## 3 Synthesis of the Initial Grasp Configuration  
In this section, we address the characterization of the object stability for a three-fingered grasp. This implies determining a force closure configuration based on the choice of three contact points from the set of all the points representing the outer 3D surface of the object. The following assumptions are considered in this procedure:  
* The use of three fingers for the grasp operation modeled as hemispheres with radius R.
* The first contacts between the fingers and the object are point contacts.
* The outer surface of the object is represented by a set of points \(\Omega\), which are described by their position vectors \(\overrightarrow{P_{i}}\) measured with respect to a reference frame located at the center of mass (\(\overrightarrow{P_{co}}\)) of the object.  
In fact, a 3-fingered grasp is more reliable in terms of stability, slip avoidance, and balance of forces when it converges towards an ideal equilateral grasp [17]. Thus, each three-fingered grasp can be characterized by a value which represents its similarity to an equilateral triangle. We suggest an algorithm based on geometric criteria to find this equilateral grasp. This algorithm determines, at first, the set of all possible grasping triangles by scanning the points belonging to the contact surface \(\Omega\). Then, using the \(Q_{1}\) criterion presented by [PERSON] [17], our algorithm compares the angle values (alpha, beta, and lambda) of these triangles with \(\pi/3\) (i.e., the angle that characterizes an equilateral triangle) in order to choose the configuration closest to an equilateral triangle (see Figure 2):  
\[Q_{1}=\frac{3}{2\pi}\left(\left|\alpha-\frac{\pi}{3}\right|+\left|\beta-\frac {\pi}{3}\right|+\left|\gamma-\frac{\pi}{3}\right|\right). \tag{1}\]  
For a target angle value of \(3/2\pi\), the minimum possible value of \(Q_{1}\) is 0 for a perfect equilateral triangle, 2 for the most unfavorable case (the triangle degenerates into a segment), and 1 for an intermediate state.  
For evaluating the angles of the triangle, we use a margin of error defined by the interval [0, 0.3]. Depending on the density of the points of the contact surface \(\Omega\), this algorithm can give several grasp configurations. Finally, a second criterion, referred to as \(Q_{2}\), is used in order to choose one between them. It measures the distance between the center of the mass of the object \(C_{O}\) and the center of the grasping triangle \(C_{T}\) (Figure 3), defined as  
\[Q_{2}=\left|C_{O}-C_{T}\right|. \tag{2}\]This criterion was developed to obtain stable grasps with respect to the torques generated by gravitational and inertial forces [17]. The complete grasp synthesis algorithm based on the these two criteria \(Q_{1}\) and \(Q_{2}\) is represented in the diagram \"Algorithm 1\":  
Grasp synthesis algorithm. Input: points \(P_{i}\) on contact surfaces \(\Omega\) with their 3D coordinates  
1. Calculation of the set \(\Gamma\) of grasp triangles defined from points \(P_{i}\).
2. Calculation of the value of the angles of the grasp triangles of the set \(\Gamma\).
3. Choice of the grasp triangles closer to an equilateral triangle by applying criterion \(Q_{1}\), the set of these triangles is named \(\Psi\).
4. Calculation of the barycenter \(C_{T}\) of the grasp triangles of the set \(\Psi\).
5. Calculation of the distance between \(C_{O}\) and the centers \(C_{T}\) of the triangles of the set \(\Psi\) by applying criterion \(Q_{2}\).
6. Choosing the triangle that minimizes \(Q_{2}\).  
Output: 3 initial contact points of the grasp configuration  
Figure 3: The distance between \(C_{O}\) and \(C_{T}\).  
Figure 2: Force arrangement criteria.