两自由度精密定位平台结构设计与运动控制
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摘要
本文研制了一款以压电陶瓷驱动的两自由度精密定位平台，并通过静力学分析、有限元分析、动力学建模、系统辨识以及时变迟滞建模等方法，较全面地研究了两自由度精密定位平台在结构设计、建模以及控制等方面的若干关键技术问题，取得了如下创新性成果：
建立了一类超静定对称柔性机构的线性柔度与转角柔度的解析模型，并根据有限元分析结果对该解析模型进行修正，极大地提高了建模精度。该类超静定对称机构理论上不存在交叉耦合，可作为移动副或转动副应用于精密定位系统。
利用超静定对称柔性机构在不同轴向上刚度差异较大的特性，设计了一款压电陶瓷驱动的两自由度精密定位平台。该定位平台最大行程范围为8μm×8μm，一阶共振频率为692Hz，且在100Hz以下驱动时两轴间交叉耦合量小于2.7%，可精确跟踪各种二维平面轨迹。
在有限元分析中利用“接触副”模拟接触表面，通过调整参数研究了接触刚度的变化对定位平台静/动力学特性的影响规律。把接触刚度作为未知参数引入系统动力学模型，并据此提出接触刚度的辨识方法，通过测量定位平台的一阶固有频率来计算接触刚度的实际值。
提出了一种新的动力学建模思路：将压电陶瓷的驱动力定义为系统的“虚拟输入”，并将压电陶瓷简化为具有一定线性刚度的弹性体。该方法可以将压电陶瓷的非线性特性与系统的线性动力学模型彻底分开，降低建模与辨识难度。将所有弹性铰链机构简化为无质量弹簧，利用状态空间法得到平台的动力学模型。
时变的Prandtl-Ishlinskii (PI)模型可精确地描述压电陶瓷驱动器的迟滞特性，但目前尚无可用于时变PI模型的通用求逆法则，而强行将现行的求逆法则应用于时变PI模型会带来较大的理论建模误差。鉴于此，提出了一种全新的直接逆模型建模法，该方法直接通过实验数据得到时变的逆PI模型，省去了不必要的求逆过程，大大提高了逆迟滞模型的建模速度与精度。
针对前馈控制，在已有迟滞补偿与蠕变补偿的基础上，增加了解耦控制器，可进一步消除定位平台两轴向间的交叉耦合；前置滤波技术可消除参考轨迹上的高频分量，抑制了定位平台的模态振型；全通滤波器可保证定位平台在运动初始阶段的平稳性。如将反馈回路作为误差补偿，可进一步提高定位平台的轨迹跟踪精度。
The dissertation focuses on the development of a2-DOF piezo-driven precisionpositioning platform. The key technological issues arising during the structure design,modeling, and control of the platform have been fully investigated. Great efforts havebeen made in the static analysis, Finite Element Method (FEM) analysis, dynamicsmodeling, system identification, and rate-dependent hysteresis compensation. Thefollowing innovative work has been completed:
The analytical linear and angular compliance models of a class of staticallyindeterminate symmetric (SIS) flexure structures have been established. Based on theFEM results, the modeling accuracy has been further improved. Theoretically, the SISflexure structures are free of parasitic motions. Thus, it can be utilized as the prismaticor revolute joint in precision positioning systems.
A prototype of the2-DOF piezo-driven precision positioning platform has beenfabricated, where the difference of the SIS flexure structure’s linear stiffness betweenaxes is utilized to build the transmission mechanism. The platform has a maximumdisplacement range of8μm×8μm, and a first natural frequency of692Hz. For inputsignals below100Hz, the cross-axis coupling ratio is less than2.7%, indicating goodapplicability in fast planar positioning applications.
The contact pair is utilized to build the Hertz contact of the platform. Therelation between the contact stiffness and the platform’s static and dynamicperformance has been investigated in FEM analyses. The contact stiffness is alsoincluded in the dynamics modeling process as an unknown parameter. Anidentification method for the contact stiffness has been proposed, which identifies thecontact stiffness through the measured first natural frequency of the platform.
A novel dynamics modeling approach is proposed, where the piezoelectricactuator’s driving force is defined as the virtual input to the system. The actuator isfurther simplified as an elastic element with constant stiffness. By doing this, thepiezoelectric actuator’s nonlinearities have been totally separated out of the lineardynamics of the system. This helps to simplify the modeling and identificationprocesses. All the flexure hinges are simplified as massless springs and the dynamicsmodel of the platform is obtained in the state space formulation.
Rate-dependent Prandtl-Ishlinskii (PI) model is powerful in describing thepiezoelectric actuator’s hysteresis. However, the valid inversion law for the rate-dependent PI model is not yet available. Simply extends the conventionalinversion law to the rate-dependent PI model will result in large theoretical modelingerror. A novel direct inverse hysteresis modeling approach is proposed herein, wherethe rate-dependent inverse PI model can be directly identified from the experimentaldata. As no inversion calculation is involved, this approach is time-efficient and themodeling accuracy can be greatly improved.
On the basis of the hysteresis and creep compensations, a new decouplingcontroller is proposed to further eliminate the cross axis coupling of the platform infeedforward applications. Prefiltering can also be incorporated to filter out all the highfrequency components of the reference trajectory. The platform’s modal vibration canthen be suppressed. An all-pass filter can be employed to guarantee the platform’stransient performance in the very beginning of the movements. If feedback is alsointegrated as an error compensator, the platform’s tracking accuracy can be furtherimproved.
建立了一类超静定对称柔性机构的线性柔度与转角柔度的解析模型，并根据有限元分析结果对该解析模型进行修正，极大地提高了建模精度。该类超静定对称机构理论上不存在交叉耦合，可作为移动副或转动副应用于精密定位系统。
利用超静定对称柔性机构在不同轴向上刚度差异较大的特性，设计了一款压电陶瓷驱动的两自由度精密定位平台。该定位平台最大行程范围为8μm×8μm，一阶共振频率为692Hz，且在100Hz以下驱动时两轴间交叉耦合量小于2.7%，可精确跟踪各种二维平面轨迹。
在有限元分析中利用“接触副”模拟接触表面，通过调整参数研究了接触刚度的变化对定位平台静/动力学特性的影响规律。把接触刚度作为未知参数引入系统动力学模型，并据此提出接触刚度的辨识方法，通过测量定位平台的一阶固有频率来计算接触刚度的实际值。
提出了一种新的动力学建模思路：将压电陶瓷的驱动力定义为系统的“虚拟输入”，并将压电陶瓷简化为具有一定线性刚度的弹性体。该方法可以将压电陶瓷的非线性特性与系统的线性动力学模型彻底分开，降低建模与辨识难度。将所有弹性铰链机构简化为无质量弹簧，利用状态空间法得到平台的动力学模型。
时变的Prandtl-Ishlinskii (PI)模型可精确地描述压电陶瓷驱动器的迟滞特性，但目前尚无可用于时变PI模型的通用求逆法则，而强行将现行的求逆法则应用于时变PI模型会带来较大的理论建模误差。鉴于此，提出了一种全新的直接逆模型建模法，该方法直接通过实验数据得到时变的逆PI模型，省去了不必要的求逆过程，大大提高了逆迟滞模型的建模速度与精度。
针对前馈控制，在已有迟滞补偿与蠕变补偿的基础上，增加了解耦控制器，可进一步消除定位平台两轴向间的交叉耦合；前置滤波技术可消除参考轨迹上的高频分量，抑制了定位平台的模态振型；全通滤波器可保证定位平台在运动初始阶段的平稳性。如将反馈回路作为误差补偿，可进一步提高定位平台的轨迹跟踪精度。
The dissertation focuses on the development of a2-DOF piezo-driven precisionpositioning platform. The key technological issues arising during the structure design,modeling, and control of the platform have been fully investigated. Great efforts havebeen made in the static analysis, Finite Element Method (FEM) analysis, dynamicsmodeling, system identification, and rate-dependent hysteresis compensation. Thefollowing innovative work has been completed:
The analytical linear and angular compliance models of a class of staticallyindeterminate symmetric (SIS) flexure structures have been established. Based on theFEM results, the modeling accuracy has been further improved. Theoretically, the SISflexure structures are free of parasitic motions. Thus, it can be utilized as the prismaticor revolute joint in precision positioning systems.
A prototype of the2-DOF piezo-driven precision positioning platform has beenfabricated, where the difference of the SIS flexure structure’s linear stiffness betweenaxes is utilized to build the transmission mechanism. The platform has a maximumdisplacement range of8μm×8μm, and a first natural frequency of692Hz. For inputsignals below100Hz, the cross-axis coupling ratio is less than2.7%, indicating goodapplicability in fast planar positioning applications.
The contact pair is utilized to build the Hertz contact of the platform. Therelation between the contact stiffness and the platform’s static and dynamicperformance has been investigated in FEM analyses. The contact stiffness is alsoincluded in the dynamics modeling process as an unknown parameter. Anidentification method for the contact stiffness has been proposed, which identifies thecontact stiffness through the measured first natural frequency of the platform.
A novel dynamics modeling approach is proposed, where the piezoelectricactuator’s driving force is defined as the virtual input to the system. The actuator isfurther simplified as an elastic element with constant stiffness. By doing this, thepiezoelectric actuator’s nonlinearities have been totally separated out of the lineardynamics of the system. This helps to simplify the modeling and identificationprocesses. All the flexure hinges are simplified as massless springs and the dynamicsmodel of the platform is obtained in the state space formulation.
Rate-dependent Prandtl-Ishlinskii (PI) model is powerful in describing thepiezoelectric actuator’s hysteresis. However, the valid inversion law for the rate-dependent PI model is not yet available. Simply extends the conventionalinversion law to the rate-dependent PI model will result in large theoretical modelingerror. A novel direct inverse hysteresis modeling approach is proposed herein, wherethe rate-dependent inverse PI model can be directly identified from the experimentaldata. As no inversion calculation is involved, this approach is time-efficient and themodeling accuracy can be greatly improved.
On the basis of the hysteresis and creep compensations, a new decouplingcontroller is proposed to further eliminate the cross axis coupling of the platform infeedforward applications. Prefiltering can also be incorporated to filter out all the highfrequency components of the reference trajectory. The platform’s modal vibration canthen be suppressed. An all-pass filter can be employed to guarantee the platform’stransient performance in the very beginning of the movements. If feedback is alsointegrated as an error compensator, the platform’s tracking accuracy can be furtherimproved.
引文
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