Figure 1.
2D illustration of (a) omniwheel mobile robot and (b) trailer
Figure 2.
Forces (a), torques (b) on trailer
Figure 3.
Forces (a), torques (b) on omniwheel mobile robot
Figure 4.
Mobile robot integrated trailer representation
Figure 5.
Cubic Bézier curve
Figure 6.
Pure pursuit algorithm
Figure 7.
Pure pursuit algorithm example for (a) low, (b) high lookahead distance
Figure 8.
Dynamic tracking control
Figure 9.
Input voltage and output velocity against time
Figure 10.
Curve fitting of the transfer function
Figure 11.
Tested motor transfer function and PID parameters
Figure 12.
PID controlled motor transfer function
Table 1.
Tuned PID Parameters
| P | I | D | N | |
|---|---|---|---|---|
| Motor 1 | 0.231 | 0.358 | 0.007 | 15796.2 |
| Motor 2 | 0.158 | 0.493 | 0.011 | 5750.9 |
| Motor 3 | 0.166 | 0.535 | 0.012 | 6962.8 |
Figure 13.
Simulation block diagram of trailer control
Figure 14.
Trailer control simulation in MATLAB Simulink
Figure 15.
Subsystem of path generation and path tracking algorithm
Figure 16.
Dynamic tracking control subsystem
Figure 17.
Subsystem of the trailer mechanical Simscape model
Figure 18.
Stopping algorithm of the trailer
Figure 19.
Simulation start, the trailer is at the beginning of the path (a) isometric view, (b) front view, (c) top view, (d) bottom view
Figure 20.
Simulation Block Diagram of the tractor robot
Figure 21.
Subsystem of dynamic tracking control of trailer towed by omniwheel mobile robot
Figure 22.
Simulation start, the mobile robot and trailer are at the beginning of the path (a) isometric view, (b) left view, (c) top view, (d) bottom view
Figure 23.
Bézier Curve path analysis with tuned parameters
Figure 24.
Linear velocity of the towed trailer
Figure 25.
Angular velocity of towed trailer
Figure 26.
Square path analysis (D=0.75 m)
Figure 27.
Linear velocity of towed trailer on square path
Figure 28.
Angular velocity of towed trailer on square path
Figure 29.
Square path analysis (D=1.25 m)
Figure 30.
Linear velocity of towed trailer on square path
Figure 31.
Angular velocity of towed trailer on square path
Figure 32.
Circular path analysis
Figure 33.
Linear velocity of towed trailer on circular path
Figure 34.
Angular velocity of towed trailer on circular path
Figure 35.
Archimedean spiral path analysis
Figure 36.
Linear velocity of towed trailer on Archimedean spiral path
Figure 37.
Simulated vs desired angular velocity of towed trailer on Archimedean spiral path
Figure 38.
Simulation block diagram including PID control
Figure 39.
Simulating the selected motors
Figure 40.
Bézier curve path analysis with implementation of PID controlled dc motors
Figure 41.
Linear velocity of towed trailer on Bézier curve path with implementation of PID controlled dc motors
Figure 42.
Angular velocity of towed trailer on Bézier curve path with implementation of PID controlled dc motors
Table 2.
Performance Comparison of Non-Tuned, Tuned, and Motor Integrated PID Tuned Controllers
| Metric | Non-Tuned | Tuned | Motor Integrated PID Tuned |
|---|---|---|---|
| Steady State Linear Velocity Error (Vess,%) | 13.29 | 6.26 | 4.32 |
| Linear Velocity Overshoot (%) | 8.68 | 17.48 | 17.17 |
| Steady State Time (Tss, s) | N/A | N/A | 61.9 |
| Maximum Angular Velocity Error | 0.104 | 0.092 | 0.117 |
| Average Angular Velocity Error | 0.010 | 0.006 | 0.008 |
| Execution Time (s) | 74.7720 | 68.7573 | 67.2488 |
| Tracking Error (Max, m) | 0.0777 | 0.0755 | 0.0756 |
| Tracking Error (RMS, m) | 0.0425 | 0.0408 | 0.0404 |
| Tracked Path Length (m) | 6.620 | 6.620 | 6.630 |
| Planned Path Length (m) | 6.627 | 6.627 | 6.627 |