In this section, a simulation is conducted to verify the performance of the proposed trajectory planning algorithm, specifically, the effectiveness of garbage collection, dynamic obstacle avoidance, and the satisfaction of the cooperation constraints in terms of speed and distance coordination between the two USVs.
4.2. The Simulation Results
The simulation results are analyzed from the following two parts. Firstly, the whole generated reference trajectories of the DUSVs are shown in
Figure 11 to visually verify effective garbage approaching, capturing, and obstacle avoidance. The effectiveness of the improved method in dynamic obstacle avoidance is stressed with results in
Figure 11 and
Table 5. Secondly, the changes of the relevant state variables over time, including the speed of the DUSVs, distances as well as distance differences to the key points between the DUSVs, and the resulting area capacities, are shown in
Figure 12, which are to verify the satisfaction of cooperation constraints.
Figure 11 shows the generated reference trajectories of the DUSVs for cooperative surface garbage cleaning in the environment with static and dynamic obstacles. The results obtained by the proposed algorithm using the IAPF method is compared with that using the traditional APF method. The trajectories are generated as discrete reference positions and orientations for the DUSVs with time intervals of 0.1 s. The triangles appearing at time intervals of 1s represent the time-varying poses of the DUSVs. In particular, the coordinates and time of the trajectories are shown when the DUSVs enter the garbage target capture area. The black dotted arrows and red arrows near the obstacles and garbage targets indicate the motion direction of the corresponding obstacles and garbage targets. The gray and pink shaded wide trails indicate the trajectories of the obstacles and the garbage targets.
This shows that the computed optimal access sequence of the garbage targets is . Both algorithms have shown their capability to generate the trajectories for the DUSVs to capture all the garbage while avoiding all the obstacles. In the beginning, the DUSVs travel towards the current garbage target, ; the distance between the DUSVs is gradually increased to ensure safety as well as to meet the current area demand of . When entering the distance of influence of the potential field of , the DUSVs pass from different sides of , so that the garbage can be captured by the floating rope while being attracted towards the next garbage, . Following that, dynamic obstacle is encountered and the DUSVs pass from the same side of the obstacle under the guidance of the repulsive potential field and the attractive potential of the current garbage target, . When the DUSVs leave the repulsive potential field of , they carry on to approach the current target, , and the distance between the two USVs is adjusted to meet the current area demand, i.e., the sum of areas of and . When is captured, the DUSVs switch to approach the next target, , and so on until they finish with the final target, , and travel to the finishing point, E. During the whole process, the speeds of the DUSVs are adjusted by a fuzzy logic-based speed controller to keep the DUSVs in parallel.
In addition, the trajectories around t = 21 s and t = 43 s are plotted separately to compare the performance for dynamic collision avoidance. It is observed that the algorithm with IAPF is effective in reducing the length of trajectories for bypassing the dynamic obstacles compared to the algorithm with traditional APF.
Table 5 presents the total trajectory length and navigation time by the two algorithms. It can be seen that the algorithm with IAPF provides shorter trajectory lengths and costs less navigation time to complete the task, which indicates its efficiency in scenarios with dynamic obstacles.
Figure 12 shows the changes in relevant state variables of the DUSVs over time to verify the effectiveness of the proposed algorithms in meeting the cooperation constraints.
Figure 12a,b show the changes in speed of the DUSVs,
and
, and distance difference,
, over time under the proposed fuzzy logic-based speed controller and simple speed compensator, respectively. For both cases, it shows that, in the beginning, the two USVs travels in parallel at the same speed, which gradually increases to the maximum 2 m/s. In the finishing period, the DUSVs also travel at the same speed, which gradually decreases until the end of the mission.
Figure 12a shows that between 32–33 s and 38–42 s, there are obvious variations in
. The distance difference,
, surpasses the threshold of 2 m and reaches 3.3 m at 38 s. This is due to the fact that USV1 turns from the outside of
to capture
and thus lags behind USV1. With the proposed speed controller, USV2 slows down and
is effectively reduced within 4 s.
Figure 12b shows that between 25–29 s, 32–35 s, and 39–47 s, there are obvious variations in
and the distance difference between the DUSVs and the target key points is greater than 2 m. Especially between 39–47 s, it takes 8 s to reduce the distance difference down below 2 m, indicating poor performance of the simple compensator. Therefore, it can be concluded that the proposed speed control strategy effectively improves the cooperation of the DUSVs.
Figure 12c shows the changes in the expected distance,
L, between the DUSVs over time. It shows that the initial expected spacing of the DUSVs is 2 m. During the whole process,
L is always between the lower bound of 2 m and the upper bound of 11 m. This effectively prevents internal collision between the DUSVs. The values of
L are marked at the particular time instants, which correspond to the marked positions of the DUSVs in
Figure 11b. It was found that
L is gradually increased to prevent any collision with the garbage targets and ensure no escape of garbage.
Figure 12d shows the changes in current area demand
and the current enclosed area of the floating rope,
S, over time. It can be seen that at current capacity,
S can always accommodate the demand,
. At 46 s, when the DUSVs start to approach the final garbage,
, the demand,
, reaches its maximum and
S is gradually increased until reaching the demand at 49 s. It should be noticed that constraint (
10) is conservative as the estimated garbage area with the circular or elliptical model is larger than the actual garbage area. This implies that the enclosed areas have well satisfied the actual area demands over time.
In summary, the above results indicate that the proposed comprehensive trajectory planner for the floating-rope-connected DUSVs for water surface garbage cleaning is satisfying in generating cooperative trajectories for the two USVS that meet the requirements for garbage collection with good coordination for preventing garbage escaping and internal collision while avoiding dynamic obstacles with a shorter travel distance