Plane Thermoelastic Waves in Infinite Half-Space Caused


FACTA UNIVERSITATIS  

Series: Electronics and Energetics Vol. 31, N
o
 1, March 2018, pp. 75 - 87 

https://doi.org/10.2298/FUEE1801075B 

 

ELECTRONIC GEARING OF TWO DC MOTOR SHAFTS 

FOR WHEG TYPE MOBILE ROBOT

 

Miloš Božić
1
, Sanja Antić

1
, Vojislav Vujičić

1
,  

Miroslav Bjekić
1
, Goran Đorđević

2
 

1
University of Kragujevac, Faculty of Technical Sciences, Ĉaĉak, Serbia 

2
University of Niš, Faculty of Electronic Engineering, Niš, Serbia 

Abstract. This paper describes the implementation of electronic gearing of two DC motor 

shafts. DC motors are drives for a mobile robot with wheels in the form of wheel - leg 

(Wheg) configuration. A single wheel consists of two Whegs (dWheg). The first DC motor 

drives one Wheg, while the second one drives another independent Wheg. One motor 

serves as the master drive motor, while the other represents the slave drive motor. As the 

motors are independent, it is necessary to synchronize the speed and adjust the angle 

between shafts. The main contribution of this paper is the implementation of control 

structure that enables the slave to follow the master drive, without mechanical coupling. 

Based on encoder measurements, the slave effectively follows the master drive for the 

given references of speed and angle. Speed and positioning loops are implemented on real 

time controller - sbRIO. The laboratory setup was created and comparison of realized and 

required angles and speeds was made. 

Key words: Electronic gearing, Master, Slave, Wheg, DC motor, sbRIO 

1. INTRODUCTION 

Coupling of motion axis in industrial and robotic applications is always a challenging 

task. It is often necessary to do coupling and synchronization of two or more axes, linear 

[1, 2], circular or other complex movements, such as curve profiles. The task can be 

performed in contact way, by using mechanical couplings like gear pairs, differential 

drive, belts, chains, etc. Another possible alternative is contactless coupling. In industrial 

applications, it is used in servo applications under the name electronic coupling, 

electronic gearing or electronic line shafting [3, 4], while in the automobile industry, this 

technology is called drive-by-wire [5, 6]. The examples of the electronic coupling can be 

found in haptic devices [7, 8]. The advantages of electronic over mechanical coupling are 

numerous. Electronic gearing is used instead of mechanical assemblies, given that the 

                                                           
Received March 14, 2017; received in revised form July 27, 2017 

Corresponding author: Miloš Božić  

University of Kragujevac, Faculty of Technical Sciences, Svetog Save 65, 32000 Ĉaĉak, Serbia 

(E-mail: milos.bozic@ftn.kg.ac.rs) 



76 M. BOŽIĆ, S. ANTIĆ, V. VUJIĈIĆ, M. BJEKIĆ, G. ĐORĐEVIĆ 

 

latter are the weakest link in the system. Electronic coupling makes the system more 

efficient and flexible. Typically, there is one master drive and one or more slave drives 

which follow references set by the master drive. Reference can be given in the form of 

torque (current), speed or position therefore the concept can be applied to a wide variety 

of applications. In this paper, the concept master-slave drive was applied to wheel leg 

drive (Wheg) [9, 10]. Wheg has a form of legged wheel with n legs (spokes). In the setup 

employed in the paper, Wheg had four spokes. Placing two Whegs next to each other and 

by independent drive of master and slave, double Wheg drive (dWheg) is obtained. 

dWheg drive allows independent control of the two Whegs. Figure 1 left illustrates all the 

seven elements of the dWheg master-slave drive wheel. 

   

Fig. 1 dWheg master-slave drive (left) and prototype of robot with rigid dWheg drives (right) 

The figure 1 left shows the main parts of the dWheg drive module. The parts are 

numbered as follows: 1) master motor, 2) slave motor, 3) coupling, 4) pulleys with a belt, 

5) bearing case and shaft 6) slave Wheg, 7) master Wheg. dWheg presents original 

authors solution [9]. This type of drive is designed to increase the efficiency of mobile 

walking robot, planned for the uneven terrain. The appearance of the prototype robot 

during the testing phase with the rigid type of dWheg drive can be seen in Figure 1 right.  

This study deals with practical issues of active control of the parameter α. By changing 

the angle α different compliance and contact area with the ground are provided. This 

allows the mobile robot to move efficiently over the flat terrain when α tends to 45° as 

well as over a rough terrain when the angle α tends to 0°. Because of the nature of the 

problem this system does not require precise angle adjustment. The response time of angle 

adjustment is not critical. It is sufficient to adjust the angle of at least one full revolution 

of dWheg. Varieties of synchronizing methods are presented in the industry [2,16,17,18]. 

In industrial drives, it is necessary to perform the synchronization speed in the shortest 

possible time and change of the speed of the master drive is allowed, so the cross coupling 

technique is commonly used. A typical example of cross coupling is the linear 

interpolation between two axes. In dWheg drive the slave drive must not have influence 

on the work speed of master drive. Slave motors represent an additional support system 

and allow efficient travel of the robot over different terrains. For example, in case of loss 

of function of slave drive in case of cross coupling techniques that would mean that 

0.5m 



 Electronic Gearing of Two DC Motor Shafts for Wheg Type Mobile Robot 77 

 

master motor should stop or slow down. So cross coupling synchronization will impact on 

mobile robot movement and it would be impossible for robot to track reference path. The 

paper presents implementation of master slave synchronization on dWheg drive type with 

additional position loop. 

In the next part of the paper the mathematical model of the dWheg drive motor is 

presented. The third part shows the control structure and the synthesis of speed and 

position loops. The fourth part of the paper shows the experimental setup and obtained 

results. The fifth part is a brief conclusion with the future steps. 

2. MATHEMATICAL MODEL OF DC MOTOR 

Electric circuit of the DC motor with permanent magnets is given in Figure 2. 

 

Fig. 2 Electric circuit of the DC motor with permanent magnets 

Mathematical model of the DC motor can be described by following equations: 

 
( )

( ) m
o

t
t

N


    (1) 

 ( ) ( ) ( )( const .)
m

m me me m me o

d
e t K K t NK t

dt


         (2) 

 ( ) ( ) ( ) ( )r
r r r r m

di
u t L t R i t e t

dt
     (3) 

 ( ) ( )
m em r

M t K i t   (4) 

 ( ) ( ) ( ) ( ) ( )
m

m c l m

d
M t M t M t J t F t

dt


       (5) 

where: 

ur(t), ir(t) – armature voltage and current, Rr, Lr – armature resistance and inductance em(t)  

armature induced emf, m – angular displacement of motor shaft; o – angular displacement of 

gearbox out; m(t) – angular speed of motor shaft [rad/s]; o(t) – angular speed of gearbox out 

[rad/s]; Kme, Kem – emf and torque constant, Mm(t) – generating motor torque; Ml(t) – load 

torque on motor side, Mc(t) – Coulomb friction; Jm, Jo – motor and load inertia; J = Jm + Jo/N
2
 – 



78 M. BOŽIĆ, S. ANTIĆ, V. VUJIĈIĆ, M. BJEKIĆ, G. ĐORĐEVIĆ 

 

total inertia on motor side; Fm, Fo – motor and load viscous friction; F = Fm + Fo/N
2
 – total 

viscous friction on motor side; N – gear ratio. 

Appropriate block diagram of the DC motor model is presented in Figure 3. 

 

Fig. 3 Block diagram of the DC motor mathematical model 

To make realistic mathematical model and to perform controller synthesis, it was 

necessary to identify relevant parameters. Parameters that were not known were determined 

experimentally [11]. Appendix provides a table with motor data. 

The measurement of the parameters showed that the mathematical model given in the 

form of block diagram in Figure 3 could be further simplified. Firstly, Coulomb friction 

was neglected because of its minor influence on the system behavior, given its low value 

[11]. The motor transfer function becomes: 

 
2 2 5 2

( ) 19.175
( )

( ) ( )( ) 2 1 2.849 10 0.019 1

m em m
m

r r r em me v r v

s K K
G s

U s L s R Js F K K T s T s s s



   

        
 (6) 

where 

em
m

r em me

K
K

R F K K



  gain, 

r
v

r em me

JL
T

R F K K



  time constant,  

2 ( )

r r
r

r r em me

FL JR

JL R F K K


 


  relative damping factor. 

Secondly, motor inductance could also be neglected. This was confirmed by the 

relatively small value of motor inductance. Therefore, the transfer function of unloaded 

drive becomes of the first order: 

 
( ) 19.175

( )
( ) 1 0.019 1

m m
m

r m

s K
G s

U s T s s


  

 
 (7) 

where 

19.175
em

m

r m em me

K
K

R F K K
 


, 19 ms,m r

m

r m em me

J R
T

R F K K
 


 (8) 

are the gain factor and time constant of the motor. 

Based on the above analysis, the first order model (7) is further used. 



 Electronic Gearing of Two DC Motor Shafts for Wheg Type Mobile Robot 79 

 

The selection of the measurement period was based on the bandwidth of the closed-

loop system. Bandwidth of the motor in a closed loop in the absence of controllers is 

determined by the equation: 

 
0

m

1 rad
105.263 .

0.5 sT
    (9) 

According to the sampling theorem, the sampling frequency s should be at least twice 

the bandwidth of the system o [13], which infers that 

 
0

0, 5 29.845 ms.
m

T Т


    


 (10) 

This value of the sampling period represents the theoretical maximum. However, more 

practical reasons require that sampling period be lower than the theoretical maximum 

allowed. The relatively large sampling period in relation to the real dynamics of the 

system can have negative impact on the closed-loop system stability [13]. Factors that 

determine the lower allowable value of sampling period are quality reference tracking, 

quality control, measured by the error in the system response due to the presence of 

external disturbance, system sensitivity to parameter variations and noise sensor-induced 

errors. In real applications, for adequate reference tracking and elimination of disturbance, 

sampling frequency selection s = (10  20)o is proposed [14]. This results in a preferred 

range of the sampling period 

  2.98ms 5.97 ms .T    (11) 

Given that the introduction of the controllers additionally increases the bandwidth of 

closed loop system, sampling periods in the speed and position loops, based on (11), were 

selected to be 1ms and 5ms, respectively. 

3. SYSTEM CONTROL STRUCTURE 

The control structure that provides electronic coupling of two motor shafts is shown in 

Figure 4. The block diagram shows two independent references, those for speed – Ωr and 

angle – α. Master drive contains only a speed loop, while slave drive has a cascade 

structure with a position and a speed loop. On the summing junction in the slave loop, 

there are two references that are summarized. The first reference is the instantaneous 

master drive speed. The second one is a speed requirement for changing the shaft angle of 

slave motor, which can be either positive or negative. PI controller is selected for speed 

loop and PD controller for positional one. The synthesis and selection of the controller 

parameters will be explained in more detail in chapters that follow. 



80 M. BOŽIĆ, S. ANTIĆ, V. VUJIĈIĆ, M. BJEKIĆ, G. ĐORĐEVIĆ 

 

 

Fig. 4 Master slave loops 

3.1. The synthesis of speed loops in Z domain 

The torque/current loop was implemented in the motor driver. In the block diagrams, 

the driver is represented only by its transfer function. The transfer function of the driver 

was obtained by recording the response of motor speed to step excitation reference 

voltage. Due to the cascade control structure, speed loop can be adjusted independently of 

the position loop. Figure 5 shows Simulink model of the speed loop. 

 

Fig. 5 Simplified model of the speed-loop 

where: T1 = 1ms is the speed loop sampling period; Kd = 8.463910
3

 is the driver constant; 

Kc = 1 / 30 is the speed translation constant from rpm to impulses per interval (imp/int); 

230
5.166 10

d m c
K K K K


  


; nr(t), n(t) are the reference and the measured speed in imp/int. 

The transfer function of the motor with the driver in  domain is: 

 
/

/

1 1
( )

1

m

m

T TsT

T T

m

e K e
G z K

s sT z e





  
  

  
  (12) 

 1

2

0.002643
( ) .

0.9488

C
G z

z z C
 

 
 (13) 

The characteristic equation is now: 

 1

2

( ) 1 ( ) 1 0
1

P I

C z
W z K K

z C z
    

 
 (14) 

i.e. 
2

1 2 1 2
[( ) 1] 0

P I P
z z K K C C C K C          



 Electronic Gearing of Two DC Motor Shafts for Wheg Type Mobile Robot 81 

 

Since the characteristic equation of a second-order system for pseudo periodic time 

response when 0 1    has the form 

 
22 2

( ) 1 2 cos( 1 ) 0n n
T T

n
W z z e T z e

  
          (15) 

it follows 
2

2

1

nT

P

C e
K

C

 


 , 
22

1

1
(1 2 cos( 1 ) )n n

T T

I n
K e T e

C

  
       

by selection different values of the damping factor  and selecting the natural frequency 

n i.e. the desired bandwidth of the closed-loop system 0 to be  n  0 = 314 rad/s 

(closed loop bandwidth without controller from (9) is 105.263 rad/ s ) different values of 

KP = 45.5984 and KI = 33.7229 were obtained, and are given in Table 1. 

Table 1 Parameters of PI regulator for 314 rad
n

   

 KP KI 

0.3 45.5984 33.7229 

0.5 82.5883 31.7538 

0.7 115.2122 29.9389 

0.9 143.9854 28.2646 

 

Digital PI parameters were also selected with Ziegler-Nichols method. 

737.3
Pkr

K  , 32 10 s
kr

T


 , 0. 3 .845 31PkrPK K   , 
1.2

199
I

P

kr

T
K

K

T


 
  

 

Fig. 6 Motor speed for different methods of digital PI parameters synthesis 

However, the aim was to achieve the aperiodic velocity response to the given reference 

with desired dynamic. So, for the regulator parameters synthesis, the method of compensation 

was applied.  

Closed loop transfer function of the velocity loop is defined with 

 
2

1 2

( )( 1)
( ) .

( ( 1) ) ( )( 1)
s

P I

z C z
W z

C K z K z z C z

 


    
 (16) 



82 M. BOŽIĆ, S. ANTIĆ, V. VUJIĈIĆ, M. BJEKIĆ, G. ĐORĐEVIĆ 

 

The parameters of the PI controller were selected in that manner so that one pole was 

equal 
1 2

0.9488z C  . In this way, the transfer function of the velocity loop was reduced 

to the first-order. Now with the selection of the second pole in order to achieve the 

aperiodic response of the given time constant 5 msT   , 
2

0.8187
T T

z e


   the 

parameters of the PI speed regulator were determined. 

2 1 2

1

65.0842
P

C z z
K

C


   and 1 2 1 2

1

1
3.5121

I

z z z z
K

C

  
  . 

PI controller was implemented within the LabVIEW code. Its discrete transfer function 

is: 

 
1( ) 65.0842 3.5121 65.0842 3512.1

1 1
r

T zz
G z

z z
   

 
 (17) 

where T1=1ms was the speed loop sampling period. 

Figure 7 shows the speed responses of Simulink and real model during PI speed 

control using compensation method. Measurement confirms the achievement of the 

desired time constant using this method. Response matching of the model and the actual 

system was obtained. 

 

Fig. 7 The speed responses during PI speed control with parameters obtained using 

compensation method: Simulink and the real system 

After the adjustment of speed loop parameters, speed loop could be presented as a 

simple first-order transfer function. Since the open loop transfer function had the first 

order astatism, the zero steady-state error is provided when the input signal has the step or 

constant input. Therefore, in order to provide the necessary dynamic characteristics of the 

response, which is preferably to have the aperiodic character, PD controller was selected.  

Desired aperiodic time response was obtained with auto tuning option with selection 

of 0.06
P

K   and 0.3
D

K  , The transfer function of PD controller in discrete time domain, 

realized in LabVIEW code, was 



 Electronic Gearing of Two DC Motor Shafts for Wheg Type Mobile Robot 83 

 

 
1 1

2

0.0015
( ) 0.06 0.3(1 ) 0.06 (1 ),

r
G z z z

T

 
       (18) 

where T2 = 5ms was the position loop sampling period. Figure 8 shows angular responses 

during PD position control. Matching of the model and the real system angular response is 

apparent. 

 

Fig. 8 The angular responses during PD position control: Simulink and the real system 

4. EXPERIMENT SETUP AND RESULTS 

In order to test the proposed control structure, experimental setup was made. The setup is 

shown in Figure 9. 

 

Fig. 9 Experimental setup for realization of electronic gearing [19,20] 

The main elements of the experimental setup were: 1 – Computer with LabVIEW 

applications for monitoring, 2 – master motor with encoder resolutions of 500 ppr, 3 – 

slave motor with encoder resolutions 500 ppr, 4 – real time sbRIO9636 controller, 5 – 

master DC driver, 6 – slave DC driver, 7 – Dual power 30VDC, 5A. The video showing 

the experimental setup in operation is available at the link given in [19, 20]. 



84 M. BOŽIĆ, S. ANTIĆ, V. VUJIĈIĆ, M. BJEKIĆ, G. ĐORĐEVIĆ 

 

 

Fig. 10 Block diagram of the experimental setup 

The block diagram of the experimental setup shows that the drivers run in open loop 

configuration. The speed loop was realized in the FPGA section of the controller, while 

the position loop was realized in the real-time part of the controller. Figures below show 

the reference and the obtained values of the input signals. Figure 11 shows angular 

tracking of the slave for the given master reference speed and angle shift. 

 

Fig. 11 Master Wheg (red) angular reference, under constant velocity,  

followed by slave Wheg (green) 

The influence of external disturbance on the Wheg drive of the master motor is shown 

in Figure 12. A satisfactory tracking of the slave drive against low resolution of the 

encoder and short duration of disturbance can be observed. 



 Electronic Gearing of Two DC Motor Shafts for Wheg Type Mobile Robot 85 

 

 

Fig. 12 Robustness demonstration on the master motor disturbance 

To test the accuracy of the system in the range of speeds and angles, whereby angle α 

was controlled, the speed–angle matrix was formed by measuring. Speed range was 

examined in the range of 0 to 3000 rpm, with a resolution of 200 rpm. The angles were 

ranging from 0° to 90° with a resolution of 5°. At higher speeds of the master a higher 

value of error in the slave was observed. The reasons for the error were noise increase at 

the encoder due to vibration and problems with fixing the encoder to the motor shaft. The 

greatest error that occurred was around 2%. 

 

Fig. 13 Angle error for range of the angle and speed values 



86 M. BOŽIĆ, S. ANTIĆ, V. VUJIĈIĆ, M. BJEKIĆ, G. ĐORĐEVIĆ 

 

5. CONCLUSION 

A functional master–slave drive structure for electronic gearing was implemented.  After 

the identification of parameters, the mathematical model of the motors was formed. Also, the 

complete synthesis of the digital control system is shown in the paper. The speed and 

position controller were designed by using the RT controller FPGA sbRIO9636. The 

efficiency of the controller was demonstrated and no notable delay in re-aligning the slave 

Wheg to the master Wheg velocity at given angle shift was observed. This practically means 

that realignment can be done within one single rotation of dWheg. This allows a robot to 

rapidly adapt to even small obstacles like stones or wet ground. Future work will be based on 

enhancement of the system performance by increasing the resolution of the encoder and the 

realization of all loops on FPGA platform. Further testing of this drive will be carried out in 

real conditions, on a treadmill belt Figure 14 left and on a rotating test station Figure 14 

right. Currently used controller and driver for rapid testing of the algorithm will be replaced 

with a cheaper microcontroller and a driver, in order to set up a complete system into the 

mobile robot. 

 

Fig. 14 Models of treadmill force plate (left) and rotating test station (right) 

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 Electronic Gearing of Two DC Motor Shafts for Wheg Type Mobile Robot 87 

 

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APPENDIX 

Table 2 Motor data  

Parameter Value 

Nominal voltage 24 V 

Nominal current 0.9 A 

Nominal torque 3.810
-2

 Nm 

Nominal speed 3600 rpm 

Friction torque at no load 0.7 10
-2

 Nm 

No load speed 4200 rpm 

Nominal power 14.3 W 

Torque constant 5.14 10
-2

 Nm/A 

Terminal resistance 5.95 Ω 

Terminal inductance 8.9 mH 

Gear ratio 6.25 

Nominal torque 40 10
-2

 Nm 

Viscous friction 6.5 10
-6

 Nm/rad/s 

Coulumb friction 4.9 10
-6

 Nm 

 

https://www.youtube.com/watch?v=DPx8kDuLsz0
https://www.youtube.com/watch?v=OSbQ7E46Yu4