**4. Soft-Collision M\* (SC-M\*)**

M\* assumes hard-collision constraint which does not apply to many real-world applications. Our contribution in this paper is to generalize M\* to soft-collision context where common resources are considered, and to introduce soft collisions as a generalized concept allows us to model different types

of collisions. In addition, we show the advantages and trade-offs of the proposed algorithm in this new scenario. The proposed SC-M\* extends M\* by changing the definition of a collision, so paths with hard collisions but with a level of dissatisfaction on resources below a given threshold are acceptable. In this section, we formulate the concept of soft collision on common resources, describe the generalized M\* (i.e., soft-collision M\*) for planning in the soft-collision context, and extend our approach to a more complex environment with multiple types of agents requesting multiple types of resources.
