NURBS曲线刀具路径实时插补技术研究
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摘要
NURBS方法以其优良的性质在CAD/CAM中得到了广泛应用,并在STEP标准中被指定为描述工业产品几何形状的唯一数学工具。在数控加工领域中,研究NURBS曲线刀具路径实时插补技术,不仅能够为现有数控系统提供扩展功能,而且还可为下一代数控系统STEP-NC的开发提供核心技术支持。以此为背景,本文开展了数控系统NURBS刀具路径实时插补技术研究,主要完成了以下几个方面的工作：
(1)研究了NURBS曲线快速求值求导计算方法。NURBS曲线插补通常需要同时计算曲线值和多阶导数值,此时传统de Boor算法综合效率较低,不利于实时计算。本文通过定义向量扩展运算方法,给出了B样条基函数快速求值算法,并实现了NURBS曲线快速求值求导计算。实例计算表明,该算法可提高NURBS曲线实时插补任务的计算效率,增强插补器实时性,提高曲线插补精度。
(2)研究了NURBS曲线插补进给速度精确控制方法。从进给速度波动产生原理分析入手,指出NURBS曲线实时插补时下一插补点的计算应以逼近插补段弦长为条件,而非弧长。以此为基础,提出了NURBS曲线插补进给速度精确控制方法。该方法采用主动控制方式,在计算量不增加或增加不大的情况下,可显著减小实时插补进给速度的波动率。
(3)研究了NURBS曲线实时插补进给速度极限分析方法。NURBS曲线实时插补进给速度极值受到了机床进给系统动力学特征和曲线几何特征的限制。通过建立直角坐标系数控机床进给系统的机电混合模型,给出了机床进给系统动力学约束条件下的速度极值解析曲线,即VLC曲线。同时还指出,当VLC曲线的最大弦高误差估计值满足NURBS曲线插补精度要求时,可不必再考虑插补弦高误差约束条件。这一结论有助于简化进给速度规划约束条件,并可为NURBS曲线实时插补进给速度规划方法研究提供理论分析依据。
(4)研究了直角坐标系下时间最短的NURBS曲线插补进给速度优化规划方法。该方法扬弃了传统的进给速度加减速段规划手段,采用最优控制思想,在机床进给系统动力学约束条件下,给出了时间最短的进给速度优化规划。所规划的优化进给速度曲线与NURBS曲线在其参数定义域内一一对应。这种基于全局信息的进给速度优化规划方法,可以为NURBS刀具路径实时插补提供时间最短的优化进给速度曲线。
(5)提出了一种基于进给速度引导曲线的NURBS曲线实时插补器结构设计及实现方法。该插补器采用具有高阶平滑特性的进给速度引导曲线,实现了实时插补进给速度规划。以此为基础,给出了基于进给速度引导的数控系统软硬件具体设计方法。试验表明：进给速度引导曲线能够很好地继承进给速度优化规划结果,使插补器可以在较短的周期内(200μs)完成NURBS曲线实时插补计算任务。
通过以上五个方面问题的研究,解决了NURBS曲线实时插补中的快速计算、进给速度波动抑制、进给速度极值曲线分析、进给速度优化规划以及实时NURBS插补器设计等问题。试验结果表明了本文方法的有效性,为NURBS曲线刀具路径实时插补提供了较为完整的解决方案。
NURBS method is widely used in CAD/CAM for its good curve and surface description properties, which is also specified by STEP standard as the unique mathematical tool for describing the geometry form of industrial production. In the field of CNC machining, the technology of NURBS tool path interpolation may be used both to extend the function of traditional CNC system and to offer key technology support for developing STEP-NC system. This dissertation mainly research on the technique of NURBS tool-paths real-time interpolation for new generation of CNC systems. The main contributions include the following aspects.
(1) A new fast algorithm to calculate NURBS curve value and derivation is researched. NURBS tool path interpolation requires computing its value and multi-order derivation at the same time. The de Boor algorithm becomes inefficient in this case. A high efficiency algorithm to calculate B-spline function is investigated by defining vector-extended operation. Based on it, a fast algorithm can be achieved for getting NURBS curve value and derivation. The application and analysis show that the given algorithm can improve the calculating efficiency and accuracy in real-time task of NURBS curve interpolation.
(2) An exact feedrate-controlled method of NURBS tool-paths interpolation is researched. Through analyzing the theory of curved-path interpolating feedrate fluctuation, it's pointed out that interpolation points should be calculated under the condition of the step chord length, not the arc. Then, a precise feedrate-controlled method of NURBS tool-paths interpolation is proposed. Adopted the proposed method, the interpolation feedrate fluctuation can be approved distinctly without more calculation time spending.
(3) A velocity limited method of NURBS tool-paths interpolation is proposed for high speed CNC. The limit feedrate of curved path interpolation is bounded by both dynamic properties of the CNC machine tool and the geometrical properties of the path. Through establishing the hybrid electromechanical model of every machine federate axis in Cartesian coordinate, the velocity limited analytical curve (VLC) is expressed under the constraints of system dynamic conditions. When the estimation of max chord error meets the interpolation accuracy requirement, the chord error conditions can be neglected. The conclusion is helpful to simplify the feedrate planning restriction conditions, and provides a theory for feedrate planning of NURBS tool-paths interpolation.
(4) A time-optimized federate planning method of NURBS tool-paths interpolation is investigated in Cartesian coordinate. The method, built upon the optimization control theory, can achieve the minimum machining time under constraint of system dynamics along the NURBS tool path, which is distinguished form the other federate planning methods. In real-time interpolating task, the given method can proved time-optimized federate curve which is built on the global information of the NURBS tool path and has the same parameter definition domain with the curved path.
(5) A new structure of real-time NURBS interpolator is presented with feedrate guide curves. With the proposed structure, the real-time federate of NURBS tool path interpolation is given by high-order smooth guided curves. Based on it, software and hardware of CNC system are designed and realized. Experiment results show that feedrate guided curves commendably inherit the results of time-optimized feedrate planning, and the real-time NURBS interpolator with feedrate guide curves can accomplish the real-time tasks in the time of 200μs for high speed CNC.
Through the research, five key problems of NURBS tool-paths interpolation are figured out reasonably, which are fast calculating, feedrate fluctuation controlling, feedrate-optimized planning, real-time NURBS interpolator realizing, and so on. The experimental results indicate that the proposed methods are effective and can be used as an integrated solution for further NURBS tool-paths real-time interpolation.
(1)研究了NURBS曲线快速求值求导计算方法。NURBS曲线插补通常需要同时计算曲线值和多阶导数值,此时传统de Boor算法综合效率较低,不利于实时计算。本文通过定义向量扩展运算方法,给出了B样条基函数快速求值算法,并实现了NURBS曲线快速求值求导计算。实例计算表明,该算法可提高NURBS曲线实时插补任务的计算效率,增强插补器实时性,提高曲线插补精度。
(2)研究了NURBS曲线插补进给速度精确控制方法。从进给速度波动产生原理分析入手,指出NURBS曲线实时插补时下一插补点的计算应以逼近插补段弦长为条件,而非弧长。以此为基础,提出了NURBS曲线插补进给速度精确控制方法。该方法采用主动控制方式,在计算量不增加或增加不大的情况下,可显著减小实时插补进给速度的波动率。
(3)研究了NURBS曲线实时插补进给速度极限分析方法。NURBS曲线实时插补进给速度极值受到了机床进给系统动力学特征和曲线几何特征的限制。通过建立直角坐标系数控机床进给系统的机电混合模型,给出了机床进给系统动力学约束条件下的速度极值解析曲线,即VLC曲线。同时还指出,当VLC曲线的最大弦高误差估计值满足NURBS曲线插补精度要求时,可不必再考虑插补弦高误差约束条件。这一结论有助于简化进给速度规划约束条件,并可为NURBS曲线实时插补进给速度规划方法研究提供理论分析依据。
(4)研究了直角坐标系下时间最短的NURBS曲线插补进给速度优化规划方法。该方法扬弃了传统的进给速度加减速段规划手段,采用最优控制思想,在机床进给系统动力学约束条件下,给出了时间最短的进给速度优化规划。所规划的优化进给速度曲线与NURBS曲线在其参数定义域内一一对应。这种基于全局信息的进给速度优化规划方法,可以为NURBS刀具路径实时插补提供时间最短的优化进给速度曲线。
(5)提出了一种基于进给速度引导曲线的NURBS曲线实时插补器结构设计及实现方法。该插补器采用具有高阶平滑特性的进给速度引导曲线,实现了实时插补进给速度规划。以此为基础,给出了基于进给速度引导的数控系统软硬件具体设计方法。试验表明：进给速度引导曲线能够很好地继承进给速度优化规划结果,使插补器可以在较短的周期内(200μs)完成NURBS曲线实时插补计算任务。
通过以上五个方面问题的研究,解决了NURBS曲线实时插补中的快速计算、进给速度波动抑制、进给速度极值曲线分析、进给速度优化规划以及实时NURBS插补器设计等问题。试验结果表明了本文方法的有效性,为NURBS曲线刀具路径实时插补提供了较为完整的解决方案。
NURBS method is widely used in CAD/CAM for its good curve and surface description properties, which is also specified by STEP standard as the unique mathematical tool for describing the geometry form of industrial production. In the field of CNC machining, the technology of NURBS tool path interpolation may be used both to extend the function of traditional CNC system and to offer key technology support for developing STEP-NC system. This dissertation mainly research on the technique of NURBS tool-paths real-time interpolation for new generation of CNC systems. The main contributions include the following aspects.
(1) A new fast algorithm to calculate NURBS curve value and derivation is researched. NURBS tool path interpolation requires computing its value and multi-order derivation at the same time. The de Boor algorithm becomes inefficient in this case. A high efficiency algorithm to calculate B-spline function is investigated by defining vector-extended operation. Based on it, a fast algorithm can be achieved for getting NURBS curve value and derivation. The application and analysis show that the given algorithm can improve the calculating efficiency and accuracy in real-time task of NURBS curve interpolation.
(2) An exact feedrate-controlled method of NURBS tool-paths interpolation is researched. Through analyzing the theory of curved-path interpolating feedrate fluctuation, it's pointed out that interpolation points should be calculated under the condition of the step chord length, not the arc. Then, a precise feedrate-controlled method of NURBS tool-paths interpolation is proposed. Adopted the proposed method, the interpolation feedrate fluctuation can be approved distinctly without more calculation time spending.
(3) A velocity limited method of NURBS tool-paths interpolation is proposed for high speed CNC. The limit feedrate of curved path interpolation is bounded by both dynamic properties of the CNC machine tool and the geometrical properties of the path. Through establishing the hybrid electromechanical model of every machine federate axis in Cartesian coordinate, the velocity limited analytical curve (VLC) is expressed under the constraints of system dynamic conditions. When the estimation of max chord error meets the interpolation accuracy requirement, the chord error conditions can be neglected. The conclusion is helpful to simplify the feedrate planning restriction conditions, and provides a theory for feedrate planning of NURBS tool-paths interpolation.
(4) A time-optimized federate planning method of NURBS tool-paths interpolation is investigated in Cartesian coordinate. The method, built upon the optimization control theory, can achieve the minimum machining time under constraint of system dynamics along the NURBS tool path, which is distinguished form the other federate planning methods. In real-time interpolating task, the given method can proved time-optimized federate curve which is built on the global information of the NURBS tool path and has the same parameter definition domain with the curved path.
(5) A new structure of real-time NURBS interpolator is presented with feedrate guide curves. With the proposed structure, the real-time federate of NURBS tool path interpolation is given by high-order smooth guided curves. Based on it, software and hardware of CNC system are designed and realized. Experiment results show that feedrate guided curves commendably inherit the results of time-optimized feedrate planning, and the real-time NURBS interpolator with feedrate guide curves can accomplish the real-time tasks in the time of 200μs for high speed CNC.
Through the research, five key problems of NURBS tool-paths interpolation are figured out reasonably, which are fast calculating, feedrate fluctuation controlling, feedrate-optimized planning, real-time NURBS interpolator realizing, and so on. The experimental results indicate that the proposed methods are effective and can be used as an integrated solution for further NURBS tool-paths real-time interpolation.
引文
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