*5.3. Comparison of SC-M\* to Baselines*

We next compared the SC-M\* to other SC-based MAPP algorithms, including SC-A\* (optimal) and SC-CBS (suboptimal), in the one-resource-one-type environment.

#### 5.3.1. Path Cost

Firstly, we compared the path cost of the three algorithms. We designed 60 planning tasks for environments with 4–6 agents (20 tasks for each), in which agents will encounter at least one collision along the individually optimal paths under the *T* = 0.05, *δ* = 1 setting. We start with small agent numbers because SC-A\* cannot handle a large number of agents.

Figure 7 shows the average difference of the three SC-based solvers relative to the individually optimal cost (i.e., the sum of the optimal cost of each agent when the agent is the only one in the system). In other words, the Y-axis represents the cost of collisions. We observe that SC-A\* and SC-CBS have the lowest and highest additional cost, respectively. SC-M\* solutions cost more than SC-A\* but noticeably less than SC-CBS.

**Figure 7.** Average cost difference of soft-collision-based multi-agent path planning (SC-based MAPP) solvers from the individually optimal cost in the one-resource-one-type context.

To be more detailed, in the experiments, we designed MAPP tasks for environments containing 4–6 agents with 20 tasks for each. All tasks were designed to encounter at least one collision along the individually optimal paths under the above-mentioned configuration. Thus, additional costs relative to the individually optimal path are expected for each of the three SC-based MAPP solvers. Table 3 compares the results of SC-M\* and SC-CBS to the optimal solutions obtained by SC-A\*. The top half of the table shows the increase in cost relative to the cost for SC-A\*; the costs for SC-A\* for all scenarios vary within a small range so the results are in absolute numbers. The bottom half shows the ratio in run time with respect to SC-A\*; the run time for SC-A\* varies greatly across the experiments so we show the cost reduction as a percentage. In the table, we observe that the additional cost of SC-M\* from the SC-A\* is consistently lower than that of SC-CBS. We also observe that SC-M\* is significantly faster than SC-A\* and competitive relative to the run time of SC-CBS. The standard deviations show the fluctuations of the solutions for SC-M\* and SC-CBS around the optimal solutions for SC-A\*.


**Table 3.** Results of the path cost experiments.

The reason for the results is that SC-A\* is an optimal solver for this type of MAPP problem because it always explores cheaper paths in the entire multi-agent joint space before considering the paths that cost more [7]. SC-M\* is suboptimal because of the process discussed in Section 4.3.2. Compared to SC-M\*, SC-CBS suffers from more path cost due to the way it collects a collision: CBS collects collisions into a *conflict tree* and arranges the collision into the form [agent *j*, vertex *v*, step *s*], indicating that agent *j* collides at vertex *v* at step *s*. In each iteration, CBS conducts decoupled planning to avoid agent *j* reaching vertex *v* at step *s*. This might lose some information in the soft-collision context because there might exist another path that leads *j* to vertex *v* at step *s* without announcing a collision, by avoiding one of the upstream vertexes involved in soft interference. In contrast, SC-M\* can explore those paths excluded by SC-CBS because it searches the entire space of the immediate colliding agents. Figure 8 provides an example to visualize the difference in planning among the three SC-based MAPP solvers.

**Figure 8.** Illustration of the difference in planning among soft-collision A\* (SC-A\*), soft-collision M\* (SC-M\*), and soft-collision conflict-based search (SC-CBS).

Figure 8 shows a two-agent MAPP problem in the soft-collision context. Agents *r*1 and *r*2 attempt to move from vertexes *a* and *f* to vertexes *e* and *g*, respectively. The individually optimal paths (shortest distance) for both agents are *a* → *b* → *c* → *d* → *e* with distance 4 for *r*1 and *f* → *b* → *c* → *d* → *g* with distance 4 for *r*2, respectively. The total cost of the joint individually optimal path is 8. *r*1 and *r*2 softly collide on the edge *b* → *c* and *c* → *d*, where *r*2 can tolerate the dissatisfying experience with distance 2. However, *r*1 can only tolerate the dissatisfying experience with distance 1 and announces a collision at the vertex *d*.

When using SC-CBS, we record the collision that occurred to *r*1 as [*r*1, *d*, 3], indicating that agent *r*1 will collide at vertex *d* at the third step. Then, SC-CSB will avoid any paths leading *r*1 to *d* at Step 3 (including *a* → *b* → *x* → *d* → *e* and *a* → *b* → *c* → *d* → *e*) and will end up with a longer detour through vertex *y*. The SC-CBS solution has a cost of 5 for *r*1 and 9 in total.

When using SC-M\*, the collision at *d* triggers the sub-dimensional expansion of the search graph in dimension 1, which includes both *x* and *y*. Thus, it can find a cheaper collision-free path through *x* and end up with a path *a* → *b* → *x* → *d* → *e* with a dissatisfying experience of distance 1 and a cost of 4.5 for *r*1 (8.5 in total). However, SC-M\* does not expand dimension 2 because no collision has been announced by *r*2.

When using SC-A\*, the joint search space of both dimension 1 and dimension 2 is expanded and searched. Instead of vertexes *x* and *y*, SC-A\* will first investigate vertex *z* in dimension 2 according to some heuristics. This process leads to another cheaper path *f* → *b* → *z* → *d* → *g* with distance 4 for *r*2 (8 in total, which is the same as the individually optimal cost) and avoids all interference by moving through this path. As a result, SC-A\* returns an optimal solution that satisfies the soft-collision constraint at the expense of search space.

The example in Figure 8 illustrates the optimality of SC-A\* and the advantage of SC-M\* in path cost over SC-CBS. To be specific, SC-M\* provides a better solution than SC-CBS by searching thoroughly through the expanded dimensions, whereas the way SC-CBS identifies collisions is inappropriate in the soft-collision context. To the best of our knowledge, no other methodology capable of dealing with the soft-collision path planning defined in Equation (11) has been developed. It is expected that, in the future, more high-performance algorithms will be developed for solving the problem.

#### 5.3.2. Run Time

Table 4 shows the average run time of the three SC-based MAPP solvers and we observe that both SC-M\* and SC-CBS are significantly faster than SC-A\* in terms of run time. This is reasonable because SC-A\* always searches the global high-dimensional joint space, which is expensive. SC-CBS is faster than SC-M\* because it always searches in one individual dimension at a time, whereas the SC-M\* needs to occasionally deal with high-dimensional space when collisions occur.

**Table 4.** Average run time of SC-based MAPP solvers in the one-resource-one-type context.

