# Singulation of Objects in Cluttered Environment Using Dynamic Estimation of Physical Properties
## 3 Impulse Modeling
### Collision Kinematics between Objects  
Consider two bodies with mass center velocities \(\mathbf{v}_{a}\), \(\mathbf{v}_{b}\) and angular velocities \(\mathbf{\omega}_{a}\), \(\mathbf{\omega}_{b}\) are approaching to each other as shown in Figure 4a. Where \((\mathbf{m}_{a},\mathbf{l}_{a})\) and \((\mathbf{m}_{b},\mathbf{l}_{b})\) denote the mass and mass moment of inertia of body 'A' and body 'B', respectively. During collision, impulsive force \(\hat{\mathbf{F}}_{ext}\) is experienced by both bodies with equal magnitude and in opposite direction. After collision, the mass center velocity of body 'A' can be expressed as follows:  
\[\mathbf{v}^{\prime}{}_{a}=\mathbf{v}_{a}+\hat{\mathbf{F}}_{ext}\mathbf{m}_{a}^{-1}, \tag{9}\]  
Similarly for body 'B':  
\[\mathbf{v}^{\prime}{}_{b}=\mathbf{v}_{b}-\hat{\mathbf{F}}_{ext}\mathbf{m}_{b}^{-1}, \tag{10}\]  
where \(\mathbf{v}^{\prime}{}_{a}\) and \(\mathbf{v}^{\prime}{}_{b}\) denote the post impact mass center velocities of body 'A' and body 'B', respectively. Figure 4: (a) Collision model of two translating and rotating bodies. (b) Object and manipulator collision model by considering friction. (c) Collision model among objects by considering friction. Similarly the post impact angular velocities can written as follows:  
\[\mathbf{\omega^{\prime}}_{a}=\mathbf{\omega}_{a}+I_{a}^{-1}(\mathbf{r}_{a}\times\hat{\mathbf{F}} _{ext}) \tag{11}\]  
and:  
\[\mathbf{\omega^{\prime}}_{b}=\mathbf{\omega}_{b}-I_{b}^{-1}(\mathbf{r}_{b}\times\hat{\mathbf{F} }_{ext}) \tag{12}\]  
for body 'A' and body 'B', respectively. The contact point velocities of both bodies is written as follows:  
\[\overline{\mathbf{v}}_{a}=\mathbf{v}_{a}+\mathbf{\omega}_{a}\times\mathbf{r}_{a}, \tag{13}\]  
and:  
\[\overline{\mathbf{v}}_{b}=\mathbf{v}_{b}+\mathbf{\omega}_{b}\times\mathbf{r}_{b}, \tag{14}\]  
where \(\mathbf{r}_{a}\), and \(\mathbf{r}_{b}\) denote the vectors directing from the center of the objects to the point of contact p.  
Finally, the Equations (9)-(14) are rearranged to find the velocity increments of both bodies in terms of external as follows:  
\[\Delta\mathbf{v}_{a}=\hat{\mathbf{F}}_{ext}m_{a}^{-1}+\Big{(}\mathbf{r}_{a}\times\hat\mathbf{{F}}_{ext}\Big{)}I_{a}^{-1}\times\mathbf{r}_{a}, \tag{15}\]  
and:  
\[\Delta\mathbf{v}_{b}=\hat{\mathbf{F}}_{ext}m_{b}^{-1}-\Big{(}\mathbf{r}_{b}\times\hat\mathbf{{F}}_{ext}\Big{)}I_{b}^{-1}\times\mathbf{r}_{b}, \tag{16}\]  
where \(\Delta\mathbf{v}_{a}=\overline{\mathbf{v}^{\prime}}_{a}-\overline{\mathbf{v}}_{a}\) and \(\Delta\mathbf{v}_{b}=\overline{\mathbf{v}^{\prime}}_{b}-\overline{\mathbf{v}}_{b}\). \(\overline{\mathbf{v}^{\prime}}_{a}\) and \(\overline{\mathbf{v}^{\prime}}_{b}\) denotes the post impact velocities at contact point of body 'A' and body 'B', respectively. Similar to Equation (8), the closed-form solution of normal impulse is obtained as follows:  
\[\hat{\mathbf{F}}_{ext}=\frac{-(1+e)(\overline{\mathbf{v}}_{a}-\overline{\mathbf{v}}_{b}) \cdot\mathbf{n}}{m_{a}^{-1}+m_{b}^{-1}+\Big{[}(\mathbf{r}_{a}\times\mathbf{n})I_{a}^{-1} \Big{]}\times\mathbf{r}_{a}+(\mathbf{r}_{b}\times\mathbf{n})I_{b}^{-1})\times\mathbf{r}_{b} \Big{]}\cdot\mathbf{n}}. \tag{17}\]  
In practical cases, for general shape objects, the impulsive force is resolved into two components: normal to the surface and tangential to the surface. For circular objects, if the friction coefficient \(\mu_{f}\) is negligible on the object surface, there will be only normal force directing toward the center of the objects. The Equations (8) and (17) denote this case. However, with consideration of the surface friction, an additional external impulse component will act along the tangential direction as shown in Figure 4b. For the case of Figure 4c, \(\hat{\mathbf{F}}_{\mathbf{n}}\) and \(\hat{\mathbf{F}}_{t}\) are normal and tangential component of impulse experienced by the object. \(\hat{\mathbf{F}}_{f}\) is the frictional impulse, which is associated with \(\mu_{f}\hat{\mathbf{F}}_{n}\).