A. Appendices

### A.1 Another solution in the theoretical analysis:

As another solution from Eq. (4), we can obtain the following condition so that R = 0 holds:

$$
\rho = \infty \tag{16}
$$

This is also a necessary and sufficient condition that the rope can move along the reference trajectory in the absence of gravity and with the constraint force R = 0. This means that the reference shape of the rope is a straight line, and therefore, it is trivial that the rope moves with straight-line motion.

Thus, we do not adopt this solution in this research.

## A.2 Case in which effects of gravity are considered

Here, we briefly explain a case in which the effects of gravity are considered in the simple deformation model and the motion planning [17].

In the calculation of the rope deformation, since the robot moves at high speed, the effects of gravity (G(t)) can be approximated as

$$G(t) = -\frac{1}{2}gt^2,\tag{17}$$

and then this term is added to z direction in Eq. (12).

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

Figure 23. Inverse problem with consideration of effects of gravity.

In the motion planning, compensation for the effects of gravity,

$$G(\Delta T - t) = \frac{1}{2}g\left(\Delta T - t\right)^2\tag{18}$$

is also considered, and this term is added to the given control points, as shown in Figure 23.

After that, the inverse kinematics are calculated in order to obtain the reference joint angles of the robot.
