Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

44, 4, pp. 929-948, Warsaw 2006

DYNAMICS AND CONTROL OF ROTARY CRANES

EXECUTING A LOAD PRESCRIBED MOTION

Wojciech Blajer

Krzysztof Kołodziejczyk

Institute of Applied Mechanics, Technical University of Radom

e-mail: w.blajer@pr.radom.pl; kkolodziejczyk@multifox.pl

Manipulating payloads with rotary cranes is challenging due to the
underactuated nature of a system in which the number of control in-
puts/outputs is smaller than the number of degrees-of-freedom. In this
paper, the outputs (specified in time load coordinates) lead to servo-
constraints on the system. A specific methodology is then developed
to solve the arising inverse dynamics problem. Governing equations are
derived as a set of index three differential-algebraic equations in state
variables and control inputs. An effective numerical code for solving the
equations, based on the backwardEulermethod, is proposed.A feedfor-
ward control law obtained this way is then enhanced by a closed-loop
control strategywith feedbackof actual errors in the loadposition topro-
vide stable tracking of the required reference load trajectory in presence
of perturbations. Some results of numerical simulations are provided.

Key words: rotary cranes, inverse dynamics control, servo-constraints

1. Introduction

Cranes are widely used in transportation and construction. In the industrial
practice, they are predominantly operated manually – the operator actuates
different joints by joysticks and/or buttons, so that to move the load from its
initial position to its desired final destination in the working space along a
trajectory, avoiding obstacles and sway. Even though almost the same paths
are often repeated, which allows the operator to ’learn’ the maneuver, the
cycle time is usually relatively large since the operator has to perform the
maneuvers slowly in order to avoid inertia-induced excitations, and a consi-
derable percentage of the time is spent on maneuvering the load close to the
target point. The latter is usually a trial-and-error process, based on feedback
provided by the operator’s own vision and assessment, and/or hand signals or



930 W. Blajer, K. Kołodziejczyk

radio communication from a designated assistant at the work zone (Rosenfeld
and Shapira, 1998). Automated cranes, after being ’taught’ a safe and efficient
route between fixed locations of the source and the target, have a potential
to play back that route much faster and more accurately than the repeated
manual cycles.

High potential of rationalization offered by automatic control systems sti-
mulatedan increasing interest andsubstantial progress in researchonmodeling
and control synthesis of cranes. A good review of the recent developments in
the field is provided byAbdel-Rahman et al. (2003). Lumped-massmodels are
most often used, in which the hoisting line is treated as a massless cable, the
payload is lumped with a hook and modeled as a point mass, and the cable-
hook-payload assembly ismodeled as a spherical pendulum, see e.g. Lee (1998)
and Ghigliazza and Holmes (2002). Automatic navigation of cranes requires
then planning of load motion aimed at minimizing the traveling time under
some kinematic limitations such as maximum velocities/accelerations/jerks
and/or frequent requirement for vanishing sway during the transfer process
and at the target position (Aschemann, 2002).

Cranes belong to a class of underactuated/underconstrained systems,
i.e. such controlled mechanical systems in which the number of control in-
puts/outputs is smaller than the number of degrees of freedom. Control of
such systems is a challenging task which has been investigated for a long time
(Spong, 1997). In the case of cranes, one of the consequences is that due to
the rope flexibility, the undesirable load swing cannot be directly actuated by
the available control, and advanced feedback control techniques are needed to
suppress the swing and assure precise load positioning (Abdel-Rahman et al.,
2003).

In this paper, the problem of dynamics and control of a rotary crane exe-
cuting a load prescribedmotion is viewed from the perspective of constrained
motion. The control outputs, expressed in terms of the system states, are tre-
ated as servo-constraints on the system (Kirgeov, 1967; Blajer andKołodziej-
czyk, 2004; Bajodah et al., 2005). It is noticed, however, that servo-constraints
differ frompassive constraints in several aspects.Mainly, they are enforced by
means of control forces which may have any directions with respect to the
manifold of servo-constraints, and in the extreme may be tangent. Such a si-
tuation arises in the load trajectory tracking control of rotary cranes, and a
specific methodologymust be developed to solve the arising ’singular’ inverse
dynamics problem.After some introductory definitions, a theoretical backgro-
und for the modeling of the partly specified/actuated motion is given. The
initial governing equations arise as index-fiveDifferential-Algebraic Equations
(DAEs), and are then transformed to a more tractable index-three form. An
effective numerical code for solving the resultant DAEs is used, based on the
backward Euler method. A feedforward control law obtained this way is then



Dynamics and control of rotary cranes... 931

enhanced by a closed-loop control strategy with feedback of actual errors in
the load position to provide stable tracking of the required reference load tra-
jectory in presence of perturbations. Some results of numerical simulations are
provided.

2. Modeling preliminaries

A rotary crane model seen in Fig.1 is considered. This is a five-degree-of-
freedom system, n = 5, whose position is described by q = [ϕ,s,l,θ1,θ2]

⊤,
where ϕ is the angle of rotation of the girder bridge, s describes the trolley
position on the girder, l is the hoisting rope length, and θ1 and θ2 are the
swing angles seen in the figure. The performance goal is a desired motion of
the load, i.e. the control outputs are specified in time by the load coordinates
x, y and z in the inertial reference frame XYZ, γd(t)= [xd(t),yd(t),zd(t)]

⊤.
The control inputs are the torque Mb regulating the bridge rotation angle ϕ,
the force F actuating the trolley position s on the bridge, and the winch
torque Mw changing the rope length l, u = [Mb,F,Mw]

⊤. In this meaning,
ϕ, s and l can be regarded as controlled coordinates, while θ1 and θ2 can be
called uncontrolled coordinates.

Fig. 1. Rotary cranemodel

The inverse dynamics control problemconsidered in this paper is following:
given a prescribedmotion of amechanical system, determine the control inputs
that force the system to complete the prescribedmotion. For the case study, the
number of control inputs u is equal to the number of outputs γ,m=3, and
it is smaller than the number of degrees of freedom of the system, m<n.We
deal thus with an underactuated system in a partly specified motion (Blajer
andKołodziejczyk, 2004). Prior todeveloping the solution to the above control



932 W. Blajer, K. Kołodziejczyk

problem, a mathematical model of the crane in the partly specified motion
needs to be formulated.
An effectivemethod for derivation of dynamic equations of the crane is the

projectionmethoddescribed byBlajer (2001). The starting point are dynamic
equations in dependent coordinates p= [ϕ,xt,yt,zt,α,x,y,z]

⊤, where ϕ, x,
y and z are as defined above, xt, yt and zt are the trolley coordinates in the
XYZ frame, and α is the winch rotation angle. Treating initially the bridge,
trolley, winch (only its rotation) and load unconstrained fromeach other, their
dynamic equations are simply

Jbϕ̈=−Dϕϕ̇+Mb Jwα̈=−Dαα̇+Mw

mt







ẍt
ÿt
z̈t






=







−Dsṡcosϕ
−Dsṡsinϕ
−mtg






+







cosϕ
sinϕ
0






F m







ẍ
ÿ
z̈






=







0
0
−mg







(2.1)

where Jb and Jw are the moments of inertia of the bridge and winch relative
to their axes of rotation, mt and m are the trolley and load masses, g is the
acceleration of gravity, and Dϕ,Dα and Ds are the respective viscousdamping
coefficients. The initial dynamic equations in p can then be gathered in the
following generic matrix form

Mp̈=f−B
⊤
u (2.2)

where M is the (diagonal) 8× 8 generalized mass matrix, f is the 8-vector

of generalized applied forces, both related to p, and B
⊤
is the 8×3 control

matrix (fc =−B
⊤
u is the 8-vector of generalized control forces related to p).

Explicit forms of M, f and B
⊤
are easy to deduce from equation (2.1).

In order to retrieve the bridge-trolley and winch-load connections, three
constraint equations Φ(p) = 0 need to be imposed on separated subsystems,
i.e.

Φ=







−xt sinϕ+ytcosϕ
−zt

√

(x−xt)
2+(y−yt)

2+(z−zt)
2− l0−rwα






=0 (2.3)

where l0 is some initial rope length and rw is the winch radius. Dynamic
equation (2.2) is modified then to the constraint reaction-induced form

Mp̈=f−B
⊤
u−C

⊤
λ (2.4)

where C(p)= ∂Φ/∂p is the 3×8 constraint matrix

C=









C11 −sinϕ cosϕ 0 0 0 0 0
0 0 0 −1 0 0 0 0

0 −
x−xt
l

−
y−yt
l

−
x−zt
l

−rw −C32 −C33 −C34









(2.5)



Dynamics and control of rotary cranes... 933

where

C11 =−xtcosϕ−yt sinϕ l=
√

(x−xt)2+(y−yt)2+(z−zt)2

The Lagrange multipliers λ = [λ1,λ2,λ3]
⊤ stand for the side and vertical

reactions between the trolley and bridge and for the tension force in the rope,
respectively.

It my be worth noting that the constraint equations given implicitly in
equation (2.3) as Φ(p)=0 (Schiehlen, 1997; Blajer, 2001) will not in fact be
involved in the subsequent formulation of the dynamic equations in q, and
are needed only if the determination of control reactions λ is required for any
good reason (for evaluation of Coulomb friction effects between the trolley
and bridge, for example). Instead, the formulation of dynamic equations in q
uses the constraint equations given explicitly. At the position, velocity and
acceleration levels these explicit constraint equations are

p= g(q) ṗ=D(q)q̇ p̈=D(q)q̈+η(q, q̇) (2.6)

where D = ∂g/∂q and η = Ḋq̇. In principle, the implicit constraint equ-
ations are satisfied identically when expressed in q, i.e. Φ[g(q)] ≡ 0 and

Φ̇=CDq̇≡0. Since q̇ are independent, the latter yieldsCD=0⇔D
⊤
C
⊤
=0,

and D is an orthogonal complement matrix to the constraint matrix C; see
e.g. Blajer (2001) for more details. For the case at hand, relationship (2.6)1
and the 8×5 matrix D are





























ϕ
xt
yt
zt
α
xl
yl
zl





























=





























ϕ
scosϕ
ssinϕ
0

(l− l0)/rw
(s+ lsinθ2)cosϕ+ lcosθ2 sinθ1 sinϕ
(s+ lsinθ2)sinϕ− lcosθ2 sinθ1cosϕ

−lcosθ2cosθ1





























(2.7)

D=





























1 0 0 0 0
−ssinϕ cosϕ 0 0 0
scosϕ sinϕ 0 0 0
0 0 0 0 0
0 0 1/rw 0 0
D61 cosϕ D63 D64 D65
D71 sinϕ D73 D74 D75
0 0 D83 D84 D85































934 W. Blajer, K. Kołodziejczyk

with

D61 =−(s+ lsinθ2)sinϕ+ lcosθ2 sinθ1cosϕ

D63 =sinθ2cosϕ+cosθ2 sinθ1 sinϕ

D64 = lcosθ2cosθ1 sinϕ

D65 = lcosθ2cosϕ− lsinθ2 sinθ1 sinϕ

D71 =(s+ lsinθ2)cosϕ+ lcosθ2 sinθ1 sinϕ

D73 =sinθ2 sinϕ− cosθ2 sinθ1cosϕ

D74 =−lcosθ2cosθ1cosϕ

D75 = lcosθ2 sinϕ+ lsinθ2 sinθ1cosϕ

D84 = lcosθ2 sinθ1

D85 = lsinθ2cosθ1

Due to complexity of the explicit formof the 8-vector η, itwill not be reported
here for shortness.
Having explicit constraint equations (2.6) defined, the n = 5 dynamic

equations of the rotary crane in q are

M(q)q̈+d(q, q̇)=f(q, q̇)−B⊤u (2.8)

where M = D
⊤
MD is the 5× 5 generalized mass matrix related to q, and

d=D
⊤
Mη, f =D

⊤
f and fu =−B

⊤u (with B⊤ =D
⊤
B) are the 5-vectors

of generalized dynamic, applied and control forces related to q. In applica-
tions, dynamic equations (2.8) need not to be obtained in an analytical form.
Symbolic computermanipulations can be used for its derivation. Here, wewill
limit ourselves to symbolic form (2.8) of these equations, and demonstrate
only the control distributionmatrix Bwhichwill be of some use in the sequel

B=







1 0 0 0 0
0 1 0 0 0
0 0 1/rw 0 0






(2.9)

It is worth noting that only M and h from initial dynamic equations
(2.2), and D and η from explicit constraint equations (2.6)2,3 are needed
to derive dynamic equations (2.8). The constraint equations in the implicit
form Φ(p) = 0 and the subsequent constraint matrix C are needed only if
the constraint reactions λ are required. The constraint reactions can be then
obtained from (Blajer, 2001)

λ(q, q̇)= (CM
−1
C
⊤
)−1C[M

−1
(f−B

⊤
u)−η] (2.10)

where C[g(q)] should be used.



Dynamics and control of rotary cranes... 935

3. Servo-constraint equations

The performance goal of the crane is to execute a desired load trajectory, i.e.
m=3 control outputs are time-specified load coordinates

γd(t)= [xd(t),yd(t),zd(t)]
⊤

The outputs, expressed in terms of the coordinates q as

c(q, t)=Φ(q)−γd(t)=0 (3.1)

canbe treated as constraints on the system, called servo-constraints (Kirgetov,
1967; Bajodah et al., 2005), also called active, program or control constraints
(Rosen, 1999; Blajer and Kołodziejczyk, 2004) as distinct from passive (con-
tact) constraints in the classical sense. After twice differentiating the initial
constraint equations with respect to time, the constraint conditions at the
acceleration level are

c̈=C(q)q̈−ξ(q, q̇, t)=0 (3.2)

where C = ∂Φ/∂q is the 3 × 5 matrix of the servo-constraints, and
ξ = γ̈d − Ċq̇ is the 3-vector of constraint specified accelerations. For the
case at hand, constraint equations (3.1) and the constraint matrix C defined
in equation (3.2) are

c=







(s+ lsinθ2)cosϕ+ lcosθ2 sinθ1 sinϕ
(s+ lsinθ2)sinϕ− lcosθ2 sinθ1cosϕ

−lcosθ2cosθ1






−







xd(t)
yd(t)
zd(t)






=0

(3.3)

C=







D61 cosϕ D63 D64 D65
D71 sinϕ D73 D74 D75
0 0 D83 D84 D85







where D61, . . . ,D85 are as defined in equation (2.7). Analytical expressions for
ξ are rather complex and will not be reported here for shortness.
Equation (3.1) ismathematically equivalent to m=3 rheonomic (or time-

dependent) holonomic constraints on the system of n=5 degrees of freedom.
Theresemblanceof the load trajectory tracking control problemto theconstra-
inedmotion casemay however bemisleading. Assumed c(q, t)=0 represents
passive constraints, the generalized actuation force fu = −B

⊤u in dynamic
equation (2.8) would be replaced by the generalized constraint reaction for-
ce fc = −C

⊤λ, where C is the constraint matrix defined in equation (3.2).
While the reactions of ideal passive constraints are by assumption orthogo-
nal to the instantaneous manifold of passive constraints in the configuration



936 W. Blajer, K. Kołodziejczyk

space related to q (Blajer, 2001), the actuating forces may have arbitrary di-
rections with respect to the instantaneous servo-constraint manifold, and in
the extreme (some of them) may be tangent. In the latter case, qualitatively,
not all of the desired outputs can directly be actuated by the control inputs.
A measure of the ’control singularity’ is the deficiency in rank of the m×m
matrix CM−1B⊤, which represents the inner product of the constrained and
controlled subspaces in the n-space of crane velocities. For a more detailed
discussion on the problem of realization of servo-constraints and the relevant
geometrical interpretations, the reader is referred toBlajer andKołodziejczyk
(2004).

In the case at hand, the realization of the three task requirements γd(t) is
enforced by the tension force in the rope, which is the sole attainable control-
induced enforcement on the load, rank(CM−1B⊤) = 1. Qualitatively, only
one task requirement can thus directly (in the orthogonal way) be actuated by
the available control, while the realization of two other requirements must be
tangent.Thevalueof the tension force canberegulatedmainlyby Mw, andthe
direct influence of F and Mb on the tension force is vanishing since of usually
small values of θ1 and θ2 rope inclination angles. On the other hand, F and
Mb can influence the load trajectory realization indirectly by actuating the
rope suspension point, which yields appropriate changes in the swing angles
θ1 and θ2, and causes that the tension force in the rope can change its space
orientation and produce required reactions on the load in all three directions.
The appropriate changes in θ1 and θ2, due to the tangent realization of servo-
constraints, can then be viewed as two additional restrictions on the crane
configuration, and in this sense the five-degree-of-freedom system is ’fully’
specified by three servo-constraints (3.1) and can explicitly be actuated by
three control inputs u.

4. Governing equations

The initial governing equations of crane motion in the case of the partly
specified motion are formed by n = 5 kinematic relations q̇ = v, n dy-
namic equations (2.8) rearranged with the use of the kinematic relation to
M(q)v̇ + d(q,v) = f(q,v)−B⊤u, and m = 3 servo-constraint equations
(3.1), which state together n+n+m=5+5+3=13 Differential-Algebraic
Equations (DAEs) in the same number of 2n state variables q and v, and
m control variables u. The problem with the DAEs formulated this way is
that their index is equal to five (Gear and Petzold, 1984; Ascher and Petzold,
1999), which is a measure of singularity/complexity of the DAE system and
determines difficulty in numerical treatment. Blajer andKołodziejczyk (2004)



Dynamics and control of rotary cranes... 937

developed then a scheme for transforming the initial index-five DAEs to an
equivalent set of numerically more tractable index-three DAEs, i.e.

q̇=v

D
⊤
Mv̇=D⊤(f −d)−D⊤B⊤u

0=CM−1(f−d)−CM−1B⊤u−ξ

0=Φ(q)−γd

⇐⇒

q̇=v

H(q)v̇=h(q, q̇,u, t)

0= b(q, q̇,u, t)

0= c(q, t)

(4.1a)

(4.1b)

(4.1c)

(4.1d)

While equations (4.1a) and (4.1d) are evident, equations (4.1b) and (4.1c)
follow from the projection of dynamic equations (2.8) respectively onto com-
plementary unconstrained (unspecified) and constrained (specified) subspaces
in the n-space related to q̇. The constrained m-subspace is defined by the
m×n (here 3×5) constraint matrix C introduced in equation (3.2), and the
unconstrained k-subspace (k = n−m) is defined by the n×k (here 5×2)
matrix D being an orthogonal complement to C, i.e. CD= 0⇔D⊤C⊤ = 0.
Thematrix D satisfying this condition can sometimes be guessed (usually for
simple cases only) or determined following a numerically oriented code like the
scheme followed fromthe coordinate partitioningmethodproposedbyWehage
andHaug (1982). Namely, assumed C is ofmaximal row-rank, rank(C)=m,
it can always be factorized to C= [U,W ], so that U and W are respectively
the m×k and m×mmatrices, and det(W) 6=0.The orthogonal complement
D to C can then be found as

D=

[

I

−W−1U

]

(4.2)

where I is the k×k (here 2×2) identity matrix.

Projection (4.1b) onto the unconstrained subspace, D⊤Mv̇ + D⊤d =
= D⊤f −D⊤B⊤u, states for k = 2 differential equations. Projection (4.1c)
onto the constrained subspace, CM−1(f−d)−CM−1B⊤u−ξ=0, represents
then m = 3 algebraic equations in the system states q and v, and control
inputs u. Due to rank(CM−1B⊤) = 1, these three algebraic equations im-
pose only one independent condition on u, however, and k = m− p = 2
additional restrictions on the crane motion, supplementary to m original re-
quirements (2.10). Therefore, as argued by Blajer and Kolodziejczyk (2004),
due to the mixed orthogonal-tangent realization of servo-constraints (2.10),
the total number of the original and supplementary motion specifications is
m+k=n=5, and thus, in this indirect way, the motion is fully specified.



938 W. Blajer, K. Kołodziejczyk

5. Dynamic analysis and synthesis of control

Similarly to the initialDAEsmentioned in Section 4, governing equations (4.1)
represent n+k+m+m=5+2+3+3=13DAEs in the samenumber of state
and control variables q, v and u, and the index of the DAEs is now equal to
three. The solution to DAEs (4.1) are variations in time of state variables for
a crane executing a load prescribedmotion, qd(t) and vd(t), and the control
ud(t) that ensures the realization of this motion.

A simple and effective code for the solution of DAEs (4.1) proposed by
Blajer and Kołodziejczyk (2004) uses the Euler backward differentiation ap-
proximationmethod, inwhich the derivatives q̇ and v̇ at time tn+1 = tn+∆t
are approximated by their backward differences, respectively (qn+1−qn)/∆t
and (vn+1−vn)/∆t, where ∆t is the integration time step. Then, given qn
and vn at time tn (note that un is not involved), the values qn+1, vn+1
and un+1 at time tn+1 = tn+∆t can be found as a solution to the following
nonlinear algebraic equations

qn+1−qn
∆t

−vn+1 =0

H(qn+1)
vn+1−vn
∆t

−h(vn+1,qn+1,un+1, tn+1)=0
(5.1)

b(vn+1,qn+1,un+1, tn+1)=0

c(qn+1, tn+1)=0

and in this way the solution can be advanced from time tn to tn+1 = tn+∆t.

In order to improve accuracy of the numerical solution, the rough Euler
scheme can possibly be replaced by a higher order backward difference appro-
ximationmethod (Gear andPetzold, 1984; Ascher andPetzold, 1999). Itmay
beworth noting however that, becouse the original servo-constraint equations
c(q, t) = 0 are involved in the governing equations, the solution obtained is
free from the constraint violation problem, and the truncation errors do not
accumulate in time. More strictly, as said before, m= 3 algebraic equations
b(v,q,u, t) = 0 impose p = 1 independent condition on u, and m− p = 2
conditions on the cranemotion, and in particular on its position q. Therefore,
the m+m=6 algebraic equations b(v,q,u, t)=0 and c(q, t)=0 represent
m−p+m=5explicit equations on thefive entries of q, and the solution qd(t)
is determined only with an accuracy error of solving algebraic equations (5.1).
Then, only vd(t) and ud(t) are determined with an error followed from the
roughbackward differencemethod,which donot accumulate as they are based
on the numerically exact solutions qd(t). The proposed simple code leads thus
to reasonable and stable solutions.



Dynamics and control of rotary cranes... 939

The inverse simulation control ud(t) obtained as a solution to DAEs (4.1)
can be used as a feedforward control for a crane executing a prescribed load
motion. It should be then enhanced by a feedback control in order to provide
stable tracking of the load trajectory in presence of external perturbations
and/or modeling inconsistencies. One possibility is to introduce, instead of
c̈=0, a stabilized formof theprogramconstraint equations at theacceleration
level, c̈+αċ+βc+γ

∫

cdt=0, where α, β and γ are the gain values. Then,
after replacing the requirement 0 = b(q,v,u, t) in equations (4.1c) by its
stabilized form

0=CM−1(f−d)−CM−1B⊤u−ξ+αċ+βc+γ

∫

c dt= bstab(v,q,u, t) (5.2)

a hybrid control can be synthesized from thus modified DAEs (4.1) by using
code (5.1). The idea for the crane control with the use of the scheme is shown
in Fig.2.

Fig. 2. Hybrid control scheme

6. Numerical simulations

6.1. Load trajectory modeling

During operation, a duty cycle of a crane consists of moving a load from
its initial position to its desired final destination in its working space along
a trajectory, avoiding obstacles and sway. This requires planning of motion
for the given load position. In the present formulation, the load coordina-
tes in the fixed Cartesian frame XYZ (Fig.1) need to be specified in time,
γd(t) = [xd(t),yd(t),zd(t)]

⊤, where xd(t), yd(t) and zd(t) are appropriate-
ly smooth functions of time. A ”rest-to-rest” maneuver is usually required
(Rosenfeld and Shapira, 1998; Abdel-Rahman et al., 2003), i.e. γ̇d(t0) =
= γ̇d(tf)= 0 and γ̈d(t0)= γ̈d(tf)= 0, where t0 and tf denote the initial and



940 W. Blajer, K. Kołodziejczyk

final times, respectively. One possibility is to synchronize the outputs using a
reference function s(t)

γd(t)=γ0+(γf −γ0)s(t) (6.1)

where t∈ 〈t0, tf〉, and γ0 = [x0,y0,z0]
⊤ and γf = [xf,yf,zf]

⊤ are the start
and target load positions at the time t0 and tf. Having s(t) and its time
derivatives, γ̇d(t) and γ̈d(t) used in themathematical model can be found as
γ̇d(t) = (γf −γ0)ṡ(t) and γ̈d(t)= (γf −γ0)s̈(t). For longer traveling distan-
ces, the maneuver is often divided into acceleration (I), steady velocity (II)
and deceleration (III) phases. The durations of phases I and III are usually
determined by some maximum acceleration/jerk limitations, while the dura-
tion of phase II may be consequent to the maximum load velocity limitation.
Following the idea of Aschemann (2002), the function s(t) used in this paper
is

sI(t)=
1

τ− τ0

(

−
5t8

2τ70
+
10t7

τ60
−
14t6

τ50
+
7t5

τ40

)

sII(t)=
1

τ− τ0

(

t−
τ0
2

)

(6.2)

sIII(t)= 1+
1

τ− τ0

(5(τ − t)8

2τ70
−
10(τ − t)7

τ60
+
14(τ − t)6

τ50
−
7(τ − t)5

τ40

)

where τ = tf−t0 and τ0 is the acceleration/deceleration time.Given τ =20s
and τ0 =5s, reference function (6.2) and their time derivatives are illustrated
in Fig.3.

Fig. 3. Reference function and its time-derivatives

Synchronized time functions (6.1) for the reference load coordinates result
in a straight line trajectory from the start to target positions. Since the tra-
jectory line cannot cross or pass too close to the tower, for tasks in which



Dynamics and control of rotary cranes... 941

the start and target load positions are at the opposite sides of the tower,
the shape function s(t) should rather specify the load cylindrical coordinates,
i.e. γd(t) = γ0 +(γf −γ0)s(t), where γ = [r,φ,z]

⊤, and r and φ are the
polar coordinates. The required γd(t), γ̇d(t) and γ̈d(t) can be then determi-
ned following xd(t) = rd(t)cosφd(t) and yd(t) = rd(t)sinφd(t) and the time
derivatives of these relations.

6.2. Inverse simulation study

The data used in computations were: m = 100kg, mt = 10kg,
Jw =0.1kgm

2, rw =0.1m, and Jb =480kgm
2. The task requirementwas to

move the load fromthe start position to the target position, γ0 = [5,0,−5]
⊤m

and γf = [−2,2,−2]
⊤m. Two ”rest-to-rest” maneuvers were considered: ma-

neuver 1 along a curvilinear trajectory followed from imposing reference func-
tion (6.2) on cylindrical coordinates as described in Section 6.1, and maneu-
ver 2 along a straight line defined in equation (6.1). After applying τ = 20s
and τ0 = 5s, the obtained reference load trajectories are seen in Fig.4. So-
me results of numerical simulations are then reported in Fig.5, obtained for
∆t = 0.01s and with no damping in the system (Dϕ = Dα = 0, Ds = 0).
While the variations of l(t) and Mw(t) are almost the same in the two ma-
neuvers, all the other motion and control characteristics are qualitatively and
quantitatively different.

Fig. 4. Maneuvers 1 and 2

6.3. Robustness of hybrid control in perturbed motion

The robustness of the hybrid control described in equation (5.2) and illu-
strated in Fig.2 was first tested by applying the inconsistent starting position
at t0 with the load placed 0.5m above its reference position.Moreover, in the
mathematical model used for the direct dynamic simulation, damping related
to s and l motions was added, respectively Ds =75Ns/m and Dα =15Ns,
not considered in themodel used for the determination of control. Themotion
disturbed this way was then stabilized along the reference motion by using



942 W. Blajer, K. Kołodziejczyk

Fig. 5. Cranemotion and control in maneuvers 1 and 2

the hybrid control. The gain values used assure the critical damping for a PID
scheme (Ostermayer, 1990), i.e. α2 =8β, 32γ =αβ, and a good choice for β
with ∆t=0.01s was β=10.

Selected simulation results formaneuver 1 as inSection 6.2 arepresented in
Fig.6. It can be seen that there is practically no difference in ϕ(t), s(t), θ1(t)
and θ2(t) between the perturbed and reference motions, and the command
Mb(t) is not changed either.The initial difference in thevertical loadposition is
quickly damped to the reference values, and this is achieved by an appropriate
abrupt change in Mw value at thebeginningof simulation.Then, themodeling
inconsistency (additional damping in s and l motions) is compensated by
appropriate differences in F(t) and Mw(t) relative to the reference control.
The differences vanish at the target pointwhen ṡ→ 0 and α̇= l̇/rw → 0 (the
additional damping vanishes).



Dynamics and control of rotary cranes... 943

Fig. 6. Simulation of motion in maneuver 1 perturbed by inconsistent initial load
position andmodeling inconsistencies

As seen from equation (5.2), the feedback control enhanced in the hybrid
control is aimed at minimizing the violations of servo-constraints. The ini-
tial inconsistency in the load vertical position is thus effectively damped to
zero. By contrast, the compensation of the modeling inconsistencies requires
some constraint violation to produce additional control commands related to
αċ+βc+γ

∫

cdt. Both, the damping features and the range of constraint vio-
lations required to overcome the modeling inconsistencies are closely related
to the values of gain coefficients. In the present experiments, the same gain
values for all three outputs were used, and as said α2 = 8β and 32γ = αβ.
In Fig.7, the effect of different gain values used, obtained for the perturbed
motion illustrated in Fig.6, is seen. The bigger the gain values, the closer
is the realization of the perturbed motion to the reference motion. On the
other hand, too large gain values lead to instability in simulation, which is a



944 W. Blajer, K. Kołodziejczyk

well-known effect in stabilization of passive constraints as well (Ostermayer,
1990). The gain values limits and/or their optimal values are dependent ma-
inly on the integration time step ∆t, and smaller integration time steps allow
for bigger gain values. They are also closely related to the system complexi-
ty/dimensionality and the type ofmotion/perturbations simulated.The choice
of appropriate gain values is often a trial-and-error process, anddisparate gain
values for particular outputs are usually involved.

Fig. 7. Differences in load position in perturbedmotion for different gain values

The other numerical experiment illustrating the robustness of the hybrid
control relates to external perturbations caused by an additional force applied
to the load in the negative sense of X direction, not considered in the model
used for the determination of control. The force time-profile, which can be
regarded as a rough model of wind blow loads on the pay-load, is seen in
Fig.8.Themotiondisturbedthiswaywas stabilized along the referencemotion
of maneuver 2 by using the hybrid control. As before, the gain values were
α2 = 8β, 32γ = αβ, and β = 30 was chosen for the integration time step
∆t=0.01s.

Fig. 8. Profile of perturbation force



Dynamics and control of rotary cranes... 945

Fig. 9. Simulation of motion in maneuver 2 perturbed by ’wind loads’

The simulation results seen in Fig.9 show that the ’wind load’ Fres causes
that the rope bends accordingly, and the trolley position and bridge rotation
angle differ slightly from the reference values so that to compensate the rope
additional bend and to execute the load prescribedmotion.The samehappens
at the target load position. The rope is out of plumb, and the final trolley
position andbridge rotation angle differ from their reference values. The ’wind
load’ effects are compensated by appropriate changes in the control commands
which cause that the servo-constraints are realized with a limited accuracy.
More strictly, the constraint violations in the Y and Z directions, ∆y and ∆z,
do not exceed ±0.0004m during the whole motion, while the violation in the
X direction, required to produce the required feedback control commands, is a
up to 3.3cm.This inaccuracy in the load position can be enlarged/diminished
by applying smaller/bigger gain values. The effect is illustrated in Fig.10.



946 W. Blajer, K. Kołodziejczyk

Fig. 10. Difference in the x-position of load in wind-affectedmotion for different
gain values

7. Summary and conclusions

The problem of control of a rotary crane executing a prescribed load trajecto-
ry has been viewed from the perspective of constrained system dynamics and
inverse dynamics control of an underactuated system. The desired performan-
ce goals (specified in time load coordinates), expressed in terms of the system
states, lead to servo-constraints on the system.Mixed orthogonal-tangent re-
alization of the servo-constraints by the available crane control is observed,
and the tangent realization yields additional conditions on the crane motion.
In this sense, themotion of the cranemodeled as a five-degree-of-freedom sys-
tem can be explicitly prescribed by three outputs, and explicitly controlled by
available three control inputs.

The governing equations for dynamics and control of a rotary crane execu-
tinga loadprescribedmotionariseas thirteen index-threedifferential-algebraic
equations in the crane state variables and control variables. An effective and
stable method for solving the governing DAEs has been proposed and tested
through numerical experiments. The solution to theDAEs aremotion charac-
teristics of the crane executing the prescribed load trajectory and the control
commands (feedforward control) ensuring the prescribed load trajectory re-
alization. The feedforward control law obtained this way was enhanced by
a closed-loop control strategy with feedback of the actual errors in the load
position.

The reported mathematical model was tested through numerical simu-
lations. The developed computational codes enable one for effective inverse
simulation studies related to a wide range of load trajectories/maneuvers.
Computational robustness of the developed control strategy in motion per-
turbed by an inconsistent initial load position, modeling inconsistencies, and
external perturbations was proved.



Dynamics and control of rotary cranes... 947

Acknowledgments

The researchwas supported by grant No. 4T12C0623̇0.

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Dybanika i sterowanie żurawiem wieżowym realizującym zadany ruch

ładunku

Streszczenie

Manewrowanie ruchem ładunku przez żurawie wieżowe jest zadaniem trudnym
między innymi ze względu na fakt, że liczba kanałów sterowania (równa liczbie regu-
lowanych współrzędnych ładunku) jest mniejsza od liczby stopni swobody żurawia.
W pracy zadane w czasie współrzędne ładunku prowadzą do sformułowania serwo-
więzów (więzów programowych) nakładanych na ruch układu. Prezentowana jest na-
stępniemetoda rozwiązania tego szczególnegozadania symulacji dynamicznej odwrot-
nej jako zadania ruchu programowego niezupełnego. Równania ruchu programowego
formułowane sąw postaci układu równań różniczkowo-algebraicznycho indeksie trzy,
względem zmiennych stanu i parametrów sterowania żurawiem. Proponowania jest
prosta i skuteczna metoda numerycznego całkowania tych równań, wykorzystująca
schemat Eulera różnic skończonych wstecznych. Otrzymywane tą drogą sterowanie
nominalne uzupełniane jest następnie sterowaniem w układzie zamkniętym zapew-
niającym stabilną realizację programowego ruchu ładunku wwarunkach ruchu zabu-
rzonego i nieścisłości modelowania matematycznego. Prezentowane są wybrane wy-
niki numerycznej symulacji ruchu i sterowania żurawiem realizującym zadany ruch
ładunku.

Manuscript received May 19, 2006; accepted for print July 10, 2006