Preliminary study on inchworm mechanism using displacement-amplified electromagnetic actuators

Abstract

Displacement-amplified electromagnetic actuators (DAEAs) have been developed for efficiently using the electromagnetic attractive force by magnifying the motion of the change of a gap in which the electromagnetic attractive force is generated. This method enables us to use the large electromagnetic force with the gap being small for a larger stroke. This study applies DAEAs to an inchworm mechanism to realize motion on waving rails and precise positioning. A prototype was developed, and a fundamental experiment revealed the performance of the proposed inchworm mechanism.

Keywords

Electromagnetic actuator displacement amplification inchworm mechanism

1. Introduction

Actuators with fast and accurate motion are urgently required in fields such as manufacturing and inspection, especially for dealing with small objects. Piezoelectric actuators have been widely investigated for such applications; however, their industrial use is limited by problems such as a weakness to humidity owing to silver migration and poor durability under mechanical shock owing to the use of brittle ceramics [1]. By contrast, electromagnetic actuators [2,3] afford advantages such as ease of use and abundant environmental devices. In particular, displacement-amplified electromagnetic actuators (DAEA) [4–6] have been developed for efficiently using the electromagnetic attractive force by magnifying the motion of the change of a gap in which electromagnetic attractive force is generated; large force is used in a small gap between an electromagnetic core and an armature, and fast reciprocated motions are achieved via fast driving [7]. This study applies DAEAs to an inchworm mechanism, a representative positioning and locomotive mechanism for submillimeter to micro stroke actuators that typically uses piezoelectric elements [8–10]. Compared to inchworm mechanisms using piezoelectric elements, the proposed inchworm mechanism is expected to afford advantages in driving on curved rails because DAEAs have structural flexibility (especially in share direction) owing to the gap for generating the electromagnetic attractive force. This feature is expected to increase the allowance of the rail accuracy, and be a first step to overcome one of the possible challenges of inchworm mechanisms, which are popular applications of those called micro actuators, realizing wide positioning range, compact size, and precise resolution. First, a simple calculation method is provided for the compliance of the DAEA in the shear direction to applications that exploit the advantages of their compliance. The obtained equation is compared with the finite element method (FEM) to confirm its validity. A prototype is developed to test the effect of a wave-shaped rail on the driving performance by comparing the driving motions with three types of rails.

Fig. 1.

Schematics of structural flexibility.

Fig. 2.

Proposed inchworm mechanism.

2. Proposed inchworm mechanism and flexibility of DAEA

While this actuator can be driven fast, its structure is flexible owing to the gaps that realize motion by the electromagnetic force, as shown in Fig. 1. This characteristic could enable an inchworm to be driven by passive deformation by an external force from unstraight walls. Figure 2(a) shows a CAD image of the proposed inchworm mechanism and, Fig. 2(b) shows the driving principle with an example of a control sequence. This inchworm mechanism mainly consists of three diamond-shaped DAEAs; their driving principle is explained in [4,5]. The middle DAEA extends in the traveling direction (direction of rail in Fig. 2(b)), and the others extend vertically toward the rail walls. By controlling each actuator in order (or in reverse order) as shown in Fig. 2(b), the inchworm can move in both directions along the rail.

Fig. 3.

Simple linkage model for shear deformation.

2.1. Shear flexibility analysis for design

This subsection analyzes the shear compliance of the DAEA by using a simple model to easily design a flexible actuator for providing a method to design the shear flexibility. Figure 3 shows a simple model for the analysis. In this analysis, h is expected to be constant owing to the small deformation, and the amplification parts shows small enough deformation to not have a large effect on the overall deformation. In this simple model, an equivalent equation for the external shear force F _s is expressed as $\begin{eqnarray}\displaystyle F_{s}=\displaystyle \frac{4k_{r}{\theta}}{h} & & \displaystyle\end{eqnarray}$ (1) k _r represents the rotational spring coefficient of the hinges, and it is calculated as follows [4,11]. $\begin{eqnarray}\displaystyle k_{r} & = & \displaystyle \displaystyle \frac{Ewt^{3}}{12}\left[l_{h}-2r+\displaystyle \frac{2r}{(2r+t)(4r+t)^{3}}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \left.\times \left\{t(4r+t)(6r^{2}+4rt+t^{2})+6r(2r+t)^{2}\sqrt{t(4r+t)}\arctan \sqrt{1+\frac{4r}{t}}\right\}\right]^{-1},\end{eqnarray}$ (2) l _h, r, t, and w indicate the length, chamfer radius, thickness, and width of the hinges, respectively. E also shows Young’s modulus of the hinges. The representative variable of the shear deformation θ is calculated from Eq. (1). $\begin{eqnarray}\displaystyle {\theta}=\displaystyle \frac{F_{s}h}{4k_{r}}. & & \displaystyle\end{eqnarray}$ (3) The relationship between θ and the shear displacement x _s is indicated as follows: $\begin{eqnarray}\displaystyle x_{s}=h\tan ({\theta}). & & \displaystyle\end{eqnarray}$ (4) The relationship between xs and F _s can be obtained from Eqs (3) and (4). $\begin{eqnarray}\displaystyle F_{s}=\displaystyle \frac{4k_{r}}{h}\arctan \left(\frac{x_{s}}{h}\right). & & \displaystyle\end{eqnarray}$ (5) Using a first order of Taylor series expansion, Eq. (6) is simplified in small deformation as follows: $\begin{eqnarray}\displaystyle x_{s}=\displaystyle \frac{h^{2}}{4k_{r}}F_{s}. & & \displaystyle\end{eqnarray}$ (6)

2.2. FEM analysis

For verifying the result of the previous analysis, an FEM analysis is conducted based on a model of a prototype whose shear compliance is designed to be much larger than the vertical compliance for comparison with the obtained theoretical equation (Fig. 4(a)). The analysis is performed by an FEM function in Autodesk Inventor (Autodesk Inc.). In the FEM analysis, shear forces of 0, 0.2, 0.4, 0.6, 0.8, and 1.0 N are applied to the surface as shown in Fig. 3, and the displacement values are obtained in the shear direction corresponding to Fig. 3. The parameters for the theoretical calculation are as follows: l _h = 2.5 [mm], r = 0.5 [mm], t = 0.15 [mm], w = 10 [mm], h = 13.5 [mm], E = 200 [GPa]. Figure 4(b) shows the results. As shown in Fig. 4(b), the FEM plots well match the theoretically obtained line. These result show that the flexibility of DAEA in the shear direction can be designed easily using Eq. (6).

Fig. 4.

FEM analysis for verification of theoretical result.

3. Fundamental verification

3.1. Prototype device

Figure 5 shows a prototype of the proposed inchworm mechanism. The sear compliance was designed high enough compared to the compliance in the driving direction. The core parts and amplification parts of DAEAs were made of electrical steel (50H270, Nippon Steel & Sumitomo Metal) and stainless steel, respectively. For the core parts, 20 magnetic steel plates with 0.5-mm thickness were stacked, they were cut by wire electric discharge machining. Each DAEA contained four coils of 50 turns each. The gap is adjusted to 50 μm using sim plates, and the amplification ratio is designed to be about four by FEM of ANSYS.

Fig. 5.

Prototype of the proposed inchworm mechanism.

Fig. 6.

Schematic of rail for driving experiment.

3.2. Driving experiment

For a fundamental evaluation to verify the concept mentioned in the introduction, a driving experiment was conducted. Three types of rails are prepared for comparing the driving characteristics of the proposed mechanism: a parallel rail with plane surfaces (called wide rail), a parallel rail with plane surfaces whose width is 50 μm narrower than the previous one (called narrow rail), and a parallel rail with waving surfaces with amplitude of 50 μm whose width is the same as that of the first one (called waving rail). The wave length, rail width, and waving amplitude were defined as show in Fig. 6. The change of width and the amplitude of wavelength were set to the same value as the gap distance in order to give large enough change compared to the actuator mechanism. Every step in the sequence takes 20 ms. Figure 7 shows the experimental results. In Fig. 7, the results of the waving and wide rail show little difference, and the results of the narrow and wide rail show significant difference. This result implies that the proposed inchworm mechanism shows much lesser influence owing to the waving of the rail than owing to the change in the width of the rail.

Fig. 7.

Experimental result.

4. Conclusion

This study develops an inchworm mechanism using DAEAs and experimentally tests its potential for driving on waving rails owing to its flexibility. First, the flexibility of the DAEA was discussed theoretically based on a simple model. Then, the theoretical result was quantitatively compared with the FEM analysis, and they were found to show good agreement. A prototype device was developed according to the proposal, and an experiment was conducted to show the applicability of this prototype to a rail that waves at the micrometer scale. In this experiment, three types of rails are developed to verify the effect on the waving of a rail by comparing the effect of the change in rail width: a wide rail, a narrow rail, and a waving rail whose width is the same as that of the wide rail. The difference in driving speed between the wide and narrow rails was far larger than the difference between the wide and waving rails. The experimental results implied that the flexibility of the actuator is effective for the motion of inchworm mechanisms on waving rails. In a future study, the effects of rails of various shapes on the driving motion should be analyzed, optimization methods should be developed for further exploring the potential of the proposed mechanism, and further applications that exploit the flexibility of DAEAs should be developed.

Footnotes

Acknowledgements

This work was partially supported by Grant-in-Aid for Young Scientists (B) 16K18001, research-aid fund of the Mitsutoyo Association for Science and Technology.

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