Abstract—To increase the effectiveness of the cargo transshipment process is one of the most important objectives for the automation of cranes. Therefore new control strategies are applied. This paper presents a nonlinear controller in order to solve the trajectory tracking and disturbance rejection problem for a boom crane. A model of the crane dynamics in radial direction is derived considering the dominant nonlinearities such as the actuator kinematics. Further on, the coupling of a slewing and luffing motion is taken into account. This coupling is caused by the centrifugal acceleration of the load in radial direction during a slewing motion. Based on the nonlinear model a linearizing and disturbance decoupling control law is derived and discussed. Measurement results from the boom crane validate the good performance of the nonlinear controller.

I. INTRODUCTION The paper addresses the problem of trajectory tracking

and disturbance rejection for the transportation of crane loads. Thinking of the requirement of a very fast and efficient transshipment of cargo in harbours, more and more automation concepts for cranes are developed and utilized. One of them is the CYCOPTRONIC system (Sawodny et al. [13], Arnold et al. [14], Neupert et al. [15]) provided by the company “Liebherr Werk Nenzing” as anti-sway control for Harbour Mobile Cranes (see Fig. 1). This kind of cranes are boom cranes which are characterized by a load capacity of up to 140 tons, a maximum outreach of 48 meters and a rope length of up to 80 meters. In case of such rotary boom cranes the slewing and luffing movements are coupled. That means a slewing motion induces not only tangential but also radial load oscillations because of the centrifugal force. This leads to the first challenge for the advancement of the existing control concept, the synchronization of the slewing and luffing motion in order to reduce the tracking error and ensure a swing-free transportation of the load. The second challenge is the accurate tracking of the crane load on the desired reference trajectory during luffing motion because of the dominant nonlinearities of the dynamic model. This nonlinear model of the crane dynamics for the radial direction excluding the influence of the centrifugal forces is derived and discussed in Neupert et al. [16]. Further on, the

Jörg Neupert, Tobias Mahl, and Prof. Oliver Sawodny are with the

Institute for System Dynamics, Universitaet Stuttgart, Stuttgart, Germany (corresponding author to provide phone: +49-711-68569899; fax:+49-711-68566371; e-mail: joerg.neupert@isys.uni-stuttgart.de).

Dr. Klaus Schneider is divisional director at the Liebherr Werk Nenzing GmbH, Postfach 10, A-6710, Austria, (e-mail: klaus.schneider@liebherr.com).

design of the flatness based controller is presented. The theoretical foundation for the design of control

structures for nonlinear systems and there analysis was introduced in numerous publications. Isidori et al. [1], [2] for example consider asymptotic output tracking of a certain class of nonlinear systems, where the reference or disturbance signals are generated by an exosystem. To calculate the feedforward trajectory partial differential equations are solved. Fliess et al. [3] discuss the differential flatness of nonlinear systems. They formulate the major property of differential flatness and propos the defect of a nonlinear system as a non-negative integer, which measures the distance from flatness. Other contributions are focused on the feedforward control for nonlinear systems. For example Hagenmeyer et al. [4] presents a flatness based design of linear and nonlinear feedforward controls. Additionally the problems of realizability of a feedforward control and the instability in case of non-minimum-phase systems are discussed.

Fig. 1. LIEBHERR Harbour Mobile Crane LHM 402

Another problem, which is addressed in the literature, is the disturbance decoupling in nonlinear systems. Andiarti et al. [5] obtain necessary and sufficient conditions for solving the disturbance decoupling problem by quasi-static and dynamic output feedback. Additional contributions on disturbance decoupling by measurement feedbacks are made by Battilotti [6] and Xia et al. [7]. If the system can be described by rational differential equations or transformed into such a description, the algorithm presented by Broecker et al. [8] solves the disturbance decoupling problem.

But there are also examples and applications of the

A Nonlinear Control Strategy for Boom Cranes in Radial Direction Jörg Neupert, Tobias Mahl, Oliver Sawodny, Klaus Schneider

Proceedings of the 2007 American Control ConferenceMarriott Marquis Hotel at Times SquareNew York City, USA, July 11-13, 2007

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nonlinear control theory. Thinking of cranes, Fliess et al. [9] for example study a two-dimensional overhead crane. The system is characterized as a differential flat system by deriving a linearizing output. Other publications from Piazzi et al. [10] and Yanai et al. [11] are also focussed on the inversion based control of overhead cranes. This is why cranes are a typical example of an underactuated mechanical system with oscillatory behaviour. Kiss et al. [12] show differential flatness for a class of cranes including overhead and rotary cranes.

In this paper the boom crane is described by a nonlinear model, which is derived in section 2. The kinematics and dynamics of the luffing motion are considered and the influence of the centrifugal acceleration is modelled as a time-varying disturbance. Based on the nonlinear dynamic model a control input, which contains a linearizing, a tracking, a stabilizing and a disturbance decoupling part, is derived in section 3. This is done by expanding the nonlinear model of section 2. Additionally the internal dynamics of the expanded nonlinear model is discussed. In section 4 measurement results are presented and analysed. The measurements are obtained by applying the control structure to the Liebherr Harbour Mobile Crane (LHM 322). In the last section concluding remarks are given.

II. NONLINEAR MODEL OF THE CRANE The performance of the crane’s control is mainly

measured by fast damping of load sway and exact tracking of the reference trajectory. To achieve these control objectives the dominant nonlinearities have to be considered in the dynamic model of the luffing motion.

The first part of this model is derived by utilizing the method of Newton/Euler. Making the simplifications

• rope’s mass and elasticity is neglected, • the load is a point mass, • coriolis terms are neglected

result in the following differential equation which characterizes the radial load sway.

( ) ( )

( ) ( )( ) 2

cossin - ...

cos... sin

SrSr Sr A

S S

SrA S Sr D

S

g rl l

r ll

ϕϕ ϕ

ϕϕ ϕ

+ = +

+

&& &&

&

(1)

As shown in Fig. 2, Srϕ is the radial rope angle, Srϕ&& the radial angular acceleration, Dϕ& the cranes rotational angular velocity, Sl the rope length, Ar the distance from the vertical axe to the end of the boom, Ar&& the radial acceleration of the end of the boom and g the gravitational constant. ZF represents the centrifugal force, caused by a slewing motion of the boom crane.

The second part of the nonlinear model is obtained by taking the actuators kinematics and dynamics into account. The dynamics of this actuator (hydraulic cylinder) can be approximated with a first order system.

LAr

Al

AϕSl

Srϕ

LmZF

GF

Ar

Dϕ& Fig. 2. schematics of the boom crane in radial direction

Considering the actuators dynamics, the differential equation for the motion of the cylinder is obtained as follows

1 VW

zyl zyl lW W zyl

Kz z uT T A

= − +&& & (2)

Where zylz&& and zylz& are the cylinder acceleration and

velocity respectively, WT the time constant, zylA the cross-

sectional area of the cylinder, lu the input voltage of the servo valve and VWK the proportional constant of flow rate to lu . In order to combine equation (1) and (2) they have to be in the same coordinates. Therefore a transformation of equation (2) from cylinder coordinates ( zylz ) to outreach

coordinates ( Ar ) with the kinematical equation

( )2 2 2

0cos arccos2

a b zylA zyl A A

a b

d d zr z l

d dα

⎛ ⎞⎛ ⎞+ −= ⎜ − ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

(3)

and its derivatives

( ) ( )( ) ( ) ( )

1

21 3

sin

sinA A A Wz A zyl

A A A Wz A zyl Wz A zyl

r l K z

r l K z K z

ϕ ϕ

ϕ ϕ ϕ

= −

= − −

& &

&& && & (4)

is necessary. Where the dependency from the geometric constants 1 2, , ,a bd d α α and the luffing angle Aϕ is substituted by 1WzK and 3WzK . The geometric constants, the luffing angle and Al , which is the length of the boom, are shown in Fig. (3).

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x

r

Aϕ

ad1α

bd

z

Zylz

2α

Zylz&

Fig. 3. cylinder kinematics

As result of the transformation, equation (2) can be displayed in outreach coordinates.

( ) {

( ) 1232 2 2

1

sin1sin

VW A A WzWzA A A l

W W zylA A Wz

ba m

K l KKr r r u

T T Al Kϕ

ϕ= − − −&& & &

1442443 144424443

(5)

In order to obtain a nonlinear model in the input affine form

( ) ( ) ( )( ) ;lx f x g x u p x w y h x= + + =& (6)

equations (1) and (5) are used. The second input w represents the disturbance which is the square of the crane’s rotational angular speed 2

Dϕ& . With the states defined as

[ ]TA A Sr Srx r r ϕ ϕ= && and the output LAy r= follow the

vector fields

( )

( ) ( )( )

( )( )

222 2

4

23 33 2 2

3 1 3

0

; 0cos( ) cos( )sin

cos( ) sin( ) 0 0 0

S S S

T

S

S

xax bx m

f x g xxg x x mx ax bxl l l

x x l xp x

l

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥− − −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥− + +⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤+= ⎢ ⎥

⎢ ⎥⎣ ⎦

(7)

and the function

( ) ( )1 3sinSh x x l x= + . (8)

III. NONLINEAR CONTROL APPROACH The following considerations are made assuming that the

right site of the differential equation for the load sway can be linearized.

( ) ( ) 21 1sinSr Sr A A S Sr DS S S

g r r ll l l

ϕ ϕ ϕ ϕ+ = − + +&& &&& (9)

In order to find a linearizing output for the simplified nonlinear system the relative degree has to be ascertained.

A. System’s Relative Degree The relative degree concerning the systems output is

defined by the following conditions

( )( )1

0 0, 2

0

ig f

r ng f

L L h x i r

L L h x x R−

= ∀ = −

≠ ∀ ∈

K (10)

The operator fL represents the Lie derivative along the

vector field f and gL along the vector field g

respectively. With the real output

1 3sin( )Sy x l x= + (11)

a relative degree of 2r = is obtained. Because the order of the simplified nonlinear model is 4, y is not a linearizing output. But with a new output

( ) 1 3Sy h x x l x∗ ∗= = + (12)

a relative degree of 4r = is obtained. Assuming that only small radial rope angles occur, the difference between the real output y and the flat output *y can be neglected.

B. Disturbance’s Relative Degree The relative degree with respect to the disturbance is

defined as follows:

( ) 0 0, 2ip f dL L h x i r= ∀ = −K (13)

Here it is not important whether dr is well defined or not. Therefore the second condition can be omitted. Applying condition (13) to the reduced nonlinear system (equations (6), (7) and simplification of equation (9)) with the linearizing output *y the relative degree is 2dr = .

C. Disturbance Decoupling Referring to Isidori [2], any disturbance satisfying the

following condition can be decoupled from the output.

( ) 0 0, 1ip fL L h x i r= ∀ = −K (14)

This means the disturbance’s relative degree dr has to be larger than the system’s relative degree. When there is the possibility to measure the disturbance a slightly weaker condition has to be fulfilled. In this case it is necessary that the relative degrees dr and r are equal. Due to these two conditions it is in a classical way impossible to achieve an output behaviour of our system which is not influenced by the disturbance. This can also easily be seen in Fig. (4),

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where the system is displayed in the Control Canonical Form with input lu , states 1 4,...,z z and disturbance Dϕ& .

4z =& K

( IV )

4z y≠& 4z y≠ &&& 3z y≠ && y&lu y

Dϕ&

y&&

,D Dϕ ϕ& &&, ,D D Dϕ ϕ ϕ& && &&& Fig.4. System in Control Canonical Form

D. Model expansion To obtain a disturbance’s relative degree which is equal to

the system’s relative degree a model expansion is required. With the introduction of 2dr r− = new states which are defined as follows,

( )

( )

5

6

2*

62

D

D

D

w xd w xdt

d w x wdt

ϕ

ϕ

ϕ

= =

= =

= = =

&

&&

&&&&

(15)

the new model is described by the following differential equations

{***

25

* *6

( )( )( )

( ) ( ) ( ) 00 0 ; ( )

0 0 1l

p xg xf x

f x p x x g xx x u w y h x

⎡ ⎤+ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= + + =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

&

1231442443

(16)

This Expansion remains the system’s relative degree

unaffected whereas the disturbance’s relative degree is enlarged by 2. The additional dynamics can be interpreted as a disturbance model. The expanded model, whose structure is shown in Fig. (5), satisfies condition (14) and the disturbance decoupling method by Isidori [2] can be used.

Disturbance Model

*( ) ( ) ( )4 Wz z z u z wα β γ= + +&

( IV )

4z y=& 4z y= &&& 2z y= &lu1z y=

I

I

I

. . .

I*w 6z 5z

Fig. 5. Extended System in Byrnes/Isidori Form

E. Input/Output Linearization Hence the expanded model has a system and disturbance

relative degree of 4 and the disturbance *w is measurable, it can be input/output linearized and disturbance decoupled with the following control input

( )( )

( )( )( )

* * *

* * * * * *

* 1 *

*, 1 * 1 * 1 *

Linearization Decoupling Tracking ... new input

4 2 2 24 5 6 3 5 3 6 4 3

3 3

( ) ( )

( ) ( ) ( )

4 2 sin..

cos cos

r rf p f

l Lin r r rg f g f g f

v

S S

L h x L L h x vu wL L h x L L h x L L h x

x x x x x x x l gx x l

mg x mg x

−

− − −= − − +

+ += − − − −

1442443 1442443 1442443

( ) ( )( )( )

( ) ( ) ( )( )( )

( ) ( )

2 2 25 3 3 5 3 2 5 6 1 6

3

4 2 21 5 3 2 2 5 1 3

3

*1 3 5

3 3

.

sin cos 4 2... ...

cos

cos - sin... ...

cos

2 ( )...

cos cos

S

S

S S S

gx x gx x x x x x x x l

mg x

x x l g x ax bx x x g x

mg x

l x l x x l vw

mg x mg x

⎛⎜⎜⎝

− − + ++

⎞+ + +⎟ −⎟⎠

⎞⎛ ⎛ ⎞+ −− +⎟⎜ ⎜ ⎟⎜ ⎜ ⎟⎟⎝ ⎝ ⎠⎠

(17)

To stabilize the resulting linearized and decoupled system a feedback term is added. The term (equation (18)) compensates the error between the reference trajectories *

refy

and the derivatives of the output *y .

( )

( )

*

* *

( )1

0, 1

iri

i reffi

l Stab rg f

k L h x yu

L L h x

−∗ ∗

=

− ∗

⎡ ⎤−⎢ ⎥

⎣ ⎦=∑

(18)

The feedback gains ik are obtained by the pole placement technique. Fig. 6 shows the resulting control structure of the linearized, decoupled and stabilized system with the complete input , ,l l Lin l Stabu u u= − . The effect caused by the usage of the fictive output in stead of the real one is discussed by Neupert et al. [19]. There it is shown that the resulting internal dynamics near the steady state is at least marginal stable. Therefore the fictive output can be applied for the controller design.

Fig. 6. control structure for luffing

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F. Internal Dynamics Another effect of the model expansion has to be

considered. Hence the system order increases from 4n = to * 6n = but the system’s relative degree remains constant,

the system loses its flatness property. Thus it is only possible to obtain an input/output linearization in stead of an exact linearization. The result is a remaining internal dynamics of second order. To investigate the internal dynamics a state transformation to the Byrnes/Isidori form is advantageous. The first 4r = new states can be computed by the Lie derivations (see equation (20)). The last two can be chosen freely. The only condition is that the resulting transformation must be a diffeomorph transformation. In order to shorten the length of the third an fourth equation, the linearizing output and its derivative have been substituted.

*

* 5

*

*1 1 1 3

*2 2 2 4

2 * 23 3 3 1

3 * 24 4 4 3 5 6 1 5 2

5 5 5

6 6 6

( ) ( )

( ) ( )

( ) ( ) - sin

( ) ( ) - cos 2

( )( )

S

Sf

f

f

z x y h x x l x

z x y L h x x l x

z x y L h x g x x z

z x y L h x x g x x x z x z

z x xz x x

φ

φ

φ

φ

φφ

= = = = +

= = = = +

= = = = +

= = = = + +

= =

= =

&

&&

&&&

(19)

With this transformation applied to the system the internal dynamics results to

5 6

*6

z z

z w

=

=

&

& (20)

which is exactly the transformed disturbance model. In our case the internal dynamics consists of a double integrator chain. This means, the internal dynamics is instable. Hence it is impossible to solve the internal dynamics by on-line simulation. But for the here given application case not only the disturbance *

D wϕ =&&& but also the new states 6 Dx ϕ= && and

5 Dx ϕ= & can be directly measured. This makes the simulation of the internal dynamics unnecessary. Stability can be guarantied because of an additional state space controller, which stabilizes the slewing motion of the crane ( 0Dϕ =& for t → ∞ ).

IV. MEASUREMENT RESULTS In this section real (non-simulated) measurement results

of the obtained nonlinear controller, which was applied to the broom crane, are presented. Fig. 7 shows a polar plot of a single crane rotation. The rope length during crane operation is 35 m. The challenge is to obtain a constant payload radius LAr during the slewing movement.

5

10

15

20

25

30

210

60

240

90

270

120

300

150

330

180 0

rLA

rLA,ref

rA

Fig. 7. payload and boom position during rotation

To achieve this aim a luffing movement of the boom has to compensate the centrifugal effect on the payload. This can be seen in Fig. 8 which displays the radial position of the load and the end of the boom over time. It can be seen that the payload tracks the reference trajectory with an error smaller than 0.7 m

0 10 20 30 40 50 6020

20.521

21.522

22.523

23.524

Time [s]

outre

ach

[m]

rLA

rA

Fig. 8. outreach of payload and boom during rotation

The second maneuver is a luffing movement. Fig. 9 shows the payload tracking a reference position, the resulting radial rope angle during this movement and the velocity of the boom compared with the reference velocity for the payload. It can be seen, that the compensating movements during acceleration and deceleration reduce the load sway in radial direction.

0 10 20 30 40 50 60 70 80 9023

26

29

32

35

38

Time [s]

outre

ach

[m]

rLA

rLA,ref

0 10 20 30 40 50 60 70 80 90-0.02

-0.01

0

0.01

0.02

Time [s]

angl

e [ra

d]

radial rope angle

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0 10 20 30 40 50 60 70 80 90

-1

-0.5

0

0.5

1

Time [s]

radi

al v

eloc

ity [m

/s]

rLA,ref,dot

rA,dot

Fig. 9. luffing movement

The next maneuver is a combined maneuver containing a slewing and luffing motion of the crane. This is the most important case at transshipment processes in harbours mainly because of obstacles in the workspace of the crane. Fig. 10 shows a polar plot where the payloads radius gets increased by 10 m while rotating the crane.

240

270

300

330

0

rLA

rLA,ref

10

20

30

Fig. 10. payloads position during the combined motion

Fig. 11 displays the same results but over time in order to illustrate, that the radial position of the load follows the reference.

0 10 20 30 40 5020

22.5

25

27.5

30

32.5

35

Time [s]

outre

ach

[m]

rLA

rLA,ref

Fig. 11. outreach of payload during combined maneuver

Comparing these results with that of the luffing motion it can be seen that the achieved tracking performance remains equal. Because of the disturbance decoupling it is possible to achieve a very low residual sway and good target position accuracy for luffing and slewing movements as well as for combined maneuvers.

V. CONCLUSION In this paper a nonlinear model for a boom crane was

derived utilizing the method of Newton/Euler. Dominant nonlinearities such as the kinematics of the hydraulic actuator (hydraulic cylinder) were considered. Additionally the centrifugal acceleration of the load during a slewing motion of the crane was taken into account. The centrifugal effect, which results in the coupling of the slewing and

luffing motion, has to be compensated in order to make the cargo transshipment more effective. This is done by first defining the centrifugal effect as a time-varying disturbance and analyzing it concerning decoupling conditions. And secondly the nonlinear model was extended by a second order disturbance model. With this extension it is possible to decouple the disturbance and to derive a input/output linearizing control law. The drawback is that not only the disturbance but also the new states of the extended model must be measurable. This is possible for the here given application case. The measurement results, obtained by real (non-simulated) experiments, validate the exact tracking of the reference trajectory with reduced load sway.

REFERENCES [1] A. Isidori, C. I. Byrnes, “Output Regulation of Nonlinear Systems”,

Transactions on Automatic Control, Vol. 35, No. 2, pp. 131-140, 1990 [2] A. Isidori, “Nonlinear Control Systems”, 3rd edition Berlin Springer

Verlag, 1995 [3] M. Fliess, J. Lèvine, P. Martin, P. Rouchon, “Flatness and defect of

nonlinear systems: introductory theory and examples”, International Journal of Control, Vol. 61, pp. 1327-1361, 1995

[4] V. Hagenmeyer, M. Zeitz, „Flachheitsbasierter Entwurf von linearen und nichtlinearen Vorsteuerungen“, Automatisierungstechnik, Vol. 52, pp. 3-12, 2004

[5] R. Andiarti, C. H. Moog, “Output Feedback Disturbance Decoupling in Nonlinear Systems”, Transaction on Automatic Control, Vol. 41, No. 11, pp. 1683-1689, 1996

[6] S. Battilotti, “A sufficient condition for nonlinear disturbance decoupling with stability via measurement feedback”, Proc. of the 36th Conference on Decision and Control, California, USA, pp. 3509-3514, 1997

[7] X. Xia, C. H. Moog, “Disturbance Decoupling by Measurement Feedback for SISO Nonlinear Systems”, Transaction on Automatic Control, Vol. 44, No. 7, pp. 1425-1429, 1999

[8] M. Bröcker, J. Polzer, M. Lemmen, “An advanced algorithm based on differential algebra for disturbance decoupling of nonlinear systems”, Proc. of the 39th Conference on Decision and Control, Sydney, Australia, pp. 4367-4372, 2000

[9] M. Fliess, J. Lèvine, P. Rouchon, “A simplified Approach of Crane Control via a Generalized State-Space Model”, Proc. of the 90th Conference on Decision and Control, Brighton, England, pp. 736-741, 1991

[10] A. Piazzi, A. Visioli, “Optimal dynamic-inversion-based control of an overhead crane”, Proc.-Control Theory Applications, Vol. 149, No. 5, pp. 405-411, 2002

[11] N. Yanai, M. Yamamoto, A. Mohri, “Feed-Back Control of Crane Based on Inverse Dynamics Calculation”, Proc. of the International Conference on Intelligent Robots and Systems, Maui, USA, pp. 482-487, 2001

[12] M. Kiss, J. Lèvine, P. Müllhaupt, “Modeling; Flatness and Simulation of a Class of Cranes”, Periodica Polytechnica, Vol. 43, No. 3 pp. 215-225, 1999

[13] O. Sawodny, H. Aschemann, J. Kümpel, C. Tarin, K. Schneider; “Anti-sway control for boom cranes”, Pro. of the American Control Conference, pp. 244-249, 2002

[14] E. Arnold, O. Sawodny, A. Hildebrandt, K. Schneider, “Anti-sway system for boom cranes based on an optimal control approach”, Proc. of the American Control Conference, pp. 3166-3171, 2003

[15] J. Neupert, A. Hildebrandt, O. Sawodny, K. Schneider, “A Trajectory Planning Strategy for Large Serving Robots”, Proc. of the SICE Annual Conference, Okayama, Japan, pp 2180-2185, 2005

[16] J. Neupert, A. Hildebrandt, O. Sawodny, K. Schneider, “Trajectory Tracking For Boom Cranes Using A Flatness Based Approach”, to be published at the SICE-ICASE Conference, Pusan, Korea, 2006

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