# Performance Analysis and Optimum Design of a Redundant Planar Parallel Manipulator

## Abstract

**:**

## 1. Introduction

## 2. Kinematics Analysis

**O**

_{0}} was fixed on the square base [23], with the origin coinciding with a vertex of the base. On the other hand, the local coordinate system {

**O**} was attached to the center of the MP. The vector

**p**= [x, y]

^{T}provided the position of the MP with respect to the global coordinate system, and θ is the corresponding rotation angle. Moreover, θ = 0° was defined as the initial assembly mode.

#### 2.1. Inverse Kinematics

_{i}in the global coordinate system can be denoted by:

**R**represents the rotation matrix of the MP,

**p**is the position vector, and ${\mathbf{a}}_{i}^{\prime}$ denotes the position vector of point A

_{i}in the local coordinate system, which can be expressed as:

_{i}can also be expressed in a form as follows:

_{i}represents the displacement of each passive prismatic joint, l

_{i}denotes the input of each active prismatic joint. Moreover,

**i**and

**j**represent the unit vectors of X

_{0}- and Y

_{0}-axes.

#### 2.2. Singularity Analysis

**J**represents the Jacobian matrix.

**J**

_{1}is the Jacobian matrix, which can be obtained as:

**J**is lower than its number of rows, namely, the determinant of

**J**

^{T}

**J**is equal to zero [24]. For the considered 4-PPR PPM, we have:

## 3. Performance Indices

#### 3.1. Dexterity Index

**J**

^{−1}represents the inverse matrix.

#### 3.2. Velocity Index

_{vi}is an eigenvalue of the matrix ${({J}^{-1})}^{T}{J}^{-1}$.

_{max}and V

_{min}can be chosen as the velocity indices, but for the 4-PPR PPM, the matrix ${({J}_{1}^{+})}^{T}{J}_{1}^{+}\in {R}^{4\times 4}$, while its rank is less than 4, this means there is always a zero eigenvalue, which leads to V

_{min}= 0, so we choose V

_{max}as the velocity index. The larger the V

_{max}is, the better the velocity performance of the manipulator is.

#### 3.3. Stiffness Index

_{si}of ${({K}^{-1})}^{T}{K}^{-1}$ is the limit value of ||Δ

**X**||

^{2}. Let k

_{1}= k

_{2}= k

_{3}= 1 and ||

**f**|| = 1, the minimum and maximum deformations can be expressed as:

_{max}is chosen as the stiffness index. The larger the deformation is, the worse the stiffness performance of the PPMs is.

## 4. Performance Evaluation and Optimum Design of the 4-PPR PPM

_{max}, and S

_{max}are all taken into consideration for optimum design. Based on the analysis above, the region with LCI > 0.1, V

_{max}> 2, and S

_{max}< 80 is defined as the optimum region. As depicted in Figure 7, it is found that the considered redundant PPM has a U-shape optimum region, which is symmetrical at about θ = 0°. Within the U-shape optimum region, the manipulator will obtain a relatively great comprehensive performance, which should be taken into consideration for mechanism design.

## 5. Dynamic Analysis

**τ**is the vector of generalized forces.

_{1}of the 4-PPR PPM is non-square, driving forces have infinite solutions. To get a unique solution, the left-inverse matrix ${J}_{1}^{+}={({J}_{1}^{T}{J}_{1})}^{-1}{J}_{1}^{T}$, as discussed in Section 3.1, is adopted to minimize the two-norm of driving forces.

_{1}= 1.7574 kg, m

_{2}= 0.8313 kg, m

_{3}= 48.4972 kg, where m

_{1}, m

_{2}are the masses of the active and passive components, and m

_{3}represents the mass of the MP. The moments of inertia of corresponding components are taken as

**I**

_{1}= 0.0043 kg·m

^{2},

**I**

_{2}= 0.0039 kg·m

^{2}, and

**I**

_{3}= 2.9431 kg·m

^{2}.

_{i}of the ith limb within the given motion can be readily obtained, as shown in Figure 8. The distribution of the four driving forces is relatively uniform, and the maximum driving forces for this PPM with the given motion trajectory is 1.12 N.

## 6. Performance Evaluation of the Non-Redundant PPMs

#### 6.1. Kinematics Analysis

#### 6.2. Performance Evaluation

#### 6.3. Dynamic Analysis

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Wu, X.
Performance Analysis and Optimum Design of a Redundant Planar Parallel Manipulator. *Symmetry* **2019**, *11*, 908.
https://doi.org/10.3390/sym11070908

**AMA Style**

Wu X.
Performance Analysis and Optimum Design of a Redundant Planar Parallel Manipulator. *Symmetry*. 2019; 11(7):908.
https://doi.org/10.3390/sym11070908

**Chicago/Turabian Style**

Wu, Xiaoyong.
2019. "Performance Analysis and Optimum Design of a Redundant Planar Parallel Manipulator" *Symmetry* 11, no. 7: 908.
https://doi.org/10.3390/sym11070908