- Email: [email protected]
Gain Scheduled Motion Control of an Overhead Traveling Crane
Copyright © IFAC Control Systems Design, Bratislava, Slovak Republic, 2000
GAIN SCHEDULED MOTION CONTROL OF AN OVERHEAD TRA VELING CRANE
H. Aschemann, S. Lahres, O. Sawodny, E.P. Hofer
Department of Measurement, Control and Microtechnology University of Ulm, Germany E-mail: [email protected] Tel.:+49 73150-26336. Fax: +4973150-26301
Abstract: This paper presents a gain scheduling approach to motion control of an overhead traveling crane using feed forward and feedback control as well as observer based disturbance rejection. All control system components are derived in symbolic form and adapted online to the varying system parameters rope length, load mass, and trolley position. The effectiveness of the developed control scheme is shown by experimental results from a 5 t - overhead traveling crane. Here, trajectory tracking performance as well as the compensation of disturbance forces caused by nonlinear friction and wind impact are presented. Copyright @2000 IFAC Keywords: Crane control, gain scheduling, decoupling control, feedforward control, disturbance rejection, trajectory planning, robotics
I.
For crane control a decentralized concept with independent control systems for each crane axis has been chosen. In this paper, the controller design for the crane x-axis is described exemplary. First, a mathematical model of this axis has to be derived. The mechanical system is modeled by an multibody sY6tem consisting of three rigid bodies: the bridge (length lB=2 11> reduced mass mB.red including the contributions from the electric drives and gears, mass moment of inertia CB with respect to the z-axis), the trolley (mass mT), and the crane load (mass md. The load centre of gravity has an offset hL in z-direction. The rope is assumed to be a massless rigid link with length IRO' For simplicity the abbreviation IR=IRo+h L for the total rope length is introduced.
INTRODUCTION
Until now industrial control systems for automated cranes only guarantee position accuracy at the target point and damping of load oscillations (Bryfors et af., 1996). In order to improve the abilities of automated cranes path planning (Sawodny et al., 1999b) combined with effective feedforward control (Deleau et af., 1998) and variable feedback controllers (Nguyen, 1998) become important. At the University of Ulm a motion control concept for an automated bridge crane has been developed that allows to track specified trajectories with high accuracy. Characteristic features of this control concept are time optimal trajectory planning, decentralized controllers for each crane axis, adaptive feedforward and feedback control, and observer based disturbance rejection to take into account disturbances. With this capability an automated crane can be used as a robot manipulator for the handling of heavy loads in a large cartesian workspace. This paper presents the modeling and the control system design for a 5 t bridge crane as well as measurement results.
2.
The movement in x-direction is fully described by three generalised coordinates: the position of the right bridge side xBr(l), the position of the left bridge side XBI(t) and the position of the load XL(t). The vector of generalised coordinates is chosen as Z=[XBr XBI xd T. According to the decentralized control concept the relative position YTO of the trolley with respect to the centre of the bridge is assumed to vary slowly with time. Therefore, the time derivatives of this quantity are neglected concerning x-axis controller design. The drive forces acting on the left and the right side of the bridge are denoted as FMB1(t) and FMBr(t), respectively. Viscous damping forces are described by the damping coefficients bB1 and bBr . The main disturbances are the nonlinear friction forces FFBI(t) and FFBr(t) acting on the drives and a possible force Fw(t) caused by wind acting on the load surface.
MULTIBODY MODEL OF CRANE X-AXIS
The overhead traveling crane consists of three actuated axes (fig. 1). The direction of crane bridge motion is denoted as x-axis. The trolley can perform a movement on the bridge, which is referred to as yaxis. The crane load can be hoisted or lowered in zdirection by means of the rope suspension.
573
ma.r
Fig. 1: Multibody model of the crane x-axis These equations can be written representation
Applying the Lagrange formalism
~(dT(Z,:i»)_ dT(z,:i) + dU(Z) =f T dt d:i dZ dZ ne '
m21
mn
0
0
r"
+ k 21 k 31
k 12
kn k 32
l"'J r k'l"'J r oo
mL
xBI +
0
xL
0
"'
FM
k 23
x Bl
k 33
xL
or
i=Ax+B u +B e =Ax+Buu+Bee, with the unity matrix I, the input vector U=[FMBr FMB1f, and the vector of disturbance forces e=[FFBr FFBI FW]T.
0
b
ml2
o T"' o J j r- ~", J b B1
xBI
0
xL
= FMB1 + - FFBI "' 0 Fw
3.
mB.red m T =- - +- -
4
4
(2)
mTYTO CB + mTY~O + ----"'-----,-:~..::... 21 1 41f
m B.red m CB + mTY~O m p = m 21 = - - - + - T -4 4 41 12
For adaptive controller design a decoupling feedback control law is derived analytically (Falb et al., 1967). The vector of output quantities y = [YI Y2]T = [~XB xd T consists of the position difference y,(t) = ~XB(t) = XBr(t) - XBI(t) and the load position Y2(t) = XL(t). The control objectives are to achieve a vanishing position difference ~XB(t) = 0, i.e. XBr(t) = XBI(t), and a perfect tracking XL(t) = XL.ret
(3) ?
m B.red m mTYTO CB + mTYTO m O? = - - - + - -T+ + 0 4 4 21 1 41] and the elements kij of K are as follows 2 2 2 k = (YTo -1 1 ) mLg k = k? = (11 - YTO)mLg 11 41 2 1 ' 12 _J 41 2 1 ' 1 R
k
-k 13 -
DECOUPLING CONTROL DESIGN
The main varying system parameters are the load mass mL, the rope length IR , and the trolley position YTO' While the load mass is constant after the picking up procedure, the rope length and the trolley position may vary during the movement on the trajectory in the workspace. The load mass is measured by a load cell at the fixed end of the rope. Also the trolley position and the rope length are available by encoder measurements. Therefore, the controller gains can easily be adapted with respect to changing operation points.
or in matrix notation Mi + D:i + Kz = f u + fe' with the mass matrix M, the damping matrix D, the stiffness matrix K, the vector of drive forces f u , and the vector of nonlinear disturbance forces f e . The elements mij of M are given by m 11
state space
(1)
with the total kinetic energy T, the potential energy U, and the vector of non conservative forces f nc , nonlinear equations of motion are computed. Because of small rope angles, the following linearized equations of motion hold
rmu
In
1 R
_(YTo-ll)mLg 21 I '
(4)
31 -
1 R
A decoupling feedback control law can be derived if the decoupling matrix Dd has full rank and no
574
instable internal dynamics exists. Therefore, the time derivatives of the output equation have to be considered. The relative degree r; with respect to the output i equals the order of the output time derivative which is affected directly by the inputs. Since
o
7 43 S +UnS +UnS- +U2Is+u20
(12)
.
T.
TA
Yl=clx=cl
..
TA 2
x'YI=cl
TAB
x+c,
The dynamics of the transfer functions on the matrix diagonal are chosen independently to achieve asymptotic stability as well as good control performance. The second-order system describes the closed-loop dynamics of the position difference XBrXBl of the bridge sides, while the fourth-order system represents the closed-loop dynamics of the load position. Hence, the coefficients alO, all and a20, a2l. a22, a23 can be derived by comparison with appropriate characteristic polynomials resulting from proposed pole configurations. This specified closedloop transfer function matrix G(s) remains constant since the feedback control law is adapted to the actual values of the varying system parameters.
(7)
uU
the input u has direct influence on the second time derivative and the relative degree of the first output YI becomes rl=2. Similarly with
the relative degree r2=4 is obtained. The total relative degree r is determined as the sum of the relative degrees rl=2 and r2=4 associated with the first and the second output, respectively r = rl + r 2 = 6 = n . As the total relative degree equals the system order, no internal dynamics occurs. The decoupling matrix is then
4. FEEDFORWARD CONTROL DESIGN The transient behavior can be essentially improved by applying feedforward control (Sawodny et aI., 1999a). Here, the feedforward control action is divided into a linear and a nonlinear part. Concerning the position difference of the bridge sides a synchronous motion with a vanishing difference is aimed at. Hence, for setpoint control 131=0 can be chosen. The trajectory planning module provides the reference trajectories for the crane load with continuous third or fourth derivatives with respect to time. Since these reference functions are available, a linear feed forward control law based on
(9)
which is non-singular in the operation range. Hence, the inverse matrix Dd-I can be computed. With the matrices Cd , L d , and Md
(13)
the decoupling control law is given by U
= -Dd1(M d +Cd)x + Dd1Ldw = uFB
is applied. Note, that the linear feedforward control part is also adapted to the varying system parameters
+uFF lin'
FF.lin = Dd1Ldw depends on the inverse decoupling matrix. Additionally, a nonlinear feedforward control part is used to compensate for nonlinear friction. The nonlinear friction characteristic of the crane x-axis, which is depicted in fig. 2, has been identified by experiments. By optimization analytical functions for the nonlinear T part UFF.nl=[UFF.nl.r UFF.nl.a with since
(11)
with Was the new input vector W=[~XB.ref XLref]T. The decoupling control law consists of two parts: the feedback part uFB = -Dd1(M d + Cd)x depending on the measured state variables x and the linear feedforward
part
uFF.lin = Dd1Ldw
which
IS
calculated with the reference trajectory and its time derivatives. All matrices are computed analytically and depend on the actual values of trolley position YTO, rope length I R and load mass mL. Therefore, the decoupling control law is adapted online by evaluating the analytical expressions. The closed - loop transfer function matrix G(s) decoupled and becomes
U
xB- ) + YIp; xB . 1 FFBp; tanh ( 0.01 I+Y2pixii
•
XB- ) + Ylni xB .1 FFBni tanh ( 0.01 I+Y2niXii
•
XB > 0
(14)
and free parameters FFBp;, FFBni, Ylpi, Yln; , Y2pi and Y2n; have been found for each drive i = r, I which describe the dependence on the bridge velocity xB .
IS
575
300
circular paths, C 3-continuous or C 4 -continuous functions are used. The modified splines are characterised by sixth degree polynomials in the first as well as in the last segment and fifth degree polynomials in the intermediate segments, Therefore, parametrised by a linear function s(t), a C 3_ continuous trajectory can be generated which interpolates given knot points. By optimization of the time history s(t) with respect to total traveling time using the time scaling technique, a wide range of different time optimal trajectories can be generated .
~~~.
200
~F;nl.i;
100 [N] 0
-100 -200
...r-"-" ~
-300 -0.6 -0.4 -0.2
~ 0
. 0.2
0.4
0.6
0.8
Bridge velocity in [ms· l ]
6. DISTURBANCE OBSERVER DESIGN
Fig. 2: Identified Friction characteristic including nonlinear friction and viscous damping
For control observer design a simplified mechanical model of the x - axis is advantageous. When an ideal synchronous movement of both bridge sides, i.e XB = XBr = XBI , is assumed, the equations of motion (2) can be simplified by adding the first and the second row. Then, the dynamics of the crane bridge is governed by the following equations of motion
The reference bridge velocity can be derived from the third row of (2) with XB,ref = XBr,ref = XBl.ref , Fw = 0, and XL = XL,ref . Differentiating once with respect to time, one yields the reference velocity of the bridge
. . lR ... XB.ref = XL,ref + - g XL .ref
(15)
which depends on time derivatives of the load reference trajectory XL,rer(t). Herewith, the nonlinear part UFF,nl is evaluated online and applied to the drives. Thus, the disturbance observer introduced below can be relieved, and a higher tracking accuracy is achieved.
where the abbreviation k = mL g / IR has been introduced. Furthermore, F MB = FMBr+FMBI is the sum of both drive forces, and FFB = FFBr+FFBI is the sum of both nonlinear friction forces. Aiming in fast observation, the disturbance observer is designed as a reduced - order observer. As disturbance models for the nonlinear friction force as well as for the wind
5. TIME OPTIMAL TRAJECTORY PLANNING A desired path can only be tracked if the kinematic and dynamic limitations of the crane system have been considered during trajectory planning. Furthermore, the performance of the crane system can be enhanced if given limitations are checked and exploited at best during path planing by means of kinematic and dynamic models of the crane system. This approach provides an important contribution to cost reducing and increasing of handling frequency (Sawodny et al., 1999b).
°
force integrator models are used FFB = ,Fw = 0. The bridge position is measured by means of an encoder connected to a pinion which rolls of an rack, while the rope angle
The trajectory planning task can be divided into two parts. The description of the spatial geometric path of the crane load qL(S)=[XL YL zLJT depending on a dimensionless path parameter s and the specification of the time history of the path parameter s(t) while tracking the geometric path. As can easily be seen from the feed forward control law (13), the generation of at least C 3-continuous reference trajectories for the crane load is necessary. Since all obstacles and objects in the workspace are assumed to be known, a collision free geometric path can be generated automatically. As important geometric path types straight-line paths, circular paths and modified fifth degree splines are available. For the description of the time history s(t) in case of straight- line paths or
where
r = [FFB Fw ] T
estimated disturbances,
denotes
y = [x B
the •
xL:
vector
xB
:
of
XL] T
the vector of measured state variables and u = FMB the input of the disturbance observer. A reasonable choice of the observer gain matrix L is
L=[0o °0 In0 10] 24
576
.
( 18)
Then, the disturbance observer matrices become
with reduced degrees of freedom. Herewith, the steady state deviation of the load position resulting from nonlinear friction forces and wind forces can be reduced. The dependence on the varying system parameters leads also to an adaptive disturbance compensation law.
'.
~~;1' =[-m,,~~mT] mL
( 19)
8. CONTROL STRUCTURE
-H
The adaptive control scheme including linear feed forward control (LFF) , non linear feed forward control (NFF), decoupling feedback control (DFB), disturbance observer (DO), and disturbance compensation (DC) is shown in fig. 3. Note that the subsystems NFF, LFF, DFB, DC, and DO are adapted to the actual system parameters rope length IR , trolley position YTO and load mass mL'
The design parameters are the observer gains 113 and 124 which can be calculated independently by pole placement according to the bandwidth fBI and f B2 for the friction force estimation and the wind force estimation, respectively. The observer gain 113 is given by 113=-21tfB1(mB,red+mT), while the observer gain 124 with respect to the wind force observation results from 124 = 21tf B2 mL'
x reference '-----' trajectory
7. DISTURBANCE COMPENSATION The appropriate gains to compensate for the observed disturbances can be determined by considering the closed-loop transfer function matrices G u (s)= C(sI - A + BuD d1 (Md +Cd»-IB u 1
Ge(s)= C(sI - A + BuD d (Md + Cd»-IB e
Fig. 3: Motion control structure (20)
9. EXPERIMENTAL RESULTS from the control input D and from disturbance input e to the output vector y according to (6), respectively. Aiming at fast compensation, a dynamic compensation law Doe = (K oo + KDls)e with elements
The efficiency of the control scheme is shown by measurements at a 5 t - bridge crane. The tracking error denotes the difference ~XL(t) = XL.ret
koo.ij+kDl.ij s is desirable. However, sensor noise is critical and only a static compensation is used. The static compensation gain K oo is calculated by demanding I
G u (s =O)K DO +Ge(s = O)~O
(21 ) 0.4
,------~--~--~--~--~---,
Irnl
This system of linear equations can be solved for the elements k oo.ij of K oo and yields
0.3 0.2
I • o. ffir(g(l, - YTO»+ITIa.red I,(g -~2IR)1 IJO=.. 2"\~I) . o 1 • ffir(g(l, + YTO)-~2IR (11 + YTO»+ ITIa.r
K
:
•
0.1
o
2 ffit.g I)
·0.1
(22)
Note, that the vector of estimated disturbances
. ]T -_[FFB FFB- F.w ]T -- -
Fw
2
2
-0.2 -0.3
(23)
.0.4 '--_ _L . -_ _- ' - -_ _- ' - -_ _- ' -_ _- ' -_ _- '
o
is calculated based on the simplified dynamic model
10
20
30
time Isl
40
Fig. 4. Comparison of tracking errors
577
50
60
300
Fig. 5 shows the same motion using the complete control structure. The maximum deviation between reference and actual load position is I~XL(t)I=IXL.ret<'t) XL(t)! ::; 5.5 cm and steady-state accuracy is achieved. The deviations in bridge position difference are below I~XBI ::; 1.5cm. The tracking error for a variation in rope length from lR = 2.85 m to IR = 4.85 m is shown in fig. 6. As can be seen, similar measurements results have be obtained for this range of variation. Furthermore, steady-state accuracy concerning the final load position is achieved. The compensation of possible wind forces acting on the load is depicted in fig. 7. A horizontal force Fw(t) applied to the load have been increased slowly from F w = 0 N to a maximum value F w.max = 280 N before decreasing to Fw = 0 N. The observed nonlinear friction force and the observed wind force are shown in the left diagrams. Compared to the case without disturbance rejection with a maximum deviation from the setpoint ~xLCt) = XL.ret<'t) - XL(t) = 0 of I~XL(t)1 < 46 cm, the maximum position error can be reduced to l~xLCt)1 < 16 cm and steady-state accuracy is guaranteed by observer based disturbance rejection.
1000 r - - - - - - - [N]
r~: L ',
bridge drive forces
500
0
-0,0 I
-1000
·0.02 -0.03
L-_______
o
20
40
60
time in 151
'
-300 0
40
~
~
~
0.3
0
0.2
0
20
60
80
40
60
80
0.1
0
20
estimated wind force 40 60 80 time in Is]
0 -01
time in (51
Bryfors, U., A. Sluteij (1996). Integrated Crane Control. In: Proceedings of the 16th Conference on Transportation Systems, Zagreb, pp. 167 - 171 Delaleau, E., J. Rudolph (1998). Control of flat systems by quasi-static feedback of generalized states. International Journal of Control, 71, pp. 745 - 765 Falb P. L., W. A. Wolovich (1967). Decoupling in the Design and Synthesis of Multivariable Control Systems. IEEE Transactions on Automatic Control, 12, pp. 651 - 659 Nguyen, H. T. (1994). State-Variable Feedback Controller for an Overhead Crane. Journal of Electrical and Electronics Engineering, 14, pp. 75 - 84 Sawodny, 0., H. Aschemann, S. Lahres, E.P. Hofer (1999a). Tracking Control for Automated Bridge Cranes. In: Advances in Manufacturing Systems (S. Tzafestas (Ed.», pp. 310 - 320, Springer, New York. Sawodny 0., H. Aschemann, S. Lahres, E.P. Hofer (1999b). A Material Handling and Logistic System Using an Automated Crane. In: Proceedings IFAC World Congress, Peking, China, Vol. B, pp. 517 - 522
60
time in 151
__ __ __ __ 10 20 30 40 50
100
40
time in 151
11. REFERENCES
bridge position difference
.
'-_~
0.5
Im]
20
In this paper a motion control concept for an overhead traveling crane using gain scheduling is presented. Based on a linearized state space model of the crane x-axis, a decoupling feedback control law is derived in symbolic form that is adapted to varying system parameters as rope length, load mass, and trolley position. Additionally, adaptive feedforward control and observer based disturbance rejection are used to improve tracking accuracy and take into account nonlinear disturbances. Measurements made at a 5 t - bridge crane show the effectiveness of the proposed control scheme with tracking errors below 8 cm and steady-state accuracy about 1 cm.
0.05
o
300
_-....----~
0
10. CONCLUSION
[m)
-0.1
. V
Fig. 7: Effects of wind impact with DC (upper figures) and without DC (lower figures) for YTO=0.35 m, mL=800 kg, IR=3.85 m, F w.max =280 N
r--~-----~--~--~----,
-0.05
-02
1Nl
-100
Fig. 5: Motion with disturbance compensation for YTO=O m, mL=800 kg, and IR=3.85 m 0.1
80
-200
,,J'.------<
20
position error \ . -0.1
-200
'-----------~
0
0
-100
-300
~
0.1
100
ol"\-),
-500
02
Iml
1Nl
~_--J
60
time in [si
Fig. 6: Tracking error for varying rope length (YTO = -2.5 m, mL=800 kg; lR = 2.85 m ... 4.85 m)
578
GAIN SCHEDULED MOTION CONTROL OF AN OVERHEAD TRA VELING CRANE
H. Aschemann, S. Lahres, O. Sawodny, E.P. Hofer
Department of Measurement, Control and Microtechnology University of Ulm, Germany E-mail: [email protected] Tel.:+49 73150-26336. Fax: +4973150-26301
Abstract: This paper presents a gain scheduling approach to motion control of an overhead traveling crane using feed forward and feedback control as well as observer based disturbance rejection. All control system components are derived in symbolic form and adapted online to the varying system parameters rope length, load mass, and trolley position. The effectiveness of the developed control scheme is shown by experimental results from a 5 t - overhead traveling crane. Here, trajectory tracking performance as well as the compensation of disturbance forces caused by nonlinear friction and wind impact are presented. Copyright @2000 IFAC Keywords: Crane control, gain scheduling, decoupling control, feedforward control, disturbance rejection, trajectory planning, robotics
I.
For crane control a decentralized concept with independent control systems for each crane axis has been chosen. In this paper, the controller design for the crane x-axis is described exemplary. First, a mathematical model of this axis has to be derived. The mechanical system is modeled by an multibody sY6tem consisting of three rigid bodies: the bridge (length lB=2 11> reduced mass mB.red including the contributions from the electric drives and gears, mass moment of inertia CB with respect to the z-axis), the trolley (mass mT), and the crane load (mass md. The load centre of gravity has an offset hL in z-direction. The rope is assumed to be a massless rigid link with length IRO' For simplicity the abbreviation IR=IRo+h L for the total rope length is introduced.
INTRODUCTION
Until now industrial control systems for automated cranes only guarantee position accuracy at the target point and damping of load oscillations (Bryfors et af., 1996). In order to improve the abilities of automated cranes path planning (Sawodny et al., 1999b) combined with effective feedforward control (Deleau et af., 1998) and variable feedback controllers (Nguyen, 1998) become important. At the University of Ulm a motion control concept for an automated bridge crane has been developed that allows to track specified trajectories with high accuracy. Characteristic features of this control concept are time optimal trajectory planning, decentralized controllers for each crane axis, adaptive feedforward and feedback control, and observer based disturbance rejection to take into account disturbances. With this capability an automated crane can be used as a robot manipulator for the handling of heavy loads in a large cartesian workspace. This paper presents the modeling and the control system design for a 5 t bridge crane as well as measurement results.
2.
The movement in x-direction is fully described by three generalised coordinates: the position of the right bridge side xBr(l), the position of the left bridge side XBI(t) and the position of the load XL(t). The vector of generalised coordinates is chosen as Z=[XBr XBI xd T. According to the decentralized control concept the relative position YTO of the trolley with respect to the centre of the bridge is assumed to vary slowly with time. Therefore, the time derivatives of this quantity are neglected concerning x-axis controller design. The drive forces acting on the left and the right side of the bridge are denoted as FMB1(t) and FMBr(t), respectively. Viscous damping forces are described by the damping coefficients bB1 and bBr . The main disturbances are the nonlinear friction forces FFBI(t) and FFBr(t) acting on the drives and a possible force Fw(t) caused by wind acting on the load surface.
MULTIBODY MODEL OF CRANE X-AXIS
The overhead traveling crane consists of three actuated axes (fig. 1). The direction of crane bridge motion is denoted as x-axis. The trolley can perform a movement on the bridge, which is referred to as yaxis. The crane load can be hoisted or lowered in zdirection by means of the rope suspension.
573
ma.r
Fig. 1: Multibody model of the crane x-axis These equations can be written representation
Applying the Lagrange formalism
~(dT(Z,:i»)_ dT(z,:i) + dU(Z) =f T dt d:i dZ dZ ne '
m21
mn
0
0
r"
+ k 21 k 31
k 12
kn k 32
l"'J r k'l"'J r oo
mL
xBI +
0
xL
0
"'
FM
k 23
x Bl
k 33
xL
or
i=Ax+B u +B e =Ax+Buu+Bee, with the unity matrix I, the input vector U=[FMBr FMB1f, and the vector of disturbance forces e=[FFBr FFBI FW]T.
0
b
ml2
o T"' o J j r- ~", J b B1
xBI
0
xL
= FMB1 + - FFBI "' 0 Fw
3.
mB.red m T =- - +- -
4
4
(2)
mTYTO CB + mTY~O + ----"'-----,-:~..::... 21 1 41f
m B.red m CB + mTY~O m p = m 21 = - - - + - T -4 4 41 12
For adaptive controller design a decoupling feedback control law is derived analytically (Falb et al., 1967). The vector of output quantities y = [YI Y2]T = [~XB xd T consists of the position difference y,(t) = ~XB(t) = XBr(t) - XBI(t) and the load position Y2(t) = XL(t). The control objectives are to achieve a vanishing position difference ~XB(t) = 0, i.e. XBr(t) = XBI(t), and a perfect tracking XL(t) = XL.ret
(3) ?
m B.red m mTYTO CB + mTYTO m O? = - - - + - -T+ + 0 4 4 21 1 41] and the elements kij of K are as follows 2 2 2 k = (YTo -1 1 ) mLg k = k? = (11 - YTO)mLg 11 41 2 1 ' 12 _J 41 2 1 ' 1 R
k
-k 13 -
DECOUPLING CONTROL DESIGN
The main varying system parameters are the load mass mL, the rope length IR , and the trolley position YTO' While the load mass is constant after the picking up procedure, the rope length and the trolley position may vary during the movement on the trajectory in the workspace. The load mass is measured by a load cell at the fixed end of the rope. Also the trolley position and the rope length are available by encoder measurements. Therefore, the controller gains can easily be adapted with respect to changing operation points.
or in matrix notation Mi + D:i + Kz = f u + fe' with the mass matrix M, the damping matrix D, the stiffness matrix K, the vector of drive forces f u , and the vector of nonlinear disturbance forces f e . The elements mij of M are given by m 11
state space
(1)
with the total kinetic energy T, the potential energy U, and the vector of non conservative forces f nc , nonlinear equations of motion are computed. Because of small rope angles, the following linearized equations of motion hold
rmu
In
1 R
_(YTo-ll)mLg 21 I '
(4)
31 -
1 R
A decoupling feedback control law can be derived if the decoupling matrix Dd has full rank and no
574
instable internal dynamics exists. Therefore, the time derivatives of the output equation have to be considered. The relative degree r; with respect to the output i equals the order of the output time derivative which is affected directly by the inputs. Since
o
7 43 S +UnS +UnS- +U2Is+u20
(12)
.
T.
TA
Yl=clx=cl
..
TA 2
x'YI=cl
TAB
x+c,
The dynamics of the transfer functions on the matrix diagonal are chosen independently to achieve asymptotic stability as well as good control performance. The second-order system describes the closed-loop dynamics of the position difference XBrXBl of the bridge sides, while the fourth-order system represents the closed-loop dynamics of the load position. Hence, the coefficients alO, all and a20, a2l. a22, a23 can be derived by comparison with appropriate characteristic polynomials resulting from proposed pole configurations. This specified closedloop transfer function matrix G(s) remains constant since the feedback control law is adapted to the actual values of the varying system parameters.
(7)
uU
the input u has direct influence on the second time derivative and the relative degree of the first output YI becomes rl=2. Similarly with
the relative degree r2=4 is obtained. The total relative degree r is determined as the sum of the relative degrees rl=2 and r2=4 associated with the first and the second output, respectively r = rl + r 2 = 6 = n . As the total relative degree equals the system order, no internal dynamics occurs. The decoupling matrix is then
4. FEEDFORWARD CONTROL DESIGN The transient behavior can be essentially improved by applying feedforward control (Sawodny et aI., 1999a). Here, the feedforward control action is divided into a linear and a nonlinear part. Concerning the position difference of the bridge sides a synchronous motion with a vanishing difference is aimed at. Hence, for setpoint control 131=0 can be chosen. The trajectory planning module provides the reference trajectories for the crane load with continuous third or fourth derivatives with respect to time. Since these reference functions are available, a linear feed forward control law based on
(9)
which is non-singular in the operation range. Hence, the inverse matrix Dd-I can be computed. With the matrices Cd , L d , and Md
(13)
the decoupling control law is given by U
= -Dd1(M d +Cd)x + Dd1Ldw = uFB
is applied. Note, that the linear feedforward control part is also adapted to the varying system parameters
+uFF lin'
FF.lin = Dd1Ldw depends on the inverse decoupling matrix. Additionally, a nonlinear feedforward control part is used to compensate for nonlinear friction. The nonlinear friction characteristic of the crane x-axis, which is depicted in fig. 2, has been identified by experiments. By optimization analytical functions for the nonlinear T part UFF.nl=[UFF.nl.r UFF.nl.a with since
(11)
with Was the new input vector W=[~XB.ref XLref]T. The decoupling control law consists of two parts: the feedback part uFB = -Dd1(M d + Cd)x depending on the measured state variables x and the linear feedforward
part
uFF.lin = Dd1Ldw
which
IS
calculated with the reference trajectory and its time derivatives. All matrices are computed analytically and depend on the actual values of trolley position YTO, rope length I R and load mass mL. Therefore, the decoupling control law is adapted online by evaluating the analytical expressions. The closed - loop transfer function matrix G(s) decoupled and becomes
U
xB- ) + YIp; xB . 1 FFBp; tanh ( 0.01 I+Y2pixii
•
XB- ) + Ylni xB .1 FFBni tanh ( 0.01 I+Y2niXii
•
XB > 0
(14)
and free parameters FFBp;, FFBni, Ylpi, Yln; , Y2pi and Y2n; have been found for each drive i = r, I which describe the dependence on the bridge velocity xB .
IS
575
300
circular paths, C 3-continuous or C 4 -continuous functions are used. The modified splines are characterised by sixth degree polynomials in the first as well as in the last segment and fifth degree polynomials in the intermediate segments, Therefore, parametrised by a linear function s(t), a C 3_ continuous trajectory can be generated which interpolates given knot points. By optimization of the time history s(t) with respect to total traveling time using the time scaling technique, a wide range of different time optimal trajectories can be generated .
~~~.
200
~F;nl.i;
100 [N] 0
-100 -200
...r-"-" ~
-300 -0.6 -0.4 -0.2
~ 0
. 0.2
0.4
0.6
0.8
Bridge velocity in [ms· l ]
6. DISTURBANCE OBSERVER DESIGN
Fig. 2: Identified Friction characteristic including nonlinear friction and viscous damping
For control observer design a simplified mechanical model of the x - axis is advantageous. When an ideal synchronous movement of both bridge sides, i.e XB = XBr = XBI , is assumed, the equations of motion (2) can be simplified by adding the first and the second row. Then, the dynamics of the crane bridge is governed by the following equations of motion
The reference bridge velocity can be derived from the third row of (2) with XB,ref = XBr,ref = XBl.ref , Fw = 0, and XL = XL,ref . Differentiating once with respect to time, one yields the reference velocity of the bridge
. . lR ... XB.ref = XL,ref + - g XL .ref
(15)
which depends on time derivatives of the load reference trajectory XL,rer(t). Herewith, the nonlinear part UFF,nl is evaluated online and applied to the drives. Thus, the disturbance observer introduced below can be relieved, and a higher tracking accuracy is achieved.
where the abbreviation k = mL g / IR has been introduced. Furthermore, F MB = FMBr+FMBI is the sum of both drive forces, and FFB = FFBr+FFBI is the sum of both nonlinear friction forces. Aiming in fast observation, the disturbance observer is designed as a reduced - order observer. As disturbance models for the nonlinear friction force as well as for the wind
5. TIME OPTIMAL TRAJECTORY PLANNING A desired path can only be tracked if the kinematic and dynamic limitations of the crane system have been considered during trajectory planning. Furthermore, the performance of the crane system can be enhanced if given limitations are checked and exploited at best during path planing by means of kinematic and dynamic models of the crane system. This approach provides an important contribution to cost reducing and increasing of handling frequency (Sawodny et al., 1999b).
°
force integrator models are used FFB = ,Fw = 0. The bridge position is measured by means of an encoder connected to a pinion which rolls of an rack, while the rope angle
The trajectory planning task can be divided into two parts. The description of the spatial geometric path of the crane load qL(S)=[XL YL zLJT depending on a dimensionless path parameter s and the specification of the time history of the path parameter s(t) while tracking the geometric path. As can easily be seen from the feed forward control law (13), the generation of at least C 3-continuous reference trajectories for the crane load is necessary. Since all obstacles and objects in the workspace are assumed to be known, a collision free geometric path can be generated automatically. As important geometric path types straight-line paths, circular paths and modified fifth degree splines are available. For the description of the time history s(t) in case of straight- line paths or
where
r = [FFB Fw ] T
estimated disturbances,
denotes
y = [x B
the •
xL:
vector
xB
:
of
XL] T
the vector of measured state variables and u = FMB the input of the disturbance observer. A reasonable choice of the observer gain matrix L is
L=[0o °0 In0 10] 24
576
.
( 18)
Then, the disturbance observer matrices become
with reduced degrees of freedom. Herewith, the steady state deviation of the load position resulting from nonlinear friction forces and wind forces can be reduced. The dependence on the varying system parameters leads also to an adaptive disturbance compensation law.
'.
~~;1' =[-m,,~~mT] mL
( 19)
8. CONTROL STRUCTURE
-H
The adaptive control scheme including linear feed forward control (LFF) , non linear feed forward control (NFF), decoupling feedback control (DFB), disturbance observer (DO), and disturbance compensation (DC) is shown in fig. 3. Note that the subsystems NFF, LFF, DFB, DC, and DO are adapted to the actual system parameters rope length IR , trolley position YTO and load mass mL'
The design parameters are the observer gains 113 and 124 which can be calculated independently by pole placement according to the bandwidth fBI and f B2 for the friction force estimation and the wind force estimation, respectively. The observer gain 113 is given by 113=-21tfB1(mB,red+mT), while the observer gain 124 with respect to the wind force observation results from 124 = 21tf B2 mL'
x reference '-----' trajectory
7. DISTURBANCE COMPENSATION The appropriate gains to compensate for the observed disturbances can be determined by considering the closed-loop transfer function matrices G u (s)= C(sI - A + BuD d1 (Md +Cd»-IB u 1
Ge(s)= C(sI - A + BuD d (Md + Cd»-IB e
Fig. 3: Motion control structure (20)
9. EXPERIMENTAL RESULTS from the control input D and from disturbance input e to the output vector y according to (6), respectively. Aiming at fast compensation, a dynamic compensation law Doe = (K oo + KDls)e with elements
The efficiency of the control scheme is shown by measurements at a 5 t - bridge crane. The tracking error denotes the difference ~XL(t) = XL.ret
koo.ij+kDl.ij s is desirable. However, sensor noise is critical and only a static compensation is used. The static compensation gain K oo is calculated by demanding I
G u (s =O)K DO +Ge(s = O)~O
(21 ) 0.4
,------~--~--~--~--~---,
Irnl
This system of linear equations can be solved for the elements k oo.ij of K oo and yields
0.3 0.2
I • o. ffir(g(l, - YTO»+ITIa.red I,(g -~2IR)1 IJO=.. 2"\~I) . o 1 • ffir(g(l, + YTO)-~2IR (11 + YTO»+ ITIa.r
K
:
•
0.1
o
2 ffit.g I)
·0.1
(22)
Note, that the vector of estimated disturbances
. ]T -_[FFB FFB- F.w ]T -- -
Fw
2
2
-0.2 -0.3
(23)
.0.4 '--_ _L . -_ _- ' - -_ _- ' - -_ _- ' -_ _- ' -_ _- '
o
is calculated based on the simplified dynamic model
10
20
30
time Isl
40
Fig. 4. Comparison of tracking errors
577
50
60
300
Fig. 5 shows the same motion using the complete control structure. The maximum deviation between reference and actual load position is I~XL(t)I=IXL.ret<'t) XL(t)! ::; 5.5 cm and steady-state accuracy is achieved. The deviations in bridge position difference are below I~XBI ::; 1.5cm. The tracking error for a variation in rope length from lR = 2.85 m to IR = 4.85 m is shown in fig. 6. As can be seen, similar measurements results have be obtained for this range of variation. Furthermore, steady-state accuracy concerning the final load position is achieved. The compensation of possible wind forces acting on the load is depicted in fig. 7. A horizontal force Fw(t) applied to the load have been increased slowly from F w = 0 N to a maximum value F w.max = 280 N before decreasing to Fw = 0 N. The observed nonlinear friction force and the observed wind force are shown in the left diagrams. Compared to the case without disturbance rejection with a maximum deviation from the setpoint ~xLCt) = XL.ret<'t) - XL(t) = 0 of I~XL(t)1 < 46 cm, the maximum position error can be reduced to l~xLCt)1 < 16 cm and steady-state accuracy is guaranteed by observer based disturbance rejection.
1000 r - - - - - - - [N]
r~: L ',
bridge drive forces
500
0
-0,0 I
-1000
·0.02 -0.03
L-_______
o
20
40
60
time in 151
'
-300 0
40
~
~
~
0.3
0
0.2
0
20
60
80
40
60
80
0.1
0
20
estimated wind force 40 60 80 time in Is]
0 -01
time in (51
Bryfors, U., A. Sluteij (1996). Integrated Crane Control. In: Proceedings of the 16th Conference on Transportation Systems, Zagreb, pp. 167 - 171 Delaleau, E., J. Rudolph (1998). Control of flat systems by quasi-static feedback of generalized states. International Journal of Control, 71, pp. 745 - 765 Falb P. L., W. A. Wolovich (1967). Decoupling in the Design and Synthesis of Multivariable Control Systems. IEEE Transactions on Automatic Control, 12, pp. 651 - 659 Nguyen, H. T. (1994). State-Variable Feedback Controller for an Overhead Crane. Journal of Electrical and Electronics Engineering, 14, pp. 75 - 84 Sawodny, 0., H. Aschemann, S. Lahres, E.P. Hofer (1999a). Tracking Control for Automated Bridge Cranes. In: Advances in Manufacturing Systems (S. Tzafestas (Ed.», pp. 310 - 320, Springer, New York. Sawodny 0., H. Aschemann, S. Lahres, E.P. Hofer (1999b). A Material Handling and Logistic System Using an Automated Crane. In: Proceedings IFAC World Congress, Peking, China, Vol. B, pp. 517 - 522
60
time in 151
__ __ __ __ 10 20 30 40 50
100
40
time in 151
11. REFERENCES
bridge position difference
.
'-_~
0.5
Im]
20
In this paper a motion control concept for an overhead traveling crane using gain scheduling is presented. Based on a linearized state space model of the crane x-axis, a decoupling feedback control law is derived in symbolic form that is adapted to varying system parameters as rope length, load mass, and trolley position. Additionally, adaptive feedforward control and observer based disturbance rejection are used to improve tracking accuracy and take into account nonlinear disturbances. Measurements made at a 5 t - bridge crane show the effectiveness of the proposed control scheme with tracking errors below 8 cm and steady-state accuracy about 1 cm.
0.05
o
300
_-....----~
0
10. CONCLUSION
[m)
-0.1
. V
Fig. 7: Effects of wind impact with DC (upper figures) and without DC (lower figures) for YTO=0.35 m, mL=800 kg, IR=3.85 m, F w.max =280 N
r--~-----~--~--~----,
-0.05
-02
1Nl
-100
Fig. 5: Motion with disturbance compensation for YTO=O m, mL=800 kg, and IR=3.85 m 0.1
80
-200
,,J'.------<
20
position error \ . -0.1
-200
'-----------~
0
0
-100
-300
~
0.1
100
ol"\-),
-500
02
Iml
1Nl
~_--J
60
time in [si
Fig. 6: Tracking error for varying rope length (YTO = -2.5 m, mL=800 kg; lR = 2.85 m ... 4.85 m)
578