4.3 Generation of ribbon shape

This section describes circular shape control of a ribbon based on the proposed method [13]. Figure 7 shows a strategy of ribbon shape generation. Using constant, high-speed motion described above and air drag, we achieve the circle shape generation as shown in Figure 7.

For this task, we extended the simplified model of the rope to a belt-like flexible object, taking into account the effects of drag and gravity [13].

Figure 7. Strategy of generation of ribbon shape.

Figure 8. Strategy of dynamic folding of cloth.

#### 4.4 Dynamic folding of a cloth

Next, we explain a dynamic folding of a cloth, as shown in Figure 8, which is an application of two-dimensional flexible object. In the initial state, two robot hands grasp a cloth. Then, based on the proposed method, we obtained the joint trajectories of the robot system in order to appropriately deform the cloth using the highspeed motion. Finally, the robot hands grasp the end of the cloth using the highspeed visual feedback.

In this task, we extended the simplified model of the rope to a sheet-like flexible object. We constructed a two-dimensional model [14].

#### 5. Rotational motion system

Here we describe the realization of additional tasks with a rotational motion system, including robotic rope insertion and pizza dough spinning. First, we briefly discuss analysis of the rotational motion system using high-speed rotational motion. Then, based on the results, we describe proposed strategies of rope insertion and pizza dough spinning using real-time visual feedback. The experimental results will be shown in Section 6.

#### 5.1 Discussion

In this section, we explain a simple theoretical analysis of the high-speed rotational motion system. Using high-speed rotational motion, the flexible characteristics of an object to be manipulated may be effectively decreased, allowing us to consider only the rigid characteristics of the object.

Considering the forces acting on a part of the object as shown in Figure 9, the following equations can be obtained:

$$\begin{cases} \ T\cos\theta = \Delta mr\omega^2, \\\ T\sin\theta = \Delta mg, \end{cases} \tag{14}$$

where T is the tension, Δm is the mass of a part of the object, r is the radius from the rotational center to the part, ω is the angular velocity, and θ is an approximate angle at the part, as shown in Figure 9. From these equations, we can get

$$
\tan \theta = \frac{\Delta mg}{\Delta m r \omega^2} = \frac{g}{r \omega^2}. \tag{15}
$$

Toward Dynamic Manipulation of Flexible Objects by High-Speed Robot System: From Static… DOI: http://dx.doi.org/10.5772/intechopen.82521

Figure 9. Model of rotational motion (side view).

From this result, we found that the angle θ decreases when the angular velocity ω increases. Thus, the deformation in the gravitational direction (z direction) can be ignored. In addition to this result, assuming that the elastic characteristics of the rope can be neglected, we can consider that the rotational system of the flexible object can be boiled down to the rotational system of a rigid body. This analysis result can be also applied to the dynamics of the flexible object in the radial direction.

## 5.2 Rope insertion

We briefly explain a rope insertion task as the first application [15]. Figure 10 shows a strategy of the rope insertion. The rope deformation is restricted to a linear shape by using high-speed rotation, and visual feedback positioning control, as shown in Figure 5, between the tip position of the rope and the position of a hole is performed using the high-speed robot system. At the time when the tip position and the hole position are the same, the rope insertion is achieved by using the motion of a linear actuator (maximum speed is 2.4 m/s).

#### 5.3 Pizza dough spinning

Next, we describe spinning of pizza dough [16]. Figure 11 illustrates a strategy of the pizza dough spinning. In this task, the pizza deformation is restricted to a

Figure 10. Strategy of rope insertion.

Figure 11. Strategy of pizza dough spinning.

planar shape by using the proposed method. Angular acceleration of the rotation was achieved by high-speed finger motion. The contact control was carried out by using high-speed visual feedback.
