Mechanical behaviors of jammable robotic structures; prediction and computation

Abstract

One of the most challenging issues to predict and measure mechanical behaviors of jammable robotic structures (when they apply for manipulating, pick-and-place, and also grasping operations) is their elastic modulus calculation for each unique considered vacuum pressure. In previous researches has been demonstrated that the elastic modulus of a trunk of a jamming robot is a fundamental function of the vacuum pressure, and must be obtained from experimental tests. Nonetheless, there was not a unique formulation, obtained from an experimental test to computation as well as prediction an accurate elastic modulus. Hence, this paper has been introduced simplified and beneficial formulation to predict the amounts of mechanical properties based on the vacuum pressure variations. The proposed procedure has been examined by performing an experimental test and results have shown the desired accuracy to predict the deflection-distance of the jammable segments. On the other hand, the necessity of identifying at least another organic material was felt because of introducing just one organic granular material (coarse coffee) as a suitable filler to employ in jamming devices at previous studies. Hence, four new organic-granular-materials were selected and tested. Consequently, a blend of Black-Pepper grains and Datura seeds has been presented as a usable filler. The blend has a lightweight body which produces the high-ratio of strength under external forces. It also provides a wide range of the stiffness which is very important for a jamming robot. Accordingly, it can be applied when high-strength, a wide range of stiffness variety, and also a lightweight jammable structure in soft robotics is required.

Appendix

1.1 Beam Bending theory

Traditional beam bending theory has been proposed for a composite beam to predict the internal stresses due to the loading of a beam-like jammable system. Figure 10a illustrates a common composite beam that is composed of two materials with Young’s Modulus E1 and E2 and subjected to a concentrated moment (M). To calculate the beam stresses, the location of the neutral axis (NA) is required. The labeled parameters in the position of the neutral axis where y = 0 have been depicted in Fig. 10b. It can determine by summing the axial forces of the beam cross-section as follow:

Fig. 10

Series of utilized parameters in composite beams

$$\mathop \sum \nolimits F_{x} = 0 = \mathop \smallint \nolimits \left( \sigma \right)_{x} dA = \int\limits_{{A_{1} }} {\left( \sigma \right)_{x,1} dA} + \int\limits_{{A_{2} }} {(\sigma )_{x,2} dA}$$

(17)

By substituting $\sigma_{x} = Ey/\rho$ into Eq. 17, where ρ is the local radius of curvature and is constant for a given cross-sectional area, it will be acquired:

$$0 = E_{1} \int\limits_{{A_{1} }} {ydA} + E_{2} \int\limits_{{A_{2} }} {ydA}$$

(18)

Therefore, the beam internal stresses resulted from the bending can be computed by summing the local moments:

$$\mathop \sum \nolimits M_{z} = 0 = M + \mathop \int \nolimits (\sigma )_{x} ydA = M + \int\limits_{{A_{1} }} {(\sigma )_{x,1} ydA} + \int\limits_{{A_{2} }} {(\sigma )_{x,2} ydA}$$

(19)

which can be rewritten as:

$$M = \frac{{E_{1} }}{\rho }\int\limits_{{A_{1} }} {y^{2} dA} + \frac{{E_{2} }}{\rho }\int\limits_{{A_{2} }} {y^{2} dA}$$

(20)

Because of each integral in Eq. 20 that is the area moment of inertia (I), for each respective cross-sectional area (A₁, A₂), Eq. 20 can be rearranged to become:

$$\rho = \frac{{E_{1} I_{1} + E_{2} I_{2} }}{M}$$

(21)

Consequently, the internal stress resulted of bending for each of materials is:

$$\sigma_{x,k} = {\raise0.7ex\hbox{${ - E_{k} y}$} \!\mathord{\left/ {\vphantom {{ - E_{k} y} \rho }}\right.\kern-0pt} \!\lower0.7ex\hbox{$\rho $}} = \frac{{ - MyE_{k} }}{{E_{1} I_{1} + E_{2} I_{2} }}$$

(22)

In Eq. 22, k = 1 and 2 are the numbers of two materials of a composite beam.