# Direction-Based Hybrid Strategy Combining Pushing and Hitting for Fast Object Singulation
## 2 Motivation and Strategy for Singulation
#### Choice of the Hitting/Pushing Directions for Singulation  
In this subsection, we describe the strategy of hitting and pushing to obtain singulation. Especially, we focus on the direction and the position of hitting or pushing. In the case of pushing, once the direction/position of pushing is chosen, the manipulator can separate the object by moving with any sufficiently slow velocity along a sufficiently large distance (e.g., in this paper, about the radius of the disk). That is the choice of the velocity, but it is not too important for the pushing. However, for hitting, the velocity should be carefully chosen. If the velocity is too small, impact will not occur. In the meanwhile, if the velocity is too big, the object may move outside the workspace of the robot. In this section, we discuss the choice of the direction/position of the hitting, and in Section 4, we discuss how to choose the velocity for hitting using a simulation. As shown in Figure 1, the impulse can be resolved into a tangential component \(\hat{\mathbf{F}}_{f}\) and normal component \(\hat{\mathbf{F}}_{n}\). In this study, we assume negligible friction between colliding bodies. The tangential component of the impulse \(\hat{\mathbf{F}}_{f}\) is due to the friction between two bodies, and if we assume that the friction between the bodies is negligible, we only need to consider the normal component \(\hat{\mathbf{F}}_{n}\) of the impulse. The desired initial direction for pushing for any two circular contacting disks is along the common tangential line of the two disks as shown in Figure 2. This is called central pushing. This is desirable because it dissipates minimal energy compared to pushing a circular disk along its center, as shown by the gray arrows in Figure 3, which dissipates high energy. Alternatively, the initial direction for hitting has been selected on the basis of central impact which will maximize the impulse. It can be simply observed from Figure 3.  
Figure 1: Impact types: (**a**) central impact contact, (**b**) oblique impact contact, and (**c**) multiple simultaneous impact contact. Figure 2: Pushing direction for two contacting circular disks. that the directions of pushing and hitting are orthogonal to each other in the sense that the pushing is trying to minimize energy for the singulation of objects, while the hitting is trying to maximize energy for the singulation of objects. The yellow and gray arrows show the directions for pushing and the red arrow shows the direction for hitting. The pushing along the yellow arrow direction will need less energy because we are just pushing the manipulator from below. For the grey arrow, we may need to push either the green or blue disk for Case A all the way up, approximately the distance equal to the radius of the disk. As a result, it will consume more energy. Figure 3 shows multiple arrows for different cases and these arrows represent the direction of hitting and pushing with respect to the manipulator. For example, in Case C if the orange circular disk is closer to the manipulator, the red arrow shows the hitting direction. Similarly, the two yellow arrows show the pushing directions. In this case, the manipulator will always choose the pushing direction arrow nearer to the manipulator. The central pushing approach of two contacting circular disks reduces the possibility of collision with other disks in the clutter, since this approach moves the disks because it only singulates by creating the minimum separation required for the desired singulation and also reduces the possibility of going beyond the workspace. This approach also helps in separating the disks at the point of contact without dragging them along other disks. Given the two disks in contact, the pushing direction is along the common tangent of the two disks at the point of contact. These two disks in Figure 2 can be separated easily by moving the manipulator by the fixed distance, along the direction shown by the arrow. The distance depends on the radius of the disks.