New Method and Formula for Degree-of-freedom Calculation of Parallel Mechanism Based on Optimal Paths

doi:10.3901/JME.2024.19.040

Abstract

Abstract: The non-instantaneous degree-of-freedom (DOF) calculation formula (Yang-Sun formula) for the general over-constraint and non-constraint parallel mechanisms (PMs) is derived for the typical PMs composed of simple branch chain in parallel form, and the optimal path is not taken into account during the DOF calculation.Therefore, when the formula is used to DOF calculation for complex PMs that contain two or more hybrid chain (including generalized hybrid loop PMs), the “calculation paths” are long, calculation process of the independent displacement equation number of loop are complex, the influence of the DOF of each loop on the previous sub-PMs is also cannot be clearly shown.To this end, a new method and DOF formula of PM based on the optimal path is proposed.This presented method is simple, less calculation, and can clearly show the specific influence of the DOF of each loop on the whole DOF of PM, which is especially suitable for the calculation of non-instantaneous DOF of general over-constraint and non-constraint PMs with two or more hybrid branch chains.

Key words: degrees of freedom, parallel mechanism, hybrid branch chain

CLC Number:

TH11

SHEN Huiping, LI Ju, ZHU Xiaorong, LI Tao, MENG Qingmei, ZHU Wei, YE Pengda. New Method and Formula for Degree-of-freedom Calculation of Parallel Mechanism Based on Optimal Paths[J]. Journal of Mechanical Engineering, 2024, 60(19): 40-52.

References

[1] IONESCU T G. Terminology for mechanisms and machine science[J]. Mechanism and Machine Theory, 2003，38(7-10)：819-825.
[2] GOGU G. Mobility of mechanisms：A critical review[J]. Mechanism and Machine Theory，2005，40(9)：1068-1097.
[3] MOROSKINE Y F. General analysis of the theory of mechanisms[M]. Moscow：Akad. Nauk，SSSR，1954.
[4] KUTZBACH K. Mechanische leitungsverzweigung，ihre gesetze und anwendungen[J]. Maschinenbau，1929，8(21)：710-716.
[5] ZHANG Q X. Study on structural theory of spatial mechanisms[J]. Chinese Journal of Mechanical Engineering，1962，9(1)：7-32.
[6] FREUDENSTEIN F，ALIZADE R. On the degree-of- freedom of mechanisms with variable general constraint[C]// Fourth World Congress on the Theory of Machines and Mechanisms. London：Mech. Eng. Publ. Ltd. 1975：51-56.
[7] ZHAO J S，ZHOU K，FENG Z J. A theory of degrees of freedom for mechanisms[J]. Mechanism and Machine Theory，2004，39(6)：621-643.
[8] DAI J S，HUANG Z，LIPKIN H. Screw system analysis of parallel mechanisms and applications to constraint and mobility study[C]// 28th ASME Biennial Mechanisms and Robotics Conference. Salt Lake City：American Society of Mechanical Engineers，2004：1569-1582.
[9] DAI J S，HUANG Z，LIPKIN H. Mobility of overconstrained parallel mechanisms[J]. ASME Journal of Mechanical Design，2005，128：220-229.
[10] DAI J S. Finite displacement screw operators with embedded chasles’ motion[J]. Journal of Mechanisms and Robotics，2012，4(4)：041002.
[11] 戴建生. 机构学与机器人学几何基础与旋量代数[M]. 北京：高等教育出版社，2014. DAI Jiansheng. Fundamentals of mechanism and robotics geometry and spinor algebra[M]. Beijing：Higher Education Press，2014.
[12] 赵景山，冯之敬，褚福磊. 机器人机构自由度分析理论[M]. 北京：科学出版社，2009. ZHAO Jingshan，FENG Zhijing，CHU Fulei. Freedom analysis theory of robot mechanism[M]. Beijing：Science Press，2009.
[13] 黄真，赵永生，赵铁石. 高等空间机构学[M]. 北京：高等教育出版社，2006. HUANG Zhen，ZHAO Yongsheng，ZHAO Tieshi. Advanced space mechanism[M]. Beijing：Higher Education Press，2006.
[14] 黄真，刘婧芳，李艳文. 论机构自由度——寻找了150年的自由度通用公式[M]. 北京：科学出版社，2011. HUANG Zhen，LIU Jingfang，LI Yanwen. On the degree of freedom of mechanism—a general formula of freedom in search of 150 years[M]. Beijing：Science Press，2011.
[15] HUANG Z，LI Q C，DING H F. Theory of parallel mechanisms[M]. Dordrecht Heideljberg：Springer，2013.
[16] KONG X W，GOSSELIN C M. Mobility analysis of parallel mechanisms based on screw theory and the concept of equivalent serial kinematic chain[C]// Proc. of the 2005 ASME Design Engineering Technical Conf. & Computers and Information in Engineering Conf. New York：American Society of Mechanical Engineers，2005：911-920.
[17] KONG X W，GOSSELIN C M. Type synthesis of parallel mechanisms[M]. New York：Springer Tracts in Advanced Robotics，2007.
[18] GOGU G. Mobility and spatiality of parallel robots revisited via theory of linear transformations[J]. European Journal of Mechanics A-Solids，2005，24：690-711.
[19] RICO J M，AGUILERA L D，GALLARDO J，et al. A more general mobility criterion for parallel platforms[J]. Journal of Mechanical Design，Transactions of the ASME，2006，128(1)：207-219.
[20] YANG T L，SUN D J. General formula of degree of freedom for parallel mechanisms and its application[C]// Proceedings of ASME 2006 Mechanisms Conference. New York：American Society of Mechanical Engineers，2006：10.
[21] 杨廷力，沈惠平，刘安心，等. 机构DOF公式的基本形式、DOF分析及其物理内涵——兼对近十年提出的DOF公式的回顾[J]. 机械工程学报，2015，41(9)：28-32. YANG Tingli，SHEN Huiping，LIU Anxin，et al. The basic form，DOF analysis and its physical connotation of mechanism DOF formula—a review of the DOF formula proposed in recent ten years[J]. Journal of Mechanical Engineering，2015，41(9)：28-32.
[22] YANG T L，SUN D J. A general DOF formula for parallel mechanisms and multi-loop spatial mechanisms[J]. ASME Journal of Mechanisms and Robotics，2012，4(1)：1-17.
[23] 杨廷力. 机器人机构拓扑结构学[M]. 北京：机械工业出版社，2004. YANG Tingli. Topological structure of robot mechanism[M]. Beijing：China Machine Press，2004.
[24] 杨廷力，刘安心，罗玉峰，等. 机器人机构拓扑结构设计[M]. 北京：科学出版社，2012. YANG Tingli，LIU Anxin，LUO Yufeng，et al. Topology design of robot mechanism[M]. Beijing：Science Press，2012.
[25] 杨廷力，沈惠平，刘安心. 机构DOF公式的统一形式及其物理内涵[J]. 常州大学学报，2013，25(4)：1-8. YANG Tingli，SHEN Huiping，LIU Anixn. Unified form of mechanism DOF formula and its physical connotation[J]. Journal of Changzhou University，2013，25(4)：1-8.
[26] YANG T L，LIU A X，SHEN H P，et al. Topology design of robot mechanisms[M]. Singapore：Springer，2018.
[27] 杨廷力. 机器人机构拓扑学及其创建历程[M]. 北京：高等教育出版社，2022. YANG Tingli. Topology of robot mechanism and its creation process[M]. Beijing：Higher Education Press，2022.
[28] ZHU X R，SHEN H P，WU C Q，et al. Computer-aided mobility analysis of parallel mechanisms[J]. Mechanism and Machine Theory，2020，148：1-26.
[29] 沈惠平. 机器人机构拓扑特征运动学[M]. 北京：高等教育出版社，2021. SHEN Huiping. Topological characteristics-based kinematics of robot mechanisms[M]. Beijing：Higher Education Press，2021.
[30] 沈惠平，朱小蓉，尹洪波，等. 并联机构的结构降耦及其设计方法[J]. 机械工程学报，2016，52(23)：102-113. SHEN Huiping，ZHU Xiaorong，YIN Hongbo，et al. Structural decoupling of parallel mechanism and its design method[J]. Chinese Journal of Mechanical Engineering，2016，52(23)：102-113.
[31] 沈惠平，李菊，王振，等. 基于结构降耦和运动降耦的并联机构拓扑结构优化及其性能改善研究[J]. 机械工程学报，2017，53(19)：176-186. SHEN Huiping，LI Ju，WANG Zhen，et al. Topology optimization and performance improvement of parallel mechanism based on structural decoupling and motion decoupling[J]. Journal of Mechanical Engineering，2017，53(19)：176-186.
[32] 沈惠平，尹洪波，王振，等. 基于拓扑结构分析的求解6-SPS并联机构位置正解的研究[J]，机械工程学报，2013，49(21)：70-80. SHEN Huiping，YIN Hongbo，WANG Zhen，et al. Topological structure analysis for positive position solution of 6-SPS parallel mechanism[J]. Journal of Mechanical Engineering，2013，49(21)：70-80.
[33] 沈惠平，何红波，李涛. 全铰两约束支链的大转角零耦合度两平移一转动并联平台：ZL 202110165364.5[P]. 2022-10-06. SHEN Huiping，HE Hongbc，LI Tac. A high-angle zero-coupling two-translation-rotation parallel platform with fully hinged two constrained branch chains：ZL 202110165364.5[P]. 2022-10-06.
[34] MENG J，LIU G F，LI Z X. A Geometric theory for analysis and synthesis of sub-6 DOF parallel manipulators[J]. IEEE Transactions on Robotics，2007，23(4)：625-649.
[35] MONSARRAT B，GOSSELIN C M. Workspace analysis and optimal design of a 3-leg 6-DOF parallel platform mechanism[J]. IEEE Transactions on Robotics and Automation，2003，19(6)：954-966.
[36] WU G L，SHEN H P. Parallel PnP robots[M]. Wuhan：Huazhong University of Science and Technology Press，Springer，2021.
[37] SHEN H P，ZHAO Y N，WU G L，et al. Kinematic design of a translational parallel mechanism based on sub- kinematic chain determined workspace superposition[J]. Proceedings of the Institution of Mechanical Engineers Part C-Journal of Mechanical Engineering Science，2022(1)：1-16.
[38] SHEN H P，ZHAO Y N，LI J，et al. A novel partially-decoupled translational parallel manipulator with symbolic kinematics，singularity identification and workspace determination[J]. Mechanism and Machine Theory，2021，164(10)：104388.
[39] SHEN H P，CHABLAT D，ZENG B X，et al. A translational three-degrees-of-freedom parallel mechanism with partial motion decoupling and analytic direct kinematics[J]. Journal of Mechanisms and Robotics，2020，12(2)：1-7.
[40] SHEN H P，YANG T L，LI J，et al. Evaluation of topological properties of parallel manipulators based on the topological characteristic indexes[J]. Robotica，2020，38(8)：1381-1399.
[41] 王恪. 基于拓扑特征的并联机构运动学与动力学集成建模与分析[D]. 常州：常州大学，2020. WANG Ke. Integrated modeling and analysis of kinematics and dynamics of parallel mechanism based on topological features[D]. Changzhou：Changzhou University，2020.
[42] 周金波. 一类两种具有符号式位置正解的2T1R并联机构的运动学及动力学分析[D]. 常州：常州大学，2020. ZHOU Jinbo. Kinematic and dynamic analysis of a class of Two 2T1R parallel mechanisms with positive symbolic position solutions[D]. Changzhou：Changzhou University，2020.
[43] 黄凯伟，沈惠平，李菊，等. 一种具有部分运动解耦和符号式位置正解的空间2T1R并联机构拓扑设计与动力学建模[J]. 中国机械工程，2022，33(2)：160-169. HUANG Kaiwei，SHEN Huiping，LI Ju，et al. Topological design and dynamics modeling of a spatial 2T1R parallel mechanism with partially motion decoupling and symbolic forward kinematics[J]. China Mechanical Engineering，2022，33(2)：160-169.
[44] 沈惠平，周金波，尤晶晶，等. 具有解析式位置正解的2T1R并联机构运动性能分析[J]. 农业机械学报，2020，51(1)：398-409. SHEN Huiping，ZHOU Jinbo，YOU Jingjing，et al. 2T1R parallel mechanism with analytic positive position solutions and its kinematic performance based optimization[J]. Transactions of the Chinese Society for Agricultural Machinery，2020，51(1)：398-409.
[45] 沈惠平，曾博雄，尤晶晶，等. 具有解析式位置正解的三平移并联机构设计与分析[J]. 农业机械学报，2020，51(2)：383-391. SHEN Huiping，ZENG Boxiong，YOU Jingjing，et al. Design and analysis of a three-translation parallel mechanism with analytic positive position solution[J]. Transactions of the Chinese Society for Agricultural Machinery，2020，51(2)：383-391.
[46] 吉昊. 具有符号式位置正解的两种新型并联机构运动学分析与性能评价及其优化[D]. 常州：常州大学，2020. JI Hao. Kinematics analysis，performance evaluation and optimization of two new parallel mechanisms with positive symbolic position solutions[D]. Changzhou：Changzhou University，2020.
[47] SHEN H P，TANG Y，WU G L，et al. Design and analysis of a class of two-limb non-parasitic 2T1R parallel mechanism with decoupled motion and symbolic forward position solution - influence of optimal arrangement of limbs onto the kinematics，dynamics and stiffness[J]. Mechanism and Machine Theory，2022，172：104815.
[48] 何洪波. 新型低耦合度并联机构的拓扑设计、刚度与逆向动力学分析[D]. 常州：常州大学，2022. HE Hongbo. Topological design，stiffness and reverse dynamics analysis of a novel low-coupling parallel mechanism[D]. Changzhou：Changzhou University，2022.
[49] 汤耀. 新型两支链三自由度并联机构的拓扑设计、运动学及其动力学性能研究[D]. 常州：常州大学，2022. TANG Yao. Topological design，kinematics and dynamic performance of a novel two-chain three-degree-of- freedom parallel mechanism[D]. Changzhou：Changzhou University，2022.
[50] 王达. 新型少自由度并联机构的拓扑设计、刚度及其动力学建模与求解[D]. 常州：常州大学，2022. WANG Da. Topological design，stiffness and dynamic modeling and solving of a novel parallel mechanism with few degrees of freedom[D]. Changzhou：Changzhou University，2022.
[51] 孔宪文，克莱门特·戈斯林. 并联机构构型综合[M]. 于靖军，周艳华，毕树生，译. 北京：机械工业出版社，2013. KONG Xianwen，GOSSELIN C M. Configuration synthesis of parallel mechanism[M]. Translated by YU Jingjun，ZHOU Yanhua，BI Shusheng. Beijing：China Machine Press，2013.