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

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.
   40   41   42   43   44   45   46   47   48   49   50