Modeling, Identification and Control, Vol. 34, No. 4, 2013, pp. 157–174, ISSN 1890–1328

Analysis of Offshore Knuckle Boom Crane — PartOne: Modeling and Parameter Identification

Morten K. Bak Michael R. Hansen

Department of Engineering Sciences, Faculty of Engineering and Science University of Agder, 4879 Grimstad,Norway. E-mail: {morten.k.bak,michael.r.hansen}@uia.no

Abstract

This paper presents an extensive model of a knuckle boom crane used for pipe handling on offshore drillingrigs. The mechanical system is modeled as a multi-body system and includes the structural flexibility anddamping. The motion control system model includes the main components of the crane’s electro-hydraulicactuation system. For this a novel black-box model for counterbalance valves is presented, which usestwo different pressure ratios to compute the flow through the valve. Experimental data and parameteridentification, based on both numerical optimization and manual tuning, are used to verify the cranemodel.The demonstrated modeling and parameter identification techniques target the system engineer and takesinto account the limited access to component data normally encountered by engineers working with designof hydraulic systems.

Keywords: Hydraulic crane, multi-body system, flexibility, directional control valve, counterbalance valve

1 Introduction

Today’s offshore drilling equipment is characterizedby high price, high level of system complexity and lowproduction numbers. For the equipment manufactur-ers, it requires a great level of skill and experience todevelop the equipment, since there are very limitedpossibilities to build prototypes for testing and verifi-cation of new designs. Increasing focus on productionand development costs adds to this challenge. Asa consequence, design engineers continuously haveto improve their procedures for decision makingregarding choice of principal solutions, componentsand materials in order to reach the best possibletrade-off between different performance criteria suchas reliability, efficiency and cost.Computer based time domain simulation and opti-mization techniques have, by far, proven themselvesas excellent tools for the challenged designer and haveover the last couple of decades increasingly been em-

ployed by drilling equipment manufacturers. However,the use of these techniques still offers a number ofchallenges both in industry as well as academia.In model based design, simulation models serve asvirtual prototypes providing information, e.g., abouta machine’s overall efficiency, stability and accuracy,enabling engineers to test, redesign and optimizethe design of the machine before it is manufactured.Model based design offers the possibility to reduceboth development time and costs while also producingmore reliable machines.The main challenge in model based design lies withinthe ability to produce simulation models that, with areasonable precision, are able to mimic the behavior ofa real system. This challenge is especially pronouncedfor hydraulically actuated machines, like many off-shore drilling applications, simply because suppliersof hydraulic component are not used to deliver allthe data needed to develop simulation models of theirproducts.

doi:10.4173/mic.2013.4.1 c© 2013 Norwegian Society of Automatic Control

Modeling, Identification and Control

An application that represents a typical piece ofhigh-end offshore equipment is the knuckle boomcrane. The ability to employ a model based approachfor design of such cranes is highly relevant.Modeling, simulation, design and control of varioustypes of cranes have been subjected to extensiveresearch. General modeling techniques and differentcontrol concepts have been presented by Hiller (1996)and Abdel-Rahman et al. (2003) and particularlymobile (truck-mounted) cranes have attracted a con-siderable amount of interest from researchers (Ellmanet al., 1996), (Mikkola and Handroos, 1996), (Esqueet al., 1999), (Hansen et al., 2001), (Nielsen et al.,2003) and (Esque et al., 2003). The dynamics ofthese types of cranes is well documented and modelingtechniques have been proven through experimentalverification.In (Than et al., 2002) and (Bak et al., 2011) off-shore boom cranes have been investigated. Modelstaking the structural flexibility into account havebeen presented, however, without any experimentalverification.Though mobile cranes are particularly flexible andbehave differently than offshore cranes, the samemodeling techniques can be used for both types ofcranes. Modeling of mechanical systems such as boommechanisms can be handled with different genericapproaches. However, the most suitable approach isnot always obvious.Modeling approaches for hydraulic components andsystems are, in general, also well-established but maycause problems when it comes to model verification.The problem is often a lack of proper model dataand/or that the physics is not fully understood.Therefore, for certain hydraulic components, theremay be a need to introduce new modeling approaches.In this paper an extensive model of an offshore knuckleboom crane is developed with a view to identify abest practice for predicting the behavior of this typeof crane. It is demonstrated how to overcome themodeling challenge by choosing an appropriate levelof modeling detail and by using experimental worktogether with parameter identification techniques.A commercially available software package,MapleSimTM, is used to develop a dynamic model andMATLAB R© is used for steady-state simulations andoptimization based parameter identification.

2 Considered System

Knuckle boom cranes are used for a wide range of off-shore and marine operations and therefore exist in dif-ferent variations. The considered crane is manufac-tured by Aker Solutions and is used on drilling rigs to

move drill pipes between the pipe deck and a trans-portation system leading to the drill floor of the rig.Prior to commissioning of a crane, it undergoes a testprocedure to verify the functionality and ensure thatthe performance corresponds to the criteria given inthe design specification. This procedure facilitates anexperimental study that can be used to calibrate andverify design models of the crane. In the following adescription of the considered crane is given along withsystem variables that have been measured and recordedduring a test procedure. The procedure itself is de-scribed in section 5 along with the model parameteridentification.The considered crane may be treated as a large multi-domain system consisting of three interacting systems:

1. A mechanical system.

2. An electro-hydraulic actuation system.

3. An electronic control system.

The main components of the cranes mechanical systemare a rotating part mounted on a pedestal, an inner jib,an intermediate jib, an outer jib and a gripping yoke,see Fig. 1.

Inner jib

Intermediate jib

Outer jib

Gripping yoke

Rotating part

Pedestal

Pipe deck

7.2

m

12.1

m2.3 m

10.4 m

Figure 1: Main components of the mechanical system.

The crane is controlled from the operators cabin (not

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shown) mounted on the rotating part, also called theking. A slewing bearing and transmission between theking and the pedestal allows for slewing of the crane.However, this degree of freedom (DOF) and the detailsof the slewing transmission are not considered here.The gripping yoke also includes a number of hydrauli-cally actuated DOFs which are not considered. Theactuation system is supplied by a hydraulic power unit(HPU) with constant supply pressure, pS = 210 bar,and return pressure, pR = 0.The considered part of the actuation system consistsof three hydraulic circuits, one for each crane jib, con-nected to the supply and return lines of the HPU.The control system includes a human-machine inter-face (HMI) which facilitates the operation of the crane,a number of sensors and instruments used for feedbackcontrol and/or monitoring and a controller where thecontrol logic is defined.When considering the actuation and control systemstogether, the three circuits of the actuation system canbe considered as three sub-systems of the motion con-trol system, including both actuation and control. Asimplified schematic of the motion control sub-systemfor the inner jib is shown in Fig. 2.

pS

pR

p2

p3

Controller

uJS

uV

lcyl

CBV

I

p1

I

IDCV

Figure 2: Simplified schematic of motion control sub-system for inner jib.

The main components of the hydraulic circuit area cylinder with integrated position sensor, a servo-type directional control valve (DCV) and an exter-nally vented (drained) counterbalance valve (CBV).The cylinder velocity is controlled via the DCV, whichcontrols the flow into either of the two cylinder cham-bers. During retraction of the cylinder (load lowering),it is exposed to negative loads (piston velocity and loadforce have the same direction) and therefore the pistonpressure, p2, needs to be controlled. This is handledby the CBV, which provides a relief valve functionalityon the outlet side assisted by the pressure, p3, on theinlet side.The DCV is controlled via a joystick (part of the HMI)from which the command signal, uJS , is fed to the con-troller and used to generate the control signal, uV , tothe valve. The signal from the position sensor in thehydraulic cylinder is also fed to the controller, whereit can be used for feedback control depending on theselected control mode. In open loop control modejoystick commands are passed through the controllerand fed directly to the DCV. In closed loop controlmode both joystick commands and cylinder positionsare used for control of the DCV.The motion control sub-systems for the intermediateand outer jibs are identical and contain the same con-trol system elements as the one for the inner jib. How-ever, as seen from Fig. 3, the elements of the actuationsystems are different.The DCV is pressure compensated and uses a loadsensing (LS) circuit and a pressure reducing valve thatmaintains a constant pressure drop across meteringedge of the main spool at any time. This makes thecontrolled flow independent of the load pressure andproportional to the spool position, i.e., the control sig-nal fed to the valve.There are two CBVs since the load force on the cylindermay act in either direction, depending on the orienta-tion of the crane jibs. The CBVs are non-vented andinclude two series connected orifices to manipulate thepilot pressures, px,1 and px,2. For practical reasons itis only possible to measure p2 and p3 this circuits.In total seven pressures, three command signals andthree position signals are measured and recorded dur-

Table 1: System variables for considered motion control sub-systems.

3 = measured Input Output State variables

7 = not measured uJS lcyl p1 p2 p3 p4 px,1 px,2

Inner jib 3 3 3 3 3 NA NA NAIntermediate jib 3 3 7 3 3 7 7 7

Outer jib 3 3 7 3 3 7 7 7

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Modeling, Identification and Control

pS

pR

pcomp pLS

p2 p3

p1 p4

Controller

uJS

uV

lcyl

px,1 px,2

CBV1 CBV2II

DCV

Figure 3: Simplified schematic of motion control sub-system for intermediate and outer jibs.

ing the experimental test procedure. Table 1 providesan overview of the system variables and the ones thatare measured for the three considered motion controlsub-systems.

3 Mechanical System Model

The mechanical system is modeled using the multi-body library in MapleSimTM. The model includes themain components shown in Fig. 1 as well as the hy-draulic cylinders for the three crane jibs.In terms of kinematic structure, the crane can betreated as a large mechanism consisting of both rigidand flexible links connected by revolute joints. Thestructural flexibility of a crane similar to the one con-sidered here has been studied by Henriksen et al. (2011)and Bak et al. (2011). It was shown that the flexibil-ity of certain structural members can have a signifi-cant influence on the overall dynamic behavior of thecrane and therefore must be accounted for in a dynamicmodel.Here the finite segment method is used to model theflexibility. The main advantage of this method is thatit is based on rigid body modeling techniques, makingit easy to implement. The method was originally devel-oped by Huston (1981) and further studied in Huston(1991), Huston and Wang (1993), Connelly and Hus-ton (1994a) and Connelly and Huston (1994b). Hansenet al. (2001) used the method for modeling of a mobilecrane and achieved encouraging results in terms of con-formity between measurements and simulations.

The masses, inertias and geometry of the mechanicalcomponents have been extracted from CAD models.

3.1 Kinematic Structure

The topology of the crane is globally an open kinematicchain, formed by the crane jibs, with locally closedchains formed by the hydraulic cylinders. With themain components and the barrels and pistons of thethree hydraulic cylinders, the model includes a total of12 bodies, see Fig 4.

Y X

Z

A

B

C D

E

F

G

H

IJ

K3

4

5

6

7

8

9

2

10

11

12

1

z 3 x 3

z4

x4

z5

x5

z6

x6

Figure 4: Topology of mechanical system model.

Since the slewing DOF of the crane is not consid-ered, the crane can be modeled as a planar mech-anism using revolute and translational joints. How-ever, only spatial multi-body elements are available inMapleSimTM, making the selection of kinematic con-straints less straight forward. To identify a suitablekinematic structure the crane is initially treated as arigid body system.The bottom of the pedestal is fixed at the global ori-gin and the king is fixed to the top of the pedestalin point A. Points B, C, D and E represent revolutejoints (RJ). The connection points of cylinder barrels,points F, H and J, are modeled as spherical joints (SJ)while the connection points for pistons, points G, I and

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K, are modeled as universal joints (UJ). The transla-tional DOFs between the cylinder barrels and pistonsare modeled as prismatic joints (PJ). This leaves fourremaining DOFs (Nikravesh, 1988):

nDOF = 6 · nbodies − 6 · nfixtures − 5 · nPJ

− 5 · nRJ − 4 · nUJ − 3 · nSJ

= 72− 12− 15− 20− 12− 9 = 4

(1)

These are the ones of the three crane jib, which are ac-tuated by hydraulic cylinders, and the rotational DOFof the gripping yoke (point E). The latter is actuatedby a hydraulic motor, which is not considered here.The DOF is included to be able to orientate and fixthe gripping in the wanted positions.

3.2 Flexibility

When applying the finite segment method to a planarmechanism, the flexible members are divided into anumber of rigid segments which are connected by rev-olute joints and rotational springs (and dampers), seeFig. 5.

kS

21

kS kS kSkC

Figure 5: Concept of the finite segment method.

The left part of the figure illustrates a model of a singlesegment. The flexibility of the segment is representedby two rotational springs, both with the stiffness kS .In the right part of the figure two segments are con-nected. The stiffness kC of the connection betweenthe segments corresponds to a series connection of thesprings for the two adjoining segments:

kC =1

1kS,1

+ 1kS,2

(2)

kS,1 is the segment stiffness of the segment 1 and kS,2is the segment stiffness of the segment 2.For bending the segment stiffness is:

kS =2 · E · IL

(3)

E is the Young’s modulus, I is the area moment of in-ertia and L is the length of the segment.The pedestal, the inner jib and the outer jib are as-sumed to dominate the overall structural flexibility ofthe crane since the remaining components are far morecompact. Fig. 6 shows the segmentation of the threeflexible members.The pedestal, inner jib and outer jib (body numbers 1,

s1(1)

s2(1)

s1(3) s2

(3)

s3(3)

s4(3)

s5(3)

s1(5)

s2(5)

s3(5)

s4(5)

kC(1)

kS,1(1)

kS,2(1)

kS,2(3)

kC,1(3)

kC,2(3)

kS,4(3)

kS,2(5)

kC(5)

kS,3(5)

Figure 6: Segmentations of flexible members.

3 and 5) are divided into two, five and four segments,respectively. The number of segments is certainly de-batable, but is kept low for computational reasons.As seen from Fig. 6, the inner and outer jibs are beam-like structures with varying cross sections. With thechosen segmentations, the cross section areas of the in-dividual segments vary linearly and continuously andtherefore average values of the segments area momentsof inertia are used to determine the segment stiffnesses,kS . The area moments of intertia at the cross sectionsbetween the segments are extracted from CAD mod-els and used to determine the average area momentsof inertia. Henriksen et al. (2011) used this approachfor a model of a single crane jib and compared bothstatic deflections and natural frequencies of a finitesegment model with results of a finite element analy-sis and achieved a remarkable conformity even withoutcalibrating the model.The segment and connection stiffnesses, determined us-ing (2) and (3), are given in Table 2. The first and lastsegments of the inner and outer jibs are assumed rigidand therefore the stiffness, kS , of the neighboring seg-ment is used as the connection stiffness, kC .The stiffnesses in the applied lumped parameter modelare indeed uncertain (soft) parameters and (2) and (3)merely represent a means to estimate those parame-ters. As argued by Shabana (1997), they can also be

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Modeling, Identification and Control

Table 2: Segment and connection stiffnesses.

Segment kS [Nm/rad] kC [Nm/rad]

s(1)1 5.75 · 109

2.88 · 109s(1)2 5.75 · 109

s(3)1 rigid

1.96 · 109s(3)2 1.96 · 109

6.03 · 108s(3)3 8.73 · 108

2.77 · 108s(3)4 4.05 · 108

4.05 · 108s(3)5 rigid

s(5)1 rigid

2.08 · 108s(5)2 2.08 · 108

8.33 · 107s(5)3 1.39 · 108

1.39 · 108s(5)4 rigid

determined by finite element analysis and parameteridentification. This is, however, not the scope of thispaper and instead the stiffnesses are tuned during thecalibration (section 5) of the entire crane model.

3.3 Damping

The dynamic characteristics of the mechanical modeldepend on the damping applied to it and thereforethe damping must be carefully considered. However,damping parameters are even more uncertain thanthose related to the flexibility. The only way to prop-erly determine the damping parameters may be tocarry out a thorough experimental study of the con-sidered structure. As in this case, time, resources andpractical circumstances seldom allow for that kind ofinvestigations.Therefore, whether working with design models ormodels for analysis and with limited possibilities forexperimental investigations, there is a need for meth-ods to determine the damping parameters with a rea-sonable accuracy. Mostofi (1999) presents a simple ap-proach that can be used for lumped parameter mod-eling techniques such as the finite segment method.For stiffness damping (damping elements in parallelwith the flexible elements), the damping of a structuralmember is:

c = βk · k (4)

k contains the spring coefficients determined in section3.2 and βk is a stiffness multiplier determined by:

βk =2 · ζSωS

(5)

ζS is the damping ratio of the structure and ωS isthe natural frequency of the structure for the consid-ered mode of motion. Representative damping ratios

for different structures are given in Adams and Aske-nazi (1999). For metal structures with joints, e.g.,weldings and bolted connections, the damping ratio isζ = 0.03 − 0.07. Naturally, this is subject to uncer-tainty but, nevertheless, better than a simple guess.To determine the damping coefficients for the mod-els of the three flexible members, simulations withoutdamping have been carried out to find the natural fre-quency of each member. In the simulations each mem-ber is fixed in one end (cantilevered) and an impulseis applied to excite oscillations from which the naturalfrequency can be observed. The natural frequenciesand corresponding range of stiffness multipliers for thethree flexible members are given in table 3.

Table 3: Natural frequencies and stiffness multipliers.

Member ωS [rad/s] βk [s]

Pedestal (with king) 94 0.0006 - 0.0015Inner jib 60 0.001 - 0.002outer jib 35 0.002 - 0.004

The stiffness multipliers are tuned together with springcoefficients during the model calibration in section 5.

4 Motion Control System Model

Whereas the model of the mechanical system, in gen-eral, is based on physical (white-box) modeling, themotion control system model is mostly based on semi-physical (grey-box) modeling. The main reason forthis is that manufacturers of hydraulic components donot provide enough and sufficiently detailed informa-tion to establish physical models. In addition, physicalmodels will quickly become too complex and compu-tational demanding for system simulation. Thereforecertain model structures have to be assumed which al-low for simplifications without ignoring or underesti-mating important physical phenomena.The motion control system is modeled using both pre-defined components from the hydraulic and the signalblock libraries in MapleSimTM and custom made com-ponents developed via MapleTM. This combination fa-cilitates both efficient model development and model-ing at a detail level that is not supported by librarycomponents.Joystick commands and control signals are representedby normalized signal, which can vary continuously be-tween -1 and 1, and component dynamics is modeledusing transfer functions.Hydraulic valves are generally modeled as variable ori-fices with linear opening characteristics:

Q = ξ · CV ·√

∆p (6)

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Here ξ is the relative opening of the valve, i.e., a dimen-sionless number between 0 and 1. It can be a functionof system pressures or controlled with an input signaldepending on the considered type of valve. Q is thevolume flow through the valve and ∆p is the pressuredrop across it.The flow coefficient in (6) can be expressed as:

CV = Cd ·Ad ·√

2

ρoil(7)

The discharge coefficient, Cd, and the discharge area,Ad, are usually not specified for a valve. Instead CV

can be obtained from characteristic flow curves givenin the datasheet of the valve. From this, a nominalflow, Qnom, corresponding to a nominal pressure drop,∆pnom, can be identified and used to derive the flowcoefficient:

CV =Qnom√∆pnom

(8)

This corresponds to the fully opened state of the valve,so with (6) it is assumed that the discharge coefficient,Cd, is constant and only the discharge area, Ad, varieswith the relative opening of the valve.This modeling approach works well for DCVs withclosed loop spool position control where dither is usedto eliminate static friction and certain design detailsare used to reduce the disturbances from flow forces.In some cases the approach may also work for pres-sure control valves like CBVs. Most often, though, at-tempting to establish physical or semi-physical modelsof such valves will encounter a number of challenges,e.g., related to friction and resulting hysteresis, non-linear discharge area characteristics, varying dischargecoefficients and varying flow forces. Therefore, a bet-ter way to model those types of valves may be to use anon-physical (black-box) approach as described in sec-tion 4.2.The following sub-sections describe models of theDCVs, the CBVs and the hydraulic cylinders.

4.1 Directional Control Valves

The main concern, when modeling a DCV, is thesteady-state flow characteristics and the dynamics(bandwidth) of the valve. For system simulations, thedesign details of the valve are usually not importantand therefore servo valves and pressure compensatedDCVs can often be represented by the same model.Breaking down the model into several elements offersflexibility and facilitates changes in the model like in-cluding or excluding a pressure compensator. The gen-eral DCV model includes a representation of the valveactuation (pilot stage) and four elements for the mainspool. For pressure compensated valves, the model

also includes the LS circuit and the pressure compen-sator. The model structure for the pressure compen-sated DCV is shown in Fig. 7. The blue lines are signallines transferring only a single state variable. The redlines transfer the two hydraulic state variable, pressureand flow, between the hydraulic components. Theseare custom made components developed in MapleTM.

pcomp

pS pR

pB pA

LS circuit

pLS

uVuspool

pressure compensator

PB BT ATPA

Figure 7: Structure of DCV model.

The valve is actuated with the control signal uV , whichis passed through a second order system representingthe dynamics of the valve:

uspooluV

=1

s2

ω2V

+ 2 · ζV · sωV

+ 1(9)

The output, uspool, is a normalized signal represent-ing the spool position, which can vary continuouslybetween -1 and 1 with 0 being the center position ofthe spool. The natural frequency, ωV , represents thebandwidth of the valve and ζV is the damping ratio.For servo valves the bandwidth can usually be identi-fied from the valves datasheet. Here the valve dynamicsis usually visualized with Bode plots for several inputamplitudes showing that the dynamics is non-linear.For system simulations, though, a linear model likethe second order system will most often be sufficientto capture the dominant dynamics.For pressure compensated DCVs there is usually no in-formation available about the bandwidth and the onlyway to identify it may be to carry out a frequency re-sponse test of the considered valve. An approach forsuch a test is described in Bak and Hansen (2012) alongwith some test results for a Danfoss PVG32, which isidentical to the DCVs used for the intermediate andouter jibs. The identified bandwidth, ωV = 30 rad/s,and damping ratio, ζV = 0.8, are therefore also usedhere. This represents the overall dynamics between thecontrol signal and the controlled flow, i.e., the dynam-ics of the pilot stage, the main spool, the LS circuitand the pressure compensator.

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Modeling, Identification and Control

The four main spool edges are modeled as variable ori-fices, according to (6), for which the relative opening,ξedge, of each spool edge is a function of the normalizedspool position signal as shown in Fig. 8.

ξedge

0.5 1-0.25-0.5

0.5

ξPB ξATξBTξPA

0.25 0.75

1

0.75

0.25

-1 -0.75

uspool

Figure 8: Opening functions for spool edges.

The spool edge openings are piecewise linear functionsthat include the overlaps of the metering edges, PB andPA, and the underlaps of the return edges, AT and BT.A more detailed description of the opening functions isgiven in Bak and Hansen (2012).When modeling pressure compensated DCVs, a prob-lem often encountered is that very little or no informa-tion about the pressure compensator is available fromthe valves datasheet. This makes it difficult to esti-mate when the pressure saturation will occur and toestablish a model of the compensator. However, if thenominal pressure drop across the main spool meteringedge (setting of compensator spring), p0, is known thecompensated pressure, pcomp, can be described as:

pcomp =

pLS + p0 for p0 ≤ pS − pLS

pS − pLS for 0 < pS − pLS < p0

pLS for pS − pLS ≤ 0

(10)

The first case describes the normal operating conditionof the compensator, maintaining the nominal pressuredrop across the main spool metering edge. The sec-ond case describes the condition where the load pres-sure is too close to the supply pressure to maintainthe nominal pressure drop. The third case describesthe build-in check valve function of the compensator,which prevents negative flow if the load pressure ex-ceeds the supply pressure.The LS circuit directing the load pressure to the pres-sure compensator is modeled as a piecewise function:

pLS =

pA for uspool < 0

pB for 0 < uspool

0 otherwise

(11)

The model parameters for the servo valve (inner jib)and the pressure compensated DCVs (intermediateand outer jibs) are given in Table 4.

Table 4: DCV model parameters.

Inner jib Int. & outer jibs

ωV 250 rad/s (40 Hz) 30 rad/s (5 Hz)ζV 0.8 0.8Overlap 5 % 10 %Underlap 0 5 %CV 111.8 l

min·bar0.5 37.8 lmin·bar0.5

p0 NA 7 bar

4.2 Counterbalance Valves

The model of the CBV consists of two components; acheck valve and a pilot assisted relief valve (the actualCBV).The check valve is modeled according to (6) with therelative opening given by:

ξCV =p1 − p2 − pcr,CV

ks,CV(12)

This is only the basic part of a piecewise function thatlimits the relative opening to the interval ξCV = [0, 1].p1 is the pressure at the port connected to the DCV andp2 is the pressure at the port connected to the cylinder.The cracking pressure, pcr,CV , is usually specified fora CBV and the normalized spring stiffness can be setto a low value of, e.g., ks,CV = 1 bar.In practice the check valve is either opened or closed.The spring stiffness is only used to handle the transi-tion between the two states, which otherwise may causecomputational difficulties.Describing the relative opening by means of a pressureequilibrium like (12) can in some cases also be used forthe CBV itself. However, for the previously mentionedreasons, it is often only valid for the condition wherethe CBV is just about to open (crack). For an exter-nally vented CBV, like the one in Fig. 2, the crackingcondition is described by:

p2 + ψ · px = pcr,CBV (13)

Once again, p2 is the pressure at the port connected tothe cylinder. px is the pilot pressure and ψ is the pilotarea ratio.For a non-vented CBV, like the one in Fig. 3 (ignoringthe two internal orifices), the cracking condition is:

p2 + ψ · px = pcr,CBV + (1 + ψ) · p1 (14)

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As for the check valve, p1 is the pressure at the portconnected to the DCV.Instead of using (6) and a pressure equilibrium tomodel the CBV it is proposed to use a black-box ap-proach based on the cracking condition in (13).For the black-box model, two new variables are intro-duced; µx and µL. The first one is the ratio betweenthe pilot pressure, px, and the load pressure, pL:

µx =pxpL

(15)

The second variable is dependent on the type of CBV.For an externally vented CBV, it is the ratio betweenthe load pressure and the cracking pressure:

µL =pL

pcr,CBV(16)

For a non-vented CBV, it is the ratio between the pres-sure drop across the valve and the cracking pressure:

µL =∆p

pcr,CBV(17)

By replacing p2 with pL in (13) it can be combined with(15) and (16) to arrive at the following expression:

µL,cr =1

1 + ψ · µx(18)

This expression, just as (13) and (14), describes thecracking condition of the CBV where there is still noflow the through the valve. It is illustrated by the solidline in Fig. 9, here for a CBV with a pilot area ratioof ψ = 3. The dashed lines above illustrate conditionswith different levels of flow through the valve.In the model the actual µx and µL for a given time stepis computed by (15) and either (16) or (17) dependingon the type of CBV. The distance, s, from the crackingline (randomly shown in Fig. 9) is then computed:

s = µL − µL,cr (19)

If the distance is negative, there is no flow through thevalve. Otherwise, the flow is computed as:

QCBV = ACBV · snCBV (20)

Here ACBV is given by:

ACBV = A0 +A1 · µx (21)

Similarly, nCBV is given by:

nCBV = n0 + n1 · µx (22)

The four parameters, A0, A1, n0 and n1 have to beexperimentally determined, ideally by a thorough

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

μx

μ L

Q = 0Q > 0

s

Figure 9: Example of the relation between µL and µx

for the CBV model.

mapping of the flow through the CBV for differentpressure combinations at the individual ports. Alter-natively, as described in section (5.2), they can bedetermined with parameter identification techniquesand suitable measurements from the system where theCBV is installed.The remaining (known) model parameters for theCBVs and the check valves for the inner jib and theintermediate and outer jibs are given in Table 5.

Table 5: CBV model parameters.

Inner jib Int. & outer jibs

CV 89.4 lmin·bar0.5 63.2 l

min·bar0.5pcr,CV 1 bar 1 barks,CV 1 bar 1 barpcr,CBV 250 bar 250 barψ 3 6

4.3 Hydraulic Cylinders

The model of the hydraulic cylinder includes the capac-itance of the chambers as well as the friction between

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the piston and the barrel. The cylinder force is:

Fcyl =

Fp − Ffr for vcyl < v0

Fp − Ffr · vcyl

v0for − v0 ≤ vcyl ≤ v0

Fp + Ffr for vcyl < −v0(23)

The pressure force is Fp = p1 ·Ap− p2 ·φ ·Ap. Here Ap

is the piston area and φ = (D2p − D2

r)/D2p, where Dp

is the piston diameter and Dr is the rod diameter. p1and p2 are the pressures in the corresponding cylinderchambers. vcyl is the cylinder piston velocity and v0is a transition velocity used to handle the change infriction force around zero velocity.The capacitance of the two chamber volumes are ac-counted for by:

p1 =βoilV1· (Q1 − vcyl ·Ap) (24)

p2 =βoilV2· (vcyl ·Aa −Q2) (25)

βoil is the effective stiffness (bulk modulus) of the hy-draulic fluid. Q1 and Q2 are volume flows in the twocylinder chambers. The chamber volumes, V1 and V2,are functions of the cylinder length.The friction in the cylinder is quite complex, especiallyaround zero velocity. As described in Ottestad et al.(2012) it consists of both static and Coulomb frictionas well as velocity dependent and pressure dependentfriction, which may be described with a model of fiveparameters. Even though the model is not very com-plex, the number of parameters represents a problembecause they cannot be determined without an exten-sive experimental study of the considered cylinder.Consequently, an even simpler model must be used andtherefore the friction force in (23) consists only of staticfriction and pressure dependent friction:

Ffr = FS + Cp · |Fp| (26)

The static friction can be set to FS = Ap ·1 ·105 m2·Pa,i.e., a pressure of 1 bar on the piston-side is required toovercome the static friction. The pressure dependentfriction may constitute 2...3 % of the hydraulic force,e.g., Cp = 0.02. The friction parameters are identifiedin section 5. The remaining (known) parameters forthe cylinders are given in Table 6.

5 Parameter Identification

For verification of the model, experiments have beencarried out where inputs, outputs and certain statevariables (given in section 2) of the real system havebeen measured and recorded. The crane used for the

Table 6: Cylinder model parameters.

Inner jib Int. & outer jibs

Dp 0.3 m 0.25 mDr 0.18 m 0.125 ml0 3.145 m 2.315 mstroke 1.755 m 1.33 mmass 1500 kg 750 kg

Figure 10: Crane used for experiments.

experimental work is shown in Fig. 10.In order to calibrate and verify the model, the inputsfrom the experiments are fed to the model and theuncertain parameters are systematically tuned (identi-fied) until both simulated outputs and state variablescorrespond to those obtained in the experiments. Themodel can then be considered as verified as illustratedin Fig. 11.

Real system

Model

Input Measured output

Simulated output

Measured state variables

Simulated state variables

� �

Figure 11: Principle of model verification.

To simplify the experiments and the following param-eter identification, the individual DOFs are considered

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0 50 100 150-1

-0.5

0

0.5

1

u JS [

-]

0 50 100 1502

3

4

5

t [s]

l cyl [

m]

Inner jibIntermediate jibOuter jib

Figure 12: Experimental input and output for motion control sub-system for outer jib.

separately and one at the time. Since the procedure foridentifying the parameters and verifying the model isthe same for all three DOFs, only the DOF of the outerjib is considered in the following. During the experi-mental procedure the crane is operated in open loopcontrol mode, i.e., joystick signals are passed directlyto the control valve of the considered DOF. The joy-stick signal and the resulting cylinder motion for theouter jib DOF are shown in Fig. 12.The calibration of the model is carried out in threesteps; first the steady-state cylinder forces are consid-ered, next the steady-state pressures and finally thesystem dynamics are considered.Both steady-state and dynamic simulation is used forthe model calibration. Steady-state simulation is car-ried out in MATLAB R©, and dynamic simulation iscarried out in Simulink R© using an S-function (com-piled C-code) generated from the MapleSimTM model.The main advantage of this approach is a significantincrease in simulation speed. Furthermore, simulatedvalues are quickly compared with measured values byimporting the latter from the MATLAB R© workspaceand plotting them together with the simulated values.The Simulink R© model is shown in Fig. 13.The pressures in the circuit of the considered DOF ismonitored and the measured pressures are comparedwith the simulated pressures. Simulated and measuredlengths of all three cylinders are monitored for verifi-

Figure 13: Simulink model of the crane.

cation, i.e., to calibrate the motion of the consideredcylinder and to ensure that the two remaining cylindersare not moving during simulation.

5.1 Steady-State Forces

A prerequisite for verification of the motion control sys-tem is to verify the mechanical model and to identifyits uncertain parameters. For this purpose a rigid bodysteady-state model of the mechanical system, as shownin Fig. 4, is used to simulate the steady-state pressureforces in the cylinders, in order to compare these with

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Modeling, Identification and Control

the measured pressure forces. The simulation is car-ried out by using the measured cylinder motion givenin Fig. 12 as input and computing the resulting steady-state force on the cylinder.A parameter identification is carried out to identify thefriction parameters for the cylinder model and to tunethe mechanical model parameters, i.e., the masses andpositions of the centers of gravity (COGs) originallyfound via CAD models. Since a CAD model seldomtakes into account all the components in a mechani-cal assembly it is reasonable to allow for a tuning ofthe mechanical properties within a given set of con-straints.The parameter identification is an optimization rou-tine, based on the fmincon function in MATLAB R©,which minimizes the squared deviation between themeasured pressure force, Fp, and the simulated pres-sure force, Fp:

minimize f = (Fp − Fp)2 (27)

The simulated pressure force is computed as:

Fp =

Fcyl − FS − Cp ·∣∣∣Fp

∣∣∣ for vcyl < 0

Fcyl + FS + Cp ·∣∣∣Fp

∣∣∣ for 0 ≤ vcyl(28)

Fcyl is the simulated steady-state force on the cylinderand vcyl is the measured cylinder piston velocity.The measured pressure force and simulated forces afterthe parameter identification are shown in Fig. 14. Theparameter values before and after the identification aregiven in Table 7. As the pedestal and the king (bodies1 and 2) are stationary, they are not included in theparameter identification. The positions of the COGsare according to the local coordinate systems in Fig. 4The steady-state levels of the measured and simulatedpressure forces are nearly equal and the mechanicalmodel along with the identified parameters are there-fore considered reliable.

5.2 Steady-State Pressures

The verification of the steady-state behavior of the con-trol system model mainly depends on the calibrationof the CBV model, i.e., identification of the four modelparameters in (21) and (22). For that the flows, Q1

and Q2, through CBV1 and CBV2 are computed fromthe measured cylinder motion shown in Fig. 12 andused in a steady-state model of the outer jib circuit,together with the two measured pressures, p2 and p3,in order to estimate the two remaining pressures, p1and p4. The measured and estimated state variablesare then used together with the CBV model to iden-tify A0, A1, n0 and n1.The two CBVs in the outer jib circuit are identical with

Table 7: Mechanical model parameters before and afterparameter identification. Positions of COGsare according to Fig. 4.

Parameter Before After

m3 5500 kg 5600 kgm4 1100 kg 1150 kgm5 2950 kg 3000 kgm6 2700 kg 2710 kg

x3 5.3 m 5.4 mz3 0.4 m 0.35 mx4 1.15 m 1.1 mz4 -0.2 m -0.15 mx5 4.9 m 4.8 mz5 -0.26 m -0.21 mx6 1 m 1.57 mz0 0 0

FS (outer jib) - 6.4 kNCp (outer jib) - 0.029

a geometric pilot area ratio of ψ = 6. During opera-tion a small amount of oil flows through the internalorifices of the active CBV, which causes a reduction ofthe actual pilot pressure, px,1 or px,2, compared to theexternal pressure, p1 or p4, at the pilot port. For prac-tical reasons, the external pressure at the pilot port isoften considered as the pilot pressure and therefore theorifices are said to lower the effective pilot area ratio.With the given sizes of the two orifices the effectivepilot area ratio is theoretically ψ = 1.95, which is rep-resented by the dashed black line in Fig. 15. However,this is only valid when the flow through the orifices isturbulent and the back pressure is zero.In practice, there is always a certain level of back pres-sure and the flow through the orifices may not followthe fully turbulent orifice equation at all times. There-fore the original cracking line for ψ = 1.95 is correctedaccording to:

µL,cr =1

1 + C1 · µx + C2 · µ2x

(29)

The corrected cracking line, the solid black line in Fig.15, is adjusted by tuning C1 and C2 until it coincideswith the actual µL values obtained with the measuredand estimated state variables.During extension of the cylinder CBV2 is active andµL is obtained by:

µL,2 =p3 − p4pcr,CBV

(30)

µx is obtained by:

µx,2 =p1p3

(31)

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Bak and Hansen, “Analysis of Offshore Knuckle Boom Crane — Part One”

0 50 100 150-300

-200

-100

0

100

200

300

400

t [s]

F [k

N]

Fp (measured)

Fcyl

(simulated)

Fp (simulated)

Figure 14: Measured and simulated forces for verification of steady-state characteristics.

During retracting of the cylinder CBV1 is the activeone and µL is obtained by:

µL,2 =p2 − p1pcr,CBV

(32)

µx is obtained by:

µx,2 =p4p2

(33)

With the computed µx and µL values, (19) and (20)can be used to simulate the steady-state flow throughthe two CBVs and compare them with the measuredflow in order to identify the four model parameters.As for the mechanical model, the parameter identifi-cation is carried out by means of numerical optimiza-tion. Also here the optimization routine is based on thefmincon function in MATLAB R©, which minimizes thesquared deviation between the measured flow, QCBV ,and the simulated flow, QCBV :

minimize f = (QCBV −QCBV )2 (34)

Fig. 16 show the measured and simulated flows afterthe parameter identification. The identified parame-ters are given in Table 8.The steady-state simulation yields good conformity be-tween the measured and simulated flows, which indi-cate that the suggested CBV model is valid and thatthe identified parameters are reliable.

Table 8: Model parameters for outer jib CBVs.

Parameter Value Parameter Value

C1 1.6267 C2 0.2452A0 27.84 A1 146.2n0 0.4707 n1 0.0359

For the actual verification of the model, a dynamic sim-ulation is carried out with the joystick signal in Fig. 12as input. The flow coefficients of the DCV are adjustedmanually in order to simulate the correct flow and ob-tain the cylinder motion shown in Fig. 17.Once the DCV model is calibrated and the simulatedcylinder motion corresponds to the measured cylindermotion, the steady-state level of the simulated pres-sures can be compared with the measured pressures.Fig. 18 shows the simulated and measured values of p2and p3.There is indeed a good conformity between steady-state levels of the measured and simulated pressures,which verifies the steady-state characteristics of theCBV model and that the identified parameters are use-ful for simulation.

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Modeling, Identification and Control

0 1 2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

μx

μ L

μL,cr

μL,cr

(adjusted)

μL,1

μL,2

Figure 15: Relation between µL and µx for the outerjib CBVs.

5.3 System Dynamics

The remaining uncertain parameters that dominate thesystem dynamics are the flexibility and damping pa-rameters of the mechanical system and the stiffness,bulk modulus, of the hydraulic oil. The actual frictionin the hydraulic cylinders, the dynamics of hydraulicvalves and other hydraulic components also influencethe overall dynamics to some extent, but they are con-sidered to be less important for the considered system.While the steady-state level of the simulated and mea-sured pressures in Fig. 18 correspond well, there is a

01

23

00.5

10

10

20

30

40

50

60

70

μx

μL

QC

BV

[l/

min

]

Q1 (measured)

Q2 (measured)

Q1 (simulated)

Q2 (simulated)

Figure 16: Measured and simulated flows through theCBVs after the parameter identification.

significant difference in dynamic response. The modelis obviously stiffer than the real crane and thereforethe flexibility parameters of the mechanical system aretuned by a scaling factor until the simulated dynamicresponse corresponds to the measured. Simultaneously,the bulk modulus is also adjusted.The main objective for the tuning is to make the fre-quency of the simulated pressure oscillations corre-spond to the measured pressure oscillations. Naturally,the simulated amplitudes should also correspond to themeasured amplitudes. However, there are more uncer-

0 50 100 1502

2.5

3

3.5

4

t [s]

l [m

]

lcyl

(simulated)

lcyl

(measured)

Figure 17: Measured and simulated cylinder motion.

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0 50 100 1500

50

100

150

200

250

t [s]

p [b

ar]

p2 (simulated)

p3 (simulated)

p2 (measured)

p3 (measured)

Figure 18: Measured and simulated pressures for verification of steady-state characteristics.

tain parameters related to the amplitude of the oscilla-tions than for the frequency. Therefore some deviationsare to be expected. Fig. 19 shows the simulated andmeasured pressures after the tuning.During extension of the cylinder, both frequency andamplitude of the simulated pressure correspond verywell with the measurements, except for the accelera-tion phase in the beginning of the sequence. The reasonfor this is most likely the un-modeled dynamics of thecylinder endstops and possibly that the CBV model isnot accurate enough for accelerating flows.During retraction of the cylinder only the frequencycorresponds to some extent in the beginning of the se-quence. The amplitudes do not correspond and thesimulated oscillations are dampened far quicker thanthe measured oscillations. Also here, the likely causesare un-modeled dynamics of the cylinder endstop andinaccuracy of the CBV model. Furthermore, as de-scribed in section 4.3, the friction in the cylinder isquite complex around zero velocity and the appliedcylinder model may be too simple to capture the realbehavior during acceleration.In general though, the simulated response correspondswell to the measurements. The observed deviations arewithin the expectations of what can be achieved witha model of the suggested detail level. The calibratedmodel is suitable for the type of simulations that canbe utilized by system engineers working with hydraulic

system design and/or control system design.To obtain the correspondence shown in Fig. 19, allthe flexibilty parameters are adjusted to 55% of theiroriginal value (Table 2) and the stiffness multipliers,βk, used to determine the damping parameters for thethree flexible members, are set to 0.002, 0.003 and0.007, respectively. Bulk modulus is set to 5500 bar,which is almost half the value of the initial guess.Immediately, it seems surprising that the estimatedstructural stiffness needs to be reduced nearly by afactor of two. However, only three of the crane’s struc-tural components are considered flexible and while theestimated compliance of these individual members maybe correct, the flexibility of the remaining componentsalso contribute to the overall dynamics. These com-ponents include the king, the intermediate jib and thefoundation on which the pedestal is mounted. In addi-tion, the connections between the individual structuralcomponents may also offer some flexibility.The identified value of bulk modules is significantlylower than the theoretical value of a typical hydraulicoil, even when accounting for a certain amount of en-trained air. However, the compliance of pipes andhoses will also lower the effective bulk modulus. Ac-cording to Merritt (1967) the effective bulk modulusis usually not more than 100,000 psi (approximately6900 bar). The identified value of 5500 bar is thereforeconsidered reliable.

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Modeling, Identification and Control

0 50 100 1500

50

100

150

200

250

t [s]

p [b

ar]

p2 (simulated)

p3 (simulated)

p2 (measured)

p3 (measured)

Figure 19: Measured and simulated pressures for verification of dynamic characteristics.

Estimating the damping of a complex system like theconsidered crane is obviously difficult. According to(5) the stiffness multipliers should be reduced alongwith the stiffness of the structural members, becausethe natural frequencies are lowered. Since the stiffnessmultipliers are actually increased, it implies that thedamping ratio of ζ = 0.03− 0.07 is too low. However,the un-modeled dynamics of structural components,other than the three considered, will also contributeto the total system damping. Furthermore the connec-tions between the structural components will also offersome damping in terms of friction.Therefore the stiffness multipliers determined in sec-tion 3.3 may be reasonable when considering thethree structural members individually. For a completemodel, though, the measurement show that additional

damping needs to be introduced.

6 Conclusions

In this paper a model of an offshore knuckle boomcrane has been presented. The model is developed inMapleSimTM and includes both the crane’s mechan-ical system and the electro-hydraulic motion controlsystem.The mechanical system is modeled as a two-dimensional multi-body system which includes thestructural flexibility and damping. The finite segmentmethod is used to model the flexibility and a pro-cedure for estimating the structural damping is pre-sented. Though these methods do not represent the

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Bak and Hansen, “Analysis of Offshore Knuckle Boom Crane — Part One”

state of the art within flexibility modeling, it is shownthat they are sufficient for the given modeling purpose.Furthermore, they are advantageous in terms of mod-eling effort and computational requirements.The motion control system is modeled using mostlysemi-physical modeling techniques in order to reducethe computation requirements without neglecting orunderestimating important physical phenomena. Formodeling of the CBVs in the hydraulic system, a novelblack-box approach is presented which uses two differ-ent pressure ratios to compute the flow through thevalve. This approach is, however, based on having acertain amount of experimental data available.The crane model is calibrated and verified with exper-imental data through three different steps:

1. Verfication of the steady-state characteristics ofthe mechanical system model by identifying thecylinder friction parameters and tuning the massesand COG positions of the bodies in the model.

2. Verification of the steady-state characteristics ofthe motion control system model, mainly by iden-tifying the unknown CBV model parameters.

3. Verification of the dynamic behavior of the cranemodel by tuning the flexibility and damping pa-rameters of the mechanical system model and thestiffness, bulk modulus, of the hydraulic oil.

For the first two steps an optimization procedure is ap-plied to efficiently identify the unknown parameters. Inthe last step the estimated parameters are simply ad-justed by a scaling factor until the simulated responsecorresponds to the measured response.The demonstrated modeling and parameter identifica-tion techniques target the system engineer by takinginto account the limited access to component data nor-mally encountered by engineers working with systemdesign. The verified crane model is an example of avirtual prototype which can be used to evaluate andimprove the design of the considered system.

Acknowledgements

The work presented in this paper is funded by the Nor-wegian Ministry of Education and Research and AkerSolutions.The authors would like to thank Morten Abusdal (AkerSolutions) for his help with the experimental work.

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