Plane Thermoelastic Waves in Infinite Half-Space Caused


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 12, N

o
 1, 2014, pp. 61 - 72 

OPTIMAL SYNTHESIS OF THE MANIPULATOR USING TWO 

COMPETITIVE METHODS


UDC (519.863, 621.86) 

Jovan Pavlović, Miomir Jovanović, Andrija Milojević 

University of Niš, Faculty of Mechanical Engineering, Serbia 

Abstract. This paper defines a program realization for finding the optimal geometry of 

a planar Z-mechanism. The paper shows the mathematical procedure for defining the 

objective function, the constraint function and the search fields which are used for 

solving the optimization problem. Starting from the solutions in practice, a numerical 

example is given to determine an optimum design with four optimization parameters. 

All the optimization parameters are geometric on the mechanism for defining the bucket 

position. The problem is solved by two different numerical methods - the method of 

formal search of hyperspace (passive scanning method) and the approximate method of 

Sequential Quadratic Programming - SQP (applying the function fminmax of the 

Matlab optimization toolbox). The verification is performed with animation using 

software for geometric modeling. The results are graphically illustrated.  

Key Words: Mining Machine, Wheel Loader Z – Mechanism, Passive Scanning, 

Optimum Design of Mechanism, Numerical Realization, SQP 

1. INTRODUCTION

For working with bulk material the wheel loaders have a mechanism that enables the 

position adjustment - an inclination angle of the bucket. At lifting, the bucket should keep 

an upward position without changing the inclination. This can be achieved by adjusting 

the geometry of the parts of the bucket movement mechanism. One of the mechanisms 

providing for it is known as the Z - mechanism (Fig. 1). The task of finding a solution for 

the special mechanism providing for a special request is known as the optimal synthesis 

according to the Theory of Mechanisms.  

Solving this problem is a common example of research and development. The 

similarity of different plane mechanisms of the machines offers the possibility of using 

the same optimization procedure. Nowadays there is a variety of mathematical methods 

Received November 14, 2013 / Accepted February 11, 2014


Corresponding author: Miomir Jovanović 

University of Niš, Faculty of Mechanical Engineering, Department of Transport and Logistics, Niš, Serbia 

E-mail: miomir@masfak.ni.ac.rs 

Original scientific paper



62 J. PAVLOVIĆ, M. JOVANOVIĆ, A. MILOJEVIĆ 

in use, from the classical one, based on the authorized procedures of differential 

programming, gradient methods, the method of polyhedra (Nelder-Mead methods) [1], 

approximate methods of programming and sequential quadratic programming [2], up to 

the modern methods based on the techniques of genetic algorithm [3], ACA algorithm 

(Ant Colony Algorithm) [4] and simulation [5].  

This paper first presents the method of formal searching for optimal solutions based 

on the computer changes of the geometric parameters of the mechanism, followed by the 

method of Sequential Quadratic Programming. Changing the parameters in the method of 

formal searching is performed incrementally by taking discrete values of independent 

parameters in an increasing order (scanning technique) [6], [7].For the evaluation of the 

effectiveness of the passive formal search method, the method of Sequential Quadratic 

Programming (SQP) is applied in this paper. 

The complexity of the general approach to the problem can be seen in the work by 

Yang Yu [8] where the optimization is performed with twenty-two parameters. Besides 

geometric optimization parameters, the synthesis can also introduce the constraints of 

internal mechanism forces. Such an approach can be found in the work of Gao Xiuhua [9] 

(derived using the Adams software package), and Rongyi Zhang [10], using the Matlab 

programming environment with the use of a combination of three optimization algorithms. 

Because of the complexity of the general approach, this paper deals with the mechanism 

synthesis based on a small number of optimization parameters. 

2. GEOMETRIC MODEL OF THE Z-MECHANISM 

A planar mechanism of a manipulator loader consists of an ar  m (1) with a bucket (4) 

and a lifting hydraulic actuator (5), Fig. 1. The arm (1) possesses a Mechanism for 

Defining the Bucket Position (MDBP) that does loading, transport position holding and 

emptying of the bucket. This mechanism consists of a tilting hydraulic actuator (6), a 

two-arm lever (2) and a coupling rod (3) (Fig. 1). 

Solving the optimization problem involves 

finding a geometry of mechanism for defining the 

bucket position to ensure minimal change in the tilt 

of the bucket when the arm of the loader is lifting 

from the minimum to the maximum slope angle β, 

min {FC}. From there, it is possible to search for 

optimization parameters in the MDBP mechanism 

geometry. The mechanism for the lifting arm is 

designed from the given operating conditions of the 

bucket position and is not subject to optimization. 

The optimization parameters can therefore be the 

position of the bearing of the hydraulic actuator (6), 

the geometry of the two-arm lever (2) and the 

length of the coupling rod (3). In this study, four 

parameters are chosen to optimize the MDBP 

mechanism related to the position of the bearing of the hydraulic actuator (6) and linkage 

O23 on the lever (2). 

 

Fig. 1 Geometric model of the  

Z – mechanism for defining 

the bucket position (MDBP) 



 Optimal Synthesis of Manuplator Using Two Competitive Methods 63 

For an efficient solution of multidimensional problems with constraints, numerical 

methods with passive searching can be used [6]. These methods use a fixed interval of 

indetermination (H), where the field of change of independent variables 1 2 3 4{ , , , }z z z z z  

is divided by the selected number of points equal to the number of sub-areas. These points 

serve as the basis for the construction of n dimensional space. The objective function 

( )FC z  is calculated at each node of the grid thus formed.With the analysis of all solutions 

the one that is the most advantageous is chosen. The advantage of the passive search 

method is in the reliability of searching. A disadvantage of multidimensional problems of 

passive searching lies in the number of calculation procedures. Therefore, this paper 

employs the procedure of choice (constriction) of the field of search based on the testing of 

the objective function. The testing determines the solution with the good initial values of 

( )FC z  based on a large interval of indetermination. After identifying the "good areas" of 

solution, the subject of solution is the grid with a fine interval of indetermination. Such a 

model with a reduced number of optimization parameters (z1, z2, z3, z4) and numerical 

implementation can give sub-optimal solutions thanks to the simplicity of the algorithm 

search. The synthesis procedure for an MDBP mechanism is illustrated in Fig. 1.  

3. MATHEMATICAL FORMULATION OF THE PROBLEM 

The formation of a mathematical model for optimization according to Fig. 1 includes 
defining a planar geometric model of the manipulator. The manipulator is a planar (2D) 
mechanism with an arm for carrying the bucket and a mechanism for defining the bucket 
position. A mechanism for defining the bucket position is labeled with darker shades in 
Fig. 1. The whole mathematical model uses the global Cartesian x-y coordinate system 
and local coordinate systems of the parts of the mechanism shown in Fig. 2. Fig. 2 shows 
four parts of the mechanism. The planar form of geometry of these two coupled 
mechanisms with the selected marked positions of parts is given in Fig. 3. 

 

Fig. 2 Parts of MDBP: a) the arm of the manipulator, b) the two-arm lever,  

c) the coupling rod, d) the bucket 



64 J. PAVLOVIĆ, M. JOVANOVIĆ, A. MILOJEVIĆ 

Let us define the geometry of the rigid mechanical system in Fig. 1:The coordinates 

of linkage O11 are given values X11, Y11 of some type of a loader. The coordinates of 

linkage O12  at the top of the arm are computed as:  
 

12 11 2 3
cos cos(90 )X X a a                                         (1)  

12 11 2 3
sin cosY Y a a                                               (2) 

 

The coordinates of bearings of the two-arm lever, the position of linkage O22 are 

computed as:  

22 11 1
cosX X a                                                     (3) 

22 11 1
sinY Y a                                                      (4)  

                       

The coordinates of linkage O21 are determined by an intersection of the two circles 

with the center in linkage O6 whose radius is c6 – the length of the hydraulic actuator (6) 

and the circle with the center in linkage O22 whose radius is equal to the length of the arm 

l1 of the lever (2). We consider that the length of the hydraulic actuator (6) is constant. 

The coordinates of linkage O21 are obtained by solving the equation: 
 

                                      

2 2 2

21 22 21 22 1

2 2 2

21 6 21 6 6

( ) ( )

( ) ( )

X X Y Y l

X X Y Y c

   

   
                                           (5) 

 

 

 

Fig. 3 Geometric model of the manipulator of the loader 

The distance between the centers of the circles is computed as: 

2 2

6 22 6 22
( ) ( )D X X Y Y                                              (6) 



 Optimal Synthesis of Manuplator Using Two Competitive Methods 65 

Value F and the coordinates of linkage O21 are computed as: 

1 6 1 6 1 6 1 6

1
( ) ( ) ( ) ( )

4
F D l c D l c D l c l D c                               (7) 

2 2

22 6 6 22 1 6 22 6
21(1,2) 2 2

( ) ( )
2

2 2

X X X X l c Y Y
X F

D D

    
  


                      (8) 

2 2

22 6 6 22 1 6 22 6
21(1,2) 2 2

( ) ( )
2

2 2

Y Y Y Y l c X X
Y F

D D

    
 


                      (9) 

Let us determine the coordinates of linkage O23. Angle η1 between direction r and d of 

the two-arm lever is computed as: 

 
3

1

2 1

l
arctg

l l
 


 (10) 

Variable angle η (between the arm and the two-arm lever), if is X22  X21 > 0 is then 

computed as: 

 21 22

22 21

Y Y
arctg

X X






 (11) 

If X22  X21 = 0, angle η = 90°, and if X22  X21<0, angle η is computed as:  

 21 22

21 22

180
Y Y

arctg
X X




 


  (12) 

Length d of the arm of the two-arm lever (Fig. 2b) is computed as: 

2 2

2 1 3
( )d l l l                                                        (13) 

Coordinates of linkage O22, are computed as: 

 
23 22 1

 cos( )X X d      (14) 

 
23 22 1

 sin( )Y Y d      (15)   

     The introduced optimization parameters z1, z2, z3, z4 correspond to the required 

geometry of the Z - mechanism by the following relations:                                                                                                                  

 
1 6

z X  (16) 

 
2 6

z Y  (17) 

 
3 2 1

z l l   (18) 

 
4 3

z l  (19) 



66 J. PAVLOVIĆ, M. JOVANOVIĆ, A. MILOJEVIĆ 

The coordinates of linkage O4 (X4, Y4) are determined using the intersection of the 

circle with the center of linkage O23 whose1. The solution of coordinates X4, Y4 is obtained 

from the equation: radius is the length of coupling rod c1 and the circle with the center of 

linkage O12 whose radius is length b: 

2 2 2

4 23 4 23 1

2 2 2

4 12 4 12 1

( ) ( )

( ) ( )

X X Y Y c

X X Y Y b

   

   
                                               (20) 

The distance between the centers of the circles and value F1 are computed as: 

2 2

1 12 23 23 12
( ) ( )D X X Y Y                                              (21) 

1 1 1 1 1 1 1 1 1 1 1 1

1
( ) ( ) ( ) ( )

4
F D c b D c b D c b c D b                             (22) 

Coordinates of linkage O4 : 

2 2

12 23 12 23 1 1 23 12
4 1(1,2) 2 2

1 1

( ) ( )
2

2 2

X X X X c b Y Y
X F

D D

    
  


                      (23) 

  
1

2 2

23 12 12 23 1 1 23 12
4 1(1,2) 2 2

1

( ) ( )
2

2 2

Y Y Y Y c b X X
Y F

D D

    
 


                        (24) 

Bucket tilt angle for the current value of arm slope β is computed as: 

 
4 12

4 12

( )
Y Y

arctg
X X

 





 (25) 

The back angle of bucket γ is determined by the position of the angle of bucket α and  

constant α4 (Fig. 2d): 

4                                                                             (26) 

The objective function is the minimum change in the slope of the bucket while the 

arm of the loader is lifting:        

1 2 3 4 max min
( , , , )FC z z z z                                            (27) 

The constraint functions are determined by inequalities:  

       

1 1 5 3

2 1 6 3

3 2 7 4

4 2 8 4

0       0     

0       0 

0      0 

0      0 

G z A G z E

G z B G z F

G z C G z U

G z D G z H

     

     

     

     

                                 (28) 

where: A, B, C, D, E, F, U, H – are the constants of the search space. 



 Optimal Synthesis of Manuplator Using Two Competitive Methods 67 

4. ALGORITHM AND PROGRAM 

Fig. 4 presents the algorithm of global numerical procedures developed for the solution 

of the optimal mechanism synthesis. The algorithm shows that the scanning method is 

realized by an increment change of the optimization parameters in a permissible four-

dimensional hyperspace. The permissible hyperspace is empirically determined. The initial 

setting of geometry of both mechanisms is provided through the files DAT1-DAT6. The 

solutions of objective function FC for discrete values of independent parameters z1 ÷ z4 are 

entered in the file DAT7. By comparing these solutions, the solution that is characterized by 

the minimum value of objective function FC is chosen. Then the obtained solution is 

checked from the point of numerical accuracy, satisfying the given nominal characteristics 

of the reach and lift height. After that, the continuity of coupling of the parts of mechanism 

for defining the bucket position is also checked. The following numerical example 

illustrates the application of this mathematical model, the efficiency of the algorithm and 

the practical realization of optimal synthesis. 

 
 

 
 

Fig. 4 The algorithm of the program for optimal numerical synthesis of MDBP 



68 J. PAVLOVIĆ, M. JOVANOVIĆ, A. MILOJEVIĆ 

5. NUMERICAL EXAMPLE 

The aim of this study is to search for the design of the mechanism for defining the 

bucket position which minimizes the angle of the bucket tilt during the bucket lifting 

operation. The initial solution is based on freely assumed geometry which is in this case 

taken from photographs and sketches [11] (Komatsu WA320). Four parameters are 

selected to optimize the synthesis: x, y coordinates of bearings of the hydraulic 

mechanism for defining the bucket position and x, y local coordinates of linkage O23 of 

the two-arm lever. The procedure for the algorithm program flow is fast and numerically 

stable. The stability is based on the simplicity of analytical modeling geometry. 

Angle β is the slope angle of the arm in relation to the horizontals and ranging, in this case, 

from -24
o 
to 62

o
. Parameters z1, z2 represent the search field of bearings position O6, where the 

geometry is in the following ranges: in the x-axis direction z1min=0.94 m; z1max=1.14 m 

(Δz1=0,2 m) (A=0.94 m; B=1.14 m) and in the y-axis direction z2min=1.645 m; z2max=1.845 m 

(Δz2=0.2 m) (C=1.645 m; D=1.845 m). The allowable field defined by constants A, B, C, D is 

chosen on the basis of the available space on the arm. By choosing M = 50 points of change of 

one independent parameter of optimization, the interval of indetermination is: 

1

1,14 0, 94
2 2 0, 00816 [m]

1 50 1

B A
H

M

 
  

 
                                  (29) 

Parameters z3, z4 represent the search field of the position of linkage O23 in the local 

coordinate system of the two-arm lever, the size of change is in the area: z3min=1.4 m; 

z3max=1.5 m (Δz3=0.1 m), (E=1.4 m; F=1.5 m) in the x-axis direction, and z4min=0.175 m;  

z4max=0.275 m (Δz4=0.1 m), (U=0.175 m; H=0.275 m) in the y-axis direction. The field is 

empirically chosen. 

Initial zeros are: z1=1.04 m, z2=1.745 m, z3=1.45 m, z4=0.225 m. These areas of 

solutions of the optimization task determined the allowable space of search-scanning.  

The dividing of the permissible field of parameters z3 and z4 by the N = 50 points is 

selected, as well as the dividing of the permissible field of parameters z1 and z2 by M = 50 

points. In this way a number of combinations of K=M 
2
•N 

2
=6250000 are obtained. From 

the file DAT-8 we can read the optimal parameters of the mechanism for defining the 

bucket position. Based on these parameters, it is possible to draw curves of objective 

function FC, depending on the angle of the slope of the arm β for the optimal and initial 

solution (Fig. 5). Starting from the fixed size of the loader mechanism given in Table 1, 

the MDBP optimal solutions are obtained, given in Table 2. The curves of the objective 

function of the required MDBP are plotted for the optimal solution. 

Table 1 Initial data of model 

a1 (m) a2(m) a3 (m) l1 (m) l2 (m) l3 (m) a1 (m) b1 (m) 

1.800 2.550 0.865 0.720 1.450 0.225 0.765 0.370 

Table 2 Values of the obtained optimal solution 

 Optimal parameters Objective function 

 z1 (m) z2(m) z3(m) z4(m) FC (˚) 

Initial zeros 1.04 1.745 1.45 0.225 7.364 
Optimal values 0.973 1.678 1.400 0.175 1.707 



 Optimal Synthesis of Manuplator Using Two Competitive Methods 69 

 

Fig. 5 Curve of initial FC0 and curve of optimal solution FC1 

The verification is performed by simulation of the working mode (meshing) of the 

mechanism, in the SolidWorks software, by using the option blocks which is suitable for 

the simulation of planar mechanisms. This achieved the verification of the behavior of the 

mechanism in successive positions, Fig. 6. It provides a functional control study of the 

mechanism and eliminates errors. 

 

Fig. 6 Simulation of the obtained optimal solution 

6. MINIMIZATION USING SQP METHOD 

In order to assess the benefits from the chosen method of minimizing the objective 

function, the method of the Sequential Quadratic Programming (SQP) is used. Minimization 

is performed by programming the objective function and constraint function in Matlab by 

using the program function (command) fminimax as part of the Matlab optimization toolbox. 

The general quadratic programming problem is defined as: 



70 J. PAVLOVIĆ, M. JOVANOVIĆ, A. MILOJEVIĆ 

1
min ( )

2

T T
f z c z z H z                                              (30)                                                 

Where: bzA  , z  0 (31) are the constraints in the form of inequality,  

c – the n-dimensional vector that describes the coefficients of linear terms of the 

objective function,  

z – the n-dimensional vector of unknown values,  

H – the (n×n) symmetric matrix that describes the coefficients of square terms of the 

objective function,  

A – the  (m × n) matrix of constraints, and, 

b – the n-dimensional vector of constraints [12]. 
 

5.1. Function minimax 

Function minimax (fminimax) is part of the Optimization Toolbox in Matlab, which 

finds the minimum of problems determined by the function: 
 

min max ( )
i

z i
F z  such that is 

( ) 0

( ) 0

c z

ceq z

A z b

Aeq z beq

lb z ub








 
  


 

                                   (32) 

 

Where the variables on the right side are the constraints:  

c(z), ceq(z) – vectors of constraints in the form of inequality and equality, 

A – matrix and b – vector are, respectively, the coefficients of constraints in the form of 

linear inequalities,  

Aeq – matrix and beq - vector are, respectively, the coefficients of linear constraints in the 

form of equity, 

lb – vector or matrix of lower bounds of constraints,  

ub – vector or matrix of upper bounds of constraints.  

In the model that is described by equations 1-28, we use only constraints  lb, ub – the 

vectors of lower and upper bounds of the search field of the optimization parameters 

solution (A, B, C, D, E, F, U, H), whose values are listed in the previous numerical 

example [13]. 

Based on the function fminimax, where it is necessary to define the objective function 

and the constraints, the desired optimal parameters are obtained: 
 

z = fminimax(fun,z0,A,b,Aeq,beq,lb,ub)        (33) 

 

where: z – required parameters, fun – objective function, z0 – initial parameters A, b, Aeq, 

beq, lb, ub – constraints. These values are defined above in equations 1÷28, since in this 

case constraints A, b, Aeq and beq are not used. They are written as an "empty" matrix. 

The inputs of constraints are in the form of the lower bounds - lb and the upper bounds - 

ub. In this case, the line of the code of function fminimax looks like: 
 

 

 

 

 

 

parameters=fminimax('Objectfun',parametri0,[],[],[],[],lb,ub)  (34) 
 



 Optimal Synthesis of Manuplator Using Two Competitive Methods 71 

SQP has performed the verification of the previous procedure and the optimization 

achieved by using the passive formal search. The results obtained by the SQP method are 

shown in the following diagram, Fig. 7, and a table of optimal solutions. 

Table 3 Values of obtained optimal solutions by using the function fminimax 

 Optimal parameters Objective function 

 z1 (m) z2(m) z3(m) z4(m) FC (˚) 

Initial zeros 1.04 1.745 1.45 0.225 7.364 
Optimal solution 1.0176 1.7188 1.4095 0.2002 0.3518 

 

Fig. 7 Curve of initial FC0 and optimal FC1solutions, obtained by using the Minimax function 

7. CONCLUSION 

1. The mathematical model of optimization operates with four independent 

parameters out of the potential eight. Even four parameters of optimization give a 

solution of objective function (change of the tilt is less than two degrees) good 

enough for practical performance. By increasing the number of independent 

parameters to possible eight, the quality of the optimal solution can be increased, 

with a complex mathematical procedure. 

2. It is recommended to use an optimal model of the MDBP mechanism for factory 

design because it is an important design requirement. 

3. Improving the model can be done by introducing the constraint function of 

internal forces in the MDBP which would require the extension of this task in the 

domain of static analysis. 

4. The SQP procedure has given a slightly more accurate solution which is a result of 

a freely adjustable step. The formal procedure limits the accuracy (quality of 

solutions) by selecting the interval of indetermination. The quality of a solution 

can be improved by increasing the number of search combinations. 

5. The computing time of implementation in each case is less than one second. 



72 J. PAVLOVIĆ, M. JOVANOVIĆ, A. MILOJEVIĆ 

Acknowledgements: The paper is a part of the research performed within the project TR 35049, 

Faculty of Mechanical Engineering. The paper is also a part of the doctoral studies of the first author. 

The authors would like to thank the Ministry of Education and Science of the Republic of Serbia.  

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Delhi, New Delhi. 

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Algorithm (ACA), Proc. Fourth International Conference on Mechatronics and Automation ICMA 2007, 
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Belgrade. 
8. Yu, Y., Lianguan, S., Mujun, L., 2010, Optimum design of working device of wheel loader, Proc. First 

International Conference Mechanic Automation and Control Engineering MACE 2010, Wuhan, China. 

9. Xiuhua, G., Yunchao, W., Erzhong, A., Jiawei, H., 2007, Optimization of the Working Device of Loader 
Based on ADAMS, Computer Simulation Journal, 1(1). 

10. Zhang, R., Qi, J., Cai, Y., 2008, Intelligent optimum design of the working mechanism of loader, Proc. 
Seventh World Congress on Intelligent Control and Automation WCICA 2008, Chongqing, China. 

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OPTIMALNA SINTEZA MANIPULATORA 

KORISTEĆI DVE UPOREDNE METODE 

U radu je data programska realizacija za traženje optimalne geometrije ravanskog Z-

mehanizma. Rad pokazuje matematičku proceduru definisanja funkcije cilja, funkcija ograničenja, 

oblasti pretraživanja kojima se rešava zadatak optimizacije. Polazeći od rešenja u praksi, dat je 

numerički primer određivanja optimalnog dizajna sa četiri parametra optimizacije. Svi parametri 

optimizacije su geometrijski na mehanizmu za određivanje nagiba kašike. Zadatak je rešavan 

dvema različitim, numerički metodama - metodom formalnog pretraživanja hiperprostora (metoda 

pasivnog skeniranja) i aproksimativnom metodom kvadratnog sekvencijalnog programiranja – SQP 

(primenom fminmax funkcije iz Matlabovog optimizacionog toolbox-a). Verifikacija rešenja je 

vršena animacijom u programu za geometrijsko modeliranje. Rezultati su grafički ilustrovani.  

Ključne reči: Pasivno traženje, Z - mehanizam, utovarivač, rudarska mašina, optimizacija 

mehanizma, numeričko rešavanje, SQP 

file:///H:/Doktorske%20studije/I%20godina/50668823/intelligent-optimum-design-of-the-working-mechanism-of-loader
http://www.komatsu.com/ce/products/pdfs/WA320-5.pdf