A Graphical Phase Plane Approach for Controlling

Cargo Transfer on Quay-Side Container Cranes with

Hoisting

Mohammed F. Daqaq∗ and Ziyad N. Masoud †

Department of Engineering Science and Mechanics, MC 0219,

Virginia Polytechnic Institute and State University, Blacksburg, VA 24060

Input-shaping is a practical open-loop strategy for the control of transient and residualoscillations on cranes, especially those having fixed cable length and predefined payloadtransfer paths. In this paper we extend the double-step input-shaping control approachto include maneuvers that involve hoisting. The approach is based on using the graphicalrepresentation of the phase plane of the payload oscillations. The phase plane is used toderive mathematical constraints to compute the switching times of a double-step acceler-ation command profile that will result in minimal transient and residual oscillations. Thecontroller design is based on an accurate two-dimensional four-bar-mechanism model of acontainer crane. For the purpose of controller design, the model is reduced to a constraineddouble pendulum with variable length link and a kinematic angular constraint. The gen-erated commands were based on both a linear and a nonlinear frequency approximationof the payload oscillation period. Results demonstrated that in contrast with the single-step input shaping controllers, which are very sensitive to frequency approximations, theproposed double-step controller is less sensitive to small variations in the frequency evenfor large commanded accelerations. Numerical simulations demonstrated that using thisapproach, oscillations during and at the end of transfer maneuvers can be reduced to lessthan 2 cm.

I. Introduction

Motion control of suspended objects has seen mounting research interest since the early 1960’s. The dy-namics of such systems is used to model the dynamics of the payload oscillations in many important industrialapplications, such as manufacturing gantry cranes, boom cranes, ship-mounted cranes, ship-telescopic cranes,quay-side container cranes, etc.

A particularly important case is that of a quay-side container crane, figure 1. Inertial forces on the payloaddue to crane commanded trajectories can cause the payload to experience large sway oscillations. To avoidexciting these oscillations, crane operators resolve to slowing down the operations such that the oscillationsdo not cause safety concerns and possible damage of the payload. However, slowing down operations increasesthe cost of loading and unloading operations.

In contrast with other types of cranes, which are usually modeled as a simple pendulum with a rigid orflexible hoisting cable and a lumped mass at the end of the cable1−7 , quay-side container cranes have asignificantly different configuration. The actual hoisting mechanism of a container crane consists typicallyof a set of four hoisting cable arrangement. The cables are hoisted from four different points on a trolleyand are attached on the payload side to four points on a spreader bar used to lift containers.

Cargo transfer controllers varies according to the container crane application. For example, in oneapplication large oscillations maybe acceptable during the transfer operation, however, the settling time,

∗PhD. Candidate, Department of Engineering Science and Mechanics, MC 0219, Virginia Polytechnic Institute and StateUniversity, Blacksburg, VA 24061.

†Assistant Professor, Department of Engineering Science and Mechanics, MC 0219, Virginia Polytechnic Institute and StateUniversity, Blacksburg, VA 24061.

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46th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics & Materials Conference18 - 21 April 2005, Austin, Texas

AIAA 2005-1841

Copyright © 2005 by Ziyad N. Masoud. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Figure 1: Typical quay-side container crane.

overshoot, and the magnitude of residual oscillations are kept minimal at the end of the transfer maneuverto allow for accurate cargo positioning. In other applications, such as nuclear reactors, or where the spacearound the crane is populated, the safety requirements are very strict. Thus large oscillations are notacceptable during and at the end of a transfer maneuver.

Input-shaping control is an old but still widely used strategy to control suspended objects in general, andcontainer cranes in particular. Its effectiveness comes from the fact that it does not require any alterationsto the original structure of the crane, or the installation of additional mechanical devices. However, thesecontrollers are not robust. Their performance is very sensitive to changes in system parameters, timedelays, external disturbances, and they require “highly accurate values of the system parameters” to achievesatisfactory system response.8,9, 10 While a good design can minimize the controller’s sensitivity to changesin the payload mass, it is much harder to alleviate the controller’s sensitivity to changes in the hoisting cablelength.

Alsop, Forster, and Holmes11 were the first to propose a controller based on input-shaping. The controllerwas used to automate ore unloader, by accelerating the trolley in steps of constant acceleration then killing theacceleration when the payload reaches zero oscillation angle (after multiples of a full period). The trolley thencoasts at constant speed along the path for a period of time necessary to complete the transfer maneuver.A replicate of the acceleration procedure is used in the deceleration stage. The switching times for theacceleration and deceleration steps were calculated using an iterative computer procedure. Alsop et al. useda linear frequency approximation of a simple pendulum model. Their results demonstrated very little residualoscillations, while transient oscillation angles were of order of 10◦ during the acceleration/deceleration stages.

Nonlinear frequency approximation of a simple pendulum was also used to improve the performance ofthe single-step controller.12,13 Numerical simulations demonstrated that an acceleration profile based on thenonlinear frequency approximation can dampen the residual oscillations two orders of magnitude more thanthat based on on a linear frequency approximation. The enhanced performance was most pronounced forlonger coasting distances and higher accelerations.

Alzinger and Brozovic14 showed that a double-step acceleration/deceleration profile results in significantreductions in travel time over a one step acceleration profile. Testing on an actual crane has shown thatthe two-step acceleration profile can deliver both faster travel and minimal payload oscillations at the targetpoint.

Starr15 used a symmetric double-step acceleration/deceleration shaped profile to transport a suspendedobject with minimal oscillations. A linear approximation of the period of the payload is used to calculatethe switching times and to generate an analytical expression for the acceleration profile. This work waslater extended by employing a nonlinear approximation of the payload frequency to generate single-step anddouble-step symmetric acceleration profiles.16

Using a new approach, Jansen, and Noakes17 showed analytically that input-shaping is equivalent toa notch filter applied to a general input signal and centered around the natural frequency of the payload.They applied a second-order robust notch filter to shape the acceleration input. Numerical simulation

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and experimental verification of this strategy on an actual bidirectional crane, moving at an arbitrary stepacceleration and changing cable length at a very slow constant speed, showed that the strategy was able tosuppress residual payload oscillation. The work was extended by developing a command shaping notch filterto reduce payload oscillation on rotary cranes excited by the operator commands. It was reported that ingeneral, there was no guarantee that applying such filter to the operator’s speed commands would result inexcitation terms having the desired frequency content, and that it only works for low speed and accelerationcommands.18 Parker et al.19 experimentally verified the numerical simulation results.

Singhose, Porter, and Seering20 developed four different input-shaping controllers. They reported thatthe best controller produced a reduction of 73% in transient oscillations over the time-optimal rigid-bodycommands. However, they noticed that “transient deflection with shaping increases with hoist distance,but not as severely as the residual oscillations”. The numerical simulations showed that “the percentagein reduction with shaping is dependent on system parameters”. As a result, the four controllers sufferedsignificant degradation in performance when applied to crane maneuvers that involved hoisting.

Daqaq, Masoud, and Nayfeh21 developed a single-step input-shaping controller for quay-side containercranes. The controller was based on a four-bar-mechanism model that was developed earlier by Masoud,and Nayfeh.22,23 The full model was simplified to a double pendulum model with kinematic constraints.The method of multiple scales was used to find an analytical nonlinear expression of the period of oscillationin both the acceleration and coast modes. They reported that a single-step input shaping controller basedon a simple pendulum fails when applied to quay-side container cranes. In fact their results showed thatsuch a controller may amplify the residual oscillations to large magnitudes. On the other hand, numericalsimulations showed that a controller based on the nonlinear frequency approximation of the simplified modelof the four-bar-mechanism-model results in a superior performance. To increase the controller performancerobustness, they combined the shaped commands with a nonlinear delayed-position feedback at the end ofthe transfer operation.

In this paper we use a graphical phase plane approach to derive geometric constraints that are used todevelop a double-step input-shaping controller that accounts for large hoisting operations. The graphicallyderived constraints are combined with physical constraints then solved numerically for the switching timesof the acceleration profile. The controller is based on a four-bar-mechanism model of the container craneand is simulated using both linear and nonlinear frequency approximations of this model.

II. Mathematical Modeling

A. Full Model

Figure 2 shows a two-dimensional side projection of a quay-side container crane. This four-bar-mechanismis developed to model the actual dynamics of the crane. The container is grabbed using a spreader bar,which is then hoisted from the trolley by means of four cables, two of which are shown. The cables arespaced a distance d at the trolley and a distance w at the spreader bar. The hoisting cables in the modelare considered as rigid massless links with variable lengths. The specific equations of these scleronomicholonomic constraints are

Φ(q, t) =

[(x + R sin θ − 1

2w cos θ − f + 12d)2 + (y − R cos θ − 1

2w sin θ)2 − L2

(x + R sin θ + 12w cos θ − f − 1

2d)2 + (y − R cos θ + 12w sin θ)2 − L2

]= 0 (1)

where q = [x, y, θ]T is the generalized coordinate vector, which represents the three degrees of freedom ofconsideration.

Using the Lagrange multipliers, one can write the set of differential-algebraic equations DAE’s24 as[M ΦT

q

Φq 0

][qλ

]=

[QA

γ

](2)

where M = diag[m,m, mk2] is the inertia matrix, m is the mass of the load and the spreader bar, k is thecombined radius of gyration of the load and spreader bar about point Q, QA = [0,−mg, 0]T is the generalized

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Figure 2: A schematic model of a container crane.

applied force vector, and

γ = −(Φqq)q − 2Φqtq − Φtt

Φq =∂Φ∂q

Φtt =∂2Φ∂t2

Φqt =∂

∂t(∂Φ∂q

)

(3)

B. Simplified Model

To better understand the dynamics of the system and to derive analytical expression for the system frequencywhich is essential to the controller design, we simplify the four-bar-mechanism model to a double pendulumsystem with a variable length cable l, a fixed length link R, and a kinematic constraint relating the anglesφ and θ as shown in figure 3.

In figure 2, point O is the mid point between points A and D, and point P is the midpoint betweenpoints B and C. The closing constraints of the loop ABPO are

l sin φ − 12w cos θ +

12d = L sin φ1 (4)

l cos φ − 12w sin θ = L cos φ1 (5)

Similarly, the closing constraints of the loop ODCP can be written as

l sin φ +12w cos θ − 1

2d = L sin φ2 (6)

l cos φ +12w sin θ = L cos φ2 (7)

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Figure 3: A schematic of a constrained double pendulum model of a container crane.

Squaring and adding Eqs. (4) and (5), and squaring and adding Eqs. (6) and (7), we can eliminate L,φ1, and φ2 from the resulting equations and obtain the following relations

θ = −φ + arcsin(

d

wsin(φ)

)(8)

l =

√L2 − 1

4(d2 + w2 − 2dw cos θ) (9)

Equation (8) represents the kinematic constraint between the angles φ and θ. The position vector to thecenter of mass of the payload of the constrained double pendulum is

r = (f + l sin φ − R sin θ)i − (l cos φ + R cos θ)j (10)

Using Eq. (10), we write the kinetic and potential energies of the constrained double pendulum as

T =12ml2φ2 +

12m(k2

Q + R2)θ2 +12mf2 − mRlφθ cos(φ + θ) + mlφf cos φ − mRθf cos θ (11)

V = −mg(l cos φ + R cos θ) (12)

Substituting the constraint Eqs. (8) and (9) into Eqs. (11) and (12) results in the elimination of θ fromthe energy of the system. To derive the equation of motion that describes the time variation of φ, we usethe Euler-Lagrange equations given by

ddt

(∂L∂φ

)− ∂L

∂φ= 0 (13)

where L = T −V . Substituting Eqs. (11) and (12) in Eq. (13) results in the full nonlinear equation of motionfor the simplified system. Due to the lengthy expressions in this equation, we only show the linear first orderexpansion

φ + 2μφ + ω2◦φ +

μ

2Lf = 0 (14)

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where

ω◦ =

√g(L + a2R) − adRL

w

L2 − 2aRL + a2(k2 + R2)μ =

L(L − aR)L2 − 2aRL + a2(k2 + R2)

a =d − w

w

(15)

In the above linear approximation, we assume that l ≈ L. For small angles of oscillation φ this assumptionresults in a maximum error of 1% in calculating the length l based on typical values of d, w, and L forcontainer cranes.

III. Controller Design

Analyzing the dynamics involved in Eq. (14) forms the basis for designing the double-step input-shapingcontroller. To find the equilibrium solution, φL, we let φ = φ = 0 in Eq. (14) to get

φL = − L − aR

g(L + a2R) − adRLw

f (16)

For constant cable length operations, L = L = 0, the linearized system has a single stationary fixed-pointgiven by

φ◦ = − L − aR

g(L + a2R)f (17)

This equilibrium solution is a marginally stable center. For constant L the solution of Eq. (14) isillustrated in the phase portraits shown in figure 4(a). The figure shows that the solution is periodic withthe payload exhibiting a limit cycle behavior. The center of the resulting limit cycle along the φaxis isdetermined by the amplitude and sign of the equilibrium solution φ◦. The radius of the resulting limit cycleis determined by the payload initial angular displacement and velocity.

For operations that include hoisting, the dynamic behavior of the payload is qualitatively different. Thevariation of L causes the equilibrium solution described by Eq. (16) to vary along the φaxis. If L is increasingthe amplitude of φL increases and vise versa. This kind of equilibrium solution is known as a non-stationaryequilibrium point. Moreover, in the case of hoisting L �= 0 and hence the term including φ in Eq. (14) is nolonger equal to zero. This term acts as a damping term with the sign of L determining the type of damping.The equilibrium solution of Eq. (14) is no longer a marginally stable center. The stability of this equilibriumsolution is now determined by the sign of L. For a positive L, the damping is positive and the equilibriumsolution is a stable focus (sink). On the other hand, for a negative L, the damping is negative and theequilibrium solution is an unstable focus (source). The qualitative dynamic behavior of the payload aroundthe non-stationary equilibrium solution is illustrated in figure 4(b) and figure 4(c).

Figure 5 shows a typical acceleration profile for a double-step input-shaping controller. The trolleyaccelerates at a constant rate of amax for time ta1 , after which the trolley coasts at a constant velocity untiltime tc1 . The trolley then accelerates again with the same acceleration amplitude until time ta2 where theacceleration phase is concluded. Afterwards, the trolley coasts until time T , then decelerates in two stepssimilar to the acceleration stage.

A shaped double-step acceleration profile like the one shown in figure 5 uses the maximum systemcapabilities to achieve the required transfer maneuver with minimal transient and residual oscillations. Togenerate the required profile, the controller calculates the switching times such that the payload dynamicsfollow the phase portrait shown in figure 6. Ideally, this phase portrait drives the dynamics of the payloadto end at the equilibrium point at the center of the phase portrait rather than a limit cycle around it.

To determine the switching times for the required phase portrait, the trolley acceleration profile mustsatisfy two sets of constraints. The first set includes the dynamic constraints that involve the amplitudes ofthe response at the beginning and at the end of each acceleration step and the physical constraints on thecrane motors.

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(a) L = 0

(b) L > 0 (c) L < 0

Figure 4: Phase portraits describing the dynamics of the crane payload.

Since L is slowly varying when compared to φ, the damping term associated with φ can be treatedas constant when analyzing the fast dynamics of the system for short periods of time. This assumptiondoes not imply slow cable hoisting. The variation is only slow compared to the fast variation of the angle φ.Hence, to derive these dynamic constraints, we will assume that the system response is exponentially damped.

Figure 5: Typical acceleration profile of a double-step input-shaping controller.

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Figure 6: Graphical representation of the controller phase portrait.

Furthermore, for typical crane parameters and operating range of the hoisting cable, the equilibrium solutionis assumed stationary.

In the first acceleration step, the trolley accelerates so that the new equilibrium solution is at point B.The response then follows the path p1 through a phase angle α1 to point A. The amplitude of the responseat that point is

b1 = beμta1 (18)

The second acceleration step tc1 to ta2 is designed to force the response of the system to home at the originof the phase portrait at the end of this acceleration step. The system response follows the path p2 throughthe phase angle α2. Going back in time, the starting amplitude of oscillation of the second acceleration stepat point C has to satisfy

b2 = be−μ(ta2−tc1 ) (19)

To guarantee the continuity of the response, points A and C are required to fall on the same coastingpath pc in the phase portrait. This can be represented by the following relation

c2 = c1eμ(tc1−ta1 ) (20)

A fourth dynamic constraint is derived from the physical constraints on the velocity and accelerationachievable by the trolley motors. This constraint can be enforced by restricting the acceleration time so thatthe trolley velocity does not exceed the maximum achievable velocity by the trolley motors.

vmax = amax(ta1 + ta2 − tc1) (21)

The second set of constraints is derived directly from the graphical representation of the controller phaseportrait, figure 6. Graphically, the coasting phase is split into two consecutive stages: β1 and β2. The firstphase ends when the response crosses the zero velocity axis on the phase portrait at time to. Considering theupper triangle OAB, and using the geometric laws of sines and cosines, the following geometric constraintsare obtained

c21 = b2 + b2

1 − 2bb1 cos α1 (22)

c1 = b1sin α1

sin β1(23)

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Similarly, considering the lower triangle OCB, two geometric constraints are also derived as

c22 = b2 + b2

2 − 2bb2 cos α2 (24)

c2 = b2sin α2

sin β2(25)

The phase angles of the controlled performance in the acceleration and coasting phases can be related tothe system frequencies and switching times by the following equations

α1 =

ta1∫0

ωadt (26)

α2 =

ta2∫tc1

ωadt (27)

β1 =

to∫ta1

ωcdt (28)

β2 =

tc1∫to

ωcdt (29)

where ωa, and ωc are the frequency of payload oscillations in the accelerations and coasting stages, respec-tively. When a linear frequency approximation is used, then ωa= ωc=ω◦.

To simplify the above eight constraint equations into four equations in terms of only the unknownswitching times ta1 , to, tc1 , and ta2 , we substitute Eqs. (18), (23), (26), and (28) into Eq. (22)

e2μta1

sin2(ta1∫0

ωadt)

sin2(to∫

ta1

ωcdt)= 1 + e2μta1 − 2eμta1 cos

( ta1∫0

ωadt

)(30)

Similarly, we substitute Eqs. (19), (25), (27), and (29) into Eq. (24) to get

e−2μ(ta2−tc1 )

sin2(ta2∫tc1

ωadt)

sin2(tc1∫to

ωcdt)

= 1 + e−2μ(ta2−tc1 ) − 2e−μ(ta2−tc1 ) cos( ta2∫

tc1

ωadt

)(31)

We also substitute Eqs. (18), (23), (19), (25), and (26)-(29) into Eq. (20)

e−μta2

sin(ta2∫tc1

ωadt)

sin(tc1∫to

ωcdt)

=sin(

ta1∫0

ωadt)

sin(to∫

ta1

ωcdt)(32)

The final four constraint equations Eqs. (21), and (30)-(32) are then numerically solved for the switchingtimes of the controller. The same approach is used for the acceleration and deceleration stages. The trolleycoasting time between the acceleration and deceleration stages is determined by the total travel distance ofthe trolley.

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0 5 10 15 20 25 30

−3

−2

−1

0

1

2

3

Time [s]

(Sw

ay [

m],

Acc

eler

atio

n [m

/s2 ])

SwayAcceleration

(a) Payload sway and trolley acceleration

−0.1 −0.05 0 0.05 0.1 0.15

−0.1

−0.05

0

0.05

0.1

0.15

φ [rad]

φ′/ω

(b) Phase portrait of the system response

Figure 7: Sway response of a container crane to shaped operator commands. Results are obtained for L = 32.5 m, S = 50 m.

IV. Numerical Simulation

To simplify the calculations, the following simulations are designed so that the trolley acceleration stagesfall within the constant hoisting speed stage. This is done to avoid discontinuities in the hoisting profilewhich results from the square hoist acceleration profile. The maximum hoisting acceleration used in thesesimulations is 0.75 m/s2 and the maximum hoisting velocity is 1.5 m/s. In the numerical simulations thefollowing crane dimensions are used: d = 2.82 m, w = 1.41 m, R = 2.5 m, and k = 1.0 m. The payload massis m = 50000 kg.

Using the linear frequency approximation Eq. (15) a shaped double-step acceleration profile is generatedfor a fixed cable-length transfer operation. The resulting profile is applied to the full model equations ofmotion Eqs. (2), and the numerical simulation is shown in figure 7.

Daqaq et. al. 21 reported that for single-step input shaping controller, a controller based on a nonlinearfrequency approximation of the payload oscillations results in a superior performance when compared tothat based on a linear approximation. To check the effect of using a nonlinear frequency approximationon the double-step controller, we will use the linear frequency approximation and the nonlinear frequencyapproximation developed earlier.21 The nonlinear approximation is dependent on the oscillation amplitudeand the trolley acceleration, hence ωa �= ωc �= ω◦.

Figure 8 illustrates a comparison between the performance of the double-step controller using the linearand nonlinear frequency approximations. The amplitude of the residual oscillation resulting from a linearfrequency approximation is larger than that resulting from a nonlinear approximation. However, in contrastwith the single-step controller where the difference is as large as 10 orders of magnitude, the difference hereis very small and can be tolerated. This is due to the fact that using the double-step profile the period overwhich large oscillations occur is so short that the effects of the difference between the linear and nonlinearfrequency approximations does not yield significant deviation from the desired system dynamics.

Two simulations are presented to demonstrate a hoisting up and lowering maneuvers in which a linearfrequency approximation is used . In the first simulation, the payload is hoisted up 15 m starting from aposition 35 m below the trolley with an acceleration of 0.75 m/s2, figure 9(a). After the hoist acceleration isconcluded in the first two seconds, the trolley starts to accelerate with a maximum acceleration of 0.5 m/s2

to reach a maximum velocity of 3 m/s. The trolley travels a distance of 50 m. The payload travel profilesis shown in figure 9(b). The switching times are listed in Table 1. The total transfer time is 25.6 seconds.The maximum payload oscillation after the acceleration stage is 15.2 mm and payload oscillation at the endof the transfer maneuver is less than 12.2 mm. Figures 9(c) and 9(d) show the phase portrait of the systemand the payload oscillation throughout the transfer maneuver.

In the second simulation, the payload is transferred 50 m starting from a position 20 m below the trolley.

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20 22 24 26 28 30 32 34 36 38 40−0.03

−0.02

−0.01

0

0.01

0.02

0.03

Time[s]

Sway

[m

]

LinearNonlinear

Figure 8: Residual oscillations of the payload resulting from a double-step input-shaping controller, L = 32.5 m, and S = 50 m.

Table 1: Controller switching times for a hoisting maneuver from Li to Lf .

Li Lf ta1 t0 tc1 ta2

35 20 2.5762 3.6751 3.9538 7.377720 20 3.0000 3.1859 3.3717 6.371720 35 3.4601 3.6677 4.7279 7.2678

Two seconds before the beginning of the deceleration stage, the payload is accelerated downward for 2 secondswith a maximum acceleration of 0.75 m/s2, figure 10(a). Motion trajectory is shown in figure 10(b). Theswitching times of the controller are listed in Table 1. The system performance is shown in figure 10(c)and 10(d). The total transfer time is 26.3 seconds. The magnitude of the payload oscillation after theacceleration stage is 12.5 mm and 11.4 mm at the end of the transfer maneuver.

V. Concluding Remarks

In this paper we successfully extend double-step input-shaping controllers to include large hoisting op-erations. This new approach is based on the graphical representation of the phase portrait that describesthe response of a container crane payload to a double-step acceleration profile. This controller is suitable forautomated crane operations, and does not require any additions to the current crane system configurations.

Since open-loop controllers require accurate system identification to ensure a good performance, we havethe controller based on a two-dimensional four-bar-mechanism model of a quay-side container crane, ratherthan the widely used simple pendulum model which does not describe the real dynamics of the actual systemand therefore provides erroneous frequency approximations.

In contrast with single-step input-shaping controllers, the controller is less sensitive to frequency ap-proximations of the simplified model, and therefore a nonlinear frequency approximation of the payloadoscillations does not have a large contribution to enhancing the performance of the controller.

The numerical simulations illustrate that with proper system identification the controller is capable ofreducing the transient and residual oscillations to very small magnitudes.

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0 5 10 15 20 25 30 35 40−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time [s]

Acc

eler

atio

n [m

/s2 ]

Trolley accelerationHoist acceleration

(a) Trolley and hoist acceleration

−10 0 10 20 30 40 50−35

−30

−25

−20

−15

x [m]

y [m

]

(b) Motion profile

0 5 10 15 20 25 30 35 40−2

−1.5

−1

−0.5

0

0.5

1

1.5

Time [s]

Payl

oad

sway

[m

]

(c) Payload sway

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

φ [rad]

φ′/ω

(d) Phase portrait

Figure 9: A transfer operation involving a 15 m hoisting action.

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0 5 10 15 20 25 30 35 40−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time [s]

Acc

eler

atio

n [m

/s2 ]

Trolley accelerationHoist acceleration

(a) Trolley and hoist acceleration

0 10 20 30 40 50 60−35

−30

−25

−20

−15

x [m]

y [m

]

(b) Motion profile

0 5 10 15 20 25 30 35 40−1.5

−1

−0.5

0

0.5

1

1.5

2

Time [s]

Payl

oad

sway

[m

]

(c) Payload sway

−0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08 0.1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

φ [rad]

φ′/ω

(d) Phase portrait

Figure 10: A transfer operation involving a 15 m lowering action.

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