Page 45 - IJEEE-2023-Vol19-ISSUE-1

Mohsin, Aldair & Al-Hussaibi | 41

x? ??00 1 0 00?? x
!%x¨ ? & ???00 10??? 0%x? 1
%¨ !(#$%&!)( %!*&! %?

= #()$%)$)%&! #()$%)$)%&!

00
!%&( %*&()$%)

#()$%)$)%&! #()$%)$)%&! (4)

+ ???#()$#%$%%%)$&&)! %&!??? U
?#()$%)$)%&!?

x
y = ;10 00 10 00= >%x? @ + ;00= u Fig.6: LQR controller for TWSBR.
(5)
IV. SIMULATION RESULTS AND DISCUSSION
%?
LQR controller is used in this modeling technique to obtain the In this section, simulation results demonstrate how the system
operates without the need for a complicated mathematical
robot's chassis and cart parameters from the Simscape model equation modeling of TWSBR by using the Simscape
analysis method. This type of analysis reduced the requirement
that is configured when PID is employed. Figure 6 illustrates for the physical system to be implemented in real life with
different control methods tested which is costly. Simulation
the LQR controller block diagram of TWSBR. The parameters studies are used to study the performance and robustness of the
proposed TWSBR model in the presence of road disturbances.
of the TWSBR platform and their numerical values are listed in Four cases are studied: The first is an open-loop system in
which the robot moves without using any controllers. The
Table II. Small matrix B value is chosen since we selected high second is a closed-loop system that uses PID controllers, and
values in matrix C to minimize the states ! and %. Resulting is the third is a closed-loop system that uses LQR controllers. The
fourth, testing the controller robustness by adding weights to
getting a stronger control signal and faster robot response. the robot's body. Simulation cases are discussed as below:

Furthermore, the weights for the cost function C and B are Case 1: open-loop system
There is no controller present in this looping system. The robot
determined through trial and error. should be able to move vertically on the road and finish the
- = 0(12([500 0 500 0]), 8 = [1] movement without falling. Unfortunately, the robot cannot
move in this case because it lost equilibrium and immediately
The following MATLAB command is used to find gain matrix: fell down to the ground when the simulation started. TWSBR is
in the initial position shown in Fig.7a when the simulation is
! = 9:;(<, =, -, 8) starting to run. Also, Fig.7b demonstrates how TWSBR is
falling down because its loop mechanism is not controlled.
MATLAB Workspace was used to export a gains matrix into a
Case 2: closed-loop using PID controller
TWSBR Simscape/Simulink model. Below are the gain values: The robot's tip is subjected to a single initial external force
! = [-22.3607 - 14.3279 56.1181 6.0326] represented by 10° input angle disturbance. As a result, the
controller is able to reveal the robot's original balancing in the
All the two controller techniques results will be discussed in the upright position and track the robot movement in one direction.
Figure 8a demonstrates how rapidly the system recovered its
simulation section. tilted angle after 0.56 seconds. The disturbance forces acting
lead the robot angle to deviate before the controller starts. Once
TABLE II the controller starts, the robot angle returns to zero because it's
beginning from zero condition and the wheel cart moves in a
Simscape Multibody Model Parameters of TWSBR forward or backward direction at a constant speed as shown in
Fig.8b.
Symbol Quantity Value The simulation results also showed the controller's ability to
M mass of the chassis 1 !2 reject multiple disturbance signals in different locations of the
same motion simulation which confirms the controller
m mass of the wheels and shaft 0.2 !2 robustness with respect to the behavior performance response
system. Figure 9 shows the self-balancing behavior of
I moment of inertia of the chassis 0.0005 simulated TWSBR model after being tilted 10° on the positive
*D/F^2

l length to chassis center of mass 0.125 F

g gravity 9.8 F/K^2

Fig.5: PID controller for TWSBR.