并联式仿生机械腿结构设计及动力学研究
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摘要
并联机构具有模块化程度高、刚度重量比大、响应速度快以及适应性强等优点,本文选用一类两转动一平动并联机构作为矿井救灾机器人机械腿的核心部分。对满足此运动要求的并联机构构型进行了综合,研究了并联式仿生机械腿相应的运动学、奇异位形、几何误差及其标定、空间运动性能优化、动力学建模以及刚度优化等关键技术,研制了一台单并联式仿生机械腿实验系统。本文的具体研究内容及主要工作可概括如下:
运用互易螺旋理论综合得到各种满足两转动一平动运动要求的并联机构构型,进而得到一类支链仅存在一线矢力的并联机构构型,并以此类构型(包括3-UCR、3-RPS等构型)做为之后研究的并联机构;以3-UCR对称并联机构为例,结合主螺旋理论与虚拟机构理论对其瞬态运动特性进行了验证,得到了此类并联机构一阶影响系数矩阵的建立与机构运动速度无关,只与机构输入参数、机构构型以及尺寸参数有关的结论。
采用Rodrigues参数对以3-RPS对称并联机构为核心的并联式仿生机械腿进行运动学建模,使得其在运动计算过程中计算参数最少、无三角函数计算、便于在线实时控制。通过机构特有的几何特性建立约束方程组,经消元处理后得到只含有一个未知数的高次输入输出方程,建立了正运动学解析模型。运用牛顿-拉夫森定理分析了机构正运动学数值解,通过数值仿真验证了所建迭代模型的正确性。根据机构空间矢量关系推导出机构速度正、逆运动学模型。分析得到机构中动平台、各支链以及末端执行杆件相应的速度、加速度解析解,为之后的机构整体动力学分析提供了运动学基础。
建立了3-RPS对称并联机构的静力学平衡方程,基于Grassmann几何从空间几何本质上对机构可能出现的正运动学奇异位形进行了研究,归纳得到各线簇秩对应的奇异位形,并通过数值仿真得到在运动空间内的各奇异轨迹,分析了Rodrigues参数随之发生的变化,从而为规划尽量避开奇异位形的运动轨迹提供了依据。
运用矩阵全微分理论,建立了并联式仿生机械腿末端位姿的几何误差模型。通过对此模型的数值仿真,可知一些结构误差使末端位姿发生了明显变化,在其制造与装配工艺性方面应相应加强;各项误差导致的动平台位置误差呈非线性变化,但变化不大,可通过软件方法进行补偿。经过归一化可达工作空间内各误差来源,得到在统计意义下的灵敏度系数数学模型,通过数值仿真可知在机构设计、加工以及装配过程中需严格控制的各类误差。通过逐次逼近迭代算法,建立了各运动副运动学参数的标定数学模型,通过数值仿真可知此逼近算法对于运动学参数标定分析比较有效,其优化迭代过程收敛速度较快。由于采用了灵敏度系数转换,所以其标定过程对灵敏度系数较高的误差来源具有突出的方向性和目标性,通过标定前后运动学参数误差值的对比,表明此标定方法可明显改善标定精度。
通过对3-RPS对称并联机构运动空间影响因素的仿真分析可知,加入末端杆件可有效增大机构末端运动空间。随着条件数取值范围的变小,其空间限制将增大;对运动空间条件数进行优化时,可取最大允许条件数为4;整个运动空间在垂直于定平台的轴上表现为一个“上大下小”的类椭圆体。矩阵奇异值在0.3≤δ≤3范围内时,可得到较大的运动空间。通过对机构全域条件数分析可知,在保证整体机构灵巧性的前提下,为了得到更大的运动空间,须使定、动平台运动副连接点结构比例系数在1.0≤η≤1.4范围内。
基于虚功原理,建立了以3-RPS对称并联机构为核心的并联式仿生机械腿的整体动力学模型,进而推导得到机构子支链、动平台连接点、定平台连接点、动平台以及末端杆件等部件的作用力与力矩的解析表达式。动力学数值仿真结果从结构上说明了支链驱动力数学模型的正确性,动平台连接点的轴向作用力、定平台连接点的法向作用力显著增加。由于垂直于定平台的轴向运动变换使得连接点所受作用力变化较快,较之对应的转动,其对连接点有着更大的影响。
运用机构能量守恒变换(CCT)的概念,兼顾到外力作用下并联式机械腿的几何形变,推导得到相应的守恒刚度映射矩阵,进而分析得出内部元素单位统一的机构刚度Jacobi矩阵。提出了机构运动空间刚度的特征值极值平均数的概念,并通过此概念优化整体机构刚度。根据分析得到的不同结构比例以及各空间剖面之间的刚度变化情况,归纳出垂直于定平台的各轴向空间截面的刚度变化趋势,得到结构比例系数1.2≤η≤1.4时机构整体刚度性能较优异的结论。
设计研制了一台高精度数显测微仪,保证了单并联式仿生机械腿实验系统的设计指标在加工、装配过程中的实现。建立了以PC主机、多轴运动控制模块、运动空间综合评价模块为核心的系统控制结构,采用了先置轨迹规划为主、批量传递为辅的数据传递模式,提高了系统的稳定性,减少了数据传递量,加快了系统响应速度。测得单并联式仿生机械腿实验系统的运动极限位置,根据人类行走的基本步态,初步规划了机械腿的运动轨迹,并对实验系统的机械结构以及控制策略等方面提出了改进意见。
Parallel manipulators have been intensively studied for over a decade. The manipulators have more advantages including good system rigidity, rapid motion velocity, not-accumulative error, high nominal load to weight ratio, and flexible end position-stance. One kind of symmetrical parallel manipulators with two rotational and one translational degree of freedom was designed as the key part of rescue robots in coal mines. After the structures of the parallel manipulators with 2R1T DOF were synthesized, the dissertation was focused on the theoretical analysis in terms of singular configures, error modeling and kinematics calibration, optimization of kinematic characters, dynamic modeling and stiffness optimization. As a result, the experimental system of single parallel bionic robot leg has been developed.
Subsequent research contents and conclusions are presented as follows:
The parallel manipulators with 2R1T DOF were synthesized by the reciprocal screw theory. Especially, the design rules and the combinations of revolute and prismatic joints with only one pure wrench force in one limb have been obtained. Based on principal screw theory and imaginary mechanism method, the kinematic characters of a three DOF symmetrical spatial parallel manipulator with three UCR limbs were validated. It is proved that the matrices of influence coefficient are only dependent on input parameters and dimension parameters, mechanical structures. When the moving platform is parallel to the base, the results of the pitches principal screws validate the kinematic characters of the parallel manipulator.
A new method with Rodrigues parameters was proposed to describe the position-stance of the parallel manipulator. By analyzing the topologic structure of the parallel manipulator, the kinematic model with Rodrigues parameters was established. By the geometrical constraints and elimination, the forward kinematic model was obtained. The numerical solutions of forward kinematic model were analyzed by using the Newton-Raphson method and the corresponding numerical simulation proved the validity of iterative steps. According to the relation of spatial vectors, the velocity and the acceleration of every part of the manipulator were deduced.
The static equilibrium equations of the parallel manipulators are developed. The forward singular configures were studied from their geometric essence by means of Grassmann theory. And singular configures of different line dimensions from 1 to 6 were concluded. The numerical simulation showed the singular trajectories and the change of every Rodrigues parameters. It is essential to design the kinematic layouts aiming at avoiding the singular configures.
By the means of the complete differential-coefficient matrix theory, we established the mechanical position-stance error model including the main errors of the parallel manipulators. By normalizing all error sources in reachable workspace, the statistical model of sensitivity coefficients was obtained. Based on successive approximation algorithm, the kinematic parameters can be calibrated. Considering the analytical results of sensitivity coefficients, the operation steps have concrete directivity.
The numerical analysis of workspace showed that the end effector can improve the workspace effectively. The whole workspace along Z axis which is perpendicular to the base can be described as a quasi-ellipsoid. The scope of singular values setting as 0.3≤δ≤3 can get more workspace. By the analysis of global condition index, in order to get more workspace and ensure dexterity of the whole manipulator, the radius ratio of the base and the moving platform can be set as 1.0≤η≤1.4.
Based on principle of virtual work, the dynamic model of the symmetrical parallel manipulators with three RPS limbs was established. Moreover, the forces and the moments of the limbs, the joints, the base, the moving platform and the end effector have been presented analytically. The joint forces of the moving platform in axial direction and the base in normal direction changed greatly. Because the motion along Z axis caused in great change of the joint forces, it is stated that the translational motion have more influence on the joints, relatively to the rotational motions.
Considering geometrical deformation of the parallel manipulator causing by outside forces, the conservative stiffness matrix was deduced on the basis of the conservative congruence transformation (CCT). Moreover, the element units of stiffness Jacobi matrix have been uniformed. The average of eigenvalue extremum of workspace stiffness was proposed for optimizing the whole stiffness of the parallel manipulator. When the radius ratio is set as 1.2≤η≤1.4, the whole performance of stiffness is better comparatively.
For the purpose of guaranteeing the design targets of the single parallel bionic robot leg, the high precision digital display instrument has been developed. The key parts of control system include PC, multi-axis control modularity and comprehensive evaluation modularity of workspace. The tracks are designed ahead for the kinematic control. The method of data transmission improves stability and expedited the response velocity of system. The position limits of the parallel bionic robot leg are measured by Optotrak Position Sensor. According to gaits of human being, the elementary track is programmed. The suggestions are proposed to improve the mechanical structure and control policy.
运用互易螺旋理论综合得到各种满足两转动一平动运动要求的并联机构构型,进而得到一类支链仅存在一线矢力的并联机构构型,并以此类构型(包括3-UCR、3-RPS等构型)做为之后研究的并联机构;以3-UCR对称并联机构为例,结合主螺旋理论与虚拟机构理论对其瞬态运动特性进行了验证,得到了此类并联机构一阶影响系数矩阵的建立与机构运动速度无关,只与机构输入参数、机构构型以及尺寸参数有关的结论。
采用Rodrigues参数对以3-RPS对称并联机构为核心的并联式仿生机械腿进行运动学建模,使得其在运动计算过程中计算参数最少、无三角函数计算、便于在线实时控制。通过机构特有的几何特性建立约束方程组,经消元处理后得到只含有一个未知数的高次输入输出方程,建立了正运动学解析模型。运用牛顿-拉夫森定理分析了机构正运动学数值解,通过数值仿真验证了所建迭代模型的正确性。根据机构空间矢量关系推导出机构速度正、逆运动学模型。分析得到机构中动平台、各支链以及末端执行杆件相应的速度、加速度解析解,为之后的机构整体动力学分析提供了运动学基础。
建立了3-RPS对称并联机构的静力学平衡方程,基于Grassmann几何从空间几何本质上对机构可能出现的正运动学奇异位形进行了研究,归纳得到各线簇秩对应的奇异位形,并通过数值仿真得到在运动空间内的各奇异轨迹,分析了Rodrigues参数随之发生的变化,从而为规划尽量避开奇异位形的运动轨迹提供了依据。
运用矩阵全微分理论,建立了并联式仿生机械腿末端位姿的几何误差模型。通过对此模型的数值仿真,可知一些结构误差使末端位姿发生了明显变化,在其制造与装配工艺性方面应相应加强;各项误差导致的动平台位置误差呈非线性变化,但变化不大,可通过软件方法进行补偿。经过归一化可达工作空间内各误差来源,得到在统计意义下的灵敏度系数数学模型,通过数值仿真可知在机构设计、加工以及装配过程中需严格控制的各类误差。通过逐次逼近迭代算法,建立了各运动副运动学参数的标定数学模型,通过数值仿真可知此逼近算法对于运动学参数标定分析比较有效,其优化迭代过程收敛速度较快。由于采用了灵敏度系数转换,所以其标定过程对灵敏度系数较高的误差来源具有突出的方向性和目标性,通过标定前后运动学参数误差值的对比,表明此标定方法可明显改善标定精度。
通过对3-RPS对称并联机构运动空间影响因素的仿真分析可知,加入末端杆件可有效增大机构末端运动空间。随着条件数取值范围的变小,其空间限制将增大;对运动空间条件数进行优化时,可取最大允许条件数为4;整个运动空间在垂直于定平台的轴上表现为一个“上大下小”的类椭圆体。矩阵奇异值在0.3≤δ≤3范围内时,可得到较大的运动空间。通过对机构全域条件数分析可知,在保证整体机构灵巧性的前提下,为了得到更大的运动空间,须使定、动平台运动副连接点结构比例系数在1.0≤η≤1.4范围内。
基于虚功原理,建立了以3-RPS对称并联机构为核心的并联式仿生机械腿的整体动力学模型,进而推导得到机构子支链、动平台连接点、定平台连接点、动平台以及末端杆件等部件的作用力与力矩的解析表达式。动力学数值仿真结果从结构上说明了支链驱动力数学模型的正确性,动平台连接点的轴向作用力、定平台连接点的法向作用力显著增加。由于垂直于定平台的轴向运动变换使得连接点所受作用力变化较快,较之对应的转动,其对连接点有着更大的影响。
运用机构能量守恒变换(CCT)的概念,兼顾到外力作用下并联式机械腿的几何形变,推导得到相应的守恒刚度映射矩阵,进而分析得出内部元素单位统一的机构刚度Jacobi矩阵。提出了机构运动空间刚度的特征值极值平均数的概念,并通过此概念优化整体机构刚度。根据分析得到的不同结构比例以及各空间剖面之间的刚度变化情况,归纳出垂直于定平台的各轴向空间截面的刚度变化趋势,得到结构比例系数1.2≤η≤1.4时机构整体刚度性能较优异的结论。
设计研制了一台高精度数显测微仪,保证了单并联式仿生机械腿实验系统的设计指标在加工、装配过程中的实现。建立了以PC主机、多轴运动控制模块、运动空间综合评价模块为核心的系统控制结构,采用了先置轨迹规划为主、批量传递为辅的数据传递模式,提高了系统的稳定性,减少了数据传递量,加快了系统响应速度。测得单并联式仿生机械腿实验系统的运动极限位置,根据人类行走的基本步态,初步规划了机械腿的运动轨迹,并对实验系统的机械结构以及控制策略等方面提出了改进意见。
Parallel manipulators have been intensively studied for over a decade. The manipulators have more advantages including good system rigidity, rapid motion velocity, not-accumulative error, high nominal load to weight ratio, and flexible end position-stance. One kind of symmetrical parallel manipulators with two rotational and one translational degree of freedom was designed as the key part of rescue robots in coal mines. After the structures of the parallel manipulators with 2R1T DOF were synthesized, the dissertation was focused on the theoretical analysis in terms of singular configures, error modeling and kinematics calibration, optimization of kinematic characters, dynamic modeling and stiffness optimization. As a result, the experimental system of single parallel bionic robot leg has been developed.
Subsequent research contents and conclusions are presented as follows:
The parallel manipulators with 2R1T DOF were synthesized by the reciprocal screw theory. Especially, the design rules and the combinations of revolute and prismatic joints with only one pure wrench force in one limb have been obtained. Based on principal screw theory and imaginary mechanism method, the kinematic characters of a three DOF symmetrical spatial parallel manipulator with three UCR limbs were validated. It is proved that the matrices of influence coefficient are only dependent on input parameters and dimension parameters, mechanical structures. When the moving platform is parallel to the base, the results of the pitches principal screws validate the kinematic characters of the parallel manipulator.
A new method with Rodrigues parameters was proposed to describe the position-stance of the parallel manipulator. By analyzing the topologic structure of the parallel manipulator, the kinematic model with Rodrigues parameters was established. By the geometrical constraints and elimination, the forward kinematic model was obtained. The numerical solutions of forward kinematic model were analyzed by using the Newton-Raphson method and the corresponding numerical simulation proved the validity of iterative steps. According to the relation of spatial vectors, the velocity and the acceleration of every part of the manipulator were deduced.
The static equilibrium equations of the parallel manipulators are developed. The forward singular configures were studied from their geometric essence by means of Grassmann theory. And singular configures of different line dimensions from 1 to 6 were concluded. The numerical simulation showed the singular trajectories and the change of every Rodrigues parameters. It is essential to design the kinematic layouts aiming at avoiding the singular configures.
By the means of the complete differential-coefficient matrix theory, we established the mechanical position-stance error model including the main errors of the parallel manipulators. By normalizing all error sources in reachable workspace, the statistical model of sensitivity coefficients was obtained. Based on successive approximation algorithm, the kinematic parameters can be calibrated. Considering the analytical results of sensitivity coefficients, the operation steps have concrete directivity.
The numerical analysis of workspace showed that the end effector can improve the workspace effectively. The whole workspace along Z axis which is perpendicular to the base can be described as a quasi-ellipsoid. The scope of singular values setting as 0.3≤δ≤3 can get more workspace. By the analysis of global condition index, in order to get more workspace and ensure dexterity of the whole manipulator, the radius ratio of the base and the moving platform can be set as 1.0≤η≤1.4.
Based on principle of virtual work, the dynamic model of the symmetrical parallel manipulators with three RPS limbs was established. Moreover, the forces and the moments of the limbs, the joints, the base, the moving platform and the end effector have been presented analytically. The joint forces of the moving platform in axial direction and the base in normal direction changed greatly. Because the motion along Z axis caused in great change of the joint forces, it is stated that the translational motion have more influence on the joints, relatively to the rotational motions.
Considering geometrical deformation of the parallel manipulator causing by outside forces, the conservative stiffness matrix was deduced on the basis of the conservative congruence transformation (CCT). Moreover, the element units of stiffness Jacobi matrix have been uniformed. The average of eigenvalue extremum of workspace stiffness was proposed for optimizing the whole stiffness of the parallel manipulator. When the radius ratio is set as 1.2≤η≤1.4, the whole performance of stiffness is better comparatively.
For the purpose of guaranteeing the design targets of the single parallel bionic robot leg, the high precision digital display instrument has been developed. The key parts of control system include PC, multi-axis control modularity and comprehensive evaluation modularity of workspace. The tracks are designed ahead for the kinematic control. The method of data transmission improves stability and expedited the response velocity of system. The position limits of the parallel bionic robot leg are measured by Optotrak Position Sensor. According to gaits of human being, the elementary track is programmed. The suggestions are proposed to improve the mechanical structure and control policy.
引文
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