This section establishes a finite element model (FEM) for the arm group mechanism to investigate the influence of group configuration, target size and velocity on the success rate of capture operation. The following assumptions are used in the model.
A single continuum arm can be simplified to an elastic rod whose bottom end fixed to the drive box while the tip end considered as a free end.
The material properties of the arm body are uniform, ignoring the internal structure of the arm.
The finite element model is established in the ABAQUS/CAE integrated environment. As mentioned above, the continuum arm is considered to be made of single material, and the equivalent elastic modulus is determined by the experiments. A DS250N material Tester is used to impose a transverse load on the free end of the arm to obtain the equivalent elastic modulus [28], according to
$$\omega = \int \int \left(\frac{M}{EI}{\text{d}}{{x}}\right){\text{d}}x+ \text{ A}x+ \text{B},$$
(1)
$$\theta = \frac{{\text{d}}\omega}{{\text{d}}{{x}}} = \int \frac{{M}}{{{EI}}}{\text{d}}x+ \text{A},$$
(2)
where \({\omega}\) is the deflection of the arm, representing the displacement of the arm along the direction of the transverse load, \(\theta\) is the angle between the tangent line of the flexure and the vertical position of the arm. A and B are the integration constants whose values are determined by the boundary conditions. In the case of a cantilever beam, we assume A = B = 0. M is the bending moment generated by the transverse load applied to the arm, which can be computed as the product of the transverse load and the length of the arm. E is the equivalent elastic modulus of the arm. I is the moment of inertia of the cross section of the arm and can be computed as \({{I}}=\frac{\pi{{D}}^{4}}{64}\), in which D is the diameter of the cross section. Through Eqs. (1) and (2), the elastic modulus of the arm can be estimated as \({E_1}=3.5\times {10^2}\text{ MPa}\).
The continuum arm has a flexible body and its cross section is circular, so we can estimate the Poisson’s ratio of the continuum arm according to the method of lateral compression test [29]. The obtained Poisson’s ratio is \({\mu_1}=0.47\).
The element type of the continuum arm is set as C3D8R in the finite element model. The size of the element is set as 3 mm, and the number of the elements is 10440. Also, friction is considered due to the surface contact between the continuum arms and the target. In our case, the materials used for the silicone sleeve and the target body are ELASTOSIL® 4601 and Somos Imagine 8000 photosensitive resin, respectively. Thus, the static friction factor is obtained as \({f_{\mu}}=0.3\) by performing sliding movement experiments between these two materials.
The material of the target ball is nylon whose modulus of elasticity is \({E_2}=8.3 \times {10^3}\text{ MPa}\) and Poisson’s ratio is \({\mu_{2}}=0.28\) [30].
The output force of the linear stepper motor 2 can reach 330 N which is considered to be evenly applied to nine driving wires 2. Therefore, the tension of each driving wire 2 is set to 36 N in the finite element model.
When t = 0 s, a target ball with different diameters, d = 30, 40, 50, 60 and 70 mm, were launched to the center of the arm group with different initial speeds, v =1, 2, 3, 4 and 5 m/s, as shown in Figure 5(a). If the ball can be stopped and grasped by the arm group eventually, Figure 5(b), we record it as a successful capture.
Here, we define a concept of average distribution radius, \({{R}}_{\text{a}}=\frac{\sum_{{i}}^{{n}}{{{R}}}_{{i}}}{{n}}\) to describe the density of the arm group, where \({{R}}_{{i}}\) (i=1, 2,…, 9) is the distance between the arm center to the group center. A large value of \({{R}}_{\text{a}}\) means a low density of the arm group, and vice versa. The clearance of the arms, C, is defined as the linear distance of adjacent arms on the same distribution circle, as shown in Figures 6, 7 and 8.
We start the tests with a simple configuration that all arms are arranged in a circle, as shown in Figure 6. The average distribution radius in Figure 6(a) is \({{R}}_{\text{a}}= 90 \,\text{mm}\) while that in Figure 6(b) is \({{R}}_{\text{a}} = 60\text{ mm}\). The diameter and velocity of the target ball resulting in successful captures are denoted by squares on a diametervelocity plane. In Figure 6(a), the successful captures are mostly located from d = 50‒70 mm, around the clearance of the arms \(C = 61.6\, \text{mm}\). In Figure 6(b), the range of d is between 30 mm and 60 mm, which is also near the clearance \(C = 41.0\, \text{mm}\). These indicate that the success rate can be increased by making the clearance of the arms close to the diameter of the target ball. The velocity range in Figure 6(a) between 1 m/s and 4 m/s is lower than that in Figure 6(b) between 3 m/s and 5 m/s. This is because the arms in Figure 6(a) has lower distribution density so it cannot provide sufficient resistance to stop the target ball with high velocity.
In addition, the continuum arms can be divided into two sets and arranged on two concentric circles with different radius, \({{R}}_{1}\) and \({{R}}_{2}\), as shown in Figure 7. Here, we set \({{R}}_{1} = 30 \text{ mm}\) and \({{R}}_{2} = 60 \,\text{mm}\). We define the incident angle \(\theta\) of the target ball as the sharp angle between its velocity direction and the x axis. Three incident angles are then tested for this configuration. In Figure 7(a), if the incident angle is 0°, the target ball will firstly hit the leftmost arm. In Figure 7(b), the incident angle is 40°, and the target ball will contact with three continuum arms at the beginning of collision. When the incident angle is 90°, the target body can contact with 5 arms at the beginning of collision, Figure 7(c). As a result, there are 7, 10 and 13 successful capture points in Figure 7(a), (b) and (c), respectively. This is because the incident angle in Figure 7(c) can make more arms to contact with the target when the collision occurs.
Based on the arm group configuration shown in Figure 7(c), we reduce the average distribution radius of the arms from 50 mm to 40 mm to carry out the simulation. The results are plotted in Figure 8. Compared with Figure 7(c) where the successful capture points are located in the area of \(d \in [40\, \text{mm}, 70\,\text{mm}]\) and \(v \in [1\, \text{m/s}, 4\, \text{m/s}]\), the successful capture points in Figure 8 moves to a new area where the target ball has lower diameter d \(\in\)[30 mm, 60 mm], but higher velocity \(v\in\)[2 m/s, 5 m/s]. These phenomena are consistent with the conclusions obtained in Figure 6.
From the above analysis, we can summarize the following three conclusions:

1)
The success rate of the capture can be improved by setting an appropriate average distribution radius, making the clearance of the arms close to the diameter of the target ball.

2)
The incident angle of the target ball has an impact on the capture performance. The incident angle which makes more arms to contact with the target ball can help to absorb the kinetic energy and improve the success rate.

3)
The density of the arm group also affects the capture performance. Increasing the density of the arm group can improve the capture ability of the target body with high velocity, and vice versa.
According to the simulated results, we implement the configuration shown in Figure 8 to our prototype to achieve high success rate because the dynamic target used in the following experiments are around 42 mm in diameter and 4 m/s in velocity.