# Grasping Assisting Algorithm in Tele-Operated Robotic Gripper
## 2 Grasp Quality Computation  
### General Grasp Formulation  
The interaction between a given object and the robotic gripper in \(\mathcal{R}^{3}\) space is analyzed through six-dimensional grasp wrench space. The object \(\mathcal{B}\)'s center of mass is locatedat \(\underline{r}_{B}\in\mathcal{R}^{3}\) in relation to the world frame coordinates and rotates around three different axes, defined by \(\underline{\theta}\in\mathcal{R}^{3}\), resulting in a total of six degrees of freedom. For our purposes we define a gripper reference frame with the origin at the gripper's wrist, and an object reference frame originating at the center of mass. These coordinate frames are noted by the indexes \(g\) and \(o\), respectively. We denote the contact points between the body and the gripper as \(\underline{p}_{i}\in\mathcal{S}\), where \(\mathcal{S}\) is the object's surface and is given in the object's coordinates. We could also define the contact points in relation to the gripper, as seen in Equation (1). The distance between the object origin and the gripper origin is donated by \(\overrightarrow{O_{g}O_{o}}\).  
\[\underline{p}_{g,i}=\overrightarrow{O_{g}O_{o}}+\underline{p}_{i} \tag{1}\]  
To calculate the grasp wrench space we need the contact forces between the gripper and the object. Using [PERSON]'s model of friction, our frictional force at the contact point is bounded by a cone centered around the normal to the contact point \(\underline{p}_{i}\). Thus, we define the _Friction Cone_ in Equation (2).  
\[FC_{i}=\left\{\begin{array}{ll}f_{n,i}&:\sqrt{f_{t,i}^{2}+f_{s,i}^{2}}\leq \mu_{s,i}f_{n,i}\lor f_{n,i}>0\\ f_{s,i}&:\sqrt{f_{t,i}^{2}+f_{s,i}^{2}}\leq\mu_{s,i}f_{n,i}\lor f_{n,i}>0\\ \end{array}\right\} \tag{2}\]  
in Equation (2), \(f_{n,i},f_{t,i}\) and \(f_{s,i}\) are the normal and tangential force components of the ith contact on the object, and \(\mu_{s,i}\) is the static friction coefficient. The normal to surface \(\mathcal{S}\) points inwards into \(\mathcal{B}\), so we demand that the normal component of the force would be definite and positive, meaning the finger pushes the object and cannot pull. Describing the forces inside the friction cone with an approximated linear combination results in the _Linearized Friction Cone_ described in Equation (3).  
\[LFC_{i}=\left\{\underline{F}_{i}:\underline{F}_{i}=\sum_{k=1}^{s}a_{ki}\hat{ f}_{ki},\forall a_{ki}\geq 0\right\} \tag{3}\]  
The vectors \(\hat{f}_{ki}\) represent the base vectors of the polytope, while \(\underline{F}_{i}\) is the total contact force at contact point \(\underline{p}_{i}\).  
Lastly, we define the wrenches exerted on the object in Equation (4).  
\[\underline{w}_{i}=\left(\begin{array}{c}\underline{F}_{i}\\ \frac{1}{\lambda}\underline{p}_{i}\times\underline{F}_{i}\\ \end{array}\right) \tag{4}\]  
The Wrench is a six-dimensional vector, containing information regarding both the force at a contact point on the object \(\mathcal{B}\) and the moment exerted by this force in relation to the center of mass. The parameter \(\lambda\) represents a relative length that converts the grasp wrench space to uniform units. Using the above _Linearized Friction Cone_, we define the primitive wrenches at point i, as described in Equation (5).  
\[\underline{w}_{i}=\left(\begin{array}{c}\underline{F}_{i}\\ \frac{1}{\lambda}\underline{p}_{i}\times\underline{F}_{i}\\ \end{array}\right)=\sum_{k=1}^{s}a_{ki}\Bigg{(}\begin{array}{c}\hat{f}_{ki} \\ \frac{1}{\lambda}\underline{p}_{i}\times\hat{f}_{ki}\\ \end{array}\Bigg{)}=\sum_{k=1}^{s}a_{ki}\widehat{w}_{ki} \tag{5}\]