# Position-Space-Based Design of a Symmetric Spatial Translational Compliant Mechanism for Micro-/Nano-Manipulation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design of a Symmetric XYZ CPM

_{ms}-, Y

_{ms}-, and Z

_{ms}-axes, but its parasitic rotations and cross-axis coupling are relatively large, which is not desired. Therefore, a symmetric XYZ CPM, with improved motion characteristics, is designed using the position space concept, based on the following steps.

- (a)
- (b)
- Further decompose each of the AMs into two DTBCMs. Figure 3c illustrates that the AM-X is decomposed into two DTBCMs: AM-X-1 and AM-X-2; the AM-Y is decomposed into two DTBCMs: AM-Y-1 and AM-Y-2; the AM-Z is decomposed into two DTBCMs: AM-Z-1 and AM-Z-2.
- (c)
- Reconfigure the AM-X by translating the AM-X-1 (within its position space) and its adjacent BSs along the X
_{ms}-axis, as shown in Figure 3d, so that the MS is located at the intermediate position between the AM-X-1 and the AM-X-2. As can be seen, a RL-X is needed to link the AM-X-1 and the AM-X-2. - (d)
- Add redundant compliant modules, AM-X-1-R and AM-X-2-R, as shown in Figure 3e, so that the AM-X is a mirror-symmetric compliant module about the MS. As studied in Section 1, a redundant copy of a compliant module can be added at any one position within the position space of the compliant module. Therefore, the positions of the AM-X-1-R and the AM-X-2-R should be within the position spaces of the AM-X-1 and the AM-X-2, respectively.
- (e)
- Add a redundant PM, PM-X-R (Figure 3e), which is the reflection of the PM-X about the MS. In this case, the PM-X cannot be reconfigured to be symmetrical about the MS, so a redundant PM is added (the redundant PM is placed within the position space of the PM). By this step, the leg of the XYZ CPM associated with the X
_{ms}-axis translation is reconfigured. - (f)
- Reconfigure the other two legs of the XYZ CPM associated with the translations along the Y
_{ms}- and Z_{ms}-axes, following the same reconfiguration process of the leg associated with the translation along the X_{ms}-axis. The resulting design can be seen in Figure 3f. - (g)
- Re-design the BSs, as shown in Figure 3g.
- (h)

## 3. Nonlinear and Analytical Kinetostatic Modelling

#### 3.1. Pre-Considerations

_{ms}-, Y

_{ms}-, and Z

_{ms}-axes are analytically modelled and verified by nonlinear finite element analysis (FEA) simulations. The nonlinear analytical models of the XYZ CPM can be used to estimate the actuation forces, and to predict the relationships between the actuation forces and the geometrical parameters, before conducting FEA simulations and experimental tests. Because the parasitic rotations (of AMs and MS) and the parasitic translations (of AMs) are much smaller than the associated primary translations (of AMs and MS), they can be ignored reasonably during the following analytical derivations of primary forces/motions [23,31], and their analytical (closed-form) models are not considered in this paper. The parasitic motions are also not modelled in this paper, because they are much smaller than the primary motions.

^{2}, and moments are normalized by EI/L. Here, E is the Young’s modulus of the material and I is the second moment inertia of cross-section area of the uniform beam [23].

#### 3.2. Closed-Form Modelling

_{cm}-, Y

_{cm}-, and Z

_{cm}-axes are 0.5 times of the reaction forces produced by the deformation of the four-beam compliant module along its X

_{cm}-, Y

_{cm}-, and Z

_{cm}-axes, respectively. Therefore, the reaction forces produced by the deformation of the TBCM along its X

_{cm}-, Y

_{cm}-, and Z

_{cm}-axes can be obtained by using the results of the reaction forces of the TBCM derived in [18], which are shown in Equations (1)–(3), respectively.

_{cm-tx}, ξ

_{cm-ty}, and ξ

_{cm-tz}are the primary translational displacements of the TBCM. ζ

_{cm-tx}, ζ

_{cm-ty}, and ζ

_{cm-tz}are the reaction forces along the X

_{cm}-, Y

_{cm}-, and Z

_{cm}-axes, respectively, produced by the TBCM due to the deformation.

_{ms}-, Y

_{ms}-, and Z

_{ms}-axes are isotropic (O

_{ms}-X

_{ms}Y

_{ms}Z

_{ms}is the global coordinate system in this section). Therefore, only the primary translations along one of the three directions need to be studied. In this paper, the derivation of the force-displacement relationship, associated with only the translations along the X

_{ms}-axis, is detailed. Given any primary displacements, ξ

_{asy}and ξ

_{asz}, of the RL-Y and RL-Z, respectively, the XYZ CPM can be simplified to the model shown in Figure 5 if only the force-displacement relationship in X

_{ms}-axis is considered.

_{ms}-, Y

_{ms}-, and Z

_{ms}-axes be δ

_{x}, δ

_{y}, and δ

_{z}, respectively, which can be written as below

_{msx}, ξ

_{msy}and ξ

_{msz}are the primary translations of the MS along the X

_{ms}-, Y

_{ms}-, and Z

_{ms}-axes, respectively. The primary translations of the RL-X, the RL-Y, and the RL-Z are represented by ξ

_{asx}, ξ

_{asy}, and ξ

_{asz}, respectively. The model, as shown in Figure 5, contains 14 TBCMs in each axis, which are termed as TBCM-1 to TBCM-14. If all the parasitic rotations and parasitic translations of the symmetric XYZ CPM are ignored, the deformation displacements of each of the TBCMs can be obtained easily according to the primary translations and lost motions. Additionally, the reaction forces of the TBCMs can also be calculated based on Equations (1)–(3). Taking the TBCM-1 as an example, the TBCM-1 is linked to the RL-X, so the deformation displacements of the TBCM-1 can be derived from the motion displacements of the RL-X. If ignoring all the parasitic rotations and parasitic translations of the RL-X, the deformation displacements of the TBCM-1 equal to ξ

_{asx}, zero, and zero along the X

_{ms}-, Y

_{ms}-, and Z

_{ms}-axes, respectively. Therefore, the reaction force, ζ

_{a}, of the TBCM-1 along the X

_{ms}-axis can be obtained, as shown in Equation (5), by substituting the deformation displacements of the TBCM-1 into Equation (2) or (3). Note that when substituting the deformation displacements into Equation (2), ξ

_{cm-tx}, ξ

_{cm-ty}, and ξ

_{cm-tz}in Equation (2) equal to zero, ξ

_{asx}, and zero, respectively; when substituting the deformation displacements into Equation (3), ξ

_{cm-tx}, ξ

_{cm-ty}, and ξ

_{cm-tz}in Equation (3) equal to zero, zero, and ξ

_{asx}, respectively. Similarly, the reaction force of the TBCM-2, TBCM-3, TBCM-4, TBCM-11, TBCM-12, TBCM-13, or TBCM-14, to the RL-X along the X

_{ms}-axis, can also be obtained as shown in Equation (5). The reaction forces of the TBCM-5, TBCM-6, TBCM-7, TBCM-8, TBCM-9, and TBCM-10, to the MS along the X

_{ms}-axis, can be derived from Equations (6) to (11), respectively, which are represented as ζ

_{b}, ζ

_{c}, ζ

_{d}, ζ

_{e}, ζ

_{f}, and ζ

_{g}, respectively, as below.

_{ms}-axis should be balanced, so Equation (12) can be obtained [34]. When substituting Equations (6)–(11) into Equation (12), Equation (13) for the lost motion along the X

_{ms}-axis can be derived. Furthermore, the actuation force, f

_{x}, should be equal to the sum of the reaction forces of all the TBCMs except TBCM-5 and TBCM-6, along the X

_{ms}-axis. Therefore, the relationship between the actuation force f

_{x}and the primary translations of the MS can be obtained, as shown in Equation (14). Similarly, the force-displacement relationships associated with the actuation forces, f

_{y}and f

_{z}, can be derived, as shown in Equations (14)–(16). Note that the actuation forces, f

_{y}and f

_{z}, are applied on the RL-Y and RL-Z, respectively.

_{x}, f

_{y}, and f

_{z}, can be obtained when specific translational displacements of the MS (output motions), ξ

_{msx}, ξ

_{msy}, and ξ

_{msz}, are required. Furthermore, the primary translational displacements of the RLs (input motions), ξ

_{asx}, ξ

_{asy}, and ξ

_{asz}, can also be obtained according to Equation (4).

#### 3.3. Quantitative Analysis and Comparisons

_{ms}-axis actuation force, can be seen in Figure 6 under the following actuation displacement conditions: (a) ξ

_{asx}varies from −0.05 to +0.05, ξ

_{asy}= 0, and ξ

_{asz}= 0; (b) ξ

_{asx}varies from −0.05 to +0.05, ξ

_{asy}= 0.05, and ξ

_{asz}= 0; and (c) ξ

_{asx}varies from −0.05 to +0.05, ξ

_{asy}= 0.05, and ξ

_{asz}= 0.05. Each of the actuation displacements is added to the simulation model by pre-setting the displacement of the center point of the outside surface of the actuated rigid stage (the point is also the one that the actuation force is applied on, as shown in Figure 5). The directions of the actuation displacements are keep the same.

_{ms}-axis actuation force match very well, with less than 2.58% difference. The difference rises with the increase of the cross-axis input displacements.

_{ms}-axis actuation force is further analyzed, which can be seen in Figure 7. It can be seen that the actuation force, f

_{x}, increases with the increase of ξ

_{asy}and ξ

_{asz}. It can also be derived that the actuation stiffness increases along the X

_{ms}-axis with the increase of the primary translations along the other directions.

## 4. Fabrication and Experimental Tests

#### 4.1. Fabrication Consideration

_{ms}-, Y

_{ms}-, and Z

_{ms}-axes with desired motion characteristics. However, the MS of the symmetric XYZ CPM is located at the center of the whole structure, and the symmetric XYZ CPM cannot be fabricated monolithically. Therefore, a practical design of the symmetric XYZ CPM is figured out in this section, and a prototype of the practical design is also presented.

#### 4.2. Prototype Testing

_{ms}- and Y

_{ms}-axes are separately measured by two digital indicators. Each indicator has a resolution of 1 µm and a very low spring force of 0.4–0.7 N (Digimatic Indicators, Mitutoyo Corporation, Kawasaki, Japan). The input displacements along the X

_{ms}-, Y

_{ms}-, and Z

_{ms}-axes are actuated by three micrometers with a resolution of 1 µm. The gravity (approximately 1.36 N) of the mobile parts of the prototype can affect slightly the input forces, especially influencing the input force along the Z

_{ms}-axis direction. The gravity can also have very small influence on the output displacement along the Z

_{ms}-axis, but it cannot affect any of the input displacements because the input displacements are directly controlled by three micrometers. Similarly, the small spring forces (0.4–0.7 N) of the digital indicators have trivial influence on the output displacements along the X

_{ms}- and Y

_{ms}-axes, but no influence on the input displacements. Because the gravity and the indicators’ spring forces are very small, their effects on the output displacements are ignored in this paper. In addition, because the parasitic motions of the input stages are very tiny, as shown earlier due to the symmetric design, the coupling errors among the input stages/micrometers can also be ignored for this experimental test.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

DOF | degree(s) of freedom |

DOC | degree(s) of constraint |

FEA | finite element analysis |

CPM | compliant parallel mechanism |

MS | motion stage |

BS | base stage |

AM | actuated compliant module |

PM | passive compliant module |

RL | rigid link |

TBCM | two-beam compliant module |

DTBCM | double-two-beam compliant module |

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**Figure 1.**Position-space-based reconfiguration of a general one degree of freedom (1-DOF) translational compliant mechanism: (

**a**) original 1-DOF translational compliant mechanism; (

**b**) decomposition of the 1-DOF translational compliant mechanism; (

**c**) change of geometrical dimension; (

**d**) change of geometrical shape; (

**e**) changes of both geometrical dimension and geometrical shape; and (

**f**) addition of redundant compliant modules (MS: motion stage, BS: base stage, RL: rigid link, compliant modules are labelled by numbers (1) to (5)).

**Figure 2.**Position-space-based reconfiguration for generating a 1-DOF symmetric translational compliant mechanism: (

**a**) decomposition of the 1-DOF translational compliant mechanism; (

**b**) translations of compliant modules; (

**c**) rotations of compliant modules; (

**d**) further translations of compliant modules; (

**e**) symmetric 1-DOF translational compliant mechanism; and (

**f**) deformation of the 1-DOF translational compliant mechanism under an actuation force.

**Figure 3.**A symmetric XYZ compliant parallel mechanism (CPM) obtained via reconfiguring a non-symmetric XYZ CPM: (

**a**) the original non-symmetric XYZ CPM [29]; (

**b**) decomposition of the non-symmetric XYZ CPM; (

**c**) further decomposition of the actuated compliant modules (AMs) of the non-symmetric XYZ CPM; (

**d**) AM-X-1 translated to a new permitted position; (

**e**) addition of redundant compliant modules (over-constraints); (

**f**) reconfiguration of the legs associated with the translations along the Y

_{ms}- and Z

_{ms}-axes; (

**g**) BS design; (

**h**) resulting symmetric XYZ CPM; and (

**i**) symmetric XYZ CPM designed by traditional approach, i.e., directly adding redundant compliant modules.

**Figure 4.**Compliant modules: (

**a**) A two-beam compliant module (TBCM) and its coordinate system and (

**b**) a four-beam compliant module and its coordinate system.

**Figure 5.**Simplified spring model of the symmetric XYZ CPM with illustrative force actuation along the X-axis (RL-X is red in color, RL-Y is green in color, and RL-Z is blue in color).

**Figure 6.**Comparison between the analytical results and the finite element analysis (FEA) results in terms of the force-displacement relationship.

**Figure 8.**Comparison of lost motions between the symmetric and non-symmetric XYZ CPMs shown in Figure 3a,h (Symbols, ‘*’, ‘△’ and ‘○’, in this figure are data points).

**Figure 9.**Comparison of input parasitic translations between the symmetric and non-symmetric XYZ CPMs shown in Figure 3a,h (Symbols, ‘*’, ‘△’ and ‘○’, in this figure are data points).

**Figure 10.**Comparison of input parasitic rotations between the symmetric and non-symmetric XYZ CPMs shown in Figure 3a,h (Symbols, ‘*’, ‘△’ and ‘○’, in this figure are data points).

**Figure 11.**Comparison of output parasitic rotations between the symmetric and non-symmetric XYZ CPMs shown in Figure 3a,h (Symbols, ‘*’, ‘△’ and ‘○’, in this figure are data points).

**Figure 12.**Coupling comparison between the symmetric and non-symmetric XYZ CPMs shown in Figure 3a,h (Symbols, ‘*’, ‘△’ and ‘○’, in this figure are data points).

**Figure 13.**Main assembling components: (

**a**) rigid cube; (

**b**) rigid washer; (

**c**) compliant passive compliant modules (PM) beam; and (

**d**) compliant AM beam.

**Figure 14.**Assembling demonstration of the practical design: (

**a**–

**e**) assembling of rigid cubes, rigid washers, compliant PM beams, and compliant AM beams; (

**f**) assembling of three RLs; (

**g**) assembling of output platform; and (

**h**) assembling of supporting seat.

**Figure 16.**Relationship between the input displacement and the output displacement along the X

_{ms}-axis.

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## Share and Cite

**MDPI and ACS Style**

Li, H.; Hao, G.
Position-Space-Based Design of a Symmetric Spatial Translational Compliant Mechanism for Micro-/Nano-Manipulation. *Micromachines* **2018**, *9*, 189.
https://doi.org/10.3390/mi9040189

**AMA Style**

Li H, Hao G.
Position-Space-Based Design of a Symmetric Spatial Translational Compliant Mechanism for Micro-/Nano-Manipulation. *Micromachines*. 2018; 9(4):189.
https://doi.org/10.3390/mi9040189

**Chicago/Turabian Style**

Li, Haiyang, and Guangbo Hao.
2018. "Position-Space-Based Design of a Symmetric Spatial Translational Compliant Mechanism for Micro-/Nano-Manipulation" *Micromachines* 9, no. 4: 189.
https://doi.org/10.3390/mi9040189