摘要
随着现代加工制造业的发展,特别是高速、高精加工的发展,对数控机床进给系统的动态性能提出了更高要求,同时直线电机驱动和并联虚拟轴驱动等形式的进给系统出现,对传统的以“伺服电机+滚珠丝杠”驱动形式的进给系统提出了挑战。本文在分析了现有进给系统优缺点的基础上指出,由于滚珠丝杠进给系统设计技术和加工工艺成熟,成本低,工作可靠,具有对外力和惯量变化不敏感等优点,如果在对其机械传动机构进行优化设计并获得其相对精确的低阶数学模型的基础上,选择适当的伺服控制方式,配置合理的伺服参数来进一步提高它的动态性能,滚珠丝杠进给系统仍是高速高精数控机床进给系统的一种理想选择。
适宜的数学模型是进行系统分析和设计的基础,是预测其工作行为的关键。由于滚珠丝杠进给系统是一个复杂的机电一体化系统,通过理论建模方法得出的数学模型的阶数一般较高,不便于进行系统分析和系统设计,模型降阶方法虽然可以实现对已有模型降阶,然而模型降阶后会降低其精度,不利于高精度系统的分析和设计；而模态分析则存在激励困难、试验结果的可信度有赖于试验人员的技术水平和经验、误差较大的不足。因此,利用理论建模和模态分析的方法很难获得滚珠丝杠进给系统不同工况下的精确数学模型。
本文围绕如何获得工况下滚珠丝杠进给系统相对精确的低阶数学模型,提出了一套基于Lyapunov稳定性理论的模型参考自适应系统的建模方法,并对其建模理论和方法进行了深入研究,主要研究内容如下：
1.对现有滚珠丝杠进给系统的理论建模和模态分析方法进行了分析,找出了相关方面存在的问题。分析指出：理论建模方法得出的滚珠丝杠进给系统数学模型阶数一般偏高,不便于进行系统分析,而模型降阶方法较少考虑实际的工程因素；实验建模方法存在施加激励困难,实验结果可信度不高等问题,因此选择或设计合适的建模方法,获取在工况下与原有系统匹配良好的低阶数学模型对系统分析和控制器设计具有重要的意义。
2.对滚珠丝杠进给系统机械传动机构和伺服控制系统各组成部分的结构特点进行了分析,用分布参数建模方法描述细长滚珠丝杠副的传递子系统,推导出了包括滚珠丝杠受扭转载荷作用下丝杠扭矩到角速度的传递函数和滚珠丝杠受轴向载荷作用下丝杠输入端应力到速度的传递函数,对滚珠丝杠和丝杠螺母的受力和变形进行了分析；建立了速度检测及控制单元、位置检测及控制单元、交流伺服电机等环节的数学模型,为分析、优化和改进滚珠丝杠进给系统的机械传动机构,合理配置伺服参数,提高动态性能奠定了理论基础。
3.对模型参考自适应系统所涉及到的理论基础进行了论述,分别研究了被控对象相对阶为1和2时的模型参考自适应系统的控制器结构和自适应律,并对其设计步骤和控制算法进行了分析。
4.论述了基于模型参考自适应系统的建模原理,对基于模型参考自适应系统的仿真建模和实验建模进行了研究,主要包括以下内容：
●对基于模型参考自适应系统的建模原理进行了研究,以被控对象相对阶数等于1为例,推导出了被控对象的模型表达式；
●研究了基于模型参考自适应系统的仿真建模方法,利用MATLAB/Simulik建立了模型参考自适应系统的仿真模型,并以获得滚珠丝杠进给系统高阶理论模型的等效二阶模型为例,验证了该方法的有效性。
●研究了基于模型参考自适应系统的实验建模方法,建立了基于标准以太网实时同步运动控制平台EtherMAC (Ethernet for Manufacture Automation Control)的模型参考自适应系统实验平台,并以获得伺服电动机系统的等效二阶模型为例,验证了该方法的有效性。
5.基于模型参考自适应系统实验平台,研究了获得工况下滚珠丝杠进给系统低阶等效数学模型的方法。把滚珠丝杠伺服进给系统看作整体并作为模型参考自适应系统的被控对象,以光栅尺的反馈数据作为系统的输出,结合滚珠丝杠进给系统的特点和控制要求,选动态品质优良的二阶模型作为参考模型,通过试验使可调系统和参考模型达到匹配,然后推导出了滚珠丝杠伺服进给系统的二阶模型。
6.对滚珠丝杠模型参考自适应系统的响应性能指标及影响因素进行了分析,从机械传动机构和控制系统两方面研究了提高进给系统动态性能的有效措施。在机械传动机构方面,运用拉格朗日方程建立了滚珠丝杠进给系统的动力学模型,借助MATLAB软件的特征值求解函数eig()对固有频率进行了求解,定量分析了工作台质量、各部件刚度和工作台位置变化对进给系统轴向一阶固有频率的影响；在控制系统方面,研究了参考模型、自适应增益和可调参数初值对滚珠丝杠进给系统动态性能的影响,并给出了相应的解决措施。
课题的研究工作,为获得工况下滚珠丝杠进给系统相对精确的低阶数学模型提供了一种新的方法,为合理配置控制系统参数,提高系统动态性能,提供了理论依据；对在设计阶段预测滚珠丝杠进给系统的动态性能,优化设计机械传动机构,降低生产成本具有重要的理论意义和工程价值。
课题中所建立的模型参考自适应控制系统和采用的模型参考自适应建模与分析方法对直线电机驱动的进给系统及其它系统的建模也具有重要的参考价值。
With the development of modern processing and manufacturing, especially in the high-speed and high-precision machining, CNC machine tools require a higher dynamical performance on feed drive systems. Meanwhile, the emergence of linear drives and virtual-axis hexapod drive takes a challenge to tranditional "Servo motor+Ball screw" feed system. Based on the analysis of the existing feed drive systems, the dissertation indicates that due to the merits of mature processing and designing technology, low cost, and insensitive to the change of force and inertia, the ball screw feed drive system is still an ideal choice if its dynamical performance is improved by choosing control system with appropriate control mode and reasonable parameters, optimizing the mechanical transmission mechanism and obtaining an accurate low-order mathematical model.
Expedient mechanical model is the basis condition for designing and analyzing a system, and is the key element to predict the working behaviors of the system. As the ball screw feed drive system is a complicated mechatronics system, the order of theoretical model, which obtained through theoretical modeling, is relatively higher which makes it difficult to analyze and design. Though model reduction can reduce the order of given model, however, its accuracy would be reduced either, which results in difficulties to analyze and design high-precision system. Moreover, modal analysis existes the difficulty of motivation and the reliability of experimental results depends on the technology and experience of the manipulator. It is even more difficult to obtain precise mathematical models of ball screw feed system in different working conditions using theoretical modeling method or modal analysis.
In order to obtain a relatively accurate low-order mathematical model of ball screw feed system in different working conditions, a model reference adaptive system based on Lyapunov stability theory is proposed and the modeling methods and theories involved are profoundly reasearched. The main research of the dissertation is as follows:
1. Analyzing the latest methods in theoretical modeling and modal analysis of the ball screw feed system, the deficiencies in relevant aspects are founded. The analysis indicates that the order of the mathematical model of ball screw feed drive system, which is obtained by theoretical modeling method, is relatively high, and which makes it difficult to analyze. Moreover, model reduction method is lack of considering practical engineering factors. The experiment modeling method also has some problems, and it exist difficulties in applying incentives, and the reliability of experimental result is not high. So selecting or designing an appropriate modeling method to obtain a lower order mathematical model, which is well-matched the original system in the working conditions, has an important significance for system analysis and controller design.
2. Based on the analysis of the components of the mechanical transmission and servo control system of the ball screw feed drive system, the ball screw is modeled by distributing parameters, the transfer functions between ball screw torque and angular velocity, and between ball screw input side stress and velocity are derived, and the force and deformation of ball screw and screw nut is analysed. The mathematical models of the speed detection and controlling unit, position detection and controlling unit as well as AC servo motor are established. All of those provide a theoretical basis to analyze and optimize the mechanical transmission mechanism, configurate servo motor parameters, impove dynamic performance of ball screw feed driving system.
3. Expound the theories of model reference adaptive system, established the adaptive controller and adaptive law whose relative order between denominator and numerator is one and two respectively. The design procedures and corresponding control algorithms are also be analyzed.
4. The modeling principle as well as the modeling methods of simulation and experiment, which is based on model reference adaptive system, was researched. The main contents are as follows:
●Studied the modeling principle based on model reference adaptive theory, taking the controlled object whose relative order is one as an example, and deduced the corresponding mathematical model.
●A simulation modeling method based on model reference adaptive system was researched, and a simulation model of the model reference adaptive system was constructed using MATLAB/Simulink. The effectiveness of the simulation modeling method was verified through a case study, in which an equivalent second order model of the high order mathematical model of ball screw feed system was obtained.
●The experiment modeling method based on model reference adaptive system was studied, and an experimental platform of the model reference adaptive system based on real-time synchronization motion control platform EtherMAC (Ethernet for Manufacture Automation Control) with standard Ethernet was built. The effectiveness of the modeling method was verified through a case study, in which an equivalent second order model of servo motor system was obtained.
5. Based on the experiment platform of model reference adaptive system, the method to obtain the equivalent low-order model of the ball screw feed drive system in the working conditions was studied. The second-order model was conducted when the adjustable system and reference model achieved an exact match through experiment, with the ball screw feed drive system serve as controlled plant, the feedback data of linear encoder as output, and a low-order model, which has excellent dynamic characteristics, as reference model.
6. Analyze the response performance indexes and influence factors of ball screw feed system, propose the effective approaches to improve the dynamic performance in both mechanical transmission mechanism and control system. In the mechanical transmission mechanism field, a dynamical model of feed drive system was established with Lagrange equation. The natural frequency of the system was solved with MATLAB software, and the influence of worktable mass, worktable position and stiffness variation of each part on axial first-order natural frequency was analyzed. In the control system field, the influence of the reference model, self-adaptive gain and initial value of adjustable parameters on the performance of feed drive system was studied, and the corresponding solutions were also provided.
The dissertation provides a new method to obtain a relatively accurate lower-order mathematical model of ball screw feed system in different working conditions, which gives a theoretical basis for configuring the parameters of control system reasonably and improving the system's dynamical performance. The achievement has important theoretical and engineering value in predicting the dynamical performance of ball screw feed drive system at the design stage, optimizing the mechanical transmission mechanism, and reducing the production costs.
In addition, the model reference adaptive control system as well as the adaptive modeling method and analysis also has important reference value for linear motor drive feed system and other systems.
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