# Optimal Design and Singularity Analysis of a Spatial Parallel Manipulator

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Dexterity Index

**J**of full rank, the corresponding condition number can be expressed as:

_{W}is equal to the number of the points uniformly distributed over the workspace. The larger the GCI is, the better the global dexterity performance of the mechanism is.

#### 2.2. Payload Index

**q**of the joints and the virtual displacement δ

**x**of the MP can be expressed as:

**τ**is the external force,

**f**is the driving force.

**q**and δ

**x**can be expressed in another form as:

**f**be a unit vector, the extremum of the norm of

**τ**can be obtained as:

_{pi}are the eigenvalues of matrix

**JJ**, and P

^{T}_{min}is the minimum external force that the mechanism can stand.

#### 2.3. Compliance Index

_{p}is a matrix describing the stiffness of the platform mounted with elastic elements. For the 3-DOF 3-PPS PM, K

_{p}= diag(k

_{1},k

_{2},k

_{3}).

_{si}of the matrix K

^{−T}K

^{−1}is the limit value of ||ΔX||

^{2}. Let k

_{1}= k

_{2}= k

_{3}=1 and ||f||=1, the maximum deformations can be expressed as:

## 3. Structure Description and Kinematic Analysis

_{i}are perpendicular to the axes of the active prismatic joints f

_{i}. Moreover, the shape of the base is parameterized, namely, the orientation of axes of the active prismatic joints is defined by pan angle δ (0° ≤ δ ≤ 90°) and tilt angle γ (0° ≤ γ ≤ 90°). On the other hand, both the MP and base platform are modeled as equilateral triangles, with side lengths a and b, respectively.

#### 3.1. Pose of the MP

_{0}} is fixed on the base, with the origin coinciding with the centroid of the base, where Y

_{0}-axis is pointed along the direction of B

_{2}B

_{3}, Z

_{0}–axis is perpendicular to the base. Local frame {O} is attached to the centroid of the MP in a similar way.

_{i}in the global frame results in:

_{i}in the local frame, $R={R}_{z}(\varphi ){R}_{y}(\theta ){R}_{x}(\psi )$ is the rotation matrix, $p={[x,y,z]}^{T}$ is the position vector.

_{i}in the base plane satisfies following equations:

_{i}in the global frame can be written as:

#### 3.2. Inverse Position and Jacobian

_{0}-axis.

_{i}can be expressed in another form as:

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

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

## 4. Results

#### 4.1. Optimal Design Based on Dexterity Performance

_{1}in the plot. The corresponding mechanism is the T-shape 3-PPS PM, as shown in Figure 4. For the T-shape 3-PPS PM, it has a partially decoupled DOF when ψ = 0°, namely, a rotation about the axis of A

_{2}A

_{3}.

#### 4.2. Optimal Design Based on Payload Performance

_{2}(δ = 60°, γ = 90°). As shown in Figure 7, the corresponding optimal mechanism is the star-shape 3-PPS PM, whose three PPS limbs are distributed symmetrically, thus the mechanism’s payload capability is improved. Note that the GPI is equal to 0 when δ = 0° or γ = 0°, which means the mechanism can no longer stand external force at these configurations.

#### 4.3. Analysis of Shape Singularity

**j**and

**k**represent the unit vectors of Y

_{0}- and Z

_{0}-axis, respectively.

**n**

_{i}=

**g**

_{i}×

**f**

_{i}.

**Q**is equal to 6, which means the manipulator is fully constrained and has no singularity. On the other hand, if shape singularity exists,

**Q**is singular.

_{0}-axis which passing through point G, even all the actuators are locked. The position vector of point G is:

_{0}-axis without actuating all the actuators.

#### 4.4. Compliance Performance of Shape Singularity and Its Application

## 5. Discussion and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Wu, X.
Optimal Design and Singularity Analysis of a Spatial Parallel Manipulator. *Symmetry* **2019**, *11*, 551.
https://doi.org/10.3390/sym11040551

**AMA Style**

Wu X.
Optimal Design and Singularity Analysis of a Spatial Parallel Manipulator. *Symmetry*. 2019; 11(4):551.
https://doi.org/10.3390/sym11040551

**Chicago/Turabian Style**

Wu, Xiaoyong.
2019. "Optimal Design and Singularity Analysis of a Spatial Parallel Manipulator" *Symmetry* 11, no. 4: 551.
https://doi.org/10.3390/sym11040551