Modeling and Control of 2D Grasping under Rolling Contact Constraints between Arbitrary Shapes: A Riemannian-Geometry Approach

Figure 6

Minimization of the squared norm $\parallel {\mathit{r}}_1-{\mathit{r}}_2{\parallel }^2$ over rolling motions is attained when the straight line $\overline{P_1{P}_2}$ connecting the two contact points becomes parallel to the vector $({\mathit{r}}_1-{\mathit{r}}_2);$ that is, $\overline{O_1{O}_2}$ becomes parallel to $\overline{P_1{P}_2}.$