## 3 Generation of Modes
### ECF-RCF Combination  
With ECFs and RCFs created, the contact states of the manipulated object are generated by matching the RCFs with ECFs. In robotic manipulations, generally, force closure is required to achieve full constraints on the targeted object, which means the non-negative combination of primitive wrenches on an object can balance any external load [23]. In the case of non-prehensile manipulation, [PERSON] et al. [19] and [PERSON] et al. [20] proposed gravity closure, which is the force closure formed by robot contacts, environmental contacts, and gravity. External loads applied on an object can be described by wrenches, and they expand the wrench vector space [24]. If sufficient wrench vector  
Figure 4: Examples of cases where (**a**) ECF is empty and (**b**) RCF is empty. bases are provided by the gravitational force and the contacts placed on an object, gravity closure can be achieved. However, the wrench vector bases given by contacts depend on the location and direction of the contacts determined by the configurations of the object and the robots, whereas these configurations are not concerned in the phase to generate the modes and their transitions. Therefore, we combine an ECF and an RCF if the possible dimension of the wrench space spanned by the ECF and the RCF and the gravity satisfies the condition necessary to realize gravity closure, through which impossible modes are omitted from the viewpoint of statics. Before calculating the dimensions of the wrench spaces spanned by the ECF and by the RCF for determining their combinations, we eliminated the infeasible combinations for a given ECF by checking whether the object's geometrical primitives were accessible to robots under any of the object's configuration. In this contact planner, we considered only the case when an object's geometrical primitives make absolute contact with the geometrical primitive of the environment; this situation would obviously disable a robot from accessing them. Therefore, for the PCs in ECF, a surface under the PC (face, face)\({}^{\text{e}}\), an edge under the PC (edge, face)\({}^{\text{e}}\), and a vertex under the PC (vertex, face)\({}^{\text{e}}\) are not accessible to robots, as shown in Figure 5. The ECF that contains a certain geometrical primitive of the object that contacts the surface of the environment will not be considered to combine with an RCF that contains the same geometrical primitive of the object that contacts with robots. With the geometrical feasibility validated, the ECF and RCF individuals are combined according to the condition for gravity closure. Here, we considered the possible dimensions of two kind of wrench vector spaces correspondingly spanned by two kinds of forces: the passive and active forces. The passive force is applied by contacts with the environment and passive robots, and the active force is applied by contacts with the active robots and gravity. The active and passive forces were derived for the given ECFs and RCFs, respectively. Based on this, we combined the ECF and RCF, by taking into account only the derived maximum dimension of the wrench vector bases of the passive and active forces to satisfy the necessary condition of gravity closure. Configurations of the objects and the robots that meet the gravity closure will be decided in our future studies after we decide the mode transitions. When force is applied on a contact point, the resultant wrench on the object is expressed as  
\[\mathbf{w}=\left[ \begin{array}{c}\mathbf{f}\\ \mathbf{p}\times\mathbf{f}\end{array} \right], \tag{1}\]  
where \(\mathbf{f}\) is the force applied to the contact, and \(\mathbf{p}\) is the position of contact with respect to the object coordinate frame, as shown in Figure 6a. Certain manipulation studies have taken into account the case when a point contact also can resist a torque, which is called as a soft finger contact. However, we have not considered this type of contact in this paper.