## 1 Introduction  
Spatial restrictions make it almost impossible to manipulate a big object in a narrow space by using big-scaled manipulators. For instance, it is impractical to carry an industrial manipulator into our house to move furniture by grasping and lifting. Owing to their small size and flexibility in motion, multiple mobile robots can be adopted to perform tasks in a narrow space [1]. These robots can move in a narrow environment to approach and manipulate objects, but these robots cannot grasp big objects as large-scale industrial manipulators. Therefore, non-prehensile methods [2, 3], which involves manipulation without grasping, is practical for such cases. For instance, a preferred way to manipulate a big object is to push it along the floor, or to pivot it with a vertex that makes contact with the floor. In certain cases, the object keeps contacting the environment in such manipulations and the restrictions on the object motion caused by the contacts complicates the kinematics in the manipulation. Furthermore, when multiple mobile robots perform a manipulation task, they themselves form a complex coordinated system [4]. Consequently, non-prehensile manipulations that use multiple mobile robots (as shown in Figure 1) require convoluted manipulation planning than standard manipulation, which comprehensively takes into account each robot's kinematics, the surrounding constraints, and their changes. Multiple robot motion planning is often faced with the high dimensional configuration space [5]. Typical planners for such problems are sample-based, such as RRT [6] and PRM [7]. However, manipulation planning problems often encounter particular _multi-modal_ structures [8], if the contacts among the robots, the manipulated objects, and the environment change during the process of manipulation. Here, a _mode_ refers to a certain set of configurations that hold the same constraints in motion (e.g., the object motion keeping a set of contacts with the environment). Possible configurations under the same constraints form sub-spaces in the configuration space. This requires the planner to be capable not only to probe the sub-spaces of each mode but also to cross among the different sub-spaces. For the application of typical sample-based methods on their original spaces, the expansiveness among the configurations of the different modes is more difficult to achieve than those of the same mode [9]. For instance, we can generate a transitable configuration of the system for a current configuration by sampling the configuration the belongs to the sub-space, while keeping the contact set among the robots, the objects, and the environment unchanged. However, in case the contact set changes, it would be necessary to check the connectivity between the different sub-spaces corresponding to the contact states, which would complicate the problem. Therefore, typical sample-based methods are usually applied to the modes' sub-spaces after splitting the configuration space into sub-spaces according to the modes. The sample-based methods become realizable by deciding the sequence of modes where connectivity is guaranteed beforehand. [PERSON] et al. [10] and [PERSON] et al. [11] sampled the manipulation states in a configuration space where a sequence of modes existed and the modes' transitions were prior defined. Some planners have been proposed for creating the modes' roadmap to guide the manipulation sequencing [8; 12; 13; 14]. In particular, [PERSON] [13] adopted a PRM-based planner [15; 16] to split the modes by comprehensively considering the multi-contacts between the object, robots, and the environment to obtain the necessary modes to pass through for a manipulation task. Mode transitions are mainly derived from the compliant transformations of contacts between the object, the environment, and the contacts of  
Figure 1: Example of non-prehensile transportation adopting multiple mobile robot: Two types of mobile robots move the refrigerator placed at the corner of a room.