# Optimization of 3-DoF Manipulators’ Parasitic Motion with the Instantaneous Restriction Space-Based Analytic Coupling Relation

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## Abstract

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## Featured Application

**Velocity-level parasitic motion optimization is performed based on the instantaneous restriction space analysis. The manipulator whose parasitic motion is successfully eliminated from the workspace has tremendous benefit for robotic assistive surgery, precision machining devices, and other applications that are critical for manipulators with parasitic motion.**

## Abstract

## 1. Introduction

#### 1.1. The Effect of Parasitic Motion on the Application of Lower Mobility Parallel Manipulators

#### 1.2. Description of Example Manipulators

## 2. Constraint Analysis at the Velocity Level

#### Analytic Constraint Equation

## 3. Analytic Coupling Relation

## 4. Optimization of Parasitic Motion

## 5. Performance Evaluation of Optimal Manipulator Designs

#### Jacobian of the Limb with Decoupled Joint

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Analytic Inversion of c_{1} Matrix

## References

- Zhou, C.C.; Fang, Y.F. Design and analysis for a three-rotational-dof flight simulator of fighter-aircraft. Chin. J. Mech. Eng. (Engl. Ed.)
**2018**, 31, 55. [Google Scholar] [CrossRef][Green Version] - Tanev, T.K. Minimally-invasive-surgery parallel robot with non-identical limbs. In Proceedings of the 2014 IEEE/ASME 10th International Conference on Mechatronic and Embedded Systems and Applications (MESA), Senigallia, Italy, 10–12 September 2014; pp. 1–6. [Google Scholar] [CrossRef]
- Wang, M.; Liu, H.; Huang, T.; Chetwynd, D.G. Compliance analysis of a 3-SPR parallel mechanism with consideration of gravity. Mech. Mach. Theory
**2015**, 84, 99–112. [Google Scholar] [CrossRef][Green Version] - Miermeister, P.; Lächele, M.; Boss, R.; Masone, C.; Schenk, C.; Tesch, J.; Kerger, M.; Teufel, H.; Pott, A.; Bülthoff, H.H. The CableRobot simulator large scale motion platform based on Cable Robot technology. In Proceedings of the IEEE International Conference on Intelligent Robots and Systems, Daejeon, Korea, 9–14 October 2016; pp. 3024–3029. [Google Scholar] [CrossRef]
- Pisla, D.; Plitea, N.; Gherman, B.; Pisla, A.; Vaida, C. Kinematical analysis and design of a new surgical parallel robot. In Proceedings of the 5th International Workshop on Computational Kinematics, Duisburg, Germany, 6–8 May 2009; pp. 273–282. [Google Scholar] [CrossRef]
- Kuo, C.H.; Dai, J.S.; Dasgupta, P. Kinematic design considerations for minimally invasive surgical robots: An overview. Int. J. Med. Robot. Comput. Assist. Surg.
**2012**, 8, 127–145. [Google Scholar] [CrossRef] [PubMed] - Jones, T.P. Kinematics, Dynamics and Design of a Spherical Positioning Robot for Satellite Tracking and Other Applications. Ph.D. Thesis, University of Canterbury, Christchurch, New Zealand, 1996. [Google Scholar] [CrossRef]
- Yu, Y.; Xu, Z.B.; Han, C.Y.; Han, H.; Wang, X.M.; Wu, Q.W. Design and testing of parallel alignment mechanism for space optical payload. In Proceedings of the 2017 2nd Asia-Pacific Conference on Intelligent Robot Systems, ACIRS, Wuhan, China, 16–18 June 2017; pp. 254–258. [Google Scholar] [CrossRef]
- Gressler, W.J. LSST telescope and site status. In Ground-Based and Airborne Telescopes VI; Hall, H.J., Gilmozzi, R., Marshall, H.K., Eds.; International Society for Optics and Photonics, SPIE: Edinburgh, UK, 2016; Volume 9906, pp. 175–189. [Google Scholar] [CrossRef]
- Jáuregui, J.C.; Hernández, E.E.; Ceccarelli, M.; López-Cajún, C.; García, A. Kinematic calibration of precise 6-DOF Stewart platform-type positioning systems for radio telescope applications. Front. Mech. Eng.
**2013**, 8, 252–260. [Google Scholar] [CrossRef] - Sun, J.; Shao, L.; Fu, L.; Han, X.; Li, S. Kinematic analysis and optimal design of a novel parallel pointing mechanism. Aerosp. Sci. Technol.
**2020**, 104, 105931. [Google Scholar] [CrossRef] - Carretero, J.A.; Podhorodeski, R.P.; Nahon, M.A.; Gosselin, C.M. Kinematic analysis and optimization of a new three degree-of-freedom spatial parallel manipulator. J. Mech. Des. Trans. ASME
**2000**, 122, 17–24. [Google Scholar] [CrossRef] - Bernabe, L.; Raynal, N.; Michel, Y. 3Pod: A High Performance Parallel Antenna Pointing Mechanism. In Proceedings of the 15th European Space Mechanisms & Tribology Symposium—ESMATS 2013’, Noordwijk, The Netherlands, 25–27 September 2013; pp. 25–27. [Google Scholar]
- Itul, T.; Pisla, D. Dynamics of a 3-DOF parallel mechanism used for orientation applications. In Proceedings of the 2008 IEEE International Conference on Automation, Quality and Testing, Robotics, AQTR 2008—THETA 16th Edition, Cluj-Napoca, Romania, 22–25 May 2008; Volume 2, pp. 398–403. [Google Scholar] [CrossRef]
- Starrag Group Receives Large Order in the US-Orizon. 2016. Available online: https://www.orizonaero.com/news/single/starrag-group-receives-large-order-us/ (accessed on 5 May 2021).
- Huang, T.; Liu, H. Parallel Mechanism Having Two Rotational and One Translational Degrees of Freedom. U.S. Patent US7793564, 14 September 2010. [Google Scholar]
- Chen, G.; Yu, W.; Li, Q.; Wang, H. Dynamic modeling and performance analysis of the 3-PRRU 1T2R parallel manipulator without parasitic motion. Nonlinear Dyn.
**2017**, 90, 339–353. [Google Scholar] [CrossRef] - Li, Y.G.; Liu, H.T.; Zhao, X.M.; Huang, T.; Chetwynd, D.G. Design of a 3-DOF PKM module for large structural component machining. Mech. Mach. Theory
**2010**, 45, 941–954. [Google Scholar] [CrossRef] - Hennes, N. Ecospeed—An innovative machinery concept for high performance 5 axis machining of large structural components in aircraft engineering. In Proceedings of the 3rd Chemnitz Parallel Kinematic Seminar, Caldes de Malavella, Spain, 24–28 June 2002; pp. 763–774. [Google Scholar]
- Tanev, T.K.; Cammarata, A.; Marano, D.; Sinatra, R. Elastostatic model of a new hybrid minimally-invasive-surgery robot. In Proceedings of the 2015 IFToMM World Congress (IFToMM 2015), Taipei, Taiwan, 25–30 October 2015. [Google Scholar] [CrossRef]
- Yaşır, A.; Kiper, G.; Dede, M.C. Kinematic design of a non-parasitic 2R1T parallel mechanism with remote center of motion to be used in minimally invasive surgery applications. Mech. Mach. Theory
**2020**, 153, 104013. [Google Scholar] [CrossRef] - Li, Q.; Chen, Z.; Chen, Q.; Wu, C.; Hu, X. Parasitic motion comparison of 3-PRS parallel mechanism with different limb arrangements. Robot. Comput.-Integr. Manuf.
**2011**, 27, 389–396. [Google Scholar] [CrossRef] - Qu, H.; Fang, Y.; Guo, S. Parasitic rotation evaluation and avoidance of 3-UPU parallel mechanism. Front. Mech. Eng.
**2012**. [Google Scholar] [CrossRef] - Liu, X.; Wu, C.; Wang, J.; Bonev, I. Attitude description method of [PP]S type parallel robotic mechanisms. Jixie Gongcheng Xuebao/Chin. J. Mech. Eng.
**2008**, 44, 19–23. [Google Scholar] [CrossRef] - Kim, D.; Chung, W.K. Analytic formulation of reciprocal screws and its application to nonredundant robot manipulators. J. Mech. Des. Trans. ASME
**2003**, 125, 158–164. [Google Scholar] [CrossRef] - Kim, D. Kinematic Analysis of Spatial Parallel Manipulators: Analytic Approach with the Restriction Space. Ph.D. Thesis, Pohang University of Science and Technology, Pohang, Korea, 2002. [Google Scholar]
- Kim, D.; Chung, W.; Youm, Y. Analytic Jacobian of in-parallel manipulators. In Proceedings of the 2000 ICRA, Millennium Conference, IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No.00CH37065), San Francisco, CA, USA, 24–28 April 2000; Volume 3, pp. 2376–2381. [Google Scholar] [CrossRef]
- Lipkin, H.; Duffy, J. The Elliptic Polarity of Screws. J. Mech. Transm. Autom. Des.
**1985**, 107, 377–386. [Google Scholar] [CrossRef] - Nigatu, H.; Yihun, Y. Algebraic Insight on the Concomitant Motion of 3RPS and 3PRS PKMs. Mech. Mach. Sci.
**2020**, 83, 242–252. [Google Scholar] [CrossRef] - Jorge Nocedal, S.J.W. Numerical Optimization; Springer Series in Operations Research and Financial Engineering; Springer: New York, NY, USA, 2006. [Google Scholar] [CrossRef][Green Version]
- Liu, H.; Huang, T.; Chetwynd, D.G. A Method to Formulate a Dimensionally Homogeneous Jacobian of Parallel Manipulators. IEEE Trans. Robot.
**2011**, 27, 150–156. [Google Scholar] [CrossRef] - Merlet, J.P. Jacobian, manipulability, condition number, and accuracy of parallel robots. J. Mech. Des. Trans. ASME
**2006**, 128, 199–206. [Google Scholar] [CrossRef]

**Figure 9.**Manipulability of PhRS Figure 5c where parasitic motion along x-axis is removed.

Variables | Value | Unit |
---|---|---|

Moving plate radius | 250 | mm |

Fixed plate radius | 350 | mm |

Leg length | 657.6473 | mm |

z position | 650 | mm |

ψ and θ | ±0.6981 | rad |

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**MDPI and ACS Style**

Nigatu, H.; Kim, D. Optimization of 3-DoF Manipulators’ Parasitic Motion with the Instantaneous Restriction Space-Based Analytic Coupling Relation. *Appl. Sci.* **2021**, *11*, 4690.
https://doi.org/10.3390/app11104690

**AMA Style**

Nigatu H, Kim D. Optimization of 3-DoF Manipulators’ Parasitic Motion with the Instantaneous Restriction Space-Based Analytic Coupling Relation. *Applied Sciences*. 2021; 11(10):4690.
https://doi.org/10.3390/app11104690

**Chicago/Turabian Style**

Nigatu, Hassen, and Doik Kim. 2021. "Optimization of 3-DoF Manipulators’ Parasitic Motion with the Instantaneous Restriction Space-Based Analytic Coupling Relation" *Applied Sciences* 11, no. 10: 4690.
https://doi.org/10.3390/app11104690