Ing. Electronico

Acta Polytechnica Hungarica

Vol. 1, No. 2, 2004

Appropriate Mathematical Model of DC Servo
Motors Applied in SCARA Robots
Attila L. Bencsik
Bánki Donát Faculty of Mechanical Engineering, Budapest Tech
Népszínház utca 8, H-1081 Budapest, Hungary
bencsik.attila@bgk.bmf.hu

Abstract: In the first part of the presentation detailed description of the modular technical
system built up ofelectric components and end-effectors is given. Each of these components
was developed at different industrial companies separately. The particular mechatronic
unit under consideration was constructed by the use of the appropriate mathematical model
of these units. The aim of this presentation is to publish the results achieved by the use of a
mathematical modeling technique invented andapplied in the development of different
mechatronic units as drives and actuators. The unified model describing the whole system
was developed with the integration of the models valid to the particular components. In the
phase of testing the models a program approximating typical realistic situations in terms of
work-loads and physical state of the system during operation was developed and applied.The main innovation here presented consists in integrating the conclusions of professional
experiences the developers gained during their former R&D activity in different
professional environments. The control system is constructed on the basis of classical
methods, therefore the results of the model investigations can immediately be utilized by the
developer of the whole complex system,which for instance may be an industrial robot.
Keywords: mechatronic, system investigation, industrial robot, DC servo motor, PID
control

1

Model of the Servo Motor

The first step was to design the model of a PM motor in order to perform the
analysis. The system model has two main parts and several smaller units.

– 99 –

A. L. Bencsik

Appropriate Mathematical Model of DC ServoMotors Applied in SCARA Robots

Figure 1

Figure 2
As a following step the control circuit was developed using the previous results.
U be ( t ) = U R a + U L a + U i

U be ( t ) = i a R a + L a

(1)

di a
+ K1
dt

Ui = KiΩ

(2)
(3)

M mot − M "oad = M accelerator = Θ
l
K 2 la − M load − B V Ω = Θ

dΩ
dt

dΩ
dt

(4)
(5)

Loop of the circle:
U be ( t ) = i a − R a +L a
K2ia = Θ

di a
+ K 1Ω
dt

dΩ
+ M load + B V Ω
dt

(6)
(7)

The system-model:
.

d ma
La
= U be ( t ) − i a R a − K 1Ω
dt
Θ

dΩ
= K 2 i a − M load + B V Ω
dt

– 100 –

(8)
(9)

Acta Polytechnica Hungarica

Vol. 1, No. 2, 2004

Figure 3
Interpretation of the blocks in system description
Out = In 1 + In 2

Out =

In
P

Out = In ⋅ P

∫

Out = In ⋅dt

Motor given by the system-model can be regarded as only unit

Parameters of the motor used in simulation 80W
Ra = 0,360 [W]

La = 0.14.10-3 [H]

Qrotor = 1,22.10-4 [kg.m2]
K1 = 50,1.10-3 [Vs]
Mn =0,301 [Nm]
K2 = 50,1.10-3 [Nm/A]

– 101 –

A. L. Bencsik

Appropriate Mathematical Model of DC Servo Motors Applied in SCARA Robots

Ube = Un = 15 V
Bv clamping coefficient(viscosity) is not given in the cataloge. Let take the value
of Bv so that in slow running, no-load current flows in the motor, I0 = 324 mA.
W0 = 286,78 rad/s. If I0 = 310 mA then Mvis = 0,05.Mn ! Then Bv = (0,05.Mn)/W0 =
5,23.10-5 [Nms]
Time constants of the motor: Tel =
Temech =

Ra ⋅Θ
K2

La
= 3,8 ⋅ 10 −4 s
Ra

= 1,7 ⋅ 10 2 s

For evaluation of the results presented in diagram.Stepping the curves (leaving them as the results):
On the left side of the diagram is the sclaling. (Ω, α, Icurrent, Uswitch, Icurrent) The first
number indicates the value belonging to the lower level of the rectangle which is
divided into parts by sections, the second number indicates the changing value
belonging to the upper level.
(In the case of those variables where the absolute value of...