertical to the centerline of the undeformed beam, E: Young's modulus, g: acceleration of gravity. and I,,, I,,: area moment of inertia about x-axis and z-axis.
This paper deals with the development of a control system to adaptive1y control the tip position in all of its three degrees of freedom (DOF) of a single-link flexible manipUlator under varying payload conditions to achieve increased load carrying capacity and minimized cycle time.
The generalized force of the system is given by, N
Qk
-
ah
EF"-a', i-I
r.
k - 1,2, .... n
(4)
where h"F, are the physical coordinates and the applied forces in terms of the generalized coordinates. The substitution of eq.(2-4) in (1) yields the dynamic equations.
2. MATHEMATICAL MODEL
From Fig. 1, a position vector on the link, R. is given by ,
The flexible manipulator studied here (see Fig. 1) carries a payload at its tip and moves in the horizontal plane. The active
R - [:uX(t)] i +w.(x,t)j -[w,(x,t)+z(t)] k
567
(5)
(J+ ..).
(13)
[ symmlttnc
2.1. Simulation Results From the simulation work. it was found that to obtain a satisfactory description of the motion only one mode shape need be included. This work has been previously reported by Sung et al. (1987). The three OOF dynamic equations are then simulated to demonstrate the general dynamic behavior of the flexible manipulator. An aluminum beam of dimensions 22" x 1 5/8" x 1/16" was used to model the flexible manipulator. An input torque in the form lOsin'(brt) for t<0.5 was applied to the system about the IJ axis without input forces in two linear directions. The simulation of eq.(12). was conducted. Fig. 2 shows the comparison of the tip angle for the three different DOF motions. One and two OOF cases resulted in the same response. The motions of these angles are independent and decoupled as suggested in the dynamic equations. However. three DOF motions result in different responses compared to one or two DOF because the r and theta motions are coupled as indicated in the dynamic equations.
payloMJ
/
A
tIP angle (8 r J
==~"'--=+---
1.2
-
~."ll='.I:..:'•.::dl
~
---- Iip(2 DOF)
--- lip(! DOF)
-6-
lip(3 OOF)
0.8
Fig.1. Flexible manipulator with three OOF
0.6 0.4
0.2
where. w(x,t) -w.(x.t)i - w,(x,t)k. w.(x,t) and w,(x,t). are obtained by the assumed mode shape method in the form of.
t
w.(x.t) -
4> :{x)q/(t), w,(x.t) -
~I
t
~l
4> ~(x)q/(t)
o ..............., -..--....-=-===--------------j .0.2 L _ _---'-
o
(6)
[X, +X(t)]i
+
y,i - [zp +z(t)]k
E 4> :{l)q/(t) - w.(l,t).
(7)
t
zp -
i-1
J
-'-
0.15
0.2
.L-_ _- l
0.3
0.25
lime(sec)
Also. the simulation work showed that the dynamic equations can be decoupled if motion is very small. Based on the observation made here. the translational motions will be added to the manipulator after the tip approaches the desircd angle for the control of tip position.
where. xp -x(l). Yp -
-'--0.1
Fig.2. Comparison of tip angle with one. two and three OOF
where the
0.05
4>:(l)q/(t) - wp,t).1I1:
3. PARAMETER IDENTIFICATION
i-I
time derivative of R is also obtained as, It is extremely difficult to sct up the mathematical model for the whole system. sincc the dynamic behavior of flexible manipulators vary with changes in configurations. loads, and speeds which are complex and nonlinear. It is therefore proposed that an equivalent linear model be set up and an on-line identification scheme. based on least squares method, be used to estimate the parametcrs of an equivalent linear model for on-line adaptive control scheme.
R-[i -w.(x.t)eJ i +[(x +X)e +w.(x.t)]i - [w,(x.t) +iJk (8) Substituting eq.(8) into eq.(2). the kmetic energy for the link can be obtained in terms of the incrtial coordinates as. I
T, -
t J[[i
-w.(x.t)6j + [(X+X)6+w.(x,t)j + [w,(x.t) +ifJlh(9) o In a similar way. the kinetic energy for the payload can be obtained as. Tp -
p
t
M ,[[i-Yp6j+ [(xp +X)6+Ypj+ [ip+ifJ
The system to be identified was assumed to be a single input. single output. deterministic. time-variant discrete system.
(10)
By substituting eqs.(3.4,9 and 10) into eq.(I). the equations of motion are derivcd in terms of the generalized coordinates.
y(z) - G(z)u(z)
The generalized forces for the clamped-free condition becomes.
Qe - T . aa and Q _ T . c36
• aa
q"
(11)
iXI i
Substituting eq.(3) and eqs.(9) through (11) into eq.(l). the dynamic equation is given by.
.
-Em,'q,I i-I
[01
101
{ol
[mj lm/'l
101
M
0
y(k) - 4>T(k-I)6(k-l)
w her e ij
T.+ ..
fq~
lE.} lE.)
-, i f
(14)
where y(z) and u(z) are the z-transform of the output and the input of the system. respectively. and G (z) is the transfer function of the system. The cquivalent linear model was used to estimate the unknown parameters of the system as shown in Fig. 3. The output is assumed to have the form.
4>(k-l) T Input
~
_
(15)
6(k-l) -[a l' a 2' ... 'n' a bl' b··· b ]T 2' , 11
fy(k-l) •...•y(k-n).u(k-l) •...u(k-n)] where n is the delay. FLEXIBLE MANIPULATOR (PROCESS)
,-mg F,+ ..
sym_tric M The governing three OOF equations (12) can be simplified by decoupling the larger amplitude rotational motion from small amplitude translational motions. Therefore. eq. (1) reduces to a one OOF equation for rotational motion only as.
IDENTIFICATION ALGORITHM
OF AN EQUIVALENT LINEAR MODEL
Fig.3. Block diagram of the idcntification process
568
and
output
The parameter vector 6 is estimated from measurements of input and output. Following Gauss, the unknown parameters of a model are chosen in such a way as to minimize the loss function,
where u(k) and y(k) are the input and output respectively and z·J is the backward shift operator. The transfer function is given by"
L
h/(k)z-I
G(Z-I)- y(k) _ B(k,z-I) _---=-1-.:..1
t
V(6,k) -
.
.
.!. ~>W
(16)
2/_ 1
u(k)
where, e(1) _y(1) - Y(i) -y(i) - cjlT(i-1)6(i-1).· Here y(i) are the observation output and y(i) are the estimated output. As a result of minimization of the loss function, the estimate parameters are,
A(k,z-I)
_
(22)
l+"to/(k)z-1 i-I
where,
a{ and h{ are the estimates of ai and hi'
From Fig. 4, one can formulate the following (Sung et al. 1987). 6(k) - p(k)[i cjl(i-1)Y(i)]
,-I
Lt
]-1
(17)
(Q(z-I)A(k.z- I ) - K(k.z-I)A(k,z-I) - H(k.z·I)B(k,z·I»Y(k) - Q(z·I)BO
(18)
(23)
To make eq.(18) computationally efficient, the least squares estimate can be shown to bc given in a rccursive form by,
Let Ajk,z·I) y(k) - B(k,z-I) w(k) be the desired closed-loop system such that the polynomial AJk,Z·I) -1 +U1dZ· 1 +auZ-2 + ... +UndZ·· has the desired set of
P(k)-
[ /-1
cjl(i-1)
6(k) - 6(k-l) + P(k)cjl(k-l)[y(k) - cjlT(k-1)6(k-l)]
(19)
P(k) _ ..!.[P(k-1) - P(k-1)cjl(k-1)T(k-1)P(k-1)] J.1 J.1 + T(k-1)P(k-1)(k-1)
(20)
poles. The coefficients of A(k.z- I ) and B(k.z- I ) are not known and are obtained as estimates from the identification algorithm. If the polynomials K(k,z·J) and H(k,z·J) are chosen to satisfy,
l
K(k,Z·I)A(k,z·l) + H(k,z·I)B(k,z-l) - Q(z-I)(A(k,z-l) -Ajk,z·l) (J4)
then, the following desired relationship would be satisfied,
Here p, the forgetting factor ranging from 0 to I, can be selected based upon spced of convergcncc dcsired. For proper identification, the input should excite all the modes of the system. This is satisfied by choosing the input signal as a periodic square wave. The two main factors that are taken into consideration in speeding up the convcrgence of the identification scheme are the initial gain matrix prO) and the forgetting factor, p. Details of this is given in Sung et al. (1987).
(25)
AJk,Z'I)y(k) - B(k,z-I)w(k).
The coefficients, K(k,z'/) and H(k,z'/) can be shown to be, 1 0 000
al °2
hi °1
0
0
... b2 bl
0
'.
Although, while the equivalent linear model has to be at least a fourth order system, identitication studies were conducted on 3'd to 6'h order equivalent linear models to achieve a better understanding of the effects of P(O) , p(O), and a on the speed of convergence. Reasonably good results were obtained when the manipulator was identified by a 3'd order model, withJ.1(O)-O.7, «-0.97, P(O)-lO"[l]and kp =lO. However, for the 4'h and 5'h order models, a had to be increased to 0.98 and 0.985 for the best convergence. See (Sung et at. 1987) for more details.
a. a._ 1 0 a.
... 0 b._I
hi b2
0
... 0
b3
0
b. b._ 1 0
(26) 0 0
By solving eq.(26), the coefficients of polynomials K(k,z·J) and H(k,z·J) are obtained. Hence, this control scheme performs satisfactorily only when the model is exact and all disturbances are also modeled. However, it is desirable for the system to be robust against minor errors in the process model. One way to eliminate this steady state error is to introduce an integrator as shown in Fig. 4. This closed·loop system is of order (n+ 1). For an overall desired pole locations, it can be shown that,
4. CONTROL METHODOLOGY A self·tuning adaptive controller was chosen to control the rotational motion in one DOF. Since, the linear motions are small a linear time-invariant PlO controller was chosen. Fig. 4 shows the block diagram of the three DOF motion control.
0 -1
0
bl
1+ 1
0 ... 0
b2
P2
b3
P3
0 ...
0
-1 1 ...
0
0
0 ...
0
0
0
0
npz
...
p.
b. -1
0
(27)
c
.+1
Thus, with a desired choice of the overall closed-loop poles given by D(z·J) and the coefficients ofB(k.z-I), the coefficients ofAAk,z' J) and c(k) , would be obtained by solving eq.(27). With
coefficients of Aik,z·J) thus obtained, eq.(26) can be solved for K(k,z'/) and H(k,z·J). It may, however, be difficult to specify all the desired closed-loop pole locations. One possibility is to specify only the dominant poles and require that the remaining poles be close to the origin (in the discrete time framework). In practice, it is often satisfactory to choose D(z·J)
Fig.4. Control structure of the three DOF motion
D(z'I)-1+PIZ-I+P2Z-2
whVt l,
where, PI - -2e"Wh coo( _(2 . . , is the natural frequency, (the damping ratio, and h is the sampling period.
4.1 Self-Tuning Pole Placement Control for rotational motion While a variety of configurations can be found in the literature (Elliot et al. 1979; Goodwin et al. 1985) for pole placement, the Luenberger observer structure was choscn. From Fig. 4, the relationship between input and output is, A(Z-I)Y(k) - B(Z-I)u(k)
(28) P2 _ e- 2,wh,
4.2 Implementation of the Self-Tuning Control Scheme The equivalent linear model is assumed to be of 4th order. The experimental setup included a rectangular cross-section
(21)
569
beam driven by a computer controlled torque motor at its hub in a horizontal plane. The tip angle was measured in real time with a video camera. The mass of payload is considered to be the main cause of change in the dynamic effccts of manipulators. By implementing the self-tuning adaptive control, a satisfactory performance can be achieved even when a high load to weight ratios are encountered. ...
.
Table 1: Fundamental Frequencies (unit: rad/sec) payload
I"
·p.ylo.d: 1.JSlb(2161.)
23.540 21.778 19.184 17.316 11.428
612.183 566.297 498.833 450.268 297.150
Table 2: Cycle Time for Various Payloads (Unit: seconds)
,_~.d A I :o~:~ \ /
Rectangular crosssectional beam(z)
5% 10% 20% 30% 100%
be,la; 0.491b
I _ lU~'
Rectangular crosssectional beam(x)
payload
5%
10%
20%
30%
cycle time
2.67
2.69
2.79
3.30
plck\lpl.1Slb
relenr
O.4r'
(9Z1)
.. l _ .. L.
J .••.1....
~1IJ.01.J
ANGLEIAAD)
0.12 r----'--'---·
I
M'_ .. J .... 1
~
•...l...
·Ht
tB lin. HU I W[ (SEC)
... 1 •.•• __.. 1.. __ •
0.1
aM. m:l
Fig.5. Commanded and actual tip angles with fixed parameters
-,
"I'"
r" ..
.,
bUII\;0.491b ~110.4: I,JSlb(V61.)
l-----------;,.-='7'-==-----
0.08 .
rei ang.
0.06 .
tip ang(3radlsec)
0.04
tip ang(2radlsec) tip ang(3.5radlsec)
0.02
oL.L--'---'-"---'---~-~--~-~---'
o
d T1MEISEC)
Fig.7. Plot of tip angles for different system frequencies
plctupl.JSlb
.....
~1."ll
L . . . L .•..
!'
_.J. ...J.
~l~."~
, __ I.
•..
,L.
_.
·Ht . .,,,
f I ",(ISEC)
_.1. _•.•. ' lj~'.H"
Large oscillation of the tip during the transition period sometimes resulted in numerical instability for higher frcquencies as shown in Fig. 7. Results indicated that the performance of the PID controller deteriorates with increasing payload. It was also found that the control of the longitudinal motion can be omitted with no loss of accuracy.
~,--,-.l
.'
1• • .1
00.nn
Fig.6. Commanded and actual tip angles with resetting parameters
5. CONCLUSIONS A scheme for the tip position control of a three degree of freedom(DOF) flexible manipulator carrying variable payload was developed and simulated. The controller included a selftuning adaptive scheme for the rotation (9), and two PID controls for the translational motions (r and z). A linearized model to describe thc dynamics of the flexible manipulator was utilized in the identification and controller design. The results of the implementation for one DOF demonstrated the ability of the controller to achieve accurate positioning under varying payloads. The simulation studies for the three DOF motion showed that, by controlling the vertical motion, settling time can be minimized so that it does not dominate the cycle time.
In this experiment, a variable payload was used to change the system parameters. After the manipulator picked up the payload, it continuously released part of the payload, and it went back to the original position after it had released all of the payload. Fig. 5 shows the tip response for fixed gain scheme. The tip response with the adaptive control is shown in Fig. 6 with resetting parameters. The manipulator had the same uniform performance regardless of the weight of the payloads. These results show that the flexible manipulator can easily handle high payload to weight ratios, about 276%, with the implementation of the adaptive control. 4.3 Simulation of Three DOF Control
6. REFERENCES
In the previous experiment, it has been assumed that there is no deflection in the vertical direction of thc manipulator by imposing limits on the payload size. However, a flexible manipulator should maintain the requircd accuracy and achieve the fast settling time while carrying heavy payloads in the work place. As a result, more than one DOF is required for more practical uses of a flexible manipulator.
Elliot, H. and Wolovich W.A. , "Parameter Adaptive Identification and Control," IEEE Trans. on Auto. ConI. AC-24, Aug. 1979, pp.592-599. Goodwin G.C. and Sin KS., "Adaptive Control of Nonminimum Phase Systems," IEEE Trans. on Auto. ConI. AC-26, April 1985, pp.478-483. Simpson, J.A., Hocken, R.J., and A1bus, J.A., "The Automated Manufacturing Research Facilities of the N.B.S.," National Bureau of Standards, Washington D.C., 1982 Sung, KH. and Yang, J.C.S., "A Dynamic Study of a Flexible One Link Manipulator: Theoreticsl and Experimental Approach," Second Int. Conf. on Robotics and Factories of the Future, San Diego, July 1987. Sung, KH., Kuduva, P., and Yang, J.C.S., "Parameter Identification of a Flexible Manipulator," IEEE Systems, Man and Cybernetics Annual Conference, October 1987. Uicker, J.J., "Dynamic Behavior of Spatial Linkages," ASME J. Eng. Ind., Feb. 1969, pp.251-265. Waiters R.G. and Byoumi M.M.,"Application of a Self-Tuning Pole-Placement Regulator to an Industrial Manipulator," IEEE Proc. on CDC, 1982, pp.323-329.
The desired position for the z-axis, always set to zero here, can be reached by moving the manipulator using a linear motor installed at the origin and controlled using a PID scheme. One more controller, in the longitudinal (r) direction, is necessary to reach the desired tip positions. (see Fig. 4) Simulation studies, using rectangular cross section beam, were conducted to evaluate the proposed controller to adaptively control the tip position of a single link flexible manipulator under varying payload conditions. Different payloads were attached to the tip of the manipulator and their weights were represented by a percentage of the weight of the manipulator. Table 1 presents the fundamental frequency of the beam carrying different payloads. The cycle time attained for various payloads are listed in Table 2.
570