基于CAD变量几何法的并联机构弹性理论研究
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摘要
并联机构的刚度和弹性理论研究逐渐成为机器人研究领域的重要课题。传统并联机构刚度和弹性理论研究方法主要包括两大类：以推导并联机构的刚度矩阵为特征的解析解法和以模型离散化为特征的有限单元法，两类方法各有优缺点。本文综合两种解法的优点，采用CAD变量几何法，对3/6-SPS和3-SPR机构进行了弹性理论研究。
本文首先采用CAD变量几何法对两机构进行位置正反解分析，用解析法验证位置反解的正确性。然后用CAD变量几何法对两机构进行静力分析，采用基于力/力矩平衡方程的求解方法，直接推导出力雅克比矩阵，在三维软件中实现力雅克比矩阵的可视化表达，由力雅克比矩阵进行了并联机构的静力分析，并用解析法和仿真方法证明求解的正确性。
根据材料力学的知识，由求得的驱动力和约束力，进一步分析单个分支的变形，得到了等效分支长度，基于CAD变量几何法，重新构造了变形之后的分支模拟机构，求解变形前后两个模拟机构的位置、姿态差值，即完成了两机构的弹性变形分析；用解析法推导出了并联机构的刚度矩阵，并以此进行并联机构的弹性变形分析；在三维实体软件中构造机构的几何模型，用有限元软件进行机构的弹性变形求解；并比较了解析法、CAD变量几何法、有限元法求解机构弹性变形的精度。
采用CAD变量几何法，本文研究了3/6-SPS和3-SPR机构在不同工况、不同位姿的弹性变形情况，发现机构的刚度和位姿、受力类型有很大关系。比较了机构在平动、转动时各向弹性变形量的变化，由此得到机构运动时各向刚度的变化情况，分析了不同方向的作用力、力矩对各个方向刚度、角刚度的影响。
Stiffness and elastic deformation theories gradually become important subjects in thestudy of Parallel Mechanisms(PMs). There are two traditional methods in stiffness solvingand elastic deformation analysis of PMs, one is analytic method characterized by deducingstiffness matrix，the other is FEA methods characterized by model discretization. The twomethods have their own merits. Based on the advantages of both methods, this thesisfocuses on an analysis of the stiffness and the elastic deformation of the 3/6-SPS PM and3-SPR PM by a novel CAD variation geometric approach.
Based on CAD variation geometric approach, inverse and forward position solutionsof 3/6-SPS and 3-SPR PM are solved. By contrasting the inverse position solutions solvedby the analytic approach, the results obtained by CAD variation geometric approach areverified correctly. Based on force-torque balancing equations, the force Jacobian matrix isderived and the force simulation mechanism is constructed in CAD software. According tothe force Jacobian matrix and the force simulation mechanism, active/constrained forcecan be solved automatically. By contrasting the force solutions solved by the analyticapproach and motion analysis module in CAD software, the results obtained by CADvariation geometric approach are verified correctly.
According to the knowledge of material mechanics, the elastic deformation of limbsare solved and the equivalent length of limbs are calculated. Based on CAD variationgeometric approach, the position mechanism of PMs after deformation is rebuilt, thus, theposition and posture changes of moving platform can be solved. The stiffness matrix isderived and the elastic deformation is solved by the analytic approach. The solid model isbuilt in CAD software, and the deformation of PMs also can be solved by FEA method.By contrasting the elastic deformation solutions solved by the analytic approach and theFEA method, the results obtained by CAD variation geometric approach are verifiedcorrectly.
Based on CAD variation geometric approach, the elastic deformation of 3/6-SPS PMand 3-SPR PM working in some different conditions under different loads is calculated,according to the deformation results, it can be find that the position and posture of moving platform and the loads have a huge influence on the stiffness of PMs. According tocomparing the deformation in different conditions, the change of stiffness can be knownand the influence of forces and torques in different directions can be solved.
本文首先采用CAD变量几何法对两机构进行位置正反解分析，用解析法验证位置反解的正确性。然后用CAD变量几何法对两机构进行静力分析，采用基于力/力矩平衡方程的求解方法，直接推导出力雅克比矩阵，在三维软件中实现力雅克比矩阵的可视化表达，由力雅克比矩阵进行了并联机构的静力分析，并用解析法和仿真方法证明求解的正确性。
根据材料力学的知识，由求得的驱动力和约束力，进一步分析单个分支的变形，得到了等效分支长度，基于CAD变量几何法，重新构造了变形之后的分支模拟机构，求解变形前后两个模拟机构的位置、姿态差值，即完成了两机构的弹性变形分析；用解析法推导出了并联机构的刚度矩阵，并以此进行并联机构的弹性变形分析；在三维实体软件中构造机构的几何模型，用有限元软件进行机构的弹性变形求解；并比较了解析法、CAD变量几何法、有限元法求解机构弹性变形的精度。
采用CAD变量几何法，本文研究了3/6-SPS和3-SPR机构在不同工况、不同位姿的弹性变形情况，发现机构的刚度和位姿、受力类型有很大关系。比较了机构在平动、转动时各向弹性变形量的变化，由此得到机构运动时各向刚度的变化情况，分析了不同方向的作用力、力矩对各个方向刚度、角刚度的影响。
Stiffness and elastic deformation theories gradually become important subjects in thestudy of Parallel Mechanisms(PMs). There are two traditional methods in stiffness solvingand elastic deformation analysis of PMs, one is analytic method characterized by deducingstiffness matrix，the other is FEA methods characterized by model discretization. The twomethods have their own merits. Based on the advantages of both methods, this thesisfocuses on an analysis of the stiffness and the elastic deformation of the 3/6-SPS PM and3-SPR PM by a novel CAD variation geometric approach.
Based on CAD variation geometric approach, inverse and forward position solutionsof 3/6-SPS and 3-SPR PM are solved. By contrasting the inverse position solutions solvedby the analytic approach, the results obtained by CAD variation geometric approach areverified correctly. Based on force-torque balancing equations, the force Jacobian matrix isderived and the force simulation mechanism is constructed in CAD software. According tothe force Jacobian matrix and the force simulation mechanism, active/constrained forcecan be solved automatically. By contrasting the force solutions solved by the analyticapproach and motion analysis module in CAD software, the results obtained by CADvariation geometric approach are verified correctly.
According to the knowledge of material mechanics, the elastic deformation of limbsare solved and the equivalent length of limbs are calculated. Based on CAD variationgeometric approach, the position mechanism of PMs after deformation is rebuilt, thus, theposition and posture changes of moving platform can be solved. The stiffness matrix isderived and the elastic deformation is solved by the analytic approach. The solid model isbuilt in CAD software, and the deformation of PMs also can be solved by FEA method.By contrasting the elastic deformation solutions solved by the analytic approach and theFEA method, the results obtained by CAD variation geometric approach are verifiedcorrectly.
Based on CAD variation geometric approach, the elastic deformation of 3/6-SPS PMand 3-SPR PM working in some different conditions under different loads is calculated,according to the deformation results, it can be find that the position and posture of moving platform and the loads have a huge influence on the stiffness of PMs. According tocomparing the deformation in different conditions, the change of stiffness can be knownand the influence of forces and torques in different directions can be solved.
引文
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[17] Lu Y, Shi Y, Hu B. A CAD Geometric Variation Approach of Solving Active/ConstrainedForces ofSome Parallel Manipulators with SPR-type Active Legs[C].ASME 2007 International DesignEngineering Technical Conferences&Computers and Information in Engineering Conference,LasVegas,Nevada,USA, 2007, 8, Part B:839-845.
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[19] Ceccarelli M, Carbone G. A Stiffness Analysis for CaPaMan(Cassino Parallel Manipulator)[J].Mechanism and Machine Theory, 2002, 37(5): 427–439.
[20] Carbone G, Ceccarelli M. A Stiffness Analysis for a Hybrid Parallel-Serial Manipulator[J].Robotica, 2004, 22(5):567–576.
[21] Joshi S, Tsai L W. A Comparison Study of Two 3-DOF Parallel Manipulators: One with Three andthe Other with Four Supporting legs[J]. IEEE Transactions on Robotics and Automation, 2003,19(2): 200-209.
[22] Yoon W.K, Suehiro T, Tsumaki Y. Stiffness Analysis and Design of a Compact Modified DeltaParallel Mechanism[J]. Robotica, 2004, 22(4): 463–475.
[23] Zhang D, Lang Sherman Y.T.Stiffness Modeling for a Class of Reconfigurable PMs withThree toFive Degree of Freedom[J].Journel of Manufacturing Systems,2004,23(4):316–327.
[24] Zhang D. On Stiffness Improvement of the Tricept Machine Tool[J]. Robotica, 2005, 23(3):377-386.
[25] D.Zhang, C.M.Gosselin. Kineto static modeling of N-DOF parallel mechanisms with a passiveconstraining leg and prismatic actuators[J]. ASME Journal of Mechanical Design, 2001, 123(3):375-384
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