2. Concept of dynamic manipulation of flexible object

The key difficulties in the dynamic manipulation of a flexible object include:

A. Deformation of the flexible object during manipulation

B. Prediction of its deformation

In order to solve these problems as easily as possible and to enable dynamic manipulation of flexible objects, we proposed an entirely new method. In the proposed method, the robot moves with a tip velocity that decreases the effect of undesirable deformation of the flexible object; as a result, the model and motion planning can be simplified. Figure 1 shows the basic concept of dynamic rope manipulation. The proposed method can be understood by picturing the manipulation of a rhythmic gymnastics ribbon, in which the ribbon deforms according to the tip motion.

From the above discussion, the rope deformation depends on the high-speed robot motion. In addition, we can assume that the rope deformation can be derived algebraically from the robot motion. Thus, the rope deformation model can be described as a simple deformation model derived from the robot motion. Also, since the rope deformation can be calculated algebraically from the robot motion in the model, the rope deformation model can be made more simple than typical models that use matrix differential equations as a multi-link model or partial differential equations as a continuous body model [9, 10].

Moreover, if the dynamic manipulation is performed in slow motion, gravity has a non-negligible effect on the rope. In that case, the algebraic equation does not hold, and we have to consider a differential equation, making dynamic

Figure 1. Basic concept. (a) Low-speed motion and (b) High-speed motion.

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

manipulation in slow motion extremely difficult as a result. Thus, high-speed robot motion is required in order to achieve dynamic manipulation.

Dynamic manipulation enables faster manipulation, high efficiency of task realization, and shorter task completion time and working time.

## 3. Dynamic manipulation method using high-speed robot motion

First, this section describes a theoretical analysis of a dynamical deformation model of flexible objects. We deal with a flexible rope as one example of such flexible objects. Then, we obtain a condition for dynamic manipulation based on the analysis. Next, we propose a simple deformation model that can be expressed by an algebraic equation. We also suggest a robot motion planning method using this simple model [11].

In the analysis, we assume that gravity is basically ignored, but we briefly describe the handling of gravity in Appendix-A.2, and we also assume that the flexible object does not exhibit elasticity.

#### 3.1 Theoretical analysis for dynamic manipulation

#### 3.1.1 Equation of motion of rope

Let us consider the equation of motion of the rope when the rope deformation is restricted to a given curve and a constraint force acts on the rope, as shown in Figure 2. The constraint force is R [N/m], the rope position is w [m], the position of the inside of the rope is σ [m], the line density of the rope is μ [kg/m], the tension is T [N], and the curvature is ρ [m]. Considering a length δσ [m] of the curve, the difference between the tensions in the tangential direction becomes δT [N]. Since the angle of the part δσ from the center of curvature is assumed to be δθ [rad], the equation δσ = ρδθ holds. Thus, the tension in the normal direction at both ends of this part can be described as:

$$(T + \delta T)\cos\frac{\delta\theta}{2} - T\cos\frac{\delta\theta}{2} \approx \delta T. \tag{1}$$

$$2T\sin\frac{\delta\theta}{2} \approx T\delta\theta = \frac{T\delta\sigma}{\rho}.\tag{2}$$

Figure 2. Rope mechanics.

Since the force R is the constraint force, the direction of the force is perpendicular to the curve. Then, the equations of motion in the tangential and normal directions of the part δσ become

$$\begin{cases} (\mu \delta \sigma) \ddot{w} = \delta T\\ (\mu \delta \sigma) \frac{\dot{w}^2}{\rho} = \frac{T \delta \sigma}{\rho} + R \delta \sigma \end{cases} \tag{3}$$

Eq. (3) leads to

$$\begin{cases} \begin{aligned} \mu \ddot{w} &= \frac{dT}{d\sigma} \\ \mu \frac{\dot{w}^2}{\rho} &= \frac{T}{\rho} + R \end{aligned} \end{cases} \tag{4}$$

Eq. (4) represents the rope dynamics under the condition that the rope behavior is restricted to the given curve. Next, we derive conditions for the robot motion in order to simplify the rope dynamics.

#### 3.1.2 Condition for restricting rope on reference trajectory

When the constraint force R is equal to zero, the rope can deform so as to track the given curve. Therefore, the condition that R = 0 be satisfied can be obtained from Eq. (4):

$$T = \mu \dot{w}^2. \tag{5}$$

This equation is a function of the rope velocity. Substituting Eq. (5) into Eq. (4) yields

$$
\mu \ddot{w} = \frac{d}{d\sigma} \left( \mu \dot{w}^2 \right). \tag{6}
$$

Since the part inside the brackets on the right-hand side is a function of the rope velocity, the right-hand size is equal to zero. This leads to

$$
\mu \ddot{w} = 0.\tag{7}
$$

As a result,

$$
\dot{w} = \text{const.}\tag{8}
$$

holds. In addition, the tension T can be obtained from Eq. (5):

$$T = \mu \dot{w}^2 = \text{const...} \tag{9}$$

And the tension T thus becomes constant. From this discussion, the 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 can be summarized as

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

$$
\dot{w} = \text{const.} \quad \text{and} \quad T = \mu \dot{w}^2 \tag{10}
$$

This result means that when the rope moves along the rope reference trajectory, the velocity in the tangential direction of the rope is constant and a uniform force is applied to each joint of the rope. On the contrary, if the condition that the velocity and tension of the rope be constant is satisfied, the rope can move along the reference trajectory of the rope. Manipulating the rope at a constant velocity can be achieved by moving the robot arm in the tangential direction at a constant velocity. It is impossible to control the tension to be constant in the case of the free end of the rope. However, assuming that the rope is sufficiently long and that the rope tracks the reference configuration, the condition that the tension be constant approximately holds.

As a result, the robot motion conditions necessary to simplify the rope model are as follows:


Thus, by manipulating the rope with this strategy, the rope can deform so as to track the robot motion. Since the robot is moved at a constant speed, each joint of the rope tracks the robot motion with a constant time delay. This time delay depends on the location of the joint.

In this analysis result, another solution (ρ = ∞) can also be obtained. The details are explained in Appendix-A.1.

#### 3.2 Simple deformation model

#### 3.2.1 Robot motion

First we consider the kinematics in order to derive the tip position of the robot arm. The joint angles and the tip position of the robot arm are defined by θ and r, respectively. In general, the relationship between the tip position and the joint angles can be obtained by the following equation:

$$r(t) = f(\theta(t)).\tag{11}$$

Although the details of the derivation are omitted, the tip position is derived by using the Denavit-Hartenberg description.

#### 3.2.2 Algebraic deformation model of rope

In general, a rope model is described by a distributed parameter system represented by partial differential equations. As another model, the rope is approximated by a multi-link system, and an equation of motion expressed by matrix differential equations is derived. In this research, we apply the multi-link system to the rope model. Then, the equation of motion can be replaced by an algebraic equation under the condition of constant, high-speed motion of the robot.

Based on the analysis described in the previous section, the following facts can be introduced in the rope deformation model:


The first assumption holds by ensuring constant, high-speed motion. This means that the rope deformation model can be described by the robot motion. The second assumption is that the rope does not have elasticity. Thus, the link distance in the multi-link model does not change.

From the above discussion, we propose a simplified rope deformation described by the following equation:

$$s\_i(t) = r(t) + \sum\_{i=1}^{N-1} l e\_i,\tag{12}$$

where t is time, i is the joint number of the rope (i = 0,1,���,N � 1), N is the number of particles in the multi-link rope model, si is the i-th joint coordinate of the rope, l is the distance between two joint coordinates (viz., the link length in the multi-link model), and ei is a unit vector that represents the direction from the (i � 1)-th joint to the i-th joint (i = 1,���,N � 1). In the case of i = 0, s0(t) = r(t) holds. This means that the location of the end point of the rope is the same as the tip position of the robot. Figure 3 shows an overview of the proposed simple model. As shown in Figure 3, the rope deforms so as to track the tip of the robot arm.

Since the proposed model does not include an inertia term, Coriolis and centrifugal force terms, or a spring term, we do not need to estimate the dynamic model parameters. The advantage of the proposed model is that the number of model parameters is lower than in typical models. Therefore, the proposed model itself is robust. Moreover, since the rope model can be algebraically calculated, the simulation time becomes much shorter.

Figure 3. Overview of simple model.

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