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TRACKING CONTROL FOR
TWO-DIMENSIONAL OVERHEAD CRANE
Feedback Linearization with Linear Observer
Tam´as R´ozsa and B´alint Kiss
Department of Control Engineering and Information Technology, Budapest University of Technology and Economics
Magyar Tud´osok krt. 2, Budapest, Hungary
Keywords:
Overhead crane, Motion planning, Tracking control, Flatness, Feedback linearization, Taylor approximation,
Linear observer.
Abstract:
A possible way to control non-linear systems is the use of exact linearization and the application of a tracking
controller to ensure exponential decay of the error along the reference trajectory. In case of overhead cranes,
it can be used if the load coordinates (or alternatively the rope angles) are known which is not the case in real
applications, where the motor axis displacements are usually measured. This paper applies the linearization
techniques such that the calculations of unmeasured states are realized with an observer, which is constructed
for the linear approximation of the dynamics along the reference trajectory. Simulation results are provided to
prove the applicability of the concept.
1 INTRODUCTION
Cranes and other types of weight handling equip-
ment are used to carry heavy loads (Gustafsson, 1996;
D. Buccieri and Bonvin, 2005; Kiss and Mullhaupt,
1999). In many cases, the load is attached to the me-
chanical structure with a rope, so the load position
cannot be directly actuated and the resulting oscilla-
tory behavior may present serious difficulties to inex-
perienced human operators. This oscillatory nature of
crane-like underactuated mechanical systems makes
them a popular benchmark application in control en-
gineering. Several tracking and sway elimination al-
gorithms are proposed in the literature (L´evine et al.,
1997; Marttinen et al., 1990; Neupert and Schneider,
2006; Overton, 1996; Hong et al., 1998), but many
of them are based on the knowledge of the load co-
ordinates (or alternatively the rope angles) which are
generally difficult to robustly measure in real applica-
tions.
The flatness property (L´evine, 2009) of the crane
models implies their feedback linearizability and it
can also be exploited for motion planning purposes.
We propose to apply a time-varying linear observer
to the linearized system dynamics along the reference
trajectory to determine the value of the unmeasured
state variables which need to be injected in the track-
ing feedback. The calculations and the simulation re-
sults will be presented on a simple, two-dimensional
overhead crane and they can be generalized for more
complex structures.
The remaining part of the paper is organized as
follows. Section 2 introduces some notations and
presents the dynamics of the two-dimensional over-
head crane. The tracking controller and the observer
design is presented in Section 3. Simulation results
of the trajectory behavior are shown in Section 4, and
our results are summarized in Section 5.
2 SYSTEM DYNAMICS
The two-dimensional overhead crane is illustrated in
Figure 1. The horizontal displacement of the cart with
mass M is denoted by R. The cart is actuated by a
motor delivering the force F which is considered to
be one of the input of the dynamics. The load with
mass m, having the coordinates (x
m
,z
m
) is accelerated
through a rope with the length of L, winched on a
drum of inertia J and radius ρ with a motor delivering
the torque T. The angle between the rope and the
vertical is denoted by θ. We assume no friction and
massless rope and due to the small value of ρ, we also
assume that the rope always connects to the winch at
point (R+ρ, 0). The resulting mechanical system has
427
Rózsa T. and Kiss B..
TRACKING CONTROL FOR TWO-DIMENSIONAL OVERHEAD CRANE - Feedback Linearization with Linear Observer.
DOI: 10.5220/0003539604270432
In Proceedings of the 8th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2011), pages 427-432
ISBN: 978-989-8425-74-4
Copyright
c
2011 SCITEPRESS (Science and Technology Publications, Lda.)