# Grasp Analysis for the Robot-Based Manipulation of Pre-Assembled Cables with Electrical Connectors
## 3 Methodology
### Contact Modeling and Grasp Force Estimation  
As a basis to estimate the required grasp force for the cable wrenches, a contact model must first be built for each of the different surface pairs. In this work, we build on the novel contact model and its related framework developed by [11], which was studied in Section 2.1, mainly due to the following reasons: First, to the best of our knowledge, it is the only contact model which can model NSCs for robotic grasping. This is required for plugs, since their exterior surfaces are commonly curved. Second, it provides the ability  
Figure 4: Visualization of sampling dynamic cable wrenches. A cable is deformed during the manipulation. This is indicated by the local curvature values of the cable during the manipulation, which are represented in color (red: high, blue: low). The varying deformations lead to different wrenches induced by the cable on the plugs at their interfaces, which are marked by black circles. The two gripper systems have to transmit these dynamic wrenches. Figure 3: Exemplary representation of an automatically generated assembly scenario in the software _IPS Cable Simulation_. The dashed arrows show the initialized trajectories along which the two plugs of one cable are manipulated by two robot systems. to evaluate whether a grasp can withstand an external disturbance. This is beneficial to determine the required grasp force for a cable wrench. The first three of the four steps described subsequently originate from the work of [11]. The description of these three steps should be regarded as a brief summary. For in-depth details, please refer to [11]. As input for this contact model, surface pairs from the CAD data of the plugs are generated. In Figure 5, an example of two different surface pairs for the plug of an EC is given. Moreover, the friction coefficient and pressure distribution for each contact surface have to be provided. The latter can be estimated based on common contact models, such as a uniform distribution or the Hertzian contact profile [32]. First, for each surface of each surface pair, the contact wrench is calculated, which is composed of a frictional wrench and a normal wrench. The latter is solely determined by the contact profile, but the former is also dependent on the relative motion between the gripper jaw and the workpiece. This relative motion is zero during a successful grasp. However, the higher an external disturbance is, the more likely a relative motion will occur. Therefore, so-called _unit twists_ are modeled, which represent instantaneous motions. A unit twist is then used to calculate the linear velocity and its direction between the workpiece and the gripper jaw at an associated contact point. This information is subsequently used to calculate the frictional wrench. The normal wrench is basically determined via a given pressure distribution [11]. Second, the unit twists are sampled to calculate all frictional wrenches which a contact can transmit. To these possible wrenches, a limit surface model is then fitted in the six-dimensional space of the six wrench components, which is introduced as the 6-D limit surface (6 DLS). As a 6 DLS, a 6-D ellipsoid is used since it outperforms the convex quartic one. This 6-D ellipsoid is built for each surface of each surface pair with their given friction coefficient and pressure distributions [11]. Third, these 6 DLS models are used to predict whether a grasp can withstand a given external disturbance. Therefore, the models are linearized for efficiency and sampled with vertices. They are then denormalized and overlaid with the associated normal wrenches. The contact wrenches generated so far in a local coordinate system are transformed into a common point of reference for all contact surfaces. Then, the GWS is calculated by computing the convex hull over the Minkowski sum of the contact wrenches. With the GWS given, it can be checked whether a given external disturbance can be withstood by the grasp.