## 3 Generation of Modes
### ECF-RCF Combination
Figure 5: Three types of principal contacts (PCs) that a robot cannot access regardless of the object’s configuration: (**a**) (face, face)\({}^{\text{e}}\); (**b**) (edge, face)\({}^{\text{e}}\); and (**c**) (vertex, face)\({}^{\text{e}}\). There are two cases when a contact resists an external force: the geometrical case and the frictional case (Figure 6a,b). The former is the force caused on the object not to penetrate into the environment or the robot. The latter is the frictional force, and there are also two types of frictional forces: static and dynamic. We ignore dynamic friction when we consider the wrench bases to span gravity closure to ensure manipulation, as explained in the section of the problem statement. For static friction on a point, the frictional cone, which is determined by the linear relationship between the geometrical force and the static frictional force, is usually considered when checking the conditions necessary for force closure. We dealt with static friction as decomposed basis independent of the geometrical force without considering the frictional cone (see Figure 7a) because our aim was to omit the modes that could not realize gravity closure regardless of the object's configuration. Therefore, the wrench space for the \(j\)th static or dynamic contact is expressed as  
\[W_{j}=\langle\mathbf{w}_{j1},\mathbf{w}_{j2},\mathbf{w}_{j3}\rangle,\quad W_{j}=\langle \mathbf{w}_{j1}\rangle, \tag{2}\]  
respectively, where \(\mathbf{w}_{j1}\) is the basis for the geometrical force, and \(\mathbf{w}_{j2}\) and \(\mathbf{w}_{j3}\) are the bases for the static frictional force. The geometric and static frictional forces applied on edge-edge-cross contact can also be represented in the same manner (see Figure 7b). To deal with edge-face contacts and face-face contacts, we consider them as two point contacts on the edge and three non-collinear point contacts on the face (such as the boundary points of the contacting area), as shown in Figure 7c,d. The wrench space spanned by the bases is calculated by summing up the bases for all PCs in the given CF as  
\[W=W_{1}\cup W_{2}\cup\ldots,\cup W_{n}, \tag{3}\]  
where \(n\) is the number of bases sets on the object defined in the above manner. Figure 6: Force applied on a point of contact. (**a**) Definition of force. (**b**) Type of forces applied on a contact when the contact is fixed. (**c**) Types of forces applied on a contact when the contact is sliding. Figure 7: Wrench bases of the force applied on the static contact in the case of (**a**) point–face contact, (**b**) edge–face contact, (**c**) face–face contact, and (**d**) edge–edge–cross contact. Then, the wrench spaces were separately determined for (a) the passive force applied by the contacts from the environment and the passive robots (denoted by \(W^{\rm ps}\)) and (b) the active force applied by gravity and the contacts from the active robots (denoted by \(W^{\rm ac}\)); see Figure 8. To achieve gravity closure, the wrench vectors consisting of the above forces must positively span \(\mathbb{R}^{6}\) as follows:  
\[\text{pos}(W^{\rm ps}\cup W^{\rm ac})=\mathbb{R}^{6}. \tag{4}\]  
The reason why we separate the space into \(W^{\rm ps}\) and \(W^{\rm ac}\) is that the force applied by the environmental contacts and the passive robots cannot realize the force closure by themselves, and the active robots or the gravitational force must act to cause internal force on the object. According to the Caratheodory Theorem, if vectors in a wrench set positively span \(\mathbb{R}^{6}\), the wrench set should contain at least seven vector frames in the six-dimensional space [24]. Thus, at least seven vector frames must exist in the wrench vector space consisting of forces applied by ECFs, RCFs, and gravity to achieve gravity closure. If the total dimension of the three kinds of wrench bases is less than seven, gravity closure would not be realized regardless of the configuration.