两维并联雕刻机工作台研究
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摘要
本文提出了两种平面两自由度并联平台：2(?)RR型和2R(?)R型(前者的驱动副为转动副，后者的驱动副为移动副)，它们来源于有较多研究的3(?)RR型和3R(?)R型并联机构，系将其中的某个自由度加以约束而得到。本文以2(?)RR型为主进行了深入研究。他们皆可应用于小型雕刻机的工作台。众所周知，运动学正解、工作空间求解和奇异性分析是并联运动学机器的三大关键问题，本文所论及的两种平台也不例外，不过由于创造性地约束了3自由度机构的转动自由度，使所得的新型2自由度机构2(?)RR的基本关键问题得到简化，从而更具实际应用价值。
运动学分析是研究工作的基础，因此成为本文的重要部分。本文得到了两种平台的正反运动学解的解析表达式。其中2(?)RR型平台的反向运动学解有4组，正向运动学解有两组；而2R(?)R型平台的反向运动学解是唯一的，正向运动学解有两组。由此可见，2R(?)R型平台较2(?)RR型平台控制更容易。
奇异性分析是并联运动学机器的又一个关键问题。本文鉴于所得到的正向和反向运动学解的解析表达式，进一步获得了2(?)RR型并联平台的完整奇异性条件，分为三种情况，由此为下一步物理样机的研制和控制奠定了坚实的基础。为了清楚地观察奇异形位的特征，本文还得到了奇异表达式的多项式形式。在奇异分析和工作空间分析的基础上，绘制了奇异曲线和平台的工作空间边界。
为掌握所研究的2(?)RR型并联平台的动态特性、使并联机器的优点得以在雕刻机中充分发挥，本文还应用拉格朗日方法建立了机构的动力学模型，并进行了初步的动力学分析。并进一步利用OpenGL和VC++软件、ADAMS虚拟样机分析软件对平面两自由度并联雕刻机进行了造型和仿真，取得了较为满意的效果。
Two types of 2-dof planar parallel platforms (2RRR and 2RPR) are addressed in this paper. They are rooted in 3-dof parallel frameworks (3RRR and 3RPR) that have been studied in many aspects and are gained by restricting its certain degree of freedom. These two platforms are lucubrated mainly on the former. They are all used in worktable of small sculptor. As most of us known,forward kinematics,workspace and singularity analysis are three pivotal problems of parallel kinematic machine. These problems are not exceptional to the two platforms discussed in this paper. But these analyses of the two platforms are simplified for restricting the revolving degree of freedom of the 3-dof frameworks in a creative way.
Kinematics analysis is basis of the research work so it becomes the most important section in this paper. An analytical form of the forward and inverse kinematics is derived. The 2RRR platform has four different branches of inverse kinematics and two of forward kinematics,while the 2RPR platform has one and only inverse kinematics and two of forward kinematics. Thus it can be seen that the control of 2RPR platform is easier than the 2RRR platform.
Singularity analysis is second pivotal problem of parallel kinematic machine. Based on analytical expressions of the forward and inverse kinematics,complete singularity conditions of 2RRR parallel platform are derived,which accordingly establish firm base for the development and control of physical prototype. To observe the character of singularity loci clearly,a polynomial form of the singularity loci are also derived. Based on the singularity and workspace analysis,the graphical representation of the complete singularity loci and workspace limits is illustrated with example.
To grasp dynamic character of the 2RRR parallel platform and to exert the advantage of parallel machine in sculptor well,dynamics model of the platform is founded by using Lagrangian approach and dynamic analysis are made primarily. By using the VC++ and OpenGL software and the ADAMS software,the simulation and modeling of the planar 2-dof parallel sculptor are completed,and comparatively satisfactory effect is obtained.
运动学分析是研究工作的基础，因此成为本文的重要部分。本文得到了两种平台的正反运动学解的解析表达式。其中2(?)RR型平台的反向运动学解有4组，正向运动学解有两组；而2R(?)R型平台的反向运动学解是唯一的，正向运动学解有两组。由此可见，2R(?)R型平台较2(?)RR型平台控制更容易。
奇异性分析是并联运动学机器的又一个关键问题。本文鉴于所得到的正向和反向运动学解的解析表达式，进一步获得了2(?)RR型并联平台的完整奇异性条件，分为三种情况，由此为下一步物理样机的研制和控制奠定了坚实的基础。为了清楚地观察奇异形位的特征，本文还得到了奇异表达式的多项式形式。在奇异分析和工作空间分析的基础上，绘制了奇异曲线和平台的工作空间边界。
为掌握所研究的2(?)RR型并联平台的动态特性、使并联机器的优点得以在雕刻机中充分发挥，本文还应用拉格朗日方法建立了机构的动力学模型，并进行了初步的动力学分析。并进一步利用OpenGL和VC++软件、ADAMS虚拟样机分析软件对平面两自由度并联雕刻机进行了造型和仿真，取得了较为满意的效果。
Two types of 2-dof planar parallel platforms (2RRR and 2RPR) are addressed in this paper. They are rooted in 3-dof parallel frameworks (3RRR and 3RPR) that have been studied in many aspects and are gained by restricting its certain degree of freedom. These two platforms are lucubrated mainly on the former. They are all used in worktable of small sculptor. As most of us known,forward kinematics,workspace and singularity analysis are three pivotal problems of parallel kinematic machine. These problems are not exceptional to the two platforms discussed in this paper. But these analyses of the two platforms are simplified for restricting the revolving degree of freedom of the 3-dof frameworks in a creative way.
Kinematics analysis is basis of the research work so it becomes the most important section in this paper. An analytical form of the forward and inverse kinematics is derived. The 2RRR platform has four different branches of inverse kinematics and two of forward kinematics,while the 2RPR platform has one and only inverse kinematics and two of forward kinematics. Thus it can be seen that the control of 2RPR platform is easier than the 2RRR platform.
Singularity analysis is second pivotal problem of parallel kinematic machine. Based on analytical expressions of the forward and inverse kinematics,complete singularity conditions of 2RRR parallel platform are derived,which accordingly establish firm base for the development and control of physical prototype. To observe the character of singularity loci clearly,a polynomial form of the singularity loci are also derived. Based on the singularity and workspace analysis,the graphical representation of the complete singularity loci and workspace limits is illustrated with example.
To grasp dynamic character of the 2RRR parallel platform and to exert the advantage of parallel machine in sculptor well,dynamics model of the platform is founded by using Lagrangian approach and dynamic analysis are made primarily. By using the VC++ and OpenGL software and the ADAMS software,the simulation and modeling of the planar 2-dof parallel sculptor are completed,and comparatively satisfactory effect is obtained.
引文
1．李桥梁，吴洪涛，朱剑英，Stewart机床发展大事记，机械设计与制造工程，1999，4，6-8
2．黄真，孔令富，方跃法，并联机器人机构学及控制，机械工业出版社，1997
3．澹凡忠，姜军生，Stewart平台机构及其应用，机械，2000年第27卷增刊，pp28
4．姚学升，相似平台6-SPS并联机床运动学分析及其仿真模型的建立，南京航空航天大学硕士学位论文，2001．3，pp．3
5．汪劲松，黄田，并联机床—机床行业面临的机遇与挑战，中国机械工程，1999年第10卷第10期
6．陈学生，陈在礼，孔民秀，并联机器人研究的进展与现状，机器人，2002年9月第24卷第5期
7．王洪斌，魏立新，王洪瑞，并联机器人的理论研究现状，自动化博览，2002年第5期
8．周凯，虚拟轴机床发展动态与新采取的对策，江苏机械制造与自动化，2000，2，7-11
9．丁学明，一种空间三自由度平动并联机床研究，南京航空航天大学硕士论文，2002．3，pp．9
10. Gosselin, C.M. and Angeles. J., 1990, "Singularity Analysis of Closed-Loop Kinematic Chains," IEEE Transactions on Robotics and Automation, Vol. 6, No.3, pp.281-290.
11．赵新华，彭商贤，并联机器人奇异位形研究，机械工程学报，第36卷第5期，pp．35-37
12. J. Sefrioui and C.M. Gosselin, Singularity analysis and representation of planar parallel manipulators, Robotics and Autonomous Systems, 10:209-224, 1992
13. C.M. Gosselin and J. Angeles, Kinematic inversion of parallel manipulators in the presence of incompletely specified tasks, Journal of Mechanical Design, Vol. 112, 1990(12), pp.494-500
14. G.R. Pennock and D.J.Kassner, Kinematic Analysis of a Planar Eight-Bar Linkage: Application to a Platform-Type Robot, Journal of Mechanical Design, Vol. 114, 1992(13), pp. 87-95
15. C.M. Gosselin and Jiegao Wang, 1997, "Singularity loci of planar parallel manipulators with revolute actuators," Robotics and Autonomous Systems 21 (1997) 377-398.
16. J. Sefrioui and C.M. Gosselin, On the quadratic nature of the singularity curves of planar 3-dof parallel manipulators, Mech, Mach. Theory, Vol.30, No.4, pp533-551, 1995.
17. G.R.Pennock and D.J.Kassner, The Workspace of a General Geometry Planar Three-Degree-of-Freedom Platform-Type Manipulator, Journal of Mechanical Design,
Vol. 115, 1993(6), pp. 269-276
18．刘辛军，汪劲松，李剑锋，高峰，一种新型空间3自由度并联机构的正反解及工作空间分析，机械工程学报，第37卷第10期，2001(10)，pp．36-39．
19. Xianwen Kong and C.M.Gosselin, Derermination of the uniqueness domains of 3-RPR planar parallel manipulators with similar platforms, Proceedings of DETC'00 ASME 2000 Design Engineering Technical conferences and Computers and Information in Engineering Conference Baltimore, Maryland, September 10-13,2000.
20. Xin-Jun Lin,Jin-Song Wang and Feng Gao, On the optimum design of planar 3-dof parallel manipulators with respect to the workspace, Proceedings of the 2000 IEEE international Conference on Robotics & Automation San Francisco, CA·April 2000
21. C.M. Gosselin and J.Angeles, The Optimum Kinematic Design of a Planar Three-Degree-of-Freedom Parallel Manipulator, Journals of Mechanisms, Transmissions, and Automation in Design, Vol. 110, 1988(3), pp.35-41
22．杨珏，何旭洪，赵昊彤，Mathematica应用指南，人民邮电出版社，1999
23．谭浩强，张基温，唐永炎，C语言程序设计教程，高等教育出版社，1998
24．黄维通，Visual C++面向对象于可视化程序设计，清华大学出版社，2000
25．向世明，OpenGL编程与实例，电子工业出版社，1999
26．徐晓刚，高兆法，王秀娟，VC++6.0入门与提高，清华大学出版社，1999
27．齐舒创作室，Visual C++6.0编程技巧与实例分析，中国水利水电出版社，1999
28．贾志刚，精通OpenGL，电子工业出版社，1998
29．郑兼荣，ADAMS—虚拟样机技术入门与提高，机械工业出版社，2002
30．袁士杰，吕哲勤，多刚体系统动力学，北京，北京理工大学出版社，1992
1、王超群，吴洪涛，一种平面并联机械手的奇异形位分析，淮海工学院学报，2003，1。
2．黄真，孔令富，方跃法，并联机器人机构学及控制，机械工业出版社，1997
3．澹凡忠，姜军生，Stewart平台机构及其应用，机械，2000年第27卷增刊，pp28
4．姚学升，相似平台6-SPS并联机床运动学分析及其仿真模型的建立，南京航空航天大学硕士学位论文，2001．3，pp．3
5．汪劲松，黄田，并联机床—机床行业面临的机遇与挑战，中国机械工程，1999年第10卷第10期
6．陈学生，陈在礼，孔民秀，并联机器人研究的进展与现状，机器人，2002年9月第24卷第5期
7．王洪斌，魏立新，王洪瑞，并联机器人的理论研究现状，自动化博览，2002年第5期
8．周凯，虚拟轴机床发展动态与新采取的对策，江苏机械制造与自动化，2000，2，7-11
9．丁学明，一种空间三自由度平动并联机床研究，南京航空航天大学硕士论文，2002．3，pp．9
10. Gosselin, C.M. and Angeles. J., 1990, "Singularity Analysis of Closed-Loop Kinematic Chains," IEEE Transactions on Robotics and Automation, Vol. 6, No.3, pp.281-290.
11．赵新华，彭商贤，并联机器人奇异位形研究，机械工程学报，第36卷第5期，pp．35-37
12. J. Sefrioui and C.M. Gosselin, Singularity analysis and representation of planar parallel manipulators, Robotics and Autonomous Systems, 10:209-224, 1992
13. C.M. Gosselin and J. Angeles, Kinematic inversion of parallel manipulators in the presence of incompletely specified tasks, Journal of Mechanical Design, Vol. 112, 1990(12), pp.494-500
14. G.R. Pennock and D.J.Kassner, Kinematic Analysis of a Planar Eight-Bar Linkage: Application to a Platform-Type Robot, Journal of Mechanical Design, Vol. 114, 1992(13), pp. 87-95
15. C.M. Gosselin and Jiegao Wang, 1997, "Singularity loci of planar parallel manipulators with revolute actuators," Robotics and Autonomous Systems 21 (1997) 377-398.
16. J. Sefrioui and C.M. Gosselin, On the quadratic nature of the singularity curves of planar 3-dof parallel manipulators, Mech, Mach. Theory, Vol.30, No.4, pp533-551, 1995.
17. G.R.Pennock and D.J.Kassner, The Workspace of a General Geometry Planar Three-Degree-of-Freedom Platform-Type Manipulator, Journal of Mechanical Design,
Vol. 115, 1993(6), pp. 269-276
18．刘辛军，汪劲松，李剑锋，高峰，一种新型空间3自由度并联机构的正反解及工作空间分析，机械工程学报，第37卷第10期，2001(10)，pp．36-39．
19. Xianwen Kong and C.M.Gosselin, Derermination of the uniqueness domains of 3-RPR planar parallel manipulators with similar platforms, Proceedings of DETC'00 ASME 2000 Design Engineering Technical conferences and Computers and Information in Engineering Conference Baltimore, Maryland, September 10-13,2000.
20. Xin-Jun Lin,Jin-Song Wang and Feng Gao, On the optimum design of planar 3-dof parallel manipulators with respect to the workspace, Proceedings of the 2000 IEEE international Conference on Robotics & Automation San Francisco, CA·April 2000
21. C.M. Gosselin and J.Angeles, The Optimum Kinematic Design of a Planar Three-Degree-of-Freedom Parallel Manipulator, Journals of Mechanisms, Transmissions, and Automation in Design, Vol. 110, 1988(3), pp.35-41
22．杨珏，何旭洪，赵昊彤，Mathematica应用指南，人民邮电出版社，1999
23．谭浩强，张基温，唐永炎，C语言程序设计教程，高等教育出版社，1998
24．黄维通，Visual C++面向对象于可视化程序设计，清华大学出版社，2000
25．向世明，OpenGL编程与实例，电子工业出版社，1999
26．徐晓刚，高兆法，王秀娟，VC++6.0入门与提高，清华大学出版社，1999
27．齐舒创作室，Visual C++6.0编程技巧与实例分析，中国水利水电出版社，1999
28．贾志刚，精通OpenGL，电子工业出版社，1998
29．郑兼荣，ADAMS—虚拟样机技术入门与提高，机械工业出版社，2002
30．袁士杰，吕哲勤，多刚体系统动力学，北京，北京理工大学出版社，1992
1、王超群，吴洪涛，一种平面并联机械手的奇异形位分析，淮海工学院学报，2003，1。