Pre-filter design for exact linearization-based tracking controllers for variable-load systems


ACTA IMEKO 
ISSN: 2221-870X 
December 2019, Volume 8, Number 4, 62 - 69 

 

ACTA IMEKO | www.imeko.org December 2019 | Volume 8 | Number 4 | 62 

Pre-filter design for exact linearisation-based tracking 
controllers for variable-load systems 

Na Wang1, Bálint Kiss1 

1 Budapest University of Technology and Economics, Műegyetem rakpart 3, 1111 Budapest, Hungary 

 

 

Section: RESEARCH PAPER 

Keywords: exact linearisation; robust design; parameter uncertainty, SCARA robot; overhead crane 

Citation: Na Wang, Balint Kiss, Pre-filter design for exact linearization-based tracking controllers for variable-load systems, Acta IMEKO, vol. 8, no. 4, 
article 11, December 2019, identifier: IMEKO-ACTA-08 (2019)-04-11 

Editor: Yvan Baudoin, International CBRNE Institute, Belgium 

Received January 16, 2019; In final form September 12, 2019; Published December 2019 

Copyright: This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, 
distribution, and reproduction in any medium, provided the original author and source are credited. 

Funding: This work was supported by the BME-Artificial Intelligence FIKP grant of EMMI (BME FIKP-MI/SC). 

Corresponding author: Na Wang, e-mail: wang@iit.bme.hu 

 

1. INTRODUCTION 

Robotic arms and cranes serve mostly as weight-handling 
devices, and they move a load to a desired target location along 
a specified path within the workspace of the equipment. A shared 
property of such weight-handling tasks is the uncertainty of the 
load mass or inertia, as load variation from one operation cycle 
to another may be high. Therefore, both for robots and for 
cranes, one of the objectives of the controller design is the 
limitation or full cancellation of the effects of this uncertainty on 
the accuracy of reference trajectory tracking. 

In contrast with fully actuated robotic arms, cranes are 
underactuated mechanisms, such that the number of 
configuration variables is larger than the number of actuators. 
Nevertheless, both robots and cranes share the formalism 
(Euler-Lagrange equations) available to obtain their dynamical 
models. 

From a control point of view, it can be observed that tracking 
precision and speed are generally conflicting requirements, 
especially if uncertainty is present for inertia values, since these 
uncertainties are amplified due to the high-motion speeds of the 

mechanism. Moreover, in addition to the load inertia, friction 
coefficients and torque constants of the (usually electric) 
actuators are also known with only a limited accuracy. 

Closed-loop control is expected to ensure robust stability and 
tracking performance in the presence of such uncertainties. 

Various methods have been already explored for robot 
controller design and implementation, including sliding mode 
control [1]-[3], neural networks [4], and fuzzy systems [5], [6], to 
name a few. A comprehensive overview of the robust control 
techniques of robot manipulators is presented in [7]. 

Similarly, several control techniques are proposed for cranes 
to ensure the tracking of a reference load trajectory. A review is 
presented in [8]. The flatness-based modelling and control of 
crane systems similar to that considered in this paper are 
addressed in [9] and [10]. The authors of [11] suggested robust 
controllers for the approximated linear crane dynamics obtained 
by (idle) setpoint linearisation. 

Setpoint linearisation remains a valid approximation for idle 
positions. This limits the applicability of the feedback to slow (or 
quasi-static) load motions. In the case of quick load trajectories, 

ABSTRACT 
Several classes of weight-handling equipment with nonlinear dynamics are exactly linearisable by feedback. This implies that a controller 
ensuring the exponential decay of the tracking error along a reference trajectory can be also designed. The control of both fully actuated 
robotic arms and underactuated crane-like systems can be addressed using this method. However, the value of model parameters such 
as the mass of the load or the friction coefficients must be known in order to calculate the actuated inputs in closed loop. The uncertainty 
of these parameters may result in unstable closed-loop dynamics or in the deterioration of the tracking performance. This paper presents 

a design procedure to take into consideration such uncertainties using a serial pre-filter, which is determined using H∞ techniques. The 
procedure is applied to a SCARA-type robotic arm and to a two-dimensional overhead crane system. The simulation results show the 
applicability of the suggested method. 

mailto:wang@iit.bme.hu


 

ACTA IMEKO | www.imeko.org December 2019 | Volume 8 | Number 4 | 63 

where the nonlinear coupling effects and the rope angles become 
large, the linear approximation is no longer valid. 

Dynamical models of robots and cranes are differentially flat 
[12]; hence, they are feedback linearisable. The linearising 
feedback is static for robots and dynamic for cranes (i.e. the 
feedback has a non-empty state vector) [9], [13]. Controllers 
exploiting the flatness property are capable of quick and precise 
reference tracking if the parameters are known with accuracy. 
Uncertainty in parameters, such as inertia, load mass, and friction 
coefficients, may destabilise the closed-loop system or 
considerably deteriorate its performance in terms of both steady-
state and transient behaviour. 

This paper addresses the problem of the uncertainty of 
parameters by the design of a serial and linear pre-filter using 

standard H∞  synthesis techniques. The need for the robust 
approach is motivated by the fact that the parameters of the real 
system and the parameters used in the linearising feedback may 
be different, especially as far as the load mass is concerned. To 
define the underlying synthesis problem, the set of systems at 
which the exact linearisation is applied with a parameter misfit to 
the robot or crane model must be covered by a standard linear 
uncertainty structure. The weighting matrices of this structure are 
calculated by solving a set of inequalities such that the linearising 
feedback is applied to a large set of robot or crane models with 
the parameter values spanning a grid of their uncertainty range. 

The idea of applying a robustifying controller together with 
exact feedback linearisation has already been presented by the 
authors in [14]. The necessary design steps to obtain the pre-filter 
are presented in Section 2. The suggested method is applied to a 

SCARA-type robot in Section 3. In Section 4, we discuss how 
the method can be used in crane control. The concluding section 
enumerates some topics subject to further investigation. 

2. THE ROBUSTIFYING PRE-FILTER AND ITS DESIGN 

Consider a nonlinear system with a finite dimensional state-

space so that its description depends on the N  elements of a 

parameter vector 1 2[ , , , ]
T

Np p p p= . The uncertainty of each 

parameter is described by a bounded real interval p
i

∈ Q
i

⊂ ℝ 

such that 1 2 NQ Q Q Q=    . The vector of the nominal 

parameter values is denoted by 
0

p Q . The nonlinear 

dynamics are described by 

( , , ) ( , )x f x u p y h x p= =  (1) 

where x ∈ ℝ𝑛  and the dimension m  of the input vector u  equal 
the dimension of the output vector y . We suppose that a 

linearising feedback can be designed in Equation (1) for all 

possible parameter values in Q , such that the linearising (or flat) 

output is y . Since the real parameter values are not known, the 

nominal values 
0

p  are used in the expressions of the feedback: 

0
( , , , )x v p  = ,  (2) 

0
( , , , )u x v p = .  (3) 

Thanks to the exact linearisability (or differential flatness 
property [12]) of the system, it is possible to express the 
derivatives of the output trajectory as the function of the full 

state vector of the closed-loop system, [ , ]
T

x  : 

 = =  − =
( ) ( )0

, ( , , ) 1, , 1
j

i

j

j j
i i j i iy j

d y
h x p

dt
y . (4) 

Let us now denote by ry  a sufficiently smooth reference 

trajectory for the output y. The tracking feedback law can be than 
set as 

( )


 
 

−
−

=

= + − = 
( ) ( 1) 0
, , , ,

1

( , , ) 1, ,
i

i i

ii i r i j i r i j

j

v y y h x p i m  (5) 

which results in an exponential decay of the tracking error, since, 

by the construction of Equation (5), ,i i r ie y y= −  satisfies 

  
  

− −
= + + ++

( ) ( 1) ( 2 )
,1 ,2 ,0

i i i

ii i i i i i i
e e e e  (6) 

where the coefficients ,i j  are design parameters. They are set 

so that the characteristic polynomial of Equation (6) is Hurwitz 
stable i.e. all its roots have negative real parts. 

The block diagram of the closed-loop system is depicted in 
Figure 1. It follows from the above discussions that the closed-

loop transfer between y
r̅
 and y, assuming p= p0, reads 

,

01
1, , ,1 ,

( )
diag ( )

( )

i

i i

i

i

i mr i i

Y s
G s

Y s s s



 




 
−

= 

  
= = 

+ ++  

. (7) 

Due to the uncertainty of the parameters, the nominal values 
p0 used in the linearising feedback (Equations (2)-(3)) may be 
different to their real values. This parameter misfit implies that 
the resulting closed-loop dynamics may still include non-
linearities. For the robotic arms and crane systems considered in 
this article, the uncertain parameters are the inertia of the moving 
parts of the mechanism, the mass of the load, and the friction 
coefficients. 

The serial pre-filter (see Figure 1) is designed using H∞ 
synthesis techniques [15]. To apply such a synthesis technique, 

the feedback structure following the controller ( )K s  in Figure 1 

 

 

Figure 1. Closed-loop dynamics with a serial pre-filter, tracking and linearising 
feedback. 

 

 

Figure 2. The controller K(s) with the augmented plant. 



 

ACTA IMEKO | www.imeko.org December 2019 | Volume 8 | Number 4 | 64 

must be replaced by an uncertainty structure as in Figure 2. A 
sufficiently smooth grid is chosen to cover the uncertainty range 
Q of the parameters. For each vertex pi of this grid, a linear 

transfer function Gi(s) is obtained between the signals ry  and y  

by a setpoint linearisation. The weighting matrices 1( )W s  and 

2( )W s  of the output multiplicative structure in Figure 2 are 

chosen so that = + 1 2 0( ) (1 ( ) ( ) ( )) ( )
i

G s W s s W s G s  holds true 

for some 


 ( ) 1s . The command ucover in Matlab 

calculates the weighting matrices accordingly if the nominal 

model and the frequency response data for each ( )
i

G s  is given. 

The H∞  design problem to obtain ( )K s  is also solved with 

Matlab, such that a standard augmented plant, denoted by 

( ),aG s  is constructed (Figure 2) with performance weighting 

transfer functions ( )M s , ( )eW s  and ( )uW s : 

 
11 12

21 22

( ) ( )
T

m u a r a

r

u
P

y e e e G s y G s
P P

P

y





 
  

= =   
   

 (8) 

with 

 

2 0
21 0

11 1 12 0
22 1

0 0

0 0

e e e

u

W G
P G

P W W W M P W G
P W I

W

   
=   

= − =
    = −
      

 (9) 

where the dependence on s  is omitted. Applying the controller 
( )K s , the closed-loop transfer matrix can be calculated using a 

lower fractional transformation 

1
11 12 21 22( , ) ( )l aF G K P P K I P K P

−
= + + . (10) 

By introducing 
1

0
−

= +( )oS I G K , we get: 

2 0 1 2 2

1 0

1

− 
 

= −
 
 − 

( , ) ( )

o o

l a e o e o

u o u o

W G KS W W G KS

F G K W S W W G KS M

W S W W KS

. (11) 

Finally, the H  optimisation consists of finding the 

controller ( )K s , which minimises the H  norm of (11). This 

optimisation process is implemented in Matlab through the 
hinfsyn command [16]. Robust stability is also guaranteed if 

2 0 1 1oW G KS W 
   (12) 

The next two sections present the application of the above 
design procedure for a four Degrees-of-Freedom (DOF) robot 
arm and for a planar crane system. 

3. CONTROL OF A SCARA-TYPE ROBOTIC ARM 

3.1. Robot modelling 

Let us consider a four-DOFs SCARA-type robotic arm with 
three rotary joints whose axes are parallel to each other and 
which have a prismatic joint. To demonstrate the feasibility of 
the proposed method, the design procedure described in the 
previous section is executed on the model of a Bosch Turbo 
SCARA SR 60-type robot [17]. The robot scheme is shown in 
Figure 3. The dynamics of a robot manipulator read 

+ + + =( ) ( , ) ( ) ( )H q q C q q q B q G q  (13) 

where ,q q  and q  are the vectors of joint angles, joint velocities, 

and joint accelerations, respectively; H(q) ∈ ℝ4×4  is a 
symmetric, positive definite inertia matrix; C(q, q̇ )q̇ ∈ ℝ4 is the 
vector of the Coriolis and centrifugal torques; B(q̇ ) ∈ ℝ4 is the 
vector of the friction torques; G(q) ∈ ℝ4  is the vector of 
gravitational torques, and the control input torque is τ ∈ ℝ4. The 
length, the position of the centre of gravity (cog), and the mass 

of the segments are denoted by il , ia , and im , respectively; and 

r  is radius of the last segment. Using standard notations in 

robotics, ijD denotes the elements of ( )H q  ( =ij jiD D ) and 

reminds us that 

4 4

1 1= =

=( , )i ijk j k
j k

C q q q D q q .  (14) 

Considering the SCARA robot, such that we set 2/i ia l= , 

the elements of the inertia matrix read 

22
2 2 41 1

11 2 2 1 1 2 2

2 2
3 4 1 2 1 2 2

2
12 2 2 1 2 2

2 2
3 4 2 1 2 2 4

2 2 2 2 2
14 4 22 2 2 3 2 4 2

33 3 4 31 32 34

1

3 3 2

( )

)( )

2

0

( 2 )

1 1

3 2

1
(

2

1 1 1

2 3

m rm l
D m l l l l C

m m l l l l C

D m l l l C

m m l l l C m r

D m r D m l m l m l r

D D Dm Dm

= + + + + +

+ + + +

= + +

+ +

= − = + + +

= +

 
 
 

 
 
 

+

 
 
 

= = =

 (15) 

and the non-zero ijkD  elements are 


= = −


= − + + 

 
112 1 1112 2 3 4 221 2 2 21

1

2
2 2D m m m l l DS D D . (16) 

 

Figure 3. A SCARA robot with RRTR (R-rotational, T-translational) joint 
configuration. 

Table 1. Uncertain parameters of the four-DOFs SCARA-type robotic arm. 

Variable Normal value Minimum value Maximum value 

m4 3 kg 2 kg 4 kg 

Fv1 0.5 kg m rad-1 s-1 0.2 kg m rad-1 s-1 0.8 kg m rad-1 s-1 

Fv2 0.5 kg m rad-1 s-1 0.2 kg m rad-1 s-1 0.8 kg m rad-1 s-1 

Fv3 0.05 kg m/s 0.02 kg m/s 0.08 kg m/s 

Fv4 0.05 kg m rad-1 s-1 0.02 kg m rad-1 s-1 0.08 kg m rad-1 s-1 
 



 

ACTA IMEKO | www.imeko.org December 2019 | Volume 8 | Number 4 | 65 

Shortcut notations S2 and C2 represent 2sin q  and 2cos q , 

respectively. Only viscous friction is considered as in [18]: 

=( )i vi iB q F q   (17) 

and only the motion of the third joint is influenced by gravity 

3 3 4 1 2 4 0 9 81
2

( ) . m/sG m m g G G G g= − + = = = = . (18) 

3.2. Pre-filter design and simulation results 

In most applications, the joint mass can only be estimated 

with limited accuracy. We assume that the parameters 1m , 2m , 

and 3m  are known with sufficient precision, and their values are 

15 kg, 12 kg, and 3 kg, respectively. The parameter m4 comprises 
the variable payload mass; hence, its value is uncertain. The 

lengths of the two links are 1 0 5. ml = , 2 0 4. ml = , and 

0 2. mr = . The friction coefficients viF  are also variable. The 

uncertain parameter ranges are given in Table 1. A grid of 50 
vertices is used to get the output multiplicative uncertainty 
structure presented in Figure 2 with a second order weighting 

matrix 1W , with the following elements: 

2 2

2 2

2 2

2 2

2

2

0.27 0.06 4.61 4 0.35 0.04 1.27 4
1,11 1,22

0.93 0.04 0.56 0.01

0.45 0.05 6.09 4 1.18 0.16 5.04 4
1,33 1,44

2.26 0.14 1.45 0.03

0.71 0.12 8.13 4
2,11

( ) ( )

( ) ( )

( )

s s e s s e

s s s s

s s e s s e

s s s s

s s e

s

W s W s

W s W s

W s

+ + − + + −

+ + + +

+ + − + + −

+ + + +

+ + −

+
=

= =

= =

2

2

2 2

2 2

0.71 0.10 2.98 4
2,22

1.94 0.07 1.06 0.02

0.45 0.05 6.09 4 0.72 0.11 3.49 4
2,33 2,44

2.26 0.14 1.07 0.02
.

( )

( ) ( )

s s e

s s s

s s e s s e

s s s s

W s

W s W s

+ + −

+ + +

+ + − + + −

+ + + +

=

= =

 (19) 

A reference model is used to define the desired closed-loop 
tracking response for each joint angle: 



  

  
=  

+ +  

2

2 2
( ) diag

2

m

m m m

M s
s s

 (20) 

with 2 rad/secm = and 0 8. ,m =  respectively. The remaining 

performance weighting transfer matrices are also diagonal (four-

by-four). Recall that uW  is linked to the actuator bandwidth. The 

weighting functions used in our design read 

 
+

+
=

2

2

40( 10 )

( 200 )
( ) diag m

m

s
u

s
W s   (21) 

and 

2000
1 1 1 1 1 2

9
( ) diag ,10 , ,10

20

m

m
e e e e e e s

W s W W W W W


+

 
= = 

 
. (22) 

The robot dynamics and the controller are simulated for 
different uncertain parameter values using Matlab and Simulink. 

The reference trajectory ry  is a step function at 1sect =  for 

each joint. The initial conditions are set to zero. Figure 4 shows 
the tracking performance without the use of the pre-filter, 
whereas Figure 5 shows the joint trajectories with the use of the 
pre-filter. For each joint, a family of curves is presented, each for 
a different vertex of the grid spanned over the uncertain 
parameter space. It can be observed that the use of the pre-filter 
reduces the effect of uncertainty and provides better 

performance. Since 2 0 1 0 5025. ,oW G KS W 
=  the robust 

stability condition (12) is satisfied. 

4. CRANE CONTROL 

4.1. Crane modelling 

We consider a 2D overhead crane [9], depicted in Figure 6. 
The load is suspended by a massless rope, which is pulled by a 

 

Figure 4. Joint trajectories without the H∞ pre-filter. 

 

Figure 5. Joint trajectories with the application of the H∞ pre-filter. 



 

ACTA IMEKO | www.imeko.org December 2019 | Volume 8 | Number 4 | 66 

winch. The suspension point is moved horizontally by a cart. The 

configuration variables are the horizontal position of the cart ,R  

the length of the rope L , and its angle with respect to (w.r.t.) the 

vertical position:  . The position of the load in the vertical 

x z−  plane is denoted by [ , ]
T

m mx z  and 

, , , , , .
T

x R L R L  =
 

 The parameters of the system are 

[ , , , ] ,
T

p m M J =  including the mass of the load, the mass of 

the cart, the inertia and the radius of the winch. Let /T  =  

denote the equivalent pulling force on the rope such that the 

system inputs are [ , ]
T

u F T= . Let us denote the rope tension 

by LT . Introducing cosC =  and sinS = , the dynamics 

read 

( )

2

2

2

2

0

0

2

.

02

M m mS mLC
R

J
mS m L

mLC mL

mL C mL S F

mL gmC T

mL L gS R S

 





 



 




 



 

 +
  
  

+ +  
  
  

 

 −  
 

 
+ − − = − 

 
 

 + −    

 (23) 

4.2. Pre-filter design and simulation results 

It is known [9], [13] that Equation (23) is differentially flat 
and hence feedback linearisable. The coordinates of the load 
form the flat output: 

1 2m my x R LS y z LC = = + + = = . (24) 

The control objective is to follow a reference trajectory 
specified for the flat output. The crane system can be 
transformed by feedback into two separated chains of integrators 

( 4 ) ( 4 )
1 1 2 2y v y v= = . (25) 

The linearising feedback has two states, 1  and 2 . These 

are the rope tension 1 LT =  and its first time derivative 

2 LT = . Let us now calculate the linearising feedback resulting 

(25). Referring to [19], the motion equations of the load read 

 = = − = = − +1 2sin cos
L L

m m

T T
y x y z g

m m
. (26) 

Let us differentiate both sides of (26) twice w.r.t. the time so 

that we use the two states of the feedback ( 1  and 2 ) as well 

( )

( )

2
2 2 1 1 1

2
2 2 1 1 2

1
sin 2 cos cos sin

1
cos 2 sin sin cos

v
m

v
m

          

          

− − − + =

− + + + =

 (27) 

and isolate 2  and   using the nominal 
0

m . We get 

1 0 2
2 1 1 1 2

0 2
1 2 1 2

2

2

S C m v S C

C S m v C S

   

   

     

    

−       − − − 
 = −        − +            

 (28) 

Equation (24) can be differentiated twice more to get 

2 1

2 1

2

2

R LS L C L C L S S
m

LC L S L S L C C g
m

    

    


  


  

+ + + + =

−
− − − = +

 (29) 

which allows us to isolate the vector of accelerations as 

1
1

0

2

2

1 1

0

2

2

R S S LC

C C LSL m

L S L C

g L C L S

  

  

 

 





 

 

−
  − −     

= +      
         

 −
+  
+ +  

 (30) 

The calculations are also illustrated in Figure 7. Blocks A, B, 
and C represent calculations Equation (28), Equation (30), and 
the first two lines of Equation (23), respectively. Recall that the 

states of the linearising feedback, 1  and 2 ,  must also be 

initialised. For the idle initial load position 
0

1 0( ) m g =  and 

2 0 0( ) = . In addition to the linearising feedback, a tracking 

controller is also designed. Suppose that a sufficiently smooth 

trajectory is given for the motion of the load as , ( )m rx t , , ( )m rz t  

and their derivatives, such that , ,[ , ]
T

m r m ry x z= . The tracking 

errors are defined as ,x m r me x x= −  and ,z m r me z z= − . The 

expression of the tracking controller can be obtained by 
specifying the desired exponential decay of the tracking errors as 

3 3
( 4 ) ( ) ( 4 ) ( )

, ,

0 0

0 0
i i

x x i x z z i z

i i

e e e e 
= =

+ = + =   (31) 

Since 
4 4 4 4

1
( ) ( ) ( ) ( )

, ,x m r m m re x x x v= − = −  and 
4 4

2
( ) ( )

,z m re z v= − , 

Equation (31) can be rewritten as 

3 3
( 4 ) ( ) ( 4 ) ( )

1 , , 2 , ,

0 0

i i
m r x i x m r z i z

i i

v x e v z e 
= =

= + = +   (32) 

The design parameters ,x i  and ,z i  0 3( , , )i =  define 

the roots of the characteristic equations of Equation (31), hence 

 

Figure 6. Overhead crane in 2D and forces acting on the load. 

 

Figure 7. Calculation blocks of the flatness-based exact linearisation of the 
crane system. 



 

ACTA IMEKO | www.imeko.org December 2019 | Volume 8 | Number 4 | 67 

the decay rate of the tracking errors. To realise Equation (32), 

one must specify how to calculate 
3 3( ) ( )

, , , , ,m m m m m mx x x z z z  from 

x  and  : 

1 1

0 0

( 3) ( 3)2 1 1 2

0 0
.

m m

m m

m m

x R LS L C z LC L S

x S z C g
m m

S C S C
x z

m m

   

 

   

 

 

     

= + + = −

−
= − = +

+ −
= − =

 (33) 

Let us emphasise that the exponential decay of the tracking 
errors satisfies Equation (30) only if the parameter values used in 
the expressions of the feedback fit the parameters of the 
controlled system. The method suggested in Section 2 is applied 
to the crane model. The parameters and their uncertainty range 
are given in Table 2. A grid of 50 vertices has been spanned over 

the uncertain parameter space Q . It is possible to choose 

2( )W s I= , and 1( )W s  reads 

 
2 2

2 2
0.5 0.005 1.9 5

0

0.32 0.01 2 6
1

14.57 0.21 0.72.7913
di( g ,a) s s e s s e

s s s s
W s + + − + + −

+ + + +
= . (34) 

The closed-loop reference model ( )M s  is set similarly to 

Equation (20), with 0 5. rad/secm =  and 0 8. .m =  The 2×2 

performance weighting transfer matrices ( )eW s  and ( )uW s  are 

set as 

2

2

8000 500( 10 )
( ) diag ( ) diag

2 ( 200 )

m m
e u

m m

s
W s W s

s s

 

 

   + 
= =   

+ +    
. (35) 

Let 0 0 0 2[ , ] [ , ]
T T

m mx z =  be the initial position of the 

reference load trajectory, which is defined as a periodic sinusoidal 
motion in both the x  and the z  directions 

1 , 1 ,( ) ( ) sin( ) ( ) ( ) 3 cos( )r m r x r m r zy t x t t y t z t t = = = = −  (36) 

with 2 rad/secx z = = . This reference trajectory results in 

sufficiently high rope angles staying away from the region of 
quasi-static motions. The derivatives of the reference trajectory 
that are required in Equation (31) are easily calculated. The 
closed-loop responses are shown in Figure 8 – Figure 12 for the 

five representative parameter values in Q . All figures show the 

transients without (top) and with (bottom) the robustifying 

controller ( )K s . We get 2 0 1 0 2021. ,oW G KS W 
= hence, the 

robust stability condition Equation (12) is satisfied. 
Figure 8 and Figure 9 show the trajectory of the load 

coordinates mz  and mx . Figure 10 depicts the rope angles. The 

input signals, namely the force applied to the cart F  and to the 
winch T , are presented in Figure 11 and Figure 12, respectively. 

The robustifying controller ( )K s  provides better performance 

and robustness. 

 

Figure 9. Horizontal displacement of the load. 

 

Figure 10. Rope angle. 

Table 2. Uncertain and fixed parameters of the 2D crane system. 

Variable Normal value Minimum value Maximum value 

m 1 kg 1 kg  1.6 kg 

M 0.3 kg 0.2 kg 0.6 kg 

J 0.001 kg m2 0.001 kg m2 0.001 kg m2 

𝜌 0.03 m 0.03 m 0.03 m 
 

 

Figure 8. Vertical displacement of the load. 

 



 

ACTA IMEKO | www.imeko.org December 2019 | Volume 8 | Number 4 | 68 

5. CONCLUSIONS 

A novel method of increasing the robustness of exact 
linearisation against parameter uncertainty is presented for two 
variable-load mechanisms, namely for a fully actuated SCARA-
type robot manipulator and for an underactuated 2D overhead 
crane. The method intends to bridge the gap between nonlinear 
control methods capable of stabilising fast load motions and 
robust synthesis techniques, ensuring robust stability and 
performance despite model uncertainties. As suggested by the 
simulation results, the proposed serial pre-filter may improve 
stability and performance robustness for quick reference 
trajectories. 

Some issues may be further investigated. A restriction of the 
presented methods is the necessity to know all elements of the 
state vector x . If these measurements are only partly available, 
some observers need to be applied [20], [21]. 

Our studies show that the choice of the nominal values of 

the parameters 
0

p  influences the uncertainty weighting 

functions 1( )W s  and 2( )W s ; hence, it affects the overall 

performance of the closed-loop system as well. The selection of 

the best 
0

p  can be investigated as an optimisation problem. 

A single ( )K s  controller may not be suitable if the 

uncertainty range Q  is too large. A possible solution for the 

large parameter uncertainty is the application of a gain scheduling 
or some adaptation techniques following the ideas presented in 
[22]-[24]. 

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Figure 12. Input F (Force acting on the cart) during the motion. 

 

Figure 11. Input T (force acting on the which) during the motion. 



 

ACTA IMEKO | www.imeko.org December 2019 | Volume 8 | Number 4 | 69 

International Conference on Informatics in Control, Automation, 
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