Stewart平台的运动学与逆动力学的基础研究
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摘要
Stewart平台并联机构由于具有刚度大、承载能力强、位置误差不累计等特点,在应用上与串联机构形成互补,已成为空间机构学的研究热点。目前,Stewart平台并联机构已经在航空、航天、海底作业、地下开采、制造装配等行业有着广泛的应用。尽管并联机构的实际应用和理论研究取得了大量的研究成果,但是在运动学、奇异性、动力学方面仍然存在一些有挑战性的问题,如少自由度并联机构雅可比矩阵的求解目前还没有1个统一的方法;寻求一种能求出并联机构位置正解的所有解的算法;寻求一种高效的逆动力学问题的解法等。本文将从空间并联机构的运动学、奇异性和逆动力学等方面对上述问题进行深入的研究。
雅可比矩阵是Stewart平台并联机构的1个主要内容,它表示由输入关节到末端执行器输出的一种映射。对于空间6自由度的Stewart平台并联机构,本文采用运动法求解其雅可比矩阵。对于少自由度并联机构,本文通过在每个支链上增加单自由度运动副,将每个支链等效转化为6自由度的支链,用运动法来求该转化机构的雅可比矩阵。该方法可以得到少自由度并联机构的完整雅可比矩阵,该方法对建立少自由度并联机构的雅可比矩阵具有通用性。
对于空间并联机构的奇异性问题。本文结合Stewart平台的位形参数与已求出的雅可比矩阵的结论,通过对该雅可比矩阵的化简求出了Stewart平台的1个奇异位形,该过程也验证了雅可比矩阵求解的正确性。当Stewart平台处于奇异位形时,动平台将得到多余的自由度并出现瞬时的螺旋运动,该螺旋运动可由动平台上不共线的3点的运动合成,并且只有当这3点速度的法平面的公共点在这3点所确定的平面内,这3点的运动才能合成1个螺旋运动。
对于Stewart平台并联机构的位置正解,本文以Stewart平台为例,建立了并联机构的约束方程组,该方程组为1组强耦合的非线性方程组,通过万能公式把该非线性方程组转化为多项式方程组的形式,最后,构造了初始方程组。本文基于同伦路径跟踪原理,提出预估——校正法,得到了求出并联机构所有位置正解的算法。通过具体的例子求出了Stewart平台位置正解的所有解,通过运动的连续性判断出其中的合理解。最后通过与其它算法结果的对比验证了该算法的正确性,有效性。
对于Stewart平台并联机构的的逆动力学问题,本文采用Kane方法,根据Stewart平台的特点采用动平台的位置和姿态参数作为广义坐标,以速度和角速度分量作为广义速度。在考虑Stewart平台的动平台和各个分支的重力和惯性力的基础上建立了完整的逆动力学模型。通过在分支的上、下连杆的质心建立局部坐标系,借助各个分支连杆的空间位置关系简化了分支连杆在惯性坐标系中惯性张量的求解。最后,对Stewart平台的逆动力学模型进行了数据仿真计算与仿真实验,同时也与现有的研究结果进行了对比实验,通过仿真实验与对比实验结果表明该模型的正确性,并且具有很高的精度。
Stewart platform is characterized by the high rigidity, large load handling capability, and non-accumulation of position error. Parallel mechanism is the supplement of the series mechanism and become the focus of the research in spatial mechanism. At present, Stewart platform extensive and important applications in many aspects such as aviation, flight, busywork in sea and land, underground exploitation, manufacturing and so on. However, despite the plentiful research achievements in the practical application and theory study in parallel mechanism, there are many challenging problems in kinematics, singularity and dynamics. For example, there is no a uniform method for solving the Jacobian matrix of the parallel mechanism which is deficient degrees of freedom, and searching for an algorithm that could solve all solutions to the forward kinematics of parallel mechanism and so on. In this dissertation, it was systematically dealed with kinematics, singularity, inverse dynamic for parallel mechanism.
Jacobian matrix is an important aspect of spatial parallel mechanism. Jacobian matrix is a mapping from input to outout. Method of motion was adopted to solve the Jacobian matrix of spatial parallel mechanism of 6 degrees of freedom. However, there is no a uniform method for solving the Jacobian matrix of the parallel mechanism which is deficient degrees of freedom. In this dissertation, the Jacobian matrix of deficient degrees of freedom parallel mechanism was established by adding imaginary kinematic pairs to each kineamtc branched-chain, which make every branched-chain to transform an imaginary mechanism with six degrees of freedom. The complete Jacobian matrix was derived from the imaginary parallel mechanism. This method is universal for deficient degrees of freedom parallel mechanism. The value of the Jacobian matrix determinant was detected in the course of motion planning based on the inverse kinematics to determine the reasonableness of the trajectory planned.
The singularity is an importan aspect that needed to be considered, and the Jacobian matrix palys an important role in this part. It could be determined the singularity of parallel mechanism by simplifiing the Jacobian matrix that had derived from the positon and configruation of the parallel mechanism. A singuilarity configuration was founded by simplifiing the Jacobian matrix, which verified the correctness of the Jacobian matrix. The moblie platform would get unexpect degrees of freedom and an instantaneous screw motion if it is in singularity configuration. The instaneous screw motion could be synthesised by the motion of three pointes that are not located in the same line on the mobile platform. The motion of the three points could synthesis a screw motin only when the intersection point of the three normal planes of velocity located in the plane that is determined by the three pointes.
As for the forward kinematics of the spatial parallel mechanism, Stewart platform was seltct as an example in this dissertation, by adopting homotype approach, an equation system of the mechanism structure constrain was established, which was a strong coupling equation system. Universial formula was used to transform nonlinear equation system into polynomial equation system. At last, the initial equation system was established. A predictor—corrector method to get the forward solutions was proposed based on the principle of homotypy, which is an method that could solve all solutions to all possible configuration of parallel mechanism. All solutions of Stewart platform was solved by this method, and all reasonable solutions were determined by considering the continuity of the motion of the platform. The comparison of the results of the homotopy and other algorithm was carried out, which verified the correctness and validity of the homotopy.
As for the inverse dynamic of the parallel mechanism, the parameters of the position and configuration were generalized coordinates according to the principle of the Kane, and the linear velocity and angular velocity were generalized velocity. The complete inverse dynamic model was built by considering the gravity and inertial of the all chains and mobile platform. The local frames of axes were located in the mass center of the each kinematic chain, which simplified the computation process inertial tensor of the every kinematic chain in the inertial frame of the axes. At last, the computer simulation was carried out to verify the correctness and reasonableness of the inverse dynamic model. The contrast test was carried out with the existing studing results, which showed that the homotopy algorithm of the inverse dynamics had high precision.
雅可比矩阵是Stewart平台并联机构的1个主要内容,它表示由输入关节到末端执行器输出的一种映射。对于空间6自由度的Stewart平台并联机构,本文采用运动法求解其雅可比矩阵。对于少自由度并联机构,本文通过在每个支链上增加单自由度运动副,将每个支链等效转化为6自由度的支链,用运动法来求该转化机构的雅可比矩阵。该方法可以得到少自由度并联机构的完整雅可比矩阵,该方法对建立少自由度并联机构的雅可比矩阵具有通用性。
对于空间并联机构的奇异性问题。本文结合Stewart平台的位形参数与已求出的雅可比矩阵的结论,通过对该雅可比矩阵的化简求出了Stewart平台的1个奇异位形,该过程也验证了雅可比矩阵求解的正确性。当Stewart平台处于奇异位形时,动平台将得到多余的自由度并出现瞬时的螺旋运动,该螺旋运动可由动平台上不共线的3点的运动合成,并且只有当这3点速度的法平面的公共点在这3点所确定的平面内,这3点的运动才能合成1个螺旋运动。
对于Stewart平台并联机构的位置正解,本文以Stewart平台为例,建立了并联机构的约束方程组,该方程组为1组强耦合的非线性方程组,通过万能公式把该非线性方程组转化为多项式方程组的形式,最后,构造了初始方程组。本文基于同伦路径跟踪原理,提出预估——校正法,得到了求出并联机构所有位置正解的算法。通过具体的例子求出了Stewart平台位置正解的所有解,通过运动的连续性判断出其中的合理解。最后通过与其它算法结果的对比验证了该算法的正确性,有效性。
对于Stewart平台并联机构的的逆动力学问题,本文采用Kane方法,根据Stewart平台的特点采用动平台的位置和姿态参数作为广义坐标,以速度和角速度分量作为广义速度。在考虑Stewart平台的动平台和各个分支的重力和惯性力的基础上建立了完整的逆动力学模型。通过在分支的上、下连杆的质心建立局部坐标系,借助各个分支连杆的空间位置关系简化了分支连杆在惯性坐标系中惯性张量的求解。最后,对Stewart平台的逆动力学模型进行了数据仿真计算与仿真实验,同时也与现有的研究结果进行了对比实验,通过仿真实验与对比实验结果表明该模型的正确性,并且具有很高的精度。
Stewart platform is characterized by the high rigidity, large load handling capability, and non-accumulation of position error. Parallel mechanism is the supplement of the series mechanism and become the focus of the research in spatial mechanism. At present, Stewart platform extensive and important applications in many aspects such as aviation, flight, busywork in sea and land, underground exploitation, manufacturing and so on. However, despite the plentiful research achievements in the practical application and theory study in parallel mechanism, there are many challenging problems in kinematics, singularity and dynamics. For example, there is no a uniform method for solving the Jacobian matrix of the parallel mechanism which is deficient degrees of freedom, and searching for an algorithm that could solve all solutions to the forward kinematics of parallel mechanism and so on. In this dissertation, it was systematically dealed with kinematics, singularity, inverse dynamic for parallel mechanism.
Jacobian matrix is an important aspect of spatial parallel mechanism. Jacobian matrix is a mapping from input to outout. Method of motion was adopted to solve the Jacobian matrix of spatial parallel mechanism of 6 degrees of freedom. However, there is no a uniform method for solving the Jacobian matrix of the parallel mechanism which is deficient degrees of freedom. In this dissertation, the Jacobian matrix of deficient degrees of freedom parallel mechanism was established by adding imaginary kinematic pairs to each kineamtc branched-chain, which make every branched-chain to transform an imaginary mechanism with six degrees of freedom. The complete Jacobian matrix was derived from the imaginary parallel mechanism. This method is universal for deficient degrees of freedom parallel mechanism. The value of the Jacobian matrix determinant was detected in the course of motion planning based on the inverse kinematics to determine the reasonableness of the trajectory planned.
The singularity is an importan aspect that needed to be considered, and the Jacobian matrix palys an important role in this part. It could be determined the singularity of parallel mechanism by simplifiing the Jacobian matrix that had derived from the positon and configruation of the parallel mechanism. A singuilarity configuration was founded by simplifiing the Jacobian matrix, which verified the correctness of the Jacobian matrix. The moblie platform would get unexpect degrees of freedom and an instantaneous screw motion if it is in singularity configuration. The instaneous screw motion could be synthesised by the motion of three pointes that are not located in the same line on the mobile platform. The motion of the three points could synthesis a screw motin only when the intersection point of the three normal planes of velocity located in the plane that is determined by the three pointes.
As for the forward kinematics of the spatial parallel mechanism, Stewart platform was seltct as an example in this dissertation, by adopting homotype approach, an equation system of the mechanism structure constrain was established, which was a strong coupling equation system. Universial formula was used to transform nonlinear equation system into polynomial equation system. At last, the initial equation system was established. A predictor—corrector method to get the forward solutions was proposed based on the principle of homotypy, which is an method that could solve all solutions to all possible configuration of parallel mechanism. All solutions of Stewart platform was solved by this method, and all reasonable solutions were determined by considering the continuity of the motion of the platform. The comparison of the results of the homotopy and other algorithm was carried out, which verified the correctness and validity of the homotopy.
As for the inverse dynamic of the parallel mechanism, the parameters of the position and configuration were generalized coordinates according to the principle of the Kane, and the linear velocity and angular velocity were generalized velocity. The complete inverse dynamic model was built by considering the gravity and inertial of the all chains and mobile platform. The local frames of axes were located in the mass center of the each kinematic chain, which simplified the computation process inertial tensor of the every kinematic chain in the inertial frame of the axes. At last, the computer simulation was carried out to verify the correctness and reasonableness of the inverse dynamic model. The contrast test was carried out with the existing studing results, which showed that the homotopy algorithm of the inverse dynamics had high precision.
引文
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