Dynamics and Control of Cranes A Review

Dynamics and Control of Cranes: A Review
EIHAB M. ABDEL-RAHMAN, ALI H. NAYFEH, and ZIYAD N. MASOUD
Department of Engineering Science and Mechanics, MC 0219, Virginia Polytechnic Institute and
State University, Blacksburg, VA 24061, USA
Abstract: We review crane models available in the literature, classify them, and discuss their
applications and limitations. A generalized formulation of the most widely used crane model is
analyzed using the method of multiple scales. We also review crane control strategies in the liter-
ature, classify them, and discuss their applications and limitations. In conclusion, we recommend
appropriate models and control criteria for various crane applications and suggest directions for
further work.
Key words: Crane, dynamics, control, stability, gantry crane, rotary crane, boom crane.
1
Figure 1: A bidirectional gantry crane.
1 Introduction
Cranes are increasingly used in transportation and construction. They are also getting larger,
faster, and higher, necessitating efficient controllers to guarantee fast turn-over time and to meet
safety requirements. The last 40 years has seen mounting interest in research on the modeling and
control of cranes. In this paper, we review this body of literature available in the English language
journals and conference proceedings.
A crane consists of a hoisting mechanism (traditionally a hoisting line and a hook) and a
support mechanism (trolley-girder, trolley-jib, or a boom). The cable-hook-payload assembly is
suspended from a point on the support mechanism. The support mechanism moves the suspension
point around the crane work space, while the hoisting mechanism lifts and lowers the payload to
avoid obstacles in the path and deposit the payload at the target point.
Cranes can be classified based on the degrees of freedom the support mechanism offers the
suspension point. The support mechanism in a gantry (overhead) crane, Figure 1, is composed of
a trolley moving over a girder. In some gantry cranes, this girder (bridge) is in turn mounted on
another set of orthogonal railings in the horizontal plane. This set-up allows the suspension point
one or two rectilinear translations in the horizontal plane. In a (tower) rotary crane, Figure 2, the
girder (jib) rotates in the horizontal plane about a fixed vertical axis. This allows the suspension
point two motion patterns in the horizontal plane, a translation and a rotation. The suspension
point in a boom crane, Figure 3, is fixed at the end of the boom. It has two motion patterns:
rotations around two orthogonal axes located at the base of the boom.
The capacity of the boom to support loads in compression (as opposed to bending) offers boom
2
Figure 2: A rotary crane.
cranes a structural advantage over the other types of cranes. As a result, boom cranes are compact
in comparison to similar capacity gantry or rotary cranes. Consequently, all mobile cranes use boom
cranes. They are mounted on ships to transfer cargo between ships and/or offshore structures and
to conduct offshore construction. Boom cranes are also mounted on trucks for use in cargo transfer
and construction sites. On the other hand, gantry and rotary cranes are used in fixed sites. Gantry
cranes are widely used in the transportation industry, mines, steel mills, and assembly lines. Rotary
cranes are mostly used in construction.
The high compliance of the cable-hook-payload assembly results in complex system dynamics.
External (base) excitations at the suspension point can produce in- and out-of-plane pendulations
as well as vertical oscillations of the payload. Even in the absence of external excitations, inertia
forces due to the motion of the crane can induce significant payload pendulations. This problem is
exacerbated by the fact that cranes are typically lightly damped, which means that any transient
motion takes a long time to dampen out. Todd et al. (1997) report that the damping of ship-
mounted boom cranes is 0.1% to 0.5% of their critical damping. Patel et al. (1987) offer a higher
estimate of 1% for the vertical motion and 5% for the lateral motions. Willemstein et al. (1986),
van den Boom et al. (1987 and 1988), Patel et al. (1987), and Michelsen and Coppens (1988) found
out, using numerical simulation, that both stationary and transient dynamic forces due to payload
motions are large enough that they need to be accounted for in the design and operation of cranes,
thus emphasizing the need to predict and control both transient and stationary responses of the
payload to excitations.
Suppression of payload pendulations/oscillations is especially important for off-shore cranes.
3
Figure 3: A boom crane.
Wave-induced motions of the platform (a crane-ship or a semi-submersible) may contain significant

energy near the natural frequency and/or twice the natural frequency of the free swinging load;
this situation could initiate an external resonance and/or a parametric resonance. Therefore, the
platform motions may induce large motions of the load directly or indirectly by creating a motion
instability. For example, the platform motions may excite a parametric instability similar in form
to that of the Mathieu instability (Nayfeh and Mook, 1979). This parametric instability has been
observed at full scale by ship-crane operators and can arise in relatively mild sea states (McCormick
and Witz, 1993).
On-shore cranes may also experience base excitations, leading to a complex dynamic response
of the free swinging load, due to a variety of reasons, such as waves breaking on the shore and the
interaction between the payload motion and the platform support system. However, this problem
is most pronounced in off-shore cranes. Assuming a work ability criterion based on the vertical
displacement of the boom tip/payload only, Rawston and Blight (1978) calculated that a crane
vessel in the North Sea could operate only for less than half of its availability time. Nojiri and
Sasaki (1983) calculated that a barge crane in the East China Sea could only be used for heavy lifts
for 34% of the time. More generally, payload pendulations/oscillations and the need to suppress
them have been identified as a bottleneck in the operations of the transportation and construction
industries even where the relatively simple gantry cranes are concerned. Pendulation suppression
is also necessary to increase the safety of operations and decrease the dynamic loads applied to the
crane structure during operations (Brkić et al., 1998). Newly designed gantry cranes are larger,
have higher lift capacities, and have greater lift heights and travel speeds, making the control of
4
load pendulations a particular challenge (Champion, 1989).
The need for payload pendulation/oscillation suppression and the progress in computing facil-
ities and sensors led to mounting interest in crane control in recent years. However, most crane
controllers developed until now have been far from satisfactory. Once tested in actual operation,
they were found to be cumbersome and ineffective and thus were left unused.
2 Modeling
Two approaches to the modeling of cranes are identified: lumped-mass and distributed-mass models.
2.1 Distributed-Mass Models

In this approach, the hoisting line is modeled as a distributed-mass cable and the hook and payload,
lumped as a point mass, are applied as a boundary condition to this distributed-mass system.
The only model available in this category is the planar model of d’Andrea-Novel et al. (1990,
1994) and d’Andrea-Novel and Boustany (1991b) for a gantry crane linearized around the cable’s
equilibrium position. They ignore the inertia of the payload and model the cable as a perfectly
flexible, inextensible body using the wave equation
∂2w

∂ ∂w
ρ 2 − τ =0 (1)
∂t ∂s ∂s
where w(s, t) is the transverse motion of the cable around its equilibrium position, s is a curvilinear
coordinate representing the arclength along the cable, ρ is the mass per unit length of the cable, τ
is the tension in the cable at equilibrium,
τ (s) = mg + sρg (2)
and m is the payload mass. The boundary conditions are

∂2w ∂w
M −τ =F at s = 0 (3)
∂t2 ∂s
∂w
=0 at s = ℓ (4)
∂s
where M is the mass of the trolley and F is the input force applied to the trolley. The boundary
condition at the payload, equation (4), subjects the motion to the constraint
∂2w
=0 at s = ℓ (5)
∂t2
thus ignoring the inertia of the payload. Joshi and Rahn (1995), Martindale et al. (1995), and
Rahn et al. (1999) extended the model of d’Andrea-Novel and coworkers to include the inertia of
the payload by changing the boundary condition at the payload to
∂2w ∂w
m 2
+τ =0 at s = ℓ (6)
∂t ∂s
5
Figure 4: Schematic and coordinate system of the lumped-mass model.
The model is valid only for a lumped mass m of the same order of magnitude as the mass of
the cable and for small trolley displacement and cable angles. So it can only be used near the end
of the trolley travel. However, even when a crane operates under no-load conditions, the mass of
the hook is typically one order of magnitude heavier than the mass of the cable. As a result, this
approach has limited practical applications.
2.2 Lumped-Mass Models

This is the most widely used approach to crane modeling. The hoisting line is modeled as a massless
cable. The payload is lumped with the hook and modeled as a point mass. The cable-hook-payload
assembly is modeled as a spherical pendulum. The resulting mathematical representation is simple
and compact while capturing the complex dynamics of the payload motion.
There are two classes of lumped-mass models, depending on the way the external excitations
are introduced to the system, namely reduced and extended models. A reduced model lumps all
external excitations into expressions representing the motion of the pendulum suspension point
(base excitations). This approach assumes that the payload motions are influenced by, but do not
have a significant influence on, the platform motion; that is, the inertial coordinates ξ, ζ, and η
of the suspension point (as shown in Figure 4) are known functions of time t. An extended model
adds the crane support mechanism and the platform to the dynamic model, thereby incorporating
the interactions among the support mechanism, the platform, and the cable-payload assembly in
the model.
All reduced models are special cases of the same classical model of a spherical pendulum under
base excitations. On the other hand, each extended model is a unique system capturing a distinct
6
set of the crane-structure dynamics. In the following, we analyze the reduced crane model and then
discuss the extended crane models available in the literature.
3 The Reduced Model

We consider a pendulum of length ℓ and mass m and express the Cartesian coordinates of the
suspension point and mass as [ξ(t), ζ(t), η(t)] and [x(t), y(t), z(t)], respectively. These coordinates
have to satisfy the constraint
(x − ξ)2 + (y − ζ)2 + (z − η)2 = (ℓ + r)2 (7)
where r is the elastic stretch in the cable. The Lagrangian of the system is
L = 21 m(ẋ2 + ẏ 2 + ż 2 ) − mgz − 21 ce r 2 (8)
where ce is the longitudinal stiffness of the cable.

To determine the equations of motion, we use equation (7) to substitute for z and ż into the
Lagrangian, apply the Euler-Lagrange equations, and obtain (Chin et al., 1998)
x−ξ x−ξ 2 x−ξ

ẍ = − √ (g + η̈) − λ̇ + λ̈ (9)
λ 4λ2 2λ
y−ζ y−ζ 2 y−ζ
ÿ = − √ (g + η̈) − λ̇ + λ̈ (10)
λ 4λ2 2λ
cr g + η̈ √ λ̇2
r̈ = − λ + λ + (11)
m(ℓ + r)2 ℓ+r 4λ(ℓ + r)
1 2 2 2

− (ℓ̇ + ṙ) − (ẏ − ζ̇) − (ẋ − ξ̇) + (ℓ + r)ℓ̈ − (y − ζ)(ÿ − ζ̈) − (x − ξ)(ẍ − ξ̈)
ℓ+r
where λ = (ℓ + r)2 − (x − ξ)2 − (y − ζ)2 . These are the exact equations of motion of the spherical
pendulum model in its most generic form.
To apply perturbation analysis to this system, Chin et al. (1998) wrote the cable length as ℓ =
ℓo +ℓc and the cable stretch as r = ro +r, where ℓo is the cable length at some reference configuration,
ℓc is the change in the cable length, and ro and r are, respectively, the static and dynamic stretches in
the cable. Assuming x, y, r to be of the same order of magnitude and considering a slow variation of
the cable length ℓc ≤ O(x2 ), Chin et al. (1998) extracted a third-order approximation of equations
7
(9)–(11), used it to model a ship-mounted boom crane, and obtained
x ¨ ω2
ẍ + ω1 2 x = − ξ̈ + ℓc − η̈ + 1 (ℓc x + r(x − ξ))
ℓcr ℓcr
2 x x
1 2 1 2
− ω1 2 r + 2 x + 2 y − 2 ẋ2 + ẏ 2
2

(12)
ℓcr ℓcr
x r̈
− 2 (x ẍ + y ÿ + rr̈) + (x − ξ)
ℓcr ℓcr
y ¨ ω2
ÿ + ω1 2 y = − ζ̈ + ℓc − η̈ + 1 (ℓc y + r(y − ζ))
ℓcr ℓcr
y y
− ω12 2 r 2 + 12 x2 + 21 y 2 − 2 ẋ2 + ẏ 2

(13)
ℓcr ℓcr
y r̈
− 2 (x ẍ + y ÿ + rr̈) + (y − ζ)
ℓcr ℓcr
ω2 2
r̈ + ω22 r = η̈ − ℓ¨c + 1 ( 21 x2 − x ξ + 12 y 2 − y ζ) − (ẋ ξ̇ + ẏ ζ̇)
ℓcr ℓcr
r (ℓcr − r) 2 2ṙ
+ω12 2 (r 2 + 12 x2 + 12 y 2 ) + 2
(ẋ + ẏ 2 ) − 2 (x ẋ + y ẏ) (14)
ℓcr ℓcr ℓcr
(ℓcr − r) r̈ 2 1
+ (x ẍ + y ÿ) − 2 (x + y 2 ) − (ξ ẍ + x ξ̈ + ζ ÿ + y ζ̈)
ℓ2cr ℓcr ℓcr
where ℓcr = ℓo + ro is the characteristic cable length, ω12 = g/ℓcr is the natural frequency of payload
pendulations, and ω22 = ce /m is the natural frequency of the longitudinal oscillations.
A special case of interest allows for the reeling and unreeling of an inextensible cable. The
equations of motion of the payload can, thus, be reduced to two equations
x ¨
ẍ + ω 2 x = − ξ̈ + ℓc − η̈
ℓo
x ω2 2
− 2 2 x + y 2 + ẋ2 + ẏ 2 + (x ẍ + y ÿ)

(15)
ℓo
y ¨
ÿ + ω 2 y = − ζ̈ + ℓc − η̈
ℓo
y ω2 2
− 2 2 x + y 2 + ẋ2 + ẏ 2 + (x ẍ + y ÿ)

(16)
ℓo
where ω 2 = g/ (ℓo + ℓc ). For the same assumptions and using a spherical body coordinate system
8
Figure 5: Spherical coordinate system.
9
attached to the suspension point, Figure 5, the exact equations of motion can be written as
!
ξ˙ η̇
θ̈ cos φ + ω 2 sin θ = 2 θ̇ φ̇ sin φ + 2 β̇ φ̇ sin φ + sin θ + cos θ
ℓ ℓ
!
ζ̇ ℓ̇
−2 γ̇ cos θ φ̇ cos φ + −2 θ̇ cos φ + β̇ cos φ + γ̇ cos θ sin φ
ℓ ℓ

2 ξ η 2 ξ 1
− β̇ cos θ − sin θ − γ̇ cos θ − 2 sin 2θ cos φ (17)
ℓ ℓ ℓ

ζ ξ η
− β̇ γ̇ sin θ + sin φ − β̈ cos φ − sin θ − cos θ
ℓ ℓ ℓ

ζ ξ̈ η̈
− γ̈ cos θ sin φ + + cos θ − sin θ
ℓ ℓ ℓ
!
2 1 2 ξ̇ η̇
φ̈ + ω cos θ sin φ = − 2 θ̇ sin 2φ − β̇ θ̇ sin 2φ − 2 cos θ sin φ + 2 sin θ sin φ
ℓ ℓ
!
ξ˙ ζ̇ ℓ̇
+2 γ̇ θ̇ cos θ cos2 φ − cos φ + sin θ sin φ − 2 φ̇ − γ̇ sin θ
ℓ ℓ ℓ

ξ η
−β̇ 2 21 sin 2φ − sin θ sin φ − cos θ sin φ
ℓ ℓ

2 1 2 ξ ζ
+ γ̇ 2 cos θ sin 2φ + sin θ sin φ + cos φ (18)
ℓ ℓ

ζ η
+ β̇ γ̇ cos θ cos 2φ − cos θ sin φ − cos φ
ℓ ℓ

ξ η ξ ζ
+ β̈ sin φ cos θ − sin θ + γ̈ sin θ − cos φ + sin θ sin φ
ℓ ℓ ℓ ℓ
ξ̈ ζ̈ η̈
− sin θ sin φ − cos φ − cos θ sin φ
ℓ ℓ ℓ
where [0, β, γ] is the rotation vector of the suspension point (β is the crane luff angle and γ is the
crane slew angle), θ is the in-plane angle, and φ is the out-of-plane angle between the equilibrium
position of the cable-payload assembly and the cable at time t.
The standard model of a spherical pendulum, however, assumes both an inextensible cable
and a constant length cable. In the following, we develop and analyze this model in details. To
determine the equations of motion to third order in x and y, we let ξ, ζ, and η be ≤ O(x, y). It
follows from equation (7) that
(x − ξ)2 + (y − ζ)2 [(x − ξ)2 + (y − ζ)2 ]2

z =η−ℓ+ + + ··· (19)
2ℓ 8ℓ3
Substituting equation (19) into equation (8), keeping up to quartic terms, and letting ω 2 = g/ℓ, we
10
obtain
1 m h ˙ + (y − ζ)(ẏ − ζ̇)
i
L= m(ẋ2 + ẏ 2 + η̇ 2 ) + η̇ (x − ξ)(ẋ − ξ)
2 ℓ
m h i2
+ 2 (x − ξ)(ẋ − ξ̇) + (y − ζ)(ẏ − ζ̇) − mg(η − ℓ) − 12 mω 2 [(x − ξ)2 + (y − ζ)2 ]
2ℓ
mω 2
− 2 [(x − ξ)2 + (y − ζ)2 ]2 + · · · (20)
8ℓ
Applying the Euler-Lagrange equations and adding linear damping ordered at µ ≤ O(x2 ) yields
(x − ξ) (x − ξ)
ẍ + 2µẋ + ω 2 x = ω 2 ξ − η̈ − ω 2 2 2

(x − ξ) + (y − ζ)
ℓ 2ℓ2
(x − ξ) h ˙ 2 + (ẏ − ζ̇)2 + (x − ξ)(ẍ − ξ̈) + (y − ζ)(ÿ − ζ̈)
i
− ( ẋ − ξ) (21)
ℓ2
(y − ζ) (y − ζ)
ÿ + 2µẏ + ω 2 y = ω 2 ζ − η̈ − ω 2 (x − ξ)2 + (y − ζ)2

ℓ 2ℓ 2
(y − ζ) h 2 2 ¨ + (y − ζ)(ÿ − ζ̈)
i
− (ẋ − ξ̇) + (ẏ − ζ̇) + (x − ξ)(ẍ − ξ) (22)
ℓ2
where the damping µ is assumed to be symmetric in both pendulation directions. The equations
are symmetric in the pairs (x, ξ) and (y, ζ), reflecting the physical symmetry of the in-plane and
out-of-plane motions of the payload. As a result, the linear natural frequencies ω of the payload
pendulations are identical. Further, the equations show that the in-plane and out-of-plane modes
are coupled by cubic terms, representing the geometric and kinetic nonlinearities in the model.
3.1 Approximate Solution of the Reduced Model

The existence of cubic nonlinearities and the symmetry between the dynamics of the in-plane and
out-plane directions produce a one-to-one internal (autoparametric) resonance, leading to complex
dynamics and energy exchange between the two modes (Nayfeh, 2000). Experiments by Todd
et al. (1997) showed that a ship-mounted boom crane exhibits this predicted dynamic behavior
under external forcing, including chaotic and/or nonplanar responses to strictly planar excitations
at frequencies near the natural frequency of pendulations.
The lateral ξ and ζ and vertical η motions of the boom tip produce external (additive) as well
as parametric (multiplicative) excitations, respectively. To examine the response of the system
to a general forcing near the resonance frequencies, we order the external excitations ξ and ζ at
≤ O(x3 ) and the parametric excitation η at ≤ O(x2 ). Expanding equations (21) and (22) and
dropping terms of order higher than O(x3 ), we obtain
ω2 x 2 x
ẍ + 2µẋ + ω 2 x + x x2 + y 2 + ẋ + ẏ 2 + xẍ + y ÿ = − ξ̈ −

2 2
η̈ (23)
2ℓ ℓ ℓ
ω2 y y
ÿ + 2µẏ + ω 2 y + 2 y x2 + y 2 + ẋ2 + ẏ 2 + xẍ + y ÿ = − ζ̈ −

2
η̈ (24)
2ℓ ℓ ℓ
We use the method of multiple scales (Nayfeh, 1973, 1981) to determine a first-order approxi-
mate solution of equations (23) and (24) for small- but finite-amplitude motions. The worst-case
11
excitation of the crane is a combination of direct excitations at the natural frequency (primary res-
onance) and parametric excitation at twice the natural frequency (principal parametric resonance).
To this end, we introduce a small dimensionless parameter ǫ as a bookkeeping device and the time
scales
T0 = t and T2 = ǫ2 t (25)
In terms of these scales, the time derivatives become
d
= D0 + ǫ2 D2 + · · · (26)
dt
d2
= D02 + 2ǫ2 D0 D2 + · · · (27)
dt2
where Dn ≡ ∂/∂Tn . We apply the displacement combination
ξ = ǫ3 u0 cos Ω1 t, ζ = ǫ3 v0 cos Ω1 t, and η = ǫ2 w0 cos Ω2 t (28)
with a primary excitation frequency

Ω 1 = ω + ǫ 2 σ1 (29)
and a principal parametric excitation frequency
Ω 2 = 2 ω + ǫ 2 σ2 (30)
where σ1 and σ2 are detuning parameters. We seek a uniform approximate solution of equations
(23) and (24) in the form
x(t; ǫ) ≃ ǫx1 (T0 , T2 ) + ǫ3 x2 (T0 , T2 ) (31)

3
y(t; ǫ) ≃ ǫy1 (T0 , T2 ) + ǫ y2 (T0 , T2 ) (32)
Substituting equations (25)–(32) into equations (23) and (24) and equating coefficients of like
powers of ǫ leads to the following problems:
Order ǫ:
D02 x1 + ω 2 x1 = 0 (33)
D02 y1 2
+ ω y1 = 0 (34)
Order ǫ3 :
4ω 2
D02 x2 + ω 2 x2 =−2D0 D2 x1 − 2µD0 x0 + ω 2 u0 cos Ω1 T0 − w0 x1 cos Ω2 T0
ℓ
−x1 (D0 x1 D0 x1 + x1 D02 x1 + D0 y1 D0 y1 + y1 D02 y1 ) (35)
4ω 2
D02 y2 + ω 2 y2 =−2D0 D2 y1 − 2µD0 y0 + ω 2 v0 cos Ω1 T0 − w0 y1 cos Ω2 T0
ℓ
−y1 (D0 x1 D0 x1 + x1 D02 x1 + D0 y1 D0 y1 + y1 D02 y1 ) (36)
12
The solutions of equations (33) and (34) can be expressed as
x1 = A1 (T2 )eiωT0 + cc and y1 = A2 (T2 )eiωT0 + cc (37)
where cc indicates the complex conjugate of the preceding terms. Substituting equations (37)
into equations (35) and (36) and eliminating the terms that produce secular terms, we obtain the
following modulation equations:
u0 w0
A′1 =− 14 iω Ā1 (A21 + 3A22 ) + 12 iωA1 A2 Ā2 − µA1 − iωeiσ1 T2 + iω Ā1 eiσ2 T2 (38)
4 ℓ
v0 w0
A′2 =− 41 iω Ā2 (A22 + 3A21 ) + 12 iωA1 Ā1 A2 − µA2 − iωeiσ1 T2 + iω Ā2 eiσ2 T2 (39)
4 ℓ
where the prime indicates the derivative with respect to the slow time scale T2 .
To determine the slow variations of the amplitudes and phases of the solution, we introduce
the polar transformation
Ak (T2 ) = 21 ak (T2 )eiβ(T2 ) , k = 1, 2 (40)
into equations (38) and (39) and obtain
1 w0
a′1 =−µ a1 − 2 u0 ω sin(β1 − σ1 T2 ) + ω a1 sin(2β1 − σ2 T2 )
ℓ
− 163
ω a1 a22 sin(2β1 − 2β2 ) (41)
u0 w0
β1′ =− ω cos(β1 − σ1 T2 ) + ω cos(2 β1 − σ2 T2 )
2 a1 ℓ
1
− 16 ω a21 + 81 ω a22 − 16
3
ω a22 cos(2 β1 − 2 β2 ) (42)
w0
a′2 =−µ a2 − 21 v0 ω sin(β2 − σ1 T2 ) + ω a2 sin(2 β2 − σ2 T2 )
ℓ
+ 163
ω a21 a2 sin(2 β1 − 2 β2 ) (43)
v0 w0
β2′ =− ω cos(β2 − σ1 T2 ) + ω cos(2 β2 − σ2 T2 )
2 a2 ℓ
+ 18 ω a1 2 − 16 1
ω a22 − 16
3
ω a21 cos(2 β1 − 2 β2 ) (44)
The cubic terms in equations (41) and (43) indicate the possibility of an exchange of energy between
the two modes. The presence of the amplitudes a1 and a2 in the denominator in equations (42)
and (44) leads to instabilities in the numerical integration of the modulation equations whenever
either amplitude approaches zero. Therefore, a Cartesian transformation instead of this polar
transformation is usually used to write the modulation equations (Nayfeh, 2000). While it is
harder to discern the system’s behavior by inspection of those equations, they do not pose any
difficulties to numerical integration.
Miles (1962, 1984) used this model to examine the response of a lightly damped, spherical
pendulum to a simple harmonic, planar displacement of the suspension point. He found out that
nonplanar motions could be excited due to the nonlinear interaction between the two modes. Chin
and Nayfeh (1996) and Chin et al. (2001) used the model to study ship-mounted crane dynamics in
two cases of harmonic base excitations at the boom tip: the case of primary resonance and the case
13
of principal parametric resonance. They found out that, while the parametric excitation exhibits
principal parametric resonance in the neighborhood of twice the natural frequency of the system,
the response is always periodic and planar. On the other hand, direct excitations produce complex
dynamics when the excitation frequency approaches the natural frequency of the system (primary
resonance). They also found out that a strictly planar excitation could produce in- and out-of-
plane pendulations and that the response may exhibit sudden jumps, modulation of the response
amplitudes and phases (quasiperiodic or two-torus motion), and chaos.
Using the method of multiple scales, Chin et al. (1998) solved equations (12)–(14) analytically
and numerically. They found out that a parametric excitation at twice the natural frequency leads
to a sudden jump in the response as the cable is unreeled. They also demonstrated that introducing
a harmonic change in the cable length at the same frequency as the excitation can suppress this
dynamic instability and result in a smooth response.
Abdel-Rahman and Nayfeh (2000) used the variable cable length model, equations (15) and
(16), to study a boom crane allowing for reeling and unreeling of an inextensible cable at a constant
speed. The analytical solution and numerical simulation show that a planar direct base excitation
near the natural frequency can produce in-plane and out-of-plane motions, sudden jumps in the
response, and a chaotic response as the cable length is changed.
Elling and McClinton (1973) were the first to examine the nonlinearities involved in the dynamic
response of a boom crane. They modeled the crane as a spherical pendulum undergoing a conical
motion (Greenwood, 1988), while the hoisting cable length is changed at a constant speed. As a
result, the model assumed the motion in one direction to be of order O(x) and in the other direction
to be of the order O(x3 ) and neglected the potential energy in the higher-order direction. In practice,
the payload pendulates around the suspension point rather than follow a prescribed circular path.
Further, whenever a nonplanar motion occurs, the motions in both directions have the same order
of magnitude and the potential energy in both directions can not be ignored. Using numerical
simulation, they solved the equations of motion subject to harmonic base excitations and found a
resonant response when the excitation frequency is near the natural frequency (primary resonance)
or one-half the natural frequency (secondary resonance) of the assembly. They also concluded that
unreeling of the cable dampens the payload motion while reeling of the cable excites it further.
3.2 Reduced Linear Model

Two special cases of the classical model are of particular interest: the linear three-dimensional
model and the nonlinear planar model. Assuming small motions, one can linearize the generalized
model around the payload equilibrium point. Dropping higher-order terms from equations (23) and
(24) results in the reduced linear model
ẍ + 2µẋ + ω 2 x = Fx (45)
2
ÿ + 2µẏ + ω y = Fy (46)
This is the widely used linear crane model. While it is quite simple, thus amenable to use as a
model of the plant in controller designs, it neglects the nonlinear terms, thus masking the dynamic
complexities arising from coupling of the two modes. Also, this approximation is only valid as long
14
as the payload motions remain small. Jones and Petterson (1988) report that, when pendulations
reach a maximum swing amplitude greater than a few degrees, the nonlinear aspects of the swinging
object must be taken into account. So the linear model breaks down whenever in-plane pendulations
grow or out-of-plane pendulation occur. Further, dropping the cubic nonlinearities on the left-hand
sides of equations (23) and (24), while retaining the direct excitations on the right-hand side, even
though both are of order O(x3 ), is inconsistent. On the other hand, introducing hard excitations
of order O(x) will produce large motions, thus requiring the inclusion of the cubic nonlinearities
on the left-hand side.
3.3 Reduced Planar Model

Setting the out-of-plane motion y equal to zero in equation (23), we obtain the nonlinear two-
dimensional model of the pendulum up to third order O(x3 ) as
ω2 3 x x
ẍ + 2µẋ + ω 2 x + x + 2 ẋ2 + xẍ = − ξ̈ − η̈

2
(47)
2ℓ ℓ ℓ
Similarly, setting the out-of-plane angle φ and the slew angle γ equal to zero in equation (17)
and assuming a constant cable length, we obtain the exact nonlinear two-dimensional model of the
pendulum in a more compact form
!
2 ξ̇ η̇ 2 ξ η
θ̈ + 2µθ̇ + ω sin θ = 2 β̇ sin θ + cos θ − β̇ cos θ − sin θ
ℓ ℓ ℓ ℓ

ξ η ξ̈ η̈
+β̈ sin θ + cos θ + cos θ − sin θ (48)
ℓ ℓ ℓ ℓ
This is the classical planar pendulum equation of motion. The higher-order terms involving β, the
luff angle, represent the dynamic effect of the rotation of the body coordinate system attached to
the suspension point. While a first-order approximation of the equation may neglect all of these
terms, that would render the equation approximate and valid only at low luffing speeds.
4 Extended Models
4.1 Gantry Cranes
The most popular extended model of gantry cranes, Figure 6, augments the planar version of the
pendulum model, equation (48), with the trolley’s planar equation of motion. The result is
M ξ¨ = F − T sin θ − f (49)
where
T = mg cos θ + mℓθ̇ 2 − mξ̈ sin θ (50)

f = µt (M g + T cos θ) (51)
15
Figure 6: Schematic of an extended gantry crane model.
T is the tension in the cable, f is the friction force between the trolley wheels and the girder, and
µt is the friction coefficient. It should be noted that, under this configuration, the linear natural
frequency of the pendulum is dependent on the trolley and payload masses; that is,
r
(M + m)g
ωp = (52)
Mℓ
Field (1961) further augmented this model with an equation describing the dynamics of the
cable hoisting motion. Auernig and Troger (1987) partially linearized the model with respect to
the pendulation angle θ.
Moustafa and Ebeid (1988) derived the only nonlinear model of a bidirectional (double girder)
gantry crane available in the literature. The model accounts for rigid-body motions of the payload,
translation of the trolley and the moving girder, the torsional stiffness and a constant hoisting
speed of the cable, and the inertia of the trolley and girder driving motors. They also presented a
version of the model linearized around the equilibrium position of the payload. Ebeid et al. (1992)
augmented the linearized model with a linear model of the trolley and girder motors accounting for
the motor dynamics. Using numerical simulation, they found out that the motors introduce linear
damping into the system response.
Zrnić et al. (1997 and 1998) presented a linear model of a bidirectional gantry crane by lumping
the bridge mass into five point masses, each of the bridge legs into a point mass, and the payload as
16
another point mass. The model accounts for linear structural stiffness and damping in the bridge
and legs and for the linear stiffness and damping in the driving motor of each leg.
4.2 Rotary Cranes

Parker et al. (1995a) linearized the spherical pendulum model, equations (23) and (24), and applied
a base excitation representing the slew motion (jib rotation) of a rotary crane, but did not apply
the trolley motion along the jib. Parker et al. (1995b) extended this model to account for a
reeling of the cable synchronized with the slew motion. The resulting linear time-varying model is
inconsistent. It neglects the cubic nonlinearities representing geometric and kinetic nonlinearities
in the model, while retaining first-order direct and second-order parametric base excitations, which
are equivalent to cubic and quartic nonlinear motion terms. Parker et al. (1996) extended the
model of Parker et al. (1995b) to account for trolley translation along the jib and dropped the
higher-order base excitations. However, the linearized model still contained direct excitations of
order O(x3 , y 3 ).
Brkić et al. (1998) presented a linear, planar model of a rotary crane by lumping the jib and
tower mass into eight point masses and each of the hoisting mechanism, trolley, and payload as
a point mass. The model accounts for linear structural stiffness in the jib and tower and for the
linear stiffness hoisting motor and cable, while neglecting the damping in the system.
4.3 Boom Cranes

Ito et al. (1978) studied a truck-mounted crane by modeling the cable-hook-payload assembly as a
spherical pendulum, the boom as a beam, and the hoisting and power lines as linearly viscoelastic
bodies subject to engine torque and friction in the bearings. They carried out numerical simulations
on a special planar case where all accelerations are set equal to zero; they also carried out exper-
iments for this case. The results show that the payload undergoes oscillations due to longitudinal
vibrations in the boom luffing line and the hoisting cable and pendulum-like payload oscillations.
The period of this oscillation is an order of magnitude larger than the periods of the longitudinal
vibrations of the lines.
Sakawa and Nakazumi (1985) augmented the spherical pendulum with two equations represent-
ing the rotational motions of the crane base (slew) and the boom (luff). Sato and Sakawa (1988)
extended this model to include an extra jib at the boom tip with a flexible joint. Both models
were partially linearized with respect to the in-plane pendulation angle. Souissi and Koivo (1992)
extended the model of Sakawa and Nakazumi (1985) by coupling the system equations with an
equation describing the reeling of the hoisting line.
Patel et al. (1987), McCormick and Witz (1993), and Witz (1995) used a linear model of a
ship-mounted crane to study the coupled motions of the ship and crane. They modeled the ship
as a rigid body, the boom as a beam under bending, the hoisting and mooring lines as elastic
bodies, and the payload as a point mass undergoing spherical pendulum-like oscillations. Added
mass and damping, due to ship-sea interaction, were also included in the model. Using computer
simulations, they solved a planar version of the model, including the surge, heave, and pitch of
the ship and the surge and heave of the payload for ship and payload motions in head seas. They
17
found out that “resultant vessel heave motions ... are not significantly affected by the vertical
dynamics” and that “pitch motions ... are only slightly affected by vertical dynamics of the hook
load.” In beam seas, they found out that “the influence of coupling on sway motions of the vessel
is very small,” and that “the hook load does not significantly affect the roll motions of the vessel.”
They concluded that coupling of ship motions with payload motions has negligible influence on ship
motions away from the natural frequency of the crane. The calculations of Nojiri and Sasaki (1983)
showed that payload pendulations near the resonance frequency of the cable-payload assembly have
a pronounced effect on the roll and pitch motions of a crane vessel in both regular and irregular
waves. Further, Patel et al. (1987), McCormick and Witz (1993), and Witz (1995) reduced a planar
model of the crane to a Mathieu equation, thus showing that the load can be parametrically excited
due to the relative motion between the boom tip and load. Based on the stability diagrams of the
Mathieu equation, they derived the operability conditions for the crane.
Schellin et al. (1989) augmented the linear planar pendulum model, equation (45), with an
equation describing the stretch motion in the hoisting line and three equations of motion describing
the planar rigid-body motion of the ship (surge, heave, and pitch). The ship and boom were modeled
as a rigid body, the mooring system as a nonlinear restoring force, and the hoisting cable was allowed
to stretch elastically. Added mass and damping due to the ship-sea interaction were also included in
the model. They found out that the natural frequency of the cable-payload assembly in stretching
is four orders of magnitude higher than that in pendulation. They also reported strong coupling
between the payload pendulation and the ship surge and pitch motions when the crane is excited
near its natural frequency. Numerical simulations of the system revealed chaos in the response of
the load to regular waves at a wave frequency near the natural frequency of “the hook load” and
period doubling at frequencies further afield from it.
Schellin et al. (1991) extended this model to three dimensions by coupling the spherical pen-
dulum model to a linear model of the ship rigid-body motions and an equation describing the cable
stretch as a viscoelastic body. They compared the simulation results of the model and those of a
linearized version of the model to the results of experiments on a ship model and found out that
the nonlinear model was “more realistic” than the linearized model near resonance where large
ship and load motions occur. Simulations of the response to wave groups show that, when the
frequencies of the component waves of the wave group are close to the natural frequency of the
cable-payload assembly, the response is chaotic. They also found out that payload motions induce
ship yaw; however “horizontal ship motions did not noticeably affect load oscillations.” There is a
need for more elaborate models to further examine the coupling between the motions of the crane
and those of the ship before this coupling can be neglected or factored into crane models.
Posiadala et al. (1990) modeled the cable-payload assembly in a truck-mounted crane as a
spherical pendulum. Base excitations due to the boom slew, luff, and telescopic (extension) motion
and forcing due to cable reeling/unreeling were introduced into the equations of motion. They used
numerical simulations to calculate the forced payload response under various motion combinations
for 10s and then its free response for the subsequent 10s. They found out that, except in the
absence of slew motions, the payload response is three-dimensional and cannot be considered as a
planar phenomenon. Posiadala et al. (1991) extended the model to account for the flexibility of the
cable being modeled as a Kelvin-Voigt body. The pendulum equations of motion were augmented
18
with an equation describing the dynamic stretching of the cable. The results show a fast-frequency
component in the tension in the cable, representing the oscillations due to the dynamic stretching
of the cable. Posiadala (1996 and 1997) extended this model to a truck crane on an elastic support.
The crane was modeled as a rigid body and the supports as elastic springs. The equations of motion
were further augmented with six equations of motion describing the rigid-body motions of the crane
and boom. Their numerical simulations show that the free response of the payload is quasiperiodic.
The slow frequency in the response (in-plane and out-of-plane pendulations) is due to the natural
frequency of the cable-payload assembly, while the fast frequency is due to the support response
and the resulting base excitations of the system at the boom tip.
Kościelny and Wojciech (1994) and Osiński and Wojciech (1994) modeled the unloading of
cargo at sea. The crane and cargo ship were modeled as rigid bodies floating on sea, the boom
as a beam undergoing bending, the cable as a viscoelastic body, and the payload as a spherical
pendulum. They wrote the equations of motion describing the planar motion of the system subject
to nonlinear constraints representing the friction between the load and the cargo ship deck and
contact between the boom and the crane ship.
Kral and Kreuzer (1995) and Kral et al. (1996), modeling a ship-mounted boom crane, aug-
mented a planar pendulum model with the equations of motion of the ship modeled as a two-
dimensional rigid body undergoing heave, surge, and pitch. The model was used to study the
influence of cable hoisting on cargo pendulations. The results show nonlinear behavior, namely
chaos at cable lengths exceeding 28m and multiple responses at the same cable length, depending
on the lowering or lifting history of the load.
Lewis et al. (1998, 1999) and Parker et al. (1999a, 1999b) modified the linear model of Parker
et al. (1996) to represent a boom crane by replacing the trolley translation in the model with boom
luff.
Towarek (1998) derived a model of a truck-mounted crane interacting with a flexible soil. The
crane platform was modeled as a rigid body undergoing small oscillations, the boom as a flexible
beam, the cable as an elastic string, the cable-payload assembly as a spherical pendulum, and
the soil as a viscoelastic Kelvin-Voigt body. Using numerical simulations, they calculated the
system response for complete revolutions of the boom slew at two different speeds. The system
response shows that the crane oscillates with a narrow band of frequencies, thereby producing base
excitations of the cable-payload assembly at the boom tip.
5 Control
Management of payload motions varies according to the particular application at hand. In some
applications, relatively large pendulations/oscillations are acceptable while the payload is en route
to target destination as long as the residual pendulations/oscillations at the target point are small
enough to allow for accurate payload positioning. In other applications, for example in a cluttered
workspace or a nuclear reactor, more stringent conditions are imposed, requiring suppression of
pendulations/oscillations along the travel path and at the target point. Considering the fact that
most payloads are heavy, payload pendulations pose a safety hazard to workers and objects in
the workspace and the structural integrity of the crane, thus more stringent motion suppression
19
requirements are the norm rather than the exception.
These unwanted motions can arise as a result of inertia forces (due to the prescribed motion
of the payload itself), base excitations (due to motions of the supporting structure), and/or wind
loads on the payload. To avoid inertia induced excitation, the crane operator has to perform
maneuvers slowly. Further, whenever unwanted payload motions develop, the operator has to
either cease operations until the motion dampens out or perform counter maneuvers to dampen
out the motion. These constraints degrade the efficiency of crane operations and lead to time delays
and high operating costs. Automatic control has gained wide interest and application as a tool to
alleviate, mitigate, or manage this problem. The increase in the payload weight, height, span, and
speed of newly designed cranes necessitates even more effective controllers.
In the following, we discuss those crane control strategies and designs available in the pub-
lished literature. Because of the structural difference between crane types, each type is discussed
separately.
5.1 Gantry Cranes

Most control strategies designed for this class of cranes assume a planar gantry crane, utilize planar,
linear models, and assume that the crane path, external forces, and control effort are all planar.
The forces they consider are exclusively inertia forces due to the acceleration and deceleration of
the trolley, and the control effort they apply is the force or torque driving the trolley. This approach
leaves the crane vulnerable to out-of-plane disturbances and the nonlinear coupling between in- and
out-of-plane motions. Crane control strategies that take into account out-of-plane contributions of
any of these factors or nonlinearities in the model are an exception rather than the rule. We will
note these exceptions wherever they exist in the strategies and designs reviewed below.
5.1.1 Open-Loop Techniques

Input-Shaping
The most advanced and practical crane controllers today are controllers based on an open-loop
approach designed to automate and/or shorten the cycle time for gantry cranes operating along
a predefined path. The most widely used of the open-loop control techniques is Input-Shaping.
Controllers using various forms of input-shaping are incorporated into gantry cranes currently
used in ports (Hubbell et al., 1992) and elsewhere. This technique is used to move a crane a
set distance along a set path. It has also been used to design “slowdown” mechanisms (covering
only the deceleration stage of trolley travel) to ensure residual pendulation-free stop at the end
of the trolley travel or for inching maneuvers in tight work spaces and near the target point. In
this technique, the acceleration profile of the trolley travel is designed to induce minimum payload
pendulation during travel and deliver the payload at the target point free of residual pendulation.
By design, the technique allows at least one-half of a pendulation cycle or integer multiples during
each of the acceleration and deceleration stages.
Alsop et al. (1965) were the first to propose a strategy to control payload pendulations. The
controller accelerates the trolley in steps of constant acceleration then kills the acceleration when the
payload reaches zero-pendulation angle (after multiples of half of the period) and let the trolley coast
20
at a constant travel speed along the path. The same procedure is replicated in the deceleration stage.
Assuming two constant acceleration/deceleration steps and a linear approximation of the cable-
payload period, Alsop et al. used an iterative procedure to calculate the acceleration profile of the
trolley. Their results showed that, although there was no residual pendulation, pendulation angles
were the order of 10◦ during the acceleration and deceleration stages. Carbon (1976) employed
one-step and two-step versions of this strategy to decelerate the trolley and eliminate residual
pendulations in the design of commercial gantry cranes used in ship unloading.
Using this strategy, Alzinger and Brozovic (1983) demonstrated, via a numerical example, that
a two-step acceleration profile results in significant reductions in the travel time over a one-step
acceleration profile. They used the two-step acceleration profile to design commercial gantry cranes
used in ship unloading. Testing on an actual crane showed that the two-step acceleration profile
can deliver both fast travel and damped payload pendulation at the target point. However, testing
also showed that significant payload pendulations, as much as 5◦ , arise and persist due to any
deviations from the prescribed acceleration profile.
Hazlerigg (1972) proposed another input-shaping strategy using a symmetric two-step constant
acceleration/deceleration profile to move the trolley to the target point in a period of time equal
to the period of the cable-payload assembly and eliminate residual pendulation. The size of each
of the two steps is determined based on the travel distance, the maximum available acceleration,
and the period of the cable-payload assembly. Experimental verifications of the strategy showed
that it dampened payload pendulations, however its performance was highly sensitive to changes
in the cable length. Kuntze and Strobel (1975) extended this strategy by introducing one or more
steps of zero-acceleration into the acceleration profile (that is; constant travel speed intervals), thus
relaxing the constraint on the optimal travel time to one period of the cable-payload assembly and
allowing for a constraint on the maximum travel speed to accommodate the capability of the trolley
motor. Numerical simulation of the strategy showed that it was very sensitive to disturbances and
parameter variations.
Yamada et al. (1983) proposed an input-shaping strategy where the acceleration profile is
generated using Pontryagin’s maximum principle to achieve minimum transfer time and no residual
pendulations at the target point. They generated an optimal profile and approximated it with a
suboptimal profile, using one or two steps of constant acceleration/deceleration along the path,
in order to simplify the control effort. The suboptimal profile was then used to generate a “data
table” containing the acceleration profiles for a grid of initial payload angular positions and speeds.
The strategy was applied to a scaled model of a gantry crane. Results showed that the payload
pendulations at the target point were less than 1.5◦ .
Jones and Petterson (1988) extended the work of Alsop et al. (1965) using a nonlinear approx-
imation of the cable-payload period to generate an analytical expression for the duration of the
coasting stage as a function of the amplitude and duration of the constant acceleration steps. This
analytical expression is then used to generate a two-step acceleration profile. Numerical simulations
using various acceleration profiles show that this technique was able to reduce the residual pendu-
lation to 0.1◦ to 0.3◦ . However, it was not able to dampen out initial disturbances of the payload
and could even amplify them. Noakes et al. (1990) and Noakes and Jansen (1990, 1992) applied
a one-step variation of this acceleration profile to an actual bidirectional crane using a constant
21
cable length and performing a U-shaped maneuver. Test results matched those of the numerical
simulations.
Dadone and van Landingham (2001b) generated a better approximation of the cable-payload
period using the method of multiple scales. Using numerical simulation, they compared the residual
pendulations due to one-step input-shaping strategies based on their nonlinear approximation, a
simplified form of that approximation, and the linear approximation of the period. They found
out a significant enhancement, of as much as two orders of magnitude, in the performance of the
nonlinear control strategies over the linear strategy. The enhanced performance of the nonlinear
strategies was most pronounced for longer coasting distances and higher accelerations.
Starr (1985) used a symmetric two-step acceleration/deceleration profile to transport a sus-
pended object with minimal pendulation. The duration of the constant acceleration steps is as-
sumed to be negligible compared to the period of the cable-payload assembly. A linear approxi-
mation of the period of the payload is used to generate analytically the acceleration profile. Strip
(1989) extended this work by employing a nonlinear approximation of the cable-payload period to
generate one-step and two-step symmetric acceleration profiles.
Karnopp et al. (1992) proposed an input-shaping strategy based on cam design techniques.
Given the distance of the desired trolley travel and the pendulation natural frequency, it produces a
prescribed input position of the trolley to deliver the payload residual-pendulation free at the target
position. However, the minimum travel time has to be at least 12 n of the payload pendulation period,
where n = 3, 5, 7, · · · . Also using the minimum travel time (that is 150% of the payload pendulation
period) results in significant intermediate pendulation; the amplitude of this pendulation is 0.096
of the travel distance divided by the cable length. While a longer travel time results in reduced
pendulation, it leads to further delays in operations and introduces higher harmonics in the motion.
Kress et al. (1994) showed analytically that input-shaping is equivalent to a notch filter applied
to a general input signal and centered around the natural frequency of the cable-payload assembly.
Based on that, they proposed a Robust Notch Filter, a second-order notch filter, applied to the
acceleration input. Numerical simulation and experimental verification of this strategy on an actual
bidirectional crane, moving at an arbitrary step acceleration and changing cable length at a slow
constant speed, showed that the strategy was able to suppress residual payload pendulation.
Optimal Control
The first to propose a control strategy to automate crane operation was Field (1961). He used
an analogue computer to simulate the dynamics of an ore unloading crane. By trial and error, he
produced optimum velocity profiles for the trolley and cable motion that will minimize the travel
time while avoiding obstacles along the path. The control strategy, however, did not attempt to
regulate payload pendulations.
Beeston (1969) used Pontryagin’s maximum principle to generate time-optimal trolley accel-
eration profiles designed to minimize the hoisting and travel time for a single set of initial crane
parameters and various target points. The strategy used bang-bang control of the trolley and
generated three switching points for each acceleration profile. He then used regression analysis to
express each of the switching points in terms of the initial trolley and payload position and velocity.
This control strategy also did not attempt to regulate payload pendulations.
22
Manson (1982) relaxed the restrictions of Hazlerigg (1972) control strategy on the travel time
and generated a time-optimal acceleration profile using Pontryagin’s maximum principle. The three
switching points on the acceleration profile and the total time were evaluated as functions of the
travel distance only where the cable length was assumed to be constant. However, these optimal
solutions were not practical to apply and were only meant as a benchmark for the performance of
other strategies.
Karihaloo and Parbery (1982) proposed a strategy to eliminate residual pendulations for a given
travel time and distance using Pontryagin’s maximum principle. The control input was expressed
as a function of the travel time and the masses of the trolley and payload.
Sakawa and Shindo (1982) applied the optimal control scheme proposed by Sakawa et al. (1981)
for a boom crane to a model of a gantry crane linearized around the payload equilibrium position.
They divided the predefined payload path into three stages: hoisting-up, travel, and hoisting-down.
The scheme was applied to each stage independently. Simulation results show that, although the
payload reaches the target point of each stage free of residual pendulations, significant pendulations,
as much as 7◦ , develop along the hoisting-up and down (first and third) stages.
Kimiaghalam et al. (1998c, 1999b) used genetic algorithms to solve the optimal control problem
formulated by Sakawa and Shindo (1982). In numerical simulation, they achieved similar results
at a shorter travel time; however, their controller calls for drive speeds higher than the constraints
on these speeds.
Auernig and Troger (1987) used Pontryagin’s maximum principle to minimize the transfer time
for a gantry crane traveling and hoisting at constant speeds subject to constraints on the maximum
speed of the trolley travel and the cable hoisting. The technique produced the profile of the force
to be applied to the trolley and/or the hoisting cable to generate the required motion profile. Their
calculations of optimal control profiles showed that the optimal path generated by the technique
was not always superior to the performance of non-optimal cranes in use at European ports. Also,
while the payload was free of residual pendulation, significant pendulations developed along the
path.
Hämäläinen et al. (1995) divided a predefined crane path into five stages: reeling in-place,
reeling and trolley acceleration, coasting at a constant travel velocity, unreeling and trolley de-
celeration, and unreeling in-place. They generated the velocity profiles of the trolley and cable
hoisting in the acceleration and deceleration (second and fourth) stages by minimizing the energy
demand on the motors using a nonlinear model of the crane. The time required for these stages
was minimized by trial and error. Numerical simulation and experimental verification on a scaled
crane model showed that there was no residual pendulations at the target point; however, pen-
dulations of as much as 6◦ developed during travel. They also found out that performance under
this control strategy was faster and smoother (that is; contained less pendulations) than that of a
skilled operator using the same crane.
Optimal control techniques and input-shaping techniques are limited by the fact that they are
extremely sensitive to variations in the parameter values about the nominal values and changes in
the initial conditions and external disturbances and that they require “highly accurate values of the
system parameters” to achieve satisfactory system response (Zinober and Fuller, 1973; Virkkunen
and Marttinen, 1988; Yoon et al., 1995). While a good design can minimize the controller’s sensi-
23
tivity to changes in the payload mass, it is much harder to alleviate the controller’s sensitivity to
changes in the cable length. In fact, Singhose et al. (1997) showed that input-shaping techniques
are sensitive to the pendulation natural frequency. As a result, they suffer significant degradation
in crane maneuvers that involve hoisting.
While closed-loop control may be used to alleviate these problems in input-shaping techniques,
it can not be used with time-optimal control techniques because it can lead to the development of
limit cycles (van de Ven, 1983). Further, the use of closed-loop control in conjunction with either
approach requires a very accurate plant model and can not therefore offer significant improvements
over open-loop control (Zinober and Yang, 1988).
All input-shaping techniques and most optimal control techniques assume an undamped crane.
The unaccounted for damping in the crane system means that the payload will not come instan-
taneously to rest at the target position as the simplified model suggests, thus producing residual
pendulations. Finally, all control strategies in this class (except for that of Hämäläinen et al.) use
a bang-bang acceleration profile. This profile applies excessive stresses on the crane structure and
is difficult to generate accurately using industrial motors.
5.1.2 Closed-Loop Techniques

While open-loop techniques are, by definition, designed to suppress pendulations due to inertia
excitations, all available closed-loop techniques are by design restricted to counter inertia excita-
tions only. In these control strategies, the control input is the force or torque applied to the trolley
and girder motor (where available) in order to suppress pendulations due to the acceleration and
deceleration of the trolley.
Linear Control
Hazlerigg (1972) was the first to propose a feedback control strategy. It employed a second-
order lead compensator to dampen the payload pendulations. Experimental verifications of the
strategy showed that, while it dampened the payload pendulations at the natural frequency of the
cable-payload assembly, it introduced pendulations at higher frequencies.
Ohnishi et al. (1981) used a two-phase strategy to dampen payload pendulations. The first
phase is a linear feedback controller designed to stabilize the payload around its equilibrium position.
To bring the payload to a stop, the trolley decelerates in two stages. The first deceleration stage is
a part of the feedback control phase. The second deceleration stage is an input-shaping technique
used to bring the load to rest over the target point. The control strategy was implemented on an
actual overhead crane in a cold strip mill. While the strategy was able to minimize the pendulation
angles, they reported that the automated system was 30% slower than the manual system it was
supposed to replace.
Ridout (1987 and 1989) proposed a feedback controller using negative feedback of the trolley
position and velocity and positive feedback of the pendulation angle to eliminate residual payload
pendulations at a constant cable length. Tests of the controller on a scaled model delivered the
payload with less than a 0.3◦ pendulation angle to the target point; however, pendulations of as
much as 10◦ developed during the travel. He also found out that the controller was insensitive to
external disturbances, changes in the payload mass, and small changes in the cable length.
24
To avoid persistent residual pendulation at the target point encountered in optimal-time con-
trol (due to unmodeled forces and disturbances), Virkkunen and Marttinen (1988) and Vähä and
Marttinen (1989) proposed a combined control strategy using Yamada et al. (1983) acceleration
profile to drive the trolley all the way until the load is close to the target point and then switching
to LQR to eliminate residual pendulations at the target point. The strategy was implemented on
a scaled model of a gantry crane and results showed that it was successful in suppressing residual
pendulations.
Moustafa and Ebeid (1988) proposed a strategy to suppress pendulations in a bidirectional
crane by controlling both the trolley and girder motors. The strategy calls for three reduced-
order feedback controllers, one for each of the acceleration, coasting, and deceleration stages of the
motions. The controllers are based on linearization of the crane model around a single payload
equilibrium position in each stage. Numerical simulations, for both the trolley travel and traversing
motion at a constant cable length, showed that this technique can dampen inertia disturbances due
to these motions. However, there were significant transient pendulations, as large as 20◦ , and
whirling motions, as large as110◦ , associated with both acceleration and deceleration.
Vähä and Marttinen (1989) and Virkkunen et al. (1990) proposed a P-controller applied to
the trolley position and the payload pendulation angle to eliminate residual pendulations in a
crane operating at a constant cable length and low travel speeds. In numerical simulation and
experimental verification on a scaled crane model, the controller was successful in eliminating
pendulations at the target point and limiting transient pendulations during travel to less than 3.5◦ .
However, the controller was sensitive to the payload initial conditions and the travel distance.
Caron et al. (1989) used a one-step acceleration profile to generate reference trajectories de-
signed to minimize payload pendulations assuming either a constant cable length or a variable cable
length. A PI controller is then used to track that path. Numerical simulations showed good track-
ing of the reference path with minimal transient pendulations of 1.7◦ . Grassin et al. (1991) used
LQR to track both of these reference trajectories. Numerical simulation, using the variable cable
length strategy, and experimental verification, using a scaled model of a crane and the constant
cable length strategy, showed smooth operation and transient pendulations less than 3.5◦ . However
both control strategies were not able to reject disturbances to the payload angular position.
Yoshida and Kawabe (1992) designed a saturation linear state feedback controller to perform
predefined maneuvers. Although the controller speeds up the travel, it incurs much larger pendu-
lations than those suffered by a traditional linear feedback controller.
Moustafa and Emara-Shabaik (1992) used the model of Ebeid et al. (1992) to design a PD
controller using the voltages of the trolley and girder motors as control input. Numerical simula-
tions representing trolley travel only showed that the controller is effective in suppressing payload
pendulations.
Moustafa (1994) designed a linear feedback controller to suppress pendulations due to trolley
motion and cable hoisting using the trolley motor force. This technique was applied to a linearized
time-varying model of the crane. Results of computer simulation show that the technique is effective
in suppressing payload pendulations, but it can develop a static error in the trolley position.
Nguyen (1994) proposed a state feedback control strategy to hoist, stabilize, and deliver the
payload. Two independent controllers are employed, one (employing gain variation with cable
25
length change) to control the trolley position and payload pendulation and another to control the
payload hoisting position. Experimental verification on a scaled model demonstrated good tracking
of the crane position and the cable length, no residual pendulations, and good damping of external
disturbances to the trolley position and payload pendulation angle. However, there were transient
pendulations of as much as 12◦ .
Yoon et al. (1995) proposed a combined control strategy in which the second acceleration
step and the coasting stage in an input-shaping two-step acceleration profile are replaced with
feedback of the payload angular velocity to dampen payload pendulations. The underlying concept
is for the feedback controller to alleviate the sensitivity of the input-shaping technique to external
disturbances and changes in the cable length. Numerical simulation and experimental verification
on a scaled crane traveling at a low speed showed that this strategy is more capable of rejecting
disturbances and adapting to changes in the cable length than pure input-shaping. However, it is
unable to reject disturbances in the deceleration stage. Further, the ability of the fixed feedback
gain to adapt to changes in the cable length is limited, and thus the strategy is unable to eliminate
residual pendulations at the target point.
Yu et al. (1995) used a perturbation technique, the method of averaging, to separate the slow
and fast dynamics of a gantry crane model. Two independent PD controllers are then applied. The
first is a slow-input controller applied to the trolley to maintain tracking of a predefined motion
profile. The second is a fast-input controller to suppress payload pendulations. Due to the approach
used to develop the model, the controller can be applied only when the payload mass is an order of
magnitude larger than that of the trolley. Simulation results showed that this control strategy can
move the payload along a predefined path smoothly with a maximum in-travel pendulation angle
of 5◦ .
Lee et al. (1997) proposed a strategy composed of a PI controller to track the trolley position
and a PD controller to dampen payload pendulations using the motion of the trolley. The control
strategy behaves as a notch filter centered around the cable-payload natural frequency. Experi-
mental verification using a scaled crane model running at a constant cable length showed transient
pendulation of 3◦ during the acceleration and deceleration stages but no residual pendulations at
the target point. Further, it showed that the control strategy is not sensitive to changes in the
payload mass because of the high gear-reduction ratio of the trolley motor. However, it also showed
that the controller damping during travel is low, thereby leaving the payload vulnerable to exter-
nal disturbance-induced pendulations. Lee (1997) refined this strategy by compensating for the
load applied to the trolley due to payload pendulations, cascading a PI trolley velocity controller
with the PI trolley position controller, and cascading a lag compensator with the PD controller to
increase the dampening of payload pendulations. He applies identical versions of this control strat-
egy independently to the in-plane and out-of-plane motions of a bidirectional crane. Experimental
verification using a scaled model showed small transient pendulations, less than 1◦ , and no residual
pendulation. The control strategy is also more effective in resisting external disturbances and offers
faster damping to payload pendulations throughout motion. The results also showed that the PD
controller is sensitive to changes in the cable length, thereby requiring adjustment of the gain to
optimize the performance.
Assuming a flexible cable and a payload mass of the same order of magnitude as that of the
26
cable, Joshi and Rahn (1995), Martindale et al. (1995), and Rahn et al. (1999) developed a linear
feedback controller (PDC) to move the trolley from rest to a desired position and stabilize the
vibrations of the cable-payload assembly at the end point of the maneuver. The controller design
was verified experimentally using a scaled model. The authors reported robust response to “wind
loading and time-varying cable length.”
Alli and Singh (1999) proposed an optimal feedback controller applied to both a model assuming
a rigid cable and another model assuming a flexible cable. The controller parameters are optimized
to minimize the integral over time of the product of time and the magnitude of the error. Computer
simulations showed good regulation of payload pendulations, however the inertia forces involved in
the simulation are minimal.
It should be noted that the underlying linearized crane model used in all of these strategies
develops significant errors as the system parameters change over time. In particular, linear control
strategies are invariably tuned to counter the effects of the natural frequency of the cable-payload
at a single cable length. As a result, they are sensitive to changes in the cable length. Therefore,
linear control imposes restrictions on raising and lowering the payload during motion and requires
low operating speeds, thus imposing unrealistic constrains on crane operations. Burg et al. (1996),
simulating a classical linear feedback controller based on pole placement, reported that the linear
controller produces large pendulations at small travel distances and complete revolutions of the
payload at larger travel distances.
Adaptive Control
To account for the sensitivity of their input-shaping strategy to initial disturbances, Kuntze and
Strobel (1975) used a linear crane model to predict the payload and trolley motions, modify the
acceleration profile accordingly, and absorb these disturbances. They also update the acceleration
profile during operation to account for parameter variations (changes in the cable length and payload
mass). Numerical simulation and experimental verification, using a scaled model, showed that the
strategy can effectively reduce the travel time and eliminate the residual pendulations.
Ackermann (1980) proposed a robust gain scheduling scheme for a linear state feedback con-
troller. The scheme is designed as a fall back controller to be activated in case of sensor failure or
large changes in the states. It schedules the feedback gains to restrict the linear system poles to a
region of stability instead of specific stable points. This control scheme reduces the performance to
cope with system emergencies and assure stability. However, the underlying linear state feedback
controller calls for a control effort to be applied to the angular velocity of the payload, but does
not propose a mechanism to apply it.
Hurteau and DeSantis (1983) proposed an adaptive control strategy applied to a linear state
feedback controller to eliminate residual pendulations. The strategy uses a gain tuning module
to choose the gains to tune a pole-placement routine to changes in the cable length. Marttinen
(1989), Salminen et al. (1990), and Virkkunen et al. (1990) proposed a similar fixed-parameters
gain tuning strategy and a time-varying parameters strategy (updated according to cable length
changes over time) to adapt the controller for changes in the cable length. Both strategies were
verified on a scaled model of a crane. The results showed that residual pendulations persist at the
target point, significant pendulations develop during travel of as much as 10◦ , and in the case of the
27
time-varying parameters strategy a steady-state error occurs in the trolley position. It also showed
that the fixed-parameters strategy, unlike the time-varying parameters strategy, is insensitive to
measurement errors.
Corriga et al. (1998) applied LQR to a linear time-varying model of a crane hoisting the payload
at a constant speed. This implicit gain-scheduling procedure produced a gain vector that was a
function of the time-varying length of the cable. Simulation results showed that while this control
strategy was effective in rejecting initial disturbances, it was excessively slow in approaching the
target point and displayed a steady-state position error.
d’Andrea-Novel and Boustany (1991a) and Boustany and d’Andrea-Novel (1992) used adaptive
control to extend the applicability of the nonlinear controller proposed by d’Andrea-Novel and
Lévine (1989) to a wider range of payload masses for the same controller parameters. However,
this control strategy is only locally stable.
Butler et al. (1991) proposed a control strategy consisting of a primary controller, employing
classical feedback control, and an adaptive controller to account for the unmodeled dynamics ne-
glected in the linear reference model used for the design of the primary controller. To account for
the unmodeled dynamics, they chose an unmodeled dynamics transfer function to minimize the
plant-model error. The control strategy was verified using a scaled model of a crane. The results
showed significant reduction in residual payload pendulations after a few cycles of trolley travel
along a predefined path.
Nguyen and Laman (1995) proposed a control strategy comprised of three independent H ∞
controllers, one for each of the trolley position, hoist position, and payload pendulation. In experi-
mental verification using a scaled model of a crane, the strategy produced small steady-state errors
in the tracking positions and good dampening of payload pendulation to external disturbances.
However, the strategy performance degrades as the acceleration applied to the plant increases.
Lee (1998) refined the control strategy of Lee (1997) by introducing a PI controller to the
hoisting motor to track the cable length and gain scheduling to adapt the fixed gains of the payload
pendulation feedback controller to slow changes in the cable length. The gains for optimum damping
at each cable length are found, then curve fitting is used to express the gains as functions of the
cable length. These functions are used to update the feedback controller gains in real time as the
cable length changes. Experimental verification using a scaled bidirectional crane model running
at low travel, traversing, and hoisting speeds showed transient pendulations less than 2◦ and no
residual pendulations and an ability to reject external disturbances.
Méndez et al. (1998 and 1999) used neural networks to enhance the performance of a state
feedback controller and to tune it online to changes in the cable length. Two neural networks are
used to model the dynamics and to generate and adjust the gains applied independently to the
states of each of the trolley and the payload. The neural networks use an LQR structure to find the
optimal gains based on the current states at each time step. Numerical simulation and experimental
verification on a scaled model showed that this strategy can produce a smooth positioning of the
trolley and suppress residual pendulations at low travel speeds.
Fuzzy Logic Control

Yasunobu and Hasegawa (1986 and 1987) and Yasunobu et al. (1987) proposed a predictive
28
fuzzy control strategy to minimize payload pendulations and travel time, while moving towards a
target point and maneuvering to avoid obstacles along the path. The strategy breaks the crane
operation into seven stages and decides which fuzzy control rule to use in each of them based on
simplified models of the trolley and payload motions. The control rules then employ feedback control
to command the trolley motion and cable hoisting. Experimental verification of the strategy using
both a scaled model and an actual crane showed that the strategy is more effective and consistent
in minimizing the travel time and payload pendulation and more accurate in stopping at the target
point than most skilled operators.
Yamada et al. (1989) used the trolley acceleration as input to move a crane at a constant
cable length and minimize residual pendulations. A fuzzy logic controller imitates the suboptimal
acceleration profile generated by Yamada et al. (1983). Using a scaled model of a crane, they
compared this strategy to the input-shaping strategy they proposed (Yamada et al., 1983). They
found out that, while the two strategies have comparable performance, the fuzzy logic strategy is
more effective in disturbance rejection. Suzuki et al. (1993) proposed a similar approach that,
in addition, is capable of suppressing pendulations along the travel path and changing the cable
length to avoid obstacles along the path. Numerical simulations showed that, at a low travel speed,
this strategy can avoid obstacles and dampen pendulations along the path to less than 1◦ .
Kim and Kang (1993) derived two fuzzy models of the crane dynamics to generate the reference
velocities of the trolley and cable and then employ two fuzzy controllers to track these velocities.
The control strategy is designed to minimize the travel time and payload pendulations while avoid-
ing obstacles along the travel path. Numerical simulations showed that the performance of the
control strategy is comparable to that of a skilled operator.
Itoh et al. (1993, 1995) proposed a control strategy imitating an input-shaping acceleration
profile with one step of acceleration and two steps of deceleration to minimize residual pendulation
and improve the accuracy of trolley positioning at the target point. Under this strategy, the cable
length is held constant throughout the motion. Experiments conducted on an actual crane showed
that this strategy is more effective in payload pendulation suppression than a skilled operator or
an input-shaping strategy.
Nalley and Trabia (1994) proposed a distributed fuzzy logic control strategy to dampen the
pendulations of a bidirectional gantry crane. They used two independent sets of fuzzy inference
engines (FIS). Each FIS set has its own rules: one FIS set tracks the desired position, while the
other corrects for payload oscillations. Each set is composed of two FIS, one for each of the two
perpendicular planes of crane motion. The outputs of the two sets of engines are added to obtain
the total control input to the motors of the trolley and girder. The controller is used to drive
the crane along a path generated by an input-shaping strategy. Simulation results showed good
damping of the pendulations.
Yoon et al. (1995) proposed a fuzzy controller designed to emulate the acceleration profile in
their combined strategy except in the deceleration stage where it emulates a target point position
feedback. Numerical simulation and experimental verification on a scaled crane traveling at low
speed showed that the strategy can suppress residual pendulation and tolerate changes in the cable
length away from the nominal value. However, external disturbances led to oscillations of the trolley
around the target position.
29
Liang and Koh (1997) used a fuzzy logic controller to eliminate residual pendulations at the
target point using a heuristic approach to minimize pendulations. The trolley decelerates as it
approaches the end point, thus producing inertia-induced pendulations. It then accelerates to bring
the trolley directly above the payload when it reaches the maximum point on its upward swing
and thus is temporarily at rest. This procedure is repeated until the payload is at rest. Computer
simulations showed that, even though a few cycles of this procedure can bring the payload to rest,
significant pendulations develop in the process. Méndez et al. (1999) proposed a similar fuzzy
controller employing the position of the trolley and the pendulation angle to eliminate residual
pendulations. Experimental verification showed that the fuzzy controller makes the trolley arrive
at the target position smoothly with no residual pendulation, however to achieve that it approaches
the target point very slowly.
Kimiaghalam et al. (1998a, b) used the model of Sakawa and Shindo (1982) to design a fuzzy
logic controller to move the payload from one side of a fixed obstacle to a known destination on
the other side without collision and in a relatively short time. The controller imitates human
decision-making process. Two designs of the controller are proposed. The first produces torques
as a function of the payload position, while the second generates desired speeds of the trolley and
hoist from which torques are computed using a PD controller. Simulation results showed that the
first design is faster, while the second is relatively slower but yields a smoother path. Using the
second design to achieve higher speeds produces larger pendulations and trolley oscillations around
the target point.
Fuzzy logic strategies are especially hard to tune. The control input is either too high, which
produces cycles of overshoot-undershoot around the target point, or too low, which produces a
very slow and time consuming approach to the target point. Further, all strategies in the literature
restrict crane operation to a predefined path.
Nonlinear Control
Zinober (1979) proposed a sliding-mode control strategy to minimize the travel time, eliminate
residual pendulations, and avoid obstacles along the travel path. The strategy is not a function of
the crane parameters and thus is not sensitive to changes in the cable length and payload mass.
It employs a linear switching function of the system states to switch up and down a bang-bang
controller of the torque applied to the trolley. A low-pass filter is then applied to the control input
to eliminate high-frequency components from the input signal. Numerical simulation showed that
the travel time is %10 longer than the optimal travel time, however the strategy is able to reject
external disturbances without degrading the system performance.
d’Andrea-Novel and Lévine (1989) showed that static state feedback linearization works only
when starting from a stable configuration and moving at slow rates and even then can only ensure
local stability. On the other hand, dynamic state feedback linearization can stabilize the system for
any initial configuration and for higher speeds. They demonstrated this result on a crane traveling
and hoisting along a predefined path at constant speeds. However, their controller is payload-mass
dependent.
Fliess et al. (1991 and 1993) proposed a nonlinear dynamic state feedback technique to linearize
the dynamics of a crane. The technique, dubbed flatness-based control, is applicable to flat systems
30
only, that is systems where the input and state variables can be expressed in terms of the output
variable and their time derivatives in closed form. Thus, based on inverse dynamics analysis of
the nonlinear planar model, they write the system inputs, hoisting and traversing accelerations, in
terms of the system outputs, payload position. Substituting the mathematical representation of
the desired trajectory into these nonlinear expressions produces the required input accelerations
and results in a linear relationship between the state and input variables. A PI controller is then
used to drive the trolley and hoist motors to track these predefined input accelerations. Computer
simulations showed an enhanced performance in the trolley and the payload positioning tasks with
improved operation time. Payload pendulations were reduced to a maximum of 1.7◦ during the
maneuver.
Bourdache-Siguerdidjane (1993 and 1995) applied a variation of the Fliess linearization tech-
nique to an extended model, including the dynamics of the trolley. The nonlinear model is first
linearized by matching it to a version of the model linearized around a single equilibrium. LQR is
then applied to the new linear system to generate the gains of the feedback controller, which drives
the motors and track the reference payload path. Simulation results showed that this strategy
eliminates the payload residual pendulation at the target point.
Maier and Woernle (1997) applied yet another variation of the Fliess linearization technique
to an extended model including the dynamics of the trolley and the hoisting motor. First, inverse
dynamics are used to linearize the model. Then cascaded feedback control using pole-placement
is applied first to the payload position and then to the trolley and hoisting motor positions in the
linear system to counteract the effect of external disturbances and unmodeled forces. Simulation
results show that this application of the Fliess linearization technique is capable of rejecting initial
disturbances to the payload position. This control strategy, however, requires an exceptionally
smooth trajectory to produce practical inputs because the inputs are functions of the trajectory
and its time derivative up to and including the fourth-order derivative.
DeSantis and Krau (1994) proposed a sliding-mode control strategy to stabilize in- and out-
of-plane pendulations of a bidirectional crane. First, two independent, planar state feedback con-
trollers estimate the control input of each motor in order to suppress inertia-induced payload
pendulations. Sliding-mode control is then applied to these estimates to produce the actual con-
trol input of the motors. In numerical simulations, they compared the sliding mode to plain state
feedback control strategies and found out that both approaches are able to stabilize the payload
motion at low trolley and girder speeds. However, sliding-mode control is more effective in coping
with changes in the crane parameters (payload mass) and external disturbances and less effective
in handling feedback delays, as compared to plain state feedback control.
Martindale et al. (1995) proposed two feedback control strategies to track a predefined path.
The first applies backstepping control, and the second adds an adaptive gain matrix to the con-
troller to account for uncertainty in the model parameters (trolley mass, payload mass, and viscous
damping applied to the trolley). Experimental verification using a scaled crane model showed that
both control strategies have the capacity to suppress pendulations at low trolley travel speeds.
It should be noted, however, that backstepping like flatness-based control uses the fourth-order
derivatives of the output, thereby requiring a smooth trajectory. Burg et al. (1996) used the Teel
saturation control approach to design a third feedback controller to minimize payload pendulations.
31
Experimental verification using the scaled crane model showed that, while the controller has a ca-
pacity to suppress pendulations at low trolley speeds, significant payload pendulations develop at
higher speeds.
Cheng and Chen (1996) proposed a control strategy which employs feedback linearization and
time delay control to move a crane along a predefined smooth path and eliminate residual pen-
dulations. Numerical simulation showed that the strategy is able to deliver the payload with no
residual pendulation and minimal transient pendulation, less than 3◦ . Further, unlike pure feedback
linearization, it is robust enough to handle changes in the payload mass and unmodeled forces. Its
performance is also better than the adaptive feedback linearization of d’Andrea-Novel and Bous-
tany (1991a) and Boustany and d’Andrea-Novel (1992) since it does not overshoot the target point.
All three feedback linearization approaches are sensitive to external disturbances, which increase
transient pendulations significantly.
Assuming a flexible cable and a payload mass of the same order of magnitude as that of the
cable, d’Andrea-Novel et al. (1990, 1994) and d’Andrea-Novel and Boustany (1991b) proposed
two embodiments of a feedback controller. In one, the dynamics of the trolley are ignored and a
nonlinear feedback law utilizing the trolley speed is proven to be able to uniformly stabilize the
cable-payload assembly. In the other embodiment, the dynamics of the trolley are included in the
model and a nonlinear feedback law utilizing the force applied to the trolley is proven to be able
to stabilize the cable-payload assembly. However, the stabilization of the system in this case is
not uniform. Computer simulations using a linear feedback law were used to demonstrate both
strategies. The results indicate successful stabilization of the cable-payload assembly.
5.2 Rotary Cranes

Rotary and boom cranes are seldom used to perform planar tasks. As a result, most control strate-
gies proposed for both crane classes are three-dimensional. On the other hand, the few rotary
crane-control strategies available in the literature deal exclusively with inertia-induced payload
pendulations even though base excitations are possible and wind-gust excitations are probable in
the operation of rotary cranes. Further, it should be noted that stabilizing the payload against
the translational motions of a gantry crane is less complicated than stabilizing the payload against
inertia-induced pendulations of the slew motion in rotary and boom cranes, which produces pen-
dulations in both the radial and tangential directions.
Gustafsson (1995) and Abdel-Rahman and Nayfeh (2001) showed that a single planar controller
can not stabilize the payload against slew-induced pendulations in a boom crane. Using an out-of-
plane controller only, Gustafsson (1995) was able to stabilize the out-of-plan motion of the payload,
but could not stabilize the in-plane motion. Thus he concluded that the control effort had to be
applied both in- and out-of-plane to completely stabilize the payload.

Parker et al. (1995a,b) applied various optimization techniques to input-shaping of the acceleration
profile of the jib in order to eliminate residual pendulations for a jib maneuver along a predefined
32
path. Experimental verification showed that significant pendulations develop during the maneuver,
reaching as much as 10◦ for the given maneuver.
Parker et al. (1996) presented another control strategy to drive both the trolley and the jib. It
uses a quasi-static notch filter to eliminate excitations of the cable-payload assembly at the natural
frequency from the slew and travel inputs of the operator. The notch location varies with the length
of the cable to filter out excitations at the current natural frequency of the cable-payload assembly.
The roll-off coefficient for the notch filter is held constant and thus is optimum only at a single cable
length, and the filter characteristics change with changes in the cable length. Experimental results
showed a significant reduction in the payload pendulations through out the maneuver. However the
filtering process imposes delays, as much as 2.5 seconds, between the operator input and the actual
filtered input to the crane. Further, the variable filter characteristics produce variable responses
for the same operator input. Also, the linear nature of the filter limits its effectiveness to low crane
speeds.

Golafshani and Aplevich (1995) used a time-optimal control scheme to generate trajectories of the
jib, the trolley, and the cable length. A sliding-mode controller is then used to track these tra-
jectories. In computer simulations, the time-optimal trajectories produced uncontrolled payload
pendulations. The constraint on the time was therefore relaxed to 110% of the optimal value,
and suboptimal trajectories satisfying a minimum payload swing energy condition were then used
instead of the optimal trajectories. Computer simulations showed that the suboptimal trajecto-
ries reduce the payload pendulations. However, significant pendulations persist throughout the
maneuver and at the end point.
Almousa et al. (2001) used two fuzzy inference engines (FIE), one for the motion in the radial
direction and another for the motion in the tangential direction, to track the position of the payload
and two other FIEs, one for each of the radial and tangential directions, to dampen the payload
pendulations. Each of the FIEs operates independently from the others. Computer simulations
showed that the fuzzy logic controller can limit in-plane and out-of-plane pendulations to small
angles throughout jib and trolley maneuvers. It can also dampen pendulations due to disturbances
to the trolley and jib positions. However, the control strategy imposes an increase in the crane
maneuver time.
Omar and Nayfeh (2001) applied two full state feedback controllers independently to the trolley
travel and the jib slew. This control strategy was effective in damping payload pendulations within
one cycle of oscillation but only when the feedback gains were tuned for a specific payload mass and
cable length. Changes in these parameters led to marked degradation in the controller efficency.
5.3 Boom Cranes

The prediction and control of payload motions in boom cranes is more complicated than it is for
other types of cranes because of the coupling between the payload response to the slew, luff, and
hoisting motions. Further, because of the mobile nature of most boom cranes, it is impossible to
isolate the crane from base excitations. Consequently, any effective control strategy for boom cranes
33
has to account for base excitations. On the other hand, it is not necessary to account for the crane
structure elasticity in the base excitations. Osiński and Wojciech (1998) used nonlinear optimization
to generate an input-shaping profile of either the moment or the velocity of the hoist motor during
the lifting of a load off a cargo ship by a boom crane. To model the plant, they simplified the model
of Kościelny and Wojciech (1994) and Osiński and Wojciech (1994) by assuming an immobile crane
ship and reducing the sea effect on the cargo ship to a harmonic heave motion. They found out
that including the elasticity of the boom had “only a small influence on load motion.”
To provide the control authority necessary to suppress base excitations, some researchers found
it necessary to use a variety of specially constructed add-on actuators, in addition to the boom
slew and luff actuators. This approach, however, makes these control strategies more expensive
and cumbersome to use.

Sakawa et al. (1981) proposed an optimization scheme to generate the torque profile necessary to
transfer a load along a predefined path while minimizing the payload pendulations during transfer
and at the target point. The transfer time is minimized by iteration. The technique was applied to
a simulated model of a boom crane slewing at a constant luff angle while reeling in the cable and
linearized around the payload equilibrium position. Simulation results showed that the payload is
free of residual pendulations at the target point, however payload pendulations develop along the
path and increase as the slewing angle increases.
Takeuchi et al. (1988) proposed an input-shaping strategy to achieve a time-optimal slew motion
only while minimizing the residual pendulations. The strategy uses a slew angle acceleration profile
similar to that generated by Yamada et al. (1983), for gantry cranes, to perform the slew motion
and control the pendulations. Numerical simulation showed that the strategy can suppress out-of-
plane pendulations but not in-plane pendulations, which persist well after the boom comes to a
stop.
Lewis et al. (1998) applied the control strategy proposed by Parker et al. (1996) to boom cranes.
Simulation results showed reductions in both the in-plane and out-of-plane payload pendulations.
However, the control strategy in this case has the same limitations observed when applied to
rotary cranes. Parker et al. (1999a and 1999b) modified this control strategy using a roll-off
coefficient linearized with respect to the natural frequency employed in the notch filter design.
Experimental verification showed an 18 dB reduction in the payload pendulations at the end of
the prescribed maneuver. Simulation results (Lewis et al. 1999) showed that, while the response
of this filter is more consistent at different cable lengths and demonstrates a slight improvement
in the pendulation reduction over that of Parker et al. (1996), it imposes more time delays and
larger changes of amplitude on the operator input. Alternatively, Lewis et al. (1999) modified the
same control strategy using a roll-off coefficient linearized with respect to the forcing (slew and luff
input velocities) in the notch filter. Simulation results did not show a significant difference in the
performance of this filter as compared to that of Parker et al. (1999a,b).
34
Sakawa and Nakazumi (1985) proposed a two-tier control strategy for a crane traversing a predefined
trajectory. An open-loop controller tracks the trajectory the boom travels, while an LQR optimized
state feedback controller employs the slew, luff, and hoisting to eliminate inertia-induced residual
pendulations at the end of the maneuver. Computer simulations showed pendulation angles during
the maneuver of as high as 21.6◦ . Sato and Sakawa (1988) extended the application of this control
strategy to a boom with a flexible jib at the boom tip.
Takeuchi et al. (1988) proposed a fuzzy logic strategy to achieve a time-optimal slew motion
only while minimizing the residual pendulations. The strategy imitates the input-shaping strategy
proposed in the same work. Numerical simulation showed that the fuzzy logic controller is unsuc-
cessful in dampening in-plane pendulations. This is expected since the control effort applied, the
slew motion, is an out-of-plane motion.
Hara et al. (1989) proposed an LQR optimized state feedback controller using the boom
telescopic motion as a control input to control planar payload pendulations due to the telescopic
motion of the boom. A saturation condition is applied to the controller input to the plant to keep it
within available control authority, thus producing a suboptimal controller. In computer simulations
and testing on an actual crane, the control strategy was successful in suppressing pendulation.
Nguyen et al. (1992) proposed a state feedback control strategy, based on a linear planar
model, to hoist and stabilize the payload and position the boom. Two independent controllers
are employed, one to control the boom luff angle and payload pendulation and another to control
the payload hoisting. Experimental verification on a scaled model showed oscillations of the boom
around the reference path and steady-state errors in the boom angle and cable length. On the
other hand, transient pendulations were contained to less than 4◦ .
Souissi and Koivo (1992) proposed a two-tier control strategy to stabilize a boom crane against
inertia-induced payload pendulations. A PID controller tracks a reference trajectory using the slew
and luff of the boom and the reeling/unreeling of the cable, while a PD controller dampens the
payload pendulations. Numerical simulation of the boom performing a predefined luffing-slewing-
luffing maneuver at a constant cable length showed significant payload pendulations, as much as
15◦ , indicating that the PD controller was not effective in damping the pendulations.
Gustafsson (1995) proposed a control strategy employing two independent, in-plane and out-of-
plane, linear position feedback controllers designed based on a partial linearization of the spherical
pendulum model to suppress inertia-induced payload pendulations. Computer-simulation results
showed stable responses for operator commanded slewing rates away from the natural frequency of
the cable-payload assembly and small pendulation angles.
Chin et al. (1998) proposed a nonlinear feedback control scheme to suppress the parametric
instabilities in payload motions due to wave-induced base excitations. They demonstrated analyt-
ically that, introducing a control effort in the form of a harmonic change in the cable length at
the same frequency as the base excitations, can suppress the parametric instability and result in a
smooth response.
Abdel-Rahman and Nayfeh (2000) used cable reeling/unreeling to avoid whirling motions and
three-dimensional responses of the payload when the frequencies of the base excitations approach
the natural frequency of the cable-payload assembly. Using numerical simulation, they demon-
35
Figure 7: A boom crane equipped with the Rider Block Tagline System.
strated that the scheme changes the underlying dynamics of the payload motion, allowing the
primary controller to exert an effort to dampen a planar motion instead of attempting to dampen
a three-dimensional harmonic or chaotic motion.
Using a planar model of a ship-mounted crane, Henry et al. (2001) developed a delayed feed-
back controller. Computer simulations and experimental results showed an effective suppression of
payload pendulations due to in-plane, roll and heave, excitations. Masoud et al. (2000) extended
this approach to the three-dimensional case. In computer simulations and experiments, the con-
troller successfully suppressed payload pendulations due to both in-plane and out-of-plane base
excitations.
5.3.3 Control Strategies Employing Modifications of Crane Structure

The most basic and the only system in practical use of this class is the Rider Block Tagline System
(RBTS), Figure 7. Under this design, a rider block is attached to the cable. Using two pulleys
and taglines, the crane operator can pull the block towards the boom and move it up and down
the cable, thereby decreasing the effective length of the cable and increasing the natural frequency
of the cable-payload assembly (Bostelman and Goodwin, 1999). This process is used to detune
the natural frequency of the cable-payload away from the excitation frequency. In practice, this
approach has proven to be cumbersome; the taglines tend to entangle with the cable or jump their
36
y
C
x M
z L4
B
T2 L2
L3
b L1 T1
A q
m1
L5 l
q+b
O m2
Figure 8: Schematic of a boom crane equipped with Maryland Rigging.
own drum, thereby necessitating re-rigging of the crane.

Lévine et al. (1997) modeled a boom crane equipped with a single pulley and tagline allowing
the operator to pull the cable towards the boom only. The model augments the planar pendulum
equation of motion with two equations of motion describing the torques of the payload and tagline
motors. It assumes viscous damping in both motors and a massless pulley. Lévine et al. (1997)
proposed an open-loop controller where Flatness-based control is used to generate the torque of the
two motors from the payload trajectory described by a smooth curve.
Ott et al. (1996) and Yuan et al. (1997) proposed a system to rig ship-mounted cranes dubbed
“Maryland Rigging,” Figure 8. Under this rigging system, the payload suspension is transformed
from a single spherical pendulum to a double pendulum system. The upper pendulum is a pulley
riding on a cable suspended from two points on the boom; the pulley is thus constrained to move
over an ellipsoid. The lower pendulum is the payload suspended by a cable from the pulley. It
continues to act as a spherical pendulum. A passive control effort is applied to the planar payload
pendulations by applying a brake system to the upper cable as it passes through the pulley. Yuan
et al. (1997) derived a planar model of this rigging and used it to investigate the system response
to periodic and chaotic roll motions. Their simulation results showed that the payload response
grows significantly when the period (dominant period in the chaotic motion case) approaches the
natural frequency of the lower pendulum. The pulley was then used as a brake to apply a constant
and continuous dry (Coulomb) friction. Simulation results showed that a constant friction at a
level equivalent to 10% of the payload weight can reduce planar payload pendulations significantly
37
even in the neighborhood of the natural frequency.
Kimiaghalam et al. (1998d) used an FIS to determine the level of Coulomb friction in the
brake of the Maryland rigging. Simulation results showed that the performance of the active friction
control is comparable to that of the original passive friction control, while the required control effort
is decreased. Kimiaghalam et al. (1999a) proposed another fuzzy logic control approach to dampen
the pendulations in a Maryland rigged crane. The FIS does not apply any friction through the
pulley, instead it changes the upper cable length to eliminate the pendulations. Simulation results
showed that this control strategy can dampen payload pendulations; however, its performance was
inferior to that of the passive controller.
Dadone and van Landingham (1999) proposed a combined control strategy to stabilize the
in-plane motions of the payload under the Maryland Rigging. A fuzzy logic inference engine is
used to determine the level of dry friction in the pulley based on the positions and velocities of
the pulley and payload. Simultaneously, the pulley velocity and acceleration are used to feedback
changes in the pulley cable to eliminate vertical oscillations of the pulley. Simulation results showed
fast damping of the payload motions, however the friction level (control effort) in the pulley, at
40% of the payload weight, was much higher than that employed by Yuan et al. (1997). Further,
the changes in the length of the pulley cable absorbed the pulley vertical oscillations, but also it
introduced horizontal oscillations in the positions of both of the pulley and payload.
Wen et al. (1999) and Kimiaghalam et al. (2000b) proposed a combined feedforward and
feedback control strategy to stabilize planar pendulations in a crane equipped with the Maryland
rigging. The feedforward law is based on the linearized planar equations of motion. It changes
the upper cable length to cancel the effects of the base excitation due to ship roll. The feedback
controller applies LQR feedback control to changes in the upper cable length to add damping to
the system. Simulation results showed that the controller can reduce the payload pendulations
to less than 3◦ for small roll motions. Kimiaghalam et al. (2000a) proposed another combined
feedforward and feedback control strategy. The feedforward controller uses boom luffing to reduce
the excursions of the equilibrium point of the pulley due to ship rolling. The feedback controller
changes the upper cable length to keep the pulley positioned directly above the payload as the ship
rolls. Simulation results showed that the combined control strategy is both effective and fast in
suppressing payload pendulations due to both ship roll and initial disturbances. However, both
feedback controllers assume full authority over the lengths of both segments of the upper cable,
and hence the pulley position. This assumption violates the pulley’s equilibrium equation.
Abdel-Rahman and Nayfeh (2001) showed that the nonlinear coupling between the in-plane
and out-of-plane motions of traditionally rigged cranes continue to exist in Maryland-rigged cranes,
leading to out-of-plane motion due to in-plane excitation and jumps in the in-plane motion, thereby
suggesting that a planar control effort can not stabilize the payload motions. Simulation results
showed that, while the control mechanism was successful in limiting the in-plane motion, it was
unable to control the out-of-plane motion and could not stabilize the overall payload motions.
Balachandran and Li (1997) and Balachandran et al. (1999) used two-dimensional and three-
dimensional (Li and Balachandran, 1999) nonlinear models of ship-mounted boom cranes to design a
nonlinear vibration absorber, a mechanical filter, to control the motions of the pivot point around
which the payload oscillates. The design modifies the boom crane configuration to suspend the
38
payload from a pivot plane, which in turn is suspended under the boom tip. The absorber has both
a passive mode and an active mode employing feedback control. Computer simulations showed that
the absorber can shift bifurcation points arising from the nonlinear dynamics of the cable-payload
assembly and suppress subcritical bifurcations. It also showed that the feedback component of
the filter can attenuate the transient and steady-state payload motions. However, in some filter
designs, suppression of the subcritical bifurcation produced a large resonance-like response around
half the natural frequency.
Iwasaki et al. (1997) and Imazeki et al. (1998) designed an active mass-damper system to
suppress payload pendulations. The system was installed on the sling of a barge-mounted boom
crane. A planar linear model of the crane was used to design a linear feedback controller. The
control effort is the acceleration applied to a 35-ton damping mass riding on the sling. A 132KW
induction motor is used to drive the damping mass. Test results showed that the sling motion was
reduced to 12 to 13 of the uncontrolled motions at the test frequency.
Dadone et al. (2001a) used a variable-geometry-truss (VGT) to suppress in-plane payload
pendulations of a ship-mounted boom crane undergoing roll and heave excitations. Actuators
embedded in the VGT apply an acceleration control effort to a control point on the cable and
constrain it to move along a straight line. LQR and fuzzy logic procedures using the positions and
velocities of the payload and the control point were designed to minimize the control effort based
on a linear, planar model of the modified crane. Simulations showed that the fuzzy logic version of
this control strategy was effective in suppressing payload pendulations throughout the bandwidth
of excitation frequencies, while the LQR version was effective only where the system behavior was
almost linear; that is at low excitation frequencies away from the natural frequencies. Comparison
of the two versions of the strategy showed that the fuzzy logic version applied larger control effort
than the LQR version.
6 Summary
6.1 State-of-the-Art
A significant research effort has been devoted over the past 25 years to the development of control
strategies to improve the efficiency and safety of cranes. Most of this research has been limited to
addressing inertia-induced pendulations in gantry cranes operating along a predefined path. Input-
shaping techniques demonstrated a potential for increasing the hoisting, travel, and traversing
speeds of gantry cranes. However, they are not robust enough to reject external disturbances or
stabilize the payload under base excitations and unmodeled forces in the plant. As a result, they
are not able to relax the operability constraints.
Linear control techniques were added to input-shaping based strategies to alleviate these short-
comings. However, they are not robust enough to allow for variations in the cable length and
payload mass, high operating speeds, and large changes in the trolley and payload positions. Fuzzy
logic and adaptive control techniques were also used to supplement input-shaping techniques. While
hybrid techniques have the potential to produce robust and efficient control strategies, experience
until now shows that the design of control strategies using these techniques is not trivial.
39
Research on boom and rotary cranes is still in the preliminary stages, as compared to research
on gantry cranes where some of the proposed strategies were put to work in the field. However, even
in this case, most of these control mechanisms have proven to be inefficient and thus were “locked
out and abandoned” by the operators (Hubbell et al., 1992). The only exception to that are input-
shaping based controllers. They are used both to operate cranes as well as to perform operator
initiated short steps (so-called inching) used in precision maneuvering near the target point and
around obstacles. The recent availability of variable-speed AC (flux vector) drives made generation
of the bang-bang acceleration profiles, typical of input-shaping strategies, feasible as demonstrated
by Noakes et al. (1993) and Kress et al. (1994) on an actual gantry crane, thus removing one of the
main hurdles to practical implementation of this class of control strategies. However, input-shaping
strategies continue to be hobbled by pendulations along the travel path, relatively slow speeds, and
the fact that most of them are designed for a particular cable length, and thus their behaviors at
other cable lengths are suboptimal. They, also, are not effective in disturbance rejection and are
sensitive to unmodeled forces, such as friction. Fuzzy logic and adaptive control strategies based
on input-shaping have proven to be more effective in disturbance rejection and less sensitive to
unmodeled forces and parameter variations than plain input-shaping.
At the root of the mismatch between the large body of research on crane controllers and those
in practical use is another mismatch between the focus of research effort and operators’ interests.
Most of the research work has been directed to crane automation. On the other hand, operators
in the field are not interested in full automation because of concerns about controller robustness,
safety restrictions, or a workplace that mandates a flexible crane allowing for a variable trajectory
from one operation cycle to the next.
6.2 Modeling
The complexities of dealing with a nonlinear model of the plant drive most of the work on crane
control to make-do with linearized approximations of the model. This simplification comes at a price
of reduced controller robustness. Burg et al. (1996) reported that the neglected nonlinearities in a
state-space model of a gantry crane may significantly impact the performance of a linear controller.
Their computer simulations show that a linear controller provides acceptable performance only
within a fixed operating range of small pendulation angles around the equilibrium point of the
payload. As a result, there has been an increasing interest in the design of crane control strategies
based on nonlinear crane models.
As demonstrated in the analysis of the spherical pendulum model and indicated by Gustafsson
(1995) and Abdel-Rahman and Nayfeh (2001), the in-plane and out-of-plane motions of the payload
are coupled. Motion in one plane interacts with and induces motion in the other plane. Whenever
large pendulations build up in-plane, any out-of-plane disturbance can give rise to out-of-plane
pendulations, and thus the planar model breaks down. Therefore, three-dimensional nonlinear
models, accounting for both geometric and kinetic nonlinearities in the payload motion, must be
adopted as the gold standard of the field.
Simplified models are still a good approximation, they are legitimate and useful under special
loading conditions. A planar model, equation (45) or (47), can be used to model a gantry crane as
long as the payload and trolley are not subject to any large and/or out-of-plane excitations (wind
40
gusts, load imbalance, or girder deviation) and the safety threshold in the particular application is
low. The nonlinear planar model, equation (47), however is superior to the linear model, equation
(45), since it is consistent and valid for small but finite pendulation amplitudes. Linear models
can also be used as long as the pendulation angles are small and the frequencies of all present base
excitations are away from the natural frequency of the cable-payload assembly.
A recent development in gantry cranes has been the introduction of multiple-point payload
suspension. This design allows for enhanced stiffness of the cable-payload system and thus more
resistance to pendulations. Further, new control schemes have been introduced to use these multiple
points of suspension to dampen payload pendulation using the differential between the tension forces
in the various suspension cables (Champion, 1989 and Hubbell et al., 1992). This control approach
mandates a model to account for the payload motion as a rigid body rather than a point mass.
As of now, we are not aware of any such model in the literature. Further, this class of control
strategies is not effective in disturbance rejection. Wind gusts and/or initial disturbances induce
payload pendulations, which cannot be stabilized with these controllers.
6.3 Control
Despite the numerous crane control strategies in the literature, very few designs have proven ap-
plicable in practice. One reason is that most of these strategies were not designed with a crane in
mind. In many cases, a crane was being used as a test-bed for a novel control concept. We propose
that a successful crane control strategy has to meet the following criteria:
- The advantages of using a crane over a robotic arm or a multiple-winch crane are flexibility, cost
efficiency, and simplicity of design and operation. Any crane control strategy has to maintain these
advantages. An appropriate guideline is for control strategies to utilize available actuators within
their existing power limitations.
- Most cranes are manually operated to maximize the flexibility, robustness, and safety of crane
operation. A control strategy designed for this class of cranes will have to be transparent to the
operator, thus precluding any control strategies that result in significant delays in response to the
operator input.
- The use of automated cranes is mostly limited to mines, factories, and similar installation where
a material handling system is required to reproduce a set sequence of motions. Automating a crane
is only feasible where the work space is well structured with constant starting and target points,
fixed positions of obstacles along the crane path, and a low safety threshold. Further, the control
strategy employed in automating the crane will have to be robust enough to handle a wide range
of cable lengths and payload weights.
- The light damping of cranes means that control strategies must be designed to take into account
both stationary and transient responses.
- The control strategy has to apply control effort in two perpendicular planes. Planar controllers
can only be used in the absence of nonplanar excitations. Even in a unidirectional gantry crane,
such a condition is not guaranteed due to load imbalance, girder deviation, and wind gusts. The
only exception under which planar crane control is safe is where the three-dimensional motion is
modeled and a safety mechanism/controller is incorporated to bring the payload motion back from
whirling to a planar motion.
41
For the same class of models, the performance of nonlinear controllers shows dramatic improve-
ment over that of linear controllers.
Acknowledgment
This work was supported by the Office of Naval Research under Grant No. N00014-96-1-1123
(MURI).
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Dynamics and Control of Cranes: A Review

EIHAB M. ABDEL-RAHMAN, ALI H. NAYFEH, and ZIYAD N. MASOUD

Wave-induced motions of the platform (a crane-ship or a semi-submersible) may contain signiﬁcant

2.1 Distributed-Mass Models

τ (s) = mg + sρg (2)

and m is the payload mass. The boundary conditions are

2.2 Lumped-Mass Models

3 The Reduced Model

(x − ξ)2 + (y − ζ)2 + (z − η)2 = (ℓ + r)2 (7)

L = 21 m(ẋ2 + ẏ 2 + ż 2 ) − mgz − 21 ce r 2 (8)

where ce is the longitudinal stiﬀness of the cable.

x−ξ x−ξ 2 x−ξ

(x − ξ)2 + (y − ζ)2 [(x − ξ)2 + (y − ζ)2 ]2

3.1 Approximate Solution of the Reduced Model

ξ = ǫ3 u0 cos Ω1 t, ζ = ǫ3 v0 cos Ω1 t, and η = ǫ2 w0 cos Ω2 t (28)

with a primary excitation frequency

x(t; ǫ) ≃ ǫx1 (T0 , T2 ) + ǫ3 x2 (T0 , T2 ) (31)

x1 = A1 (T2 )eiωT0 + cc and y1 = A2 (T2 )eiωT0 + cc (37)

3.2 Reduced Linear Model

3.3 Reduced Planar Model

T = mg cos θ + mℓθ̇ 2 − mξ̈ sin θ (50)

4.2 Rotary Cranes

4.3 Boom Cranes

5.1 Gantry Cranes

5.1.1 Open-Loop Techniques

5.1.2 Closed-Loop Techniques

Fuzzy Logic Control

5.2 Rotary Cranes

5.2.1 Open-Loop Techniques

5.2.2 Closed-Loop Techniques

5.3 Boom Cranes

5.3.1 Open-Loop Techniques

5.3.3 Control Strategies Employing Modifications of Crane Structure

Figure 8: Schematic of a boom crane equipped with Maryland Rigging.

own drum, thereby necessitating re-rigging of the crane.

Bourdache-Siguerdidjane, H., 1993, “Application of Fliess linearization to multiple output non-

Bourdache-Siguerdidjane, H., 1995, “Optimal control of a container crane by Fliess linearization,”

Ebeid, A. M., Moustafa, K. A. F., and Emara-Shabaik, H. E., 1992,“Electromechanical modelling

Greenwood, D. T., 1988, Principles of Dynamics, Prentice-Hall, New Jersey.

Nayfeh, A. H., 1981, Introduction to Perturbation Techniques, Wiley, New York.

Zinober, A. S. I. and Yang, X. H., 1988, “Continuous self-adaptive control of a time-varying

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