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Robust Tool Point Control for Offshore

Knuckle Boom Crane

M. K. Bak M. R. Hansen H. R. Karimi

Department of Enginering, Faculty of Engineering and Science, University of Agder, NO-4898 Grimstad, Norway (e-mails:[email protected];[email protected];[email protected])

Abstract: This paper considers the design of an H ∞ controller for tool point control of ahydraulically actuated knuckle boom crane. The paper describes the modelling of the crane’smechanical and hydraulic systems and a disturbance model. These are linearised and combinedin a state-space model used for the controller design. The controller synthesis problem is todesign (if possible) an admissible controller that solves the problem of robust regulation againststep inputs with an H ∞ constraint based on the internal model principle. Simulation results are

given to show the effectiveness of the method.Keywords: Multi-input/multi-output systems, state feedback, robust control.

1. INTRODUCTION

This paper describes a design approach for developmentof an H ∞ controller for tool point control for an offshoreknuckle boom crane used for pipe handling on a drillingrig. The purpose is to achieve robust regulation againststep inputs based on the internal model principle.Offshore equipment in general is constantly exposed toincreasing demands regarding productivity, reliability and

safety in order to maximise capacity and uptime and toreduce risks of harming equipment, personal and environ-ment. For the considered type of crane, one consequence of this is generally a demand for efficient tool point control.Tool point control, i.e. direct control of the tool pointby automatic coordinated control of the individual DOF(degrees of freedom) as opposed to direct control of eachDOF to achieve control of the tool point, has been sub- jected to research for several years. In the early ninetiesa simple vector control strategy for a two-DOF crane,like the knuckle boom type, was developed by Krus andPalmberg (1992). Mattila and Virvalo (2000) presented amore advanced control strategy for a similar crane, whichreduces energy consumption by reducing pressures levelsin the hydraulic system.Munzer (2003) developed a control scheme for exiblecranes with redundant DOF. Since then, work on simi-lar cranes has been followed by several others. Ebbesenet al. (2006) presented a control strategy based on optimalcylinder velocities determined by minimisation of energyconsumption. Recently Pedersen et al. (2010) developed acontrol scheme by means of dynamic real-time simulationbased on a pseudo inverse approach.These strategies are either purely feedforward based, whichyield fast yet still stable response, but in some cases sac-rices tracking performance. Or they involve feedback of several measured signals and need for certain components,

which may result in expensive systems.The work presented in this paper is funded by the Norwegian

Ministry of Education and Research and Aker Solutions.

The goal of the work presented in this paper is to investi-gate if an H ∞ controller, based on measured signals, candeliver satisfying performance regarding speed and track-ing and at the same time remain robust in the presence of disturbances.H ∞ based design approaches for tracking problems hasbeen developed by Abedor et al. (1995) and furthermoreH ∞ control has successfully been applied to numerouscrane control problems. Recent examples are control sys-

tem design for gantry cranes, presented by Burul et al.(2010), controller design for rotary crane, presented byIwasa et al. (2010), and control of an overhead crane byMoradi et al. (2009).The theory of H ∞ control is already well established andtherefore the work presented in this paper is focused onmodelling of the physical system and applying the con-troller design approach.The paper is organised in the following way: rst theknuckle boom crane and functionality is described. Afterthis the relevant models are established, then linearisedand formulated as a state-space model. Next the controllerdesign is presented and nally performance is evaluated bya presentation of simulation results.Model parameters and comprehensive expressions aregiven in the appendix.

2. CONSIDERED SYSTEM

The considered knuckle boom crane is used on a semi-submersible drilling rig, i.e. a oating rig that is partiallysubmersed for increased stability. Because the rig is oat-ing its movements act as disturbances on the equipmenton board.The crane is used to move drill pipes between the pipe deckof the rig and a tubular feeding machine, which handlesthe further transportation of the drill pipes. As illustrated

in Fig. 1, the crane consists of ve main parts; a pedestal,a rotating part for slewing of the crane, an inner jib, anouter jib and a gripping yoke.

Preprints of the 18th IFAC World CongressMilano (Italy) August 28 - September 2, 2011

Copyright by theInternational Federation of Automatic Control (IFAC)

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Pedestal

Rotating part

Inner jib

Outer jib

Gripping yoke

Pipe deck

23 . 5 m

16 m

X

Y

Fig. 1. Offshore knuckle boom crane for pipe handling.

The controller design will focus only on the two-dimensionalproblem of tool point control, i.e. controlling the planarmotion of the tool point. The functionalities of the rotatingpart and the gripping yoke is not considered.

The inner and outer jibs are actuated by hydraulic cylin-ders, which must be controlled in a coordinated mannerin order to achieve X-Y-motion control of the tool point.The cylinders are controlled as illustrated by Fig. 2, bytwo pressure compensated proportional directional controlvalves and counter balance valves for load lowering.

p S

p R

Fig. 2. Hydraulic system for control of the crane jibs.

The system is supplied from a ring-line with constantsupply and return pressures pS and pR .

3. MODEL DESCRIPTION

The mechanical structure is modelled according to Fig. 3.

The global rotations φ1 and φ2 of the inner and outer jibsand the velocities φ1 and φ2 are chosen as state variablesfor the mechanical system. For convenience the rotationsof the hydraulic cylinder, θ1 and θ2 , are introduced. Theseare dependent coordinates, which can be determined whenφ1 and φ2 are known.The dotted red line between the points P A and P B rep-resents the operation to be considered - lifting a payloadfrom the pipe deck in a straight vertical motion. Whenconsidering this operation, the hydraulic system can besimplied and modelled according to the circuit in Fig. 4.

Cylinder 1 must extend in order to raise the inner jib and

cylinder 2 must retract in order to keep point C movingalong a straight path. For the given geometric congura-tion the load force and the velocity of cylinder 2 has the

A

B

C

D

E

F

G

θ 1

φ 1

θ 2 φ 2

Load δh = 10 m

P A

P B

Pipe deck

X

Y

Fig. 3. Simplied geometry of the crane.

p S

p R

p 1 p r 1 p r 2 p 2

Q 1 Q 2

V 1 V 2v 1 v 2

F 1 F 2

Fig. 4. Simplied hydraulic circuit.same direction. In practice a certain outlet pressure, pr 2 ,will be present.The state variables for the hydraulic system are the pres-sures p1 and p2 . The outlet pressures, pr 1 and pr 2 , can beassumed constant during the operation.The state variables may be assembled in the state vector:

x = φ1 φ1 φ2 φ2 p1 p2T

(1)

3.1 Mechanical System

Since only the state variables and their time derivativesare of interest, the most convenient way to model themechanical system is by means of Lagrange’s equation:

ddt

∂L∂ q i

− ∂L∂q i

= QNC i (2)

where L = T − V , T being the kinetic energy and V beingthe potential energy of the system.In (2) q i and q i are the generalised coordinates and their

time derivatives. In this case φ1 and φ2 are the generalisedcoordinates.QNC

i are the non-conservative generalised forces. In thiscase they correspond to the moments around points A andB produced by the cylinder forces. They are given by:

QNC i =

n

j =1

F NC j ·

∂ r j

∂q i(3)

F NC j is the force vector at the j’th connection point and

r j is the position vector to the same point.Comprehensive expressions for the components of (2)-(3)are given in appendix A.1, A.2 and A.3.Obtaining the equations of motion yields the followingequation system:

a 11 a12

a 21 a22· φ1

φ2 = b1

b2 (4)


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where a 11 and a 22 are constant, a 12 and a 21 are functionsof φ1 and φ2 and b1 and b2 are functions of all six statevariables. Expressions for these components are given by(A.39)-(A.44) in appendix A.4.φ1 and φ2 can be expressed independently by applyingCramer’s rule:

φ1 = b1 · a 22 − a 12 · b2

D (5)

φ2 = a 11 · b2 − b1 · a 21

D (6)

whereD = a11 · a 22 − a 12 · a21 (7)

3.2 Hydraulic System

By means of the ow continuity equation:

Q in − Qout = V + V β

· ˙ p (8)

the pressure gradients can be expressed as:

˙ p1 = β e · (Q1 − v1 · A p 1 )

V 1(9)

˙ p2 = β e · (Q2 + v2 · CR 2 · A p 2 )

V 2(10)

where β e is the effective stiffness of the hydraulic uid andA p 1 and A p 2 are the piston areas for the cylinders.v1 and v2 are the cylinder piston velocities, which arefunctions of the mechanical state variables. Expressions forthe two piston velocities are given by (A.15) and (A.18) inappendix A.1.Q1 and Q2 are the controlled ows given as:

Q1 = u1 · Qmax (11)Q2 = u2 · Qmax (12)

where u1 and u2 are the control signals. Qmax is themaximum ow available for each control valve.V 1 and V 2 are the actual sizes of the control volumes andare given by:

V 1 = A p 1 · (lDE − l1 ,min ) (13)V 2 = CR 2 · A p 2 · (h2 − (lF G − l2 ,min )) (14)

where lDE and lF G are the actual lengths of the cylinders,

derived from r DE and r F G given by (A.13) and (A.16) inappendix A.1. l1 ,min and l2 ,min are the collapsed lengths.CR 2 is the ratio between piston-side-area and the rod-side-area for cylinder 2 and h2 is the cylinder stroke.

3.3 Wave Disturbance

The disturbance model describes the heave motion of the rig where the crane is located. The roll and pitchmotions are not considered since the effect of these areless signicant. The heave disturbance is described as aharmonic motion:

yh (t) = − ω2h · Ah · sin(ωh t) (15)

where yh is the heave acceleration, Ah is the heave ampli-tude and ωh is the heave frequency.The heave acceleration acts on the system masses and

yields a disturbance force for each mass, which can bedescribed as generalised forces by means of (3). In thisway the angular accelerations of the two crane jib, causedby the disturbance can be described as:

a 11 a12

a 21 a22· δ φ1

δ φ2 = δb1

δb2 (16)

where δ φ1 and δ φ2 are the accelerations caused by gener-alised disturbance forces δb1 and δb2 , which are given by(A.54)-(A.55) in appendix A.5. a11 , a12 , a21 and a22 arethe same as in (4).

3.4 State-Space Representation

In order to establish a state-space model for the controllerdesign, the mechanical and hydraulic models must belinearised.The linearised equations of motion can be written as:

φ1 =n

i =1

∂ ∂x i

b1 · a 22 − a 12 · b2

Dx ss

· x∗

i (17)

φ2 =n

i =1

∂ ∂x i

a11 · b2 − b1 · a 21

Dx ss

· x∗

i (18)

where xi is the i’th state variable, x ss are the steady-state values of the state variables in the selected operatingpoint. x∗

i = xi − x i,ss which means that the linearisedmodel relates to deviations of the state variables aroundthe operating point.The linearised ow continuity equations can be written as:

˙ p1 = β eV 1

· Qmax · u1 −n

i =1

β eV 1

· A1 · ∂v1

∂x ix ss

· x∗

i (19)

˙ p2 = β eV 2

· Qmax · u2 −n

i =1

β eV 2

· A2 · ∂v2

∂x ix ss

· x∗

i (20)

As seen from (13)-(14), V 1 and V 2 are also non-linearterms. Instead of linearising these, steady-state values willbe calculated, which is assumed to be constant within acertain range around the selected operating point.For the controller design the disturbance will represent aworst case situation by setting Ah = 2m and ωh = 0 .5Hz

giving a maximum heave acceleration of ¨ yh = 0 .5m/s2

.The angular accelerations caused by the disturbance canthen be described similar to (5)-(6) in the following way:

δ φ1 = δb1 · a 22 − a12 · δb2

D (21)

δ φ2 = a 11 · δb2 − δb1 · a 21

D (22)

which are constant disturbances that can be scaled by anexogenous input.The performance of the system is expressed by the exoge-nous output z, which is the position, X tp and Y tp , of thecrane’s tool point and the control signals u1 and u2 . Thetool point position, denoted z1 and z2 , is dependent on φ1

and φ2 , a non-linear relation which needs to be linearised:


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angles φ1 and φ2 in order to track the tool point positiongiven by z . The control problem is illustrated by Fig. 5.

P

K

u y

w z

φ 1 , φ 2

Fig. 5. P-K structure illustrating the control problem.

In (31) L is the unique solution to the Lyapunov equation:L A − AL = B (34)

where A = A + ( γ − 2 B 1 B T 1 − B 2 B T

2 )X . γ is a tuningparameter and X is the positive semidenite stabilisingsolution to the Riccati equation:

A T X + XA + X (γ − 2 B 1 B T 1 − B 2 B T

2 )X + C 1 C T 1 = 0 (35)

In (31) W is the positive-denite solution satisfying theLyapunov inequality:

AW + W AT

+ γ − 2 LB 1 B T 1 L T − LB 2 B T

2 L T < 0 (36)

5. SIMULATION

The performance of the system with the implementedcontroller is simulated with MATLAB/Simulink R . Thesimulation shows the performance of the controller in thesituation where the tool point is to move up 0 .5m ina straight line. The state-space model is assumed to berepresentative during the whole operation.The reference signals are calculated by inverse kinematicsfor the new point ( X tp , Y tp ) = (25 m, 0.5m) which givesthe new jib angles φ1 = 0.1315rad and φ2 = − 1.4786rad .Since the state-space model relates to deviations of thestate variables, the reference signals are determined bysubtracting the angles for initial point from the angles forthe new point:

φ1 ref = φ1 new − φ1 ini = 0.0212rad (37)φ2 ref = φ2 new − φ2 ini = 0.0033rad (38)

Setting the tuning parameter γ = 1 the controller is foundto be:

K ∼A K B K C K D K

(39)

where

A K =0 0

0 0, B K =

1 0 0 0 0 0

0 1 0 0 0 0, C K =

− 0 . 0021 0

− 0 . 0005 0(40)

D K =− 22 . 757 − 0 . 094 − 3 . 949 − 0 . 221 − 0 . 0004 0 . 0004

− 5 . 643 0 . 108 17 .797 0 . 639 0 . 0002 − 0 . 0016(41)

In the simulation the reference is given as a step input,which the system should track as fast and precise as

possible without introducing instability.Tracking of the relative tool point position is illustrated inFig. 6 both with and without the heave disturbance.

0 2 4 6 8 100

0.2

0.4

0.6

t [sec]

P o s i t i o n

[ m ]

Relative position of tool point

Xtp

Ytp

0 2 4 6 8 100

0.2

0.4

0.6

t [sec]

P o s i t i o n

[ m ]

Relative position of tool point during disturbance

Xtp

Ytp

Fig. 6. Tracking of tool point.

0 2 4 6 8 100

0.2

0.4

0.6

t [sec]

C o n

t r o l s i g n a l

Control signals

u1

u2

0 2 4 6 8 100

0.2

0.4

0.6

t [sec]

C o n t r o l s i g n a l

Control signals during disturbance

u1

u2

Fig. 7. Control signals.

The designed controller yields good tracking and distur-

bance attenuation, however the response is slow.The usage of H ∞ control signal is illustrated in Fig. 7.This reveals the reason for slow response, since only half of the available ow is utilised. The controller guarantiesstability, however in this case the controller design mightbe too conservative, sacricing the system performance.To attenuate the disturbance the control signal becomesnegative in certain periods. For this model this is not phys-ical meaningful, since the ow control valves in Fig. 4 isunidirectional. To deal with this phenomena, a saturationof the control signal must be introduced or the hydraulicmodel must be extended to include more of the physicalsystem’s capabilities.

6. CONCLUSIONS

The purpose of the presented work has been to investigateif an advanced control method can be applied to achievesatisfactory performing tool point control for a hydrauli-cally actuated knuckle boom crane. The designed con-troller shows good tracking performance and robustness,but is rather slow reacting. Compared to other and moreconventional methods for tool point control, this is notsatisfactory. However by including additional techniques,such as loop shaping, in the design approach, the perfor-mance is likely to be improved. This is one relevant taskfor future work.

Other targets for future work is to extend the hydraulicmodel to include more of the functionalities of the physicalsystem, and thereby improving the capabilities of the con-


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troller. Finally it must be possible to handle more completeduty cycles than the one the controller has been designedfor. As mentioned in section 4, this may be solved by mak-ing a gain schedule from several models and controllers fora number of operating points representing the operationto be considered. Another approach is to reformulate thecrane model as a linear parameter varying (LPV) modeland apply the presented design approach to this model.If the controller performance can be improved, the LPVapproach is indeed worth investigating, since it offers asmooth variation of controller gain during operation of thecrane.Alternatively, other control strategies, such as feedbacklinearisation or sliding-mode control, may be considered.Both strategies have succesfully been applied in othercrane control problems by Park et al. (2007) and Ngo andHong (2011), respectively.

REFERENCES

Abedor, J., Nagpal, K., Khargonekar, P.P., and Poolla,K. (1995). Robust regulation in the presence of norm-bounded uncertainty. IEEE Transactions on Automatic Control , 40(1), 147–153.

Burul, I., Kolonic, F., and Matusko, J. (2010). The controlsystem design of a gantry crane based on h-innitycontrol theory. MIPRO, 2010 Proceedings of the 33rd International Convention, Opatija, Croatia , 183–188.

Ebbesen, M.K., Hansen, M.R., and Andersen, T.O. (2006).Optimal tool point control of hydraulically actuatedexible multibody system with an operator-in-the-loop.III European Conference on Computational Mechanics,Solids, Structures and Coupled Problems in Engineering,Lisbon, Portugal , 568–568.Iwasa, T., Terashima, K., Jian, N.Y., and Noda, Y. (2010).Operator assistance system of rotary crane by gain-scheduled h-innity controller with reference governor.2010 IEEE International Conference on Control Appli-cations, part of 2010 IEEE Multi-Conference on Sys-tems and Control, Yokohama, Japan , 1325–1330.

Krus, P. and Palmberg, J.O. (1992). Vector controlof a hydraulic crane. Procedings of the International Off-Highway and Powerplant Congress and Exposition,Millwaukee, USA.

Mattila, J. and Virvalo, T. (2000). Energy-efficient motioncontrol of a hydraulic manipulator. Procedings of the 2000 IEEE International Conference on Robotics and Automation, San Fransisco, USA , 3000–3006.

Moradi, H., Hajikolaei, K.H., and Bakhtiari-Nejad, F.(2009). Robust h-innity control of an over-head cranesystem with parametric uncertainties. Proceedings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engi-neering Conference, San Diego, USA , 377–384.

Munzer, M.E. (2003). Resolved motion control of mobilehydraulic cranes. Ph.D. Thesis, Aalborg University .

Ngo, Q. and Hong, K.S. (2011). Sliding-mode antiswaycontrol of an offshore container crane. IEEE/ASME Transaction on Mechatronics , 16.

Park, H., Chwa, D., and Hong, K.S. (2007). A feedback

linearization control of container cranes: varying ropelength. International Journal of Control, Automation and Systems , 5(4), 379–387.

Pedersen, M.M., Hansen, M.R., and Ballebye, M. (2010).Developing a tool point control scheme for a hydrauliccrane using interactive real-time dynamic simulation.Modeling, Identication and Control , 31(4), 133–143.

Appendix A. MODEL DESCRIPTION

The appendix presents comprehensive expressions of theequations derived to establish the model of the crane’smechanical systems. It also contains expressions for thedisturbance model and tool point kinematics. To view theappendix please go to:

http://www.uia.no/en/portals/about_the_university/engineering_and_science/-_engineering/-_-_mechatronics/bak

and select the link for pulication [C3] on the list.


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Bak 2011 Robust