Author Contributions
Conceptualization, A.Š., M.S. and I.P.; methodology, A.Š. and M.S.; software, A.Š.; validation, A.Š; formal analysis, A.Š.; investigation, A.Š.; resources, A.Š.; data curation, A.Š.; writing—original draft preparation, A.Š., M.S. and I.P.; writing—review and editing, A.Š. and M.S.; visualization, A.Š.; supervision, M.S. and I.P.; project administration, M.S. and I.P.; funding acquisition, I.P. All authors have read and agreed to the published version of the manuscript.
Figure 1.
A theoretical framework for robot motion planning.
Figure 1.
A theoretical framework for robot motion planning.
Figure 2.
Control architecture of a differential-drive mobile robot.
Figure 2.
Control architecture of a differential-drive mobile robot.
Figure 3.
The pipeline of the smooth patrolling scheme.
Figure 3.
The pipeline of the smooth patrolling scheme.
Figure 4.
Two robots’ positions along with real obstacles (green dots), the patrolling route (blue line), the smoothed patrolling path (red line), and the tracked trajectory (green line).
Figure 4.
Two robots’ positions along with real obstacles (green dots), the patrolling route (blue line), the smoothed patrolling path (red line), and the tracked trajectory (green line).
Figure 5.
Orientation alignment is based on a golden ratio, where angle is the divergence between the robot’s current orientation and the direction of the first segment on the replanned TWD* path.
Figure 5.
Orientation alignment is based on a golden ratio, where angle is the divergence between the robot’s current orientation and the direction of the first segment on the replanned TWD* path.
Figure 6.
Clothoid with a length of 10 m interpolated via circular interpolation with the sampling interval = 0.1 and interpolation error for different values of the sampling interval . (a) Clothoid interpolated via circular interpolation. (b) Interpolation error e for different values of sampling interval in the lookup table.
Figure 6.
Clothoid with a length of 10 m interpolated via circular interpolation with the sampling interval = 0.1 and interpolation error for different values of the sampling interval . (a) Clothoid interpolated via circular interpolation. (b) Interpolation error e for different values of sampling interval in the lookup table.
Figure 7.
The comparison of the patrolling route (blue line) with the original patrolling method and the proposed smooth method, smoothed patrolling path (red line), the tracked trajectory (green line), the reference velocity profile (blue line), and the actual velocity profile (red line). (a) Patrol carried out using the original patrolling method. (b) Patrol carried out using the proposed smooth method. (c) Velocity profile yielded when using the original patrolling method. (d) Velocity profile yielded when using the proposed smooth method.
Figure 7.
The comparison of the patrolling route (blue line) with the original patrolling method and the proposed smooth method, smoothed patrolling path (red line), the tracked trajectory (green line), the reference velocity profile (blue line), and the actual velocity profile (red line). (a) Patrol carried out using the original patrolling method. (b) Patrol carried out using the proposed smooth method. (c) Velocity profile yielded when using the original patrolling method. (d) Velocity profile yielded when using the proposed smooth method.
Figure 8.
The curvature profile of the smooth patrolling path from
Figure 7b.
Figure 8.
The curvature profile of the smooth patrolling path from
Figure 7b.
Figure 9.
The smoothed path in a static environment for the proposed method and other state-of-the-art smoothing methods.
Figure 9.
The smoothed path in a static environment for the proposed method and other state-of-the-art smoothing methods.
Figure 10.
Curvature profiles and tangent angle change on a smooth path in a static environment with the proposed method and other state-of-the-art smoothing methods.
Figure 10.
Curvature profiles and tangent angle change on a smooth path in a static environment with the proposed method and other state-of-the-art smoothing methods.
Figure 11.
Velocity profiles on a smooth path in a static environment for the proposed method and other state-of-the-art smoothing methods.
Figure 11.
Velocity profiles on a smooth path in a static environment for the proposed method and other state-of-the-art smoothing methods.
Figure 12.
The patrolling route (blue line) during the execution of replanning, smoothed patrolling path (red line), the tracked trajectory (green line), the reference velocity profile (blue line), and the actual velocity profile (red line).
Figure 12.
The patrolling route (blue line) during the execution of replanning, smoothed patrolling path (red line), the tracked trajectory (green line), the reference velocity profile (blue line), and the actual velocity profile (red line).
Figure 13.
The step-by-step local path-planning scheme in a dynamic environment presenting the travel trajectories of robot1 and robot2 (cyan dashed lines) toward goal1 and goal2, respectively: the STWD* (green line), APF (red line), DWA (magenta line), and MPC (blue line) travel trajectories of robot0 from start to goal0.
Figure 13.
The step-by-step local path-planning scheme in a dynamic environment presenting the travel trajectories of robot1 and robot2 (cyan dashed lines) toward goal1 and goal2, respectively: the STWD* (green line), APF (red line), DWA (magenta line), and MPC (blue line) travel trajectories of robot0 from start to goal0.
Figure 14.
Velocity profiles of compared local path-planning algorithms executed in a dynamic environment: the STWD* (green line), APF (red line), DWA (magenta line), and MPC algorithms (blue line).
Figure 14.
Velocity profiles of compared local path-planning algorithms executed in a dynamic environment: the STWD* (green line), APF (red line), DWA (magenta line), and MPC algorithms (blue line).
Figure 15.
Experimental setup for smooth autonomous patrolling using the Husky mobile robot.
Figure 15.
Experimental setup for smooth autonomous patrolling using the Husky mobile robot.
Figure 16.
A comparison of patrolling route (blue line) traveled using the original and the proposed smooth methods: D* path (yellow line), smoothed patrolling path (red line), the tracked trajectory (green line), the reference velocity profile (blue line), and the actual velocity profile (red line).
Figure 16.
A comparison of patrolling route (blue line) traveled using the original and the proposed smooth methods: D* path (yellow line), smoothed patrolling path (red line), the tracked trajectory (green line), the reference velocity profile (blue line), and the actual velocity profile (red line).
Figure 17.
The step-by-step patrolling task executed in a static environment where the real obstacles are presented as extracted 2D laser range data from the Velodyne point cloud data (green dots), the patrolling path (blue line), the smooth patrolling path (red line), replanned TWD* path (black line), and tracked trajectory from start to goal (green line).
Figure 17.
The step-by-step patrolling task executed in a static environment where the real obstacles are presented as extracted 2D laser range data from the Velodyne point cloud data (green dots), the patrolling path (blue line), the smooth patrolling path (red line), replanned TWD* path (black line), and tracked trajectory from start to goal (green line).
Figure 18.
Velocity profile for the proposed method during the execution of the patrolling task in a static environment, presenting the reference velocity profile (blue line) and the actual velocity profile (red line).
Figure 18.
Velocity profile for the proposed method during the execution of the patrolling task in a static environment, presenting the reference velocity profile (blue line) and the actual velocity profile (red line).
Figure 19.
The execution of the step-by-step patrolling task in a dynamic environment, where the real obstacles are presented as extracted 2D laser range data from the Velodyne point cloud data (green dots), along with the patrolling path (blue line), the smooth patrolling path (red line), replanned TWD* path (black line), tracked trajectory from start to goal (green line), and the dynamic obstacle trajectory (magenta line).
Figure 19.
The execution of the step-by-step patrolling task in a dynamic environment, where the real obstacles are presented as extracted 2D laser range data from the Velodyne point cloud data (green dots), along with the patrolling path (blue line), the smooth patrolling path (red line), replanned TWD* path (black line), tracked trajectory from start to goal (green line), and the dynamic obstacle trajectory (magenta line).
Figure 20.
Velocity profile for the proposed method during the execution of the patrolling task in a dynamic environment, presenting the reference velocity profile (blue line) and the actual velocity profile (red line).
Figure 20.
Velocity profile for the proposed method during the execution of the patrolling task in a dynamic environment, presenting the reference velocity profile (blue line) and the actual velocity profile (red line).
Table 1.
Table comparing paths followed with and without smooth algorithm.
Table 1.
Table comparing paths followed with and without smooth algorithm.
| Original Method | Proposed Method |
---|
L [m] | 25.21 | 22.55 |
T [s] | 61.8 | 51.6 |
[m] | 4.32 | 3.67 |
[m/s] | 0.24 | 0.08 |
[m] | 0.94 | 0.24 |
[ms] | 407.56 | 6.27 |
Table 2.
Smoothing algorithm comparison table.
Table 2.
Smoothing algorithm comparison table.
Smoothing Method | L [m] | T [s] | [m] | [m] | BE | CVE |
---|
Clothoid | 22.55 | 47.16 | 3.48 | 0.25 | 0.14 | 0.25 |
B-spline | 27.79 | 60.59 | 6.05 | 0.84 | 5.55 | 20.21 |
Bézier curve | 25.97 | 54.66 | 8.25 | 0.44 | 0.28 | 1.67 |
Cubic spline | 24.93 | 51.87 | 3.82 | 0.29 | 0.13 | 0.11 |
Cubic Hermit spline | 25.28 | 53.96 | 3.48 | 0.40 | 0.35 | 4.16 |
Dubins curve | 24.98 | 54.18 | 4.28 | 0.68 | 0.24 | 2.88 |
Table 3.
Comparison table of local path-planning algorithms.
Table 3.
Comparison table of local path-planning algorithms.
Method | L [m] | T [s] | [m/s] | [m] | [ms] |
---|
STWD* | 10.70 | 24.20 | 0.13 | 0.47 | 369.15 |
APF | 13.17 | 26.72 | 0.13 | 0.55 | 24.58 |
DWA | 12.07 | 24.45 | 0.12 | 0.39 | 49.45 |
MPC | 10.93 | 24.40 | 0.08 | 0.65 | 27.35 |
Table 4.
Table comparing the results with and without using the smooth algorithm.
Table 4.
Table comparing the results with and without using the smooth algorithm.
| Original Method | Proposed Method |
---|
L [m] | 14.90 | 13.77 |
T [s] | 61.18 | 40.07 |
[m] | 1.89 | 0.68 |
[m/s] | 0.27 | 0.25 |
[m] | 0.58 | 0.49 |
[ms] | 46.26 | 4.27 |