Simulation for the merging maneuver of one merging vehicle and mul-

All the simulation, show expected result. Therefore the it is concluded that the proposed method is eective for cooperative and mild motion optimizing for vehicles during merging.

To be universal formulation method, the proposed method should be usable for the merging of one merging vehicle with multiple main lane vehicles. The next section will investigate whether the proposed method can be used in that case.

6.2.2 Simulation for the merging maneuver of one merging vehicle

0 50 100 150 200 250 300 0

10 20

l₄

l₃₃ l₅

Y

X

Figure 6.2.10: Approximated lanes and available moving area.

x_2x=110.9 m,x_3x=34.4 m,v_1x=12.1 m/s, v_2x=20.1m/s, v_3x=18.0 m/s.

In consideration of the fact that, in a merging maneuver the driver of the merging ve-hicle is more willing to produce acceleration or deceleration than the drivers of the main lane vehicles, the bounds of the accelerations were set as follows: a_1xmin=a_bmin=−4.0 m/s², a_1xmax=a_bmax=4.0 m/s², a_2xmin=a_3xmin=−3.0 m/s², a_2xmax=a_3xmax=3.0 m/s². The parame-ters in the penalty functions were set so that the simulation results can be similar to the actual merging results. The resultant weights and parameters in the penalty functions are as follows: ω₁₂=1.0, ω₁₃=1.0, ω₂₃=1.0, ω_lb=0.01, ω_rb=0.01, ω_v1=0.008, ω_v2=0.01, ω_v3=0.01, ω_a1=0.01,ω_a2=0.015,ω_a3=0.018,ω_ab=0.01. a_r12=a_r13=a_r23=16.0m,b_r12=b_r13=b_r23=1.75m.

The sampling time h is 0.01 s. Prediction horizon was set as T=3.0s.

In this simulation, gures are shown in the following manners: The positions of the relevant vehicles at some instants are plotted in the merging trajectories. The positions of Vehicle1 are plotted with `⃝', Vehicle2 with `□', and Vehcile3 `×'. l₄ and l₅ are also shown in gures of the generated merging paths. To ensure there is no excessive accelerations, a1y

is also calculated and shown in the simulation results. The unit for all the speeds is m/s.

The unit for the accelerations is m/s². The unit for b, r, the positions and the distances is m. Three situations were simulated in the following subsections.

Reproduce of merging maneuver produced by human driver

In this section, the relative positions of the relevant vehicles in the actual merging scene were reproduced. According to the time history of this merging maneuver the desired speeds of the vehicles were set in such a way that the time history of the relevant vehicles would be similar to actual data. They were as follows: v_1d=16.7 m/s, v_2d=v_3d=21.0 m/s. In the simulation, the initial conditions were set to be the same with the helicopter-short merging as follows: x_1x=120.9 m, x_2x=110.9 m, x_3x=34.4 m, b=0 m, v_1x=12.1 m/s, v_2x=20.1 m/s, v_3x=18.0 m/s, v_b=0 m/s. The actual merging path of the helicopter-shot merging scene is shown in Fig 6.2.11. Fig. 6.2.12 shows the merging path generated with the proposed method. Fig. 6.2.13 shows the time history of the variables in the simulation and the actual time history of the relative distances between the relevant vehicles.

By comparing Fig. 6.2.11 and Fig. 6.2.12, it can be concluded that, with the above parameters, the proposed method can reproduce the actual relative positions of the relevant vehicles in the merging maneuver conducted by a human driver. The positional order of Vehicle1, Vehicle2 and Vehicle3 in the actual merging were the same with those in the simulation results. This means, in the simulation, the merging vehicle merged into the same gap as in the actual merging. In Fig 6.2.13 the thin lines are the simulation results. The thick lines in the diagram of the relative distances are actual data derived from analysis of the actual merging maneuver. It can be seen that the trends of the relative distances in the actual merging scene and that of the simulation results were almost the same. Fig. 6.2.13 shows that the accelerations of the relevant vehicles were all kept in their specied ranges during the merging respectively. Since in actual merging, drivers pay more attention to the relative distances rather than the desired speeds, the weights of the terms concerned with the desired speeds, ω_v1, ω_v2, and ω_v3, were set to be quite small. As a result, as shown in

Figure 6.2.11: Merging path of the helicopter-shot trac scenes.(The pictures are aected by the variation of the helicopter.)

Fig. 6.2.13, the three vehicles varied their speeds during merging, and their speeds did not tended to their desired speeds att=18s. It can be seen from Fig. 6.2.13 thatb varied a little during merging, but the amplitude was not very large. Since the gap between Vehicle2 and Vehicle3 was long enough, Vehicle1 did not have to modify its merging path very much. As a result it can be concluded from this case that, the proposed method can reproduce the actual merging maneuver.

The validation of the weights under a trivial setting

To investigate whether appropriate results can be generated in a trivial case without read-justment of weights, the simulation of this case was conducted. In order to make the merging

Figure 6.2.12: Generated merging path with the proposed method

results predictable, in this case the desired speeds were set as: v1d=v2d=v3d=16.7 m/s. To this end, the initial conditions were set as: x_1x=75.0 m, x_2x=150.0 m, x_3x=0 m, b=0 m, v_1x=16.7 m/s, v_2x=16.7m/s, v_3x=16.7 m/s, v_b=0 m/s. Since the initial speeds of the three vehicles were already the desired speeds, the length of the gap between Vehicle2 and Vehi-cle3 was 150.0 m which was long enough for a safe merging, and the relative positions of the three vehicles were very appropriate for merging, Vehicle1 could merge into the gap between Vehicle2 and Vehicle3 easily. What's more, it can be expected that the speeds of the relevant vehicles did not have to change very much. Fig. 6.2.14 and Fig. 6.2.15 show the simulation results. Fig. 6.2.14 shows the merging path of this case, and Fig. 6.2.15 shows the variation of variables in the simulation.

Fig. 6.2.14 shows that the merging conducted successfully, and Vehicle1 merged into

Figure 6.2.13: Time history of the variables in the simulation and actual relative distances

Figure 6.2.14: Generated merging path in the trivial setting.

the gap between Vehicle2 and Vehicle3. Fig. 6.2.15 shows that the speeds of the three vehicles did not change apparently during the merging maneuver. The specied ranges of the accelerations were kept. The relative distances between the vehicles were appropriate during merging. The merging was done as expected with the same weights with Section 3.1.

The collision avoidance function of the proposed method

To investigate the eectiveness of the proposed method and the weights when without control Vehicle1 and Vehicle3 would collide, I conducted the simulation of this case. If the initial motions of Vehicle1 and Vehicle3 in the X-axis direction were the same, since the path of Vehicle1 converges to the path of Vehicle3, without control the two vehicles would collide with each other. To this end, the desired speeds were set as: v_1d=v_2d=v_3d=16.7m/s, and the initial conditions were set as: x_1x=70.0m,x_2x=170.0m,x_3x=70.0m,b=0m,v_1x=16.7m/s, v_2x=16.7 m/s, v_3x=16.7 m/s, v_b=0 m/s. In actual situation, three possible results can be considered for the merging. 1) Vehicle1 merges into the gap in front of Vehicle2; 2) Vehicle1

Figure 6.2.15: Time histories of the variables in the trivial setting.

merges into the gap between Vehicle2 and Vehicle3; 3) Vehicle1 merges into the gap behind vehicle3. Fig. 6.2.16 and Fig. 6.2.17 show the simulation results. Fig. 6.2.16 is the merging path in the simulation and Fig. 6.2.17 shows the time histories of the variables.

Figure 6.2.16: Generated merging path in the collision case.

Fig. 6.2.16 shows that Vehicle1 merged into the gap behind Vehicle3. Fig. 6.2.17 shows that Vehicle1 decelerated, and as a result the relative distance between Vehicle1 and Vehicle3 increased. Vehicle3 accelerated cooperatively to increase the relative distance with Vehicle1 too. This clearly showed the cooperative function of the proposed method. On the contrary, the speed of Vehicle2 did not change apparently. This result is motivated by setting ω_a2 larger than ω_a3 and ω_a1. Since in actual merging the front vehicle seldom considers the motion of the merging vehicle, this result is reasonable. All the speeds tended to the desired speeds after 18 s. The accelerations are all kept in the specied ranges. The shortest relative distance between Vehicle1 and Vehicle3 was 2.4 m. But at that time, the two vehicles are running parallels, so the distance, 2.4 m, is considered to be appropriate. It can also be seen from Fig 6.2.17 that b varied apparently in this case. To show the inuence of the variation

Figure 6.2.17: Time histories of the variables in the collision case.

of b, the zoomed merging path is shown in Fig 6.2.18.

Figure 6.2.18: The zoomed merging path in the collision case.

Apparently, Vehicle1 departed from l₃ and move to the right side of l₃ in the available moving area. This clearly showed the path modication function of the proposed method.

As a result, the merging maneuver conducted successfully and Vehicle1 always moved in the available moving area during merging.