基于圆柱螺旋样条的刀具轨迹模型研究
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摘要
曲面的高精度高速度数控加工,要求刀具轨迹的插补算法稳定且计算量小。用NURBS曲线描述曲面的刀具轨迹时,插补点的位置、插补误差、进给速度等关键的插补数据与曲线的弧长之间没有直接的显式关系,插补计算量大,不适合高速实时运算;而用小直线段逼近曲面加工的刀具轨迹时,数据量大,段长小且不可避免频繁的跨段转接,速度波动和插补误差均难以控制。圆弧样条、圆柱螺旋样条、圆锥螺旋样条这一类曲线,不仅能以弧长为参数进行精确插补计算,同时相对小直线段而言,可以用较少的段数逼近刀具轨迹。因此本文提出用螺旋线段描述曲面数控加工的刀具轨迹,研究相应的刀具轨迹拟合及插补方法。
研究了空间双螺旋线、单螺旋线插值以及单螺旋线与圆弧的混合插值三种插值算法,仿真验证了三种算法的有效性。提出了圆柱螺旋样条的试探插值逼近构造算法,该算法综合利用了插值、逼近的优点。分别用小直线段和圆柱螺旋样条拟合空间3次B样条,结果圆柱螺旋样条所用段数少,光滑程度高。
研究了圆柱螺旋样条轨迹的连续插补以及速度控制技术,得到了圆柱螺旋线以弧长为参数的插补点坐标递推计算公式,公式简洁、精确;对插补弦高误差进行了理论分析和数值计算,找到了插补误差关于弧长的近似计算公式;从曲线的自然坐标系出发,推导了段内位移、速度、加速度的计算公式;研究了圆柱螺旋样条跨段插补时进给速度与转接误差控制以及确定转接点的算法,由于插补点与弧长一一对应,算法简单、稳定。
通过大型船舰用螺旋桨叶片实例,分析、验证圆柱螺旋样条轨迹的建模以及插补算法。叶片用NURBS曲面表达,在每片叶片的10截面上,给定允许逼近误差0.001时分别得到2234段螺旋段以及6010段直线段;两种轨迹分别进行插补分析,圆柱螺旋样条每插补一个点运算耗时6.1μs,直线段耗时4.2μs,但是插补完同样长度的曲线段时螺旋样条耗时少;两种插补产生的轮廓误差相当,都小于一个脉冲当量;从两种插补的进给速度、最大偏差以及方差值三个方面比较发现螺旋样条插补的速度更平滑。对叶片截面上NURBS曲线直接插补,每插补一个点的运算时间为42.6μs,比圆柱螺旋样条和直线段插补计算时间长。
设计了圆柱螺旋样条插补实验平台并验证了圆柱螺旋样条连续插补算法的有效性,与华中数控世纪星现有的小线段连续插补做了对比,实验结果与理论分析一致。通过非圆齿扇插齿、冰刀刃磨、圆柱和圆锥立铣刀刃磨三个实例探讨了圆弧样条、圆柱螺旋样条的应用。
To realize the high accuracy and high speed NC machining of surface, the tool-path interpolating algorithm should be simple and stable. When the tool-path of surface is represented with NURBS curves, the important interpolating data, such as the position of interpolating point, the interpolating error, the feed speed etc. have not direct and apparent relationship with the arc length of curves. In this case, the amount of calculation is too large to meet the real time interpolating computation. While the tool-path for surface machining are approximated by small straight line segments, the number of segments is large, the length of segments are short, and it is inevitable to transfer cross segments frequently in the continuous interpolation. So, it is difficult to control the feedrate waving and interpolating error. Circular arcs, cylindrical helixes and conic helixes splines etc. are not only real and accurate arc-length parameter curves but also tool-path can be approximated by their less segment number. Based on this kind of characteristics, a new tool trajectory expressed by cylindrical helix splines has been presented in NC surface machining, the methods of fitting and interpolating on cylindrical helix have also discussed in this dissertation.
The interpolation method of spatial double cylindrical helix, single cylindrical helix and the blending interpolation between single cylindrical helix and arc have been studied, and the efficiencies of three methods have been tested by simulation. In order to construct the cylindrical helix splines, a new algorithm‘Try-Approximation-Interpolation’which assembles the strong points of interpolation and approximation has been presented. Approximated the cubic B splines with the little straight line segments and cylindrical helix segments by this method, the cylindrical helix splines have less segments and more smoothness.
The method of real-time trajectory generation in NC machining and feed rate control in the continuous interpolation of cylindrical helix splines have been researched, and the recursive de-coupling interpolation formula has been deduced, the method is simple and efficient, and the chord error of real-time interpolation segment has also been studied theoretically and calculated numerically, we acquired an effective approximate formula. The formulae and profiles of acceleration/deceleration, velocity and displacement in one segment can be gained. A new control algorithm of corner speed within the given error and the computation of transfer point in the across segment interpolation has been presented. Because the interpolating points have close relationship with arc length, the algorithm is simple and stable.
The methods of modeling and interpolating of helix splines have been researched and experimented by the large-scale marine propeller. Given the error of 0.001mm, we can get the helix splines of 2234 segments and the linear splines of 6010 segments on the ten sections of every piece blade. By contrast the interpolating calculation of two kinds of tool-path, we can find that every one interpolating point need 6.1μs for helix splines, and 4.2μs for linear segments, but helix splines consumes less time wholely. The contour errors made by two interpolations are equivalent, and less than one pulse equivalent. We can also find that the oscillation of feed rate of helix splines is smaller than linear segments from the maximum deviation and standard deviation. To interpolate this B-splines segment directly, every one interpolation point costs 42.6μs, it consumes much more.
An experiment platform for validating the algorithm of space cylindrical helix splines interpolation has been designed. The efficiency of the interpolation algorithm for cylindrical helix splines has been testified. Comparation with the existed little straight lines segments continuous interpolation of Huazhong Century Star CNC from the consuming of interpolation time and velocity oscillation, the experiment results show the cylindrical helix splines is better. The practical applications of arc splines and cylindrical helix splines have been discussed, three practical examples, i.e. machining of non-circular gear sector, grinding of the blade curve of short track skates, the grinding of cylindrical and taper milling cutter show that cylindrical helix splines has a good application foreground.
研究了空间双螺旋线、单螺旋线插值以及单螺旋线与圆弧的混合插值三种插值算法,仿真验证了三种算法的有效性。提出了圆柱螺旋样条的试探插值逼近构造算法,该算法综合利用了插值、逼近的优点。分别用小直线段和圆柱螺旋样条拟合空间3次B样条,结果圆柱螺旋样条所用段数少,光滑程度高。
研究了圆柱螺旋样条轨迹的连续插补以及速度控制技术,得到了圆柱螺旋线以弧长为参数的插补点坐标递推计算公式,公式简洁、精确;对插补弦高误差进行了理论分析和数值计算,找到了插补误差关于弧长的近似计算公式;从曲线的自然坐标系出发,推导了段内位移、速度、加速度的计算公式;研究了圆柱螺旋样条跨段插补时进给速度与转接误差控制以及确定转接点的算法,由于插补点与弧长一一对应,算法简单、稳定。
通过大型船舰用螺旋桨叶片实例,分析、验证圆柱螺旋样条轨迹的建模以及插补算法。叶片用NURBS曲面表达,在每片叶片的10截面上,给定允许逼近误差0.001时分别得到2234段螺旋段以及6010段直线段;两种轨迹分别进行插补分析,圆柱螺旋样条每插补一个点运算耗时6.1μs,直线段耗时4.2μs,但是插补完同样长度的曲线段时螺旋样条耗时少;两种插补产生的轮廓误差相当,都小于一个脉冲当量;从两种插补的进给速度、最大偏差以及方差值三个方面比较发现螺旋样条插补的速度更平滑。对叶片截面上NURBS曲线直接插补,每插补一个点的运算时间为42.6μs,比圆柱螺旋样条和直线段插补计算时间长。
设计了圆柱螺旋样条插补实验平台并验证了圆柱螺旋样条连续插补算法的有效性,与华中数控世纪星现有的小线段连续插补做了对比,实验结果与理论分析一致。通过非圆齿扇插齿、冰刀刃磨、圆柱和圆锥立铣刀刃磨三个实例探讨了圆弧样条、圆柱螺旋样条的应用。
To realize the high accuracy and high speed NC machining of surface, the tool-path interpolating algorithm should be simple and stable. When the tool-path of surface is represented with NURBS curves, the important interpolating data, such as the position of interpolating point, the interpolating error, the feed speed etc. have not direct and apparent relationship with the arc length of curves. In this case, the amount of calculation is too large to meet the real time interpolating computation. While the tool-path for surface machining are approximated by small straight line segments, the number of segments is large, the length of segments are short, and it is inevitable to transfer cross segments frequently in the continuous interpolation. So, it is difficult to control the feedrate waving and interpolating error. Circular arcs, cylindrical helixes and conic helixes splines etc. are not only real and accurate arc-length parameter curves but also tool-path can be approximated by their less segment number. Based on this kind of characteristics, a new tool trajectory expressed by cylindrical helix splines has been presented in NC surface machining, the methods of fitting and interpolating on cylindrical helix have also discussed in this dissertation.
The interpolation method of spatial double cylindrical helix, single cylindrical helix and the blending interpolation between single cylindrical helix and arc have been studied, and the efficiencies of three methods have been tested by simulation. In order to construct the cylindrical helix splines, a new algorithm‘Try-Approximation-Interpolation’which assembles the strong points of interpolation and approximation has been presented. Approximated the cubic B splines with the little straight line segments and cylindrical helix segments by this method, the cylindrical helix splines have less segments and more smoothness.
The method of real-time trajectory generation in NC machining and feed rate control in the continuous interpolation of cylindrical helix splines have been researched, and the recursive de-coupling interpolation formula has been deduced, the method is simple and efficient, and the chord error of real-time interpolation segment has also been studied theoretically and calculated numerically, we acquired an effective approximate formula. The formulae and profiles of acceleration/deceleration, velocity and displacement in one segment can be gained. A new control algorithm of corner speed within the given error and the computation of transfer point in the across segment interpolation has been presented. Because the interpolating points have close relationship with arc length, the algorithm is simple and stable.
The methods of modeling and interpolating of helix splines have been researched and experimented by the large-scale marine propeller. Given the error of 0.001mm, we can get the helix splines of 2234 segments and the linear splines of 6010 segments on the ten sections of every piece blade. By contrast the interpolating calculation of two kinds of tool-path, we can find that every one interpolating point need 6.1μs for helix splines, and 4.2μs for linear segments, but helix splines consumes less time wholely. The contour errors made by two interpolations are equivalent, and less than one pulse equivalent. We can also find that the oscillation of feed rate of helix splines is smaller than linear segments from the maximum deviation and standard deviation. To interpolate this B-splines segment directly, every one interpolation point costs 42.6μs, it consumes much more.
An experiment platform for validating the algorithm of space cylindrical helix splines interpolation has been designed. The efficiency of the interpolation algorithm for cylindrical helix splines has been testified. Comparation with the existed little straight lines segments continuous interpolation of Huazhong Century Star CNC from the consuming of interpolation time and velocity oscillation, the experiment results show the cylindrical helix splines is better. The practical applications of arc splines and cylindrical helix splines have been discussed, three practical examples, i.e. machining of non-circular gear sector, grinding of the blade curve of short track skates, the grinding of cylindrical and taper milling cutter show that cylindrical helix splines has a good application foreground.
引文
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