平面广义Stewart平台位置正解求解策略及仿真
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摘要
与六自由度并联机构相比,三自由度并联机构结构简单、易于控制、制造成本低,发展前景广阔。并联机构位置正解研究可为机构的设计提供必要的依据,然而到目前为止,Stewart平台位置正解的求解问题还未彻底解决。采用基于图论的自由度分析法和基于吴消元法的符号计算方法,研究了平面广义Stewart平台的位置正解求解策略。先将平面广义Stewart平台的位置正解问题转化为约束图,再对约束图进行完整约束完备化,生成构造序列,降低同时求解的方程组规模,并用吴消元法求得方程组对应的闭形解。最后对平面广义Stewart平台位置正解进行了基于MapleSim软件的仿真,用实例验证了方法的有效性。
The three-degrees-of-freedom parallel mechanism has broad prospects for development due to its simple structure,easy control and low manufacturing cost compared with the six-degrees-of-freedom parallel mechanism.The direct kinematics of a parallel mechanism is one of the most important issues for parallel kinematic mechanisms. However,to date,the problem of direct kinematics of a Stewart platform has not been completely solved. In this work,strategies for applying direct kinematics to a planar generalized Stewart platform have been illustrated according to graph theory with analysis of degrees of the freedom and a symbolic computation method based on Wu's elimination. The problem of direct kinematics for a planar generalized Stewart platform is transformed into a constraint graph. In order to reduce the scale of solving equations simultaneously,the under-constrained constraint graph is completed to make it well-constrained and acquire a construction sequence. Subsequently,the equation system of each construction sequence is solved with Wu's elimination method to obtain the closed form solution. Finally,the corresponding simulation is performed using Maple Sim software. The effectiveness of the method is verified by experiments.
The three-degrees-of-freedom parallel mechanism has broad prospects for development due to its simple structure,easy control and low manufacturing cost compared with the six-degrees-of-freedom parallel mechanism.The direct kinematics of a parallel mechanism is one of the most important issues for parallel kinematic mechanisms. However,to date,the problem of direct kinematics of a Stewart platform has not been completely solved. In this work,strategies for applying direct kinematics to a planar generalized Stewart platform have been illustrated according to graph theory with analysis of degrees of the freedom and a symbolic computation method based on Wu's elimination. The problem of direct kinematics for a planar generalized Stewart platform is transformed into a constraint graph. In order to reduce the scale of solving equations simultaneously,the under-constrained constraint graph is completed to make it well-constrained and acquire a construction sequence. Subsequently,the equation system of each construction sequence is solved with Wu's elimination method to obtain the closed form solution. Finally,the corresponding simulation is performed using Maple Sim software. The effectiveness of the method is verified by experiments.
引文
[1]廖启征.连杆机构运动学几何代数求解综述[J].北京邮电大学学报,2010,33(4):1-11.LIAO Q Z.Geometry algebra method for solving the kinematics of linkage mechanisms[J].Journal of Beijing University of Posts and Telecommunications,2010,33(4):1-11.(in Chinese)
[2]STEWART D.A platform with six degrees of freedom[J].Proceedings of the Institution of Mechanical Engineers,1965,180(1):371-386.
[3]韩方元.并联机器人运动学正解新算法及工作空间本体研究[D].长春:吉林大学,2011.HAN F Y.The research on novel forward kinematics numerical algorithm and workspace ontology for parallel robot[D].Changchun:Jilin University,2011.(in Chinese)
[4]黄昔光,黄旭.利用共形几何代数的平面并联机构位置正解求解方法[J].西安交通大学学报,2018,52(3):76-82.HUANG X G,HUANG X.Direct kinematics of planar parallel mechanisms based on conformal geometric algebra[J].Journal of Xi'an Jiaotong University,2018,52(3):76-82.(in Chinese)
[5]董倩文.3-PRP平面并联机构的运动与刚度性研究[D].太原:中北大学,2015.DONG Q W.Research on kinematics and stiffness characteristics of 3-PRP planar parallel mechanism[D].Taiyuan:North University of China,2015.(in Chinese)
[6]PENNOCK G R,KASSNER D J.Kinematic analysis of a planar eight-bar linkage:application to a platform-type robot[J].Journal of Mechanical Design,1992,114(1):87-95.
[7]MERLET J P.Direct kinematics of planar parallel manipulators[C]∥Proceedings of the 1996 IEEE International Conference on Robotics and Automation.Minneapolis,1996:3744-3749.
[8]刘惠林,张同庄.3-RPR平面并联机构正解的吴方法[J].北京理工大学学报,2000,20(5):565-569.LIU H L,ZHANG T Z.Forward solutions of the 3-RPR planar parallel mechanism with Wu's method[J].Transaction of Beijing Institute of Technology,2000,20(5):565-569.(in Chinese)
[9]吴文俊.几何定理机器证明的基本原理:初等几何部分[M].北京:科学出版社,1984.WU W J.The basic principles of geometric theorem proving:elementary geometry[M].Beijing:Science Press,1984.(in Chinese)
[10]GAO X S,LEI D L,LIAO Q Z,et al.Generalized Stewart-Gough platforms and their direct kinematics[J].IEEE Transactions on Robotics,2005,21(2):141-151.
[11]TOZ M,KUCUK S.Parallel manipulator software tool for design,analysis,and simulation of 195 GSP mechanisms[J].Computer Applications in Engineering Education,2015,23(6):931-946.
[12]ZHANG G F,ZHANG J W,ZHAO L N.Strategies on direct kinematics to generalized Stewart platforms[C]∥2016 International Conference on Electrical Engineering and Automation.Xiamen,2016:451-456.
[13]ZHANG G F,GAO X S.Planar generalized Stewart platforms and their direct kinematics[M]∥HONG H,WANG D.Automated deduction in geometry.Berlin:Spinger,2006:198-211.
[14]ZHANG G F.Classification of direct kinematics to planar generalized Stewart platforms[J].Computational Geometry:Theory and Applications,2012,45(8):458-473.
[15]ZHANG G F,GAO X S.Well-constrained completion and decomposition for under-constrained geometric constraint problems[J].International Journal of Computational Geometry&Applications,2006,16(5/6):461-478.
[16]Maple Sim_符号方式求解Stewart平台的逆运动学问题[EB/OL].(2017-11-01).http:∥www.cybernet.sh.cn/solution/s19.html.Maple Sim_solving inverse kinematics to Stewart platform with symbolic method[EB/OL].(2017-11-01).http:∥www.cybernet.sh.cn/solution/s19.html.(in Chinese)
[2]STEWART D.A platform with six degrees of freedom[J].Proceedings of the Institution of Mechanical Engineers,1965,180(1):371-386.
[3]韩方元.并联机器人运动学正解新算法及工作空间本体研究[D].长春:吉林大学,2011.HAN F Y.The research on novel forward kinematics numerical algorithm and workspace ontology for parallel robot[D].Changchun:Jilin University,2011.(in Chinese)
[4]黄昔光,黄旭.利用共形几何代数的平面并联机构位置正解求解方法[J].西安交通大学学报,2018,52(3):76-82.HUANG X G,HUANG X.Direct kinematics of planar parallel mechanisms based on conformal geometric algebra[J].Journal of Xi'an Jiaotong University,2018,52(3):76-82.(in Chinese)
[5]董倩文.3-PRP平面并联机构的运动与刚度性研究[D].太原:中北大学,2015.DONG Q W.Research on kinematics and stiffness characteristics of 3-PRP planar parallel mechanism[D].Taiyuan:North University of China,2015.(in Chinese)
[6]PENNOCK G R,KASSNER D J.Kinematic analysis of a planar eight-bar linkage:application to a platform-type robot[J].Journal of Mechanical Design,1992,114(1):87-95.
[7]MERLET J P.Direct kinematics of planar parallel manipulators[C]∥Proceedings of the 1996 IEEE International Conference on Robotics and Automation.Minneapolis,1996:3744-3749.
[8]刘惠林,张同庄.3-RPR平面并联机构正解的吴方法[J].北京理工大学学报,2000,20(5):565-569.LIU H L,ZHANG T Z.Forward solutions of the 3-RPR planar parallel mechanism with Wu's method[J].Transaction of Beijing Institute of Technology,2000,20(5):565-569.(in Chinese)
[9]吴文俊.几何定理机器证明的基本原理:初等几何部分[M].北京:科学出版社,1984.WU W J.The basic principles of geometric theorem proving:elementary geometry[M].Beijing:Science Press,1984.(in Chinese)
[10]GAO X S,LEI D L,LIAO Q Z,et al.Generalized Stewart-Gough platforms and their direct kinematics[J].IEEE Transactions on Robotics,2005,21(2):141-151.
[11]TOZ M,KUCUK S.Parallel manipulator software tool for design,analysis,and simulation of 195 GSP mechanisms[J].Computer Applications in Engineering Education,2015,23(6):931-946.
[12]ZHANG G F,ZHANG J W,ZHAO L N.Strategies on direct kinematics to generalized Stewart platforms[C]∥2016 International Conference on Electrical Engineering and Automation.Xiamen,2016:451-456.
[13]ZHANG G F,GAO X S.Planar generalized Stewart platforms and their direct kinematics[M]∥HONG H,WANG D.Automated deduction in geometry.Berlin:Spinger,2006:198-211.
[14]ZHANG G F.Classification of direct kinematics to planar generalized Stewart platforms[J].Computational Geometry:Theory and Applications,2012,45(8):458-473.
[15]ZHANG G F,GAO X S.Well-constrained completion and decomposition for under-constrained geometric constraint problems[J].International Journal of Computational Geometry&Applications,2006,16(5/6):461-478.
[16]Maple Sim_符号方式求解Stewart平台的逆运动学问题[EB/OL].(2017-11-01).http:∥www.cybernet.sh.cn/solution/s19.html.Maple Sim_solving inverse kinematics to Stewart platform with symbolic method[EB/OL].(2017-11-01).http:∥www.cybernet.sh.cn/solution/s19.html.(in Chinese)