Introduction
Proportional–integral–derivative (PID) control method
Bacterial foraging optimization (BFO) algorithm
Fuzzy logic control (FLC) method
Research objective and paper organization
Two-wheeled robotic machine system description
-
The attached payload linear displacement in vertical direction (h1).
-
The attached payload linear displacement in horizontal direction (h2).
-
The angular displacement of the angular rotation of the right wheel (δR).
-
The angular displacement of the angular rotation of the left wheel (δL).
-
The tilt angle of the intermediate body around the vertical Z-axis (θ).
TWRM mathematical modeling
Terminology | Description | Value | Unit |
---|---|---|---|
θ
| Tilt angle of the intermediate body around the vertical Z-axis | – | ° |
δ
R
, δ
L
| Angular displacement of right and left wheels | – | m |
h1, h2 | Vertical and horizontal linear link displacement | – | m |
F1, F2 | Force generated by the vertical and horizontal linear actuators | – | N |
τ
R
, τ
L
| Right and left wheels torque | – | N/m |
m
1
| Mass of the chassis | 3.1 | kg |
m
2
| Mass of the linear actuators | 0.6 | kg |
m
w
| Mass of wheel | 0.14 | kg |
R
| Wheel radius | 0.05 | m |
J
1
| Chassis moment of inertia | 0.068 | kg m2 |
J
2
| Moving mass moment of inertia | 0.0093 | kg m2 |
J
w
| Wheel moment of inertia | 0.000175 | kg m2 |
Ɩ
| Distance of chassis’ center of mass for wheel axle | 0.14 | m |
µ
1
| Coefficient of friction of vertical linear actuator | 0.3 | Ns/m |
µ
2
| Coefficient of friction of horizontal linear actuator | 0.3 | Ns/m |
µ
w
| Coefficient of friction between wheel and ground | 0 | Ns/m |
µ
c
| Coefficient of friction between chassis and wheel | 0.1 | Ns/m |
g
| Gravitational acceleration | 9.81 | m/s2 |
Control system design
PID control design
BFO-PID control design
Parameter symbol | Description |
---|---|
p
| Search space dimension |
S | Total number of bacteria in the population |
N
s
| Number of bacteria swims in the same direction |
N
c
| Number of chemotactic steps |
N
re
| Number of reproduction steps |
N
ed
| Number of elimination and dispersal events |
P
ed
| Probability of the elimination and dispersal of bacterium |
C
| Step size of the bacterium tumble |
J
| Cost function value |
-
Mean of the squared error (MSE).
-
Integral of time multiplied by absolute error (ITAE).
-
Integral of absolute magnitude of the error (IAE).
-
Integral of the squared error (ISE).
-
Integral of time multiplied by the squared error (ITSE).
Controlled parameter | Gain parameter | Upper boundary | Lower boundary |
---|---|---|---|
Loop 1
| |||
δ
R
|
Kp
1
| 20 | − 20 |
Kd
1
| 20 | − 20 | |
Ki
1
| 0.1 | − 0.1 | |
Loop 2
| |||
δ
L
|
Kp
2
| 20 | − 20 |
Kd
2
| 20 | − 20 | |
Ki
2
| 0.1 | − 0.1 | |
Loop 3
| |||
θ
|
Kp
3
| 50 | − 50 |
Kd
3
| 10 | − 10 | |
Ki
3
| 0.1 | − 0.1 | |
Loop 4
| |||
h
1
|
Kp
4
| 20 | − 20 |
Kd
4
| 10 | − 10 | |
Ki
4
| 0.1 | − 0.1 | |
Loop 5
| |||
h
2
|
Kp
5
| 60 | − 60 |
Kd
5
| 50 | − 50 | |
Ki
5
| 0.1 | − 0.1 |
PD-FLC control design
Error | Change of error | ||||
---|---|---|---|---|---|
NB | NS | Z | PS | PB | |
NB | NB | NB | NB | NS | Z |
NS | NB | NB | NS | Z | PS |
Z | NB | NS | Z | PS | PB |
PS | NS | Z | PS | PB | PB |
PB | Z | PS | PB | PB | PB |
Comparison between implementation of PID, BFO-PID, and PD-FLC
Output parameter | Gain parameter | PID | BFO-PID | PD-FLC |
---|---|---|---|---|
Loop 1
| ||||
δ
R
|
Kp
1
| 80 | 10.255 | 7 |
Kd
1
| 75 | 0.016 | 3.5 | |
Ki
1
| 0.05 | 15.05 | 0 | |
Loop 2
| ||||
δ
L
|
Kp
2
| 80 | 10.255 | 7 |
Kd
2
| 75 | 0.016 | 3.5 | |
Ki
2
| 0.05 | 15.05 | 0 | |
Loop 3
| ||||
θ
|
Kp
3
| 80 | − 1.733 | 7 |
Kd
3
| 9 | − 0.0693 | 1.5 | |
Ki
3
| 0.02 | 0.0835 | 0 | |
Loop 4
| ||||
h
1
|
Kp
4
| 8 | 10.3279 | 4.5 |
Kd
4
| 10 | 7.3378 | 6 | |
Ki
4
| 0.01 | 0.013 | 0 | |
Loop 5
| ||||
h
2
|
Kp
5
| 27 | 50.1502 | 14 |
Kd
5
| 32 | 30.7237 | 16 | |
Ki
5
| 0.05 | 0.027 | 0 |
Control method | Percent overshoot (%) | Settling time (s) | Rise time (s) | Peak time (s) |
---|---|---|---|---|
PID | 48.1 | 2.2870 | 0.2790 | 0.5710 |
PID-BFO | 27.9 | 0.7800 | 0.2300 | 0.4400 |
PD-FLC | 38.6 | 1.4410 | 0.2170 | 0.4070 |