## 1 Introduction
the given robots; two robot contacts in the examples [13] were adopted to realize the subsequent mode transitions. However, when multiple mobile robots manipulated objects by using non-prehensile methods, mode transitions became more complex, because besides considering the mode transitions caused by the changes in the environmental contact, robot contacts also had to be considered. One way of addressing this complexity is by eliminating unfeasible modes in statics from the enormous combinations of contacts among them before considering the transitions among modes. An object should be under sufficient constraints in each mode; otherwise, the robots will fail to manipulate the object (e.g., the object drops down in an unexpected direction because of the lack of constraints). This means that the contacts from the environment and the robots should be able to form full constraints on the objects for non-prehensile manipulations. If we consider the environmental contact alone, the resultant constraints would vary in different contact states. For example, when an object-floor contact state changes from face-face contact into vertex-face contact, the constraints exerted by the floor would reduce. This results in various least requirements of robot constraints under different environmental constraints. Sometimes, we require the object to keep stationary contact with the environment, so that the robot can use friction to move the object (e.g., inclining a box); sometimes, we require the robots to manipulate the object to slide along the contacting part (e.g., sliding a box on a floor). To distinguish these manipulations, the environmental contact should be identified into the fixed contact and the sliding contact. The restraint placed on the object's degrees of freedom by a contact is different between the cases when it is fixed or sliding. Thus, we consider the necessary amount of robot for manipulations, both when relative sliding happened and did not happen. Therefore, a proper consideration of the individual robot's kinematics and the changes in the environmental constraints is essential when splitting the modes. Furthermore, a robot contact changes when the robot makes or breaks contact with the object either actively or passively (by the object's motion). For example, in Figure 1, a large robot with a manipulator actively makes contact with the object and pushes it over the small robots, and the small robots passively make contact with the object by the action of the large robot. The distinction between the active and passive action appears in many manipulation tasks, and the possible sequence of actions depends on this distinction of actions. Thus, the planner should be able to reason about all such possible mode transitions. By addressing the problems peculiar to non-prehensile manipulations using the mobile robots described above, we propose a method to generate the modes and transitions between the modes for contact planning. In our method, for a given set of contacts, we identified the modes by analyzing its constraints and least requirements to constrain the object's motion. We determine the mode transitions based on how the robot contact influenced the contact state of a targeted object. The mobile robots were divided into active robots and passive robots according to how their contacts changed in the state transition. Finally, the manipulation actions were determined by sequencing a series of modes. We applied the same concept to the limited cases also [17]. In this paper, we propose a more generalized framework to determine the action sequence including the distinction between the fixed and the sliding contact states. In the second section, we describe the problem statement, and in the third section, we introduce the generation of the contact state. In the fourth section, the state transition is investigated to sequence the action series. In the fifth section, we describe the applications of our methodology by conducting two simulations of the specified manipulation cases.