Mathematical Modeling of Robot Manipulator the with Four … · 2016. 10. 31. · Mathematical...

Global Journal of Pure and Applied Mathematics.

ISSN 0973-1768 Volume 12, Number 5 (2016), pp. 4419-4429

© Research India Publications

http://www.ripublication.com/gjpam.htm

Mathematical Modeling of the Robot Manipulator

with Four Degrees of Freedom

Tatiana Victorovna Zudilova

ITMO National Research University (ITMO University), Head of Software Development Chair, 197101 Saint Petersburg, 12 Gastello Str., office 311,

Russian Federation.

Sergei Evgenievich Ivanov

ITMO National Research University (ITMO University), Department of Information Systems, 197101 Saint Petersburg, 49 Kronverksky Pr., Russian Federation.

Abstract

The nonlinear mathematical model of an industrial robot manipulator with

four degrees of freedom is developed. For creation of the equations of robot

manipulator is applied the matrix method and Lagrange equations of the

second kind.

The robot manipulator is designed to create automated complexes for service

of devices, for removing and installing the equipment, change of details and

tools.

The kinematic structure of the robot manipulator and the dynamic

characteristics of motion for robot manipulator are defined. The mathematical

model is presented by nonlinear system of four ordinary differential equations

of second order. The analytical solution of the nonlinear system is obtained by

method of polynomial transformations. The power of actuators and

generalized forces for move the gripping device at a given point of space for a

certain period of time are defined.

Keywords: the nonlinear mathematical model, the industrial robot

manipulator, the dynamic characteristics of the robot manipulator.

4420 Tatiana Victorovna Zudilova & Sergei Evgenievich Ivanov

INTRODUCTION

The robots manipulators are widely used in many fields and spheres of human activity:

assembly production, machining, heat treatment.

The industrial robot manipulator is an automatic machine, which includes the

manipulator with four degrees of freedom. The robot manipulator performs functions

similar to the human hand and is controlled by operator or automatically. In the

industrial production are used the technological manipulators for operations of

assembly and welding. The auxiliary robots manipulators are used for maintenance of

the main technological equipment for lifting and transporting [1-4].

The structure of robot manipulator consists of main elements, connected with each

other: the base, the stand, the mechanical arm, the gripping device. The working body

of the manipulator is the gripping device. The elements connecting the base with the

working body constitute a kinematic chain of the robot manipulator. The two adjacent

elements constitute a kinematic pair.

The number of degrees of freedom provides the movement of the device of gripping

in any point of the working area. The main function of the robot manipulator is

determined by kinematic scheme.

NONLINEAR MATHEMATICAL MODEL OF THE INDUSTRIAL ROBOT

MANIPULATOR

The industrial robot manipulator M20P for robotic complex is considered. The robot

manipulator is shown in Figure 1 consists of a base, the stand, hands, the gripping

device and actuators for moving and turns.

Figure 1: The industrial robot manipulator M20P

The industrial robot manipulator M20P has two translational and two rotational

kinematic pairs. The elements of manipulator are numbered starting with fixed

Mathematical Modeling of the Robot Manipulator with Four Degrees of Freedom 4421

element - the base with number of zero. The motion of clamp-unclamp for the

gripping device is not considered.

The kinematic scheme for industrial robot manipulator is shown in Figure 2 and

consists of four moving parts.

We introduce the relative coordinate system associated with elements, with the origin

at the points: 1 2 3 4, , ,O O O O . The initial coordinate system 0O we correlate with the

fixed element.

For generalized coordinates we accept: the angle of rotation around the rack - 1q ,

lifting height - 2q , the length of arms - 3q , the angle of gripping device - 4q .

The system of dynamic equations of industrial robot manipulator is obtained.

We apply the matrix method and dynamic Lagrange equations in matrix form to

produce the equations of motion [5-7].

The transition from the coordinate system 0O to 1O occurs by rotation around z

axis at an angle 1q .

The transition from the coordinate system 1O to 2O occurs by rotation around z

axis at an angle , by displacement along z axis on 2q and by rotation around x axis

at an angle / 2 .

The transition from the coordinate system 2O to 3O occurs by displacement along z

axis on 3q . The transition from the coordinate system 3O to 4O occurs by rotation

around z axis at an angle 4q , by displacement along z axis on s and by rotation

around x axis at an angle / 2 .

We introduce the radius vector of points 1,2, ,3,( )4i iO in the i-local coordinate

system:

1T

i i i iR x y z .


Figure 2: The kinematic scheme

The communication of radius vectors in the coordinate system i-1 and i by means of

the transition matrix 1,i iA : 1 1, .i i i iR A R

The transition matrix from point 0O to point 1O is of the form:

1 1

1 1

01

cos sin 0 0

sin cos 0 0

0 0 1 0

0 0 0 1

q qq q

A

The matrices of the transition from point 1O to point 2O and of the transition from

point 2O to point 3O are of the form:

12

2

1 0 0 0

0 0 1 0

0 1 0

0 0 0 1

Aq

, 23

3

1 0 0 0

0 1 0 0

0 0 1

0 0 0 1

Aq


The transition matrix from point 3O to point 4O is of the form:

4

34

4

4 4

cos 0 sin 0

sin 0 cos 0

0 1 0

0 0 0 1

q

A

qq q

s

The transition matrix from point 0O to point 2O is defined as the product of

matrices: 02 01 12A A A ,

1

02

1

1 1

2

cos 0 sin 0

sin 0 cos 0

0 1 0

0 0 0 1

A

q qq q

q

The transition matrix from point 0O to point 3O is obtained as the product of

matrices: 03 01 12 23A A A A ,

1 1 3

03

1

1 1 3 1

2

cos 0 sin sin

sin 0 cos cos

0 1 0

0 0 0 1

q q q qq q

Aq q

q

The transition matrix from point 0O to point 4O is defined as the product of

matrices: 04 01 12 23 34A A A A A ,

1 4 1 1

0

4 3 1 1

4 1 1 1 4 3 1 1

4 4 2

4

cos cos sin cos sin sin sin

cos sin cos sin sin cos cos

sin 0 cos

0 0 0 1

q q q q q q q s qq q q q q q q s q

q qA

q

The origin of coordinates 4O associated with gripping device in the fixed coordinate

system 0O associated with a base of rack is defined by coordinates:

4 4 43 23 1 1 1 1sin( ) sin( ) cos( ) cos( ,),O O Oq q s q q q s qx y z q .

We denote the coordinates of the gripper fingers in a coordinate system 4O

as 4 4 4( , , )x y z , then in a fixed coordinate system 0O the coordinates of the gripper


fingers can be represented in the form of:

3 1 4 1 4 4 1 4 1 4 1

04 3 1 4 1 4 4 1 4 1 4 1

04 2 4 4 4

04

4

sin( ) cos( )cos( ) cos( )sin( ) sin( ) sin( )

cos( ) sin( )cos( ) sin( )sin( ) cos( ) cos( )

sin( ) cos( )

q q x q q z q q s q y qy q q x q q z q q s q y qz q x q q

x

z

The coordinates of the point 0 0 0, ,x y z , given in a fixed coordinate system 0O in the

coordinate system 4O associated with a rack, determined in the form of:

2 2

2 1 4 2 1 4 1 44 2 2 2 2

1 1 4 4

2 2

1 4 1 4 1 4

2 2 2 2

1 1 4 4

sin ( )sin( ) cos ( )sin( ) cos( )cos( )

sin ( ) cos ( ) sin ( ) cos ( )

sin( )cos( ) sin ( )sin( ) cos ( )sin( ),

sin ( ) cos ( ) sin ( ) cos ( )

q q q q q q x q qxq q q q

y q q z q q z q qq q q q

2 2 2 2 2 2

3 1 4 3 1 4 1 44 2 2 2

1 4 4

2 2 2 2

1 4 1 1 4 1 1 4

2 2 2

1 4 4

2

1 4

tan ( )sin ( ) tan ( )cos ( ) tan ( )sin ( )

tan ( ) 1 sin ( ) cos ( )

tan ( )cos ( ) tan( )sec( )cos ( ) tan( )sec( )sin ( )

tan ( ) 1 sin ( ) cos ( )

sec( )cos ( )

q q q q q q s q qyq q q

s q q x q q q x q q qq q q

y q q y

2 2 2 2 2

1 4 3 4 3 4 4 4

2 2 2

1 4 4

sec( )sin ( ) sin ( ) cos ( ) sin ( ) cos ( ),

tan ( ) 1 sin ( ) cos ( )

q q q q q q s q s qq q q

2 2

2 1 4 2 1 4 1 4 44 2 2 2

4 1 1

2 2

1 4 4

2 2 2

cos ( )sec( ) sin ( )sec( ) cos( ) tan( )sec( )

tan ( ) 1 sin ( ) cos ( )

sin( ) tan( )sec( ) cos ( 1)sec( 4) sin ( 1)sec( 4).

tan ( 4) 1 sin ( 1) cos ( 1)

q q q q q q x q q qzq q q

y q q q z q q z q qq q q

We calculate the kinetic energy of each unit and the total energy.

For determine the kinetic energy of i-unit we use the matrix formula with transition

matrices:

0 0

1( )

2

Ti i i iT tr A H A , (1)

where iH - the inertia matrix of i - unit,


ixx ixy ixz i i

iyx iyy iyz i ii

izx izy izz i i

i i i i i i i

J J J m xJ J J m y

HJ J J m zm x m y m z m

,

where 1 2 3, ,m m m - mass of units,

, ,i i ix y z - coordinates of the centers of gravity of unit in the local coordinate system,

, ,ixx iyy izzJ J J - elements of the inertia tensor of unit, a relatively their own axes,

, ,xi yi ziJ J J - moments of inertia of unit relative to the axes.

The kinetic energy for units is determined by a matrix formula (1):

2 2 2

1 1 1 2 1 2 2 2 2

2 2 2

3 2 3 3 3 1 3 3 3 3 3 3

1 (0.5 0.5 ), 0.5 0.5 0.5 ,

0.5 0.5 0.5 0.5 0.5 ,

xx yy xx zz

xx zz

q J J T q J J q m

T q m q m q J J m q

T

q z

2 2 2

1 4 4 4 4 3 4 3 4

2 2 2 2

1 4 4 4

4

4

0.25 0.5 0.25 0.5 0.5

(0.25 4 0.25 4 )cos(2 ) 0.5 2 4 0.5 3 4 (0.5 0.5 ).

xx yy zz

xx zz

q J J J m q m q s m s

q J xx J zz q dq m dq q J

T

m J

The total kinetic energy is equal to the sum of kinetic energies for units, taking into

account the equalities:

, ,iyy izz xi ixx izz yi ixx iyy ziJ J J J J J J J J

2 2 2 2

1 4 3 3 3 3 1 2

2 2 2 2

2 3 4 2 3 3 4

1 2 3 4

3 4 4

0.5 0.5 0.0025 0.5 0.0025 0.0025 0.0025

0.5 0.5 0.5 0.5 0.5 0.0 .025

q m sq q s m q m m

m m m q m q m q m q

T T T T T

We calculated the potential energy for each unit and the total energy.

In matrix form, the potential energy of unit is defined in the form of: T

i i i iP m G AR ,

where 1T

i i i iR x y z - column matrix of the coordinates of the center of

gravity unit,

0 0 0TiG g - matrix row for the acceleration of free fall.

The total potential energy is defined by the form: 2 2 3 4q m mP g m .

We apply Lagrange equations in matrix form for equations of motion:


ii i i

d T T P Qdt q q q

, (2)

where 1 2 3 4, , ,Q Q Q Q - generalized forces created for unit.

By substituting the kinetic energy, the potential energy and the generalized forces in

the Lagrange equations (2), we obtain equations for robot manipulator with four

degrees of freedom:

2 2 2

1 4 3 3 3 3 1 2

1 3 3 4 3 4 1

( ) 2 ( ) ( ) 0.005 ( ) 0.005 0.005 0.005

( ) ( ) 2 2 ( ) 2 0,

q t m sq t q t s m q t m m

q t q t m m q t m s Q

2 3 4 2 2

2 2

4 3 1 3 4 3 1 3 3 3

4 4 4

( ) ,

( ) ( ) ( ) ( ) ( ) ,

0. .005 ( )

m m m q t g Q

m q t sq t m m q t q t m q t Q

m q t Q

For the analysis of nonlinear mathematical models apply different analytical methods

[8-11]: averaging method, small parameter method, harmonic balance method, Van

der Pol method, Krylov-Bogolyubov method, the Poincare perturbation method, the

polynomial transformation method.

We obtained the analytical solution of equations by polynomial transformations

method:

21 4

1

1 2 3 4 3 4

22 2 3 42

2 3 4

( )2 36

(( )

2(

,

),

)

Q m sq t tm m m m m m

Q g m m mq t tm m m

3 323 4

3 3

3 4 3 4

244

4

( )2 18

100(

,

.)

Q m sq t tm m m m

Qq t tm

(3)

Figure 3 shows graphs of the generalized coordinates ( 1q -thick line, 2q -dashed line,

3q -dot-dashed line, 4q -dotted line) with the following parameters for robot

manipulator:

1 2 3 4 1 2 3 4100, 20, 10, 40, 0.5, 8, 700, 5, 0.02m m m m s Q Q Q Q


1 2 3 4 5t

0.5

1.0

1.5

2.0

q

Figure 3: The dynamics of the generalized coordinates: 1 2 3 4, , ,q q q q

We denote the coordinates of the object for gripping in a fixed coordinate system 0O

for ( , , )x y z . For bring the object to the point ( , , )x y z necessary that the generalized

coordinates are:

2 21

1 22 tan , ,x y yq q z

x

2 2

3 4, .q s x y q (4)

We equate the solution (3) and the expression for the generalized coordinates (4), we

obtain the system for determining the generalized forces: 1 2 3 4, , ,Q Q Q Q .

We identified the necessary generalized forces 1 2 3 4, , ,Q Q Q Q for moving the gripper

to the specified point ( , , )x y z in the time interval kt :

2 214

1 1 2 3 4 2

3 4

2 2 3 4 2 3 42

4tan ,

18

2( () ),

k

k

x y ym sQ m m m mm m t x

zQ m m m g m m mt


2 2 3 3

4 43 3 4 422 2

3 4

2 , .1009k k

s x y m s mQ m m Qt tm m

The powers of actuators 1 2 3 4, , ,F F F F for move the gripping device at a given point

( , , )x y z of space for a certain period of time kt are defined.

2 214

1 2 3 4 2

3 4

2 214

2

3 4

1

4tan

9

4tan ,

18

k

k

k

x y ym stm m mF mm m t x

x y ym sm m t x

2

2 3 4 32

2 4,

k k

gz zm m mt t

F

2 23 3

2 2 43 4 3 33

3 4

42,

9 kk

s x ym sm m s x ytt m m

F

2

44 350 k

mFt

CONCLUSION

The mathematical model of industrial robot manipulator with four degrees of freedom

is developed. The robot manipulator is designed to create automated systems for

service devices, for removing and installing the equipment, change details and tools.

For creation of the equations of robot manipulator we used the method of matrices

and matrix Lagrange equations of the second kind. The mathematical model is

presented nonlinear system of four ordinary differential equations of second order.

The analytical solution of the nonlinear system is obtained by polynomial

transformations method. The kinematic structure and dynamic characteristics of robot

manipulator are defined. The power of actuators and generalized forces generated by

actuators for moving the gripper at a given point in space are found.

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