# Elastic Cube Actuator with Six Degrees of Freedom Output

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

**Figure 1.**Example applications for the cube dielectric elastomer actuator (DEA) with six degrees of freedom (6-DOF) output: (

**a**) lightweight haptic device; (

**b**) active vibration stabilisation of camera on a mobile quadrotor robot; (

**c**) 6-DOF manipulator (where grasping mechanism is driven by separate 1-DOF actuator inside cube DEA).

## 2. Principle of Operation

_{0}is the permittivity of free space (8.85 × 10

^{−12}F/m) and ε

_{r}is the relative dielectric constant of the elastomer. An unconstrained DEA membrane produces biaxial actuation strain, so in order to produce a usable output in a specific DOF it is necessary to introduce some form of constraint. This can be achieved by attaching a single membrane to a rigid ring, so that all active strain is directed inwards towards a rigid central inclusion (Figure 2a). Circumferential strain is largely constrained so applying an electric field results in a radial strain that pushes the central effector. By segmenting the electrodes, so that different regions of the membrane can be selectively actuated, 2-DOF actuation is produced by the single membrane (Figure 2b). The actuator concept presented here combines four of these DEA membranes into a cubic frame, with a central cross rod to couple the output of each membrane to an external load (Figure 2c). By independently controlling the response of each electrode quadrant, the cube DEA is capable of producing full 6-DOF actuation.

**Figure 2.**Schematic of a single DEA membrane with four electrodes (shaded) in: (

**a**) passive state; and (

**b**) active state with two electrodes actuated producing output displacement, δ. (

**c**) Cube DEA design with the output of four membranes coupled using a cross rod. The characteristic dimension, L, of the cube DEA is the distance between each pair of parallel membranes.

**Figure 3.**Translational (

**a**,

**b**,

**c**) and rotational (

**d**,

**e**,

**f**) displacement generated by selective activation of electrodes. Active quadrants are highlighted and passive quadrants are black. Note that actuation along and about the x and y axes requires two active DEA membranes (m = 2), while actuation along and about the z axis requires four active DEA membranes (m = 4).

## 3. Hyperelastic Electro-Mechanical Model

#### 3.1. General DEA Model

_{1}and λ

_{2}and thickness stretch, λ

_{3}. The elastomer is assumed to be incompressible (λ

_{1}λ

_{2}λ

_{3}= 1). The equations of state for an ideal dielectric elastomer are given as [16,17,18,24,25]:

_{1,2,3}are the true principal stresses, V is the applied voltage (V), T is the nominal membrane thickness (m) and W is the hyperelastic strain energy density. For the DEA material used in this work, 3 M VHB 4905, we assume ε

_{r}= 4.5 (based on the mean of ε

_{r}values in [3]).

_{3}= 0) produces Equations (5) and (6) describing general biaxial DEA behaviour:

#### 3.2. Cube DEA Model

_{r}, circumferential, λ

_{θ}and thickness, λ

_{t}, directions. Each DEA membrane quadrant, when active, will act to apply a radial force to the rigid central effector. Therefore it can be assumed that λ

_{r}will be greater than λ

_{θ}during actuation and hence the analytical model of the global membrane behaviour can be approximated by treating λ

_{θ}as a constant and λ

_{r}as a dependent variable. This approximation has been applied and experimentally validated in previous analytical models of circular membrane DEAs with rigid central inclusions [15,28], but the assumption of constant λ

_{θ}limits the applicability of the model to experimental parameters where localized increases of λ

_{θ}and hence E and P are negligible relative to λ

_{r}.

_{θ}, is given by (R – r) and the actuated length, l

_{θ}’, is given by (λ

_{r}/λ

_{r,pre})(R – r), where λ

_{r,pre}is the pre-stretch applied to the elastomer in the radial direction prior to its attachment to the rigid frame. Hence, λ

_{r}is described by considering the relation between l

_{θ}’ and δ and θ:

_{r,pre}. Using Equations (5) and (7), the radial stress, σ

_{r}, can be found for arbitrary values of θ, δ and V. A force component, dF’, acts on the central effector due to σ

_{r}. dF’ is found by multiplying σ

_{r}by area, which is equal to an infinitesimal section of the central effector, 2πr(dθ/2π), multiplied by the deformed thickness, T/λ

_{r}λ

_{θ}, of the elastomer:

**Figure 4.**(

**a**) Representation of change in elastomer length, from l

_{θ}to l

_{θ}’, along the first principal axis (radial) due to output displacement, δ. The elastomer is contained within a rigid boundary with radius R and contains a rigid inclusion with radius r. (

**b**) Electrodes B and D are activated to induce δ and the axis of actuation, and line of symmetry, is the line from θ = 0 to θ = π.

_{ext}is an externally applied load (with the same orientation as the actuation axis of interest). In Equation (9), it is assumed that two electrode quadrants per membrane are active as shown in Figure 2b. One quadrant can be neglected from consideration due to symmetry (the factor of two in each term in Equation (9) accounts for this), so that 0 < θ < Θ

_{1}represents the first passive region, Θ

_{1}< θ < Θ

_{2}represents the active electrode quadrant and Θ

_{2}< θ < π represents the remaining passive region. Hence each electrode quadrant is assumed to be equal to (Θ

_{2}– Θ

_{1}), which for the design presented in this work will be slightly less than 90° due to the passive gaps between quadrants. Note that if different electrode geometries are used or if non-equal voltages are applied to the active electrode constants, then the integration intervals in Equation (9) must be adapted accordingly and the assumption of symmetry about the plane of axis of actuation may no longer hold. For linear actuation along the x and y-axes, Equation (9) is numerically solved for a state of equilibrium with m = 2 (see Figure 3a,b), assuming single layers DEAs. Similarly, for linear actuation along the z-axis, Equation (9) is solved with m = 4 (see Figure 3c), assuming single layers DEAs. To predict rotational actuation, Equation (10) includes an additional design constant, the cube length halved, L/2, to convert membrane force into a moment:

_{ext}is an externally applied load (with the same orientation as the actuation rotation of interest). For rotational actuation about the x and y-axes, m = 2 (see Figure 3d,e), and about the z-axis, m = 4 (Figure 3f). It is assumed in Equations (9) and (10) that the cross rod is kinematically free to slide and rotate through the rings attached to each membrane, whereas in practice friction may contribute an additional force. Equations (9) and (10) were numerically solved in Matlab (Mathworks) to predict performance of the prototype cube DEA described in Section 4 (r = 5 mm, R = 25 mm, L = 30 mm, Θ

_{1}= 5°, Θ

_{2}= 85°, µ = 80 kPa, J = 90, λ

_{1,pre}= λ

_{2,pre}= 3).

## 4. Materials and Methods

_{r,pre}= λ

_{θ,pre}= 3) prior to being attached to the frame, which resulted in the nominal thickness, T, being reduced from 500 µm to a pre-stretched thickness, λ

_{3}T, of 55.6 µm. The maximum applied biaxial stretch before mechanical failure for VHB 4905/4910 has been experimentally determined to be λ

_{r,pre}= λ

_{θ,pre}= 36 [29,30]. Therefore, by selecting λ

_{r,pre}= λ

_{θ,pre}= 3, and considering that λ

_{r}≤ 2λ

_{r,pre}, the cube DEA should be fail-safe in its passive state.

^{2}fixed to one end of the connecting cross rod, with a sampling rate of 200 Hz. Force measurements were taken using a load cell (LMA-A-10N, Kyowa, Japan) aligned with the flat plastic marker and a signal amplifier (DPM-712B, Kyowa, Japan).

## 5. Results and Discussion

**Figure 5.**(

**a**) Cube DEA prototype with 50 mm diameter membranes capable of 6-DOF actuation. Activated membranes shown with single active electrode (

**b**); and with two active electrodes (

**c**).

**Figure 6.**(

**a**) Lateral displacement in x, y and z-axes; and (

**b**) rotational output about z-axis against voltage.

_{θ}. Any increase of λ

_{θ}will concomitantly increase the electric field and hence P, which will generate a greater blocking moment. Electromechanical membrane wrinkling and increased λ

_{2}is not captured in the current analytical model so is the likely sources of error. In summary the model has been shown to predict the general response but more computationally involved methods such as finite element modelling are required for detailed design.

Output | Scaling Relationship | |
---|---|---|

linear displacement, δ_{x,y,z} | δ_{x,y,z} ∝ 1/R | actuation stretch is dimensionless |

rotational displacement, φ_{x,y,z} | independent of independent of R | actuation stretch is dimensionless |

blocking force, F_{b} | F_{b} ∝ 1/R | Equation (8): dF’ ∝ r and r ∝ R |

blocking moment, M_{b} | M_{b} ∝ 1/R^{2} | Equation (8): dF’ ∝ r and r ∝ R Equation (10) dF’ ∝ L and L ∝ R |

## 6. Conclusion

## Acknowledgements

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Wang, P.; Conn, A.T. Elastic Cube Actuator with Six Degrees of Freedom Output. *Actuators* **2015**, *4*, 203-216.
https://doi.org/10.3390/act4030203

**AMA Style**

Wang P, Conn AT. Elastic Cube Actuator with Six Degrees of Freedom Output. *Actuators*. 2015; 4(3):203-216.
https://doi.org/10.3390/act4030203

**Chicago/Turabian Style**

Wang, Pengchuan, and Andrew T. Conn. 2015. "Elastic Cube Actuator with Six Degrees of Freedom Output" *Actuators* 4, no. 3: 203-216.
https://doi.org/10.3390/act4030203