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Control of a Suspended Load Using Inertia Rotors
with Traveling Disturbance Yasuo Yoshida
Department of Mechanical Engineering Chubu University
1200 Matsumoto-cho, Kasugai-shi, Aichi 487-8501, Japan yyoshida@isc.chubu.ac.jp
Abstract
Rotational control and swing suppression of a crane suspended load model with traveling disturbance are
studied. A rotational free rigid body suspended by a
single rope is controlled using three inertia rotors. The
end-supporting-point of the single rope is forced to be traveled as disturbance. Control angles and angular
velocities are derived from measured data of fiber optic
gyros installed on the suspended load. Rotor angular velocities controlling the suspended load are obtained by
integrating the computed digital sliding-mode control
feedback accelerations based on the coupling system’s
dynamics. Experiments and simulations investigate simultaneous control of the load’s rotational orientation
and swing suppression for traveling disturbance.
1. Introduction A crane suspended load in a construction work is
frequently rotated and swung by wind pressure or inertia
force accompanied with movement of the crane. Most studies to control the suspended load have focused on
suppressing swing of the load as a lumped mass and
manipulating the crane. For example, Sakawa and
Nakazumi [1] treated a rotary crane and Lee [2] discussed the simultaneous traveling and traversing
trolley. The suspended load is actually a rigid body
different from above-mentioned a lumped mass model
and possible to be rotated. Kanki et al. [3] developed an active control device using gyroscopic moment to
control the rotation of a crane suspended load.
There are few studies to control both rotation and
swing of the suspended load as a rotational free rigid
body. Yoshida and Mori [4] studied simultaneous control
of rotation and swing of a pendulum having a rotational
free rigid body using inertia rotors and recently Yoshida and Yajima [5] studied for a rotational free rigid body
model suspended by a single rope with initial swing
disturbance.
This paper presents rotational control and swing suppression of a suspended load model with traveling
disturbance. The rotational free rigid body is suspended
by a single rope and controlled using three inertia rotors, where the end-supporting-point of the single rope is
forced to be traveled as disturbance. The suspended load
as a rigid body has non-actuating (passive) three-degree-
of-freedom and three inertia rotors have individually actuating (active) one-degree-of-freedom. The model
system has six-degree-of-freedom and passive degrees of
freedom are controlled by active degrees of freedom
based on the coupling system’s dynamics.
2. Dynamic Model and Controller
Figure 1 shows static state of the suspended load
model, where the coordinate x y z0 0 0 is nonmoving
base frame and xyz is moving frame fixed with the
load. The rope-end-point travels to x0 direction. Three
inertia rotors installed on the load. The purpose of this
study is to control the motion of the load using inertia
rotors. Fig.2 shows moving state, where the suspended
load is represented with the rope-end-point displacement
x , zxy Euler angles θ θ θ1 2 3, , and zxz Euler
angles θ θ θ1 2 3, , . Angular velocities of the load are
ω ω ωx y z, , with respect to ,x y and z axes.
Angles and angular velocities of inertia rotors are
Proceedings of the 2001 IEEE International Conference on Robotics & Automation
Seoul, Korea • May 21-26, 2001
0-7803-6475-9/01/$10.00© 2001 IEEE
θ θ θ4 5 6, , and ω θ ω θ ω θz x y+ + +, ,4 5 6 , and torques
are τ τ τ1 2 3, , with respect to ,z x and y axes.
Based on zxy Euler angles θ θ θ1 2 3, , of the
suspended load and θ θ θ4 5 6, , of inertia rotors, the
equation of motion can be written as
M M
M M
q
q
h
h
Xx
c c c f
f c f f
c
f
c
f
+
+
=
0
0
τ, (1)
where
M
m m m
m m
m mcc =
11 12 13
12 22
13 33
0
0
, M M
m m m
m m
mcf fc
T= =
14 15 16
24 25
36
0
0 0
,
M
I
I
Iff
R
R
R
=
3
1
2
0 0
0 0
0 0
,
[ ] [ ]q qc
T
f
T= =θ θ θ θ θ θ1 2 3 4 5 6, ,
[ ] [ ]h h h h h h h hc
T
f
T= =1 2 3 4 5 6, , [ ]X x x xT= 1 2 3 ,
[ ] [ ]0 0 0 0 1 2 3= =T T, .τ τ τ τ
qc shows controlled (non-actuating) angle state vector
and q f free (actuating) angle state vector. Mass matrix
M M Mcc cf fcT, ( )= are function of θ θ2 3, . h hc f, are
vector of Coriolis, centrifugal and gravity terms and
function of θ θ θ2 3 1 6), , ( ~i i = . X is vector of
traveling disturbance and function of θ θ θ1 2 3, , .
The system equation of controlled zxy Euler angle’s
state vector qc is linearized by using the control input
vector uc as
[ ]q u u u uc c
T= = 1 2 3. (2)
A digital sliding-mode controller by which the
controlled angles θ θ θ1 2 3, , of the suspended load
follow the desired values θ θ θ1 2 3d d d, , is designed from
equation (2). Digital state equation of control angle error
is obtained using sampling time T and time steps
k ( , , )=1 2 as
E pE q u ii k ik ik, ( ~ )+ = − =1 1 3 (3)
Ee
ep
Tq
T
Tik
ik
ik
=
=
=
, ,
.1
0 1
05 2
e e u uik idk ik ik idk ik ik ik idk= − = − = −θ θ θ θ θ, ,
Rope- end Traveling Direction
Rotor for yaw rotation
Rotor for roll swing Rotor for pitch swing
Suspended load
Rope
x
y
z
G
O
x0
y0z0
x
Fig.1 Suspended load model using inertia rotors with
traveling disturbance.
θ1
θ2
θ3
θ1
θ2
θ3
τ ω θ3 6, y
+
τ ω θ2 5, x
+
τ ω θ1 4, z
+ θ3
θ1
θ1
x
z
y
x
g
x0 y0
z0
( )z
( )x
( )y
Fig.2 Coordinates and angles of the suspended load
model.
Switching function σ ik is defined with scalar
vector ][S s s= 1 2 as follows
σ ik ikSE i= =( ~ )1 3 , (4)
where, σ ik = 0 gives switching line. Sliding-mode
control input uik can be represented as follows, where
u eqik is equivalent input on switching line and u nlik
is nonlinear input acting to reach switching line,
u u uik eqik nlik= + , (5)
u Sq S p I Eeqik ik= −−( ) ( )1 ,
u Sq fornlik ik= < <−η σ η( ) 1 0 2 .
From equation (2) to (5), sliding-mode controller is
obtained as
u K e K e for iik ik ik idk= + + =1 2 1 3( ~ )θ , (6)
KT
T s sK
T s s
T s s12 1
22 1
2 105
1
05=
+=
++
( / )
. ( / ),
( / )( / )
. ( / )
η η .
Estimated accelerations of inertia rotor’s angles q f
are taken as
[ ]q u u u uf f k k k
T= = 4 5 6. (7)
After putting equation (2), (7) into equation (1),
upper part of the equation of motion (1) becomes as
M u M u h Xxcc c cf f c+ + + = 0 . (8)
Above equation (8) shows the dynamic coupling
between controlled angles qc and free angles q f .
Then, estimated accelerations u f are given as
( )u M M u h Xxf cf cc c c= − + +−1 . (9)
By integrating equation (9) with the sampling time
T , manipulating velocities of inertia rotors q vf k=
[ ]= v v vk k k
T
4 5 6 are obtained as
v v u Tk k f= +−1. (10)
Fiber optic gyros installed on the suspended load
can measure the angles θ θ θ θ θ13 1 3 2 3( ), ,≡ + and the
angular velocities ω ω ωx y z, , .
zxy Euler angles θ θ θ1 2 3, , and zxz Euler
angles θ θ θ1 2 3, , are mutually convertible. zxz Euler
angles are obtained by measured angles as
θ θ θθ
θ
1 13 3
21
2 3
3 2 3 22
= −== −
−
,
cos ( ),
tan ( , ).
C C
A C S S
(11)
Under the condition that θ θ2 3, are directly
measured by fiber optic gyros, θ1 must be calculated
based on equation (11) as
θ1 1 3 1 2 3 1 3 1 2 32= + − +A C S S C C S S C C Ctan ( , ) , (12)
where Ci and Si
are defined as cosθi and
sinθi for i = 1 2 3, , , respectively.
Angular velocities , ,θ θ θ1 2 3 are obtained using
ω ω ωx y z, , and θ θ2 3, as
/ /θ
θθ
ωωω
1
2
3
3 2 3 2
3 3
2 3 2 3
0
0
1
=−
−
S C C C
C S
T S T C
x
y
z
, (13)
where T2 is tanθ2 . Fig.3 shows the control system.
θθθωωω
13
2
3
k
k
k
xk
yk
zk
θθθ
θ
θ
θ
1
2
3
1
2
3
k
k
k
k
k
k
θθθ
1
2
3
k
k
k
θ1dk
[ ]u K e K e
u u u u
ik i ik i ik idk
c k k k
T
= + +
=1 2
1 2 3
θ
[ ]( )u M M u h Xx
u u u
f cf cc c c
k k k
T
= − + +
=
−1
4 5 6
[ ]v v u T
v v v
k k f
k k k
T
= +
=
−1
4 5 6
i =1 3~
+
−
gyro
data
zxz
Euler
angle
zxy
Euler
angle
t etargdisturbance x
suspended load
Fig.3 Control system.
Simulation is performed for the sampling time
between t (=step k) and t+T (=step k+1). Following
simultaneous differential equations are solved using
Runge-Kutta numerical integration.
( )q
q
M M u h X xc
f
cc cf f c
=
− + +
0
, (14)
where u v v Tf k k= − −( ) /1.
Since the manipulating velocities are held and constant at
the step k, angular accelerations of free angle q f are
zeros in the time between sampling time T.
However, the manipulating velocities change at each
time step and therefore accelerations of free angle are
approximately taken as u f that is the change of the
manipulating velocity between k-1 and k steps. u f
assures the dynamic coupling in the simulation.
3. Experimental Results
Fig.4 shows the experimental suspended model
photograph. Length between support point and the load’s
center of gravity is l m= 1 22. , mass of the load including
inertia rotors m kg= 20 9. , moments of inertia of the load
I kgmx = 136 2. , I kgmy = 0 676 2. , I kgmz = 117 2. and inertia
rotors I I I kgmR R R1 2 320 0377= = = . . Maximum angular
velocities of servomotors driving inertia rotors are
θ4 173= rpm , θ5 184= rpm , θ6 168= rpm . Sliding-mode
digital gains are K K1 280 16= =, using sampling time
T = 0 05. sec and η = =05 012 1. , / .s s , selected from
simulation results using 2~0=η , 10~0/ 12 =ss . And,
digital poles of closed system obtained from equation (3)
Fig.4 Experimental suspended model’s photograph.
and (6) using these digital gains are 5.0=z and 6.0 ,
therefore the control system is stable. Simulation is used
by Runge-Kutta numerical integration in time by steps of
size 0 005. sec . Fig.5 shows the linear traveling of the
end-supporting-point of the single rope as disturbance
for the suspended load. Fig.5(a) is traveling displacement
moving 0 25. m within 15. sec . Fig.5(b) shows velocity
and acceleration, that is, accelerating to 0.5 sec, uniform
velocity to 1.3 sec and decelerating to 1.5 sec. Initial
angles of the suspended load are manually positioned
and it is difficult to give the accurate same initial angles
to the experiments of both without and with controls. But
the differences of initial zxy Euler angles are small
between θ1 25= , θ2 1= , θ3 0= in the case of
experimental values without control and θ1 27= ,
θ2 0= , θ3 0= with control. Zero desired values
θ θ θ1 2 3d d d, , are given for controlled angle θ θ θ1 2 3, , .
0 1 2 3 4 5-0.3
-0.2
-0.1
0.0
Linear Traveling
t (s)
x (m
)
(a) Traveling displacement.
0 1 2 3 4 5-1.0
-0.5
0.0
0.5
1.0
..
x .
x
...
x (m
/s2 )
x (m
/s),
Linear Traveling
t (s)
(b) Traveling velocity and acceleration.
Fig.5 Linear traveling disturbance.
3
Fig.6 shows the experimental time response of
controlled rotational angle θ1 within 30 sec. Thin line
indicates the case of without control and thick line with
control, and the following figures show in similar ways.
Thin line of without control moves from initial angle
25 to maximum − 66 at 12 sec and thereafter largely
fluctuates. This phenomenon is explained by Yoshida
and Mori [5] that both centrifugal force caused by swing
and the difference between moment of inertia Ix and
I y generate yaw rotation torque. Thick line of with
control moves rapidly from initial angle to zero desired
value within 2 sec and hereafter keep desired value.
Fig.7 shows the experimental swing angle’s
components. Fig.7(a) shows controlled angle θ2 . Thin
line of without control enlarges vibration amplitude to
maximum 6 between 5 sec to 15 sec and and thick line
with control keeps within 15. value from initial time to
30 sec. Fig.7(b) shows controlled angle θ3 . Both cases
of without and with control are same till the time 15 sec,
but after that time thin line of without control remains
4 constant vibration amplitude and thick line with
control damps to zero desired value.
Fig.8 shows the response of swing angle ( zxz Euler
angle) θ2 . The amplitudes in the case of with control
are suppressed under half of those of without control.
Fig.9 shows swing displacement x y, for the fixed
coordinate when the load is at rest. The vibration
displacement x is induced by travel disturbance.
0 5 10 15 20 25 30-80
-60
-40
-20
0
20
40Exp.
With Conrol
Without Conrol
t (s)
θ 1 (de
g)
Fig.6 Experimental time response of
rotational angle θ1 .
Thin line of x without control leaves vibration
amplitude at 30 sec, but thick line of x with control
damps to zero desired value.
0 5 10 15 20 25 30-8
-4
0
4
8
Exp.Without Conrol
With Conrol
t (s)
θ 2 (de
g)
(a) Time response of θ2 .
0 5 10 15 20 25 30-8
-4
0
4
8Exp.
With Conrol
Without Conrol
t (s)
θ 3 (de
g)
(b) Time response of θ3 .
Fig.7 Experimental swing angle’s components.
0 5 10 15 20 25 300
2
4
6
8
With Conrol
Without ConrolExp.
t (s)
- θ 2 (de
g)
Fig.8 Experimental swing angle θ2 .
0 5 10 15 20 25 30-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1 Exp.
yy
xx With ConrolWithout Conrol
t (s)
x(m
), y
(m)
Fig.9 Experimental swing displacements x y, .
0 5 10 15 20 25 30-40
-20
0
20
40
With Conrol
Cal.
Without Conrol
t (s)
θ 1 (de
g)
Fig.10 Simulation time response of
rotational angle θ1 .
0 5 10 15 20 25 30-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
y
xxy Without Conrol With Conrol
Cal.
t (s)
x(m
), y
(m)
Fig.11 Simulation swing displacements x y, .
Fig.10 shows simulation time response of rotational
angle θ1 corresponding to experimental result of Fig.6.
Torsional spring coefficient of the rope is considered as
01. /Nm rad in simulation. Fluctuation tendency of thin
line without control shows the same pattern of
experiment and thick line with control moves rapidly
from initial angle to zero value same as experiment.
Fig.11 shows simulation swing displacements x y,
corresponding to experimental result of Fig.9 and these
are similar as experimental ones.
4. Conclusions A rotational free rigid body suspended by a single rope model with traveling disturbance is controlled using
three inertia rotors. Velocity-command-type control
system is developed for inertia rotors by integrating the
computed feedback accelerations of digital sliding-mode control, based on the coupling system’s dynamics. The
experimental and simulation results show that inertia
rotors can control the rotation and swing of the
suspended load for traveling disturbance.
References [1] T. Sakawa and A.Nakazumi, “Modeling and Control
of a Rotary Crane”, ASME Journal of Dynamic Systems, Measurement, and Control, Vol.107, pp.200-205, 1985.
[2] H. H. Lee, “Modeling and Control of a Three-Dimensional Overhead Crane”, ASME Journal of Dynamic Systems, Measurement, and Control, Vol. 120, pp.471-476, 1998.
[3] H. Kanki and Y. Nekomoto et al., “Development of Suspender Controlled by CMG (in Japanese with English abstracts)”, Proceedings of JSME Dynamics and Design Conference ’95, No.95-8 (1), B, pp.34-37, 1995.
[4] Y. Yoshida and K. Mori, ”Simultaneous Control of Attitude and Swing of a Pendulum Having a Rotational Free Body (in Japanese with English abstracts)”, Trans. of JSME, Series C, Vol.64, No.628, pp.4660-4665, 1998.
[5]Y. Yoshida and M. Yajima, “Control of a Suspended Load Using Inertia Rotors”, Proceedings of the 1999 ASME Design Engineering Technical Conferences, Symposium on Motion and Vibration, DETC99/MOVIC-8418, pp.1-6, Las Vegas, 1999.