六自由度运动模拟平台的分析及结构参数的优化
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摘要
相对于串联机器人而言，Gough-Stewart平台具有精度高和承载能力强等优点，被广泛用作各种六自由度运动模拟平台。由于虎克铰比球铰的转角范围大、且比球铰能承受更大的拉力，从而六自由度运动模拟平台一般用虎克铰作为被动副，把液压缸或电动缸连接于动平台和静平台上。由于液压缸和电动缸中的活塞以及活塞杆不仅沿轴线方向作直线主动运动，还能绕轴线方向被动地转动，即为圆柱副，而不是移动副，从而实际上六自由度运动模拟平台是6-UCU（虎克铰-圆柱副-虎克铰）并联机器人，而不是通常所说的6-UPS（虎克铰-移动副-球铰）并联机器人。
为了得到性能优良的六自由度运动模拟平台，首先需要对其运动学和动力学特性进行分析，然后需要通过优化设计得到性能优良的结构参数。
六自由度运动模拟平台的运动学反解分析和动力学反解分析是其机械结构进行设计的基础。基于上、下铰采用虎克铰、中间采用圆柱副的实际结构形式，在建立完整运动学反解的基础上，本文利用牛顿——欧拉方程方法与达朗贝尔原理，建立了六自由度运动模拟平台的完整动力学反解模型。完整运动学反解分析和完整动力学反解分析分别得到了六自由度运动模拟平台各组成部分的运动状况和受力状况。在分析过程中得到了两种导致分母为零的特殊的虎克铰轴线方向布置形式。最后通过仿真实例验证了本文所建立的完整运动学反解模型和完整动力学反解模型的正确性。
当六自由度运动模拟平台在工作空间内存在奇异时，其动静态特性将发生变化。为了得到性能优良的六自由度运动模拟平台，需要在设计过程中进行奇异性分析和奇异性检测，使设计得到的机械结构参数在所需要的工作空间内不存在奇异位姿。由于虎克铰轴线布置方向会影响六自由度运动模拟平台的奇异性，从而本文考虑主动移动副和被动副虎克铰的影响对其奇异性进行分析。首先根据造成六自由度运动模拟平台奇异的不同原因，把奇异性分为两类：支路奇异与驱动奇异，然后应用螺旋理论分析得到了产生两种奇异类型的条件。通过分析得到了两种导致六自由度运动模拟平台处于支路奇异位姿的情况，即当虎克铰固定于动平台或固定于静平台上转轴的轴线与主动副的轴线共线时，六自由度运动模拟平台处于支路奇异位姿。为了对六自由度运动模拟平台的全部类型的奇异都能检测，本文相应地提出了可直接检测得到六自由度运动模拟平台在整个六维给定工作空间或六维可达工作空间内是否存在奇异位姿结论的奇异性检测算法。通过实例分析验证了本文所提出的奇异性检测算法的有效性。
本文对六自由度运动模拟平台的两个常用性能指标函数——基于运动学雅克比矩阵的条件数和可操作度进行了介绍。由于基于运动学传统雅克比矩阵的可操作度和条件数数值随着传统雅克比矩阵中元素表示单位的不同而发生变化，从而也对两种常用构造量纲统一的新雅克矩阵的方法——特征长度法和三点坐标法的具体推导过程进行了叙述。为了度量各个作动器动态特性的一致性，本章在铰点工作空间内定义了一个新的性能指标函数——基于广义惯量矩阵的条件数，并且通过仿真实例验证了新性能指标函数不随广义惯量矩阵中元素表示单位的不同而发生变化。
为了得到性能优良的六自由度运动模拟平台，需要对其结构参数进行优化。实际上机器人的设计通常不是一步到位的，而是多步迭代的过程，从而需要在设计的初始阶段后能为设计者提供多个备选方案。六自由度运动模拟平台的设计可以分为两步，其中第一步设计过程中需依据运动学的要求进行结构参数的设计以满足用户对工作空间的需求。本文首先根据应用需求的不同，把六自由度运动模拟平台设计问题分为3类，然后提出了相应的结构参数优化设计算法。由于基于传统运动学雅克比矩阵的条件数与可操作度能很好地表征六自由度运动模拟平台的性能特性，它们被选作目标函数，然后应用多目标进化算法NSGA-II同时对这两个目标函数进行寻优。寻优结果能得到多组优化解，从而能为设计者在第二步设计过程中提供多个备选方案。本文通过3个设计实例，验证了本文所提出的结构参数优化算法的有效性。
Compared with serial robot, Gough-Stewart platform has been widely used as motion simulators for its higher accuracy and higher load capability, etc. Compared with spherical joint, universal joint can bear more tension and rotate in larger angular range, and then universal joint has been extensively used as the passive joints of6-DOF (degree of freedom) motion simulation platforms to connect hydraulic cylinder or electric cylinder to the moving platform and fixed base. The piston and piston rod of the hydraulic cylinder and electric cylinder not only retract (or extend) along the axis of the cylinder actively, but also can rotate along the axis passively, so it is a cylinder joint, and then these6-DOF motion simulation platforms are6-UCU parallel manipulator, not the6-UPS parallel manipulator, where U stands for the universal joint, C for the cylinder joint, P for the prismatic joint, and S for the spherical joint.
In order to derive good performance of the6-DOF motion simulation platform, its kinematic characteristics and dynamic characteristics are needed to be analyzed firstly, and then the good performance structural parameters are needed to be derived through optimal design.
The inverse kinematic and dynamic analyses are the mechanical structure design bases for the6-DOF motion simulation platforms. In practice, the real structure of the6-DOF motion simulation platforms is using universal joints as the lower and upper passive joints of each leg, and using cylinder joint in the middle of each leg. Based on the real structure, the complete inverse kinematic modeling of the6-DOF motion simulation platforms is obtained firstly, and then the complete inverse dynamics of the6-DOF motion simulation platforms is analyzed by using the Newton-Euler method and D'Alembert principle. The movements and forces of various components of6-DOF motion simulation platforms can be derived through the complete inverse kinematics analysis and the complete inverse dynamics analysis respectively. Two special universal joint axis arrangements, which cause denominator to be zero, are derived through the kinematic analysis process and dynamic analysis process. The correctness of the complete inverse kinematic model and complete inverse dynamic model of the6-DOF motion simulation platforms is confirmed through a case study.
If the6-DOF motion simulation platforms are in a singular pose in the required workspace, their kinestatic characteristics would be changed. In order to derive good performance of the6-DOF motion simulation platform, the required workspaces for the optimized mechanical structure parameters are needed with no singular posture, and then singularity analysis and singularity detection are needed in the design process. The singularities considering the impacts of the active prismatic joint and the passive universal joint are analyzed in this thesis as the universal joint axis arrangement would affect the singularities of the6-DOF motion simulation platforms. Two singular types are defined as leg singularity and actuator singularity according to the different causing reasons. The conditions caused these singularities are derived by using screw theory. Two cases of leg singularities of the6-DOF motion simulation platforms are found in this thesis, namely, when the rotational joint axis fixed on the base or fixed on the moving platform of the universal joints and the axial direction of the actuator joint of the same leg are collinear, the6-DOF motion simulation platform is in the leg singularity posture. In order to detect all the singularities of the6-DOF motion simulation platforms, singularity detection procedures are proposed correspondingly, which can directly determine whether there are singularities within a6-DOF given workspace or6-DOF reachable workspace or not. The effectiveness of the proposed singularity detection procedures are confirmed through case studies.
Two commonly used performance index functions of the6-DOF motion simulation platform, condition number and manipulability index based on the kinematic traditional Jacobian matrix, are introduced in this thesis. Because the values of condition number and manipulability index based on the kinematic traditional Jacobian matrix are changed as the matrix elements in different units, then the concrete derivations of the two common homogeneous matrix formulation methods, characteristic length method and three end-effector point method, are presented. In order to measure the consistency of each actuator dynamics, a new condition number performance index function based on the generalized inertia matrix in the joint workspace is proposed. The invariant of the new condition number as the elements of the generalized inertia matrix in different units is confirmed through a case study.
In order to get good performance of the6-DOF motion simulation platforms, it is needed to optimize in the design procedure. In practice, there is a multi step process to derive the final parameters of the robot not with only one step, and then multiple alternatives are needed to provide for designers after the primary design stage. The6-DOF motion simulation platform design procedure can be divided into two stages. In the first design stage, structural parameter design is needed to meet the workspace requirements of users. The6-DOF motion simulation platform design issues are divided into three categories according to different application requirements, and then the corresponding structural parameter optimization design algorithms are put forward. Because condition number and manipulability index based on the kinematic traditional Jacobian matrix are consistent with the performances of the6-DOF motion simulation platform, they are chosen as optimization objective functions. Multi-objective evolutionary algorithm NSGA-II is applied to optimize the two objective functions simultaneously to get multiple optimal solutions, and then multiple alternatives can be provided for designers in the secondary design stage. The effectiveness of the proposed structural parameter optimization design procedures is confirmed through three design case studies.
为了得到性能优良的六自由度运动模拟平台，首先需要对其运动学和动力学特性进行分析，然后需要通过优化设计得到性能优良的结构参数。
六自由度运动模拟平台的运动学反解分析和动力学反解分析是其机械结构进行设计的基础。基于上、下铰采用虎克铰、中间采用圆柱副的实际结构形式，在建立完整运动学反解的基础上，本文利用牛顿——欧拉方程方法与达朗贝尔原理，建立了六自由度运动模拟平台的完整动力学反解模型。完整运动学反解分析和完整动力学反解分析分别得到了六自由度运动模拟平台各组成部分的运动状况和受力状况。在分析过程中得到了两种导致分母为零的特殊的虎克铰轴线方向布置形式。最后通过仿真实例验证了本文所建立的完整运动学反解模型和完整动力学反解模型的正确性。
当六自由度运动模拟平台在工作空间内存在奇异时，其动静态特性将发生变化。为了得到性能优良的六自由度运动模拟平台，需要在设计过程中进行奇异性分析和奇异性检测，使设计得到的机械结构参数在所需要的工作空间内不存在奇异位姿。由于虎克铰轴线布置方向会影响六自由度运动模拟平台的奇异性，从而本文考虑主动移动副和被动副虎克铰的影响对其奇异性进行分析。首先根据造成六自由度运动模拟平台奇异的不同原因，把奇异性分为两类：支路奇异与驱动奇异，然后应用螺旋理论分析得到了产生两种奇异类型的条件。通过分析得到了两种导致六自由度运动模拟平台处于支路奇异位姿的情况，即当虎克铰固定于动平台或固定于静平台上转轴的轴线与主动副的轴线共线时，六自由度运动模拟平台处于支路奇异位姿。为了对六自由度运动模拟平台的全部类型的奇异都能检测，本文相应地提出了可直接检测得到六自由度运动模拟平台在整个六维给定工作空间或六维可达工作空间内是否存在奇异位姿结论的奇异性检测算法。通过实例分析验证了本文所提出的奇异性检测算法的有效性。
本文对六自由度运动模拟平台的两个常用性能指标函数——基于运动学雅克比矩阵的条件数和可操作度进行了介绍。由于基于运动学传统雅克比矩阵的可操作度和条件数数值随着传统雅克比矩阵中元素表示单位的不同而发生变化，从而也对两种常用构造量纲统一的新雅克矩阵的方法——特征长度法和三点坐标法的具体推导过程进行了叙述。为了度量各个作动器动态特性的一致性，本章在铰点工作空间内定义了一个新的性能指标函数——基于广义惯量矩阵的条件数，并且通过仿真实例验证了新性能指标函数不随广义惯量矩阵中元素表示单位的不同而发生变化。
为了得到性能优良的六自由度运动模拟平台，需要对其结构参数进行优化。实际上机器人的设计通常不是一步到位的，而是多步迭代的过程，从而需要在设计的初始阶段后能为设计者提供多个备选方案。六自由度运动模拟平台的设计可以分为两步，其中第一步设计过程中需依据运动学的要求进行结构参数的设计以满足用户对工作空间的需求。本文首先根据应用需求的不同，把六自由度运动模拟平台设计问题分为3类，然后提出了相应的结构参数优化设计算法。由于基于传统运动学雅克比矩阵的条件数与可操作度能很好地表征六自由度运动模拟平台的性能特性，它们被选作目标函数，然后应用多目标进化算法NSGA-II同时对这两个目标函数进行寻优。寻优结果能得到多组优化解，从而能为设计者在第二步设计过程中提供多个备选方案。本文通过3个设计实例，验证了本文所提出的结构参数优化算法的有效性。
Compared with serial robot, Gough-Stewart platform has been widely used as motion simulators for its higher accuracy and higher load capability, etc. Compared with spherical joint, universal joint can bear more tension and rotate in larger angular range, and then universal joint has been extensively used as the passive joints of6-DOF (degree of freedom) motion simulation platforms to connect hydraulic cylinder or electric cylinder to the moving platform and fixed base. The piston and piston rod of the hydraulic cylinder and electric cylinder not only retract (or extend) along the axis of the cylinder actively, but also can rotate along the axis passively, so it is a cylinder joint, and then these6-DOF motion simulation platforms are6-UCU parallel manipulator, not the6-UPS parallel manipulator, where U stands for the universal joint, C for the cylinder joint, P for the prismatic joint, and S for the spherical joint.
In order to derive good performance of the6-DOF motion simulation platform, its kinematic characteristics and dynamic characteristics are needed to be analyzed firstly, and then the good performance structural parameters are needed to be derived through optimal design.
The inverse kinematic and dynamic analyses are the mechanical structure design bases for the6-DOF motion simulation platforms. In practice, the real structure of the6-DOF motion simulation platforms is using universal joints as the lower and upper passive joints of each leg, and using cylinder joint in the middle of each leg. Based on the real structure, the complete inverse kinematic modeling of the6-DOF motion simulation platforms is obtained firstly, and then the complete inverse dynamics of the6-DOF motion simulation platforms is analyzed by using the Newton-Euler method and D'Alembert principle. The movements and forces of various components of6-DOF motion simulation platforms can be derived through the complete inverse kinematics analysis and the complete inverse dynamics analysis respectively. Two special universal joint axis arrangements, which cause denominator to be zero, are derived through the kinematic analysis process and dynamic analysis process. The correctness of the complete inverse kinematic model and complete inverse dynamic model of the6-DOF motion simulation platforms is confirmed through a case study.
If the6-DOF motion simulation platforms are in a singular pose in the required workspace, their kinestatic characteristics would be changed. In order to derive good performance of the6-DOF motion simulation platform, the required workspaces for the optimized mechanical structure parameters are needed with no singular posture, and then singularity analysis and singularity detection are needed in the design process. The singularities considering the impacts of the active prismatic joint and the passive universal joint are analyzed in this thesis as the universal joint axis arrangement would affect the singularities of the6-DOF motion simulation platforms. Two singular types are defined as leg singularity and actuator singularity according to the different causing reasons. The conditions caused these singularities are derived by using screw theory. Two cases of leg singularities of the6-DOF motion simulation platforms are found in this thesis, namely, when the rotational joint axis fixed on the base or fixed on the moving platform of the universal joints and the axial direction of the actuator joint of the same leg are collinear, the6-DOF motion simulation platform is in the leg singularity posture. In order to detect all the singularities of the6-DOF motion simulation platforms, singularity detection procedures are proposed correspondingly, which can directly determine whether there are singularities within a6-DOF given workspace or6-DOF reachable workspace or not. The effectiveness of the proposed singularity detection procedures are confirmed through case studies.
Two commonly used performance index functions of the6-DOF motion simulation platform, condition number and manipulability index based on the kinematic traditional Jacobian matrix, are introduced in this thesis. Because the values of condition number and manipulability index based on the kinematic traditional Jacobian matrix are changed as the matrix elements in different units, then the concrete derivations of the two common homogeneous matrix formulation methods, characteristic length method and three end-effector point method, are presented. In order to measure the consistency of each actuator dynamics, a new condition number performance index function based on the generalized inertia matrix in the joint workspace is proposed. The invariant of the new condition number as the elements of the generalized inertia matrix in different units is confirmed through a case study.
In order to get good performance of the6-DOF motion simulation platforms, it is needed to optimize in the design procedure. In practice, there is a multi step process to derive the final parameters of the robot not with only one step, and then multiple alternatives are needed to provide for designers after the primary design stage. The6-DOF motion simulation platform design procedure can be divided into two stages. In the first design stage, structural parameter design is needed to meet the workspace requirements of users. The6-DOF motion simulation platform design issues are divided into three categories according to different application requirements, and then the corresponding structural parameter optimization design algorithms are put forward. Because condition number and manipulability index based on the kinematic traditional Jacobian matrix are consistent with the performances of the6-DOF motion simulation platform, they are chosen as optimization objective functions. Multi-objective evolutionary algorithm NSGA-II is applied to optimize the two objective functions simultaneously to get multiple optimal solutions, and then multiple alternatives can be provided for designers in the secondary design stage. The effectiveness of the proposed structural parameter optimization design procedures is confirmed through three design case studies.
引文
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