# A Decoupled 6-DOF Compliant Parallel Mechanism with Optimized Dynamic Characteristics Using Cellular Structure

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

_{Z}) CPMs [9,10,11,12,13] and three-DOF out-of-plane-motion (θ

_{X}-θ

_{Y}-Z) CPMs [14,15,16,17,18] because it offers stable structure and performs lower actuating stiffness as compared to the four-legged counterpart, which is often used to design two-DOF planar-motion (X-Y) CPMs [4,5,6]. Furthermore, the three-legged configuration has also been employed to create six-DOF CPM [19,20,21,22,23], a complicated design due to the challenges in synthesizing six individual motions based on parallel spring-systems. For six-DOF positioning stages, the motion decoupling capability, demonstrated by the elimination of parasitic motions [24], is always desired in order to produce highly accurate output motions. Referring to previous works [19,20,21,22,23], the motion decoupling of CPMs has not been clearly discussed even though it is important. Another drawback that could limit the application range of existing six-DOF CPMs is the small workspace, i.e., <1 mm for translations and <1° for rotations. In addition, the dynamic behaviors, which are essential for precise motion systems, have not been considered so that existing designs are unable to achieve targeted dynamic responses.

## 2. Synthesis of the Six-DOF CPM

^{2}. As mentioned above, the dimension along the Z axis is eliminated since the center curves of beams are located in the XY plane to ensure the motion decoupling capability of the entire CPM.

**K**

^{e}and

**M**

^{e}be the [12 × 12] stiffness and mass matrices of each beam element respectively,

**K**

^{s}and

**M**

^{s}be the [s × s] stiffness and mass matrices of the entire CPM respectively, the formulations of

**K**

^{s}and

**M**

^{s}are expressed as

**K**, represents the stiffness characteristics in the six DOF of the CPM can be obtained by the condensation of

**K**

^{s}. Details of the matrix condensation are described in [26].

_{i}is the displacement at the center of the end effector caused by the load P

_{i}(i = 1, 2, …, 6 represent the six DOF Δ

_{X}, Δ

_{Y}, Δ

_{Z}, θ

_{X}, θ

_{Y}, and θ

_{Z}, respectively); the work, W

_{i}, done by P

_{i}can be addressed as

_{ii}denotes the ith diagonal component in

**K**. As the beam-based optimization method is developed based on finite element method and the CPM is constructed by a number of beam elements, the expression of K

_{ii}is in complicated numerical form with the design variables of the stiffness optimization included and not practical to be fully expressed here.

_{i}, κ is a constant factor and can be eliminated from the objective function. As a result, Equation (4) can be simplified as

_{ii}that satisfy Equation (5) are obtained. The optimal geometry of two reflecting curved beams in a leg is illustrated in Figure 3a. Note that the optimized orientation of the curved beams is 90°, which means that the beams are perpendicular to the XY plane as shown in Figure 3b.

**ω**and

**F**

^{s}are the bandwidth and natural frequency vectors of the CPM respectively; F

_{i}, the ith component in

**F**

^{s}, represents the natural frequency of the ith vibration mode of the CPM.

_{1}, of the CPM to the desired value, F

_{d}= 100 Hz, and maximize the difference between two neighbor modes. In addition, the high flexibility of the entire CPM is maintained by the second equation.

^{2}and 1.20 × 12.44 mm

^{2}, respectively, the size of the end effector is defined by the distance of 35.02 mm from its center to the moving end of each leg. Six primary resonance frequencies of the CPM, which are the first six components in

**F**

^{s}, are represented by the frequency vector,

**F**. The stiffness characteristic of the CPM is represented by the compliance matrix,

**C**, which is the inverse of the stiffness matrix (

**K**). The obtained results of

**C**and

**F**are given in Equations (8) and (9), respectively.

^{3}and yield stress of 950 MPa. The obtained results can be verified by substituting the optimized geometrical parameters of the curved beams shown in Figure 3a and the optimized values of design variables given above into the analytical approach of the beam-based method, which is presented in [25].

**F**is the resonance frequency of the translation along the Z axis, the second and third modes (148 Hz) are the translation along the X and Y axes, respectively, the resonance frequency of the rotation about the Z axis is 169 Hz and the last two modes shape (340 Hz) are the rotations about the X and Y axes. The elastic deformation of the CPM in each direction is demonstrated in Figure 4. The simulation results via ANSYS show that the optimized CPM can produce a large workspace of ±3 mm × ±3 mm × ±6.5 mm × ±6° × ±6° × ±7.5°.

## 3. Improvement of Dynamic Property by Employing Cellular Structure

^{2}respectively. Note that each beam has a square cross section. The relationship between a and b is given as

_{cellular}, is expressed as

_{R}, representing the ratio between the volumes of cellular structure (V

_{cellular}) and solid structure (V

_{solid}) is

_{R}and Δ. It is observed that the volume of cellular structure is equal to zero when Δ = 0 and equal to a solid structure when Δ = 0.5. This represents the CPM can achieve faster dynamic response with the smaller mass/volume of the end effector. However, V

_{R}cannot be too small since it makes the cellular structure weaker and as a result, the end effector can be easily deformed and buckled under external loads. Many simulations have been done and the results suggest that V

_{R}= 0.2 (corresponding to Δ = 0.144) is small enough to maintain the rigidity of the end effector and prevent the buckling failure while reducing its mass up to 80%. The CPM is reanalyzed with the cellular end-effector and the updated frequency vector,

**F**

_{cellular}, is shown in Equation (13). Comparing Equation (13) to Equation (9), it is observed that the first resonance frequency has increased 33% and the total difference between neighbor modes is also improved.

## 4. Experimental Investigation and Results

#### 4.1. Evaluation of Stiffness Property

#### 4.2. Evaluation of Decoupled Motions

_{i}, can be expressed as

_{i}is the combination of the energy of desired motion (defined by P

_{i}and ${k}_{ii}^{m}$) and the energies of undesired parasitic motions (defined by P

_{i}and ${k}_{ij}^{m}$ with j ≠ i). A motion can be considered as decoupled if the energies caused by parasitic motions have minor contributions over the total energy. Based on Equation (16), when applying a force along the X axis, two parasitic motions are generated, i.e., the rotations about the X and Y axes represented by ${k}_{41}^{m}$ and ${k}_{51}^{m}$ respectively. The bending moments about the X and Y axes were measured by the F/T sensor with the same setup as shown in Figure 7a. The experiment was conducted five times and the graph shows the ratios between energies of parasitic motions and the total energy along the X axis is plotted in Figure 9a. With similar experimental setups being adopted, the energy ratios along the Y and Z axes are illustrated in Figure 9b,c, respectively.

#### 4.3. Evaluation of Dynamic Property

## 5. Conclusions

_{R}in this work, can be considered in the design process of CPMs in order to achieve desired dynamic behaviors. Moreover, the merits of 3D printing technologies, i.e., the ability to build complex structures and fabricate multiple materials in a single part, will be further investigated to develop new compliant devices with enhanced performances through lattice flexures, free-form flexible structures, multi-material CPMs, etc. The prediction and compensation models for manufacturing errors of 3D printing techniques should also be explored.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**General model of 3-legged compliant parallel mechanism (CPM) synthesized by beam-based method: (

**a**) overall structure of the CPM; (

**b**,

**c**) structures of the 90°-oriented and 0°-oriented curved beams, respectively.

**Figure 3.**Obtained result after optimization process: (

**a**) Optimal geometry of the center curve of curved beams in each leg; (

**b**) structure of the optimized CPM; (

**c**) Ti6Al4V 3D-printed prototype.

**Figure 4.**Illustration of six degrees-of-freedom (DOF) of the CPM: (

**a**–

**c**) Translations along the X, Y, and Z axes, respectively; (

**d**–

**f**) Rotations about the X, Y, and Z axes, respectively.

**Figure 7.**Experimental setups to measure the compliance: (

**a**) along the X axis; (

**b**) about the Y axis; (

**c**) about the Z axis.

**Figure 8.**Experimental results of the 3D-printed CPM: (

**a**–

**c**) translational compliance along the X, Y, and Z axes, respectively; (

**d**–

**f**) rotational compliance about the X, Y, and Z axes, respectively.

**Figure 9.**Ratios of energies caused by parasitic motions and desired motions: (

**a**) along the X axis; (

**b**) along the Y axis; (

**c**) along the Z axis.

**Figure 10.**Experimental setup to measure the dynamic response: (

**a**) along the Z axis; (

**b**) along the X axis; (

**c**) about the Z axis; (

**d**) about the Y axis.

**Figure 11.**Experimental dynamic response of the CPM: (

**a**–

**c**) along the Z, X, and Y axes, respectively; (

**d**–

**f**) about the Z, X, and Y axes, respectively.

Compliance | Predicted | Experiment | Deviation |
---|---|---|---|

along the X axis (m/N) | 2.77 × 10^{−5} | 3.02 × 10^{−5} | 9.03% |

along the Y axis (m/N) | 2.77 × 10^{−5} | 2.73 × 10^{−5} | 1.44% |

along the Z axis (m/N) | 5.92e × 10^{−5} | 6.54 × 10^{−5} | 10.47% |

about the X axis (rad/Nm) | 3.09 × 10^{−2} | 2.93 × 10^{−2} | 5.18% |

about the Y axis (rad/Nm) | 3.09 × 10^{−2} | 2.80 × 10^{−2} | 9.39% |

about the Z axis (rad/Nm) | 2.75 × 10^{−2} | 2.65 × 10^{−2} | 3.64% |

**Table 2.**Deviations between the experimental dynamic responses compared against the predicted values.

Mode | Resonance Frequency | Predicted | Experiment | Deviation |
---|---|---|---|---|

1 | Translation along the Z axis | 143 Hz | 138 Hz | 3.50% |

2 | Translation along the X axis | 191 Hz | 175 Hz | 8.38% |

3 | Translation along the Y axis | 191 Hz | 181 Hz | 5.24% |

4 | Rotation about the Z axis | 207 Hz | 211 Hz | 1.93% |

5 | Rotation about the X axis | 361 Hz | 351 Hz | 2.77% |

6 | Rotation about the Y axis | 361 Hz | 353 Hz | 2.22% |

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**MDPI and ACS Style**

Pham, M.T.; Yeo, S.H.; Teo, T.J.; Wang, P.; Nai, M.L.S.
A Decoupled 6-DOF Compliant Parallel Mechanism with Optimized Dynamic Characteristics Using Cellular Structure. *Machines* **2021**, *9*, 5.
https://doi.org/10.3390/machines9010005

**AMA Style**

Pham MT, Yeo SH, Teo TJ, Wang P, Nai MLS.
A Decoupled 6-DOF Compliant Parallel Mechanism with Optimized Dynamic Characteristics Using Cellular Structure. *Machines*. 2021; 9(1):5.
https://doi.org/10.3390/machines9010005

**Chicago/Turabian Style**

Pham, Minh Tuan, Song Huat Yeo, Tat Joo Teo, Pan Wang, and Mui Ling Sharon Nai.
2021. "A Decoupled 6-DOF Compliant Parallel Mechanism with Optimized Dynamic Characteristics Using Cellular Structure" *Machines* 9, no. 1: 5.
https://doi.org/10.3390/machines9010005