# Dynamic Modeling and Frequency Characteristic Analysis of a Novel 3-PSS Flexible Parallel Micro-Manipulator

^{*}

## Abstract

**:**

_{bm}of flexible spherical hinge and the difference in radius E

_{r}between static platform and mobile platform, while decrease with the increase of the length l of the link rod and the masses of the main components of mechanism. Besides, it can be further drawn from these obtained results that the natural frequencies increase with the increase of the angle θ

_{l}between the link rod and the z axis of reference coordinate system. Considering that the increase of the stiffness k

_{bm}and the angle θ

_{l}will reduce the scope of working space, it is recommended in designing the structure to choose a set of larger stiffness k

_{bm}and angle θ

_{l}as much as possible under the premise of satisfying the working space.

## 1. Introduction

## 2. Structure of Micro-Manipulator

_{ij}P

_{ij}(i = 1, 2 and 3; j = 1, 2), which are respectively connected to the slider and the mobile platform through flexible spherical hinges. Three sliders are respectively fixedly connected with the moving stages of the piezo stages, and move with them. The fixed frame of the piezo stage is connected with the vertical rail on frame of the mechanism.

## 3. Inverse Kinematic Analysis

_{ij}P

_{ij}. Then, the pseudo rigid body (PRB) model of this micro-manipulator can be established, as is shown in Figure 2b. Considering that the motion characteristics of the two link rods B

_{ij}P

_{ij}of the same limb have identical motion, each limb can be regarded as one rigid link rod B

_{i}P

_{i}(hereinafter referred to as “equivalent rod B

_{i}P

_{i}”) for the sake of simplicity of the analysis, as can be seen in Figure 2c. Obviously, the length and motion characteristics of the equivalent rods B

_{i}P

_{i}are the same as the original link rod B

_{ij}P

_{ij}, and the mass is twice that of the original link rod B

_{ij}P

_{ij}. And the two flexible spherical hinges at each end of the original limb can be equivalent to one spherical hinge (referred to as “equivalent spherical hinge”), whose deformation is equal to that of the original flexible spherical hinge. The only difference is that the bending stiffness of the equivalent spherical hinge is equal to twice that of the original flexible spherical hinge. In order to facilitate the analysis, the following inverse kinematics analysis can be carried out based on the simplified pseudo-rigid body (PRB) model in Figure 2c.

_{i}(i =1, 2, 3) of the three sliders is defined as the static platform of the mechanism. The reference coordinate system O{x, y, z} is attached at the center point O of the static platform. The moving coordinate system P{x

_{p}, y

_{p}, z

_{p}} is established at the center point P of the mobile platform, whose x-axis is parallel to the x-axis of the reference coordinate system, and its z-axis coincides with the z-axis of the reference coordinate system. The radius of the static platform and the mobile platform are r

_{a}and r

_{p}, respectively. In the initial state, the vertical distance between the mobile platform and the static platform is h. The length of each link rod B

_{i}P

_{i}is l, where the point B

_{i}and the point P

_{i}respectively represent the center points of the flexible spherical hinges at two ends of the link rod. It should to be noted that the centroid of lower flexible spherical hinge coincides with the centroid of the slider, so A

_{i}B

_{i}represents the input displacement of the i-th slider. OA

_{i}is the position vector from the center point O of the static platform to the initial centroid point A

_{i}of the slider, and φ

_{i}(φ

_{1}= 0, φ

_{i}

_{+ 1}= φ

_{i}+ 2/3π) is the angle between OA

_{i}and the x-axis of the reference system. The angle between the axis of link rod B

_{i}P

_{i}and the z-axis of the reference coordinate system is defined as θ

_{l}, as shown in Figure 3.

#### 3.1. Jacobian Matrix

_{p}, y

_{p}, z

_{p}} relative to the reference coordinate system O{x, y, z} can be expressed as

_{i}represents sinφ

_{i}, cφ

_{i}represents cosφ

_{i}, and the others are the same. The b

_{i}represents the input distance of i-th slider.

**J**is the 3 × 3 Jacobian matrix of the 3-PSS flexible parallel micro-manipulator.

_{i}and output displacement x, y, z of this micro-manipulator is micrometers, where the unit of E

_{r}and h is millimeters. Thus, the Jacobian matrix can be approximately expressed as

#### 3.2. The Relationship between the Micro Angular Deformation of the Flexible Spherical Hinge and the End Pose of Mobile Platform

_{1}, y

_{1}, z

_{1}}, ${B}_{i}^{2}${x

_{2}, y

_{2}, z

_{2}} and ${B}_{i}^{3}${x

_{3}, y

_{3}, z

_{3}} are assigned to the center point B

_{i}of the flexible spherical hinges at the lower ends equivalent rod B

_{i}P

_{i}. The coordinate system ${B}_{i}^{1}$ {x

_{1}, y

_{1}, z

_{1}} is obtained by translating the reference coordinate system O{x, y, z} to point B

_{i}. The coordinate system ${B}_{i}^{2}${x

_{2}, y

_{2}, z

_{2}} is obtained through a series of coordinate transformations of the local coordinate system ${B}_{i}^{1}${x

_{1}, y

_{1}, z

_{1}}, and its z-axis direction is the same as the initial vector direction of the equivalent rod B

_{i}P

_{i}. In the initial state of the micro-manipulator, the coordinate system ${B}_{i}^{3}${x

_{3}, y

_{3}, z

_{3}} that changes with the motion of the flexible spherical hinge coincides with the coordinate system ${B}_{i}^{2}${x

_{2}, y

_{2}, z

_{2}}.

_{1}, y

_{1}, z

_{1}} relative to reference coordinate system O{x, y, z} is

_{2}, y

_{2}, z

_{2}} respect to the coordinate system ${B}_{i}^{1}${x

_{1}, y

_{1}, z

_{1}}. It can be obtained by twice coordinate transformations of the coordinate system ${B}_{i}^{1}${x

_{1}, y

_{1}, z

_{1}}. The two coordinate transformations are the coordinate system ${B}_{i}^{1}${x

_{1}, y

_{1}, z

_{1}} first rotate around the y

_{2}axis by -θ

_{l}angle, and then around the z

_{2}axis by φ

_{i}angle. Therefore, the matrix ${}_{{B}_{i}^{2}}^{{B}_{i}^{1}}T$ can be expressed as

_{i}are α

_{i}and β

_{i}, respectively. Since the angular displacement of the flexible spherical hinge is small, sα ≈ α, cα ≈ 1, sβ ≈ β, cβ ≈ 1, and the infinitesimal terms are ignored. Thus, the coordinate transformation of coordinate system ${B}_{i}^{3}${x

_{3}, y

_{3}, z

_{3}} relative to coordinate system ${B}_{i}^{2}${x

_{2}, y

_{2}, z

_{2}} can be expressed as

_{3}, y

_{3}, z

_{3}} relative to reference coordinate system O{x, y, z} is

_{i}in the local coordinate system ${B}_{i}^{3}${x

_{3}, y

_{3}, z

_{3}} is ${P}_{i}^{{B}_{3}}=\left(\begin{array}{ccc}0& 0& l\end{array}\right)$. Thus, in the reference coordinate system O{x, y, z}, the coordinate of point P

_{i}can be expressed as

_{i}in the moving coordinate system P{x

_{p}, y

_{p}, z

_{p}} is ${P}_{i}^{P}=\left(\begin{array}{ccc}{r}_{p}c{\phi}_{i}& {r}_{p}s{\phi}_{i}& 0\end{array}\right)$. Then, the coordinate of point P

_{i}in the reference coordinate system O{x, y, z} can also be expressed as

_{i}and β

_{i}) of the flexible spherical hinge can be obtained.

## 4. Dynamic Modeling

_{b}is the mass of the slider, and g is the acceleration of gravity.

_{p}is the mass of mobile platform.

_{ij}P

_{ij}is evenly distributed to its two ends. In this way, it can be considered that the motion of distributed masses at the upper and lower ends of the link rod are consistent with the mobile platform and the slider, respectively. Therefore, the total kinetic energy and potential energy of all the link rods can be respectively expressed as

_{c}is the mass of one link rod.

**K**

_{b}represents the stiffness matrix of flexible spherical hinge. k

_{bm}is bending stiffness of the flexible spherical hinge, and it is determined by structural parameters of the flexible spherical hinge including minimum thickness t

_{b}, cutting radius R and maximum cutting angle θ

_{m}[26]. Since the stiffness of the flexible spherical hinge is not the focus of this paper, expression of k

_{bm}will not be expressed in detail here.

_{i}represents i-th generalized coordinate, F

_{i}is i-th actuation force.

**M**is the mass matrix,

**K**is the stiffness matrix,

**G**is the gravity force vector matrix, and

**F**is the actual force matrix.

## 5. Examples and Simulation Analysis

#### 5.1. Numerical Examples and Modal Analysis

^{3}. Dimension parameters of the micro-manipulator are shown in Table 1.

#### 5.2. The Influence of Basic Structural Parameters Changes on Natural Frequencies

_{bm}of the flexible spherical hinge, the masses of main components (sliders, link rods and mobile platform), the difference in radius E

_{r}between the static platform and mobile platform and the length l of the link rod.

#### 5.2.1. Influence of the Stiffness k_{bm} of Flexible Spherical Hinge on the Natural Frequency

_{bm}. However, it should be noted that increase of the stiffness of the flexible spherical hinge k

_{bm}will reduce the working space of the mechanism [27,28].

#### 5.2.2. Influence of the Mass of Main Component on the Natural Frequency

^{−3}Kg) respectively to the link rod, the slider and the mobile platform, the corresponding changes of natural frequency can be obtained according to Equation (26), as shown in Figure 7. Obviously, when the same mass Δm is added to different component, the link rod has the greatest impact on the natural frequency, followed by the mobile platform, and smallest the slider.

#### 5.2.3. The Influence of the Variations of the Length l of Link Rod and the Difference in Radius E_{r} on the Natural Frequency

_{r}between the static platform and mobile platform on the natural frequencies, it is assumed that the radius r

_{p}of the mobile platform is kept unchanged. In addition, it is assumed that the mass m

_{c}of the link rods remains unchanged when the length l of the link rods changes. Variation ranges of the difference in radius E

_{r}and the length of the link rod are set to be 20–60 mm and 65–100 mm, respectively. Corresponding variation of the natural frequency can be obtained according to Equation (26), as shown in Figure 8. It can be seen from Figure 8 that natural frequencies of the mechanism increase as the difference in radius E

_{r}becomes larger, while decrease as the length l of the link rods increases.

_{r}and the length l of the link rod determine the angle θ

_{l}between the link rod and the z-axis of the reference coordinate system, the influence of change of the angle θ

_{l}on the natural frequency is further analyzed. According to Figure 8, the influence of variation of the angle θ

_{l}on the natural frequency under different link rod lengths and different radius difference can be obtained, as shown in Figure 9a,b, respectively. Both Figure 9a,b show that the natural frequencies increase with the increase of the angle θ

_{l}under the condition of any given link rod length l or radius difference E

_{r}. Obviously, increasing the angle θ

_{l}is a good way for increasing the natural frequencies of the structure in structural design. However, it should be noted that increase of θ

_{l}will reduce the working space of the mechanism [28]. Furthermore, it can be seen in Figure 9a that under different link rod length conditions, the angle change has approximate effect on the change of the natural frequency. And Figure 9b shows that the smaller the radius difference E

_{r}, the greater the influence of the angle change on the natural frequency. To further verify the correctness of the above conclusions, modal analysis is carried out on the different simulated models with different structural parameters collected from Figure 9. For the sake of brevity, only six points are collected from Figure 9a. According to the structural parameters corresponding to the six points, six structural models can be established and modal analysis can then be carried out by ANSYS software. Since the frequencies of the first-order models of the modal analysis of different models are all 0 Hz, and the frequencies of the second-order and third-order models are approximately equal. For simplicity, the second-order mode shape is given here. The frequency and mode shape results are shown in Figure 10.

_{l}from 45° to 70° are relatively close, all about 5 Hz. This also verifies the correctness of the previous conclusions that “under different link rod length conditions, the angle change has approximate effect on the change of the natural frequency.”

## 6. Conclusions

_{bm}of the flexible spherical hinge and the difference in radius E

_{r}between the static platform and the mobile platform, while decrease with the increase of the length l of link rod and the mass of the main components of mechanism increase. In addition, it can be further inferred from the obtained results that the natural frequencies increase with the increase of the angle θ

_{l}between the link rod and the z-axis of the reference coordinate system.

_{bm}and the angle θ

_{l}will reduce the scope of the working space, it is recommended in designing the structure to choose a larger θ

_{l}as much as possible under the premise of satisfying the working space. Therefore, during structural design, it is recommended to choose a relatively larger stiffness k

_{bm}and angle θ

_{l}under the premise of ensuring working space requirements. Research of this paper can provide a theoretical basis for the structural optimization design of the 3-PSS flexible parallel micro-manipulation.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**The equivalent process of the motion diagram. (

**a**) Motion diagram. (

**b**) PRB model. (

**c**) Simplified PRB model.

**Figure 4.**The first six modal shapes of the micro-manipulator. (

**a**) Modal shape 1. (

**b**) Modal shape 2. (

**c**) Modal shape 3. (

**d**) Modal shape 4. (

**e**) Modal shape 5. (

**f**) Modal shape 6.

**Figure 5.**The influence the stiffness of flexible spherical hinge on the natural frequency of the micro-manipulator.

**Figure 6.**The influence of the variation of the mass proportional of different components on the natural frequency. (

**a**) Proportionally increasing the mass of the link rod and the slider. (

**b**) Proportionally increasing the mass of the slider and mobile platform. (

**c**) Proportionally increasing the mass of link rod and mobile platform.

**Figure 7.**The influence of the absolute change of mass of different components on the natural frequency.

**Figure 8.**The influence of the variations of the length l of link rod and the difference in radius E

_{r}on the natural frequency.

**Figure 9.**The change of natural frequency with variation of angle θ

_{l}

_{.}(

**a**) Under the condition of different link rod length l. (

**b**) Under the condition of different radius difference E

_{r}.

**Figure 10.**Modal analysis of models with different length l and angle θ

_{l}. (

**a**) l = 65 mm, θ

_{l}=45°. (

**b**) l = 70 mm, θ

_{l}=45°. (

**c**) l = 75 mm, θ

_{l}=45°. (

**d**) l = 65 mm, θ

_{l}=70°. (

**e**) l = 70 mm, θ

_{l}=70°. (

**f**) l = 75 mm, θ

_{l}= 70°.

Parameter | r_{p}/mm | r_{a}/mm | l/mm | t_{b}/mm | R/mm | θ_{m}/° |
---|---|---|---|---|---|---|

Numerical value | 45 | 25 | 65 | 1 | 2.5 | 60 |

No | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

Natural Frequencies f/Hz | 0 | 57.681 | 57.682 | 353.35 | 708.63 | 708.67 |

No | Natural Frequencies f/Hz | Relative Error/% | |
---|---|---|---|

Theoretical Value | Simulation Value | ||

1 | 0 | 0 | 0 |

2 | 56.16 | 57.681 | 2.71 |

3 | 56.16 | 57.682 | 2.71 |

**Table 4.**Comparison of numerical calculation results and finite element simulation of six models with different parameters.

Parameters | l = 65 mm | l = 70 mm | l = 75 mm | ||||
---|---|---|---|---|---|---|---|

f/Hz | θ_{l} = 45° | θ_{l} = 70° | θ_{l} = 45° | θ_{l} = 70° | θ_{l} = 45° | θ_{l} = 70° | |

Numerical results/Hz | 58.74 | 63.81 z | 54.55 | 59.27 | 50.91 | 55.3 | |

Simulation results/Hz | 60.88 | 65.186 | 56.004 | 60.028 | 51.808 | 55.533 | |

Relative error/% | 3.64 | 2.15 | 2.66 | 1.27 | 1.76 | 0.42 |

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**MDPI and ACS Style**

Ren, J.; Cao, Q.
Dynamic Modeling and Frequency Characteristic Analysis of a Novel 3-PSS Flexible Parallel Micro-Manipulator. *Micromachines* **2021**, *12*, 678.
https://doi.org/10.3390/mi12060678

**AMA Style**

Ren J, Cao Q.
Dynamic Modeling and Frequency Characteristic Analysis of a Novel 3-PSS Flexible Parallel Micro-Manipulator. *Micromachines*. 2021; 12(6):678.
https://doi.org/10.3390/mi12060678

**Chicago/Turabian Style**

Ren, Jun, and Qiuyu Cao.
2021. "Dynamic Modeling and Frequency Characteristic Analysis of a Novel 3-PSS Flexible Parallel Micro-Manipulator" *Micromachines* 12, no. 6: 678.
https://doi.org/10.3390/mi12060678