www.ijecs.in
International Journal Of Engineering And Computer Science ISSN:2319-7242
Volume 3 Issue 9, September 2014 Page No. 8399-8417
New refined Model for Mechatronics design of solar differential
drive mobile robotic platforms
Farhan A. Salem 1,2
1
Dept. of Mechanical Engineering, Mechatronics Engineering Prog., College of Engineering, Taif University, 888, Taif, Saudi Arabia.
2
Alpha center for Engineering Studies and Technology Researches, Amman, Jordan.
Email: salem_farh@yahoo.com
Abstract: This paper proposes a new generalized and refined model for solar electric differential drive Mobile Robotic platforms
(SEDDMRP) and some considerations regarding design, modeling and control solutions. The proposed system design consists of seven main
subsystems, each subsystem, is mathematically described and corresponding Simulink sub-model is developed, then an integrated generalized
overall model of all subsystems is developed. The proposed whole SEDDMRP system model is developed for research purposes and
application in educational process, to help in facing the main challenges in developing Mechatronics SEV systems; early identifying system
level problems and ensuring that all design requirements are met, also, it is developed to allow designer to have the maximum output data to
select, evaluate and control the overall SEDDMRP system and each subsystem outputs characteristics and response, for desired overall
and/or either subsystem's specific outputs, under various PV subsystem input operating conditions, to meet particular SEDDMRP system
requirements and performance. The obtained results show the simplicity, accuracy and applicability of the presented models in Mechatronics
design of SEDDMRP system application.
Keywords: Mechatronics, Mobile Platform, differential drive, PV Panel, Modeling/simulation.
[1] Introduction
Modeling, simulation, analysis and evaluation process in
Mechatronics design consists of two levels; sub-systems
models and whole system model with various sub-system
models interacting similar to real situation, the subsystems
models and the whole system model, are to be tested and
analyzed for desired system requirements and performance [1].
Mobile robot is a platform with a large mobility within its
environment and have potential application in industrial and
domestic applications. One of the simplest and most used
structures in mobile robotics applications, are the two-wheel
differential drive mobile robots shown in Figure 1(a)(b), it
consists of a chassis with two fixed and in-line with each other
electric motors and usually have one or two additional third (or
forth) rear wheel(s) as the third fulcrum, in case of one
additional rear wheel, this wheel can rotate freely in all
directions, because it has a very little influence over the robot’s
kinematics, its effect can be neglected [2]. Solar electric
mobile platform (SEMRP) system is relatively new field of
mobile robots and new research area, different researches on
Mobile robots fundamentals, applications, mathematical and
Simulink models can be found in different texts including but
not limited to in [1-14], most of it study separate specific
system or subsystem design, dynamics analysis, control or
specific application. A refined model of solar electric
differential drive mobile robotic platforms (SEDDMRP)
system that can represent the actual system kinematics and
dynamics, and that can be used in Mechatronics SEDDMRP
system design is of concern.
To help in facing the main challenges in developing
Mechatronics SEDDMRP systems; early identifying system
level problems and ensuring that all design requirements are
met, this paper extends writer's previous works [1-5][15-18]
and proposes a new generalized and refined model for
SEDDMRP systems and some considerations regarding design,
modeling and control solutions. The proposed SEDDMRP
system model consists of seven main subsystems, including;
PV panel, DC/DC converter, PWM, battery, mobile platform,
DC machine, control unit and differential drive kinematics and
dynamics subsystems, each subsystem, is to be mathematically
described and corresponding Simulink sub-model is to be
developed, then an integrated generalized model of all
subsystems is to be developed, the sub-models and whole
system model, are to be developed to allow designer to have
the maximum output data to select, evaluate and control the
overall SEDDMRP system and each subsystem outputs
characteristics and response, for desired overall and/or either
subsystem's specific outputs, under various PV subsystem
input operating conditions, to meet particular SEDDMRP
system requirements and performance. The block diagram
representation of proposed system configurations and control is
shown in Figure 1(a)(b).
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
Page 8399
these two driven wheels. When a wheel movement is restricted
to a 2D plane (x,y), and the wheel is free to rotate about its
axis (x axis), the robot exhibits preferential rolling motion in
one direction (y axis) and a certain amount of lateral slip (
Figure 2(a)(b)), the wheel movement (speed) is the product of
wheel's radius and angular speed and is directly proportional
by the angular velocity of the wheel, and given by Eq.(1):
x  r    r 
(1)
Where: φ: wheel angular position. Referring to Figure 2(c),
while the wheel is following a path and having no slippery
conditions, the velocity of the wheel at a given time, has two
velocity components with respect to coordinate axes X and Y
and given as by Eq.(2):
r   x sin    y cos 
0  x cos    y sin 
Figure 1(a) The block diagram representation of proposed
system configurations and control
DC motor
Right wheel
Tacho
Tacho
Left wheel
Electronic components; Controller , drive, Power supply
Mobile Platform
Solar panel
Gears
Figure 1(b) Components layout of SEDDMRP system
2. SEDDMRP system modeling
The proposed SEDDMRP system model consists of seven
main subsystems, including; PV panel, DC/DC converter,
battery, mobile platform, DC machine, control unit and
differential drive kinematics and dynamics subsystems, each
subsystem, is to be mathematically described and
corresponding Simulink sub-model is to be developed, then an
integrated generalized model of all subsystems is to be
developed and tested . In [1-5][15-18] the refined mathematical
models and corresponding Simulink sub-models of main
subsystems are developed and tested, the modified versions of
these sub-models will be used to develop the integrated and
new generalized SEDDMRP system model .
2.1 Differential drive Kinematics and dynamics
modeling.
Different resources can be found on modeling differential drive
Kinematics and dynamics, including [2][19-22]. In [2]
mathematical description of Differential drive Kinematics and
dynamics is derived, Simulink model is developed and tested.
To characterize the current localization of the mobile robot in
its operational space of evolution in a 2D plane (x,y), first we
must define its position and its orientation. Assuming the
angular orientation (direction) of a wheel is defined by angle θ,
between the instant linear velocity of the mobile robot v and
the local vertical axis as shown in Figure 2(a). The linear
instant velocity of the mobile robot v, is a result of the linear
velocities of the left driven wheel vL and respectively the right
driven wheel vR . These two drive velocities vL and vR are
permanently two parallel vectors and in the same time, they are
permanently perpendicular on the common mechanical axis of
(2)
Assuming the mobile robot follows a circular trajectory shown
in Figure 3, and Δs , Δθ and R are the arc distance traveled by
the wheel, and its respective orientation with respect to the
global coordinates, and R is the circumference radius of the
wheel, then the linear and angular velocities of the mobile
robot are given by Eq.(3):
  s / t     / t
(3)
The arc distance Δs traveled in time equal to Δt, is given
by s  R  , and the curvature, that is the inverse of the
radius R, is given by   1 / R , referring to Figure 3, the
movement equation in the initial position are given by Eq.(4),
these coordinate equations can be extended by their rotating,
which result in Eq.(5),assuming the time is so small, then we
have; cos( )  1 and sin( )   , substituting in
Eq.(5), will result in Eq.(6):
x  R  cos( )  1
y  R (sin( ))
(4)
x  R  cos( )  1 cos   R sin( )sin 
y  R (sin( ))  1)sin   R sin( ) cos 
(5)
x  R  sin( )
y  R  cos( )
(6)
Substituting
R,
from
the
arc
distance,
s  R   R  s /  , and dividing both sides by ∆t, we
have Eq.(7):
x s sin( )

 x   sin( )
t
t
y s cos( )
y  s cos( ) 

  y   cos( )
t
t
(7)
x  s sin( ) 
2.1.1 The instantaneous center of curvature (ICC), (shown
in Figure 2(a)) is the point the robot must rotate around it, to
avoid slippage and have only a pure rolling motion, ICC lies on
the common axis of the two driving wheels. In case of
differential drive, by changing the velocities υL and υR of the
two Left and Right wheels, the ICC of rotation will move and
different trajectories will be followed, at each time Left and
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
Page 8400

d
  R  mob
dt
(8)
Referring to Figure 2(a) The right and left wheels linear
velocities in terms of mobile angular speed are given by
Eq.(9):
L
L




(9)
L  mob  R    R  mob  R  
2
2


Where: R is the distance from the ICC point to the midpoint P,
between the two wheels, and can be found by Eq.(10):
  L L
R  R
*
R  L 2
(10)
Because the vectors for linear speed of wheels υL and υR are
orthogonal on the common axis of the driven wheels, we can
write an equation to represent the angular velocity of the robot
to be given by Eq.(11):
  L R  L  r
 R

L
L
(11)
The instant linear velocity of the mobile robot vmob is attached
and defined relative to the characteristic point P, this velocity
is a result of the linear velocities of the left driven wheel υL and
respectively the right driven wheel υR. These two drive
velocities υL and υR are permanently two parallel vectors and, in
the same time, they are permanently perpendicular on the
common mechanical axis of these two driven wheels [23], and
given by Eq.(12):
mob 
R  L
2

R  L  r
(12)
2
A differential drive mobile robot is very sensitive to the
relative velocity of the two wheels, where: for R  L : then
the radius R is infinite and the robot moves in a straight line. If
R  L :then the radius R is zero and the robot rotates around
robot center point P (it rotates in place) . If R  L : The robot
follows a curved trajectory around an ICC point located at a
distance R from robot center point P.
The curvature, which is the inverse of the radius R is given by
Eq.(13):

To derive the expressions for the actual position of the robot,
referring to Figure 2(a), lets suppose that a differential drive
robot is rotating around the point ICC with an angular velocity
ω(t). During the infinite short time dt the robot center will
travel the distance from the point P(t) to P(t+dt) with a linear
velocity Vmob(t). For infinite short time we can assume that the
robot is moving along a straight line tangent in the point P(t) to
the real trajectory of the robot. Based on the two components
of the velocity Vmob(t), the traveled distance in each direction
can be calculated [20]
dx  x (t )dt
(16)
dy   y (t )dt
Substituting υx and υy from Eq.(7) , gives Eq.(17) :
dx   sin( (t ))dt
dy   cos( (t ))dt
(17)
Similarly, the angular position can be found to be :
d    (t )dt
(18)
Integrating Eqs.(16,17) , substituting Eq.(7) and manipulating,
we have Eq.(19) :
x (t )    sin( (t ))dt  x 0
y (t )    cos( (t ))dt  y 0
(19)
 (t )    (t )dt   0
Substituting Eqs.(12 and 14) and manipulating, we have
Eq.(20) :
   L 
x (t )    R
 sin( (t ))dt  x 0
 2 
   L 
y (t )    R
 cos( (t ))dt  y 0
 2 
  L
 (t )   R
dt   0
L
(20)
Y axis
Right wheels, moves around the ICC with the same angular
speed rate, given by Eq.(8):
θ
1

2   L 

 * R
R mob L R  L 
(13)
Now, substituting Eq.(12) in Eq.(7) , gives:
R  L
y  
2
sin( )
R  L
2
X axis
Y axis
x 
cos( )
P
The arc length, distance, traveled by the mobile robot,
(point P) is the average of arcs lengths traveled by the two
driven wheels and given by Eq.(14): :
S L   R  L / 2  
S R   R  L / 2    S   S L  S R  / 2
(14)
Correspondingly, the center orientation angle of the
trajectory is given by Eq.(15) :
   S L  S R  / R
ICC
(15)
X axis
Fig.2(a) The differential drive motion
2.1.2 Odometry for differential drive
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
Page 8401
Eq.(30). Based on fundamental principle of dynamics the
acceleration of the vehicle is given by Eq.(31):
sun
r
φ
Motion
Solar panel
Y axis
X axis
(b) Wheel
(c)
M*g
r =ν/ω
α
Road incl.
Figure 4. Forces acting on moving solar electric mobile robotic
platform.
FTotal  Fa  FR  FC  FLin_a  Fang_a
 MgCr cos( )  rwheel
Tclimb  Tslope =  Mg sin( )  rwheel
Trolling 
(d)
Figure 2(a)(b)(c) Wheel movement kinematics [2]
Finertia  Fslope  M
d
dt
Different resources can be found on modeling of mobile
platform dynamics, including [2,3,18][20-23]. In [2,3,18] a
detailed derivation of refined mobile platform mathematical
model and Simulink model considering most possible acting
forces, are introduced, tested and verified.
The disturbance torque to mobile platform is the total resultant
torque generated by the acting forces and given by Eq.(21),
where and refereeing to Figure 4, the acting forces and torques
on mobile platform include; the rolling resistance torque given
by Eq.(22), The hill-climbing resistance, slope, torque, given
by Eq.((23), the total inertia force of the mobile platform given
by Eq.(24), the aerodynamics torque given by Eq.(25),the
aerodynamics lift force, Flift; given by Eq.(26), the angular
acceleration force, the force required by the wheels to make
angular acceleration and is given by Eq.(27), the linear
acceleration force Facc , the force required to increase the speed
of the SMEV and can be described as a linear motion given by
Eq.(28).
To determine the electric battery capacity, and correspondingly
design the Photovoltaic panel with series and parallel
connected cells, it is required to estimated energy required by
platform, the required power in kW that platform must develop
at stabilized speed can be determined by multiplying the total
force with the velocity of the SMEV, and given by Eq.(29):
Electrical power (in watts) in a DC circuit can be calculated by
multiplying current in Amps and V is voltage, and given by
(23)
(24)
(25)
2
Flift  0.5C L B vehicle
(26)
n
2
2
rwheel
a
J
d 
 d
  M  wheel

dt 
r 2  dt
d
T
Facc  M * a  M
M
dt
J
Facc  M * a  M
2.2 Modeling of mobile platform dynamics
(22)
1

Taerod    ACdvehicle 2  rwheel
2

Facc _ angle  J
Figure 3 circumference movement of mobile robot [2]
(21)
(37)
(28)
PTotal  ( F ) *  FTotal *
(29)
P= I x V
(30)

Pm  Ptotal
M *
(31)
2.3 Actuator and Sensor subsystems modeling
In order to drive a SEMRP, induction motors, reluctance
and permanent magnet motors can be used, the actuating
machines most used in Mechatronics motion control
applications are DC machines (motors), based on this, the
SEDDMRP system motion control can be simplified to a DC
machine motion control. In this paper, PMDC motor is
considered as SEDDMRP electric actuator, depending on
application requirements, any other actuating machine can also
be used to replace DC motor to develop a generalized model
[18]. In [1-4][14] introduced, tested and verified a detailed
derivation of refined DC machine and tacho sensor systems'
mathematical model and corresponding Simulink model, based
on this, the open-loop transfer function of the DC machine is
given by Eq.(31), In the following calculation the disturbance
torque, T, is all torques including coulomb friction, and given
by (T=Tload+Tf). Dynamics of tachometer can be represented
using Eq.(32),
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
Page 8402
Gopen (s ) 
 platform (s )

V in (s )
(31)
Kt / n

2
 La s  R a  (J equiv s  bequiv s )   Las  R a T   K b K t
d  t 

dt
 K tach  
V out t   K tach
V out t 
K tach 
3. Proposed generalized SEDDMRP system model
(32)
V out  s 
V
and , 
 s 
r
2.4 Photovoltaic Panel and Converter (PVPC)
subsystems modeling
In [15-16] a detailed derivation of mathematical model and
corresponding Simulink model of both PV system and DC/DC
converter systems are developed, tested and verified. In [16] a
generalized Photovoltaic Panel –Converter (PVPC) subsystem
model is developed, tested and verified, based on these
references, the output net current I, of PV cell and the V-I
characteristic equation of a PV cell are given by Eq.(33), it is
the difference of three currents; the light-generated
photocurrent Iph, diode current Id and the shunt current IRSH .
The output voltage, current and power of PV array vary as
functions of solar irradiation level β, temperature T , cell
voltage V and load current I.
Duty cycle is the ratio of output voltage to input voltage is
given by Eq.(34), and defined as the ratio of the ON time of the
switch to the total switching period, where: Iout and Iin, : the
output and input currents. In this paper, the PWM generator is
assumed as ideal gain system, the duty cycle of the PWM
output will be multiplied with gain Kv= KD
I  I ph  I d  I RSH
 q (V  IRS )  V  R S I
I  I ph  Is  e NKT  1 
R sh


I   I sc  K i T T ref

(33)
 q (V  IRS )  V  R S I
 Is  e NKT  1 
1000
R sh



V out
Ton
I
 D  in V out  D *V in  D 
V in
I out
Ton Toff

K 
K P s  I 
 K s  K I    K P   K P  s  Z o  (35)
K
G PI (s )  K P  I  P
s
s
s
s


(T s  1)
1
G PI (s )  K PI * I
 K PI * 1 

TI s
T

Is 
(34)
2.5 Control algorithm subsystem selection and
modeling
Different control approaches can be proposed to control
the overall SEDDMRP system output performance in terms of
output speed, as well as, controlling output characteristics and
performance of PVPC subsystem to meet desired output
voltage or current under input working operating conditions.
Because of its simplicity and ease of design, PI controller is
widely used in variable speed applications and current
regulation, in this paper, different and separate PI controllers
configurations will be applied for achieving desired outputs
characteristics of both PVPC subsystem and SEDDMRP , and
meeting desired output speed, voltage and load currents for
particular SEDDMRP system application. The PI controller
transfer function in different forms is given by Eq.(35).
The proposed generalized SEDDMRP system model shown in
Figure 5 is developed by integrating all seven subsystems’ submodels to result in overall SEDDMRP system model, each
subsystem’s sub-model is designed and developed to return
maximum output data (numerical visual and graphical) that can
be used to select, evaluate and integrate sub-models output
characteristics and performance as a part of the whole system
performance, as well as, overall system performance. The
proposed generalized model can have the design shown in
Figure 5(b) where PV panel, converter and DC machine are
modeled to represent platform's left side and wheel, and
similarly, identical block to represent right side and wheel, the
PV panel is modeled as two separate models to generate
output voltage of 12 V.
The proposed generalized model is to be tested for straight
line, curvature, circular and motion profile motions. to result
in desired motion, a corresponding input signals must be
applied to each or both actuators, for instant, for straight line
motion the same input is to be applied to both actuator. To
simulate different input values to both actuators, one (Figure
5(c)) or two ((Figure 5(b))) PV panels sub-model can be used
in simulation (). To represent different motion types, a gain
reduction or increasing Simulink block is to be used to send
different inputs to different actuators, or to reduce output
speed. It is important to notice that in microcontroller based
mobile robots, a drive is used to drive DC motor and according
to control program and applied PWM, each motor can be
controlled separately, to move with specific speed and
direction.
The desired output linear speed of SEDDMRP system is
selected to be 1 m/s, for wheel radius r = 0.075 m, the selected
DC actuator is running at 12 DC input voltage , the tachometer
constant Ktach , given by Eq.(32) and ω=ν/r=1/0.075=13.33
rad/s, correspondingly Ktach=12/13.33=0.9.
The DC machine actuator sub-model is shown in Figure
6(a)(b), with corresponding PI-armature-load-current
controller configurations and corresponding load torques given
by Eqs.(21-29)
The PV panel subsystem sub-model is shown in Figure 7,this
model can be used to represent a single PV cell, module, panel
or array, it is required to define number of corresponding cells
in series and parallel. PV panel can be modeled as two
separate panels or one single panel, in case of single PV panel,
it is to be designed to generate 24 V, where 12 V is required
for each actuator, with series cells of NS=96 and parallel cell of
NP=30,
The DC/DC buck converter subsystem sub-model is shown in
Figure Figu 8(a)(b), the desired output voltage is 24 V, half
this value will be send to each DC machine. The duty cycle is
calculated automatically by Eq.(36), both sub-models of PV
panel and DC/DC converter are integrated in one PV panelconverter (PVPC) subsystem sub-model as shown in Figure
8(c).
D
V Conv __ out _ desired
V Panel _ out
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
(36)
Page 8403
Control systems selection, design and simulation; Three
separate PI control subsystems are used to control each of the
following subsystems characteristics; PVPC subsystem output
characteristics and platform subsystem output linear speed.
For controlling the electric machine current and voltage a
cascade starategy is applied where two seperate current and
speed controllers are used: First PI speed controller is used to
control the platform linear speed to meet desired speed, and
shown in Figure 6(b). The generalized model is also supported
with other control algorithm including; PID, PD,PI with
deadbeat response, and can be modified to include any other
control strategy. Second separate PI current controller shown
in Figure 6(a) is used to match motor-current with required
Load-Torque (Variations) current to overcome, where both
currents are compared and the difference is fed to PI controller
to generate output converter current to match required load
current.
Third PI voltage controller shown in
Figure 8(d), used to
control the converter's output voltage to match desired output
voltage, by comparing desired converter output voltage (12V)
and actual converter's voltage, the difference is fed to PI
controller to control the buck converter MOSFET switch
according calculated duty cycle to result in desired voltage.
Linear speed of Robot'
V_Robot
1
[V_R]
,
mobile2.mat
Linear speed Robot,
;
Turning Radius
inf
of Mobile robot1
Divide1
Curvature
mobile1.mat
-T-
0.5
mobile6.mat
Wheel Velocity
Gain
Conv Iout
Goto2
12.12
Wheel Position1
-T-
Curvature1
Turning Radius of Mobile robot1
,.
-K- Dist_wheels
Turning Radius
T
B
Conv Vout
T
B.
Subsystem of left motor & wheel
Conv. I out
V
V
A
Cell surface area
12.12
(Dist_wheels)/2
Linear speed Robot.1
''1
-T-
1
Ns
Angular speed
0
of Robot.
Nm
Np
Omega=
(Vl-Vr)/Distanc
[V_L]
Conv Vout
0.5
Conv Iout
Motor current
Gain1
'2
'1
PV panel Subsystem
Wheel Position1
S_left
mobile3.mat
Curv. Raduis
of L & R wheels
and mobilr robot
[theta]
Theta, Robot
0
orientation-Dirction1
Omega1
mobile5.mat
Turning Radius of Mobile robot.
inf
Theta, Robot orientation-Dirction
...
X
1
mobile.mat
S_right
12
Conv Vout
Curv. Raduis
of L vs R wheels
Angular speed of Robot.
,.1
1
s
1
s
0.5
Wheel Velocity
-KCurv, radius of R_ WHEEL
''
Omega
1/(Dist_wheels)
0.5
-K-
Ns
-KCurv. radius of L_ WHEEL
..
-C-
nan
Motor current
V_Robot
Y
XY Graph
Motor current
Subsystem
2
Curvatures covered
0
distance
mobile4.mat
Curvature covered distance
,.3
Subsystem of right motor & wheel
Figure 5(a) The proposed generalized SEDDMRP system model, with main subsystems; one PV panel, left motor, right motor and
differential drive Kinematics and dynamics modeling
Linear speed of Robot'
V_Robot
T
T
B
B.
V
V
[V_R]
2
mobile2.mat
Linear speed Robot,
,
;
-T-
0.5
Wheel Velocity
mobile6.mat
Nm
Gain
Goto2
Turning Radius
of Mobile robot1
Cell surf ace area
A
Ns
Ns
Np
-T-
Wheel Position1
Turning Radius
(Dist_wheels)/2
Linear speed Robot.1
''1
-T-
1/(Dist_wheels)
Angular speed
of Robot.
B.
V
Nm
Omega=
(Vl-Vr)/Distanc
[V_L]
1
V
Gain1
Cell surf ace area
A
Ns
Ns
Np
1
s
0.5
Wheel Velocity
S_lef t
Wheel Position1
Subsystem of Right motor & wheel
-KCurv, radius of R_ WHEEL
''
Omega
-K-
B
-KCurv. radius of L_ WHEEL
..
-C-
1
T
Curvature1
Turning Radius of Mobile robot1
,.
-K- Dist_wheels
Subsystem of left motor & wheel
T
Divide1
Curvature
mobile1.mat
'1
mobile3.mat
Curv. Raduis
of L & R wheels
and mobilr robot
Angular speed of Robot.
,.1
1
s
'2
[theta]
Theta, Robot
orientation-Dirction1
Omega1
mobile5.mat
Theta, Robot orientation-Dirction
...
X
Curv. Raduis
of L vs R wheels
Turning Radius of Mobile robot.
1
mobile.mat
S_right
V_Robot
Y
XY Graph
Subsystem
Curvatures covered
distance
2
mobile4.mat
Curvature covered distance
,.3
Figure 5(b) The proposed generalized SEDDMRP system model, with two mask sub-models, each with separate PV panel and
converter connected to DC machine
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
Page 8404
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
Nm
Ns
A
V
B
T
Conv . I out
Conv Vout
PVPC Subsystem
Motor current
Np
Ns
Cell surf ace area
V
B.
T
12
0.5
12.12
Motor current
Wheel Position1
Wheel Velocity
Motor current
Wheel Position1
Wheel Velocity
0.5
Subsystem of right motor & wheel
Conv Vout
Conv Iout
12.12
Subsystem of left motor & wheel
Conv Vout
Conv Iout
[V_L]
,
[V_R]
1
1
Left wheel
Subsystem
Y
X
Right wheel
wheels linear speeds
V_Robot
S_right
S_lef t
-T-
Omega
'1
1
s
''1
-T-
mobile11.mat
mobile10.mat
XY Graph
1
s
1
nan
Omega1
mobile5.mat
mobile4.mat
wheels linear positions
14.15
14.15
Curvature covered distance
,.3
Curvatures covered
0
distance
Theta, Robot orientation-Dirction
...
Theta, Robot
0
orientation-Dirction1
[theta]
inf
R_wheel
L_wheel
mobile13.mat
mobile12.mat
inf
Curvature
Platform
curvature
Platform
acceleration
mobile8.mat
mobile7.mat
Turning Radius RIGHT
4inf
inf
Turning Radius LEFT
mobile.mat
inf
Curv. Raduis
of L & R wheels
and mobilr robot
Curv. Raduis of L vs R wheels
Platform Turning Radius
-KCurv, radius of R_ WHEEL
''
mobile3.mat
Angular speed of Robot.
,.1
Angular speed
0
of Robot.
(Dist_wheels)/2
-C-
mobile6.mat
0
7.228e-006
-K- Curvature Radius of wheels
Curv. radius of L_ WHEEL
..
Divide1 Curvature
mobile1.mat
;
Curvature3
mobile2.mat
mobile9.mat
Turning Radius of Mobile robot1 ,.
Turning Radius
inf
of Mobile robot1
Linear speed Robot,
Linear speed Robot.1
'2
Turning Radius
Omega=
(Vl-Vr)/Distanc
Gain1
0.5
-K-
-TGoto2
1/(Dist_wheels)
1
-K- Dist_wheels
Gain
0.5
V_Robot
du/dt
www.ijecs.in
International Journal Of Engineering And Computer Science ISSN:2319-7242
Volume 3 Issue 9, September 2014 Page No. 8399-8417
Page 8405
Figure 5(c) The proposed generalized SEDDMRP system model, with shown subsystems; one PV panel, left motor, right motor
and differential drive Kinematics and dynamics modeling
Rou: The invironment ( air) density (kg/m3) 17
A:Cross-sectional area of mobile robot, where it is the widest, (m2) 16
aerodaynamic torque,
0.5*p*A*Cd*v^2
Divide14
Cd : Aerodynamic drag coefficient 15
Divide3
Divide4
Divide5
0.5
aerodynamics
. lift force
0.5
sin(u)
45
cos(u)
Inclination SinCos.
angle
10
,
1
. r/2
.
-K-
Rolling resistance force
M*g*Cr*cos()
Divide6
-K-
Divide7
M : The mass of the mobilr robot
18
B : mobile robot underside area 13
CL: The coefficient of lift,
12
( CL to be 0.10 or 0.16)
Divide17 Divide18
Divide15
du/dt
Divide16
Divide8
0.5
Cr: The rolling resistance coefficients
11
r*m/2
Divide9
g: The gravity acceleration (m/s2).
Divide11
Divide10
Torque, T 1
Kp
Controller 20
Armature
6
Inductance, L
Torque constant ,Kt 8
current
PI Controller ,1
1
s
Manual
Switch
1
Saturation1
4
Add3
2
5
Divide1
0.5
1
9
Inertia
wheel radius, r
(motor+ load)
1
s
21
Converter I out
Divide27
1
1
6 DC machine Current, I
Divide26 Integrator12
19 Prefilter
wheel radius,
V=W*r1
-K-
s
1.
Coloum Friction
Divide12
mobile_platform_.mat
s+Zo
1
bm
12.12
Divide30 Integrator9
[motor_current]
7
12.12
Divide29
Out1
Divide,1
3
speed in mps
rads2mps=
R_wheel*(2*pi)/(2*pi).
2
output angual speed, Omega
Divide22
Load Current, I 7
Armature
Resistance , R
angular
speed
5 All viscous damping
2
Input Volt,(0 :12) Out2
Divide21
EMF constant, Kb 14
Divide2
3 Gear ratio,n
4 Tachometer constant , Ktac
Figure 6(a) The DC machine sub-model, with PI-armature-load-current controller configurations, all shown as sub-model of
generalized SEDDMRP model
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
Page 8406
Figure 6(b) The DC machine mask sub-model, shown as sub-model of generalized model
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
Page 8407
Kp
s
s+Zo
PI Controller
PI(s)
PD Controller
PD(s)
s+Zo
Prefilter
1
1
Prefilter.
PID Controller1
PID(s)
Zo
Vin 12 V
,1
PI Controller
Motion profile
Out1
Signal Builder
Signal 1
2
Conv Vout
n
n
Cr
Cd
CL
Ra
g
r
Kt
m
Kb
Conv Iout
1
reference_area
La
front_area
Damp.
bequiv
rou
Ktach
Ktach
Jequiv
Conv erter I out
Controller
Pref ilter
Mobile platform and load Subsystem
Cr: The rolling resistance coef f icients
Rou: The inv ironment ( air) density (kg/m3)
A:Cross-sectional area of mobile robot, where it is the widest,
Cd : Aerody namic drag coef f icient
EMF constant, Kb
B : mobile robot underside area
CL: The coef f icient of lif t, ( CL to be 0.10 or 0.16)
g: The grav ity acceleration (m/s2).
M : The mass of the mobilr robot
wheel radius, r
Torque constant ,Kt
Armature Resistance , R
Armature Inductance, L
All v iscous damping
Tachometer constant , Ktac
Gear ratio,n
Input Volt,(0 :12)
Inertia (motor+ load)
Load Current, I
DC machine Current, I
Out2
Out1
speed in mps
output angual speed, Omega
Torque, T
1
Wheel Position1
2
Wheel Velocity
mobile_platform3.mat
12.12
Load Current
[motor_current]
Motor current
3
mobile_platform.mat
Motor current
12.12
1
s
mobile_platform41.mat
1.00
Output linear speed
mobile_platform2.mat
13.33
Platform angular speed
mobile_platform1.mat
Torque
14.4
[P]
8
From
Fill Factor
Isc
.10
5
Vo
Ns
eu
.
48
''4
PV module output voltage
,
.1
1
1
.3
N
Module V
Is
PV.mat
V.mat
'
Tref
K
1
''
.'
To File
0.5
B
.8
T
B0
Cell Vout
P
[P]
B
.2
PV cell voltage
3
V
Goto
.4
q
Ki
.7
.
Iph
2
Panel V out
Cell P-V
1000
6
V
V1
3
P
eu
V
I
Cell Power out
41.01
P.mat
To File1
,.
,
.5
I
Id
4
Cell I out
1
''3
V
Rs
1
''1
6
I
Cell I-V
Np
''2
2
I12.mat
Rsh
Panel I out
Ish
To File2
.6
1.367
cell current
4
5
A
Cell Power in
.9
1
.11
Figure 7 The PV panel subsystem sub-model
7
Cell Efficiency
100
24.1
2
1/L
Vin
1
1
s
Product
Duty cycle, D
2
IL
((RL+Ron)+((R*Rc)/(R+Rc)))/L
Duty cy cle, D switchin signal (0,1) PWM
(R*Rc)/(R+Rc)
3
Vo
R/(C*(R+Rc))
PWM Generator Subsystem
load current
3
1
s
R/(R+Rc)
1/(C*(R+Rc))
1
Vc
R/(L*(R+Rc))
Figure 8(a ) Buck converter subsystem sub-model, with PWM sub-model [24]
Duty cy cle, D
1
Duty cy cle, D
Vc
T ermi nator
switchin signal (0,1) PWM
Duty cycl e
PWM Generator Subsystem
Vin
2
T
Converter current out
Pout
Panel V out
5
T
PV panel
Panel I out
3
Irradi ati on, B
Vol t out
4
load current
Vo
Product
4
Converter Power out
6
PV panel current out
B
Cell Vout
2
Buck converter Subsystem
7
Converter vol tage out
3
PV cel l Vol t out
Vol atges compari si on
V
13
V
Cell I out
8
PV cel l current out
5
Cel l surface
area A
1
IL
8
l oad current
Cell Power in
A
PV panel Power out
''1
9
PV cel l Power i n
6
Cell Power out
Ns
10
PV cel l Power out
Ns
7
Cell Ef f iciency
Np
11
PV cel l effi ci ency
Np
Fill Factor
12
Fi l l Factor
PV Panel Subsystem1
Figure 8(b) Mask sub-model of both PV panel and DC/DC converter sub-models
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
Page 8408
12.12
3
PV_con.mat
Converter I out
Conv erter current out
1
24
Converter current Iout
Duty cy cle
PV_con1.mat
Duty cycle
4
Conv erter v oltage out
Converter V out
Converter Volt Iout
24
2
T
Volatges comparision
Vout
48
Vin
System Vin-Vout comparision
T
290.9
PV_con2.mat
Conv erter Power out
48
3
Converter Power out
PV_con3.mat
Irradiation, B
Irradiation, B
2
PV panel Volt out
[panel_Vout]
PV cell output current
PV panel Volt out
41.01
PV_con4.mat
PV panel current out
4
V
1
0.5
V
PV_con5.mat
PV cell Volt out
PV panel I out
PVPC Subsystem
41.01
1.367
[control] PV_con6.matDuty cy cle
[Cell_Vout]
PV cell output volt
5
PV panel current out
[D] output current
PV panel
[panel_Iout]
PV panel current out
[Cell_Iout]
Cell surf ace area A
PV cell current out
Cell surface
area A
1
0.5
PV_con7.mat
PV cell output
T current
PV cell Power in
6
Ns
3
PV cell Power out
48
PV panel Volt out
[panel_Vout]
PV_con8.mat
V
Cell I-V
Converter I out
PV cell output power
PV_con9.mat
PV cell ef f iciency
4
Conv. I out
Cell surface area
PV cell efficiency
5
Conv erter I out
PV_con10.mat
Ns
[Conv_current]
Ns
Fill Factor
1969
Converter V out
PV Fill factor
PV_con12.mat
load current
6
24
Np
Np
PV panel Power out
Conv erter V out
[Conv_Vout]
Load current
PV panel Power out
PV-Converter Subsystem
Figure 8(c) Mask sub-model of both PV panel and DC/DC converter sum-models
[D]
1
12.12
Cell surf ace area A
0.1374
Np
Load current
Cell P-V
Irradiation, B
V
0.7315
8
PV panel V out
,.
0.6835 B.
PV panel power in
Np
Cell power
[Cell_Vout]
2
Ns
7
T
PV panel I out
PVPC Subsystem
2
Conv Vout
[Conv_current]
PI(s)
41.01
Duty cy cle
[control]
PI Controller2
7
[panel_Iout]
PV panel current out
1
Motor current
T
-2.532e-007
PV panel V out
T
2
48
Irradiation, B
B.
PV panel Volt out
3
[panel_Vout]
[Conv_Vout]
V
1.004
0.5
V
Converter I out
4
1
12.12
Cell surf ace area A
[panel_Vout]
Conv. I out
Cell surface area
PI Controller1
Conv erter I out
5
Ns
PV_con11.mat
0.5
24
Np
Np
[control]
4
V_out_desired
Converter V out
6
[D]
[Conv_current]
Ns
PI(s)
D calculated
Vout desired
D desired
Conv erter V out
[Conv_Vout]
Load current
2
Conv Vout
[Conv_current]
PI(s)
PI Controller2
7
Motor current
-2.532e-007
Figure 8(d) sub-models of; PV panel, DC/DC converter and PWM sub-models, with both PI controller for current and voltage
control
[Conv_Vout]
1.004
0.5
[panel_Vout]
D calculated
PI(s)
PI Controller1
[control]
PV_con11.mat
4. Testing, analysis and evaluation
[D]
Farhan A. Salem , IJECS Volume
3Issue 9 September 2014 Page No.8399-8417
4
V_out_desired
0.5
Page 8409
Figure 6 (b) P-V Characteristics for β=200, and T=50
SEDDMRP curvature and linear speed:(sraight line motion)
1.2
Platform linear speed
1
0.8
Magnitude
The proposed SEDDMRP system model is to be tested for
output straight line, curvature, circular and motion profile
motions, for defined subsystems parameters given in Table-3,
and under PV panel input operating conditions of irradiation
β=200, T=75, number of series and parallel cells Ns=96 , Nm=
30, cell surface area A=0200.0 m2, and inclination of 45
degree.
The proposed model, is developed to return maximum
required output numerical and graphical data of each
subsystem and overall SEDDMRP system to be used to test,
analyze and evaluate the SEDDMRP system performance and
outputs characteristics including; Differential drive Kinematic
and dynamic data; platform output linear speed, each wheel
speed, position, acceleration, curvature, turning radius. DC
machine outputs including; linear speed, acceleration, motor
current, motor torque, load torque. PV cell-panel outputs,
including; I-V, and P-V characteristics, current, voltage,
power, efficiency, fill factor, and finally, DC/DC converter
output including: current, voltage and Duty cycle value,
tachometer constant.
0.6
0.4
4.1 Testing proposed generalized SEDDMRP system
model for straight line motion,
0.2
curvature
0
-0.2
0
5
10
Time(s)
15
20
Figure 9 (c-1)
SEDDMRP (2) Kinematics and dynamics
1
0.5
Magnitude
Testing SEDDMRP model for straight line motion, will result
in values numerically shown in each subsystem sub-model in
all up Figures and listed in Table-1(a)(b). The output
graphical data plots of PV panel output I-V, and P-V
characteristics are shown in Figure 9(a)(b).The differential
drive Kinematics and dynamics data including, platform and
counterpoint linear speed of 1 m/s, platform linear
acceleration, curvature, curvature covered distance, turning
radius and wheels position, for straight line are shown in
Figure 9(c)(d)(e). Left and right motors; linear speed, angular
speed, torque and current, (both are identical) are shown in
Figure 9 (d). PV panel-converter output voltages and output
currents are shown in Figure 9(f). The control signals of PI
speed and voltage controllers are shown in Figure 9(g),
platform center point straight line motion is shown in Figure
9(e)
The proposed model is support with different control
algorithms, any can be selected using manual switches, tested
and evaluated. Selecting control PI-control algorithm with
deadbeat response, designed according to design procedure
given in [3-4], for desired output linear speed of 0.5 m/s, and
straight line motion, will result in response curve shown in
Figure 10.
Platform curvature, curvature covered
distance & Platform turning radius all =0
0
-0.5
-1
0
5
10
15
Time(s)
Figure 9 (c-2)
SEDDMRP:Kinematics and dynamics;(Linear speed)
2
1.5
Left and right wheel positions
Platform linear speed
Magnitude
1
0.5
Platform linear acceleration
0
-0.5
-1
Figure 9(a) V-I Characteristics for β=200, and T=50
0
2
4
6
Time(s)
8
10
Figure 9 (c-3)
Figure 9 (c) Platform kinematics and dynamics ; linear speed,
Platform's center point curvature, curvature covered distance
and turning radius (zero) of SEDDMRP system for straight
line motion
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
Page 8410
PV Panel-converter outputs
LEFT motor outputs
16
LEFT Motor torque
LEFT motor angular speed
LEFT motor Load current
14
12
50
X: 5.453
Y: 48
PV Panel Iout
40
Magnitude
10
Mgnitude
PV Panel Vout
8
30
Converter Vout
20
6
Converter Iout
4
10
2
LEFT motor linear speed
0
0
-2
0
5
10
0
2
4
6
Time (s)
8
10
Figure 9(f) Panel-converter output voltages and output
currents
15
Time (s)
Control signals
Figure 9(d-1)
16
RIGHT motor outputs
Motor PI controller signal
14
16
RIGHT Motor torque
RIGHT motor angular speed
RIGHT motor Load current
12
12
Magnitude
14
Mgnitude
10
8
8
6
4
6
X: 0.5196
Y: 1.001
2
4
0
2
RIGHT motor linear speed
0
-2
10
0
Voltage PI controller signal
2
4
Time (s)
6
8
Figure 9(h) The control signals of PI speed and voltage
controllers
0
5
10
15
Time (s)
Figure 9(d-2)
Figure 9(d) left and right motors; linear speed, angular speed,
torque and current, for straight line motion, (both are
identical)
Figure 9(e) platform center point; Straight line motion
Table 1(a) Simulation results of each subsystem and whole SEDDMRP system for straight line motion
PVPC system
PV cell outputs
PV Panel outputs Converter outputs
Both DC machines
inputs
outputs
β
200
Voltage
0.5 V
Voltage
48 V
Voltage
24
Input voltage
12 V
T
75
Current
1.37 A
Current
41.01
Current
12.12
D
0.5
Fill factor
0.1374
1.277
Power out
290.9
13.33
rad/s
14.4 N/m2
A
Ns
NP
0.0025
96
30
Power out
Power in
Efficiency
0.6835
0.5
0.7315
Cell
efficienc
y
FF
Shaft Angular.
Speed.
Motor Torque
Motor current
Load current
12.12 A
12.12 A
0.1374
Table 1(b) Simulation results of each subsystem and whole SEDDMRP system for straight line motion
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
Page 8411
Right Wheel outputs
left Wheel outputs
Platform outputs
Linear Speed
1 m/s
Linear Speed
1 m/s
Linear speed
1 m/s
Curv. radius
0
Curv. radius
0
Curvature
0
Curvature
0*
Curvature
0*
Angular. Speed.
0
Lin. Position
13.9
Position
13.9
Orient. direction
0
Turning radius
Curvature covered
distance
0
0
4.2 Testing for circular motion
Response applying PI w ith deadbeat response
0.7
0.6
Testing SEDDMRP model for circular motion, will result in
output numerical and graphical outputs values of each
subsystem sub-model and overall system response shown in
Figures 11(a), and listed in Table-2(a)(b), as well as, graphical
outputs and responses shown in Figures 11, including outputs
of ; PV panel, converter, DC machines, right and left motors,
and overall SEDDMRP system.
The output graphical data plots of PV panel output I-V, and PV characteristics are the same as shown in Figure 11(b).The
differential drive Kinematics and dynamics data including,
platform and counterpoint linear speed of 1 m/s, platform
linear acceleration, curvature, curvature covered distance,
turning radius and wheels position, for straight line are shown
in Figure 11(b-i). Left and right motors; linear speed, angular
speed, torque and current, are shown in Figure 11(f). PV
panel-converter output voltages and output currents are shown
in Figure 11(g). The control signals of PI speed and voltage
controllers are shown in Figure 11(h), platform center point
circular motion is shown in Figure 11(i). Testing SEDDMRP
model for double curvature motion is shown in Figure 12.
System:
untitled1
System:
untitled1
Time
(sec):
3.89
Time
(sec):
3.78
Amplitude:
Amplitude:
0.50.5
0.5
M
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Time (sec)
Figure 10 linear speed vs. time applying PI-control algorithm
with deadbeat response, for desired output linear speed of 0.5
m/s, and straight line motion
du/dt
V_Robot
0.7454
[V_R]
mobile9.mat
0.003346
Curvature3
mobile2.mat
1.667
Linear speed Robot,
,
Platform
curvature
;
-T-
0.5
Platform
acceleration
mobile12.mat
mobile6.mat
Wheel Velocity
Gain
Conv Iout
Goto2
mobile13.mat
Turning Radius
0.6
of Mobile robot1
Wheel Position1
12.06
-T-
Divide1 Curvature
mobile1.mat
Curvature
Turning Radius of Mobile robot1 ,.
-K- Dist_wheels
0.4
L_wheel
0.8
R_wheel
Turning Radius
T
-C-
Conv Vout
T
-K- Curvature Radius of wheels
Curv. radius of L_ WHEEL
..
0.7454
Motor current
(Dist_wheels)/2
B
B.
Subsystem of left motor & wheel
Conv . I out
V
A
V
''1
-T-
1
12.06
Linear speed Robot.1
Omega
1/(Dist_wheels)
0.5
Cell surf ace area
Angular speed
1.242
of Robot.
-KNs
Ns
Nm
Np
Omega=
(Vl-Vr)/Distanc
[V_L]
Conv Vout
0.5
Conv Iout
Motor current
Gain1
S_lef t
mobile3.mat
1
s
'2
'1
[theta]
Theta, Robot
6.601
orientation-Dirction1
PVPC Subsystem
Wheel Position1
Curv. Raduis
of L & R wheels
and mobilr robot
0.6
Omega1
mobile5.mat
Theta, Robot orientation-Dirction
...
X
Platform Turning Radius
mobile.mat
Turning Radius LEFT
S_right
12
Conv Vout
Curv. Raduis of L vs R wheels
Angular speed of Robot.
,.1
1
s
0.5
Wheel Velocity
-KCurv, radius of R_ WHEEL
''
V_Robot
Y
XY Graph
Motor current
Subsystem
Subsystem of right motor & wheel
mobile10.mat
0.9938
Curvatures covered
1.32
distance
mobile4.mat
Curvature covered distance
,.3
0.4
40.8
mobile8.mat
Turning Radius RIGHT
5.281
Left wheel
0.4969
Right wheel
wheels linear speeds
mobile7.mat
2.64
mobile11.mat
wheels linear positions
Figure 11(a) Testing model for circular motion
Table 2(a) Simulation results of each subsystem and whole SEDDMRP system for straight line motion
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
Page 8412
PVPC system
inputs
PV cell outputs
PV Panel outputs
Converter outputs
Right DC machine
outputs
β
200
Voltage
0.5 V
Voltage
48 V
Voltage
24
Input voltage
12 V
T
75
Current
1.37 A
Current
41.01
Current
12.12
0.5
0.0025
96
30
Fill factor
Power out
Power in
Efficiency
0.1374
0.6835
0.5
0.7315
Power out
290.9
Shaft Angular.
Speed.
Motor Torque
Motor current
Load current
13.33
rad/s
14.4 N/m2
12.12 A
12.12 A
D
A
Ns
Np
Table 2(b) Simulation results of each subsystem and whole SEDDMRP system for straight line motion
Left DC machine outputs
Right Wheel outputs
left Wheel
Platform outputs
outputs
Input voltage
12 V
Linear Speed
0.5 m/s
Linear
1
Linear speed
0.75 m/s
Speed
m/s
Shaft Angular.
6.665
Curv. radius
0.4
Curv.
0.8
Curvature
1.667
Speed.
rad/s
radius
Motor Torque
14.4 N/m2
Angular. Speed.
1.25 rad/s
Motor current
12.12 A
23.22
Load current
12.12 A
Orient. direction, θ *
for t
Turning radius
Curvature covered
distance * for t
4.655 m
SEDDMRP:Kinematics and dynamics;(Linear speed)
1.2
LEFT motor linear speed
1
Magnitude
0.8
0.6
Figure 11(b-2)
Figure 11(b) Platform Kinematic and dynamic analysis ,
including ; curvature, turning radius, covered curvature
distance , platform linear speed and speed of right and left
wheels
Platform linear speed
0.6
RIGHT motor linear speed
SEDDMRP WHEELS Kinematics
0.4
1
0.2
0.9
RIGHT wheel turning radius
0.8
0
-0.2
0
5
10
Time(s)
15
20
Figure 11(b-1)
SEDDMRP (2) Kinematics and dynamics
0.6
0.5
LEFT wheel turning radius
0.4
4.5
0.3
4
0.2
3.5
0.1
Curvature covered distance
3
Magnitude
Magnitude
0.7
0
2.5
0
2
4
6
Time(s)
2
Figure 11(c-1)
Platform curvature
1.5
1
Platform turning radius
0.5
0
-0.5
0
5
10
Time(s)
15
20
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
Page 8413
SEDDMRP:Kinematics and dynamics (Angular speed)
25
Orientation angle
Figure 11(d) Platform Kinematic and dynamic analysis; left
and right wheel positions, platform curvature, platform
turning radius, curvature covered distance
20
SEDDMRP WHEELS Kinematics
1
Magnitude
15
0.9
LEFT motor angular speed
RIGHT wheel turning radius
0.8
10
0.7
5
Platform angular speed
Platform turning radius
0.6
Magnitude
RIGHT motor angular speed
0.5
LEFT wheel turning radius
0.4
0
0.3
0.2
-5
0
5
10
Time(s)
15
0.1
20
0
Figure 11(c-2)
Figure 11(c) Platform Kinematic and dynamic analysis; left
and right motor turning radius, angular speed, orientation
angle, and platform angular speed
0
1
2
4
5
6
Figure 11(e) Platform Kinematic and dynamic analysis, right
and left wheels and platform turning radius
LEFT motor outputs
SEDDMRP (2) Kinematics and dynamics
16
4
LEFT Motor torque
LEFT motor angular speed
LEFT motor Load current
14
3.5
12
Curvature covered distance
3
3
Time(s)
10
2
Mgnitude
Magnitude
2.5
Platform curvature
8
6
1.5
4
X: 15.75
1
Platform turning radius
0.5
0
0
-0.5
Y: 1
LEFT motor linear speed
2
-2
0
5
10
Time(s)
15
0
20
10
Time (s)
15
20
Figure 11(f-1)
Figure 11(d-1)
RIGHT motor outputs
SEDDMRP:Kinematics and dynamics;(wheels position)
16
16
RIGHT Motor torque
14
14
RIGHT motor Load current
12
12
Left wheel position
10
Mgnitude
10
Magnitude
5
8
6
Right wheel position
RIGHT motor angular speed
6
4
4
X: 15.36
Y: 0.5
2
Platform curvature
2
8
RIGHT motor linear speed
0
0
-2
-2
0
5
10
Time(s)
Figure 11(d-2)
15
20
0
5
10
Time (s)
15
20
Figure 11(f-2)
Figure 11(f) Right and left motor outputs; linear speed,
torque, and angular speed
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
Page 8414
PV Panel-converter outputs
50
PV Panel Iout
PV Panel Vout
Magnitude
40
30
Converter Vout
20
Converter Iout
10
0
0
2
4
6
Time (s)
8
10
Figure 11(g) Panel-converter output voltages and output
currents
Control signals
15
Motor PI controller signal
Magnitude
10
5
Voltage PI controller signal
0
0
5
10
Time (s)
Figure 11(h) Control signal
15
Figure 12 Double curvature motion
Conclusion
A new generalized and refined model for solar electric
differential drive Mobile Robotic Platforms (SEDDMRP) and
some considerations regarding design, modeling and control
solutions are proposed. Each subsystem of the proposed
SEDDMRP system's
seven
main
subsystems
is
mathematically described and corresponding Simulink submodel is developed, then an integrated generalized model of
all subsystems is developed. The proposed model is developed
to help in facing the two top challenges in developing
Mechatronics SEDDMRP systems; early identifying system
level problems and ensuring that all design requirements are
met, where it is developed to allow designer to have the
maximum output data to select, evaluate, integrate and
control the overall SEDDMRP system and each subsystem
outputs characteristics and response, for desired overall and/or
either subsystem's specific outputs, under various PV
subsystem input operating conditions, to meet particular
SEDDMRP system requirements and performance.
The obtained results of testing the model for straight line,
circular and double curvature motions, show the simplicity,
accuracy and applicability of the presented models in
Mechatronics design of SEDDMRP system application, as
well as application in educational process.
Table- 3 Nomenclature and nominal characteristic of
SEMRP subsystems
DC machine parameters
Figure 11(i) Resulted circular motion
Vin=12 V
Input voltage to DC machine
Kt=1.188 Nm/A
Motor torque constant
Ra = 0.156 Ω
Motor armature Resistance
La=0.82 MH
Motor armature Inductance,
Jm=0.271 kg.m2
Geared-Motor Inertia
bm=0.271 N.m.s
Viscous damping
Kb=1.185 rad/s/V
Back EMF constant,
n=1
Gear ratio
r=0.075 m,
Wheel radius
Jequiv kg.m2
The total equivalent inertia,
bequiv N.m.s
The total equivalent damping,
L=0.4 m
The distance between wheels
centers
Tachometer constant,
Ktac=1.8 rad/s
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
Page 8415
ω=speed/r, rad/s
Tshaft
=0.5/0.075=6.667 ,also
1/0.075=13.3333
The torque produced by motor
η
The transmission efficiency
Tshaft
The torque, produced by the
driving motor
Nominal values for Mobile platform
M,m, Kg
The mass of the mobile
platform
Cd =0.80
Aerodynamic drag coefficient
CL
ρ , kg/m3
The coefficient of lift, with
values( CL to be 0.10 or 0.16),
The rolling resistance
coefficient
The air density at STP, ρ =1.25
a, m/s2
Platform linear Acceleration
G, m/s2
The gravity acceleration
Ν, m/s
The vehicle linear speed.
α , Rad
L
Road slope or the hill climbing
angle
Mobile platform's reference
area
lift,
Af
Platforms frontal area
KP
Proportional gain
KI
Integral gain
Z0
PI controller zero
Pm
The power available in the
wheels of the vehicle.
The total resistive torque, the
torque of all acting forces.
Cr=0.5
B
TTotal
Solar cell parameters
Isc=8.13 A , 2.55 A ,
3.8
IA
Iph A
Eg : =1.1
V t  KT / q
Is ,A
Rs=0.001 Ohm
Rsh=1000 Ohm
V
q=1.6e-19 C
Bo=1000 W/m2
β =B=200 W/m2
Ki=0.0017 A/◦C
The short-circuit current, at
reference temp 25◦C
The output net current of PV
cell (the PV module current)
The light-generated
photocurrent at the nominal
condition (25◦C and 1000
W/m2),
The band gap energy of the
semiconductor
The thermo voltage of cell. For
array :(V t  N s KT / q )
The reverse saturation current
of the diode or leakage current
of the diode
The series resistors of the PV
cell, it they may be neglected to
simplify the analysis.
The shunt resistors of the PV
cell
The voltage across the diode,
output
The electron charge
The Sun irradiation
The irradiation on the device
surface
The cell's short circuit current
temperature coefficient
Vo= 30.6/50 V
Ns= 48 , 36
Open circuit voltage
Series connections of cells in
the given photovoltaic module
Nm= 1 , 30
Parallel connections of cells in
the given photovoltaic module
K=1.38e-23 J/oK;
The Boltzmann's constant
N=1.2
The diode ideality factor, takes
the value between 1 and 2
T= 50 Kelvin
Working temperature of the p-n
junction
Tref=273 Kelvin
The nominal reference
temperature
Buck converter parameters
C=300e-6; 40e-6 F
Capacitance
L=225e-6; .64e-6H
Rl=RL=7e-3
rc= RC=100e-3
Vin= 24 V
R=8.33; 5 Ohm;
Ron=1e-3;
Inductance
Inductor series DC resistance
Capacitor equivalent series
resistance, ESR of C ,
Input voltage
Resistance
Transistor ON resistance
KD=D= 0.5, 0.2,
Tt=0.1 , 0.005
VL
IC
Duty cycle
Low pass Prefilter time constant
Voltage across inductor
Current across Capacitor
References
[2]
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pp 12-39,2013.
[3]
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pp253-263 (June 2013).
[4]
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[5]
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[6]
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Author Profile
Farhan Atallah Salem : Now with Taif University, Mechanical
engineering Dept, Mechatronics engineering track.
Farhan A. Salem , IJECS Volume 3Issue 9 September 2014 Page No.8399-8417
Page 8417