# The Effect of the Optimization Selection of Position Analysis Route on the Forward Position Solutions of Parallel Mechanisms

^{*}

## Abstract

**:**

## 1. Introduction

^{nd}, …, v

^{-th}independent loops, especially the first loop, will become the key to solving FPS. This problem will directly affect whether the kinematics modeling and FPS can be carried out smoothly.

## 2. The Optimization Selection Criteria for the Position Analysis Route (PAR)

#### 2.1. Optimization Criteria for PAR

_{min}and the minimum number of independent displacement equations (NIDE) ξ

_{min}should be satisfied at the same time when selecting an optimal PAR. Once choosing the optimal PAR correctly, the FPS can be carried out efficiently, and its symbolic position solutions can be obtained as much as possible.

#### 2.2. Procedures of Optimization Selection of PAR

- i.
- ii.
- If there are several optional first loops with the same minimum constraint degree Δ, the loop with the smallest of the NIDE ξ
_{min}should be selected as the first loop. In this way, the number of position equations required to solve the loop positions can be minimized, which is exactly equal to ξ_{min}. - iii.
- If there are both planar SKC
_{(s)}and space SKC_{(s)}in a PM, the FPS should be started from the planar SKC_{(s)}first, and then the space SKC_{(s)}should be analyzed. This is because the NIDE of the planar mechanism loop is always the smallest, i.e., ξ = 3.

## 3. Basic Formulas for Topological Characteristics Analysis

#### 3.1. Analysis of the POC Set

- ${M}_{Ji}$—POC set generated by the ith joint.
- ${M}_{bi}$—POC set generated by the end link of the ith chain.
- ${M}_{Pa}$—POC set generated by the moving platform of PM.

#### 3.2. Determining the DOF

- F—DOF of PM.
- f
_{i}—DOF of the ith joint. - v—number of independent loops, and v = m − n + 1.
- m, n—number of all joints and links of the whole PM, respectively.
- ${\xi}_{{\mathrm{L}}_{j}}$—number of independent displacement equations (NIDE) of the jth loop.
- $\underset{i=1}{\overset{j}{{\displaystyle \cap}}}{M}_{{\mathrm{b}}_{i}}$—POC set generated by the sub-PM formed by the former j branches.
- ${M}_{\mathrm{b}(j+1)}$—POC set generated by the end link of (j + 1)
^{th}sub-chains.

#### 3.3. Determining the Coupling Degree

_{j}, of the jth SOC is defined by:

- ${m}_{j}$—number of joints contained in the jth SOC
_{j}. - ${I}_{j}$—the number of actuated joints in the jth SOC
_{j}.

^{−}, will apply $\left|{\Delta}_{j}^{-}\right|$ constraint equations to a mechanism, and the number of DOF of the mechanism will be decreased by DOF

_{s}of $\left|{\Delta}_{j}^{-}\right|$.

^{+}, will increase the number of DOF of the mechanism by ${\Delta}_{j}^{+}$. Therefore, its forward kinematics solutions could not be solved immediately. Its assembly can be determined only on the condition that ${\Delta}_{j}^{+}$ virtual variables are assigned. When the number of the virtual variables is equal to the number of $\left|{\Delta}_{j}^{-}\right|$ constraint equations applied in SOC

^{−}, i.e., $k=\left|{\Delta}_{j}{}^{-}\right|={\Delta}_{j}{}^{+}$, the motion of the mechanism is defined, and its forward kinematics can be obtained.

^{0}, does not affect the DOF. Its forward kinematics solutions can be obtained immediately without assigning virtual variables.

## 4. Case Studies

#### 4.1. Three-Translation PM (3T-CU)

_{11}//R

_{21}//C

_{31}, which is connected to the base platform 0 through the revolute joint R

_{11}and connected to the moving platform 1 through the cylindrical joint C

_{31}. The second branch is R

_{12}-U

_{22}-U

_{32}, which is connected to the base platform 0 through the revolute joint R

_{12}and connected to the moving platform 1 through the moving joint U

_{32}. The third branch is a hybrid branch chain, which includes a parallelogram, and its two ends are respectively connected to the base platform 0 and the moving platform 1 through revolute joints R

_{13}and R

_{33}. Here, R

_{11}, R

_{12}, and R

_{13}on the base platform 0 could be the actuated joints.

#### 4.1.1. Topology Analysis

_{1}:{R

_{11}//R

_{21}//C

_{31}}, SOC

_{2}:{R

_{12}-U

_{22}-U

_{32}}, SOC

_{3}:{R

_{13}//R

_{23}(P

_{a}

^{(4R)})//R

_{33}}.

_{S}are determined according to Equation (1) as follows, respectively.

_{11}//R

_{21}//C

_{31}-R

_{33}//(P

_{a}

^{(4R)})R

_{23}//R

_{13}, the number of independent displacement equations (NIDE) is obtained by Equation (4)

^{st}sub-PM) is:

_{12}-U

_{22}-U

_{32}-R

_{33}//(P

_{a}

^{(4R)})R

_{23}//R

_{13}, the NIDE is obtained by Equation (4):

^{st}sub-PM) is:

^{st}sub-PM and the first branch chain. From Equation (4), the number of independent displacement equation is:

_{11}, R

_{12}, and R

_{13}on the base platform 0 are selected as the actuated joints, the moving platform 1 can realize 3T motion outputs.

_{L1}= ξ

_{L2}= 5, but their constraint degree ∆ (or coupling degree κ) is different, i.e., κ of the case ① is one, while κ of the case ② is two. Therefore, according to the optimization criteria for the PAR, the case ① that has the smallest constraint degree value (∆

_{min}= 1) and the minimum NIDE (ξ

_{min}= 5) should be used to solve the FPS. The details are described below.

#### 4.1.2. Position Analysis

_{2}. Let moving platform 1 be an equilateral triangle with a circle radius r, and select the O’ point on the moving platform as the origin of the moving coordinate system. The x’- and y’-axis are perpendicular and parallel to the line O’C

_{2}. The z and z’ axis are determined by the Cartesian coordinate rule.

_{i}between vectors A

_{i}B

_{i}and A

_{i}O be the input angle, and the length of the line A

_{i}B

_{i}and B

_{i}C

_{i}(i = 1–3) is equal to l

_{1}and l

_{2}, respectively.

_{1}, θ

_{2}, and θ

_{3}.

_{i}and B

_{i}(i = 1–3) are easily known as:

_{1}-B

_{1}-C

_{1}-C

_{3}-B

_{3}-A

_{3}) with a positive constraint (∆

_{1}= 1)

_{1}between B

_{1}C

_{1}and B

_{1}D

_{1}be the virtual variable, where B

_{1}D

_{1}//A

_{1}O. It is easy to know that the coordinates of point C

_{1}are below:

_{i}(i = 1–3) in the base coordinate system are calculated as:

_{3}C

_{3}= l

_{2}, we can get:

_{1}and C

_{3}can be obtained.

_{2}: A

_{2}-B

_{2}-C

_{2}) with negative constraint (∆

_{2}= −1)

_{2}are:

_{1}C

_{1}= l

_{2}, B

_{3}C

_{3}= l

_{2}, ${y}_{{C}_{3}}={y}_{{C}_{1}}$; (2) two topological constraint equations introduced by the POC feature (3T): ${z}_{{C}_{2}}={z}_{{C}_{1}},{z}_{{C}_{3}}={z}_{{C}_{1}}$, which are exactly equal to the NIDE ξ

_{L}= 5, and we ensure that the positions of the PM can be solved.

_{1}= 40, l

_{2}= 40(unit: mm), and the input angles of the three actuated joints are θ

_{1}= 30°, θ

_{2}= 60°, and θ

_{3}= 60°.

#### 4.2. Three-Translation PM (Delta-CU)

_{11}and R

_{13}, and they are connected to the moving platform 1 through the revolute joints R

_{31}and R

_{33}, respectively. The second branch is R

_{12}-U

_{22}-U

_{32}, and its two ends are connected to base platform 0 and moving platform 1 through R

_{12}and U

_{32}, respectively. The PM is called Delta-CU, i.e., 2-R//R//P

_{a}//R+R-U-U.

#### 4.2.1. Topological Analysis

_{i}{R

_{1i/}/R

_{2i/}/Pa

^{(4R)}-R

_{3i}}(i = 1, 3), while the second branch is denoted as SOC

_{2}{R

_{12}-U

_{22}-U

_{42}}.

_{S}are determined according to Equation (1) as follows, respectively:

_{12}-U

_{22}-U

_{32}) with 5-DOF.

_{11}, R

_{12}, and R

_{13}on the base platform 0 are selected as the actuated joints, the moving platform 1 can realize 3T motion outputs.

_{11}, R

_{12}, and R

_{13}on the base platform are actuated joints, the moving platform 1 can realize 3T motion outputs.

_{1}= 1 and Δ

_{2}= −1, and the NIDEs are ξ

_{L1}= 6, ξ

_{L2}= 4, respectively, which means that when performing the FPS, six position constraint equations must be found in the first loop. Obviously, the difficulty will increase, and it will even not be solved. On the contrary, if the position analysis is carried out according to Case ①, the constraint degrees of the two loops are Δ

_{1}= 1, Δ

_{2}= −1, and the NIDEs are ξ

_{L1}= 5 and ξ

_{L2}= 5, respectively, which means that when performing the FPS, only five position constraint equations are needed to be found in the first loop, which is obviously easier.

#### 4.2.2. Position Analysis

_{2}. Let moving platform 1 be an equilateral triangle with a circle radius r, and select the O’ point on the moving platform as the origin of the moving coordinate system. The x’- and y’-axis are perpendicular and parallel to the line O’C

_{2}. The z- and z’-axis are all determined by the Cartesian coordinate rule. Let the angle θ

_{i}between vectors A

_{i}B

_{i}and A

_{i}O be the input angle, and the lengths of the lines A

_{i}B

_{i}and B

_{i}C

_{i}(i = 1–3) are equal to l

_{1}and l

_{2}, respectively. Let the coordinates of the origin of the moving coordinate system be O’ (x, y, z).

_{2}, θ

_{2}, and θ

_{3}.

_{1}= 1)

_{i}point (i = 1–3) on the moving platform are given, respectively.

_{i}C

_{i}= l

_{2}(i = 1, 3), we can get:

_{2}= −1)

_{2}C

_{2}= l

_{2}, we can get:

_{1}C

_{1}= l

_{2}and B

_{3}C

_{3}= l

_{2}, and ③ the position of C

_{2}is also constrained by the length constraint equation B

_{2}C

_{2}= l

_{2}inside the second loop. In this way, there are five position constraint equations, which are equal to the NIDE ζ

_{L}= 5. Therefore, the position of the first loop is guaranteed to be solvable, since these position equations are nonlinear ones and simper, symbolic solutions can be easily obtained.

## 5. Conclusions

_{min}) and minimum number of independent displacement equations (ξ

_{min})” in order to effectively perform FPS and obtain the symbolic solutions to the greatest possible extent. Otherwise, the FPS may be difficult, complicated, or symbolic solutions cannot be obtained. Two examples are used to illustrate the procedures of how to select the PAR.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Kinzel, G.L.; Chang, C. The analysis of planar linkage using a modular approach. Mech. Mach. Theory
**1984**, 19, 165–172. [Google Scholar] [CrossRef] - Kong, X.; Gosselin, C.M. Generation and forward displacement analysis of RPR-PR-RPR analytic planar parallel manipulators. J. Mech. Des.
**2001**, 8, 2195–2304. [Google Scholar] - Pennock, G.R.; Hasan, A. A Polynomial Equation for a Coupler Curve of the Double Butterfly Linkage. J. Mech. Des.
**2002**, 124, 39–46. [Google Scholar] [CrossRef] - Husain, M.; Waldron, K.J. Direct Position Kinematics of the 3-1-1-1 Stewart Platforms. J. Mech. Des.
**1994**, 116, 1102–1107. [Google Scholar] [CrossRef] - Wampler, C.W. Forward displacement analysis of general six-in-parallel sps (Stewart) platform manipulator using soma coordinates. Mech. Mach. Theory
**1996**, 31, 311–337. [Google Scholar] [CrossRef] - Husty, M.L. Algorithm for Solving the direct kinematic of Stewart-Gough-Type platforms. Mech. Mach. Theory
**1996**, 31, 365–380. [Google Scholar] [CrossRef] - Rojas, N.; Thomas, F. On closed-form Solutions to the position analysis of Baranov trusses. Mech. Mach. Theory
**2012**, 50, 179–196. [Google Scholar] [CrossRef] [Green Version] - Yang, T.; Liu, A.; Shen, H.; Hang, L.; Luo, Y.; Jin, Q. Theory and Appication of of Robot Mechanism Topology; Science Press: Beijing, China, 2012. [Google Scholar]
- Yang, T.; Liu, A.; Shen, H.; Hang, L.; Luo, Y.; Jin, Q. Topology Design of Robot Mechanisms; Springer: Singapore, 2018. [Google Scholar]
- Shen, H. Research on Forward Position Solutions for 6-SPS Parallel Mechanisms Based on Topology Structure Analysis. Chin. J. Mech. Eng.
**2013**, 49, 70–80. [Google Scholar] [CrossRef] - Shen, H.; Chablat, D.; Zen, B.; Li, J.; Wu, G.; Yang, T. A Translational Three-Degrees-of-Freedom Parallel Mechanism with Partial Motion Decoupling and Analytic Direct Kinematics. J. Mech. Robot.
**2020**, 12, 1–7. [Google Scholar] [CrossRef] [Green Version] - Shen, H.; Yang, T.; Li, J.; Zhang, D.; Deng, J.; Liu, A. Evaluation of Topological Properties of Parallel Manipulators Based on the Topological Characteristic Indexes. Robotica
**2020**, 38, 1381–1399. [Google Scholar] [CrossRef] - Shen, H.; Xu, Q.; Li, J.; Yang, T. The Effect of the Optimal Route Selection on the Forward Position Solutions of Parallel Mechanisms. In Advances in Mechanism and Machine Science; Springer: Cham, Switzerland, 2020; pp. 450–458. [Google Scholar]
- Li, J.; Shen, H.; Meng, Q.; Deng, J. A delta-CU—Kinematic analysis and dimension design. In Proceedings of the 10th International Conference on Intelligent Robotics and Applications, Wuhan, China, 16–18 August 2017; Springer: Cham, Switzerland, 2017; pp. 371–382. [Google Scholar]
- Shen, H.; Wang, Y.; Wu, G.; Meng, Q. Stiffness Analysis of a Semi-symmetrical Three-Translation Delta-CU Parallel Robot. In Proceedings of the 6th IFToMM International Symposium on Robotics and Mechatronics, ISRM 2019, Taipei, Taiwan, 28 October–3 November 2019. [Google Scholar]
- Shen, H.; Xu, Q.; Li, J.; Wu, G.; Yang, T.-L. The Effect of Selection of Virtual Variable on the Direct Kinematics of Parallel Mechanisms. In New Trends in Mechanism and Machine Science; Springer: Cham, Switzerland, 2020. [Google Scholar]
- Shen, H.; Ji, H.; Xu, Z.; Yang, T. Design, Kinematic Symbolic Solution and Performance Evaluation of a New Three Translation Mechanism. Trans. Chin. Soc. Agric. Mach.
**2020**, 51, 397–407. [Google Scholar] - Shen, H.; Zhu, Z.; Meng, Q.; Wu, G.; Deng, J. Kinematics and Stiffness Modeling Analysis of a Spatial 2T1R Parallel Mechanism with Zero Coupling Degree. Trans. Chin. Soc. Agric. Mach.
**2020**, 51, 411–420. [Google Scholar]

**Figure 1.**Three-translation parallel mechanisms (PM) (3T-CU). (

**a**) Topology structure; (

**b**) Kinematic modeling of the 3T PM.

**Figure 2.**Three-translation PM (Delta-CU). (

**a**) Topology structure; (

**b**) Kinematic modeling of the 3T PM.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shen, H.; Xu, Q.; Li, J.; Yang, T.-l.
The Effect of the Optimization Selection of Position Analysis Route on the Forward Position Solutions of Parallel Mechanisms. *Robotics* **2020**, *9*, 93.
https://doi.org/10.3390/robotics9040093

**AMA Style**

Shen H, Xu Q, Li J, Yang T-l.
The Effect of the Optimization Selection of Position Analysis Route on the Forward Position Solutions of Parallel Mechanisms. *Robotics*. 2020; 9(4):93.
https://doi.org/10.3390/robotics9040093

**Chicago/Turabian Style**

Shen, Huiping, Qing Xu, Ju Li, and Ting-li Yang.
2020. "The Effect of the Optimization Selection of Position Analysis Route on the Forward Position Solutions of Parallel Mechanisms" *Robotics* 9, no. 4: 93.
https://doi.org/10.3390/robotics9040093