三自由度并联角台机构运动性能的理论分析及仿真
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摘要
现代机构学研究的主要任务,是对适应现代化生产的新机构进行有效的理论分析,从而为其应用提供明确的目标,减少新机构创新的盲目性。用已有特定性能的机构制造新机器,可以有效地开发资源、利用资源,为社会生产和生活服务。本文在国家自然科学基金的支持下,主要进行了三自由度并联角台机构的运动性能分析及仿真研究,对其中典型的3-RPS并联角台机构和3-(RP)S并联角台机构进行了分析和仿真,包括以下几个方面:
首先分析了3-RPS并联角台机构具有三个转动自由度:初始位置时具有绕确定轴的瞬时转动,以及离开初始位置后绕不确定轴的连续转动,且三个转动轴线在空间不相交。判断了机构输入选取的合理性。建立了角台机构的反解方程,通过确定结构约束和齐次变换矩阵,解得角台机构的三个移动副输入参数。角台机构和其他并联机构一样,正解是难点。根据空间几何条件并通过变量代换,确定了机构角台的位置和姿态。给出数值算例求得机构在给定条件下的正、反解。
把平面卡当(CARDAN)运动进行推广,定义这种三维空间中类似于小齿轮在大齿轮内表面的运动为空间卡当运动。从空间解析几何的角度,详细分析了3-RPS角台机构在卡当运动过程中,动角台各特征点的运动轨迹。文中分析、证明了3-RPS并联角台机构能实现空间三维卡当运动,其三维卡当运动在空间的运动轨迹为三坐标方向的椭圆簇。
用MATLAB对3-RPS角台机构的运动进行了仿真,结果表明:3-RPS并联角台机构是众多并联机构中较特殊的一种,该机构能实现空间的卡当运动和沿主对角线的螺旋运动;由于机构的三个转动轴线不相交,加工、制造较三个转动轴线交于一点的球面并联机构容易;由于转动的轴线不固定,因此适用于大角度、绕不确定轴转动的场合。
引进了虚拟机构法,成功地建立了3-(RP)S角台机构的Jacobian矩阵。基于二次曲线和二次曲面分解理论,对3-(RP)S角台机构的瞬时运动进行了分析,识别了该机构在三种位形下的主螺旋。研究了在每种位形下输出运动螺旋的节距与输入速度之间的关系以及具有相同节距的所有运动螺旋轴线在每种位形下的空间分布。并得出结论:此机构在起始位置和三杆等距伸长时属于第一种特殊的螺旋三系,在一般位形下属于Hunt提出的一般螺旋三系,且在任意位形下对应于相同节距的所有螺旋轴线都分布在同一个单叶双曲面上。上述分析有助于从整体上把握3-(RP)S角台机构的瞬时运动特性,为进一步研究其它角台机构的运动规律,促进该类机构的应用奠定了良好的基础。
The modern mechanics plays a dominating role in analyzing new machines applicable for the modernization production, which is useful for clear orientation of the applications and innovations. The new machines are often made of mechanisms with inherent and specific features, which is favorable to develop resources so as to better serve the social life. In the paper, the kinematic performance of 3-DOF parallel pyramid mechanisms, especially the 3-RPS and 3-(RP)S pyramid mechanisms are analyzed and simulated under the support of National Nature Science Foundation. The main contents of this paper are as follows:
In this paper, the mobility of the 3-RPS pyramid mechanism is firstly analyzed. This mechanism has three rotations, the axes of which are fixed under the initial displacement while that is always mutative under any other configuration. The paper also judges the rationality of input selection. Then it builds the equations for inverse problem and obtains the three input parameters according to the structure constraints and homogeneous transforming matrixes. Similar to other parallel mechanisms, it is also difficult to get the direct problem of the 3-RPS pyramid mechanism. The orientation is successfully determined by coordinate transformations in terms of the special geometric conditions. Furthermore, the numerical values of direct and inverse problems under certain condition are calculated.
The paper extends the plane CARDAN motion which is similar to the inside mesh movement of two gears. The kinematic loci of the representative points are analyzed in detail. It is proved that the 3-RPS pyramid mechanism also has the three-dimensional CARDAN motion and the kinematic trajectory is an ellipse cluster.
The motions of the 3-RPS pyramid mechanism are simulated using MATLAB, which indicates that this mechanism is the special one among diverse mechanisms. The 3-RPS pyramid mechanism has three-dimensional CARDAN and helical motions and its workspace is larger than that of the platform manipulator. Since the axes of three rotations are not intersecting, it is comparatively easy to be manufactured than that of the spherical parallel mechanisms. In addition, the mechanism can be applied to lange-angle rotations because of the various axes of motions.
The instantaneous motions of the 3-(RP)S pyramid mechanism are analyzed and the principal screws under three different configurations are identified by means of the conic section and the quadric degenerating theory, respectively. The relations between the pitches of the output twists and hree velocity inputs are described, and the spatial distribution of the axes of all the twists with the same pitch is illustrated under each configuration. It is concluded that the motions of this mechanism belong to the first special three-system under the first two configurations, while it belongs to a general third-order screw system under a general configuration. The axes of all the screws with the same pitch lie in a hyperboloid. The above study helps us comprehend the instantaneous kinematic property of the 3-(RP)S pyramid mechanism on the whole, and lays a foundation of the research on the kinematic laws and applications of other pyramid mechanisms.
首先分析了3-RPS并联角台机构具有三个转动自由度:初始位置时具有绕确定轴的瞬时转动,以及离开初始位置后绕不确定轴的连续转动,且三个转动轴线在空间不相交。判断了机构输入选取的合理性。建立了角台机构的反解方程,通过确定结构约束和齐次变换矩阵,解得角台机构的三个移动副输入参数。角台机构和其他并联机构一样,正解是难点。根据空间几何条件并通过变量代换,确定了机构角台的位置和姿态。给出数值算例求得机构在给定条件下的正、反解。
把平面卡当(CARDAN)运动进行推广,定义这种三维空间中类似于小齿轮在大齿轮内表面的运动为空间卡当运动。从空间解析几何的角度,详细分析了3-RPS角台机构在卡当运动过程中,动角台各特征点的运动轨迹。文中分析、证明了3-RPS并联角台机构能实现空间三维卡当运动,其三维卡当运动在空间的运动轨迹为三坐标方向的椭圆簇。
用MATLAB对3-RPS角台机构的运动进行了仿真,结果表明:3-RPS并联角台机构是众多并联机构中较特殊的一种,该机构能实现空间的卡当运动和沿主对角线的螺旋运动;由于机构的三个转动轴线不相交,加工、制造较三个转动轴线交于一点的球面并联机构容易;由于转动的轴线不固定,因此适用于大角度、绕不确定轴转动的场合。
引进了虚拟机构法,成功地建立了3-(RP)S角台机构的Jacobian矩阵。基于二次曲线和二次曲面分解理论,对3-(RP)S角台机构的瞬时运动进行了分析,识别了该机构在三种位形下的主螺旋。研究了在每种位形下输出运动螺旋的节距与输入速度之间的关系以及具有相同节距的所有运动螺旋轴线在每种位形下的空间分布。并得出结论:此机构在起始位置和三杆等距伸长时属于第一种特殊的螺旋三系,在一般位形下属于Hunt提出的一般螺旋三系,且在任意位形下对应于相同节距的所有螺旋轴线都分布在同一个单叶双曲面上。上述分析有助于从整体上把握3-(RP)S角台机构的瞬时运动特性,为进一步研究其它角台机构的运动规律,促进该类机构的应用奠定了良好的基础。
The modern mechanics plays a dominating role in analyzing new machines applicable for the modernization production, which is useful for clear orientation of the applications and innovations. The new machines are often made of mechanisms with inherent and specific features, which is favorable to develop resources so as to better serve the social life. In the paper, the kinematic performance of 3-DOF parallel pyramid mechanisms, especially the 3-RPS and 3-(RP)S pyramid mechanisms are analyzed and simulated under the support of National Nature Science Foundation. The main contents of this paper are as follows:
In this paper, the mobility of the 3-RPS pyramid mechanism is firstly analyzed. This mechanism has three rotations, the axes of which are fixed under the initial displacement while that is always mutative under any other configuration. The paper also judges the rationality of input selection. Then it builds the equations for inverse problem and obtains the three input parameters according to the structure constraints and homogeneous transforming matrixes. Similar to other parallel mechanisms, it is also difficult to get the direct problem of the 3-RPS pyramid mechanism. The orientation is successfully determined by coordinate transformations in terms of the special geometric conditions. Furthermore, the numerical values of direct and inverse problems under certain condition are calculated.
The paper extends the plane CARDAN motion which is similar to the inside mesh movement of two gears. The kinematic loci of the representative points are analyzed in detail. It is proved that the 3-RPS pyramid mechanism also has the three-dimensional CARDAN motion and the kinematic trajectory is an ellipse cluster.
The motions of the 3-RPS pyramid mechanism are simulated using MATLAB, which indicates that this mechanism is the special one among diverse mechanisms. The 3-RPS pyramid mechanism has three-dimensional CARDAN and helical motions and its workspace is larger than that of the platform manipulator. Since the axes of three rotations are not intersecting, it is comparatively easy to be manufactured than that of the spherical parallel mechanisms. In addition, the mechanism can be applied to lange-angle rotations because of the various axes of motions.
The instantaneous motions of the 3-(RP)S pyramid mechanism are analyzed and the principal screws under three different configurations are identified by means of the conic section and the quadric degenerating theory, respectively. The relations between the pitches of the output twists and hree velocity inputs are described, and the spatial distribution of the axes of all the twists with the same pitch is illustrated under each configuration. It is concluded that the motions of this mechanism belong to the first special three-system under the first two configurations, while it belongs to a general third-order screw system under a general configuration. The axes of all the screws with the same pitch lie in a hyperboloid. The above study helps us comprehend the instantaneous kinematic property of the 3-(RP)S pyramid mechanism on the whole, and lays a foundation of the research on the kinematic laws and applications of other pyramid mechanisms.
引文
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