# Grasping Assisting Algorithm in Tele-Operated Robotic Gripper
## 2 Grasp Quality Computation
### Grasp Quality Measure  
Many grasps can be used to reach force-closure on the body, and so we wish to find the best grasps. For this purpose, we define the grasp quality measure. The grasp quality measure is calculated based on the properties of the grasps, and it rates a specific grasp relatively to other grasps. Using the process described by [PERSON] and [PERSON] [11] and a similar concept in [29], we found the quality measures that best fit for our purposes. The connection between the robot motor torques and the end-point force (fingertip force) is described by the Jacobian matrix \(J_{i}\). Our gripper has several fingers, so we describe the connection between all the motor's torques and all end-effectors' forces with the hand Jacobian matrix \(J_{h}=diag\left(J_{1}\quad J_{2}\quad\ldots\quad J_{n}\right)\), and the total result is presented in Equation (12). \[\underline{T}=J_{h}^{\mathsf{T}}\underline{f}_{\mathsf{T}} \tag{12}\]  
The vector \(\underline{f}_{\mathsf{T}}\) is given by the xyz components of all the finger forces we described earlier, namely \(\underline{f}_{\mathsf{T}}=\left(f(n,1)\quad f(t,1)\quad f(s,1)\quad\ldots \quad f(s,n)\right)^{T}\) equally, and the connection between the motor velocities \(\underline{\phi}\) and end-effector velocity \(\underline{v}\) is given in Equation (13). \[\underline{v}=J_{h}\underline{\phi} \tag{13}\]  
The connection between the forces acting at the contact points between the gripper and the object, to the net wrench acting on the object, is given by the grasp matrix G presented in Equation (14). \[\underline{w}=G\underline{f}_{\mathsf{T}} \tag{14}\]  
Additionally, the connection between the fingers' velocities and the object velocity is given in Equation (15). \[\underline{v}=G^{T}\dot{\underline{x}} \tag{15}\]  
From these definitions we receive the connections in Equation (16). \[\left\{ \begin{array}{l}\underline{T}=J_{h}^{\mathsf{T}}G^{-1}\underline{w} \\ \dot{\underline{x}}=G^{-T}J_{h}\underline{\phi}\end{array} \right.\rightarrow \left\{ \begin{array}{l}\underline{T}=H^{\mathsf{T}}\underline{w}\\ \dot{\underline{x}}=H\underline{\phi}\end{array} \right. \tag{16}\]  
The matrix \(H\) is called the hand object Jacobian, and its dimensions are \(H\in\mathcal{R}^{d\times nm}\), where \(d=6\) is the dimension of the grasp wrench space, \(n\) is the number of contact points and \(m\) is the number of motors in each finger. Under the assumption that a boundary to the size of torques exists and is a function of the power \(P\), we write Equation (17). \[\|\underline{T}\|\leq f(P) \tag{17}\]  
Substitution of \(\underline{T}\) with \(H^{\mathsf{T}}\underline{w}\) results in limitation in the wrench space, as described in Equation (18). \[\underline{w}^{\mathsf{T}}\Big{(}HH^{\mathsf{T}}\Big{)}\underline{w}\leq f(P) \tag{18}\]  
The above transformation matrix transforms a sphere in \(nm\times nm\) torque space with radius \(r=f(P)\) to an ellipsoid in \(d\times d\) wrench space. The same process can be applied to object \(\mathcal{B}\) velocity space resulting in the connection described in Equation (19). \[\dot{\underline{x}}^{\mathsf{T}}\Big{(}HH^{\mathsf{T}}\Big{)}^{-1}\dot{ \underline{x}}\leq g(P) \tag{19}\]These two transformations have the same eigenvalues and same eigenvectors but in inverse proportion; when we apply the maximal wrenches on the body, we get the minimal velocity and vice versa. While the primitive wrenches are not bound by the ellipsoid we have obtained, and consequently our convex hull, _CH(GWS)_ might intersect the boundaries of the ellipsoid, the net wrench exerted on the body is bounded by the ellipsoid, and that is because we wish to maintain static equilibrium in the force-closure grasp. Following the above conclusions, we wish to define a grasp quality measure for two cases. The definitions from [1,2] will be used to define these types of grasps:  
* **Power grasp--the goal is to hold the object as firmly as possible, preventing movement, resisting the biggest external loads and exerting the maximal contact forces.