Application of Simpson Integration Method in Dual-NURBS Curve Follow-up Interpolation
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摘要: 提出一种精确计算插补步长的双NURBS曲线随动插补算法。首先由曲面数控加工的离散刀位数据分别拟合出刀尖点和刀轴点NURBS曲线,并建立两条曲线插补参数间的随动关系模型; 然后采用辛普森积分法计算出曲线的总弧长,进行插补运动的加减速规划; 再以刀尖点NURBS曲线为基准确定插补参数,采用辛普森法确定各插补周期的进给步长及插补点坐标; 最后依据随动关系模型获得刀轴点NURBS曲线对应的插补参数,完成曲面加工刀路规划的刀具位姿插补。仿真实验表明,与同一参数插补法相比,参数随动法可以获得更加稳定的等距效果,便于实时控制插补过程中的刀轴位置和姿态。Abstract: A following-up interpolation algorithm for dual-NURBS (Non-uniform rational B-spline) curve was proposed based on precise calculation of interpolation step-length. First, tool-tip and tool-axis NURBS curves are fitted out from the discrete cutter location data for curved surface NC machining, respectively, and after it the follow-up relation model between the two curve interpolation parameters was established. Then, Simpson integral method is used to calculate the total arc length of the curve, and the acceleration/deceleration planning is conducted for interpolation motion. And next, taking tool-tip NURBS curve interpolation parameters as benchmark, and using Simpson method to determine feeding step-length in each interpolation cycle, and the interpolation point coordinates are subsequently gained. The corresponding interpolation parameters of the tool-axis NURBS curve is acquired according to follow-up relationship model, and the tool pose interpolation of the tool-path planning of surface machining is achieved. Simulation experiment shows that compared with the same parameter interpolation method, the follow-up parameters method can obtain more stable isometric effect, which is conducive to real-time control the tool position and posture in the process of interpolation.
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Key words:
- Simpson integration /
- dual-NURBS /
- parameter follow-up /
- interpolation algorithm
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表 1 刀尖点及法向矢量数据
序号 刀尖点{Oi} 法向矢量{Ni} x y z i j k 1 0 4.65 4.97 0.14 0.33 0.93 2 1.97 4.65 4.79 0.02 -0.04 0.99 3 3.96 4.65 4.91 -0.13 -0.29 0.94 4 5.91 4.65 5.38 -0.29 -0.40 0.86 5 7.78 4.65 6.05 -0.01 -0.43 0.90 6 9.50 4.65 5.65 0.34 -0.29 0.89 7 11.39 4.65 4.80 0.44 -0.10 0.88 8 12.96 4.65 3.84 0.57 0.01 0.81 9 14.77 4.65 3.08 0.16 0.08 0.98 10 16.76 4.65 3.25 -0.30 0.07 0.94 11 18.51 4.65 4.14 -0.56 0.01 0.82 12 20.00 4.65 5.36 -0.68 -0.07 0.72 表 2 刀轴点及双NURBS曲线的控制顶点
序号 控制顶点{Pi} 刀轴点{Qi} 控制顶点{Ri} x y z x y z x y z 1 0 4.65 4.97 0.29 5.31 6.83 0.29 5.31 6.83 2 2.28 4.65 4.68 2.01 4.56 6.78 2.20 4.42 6.78 3 3.89 4.65 4.88 3.68 4.05 6.80 3.59 4.03 6.74 4 5.94 4.65 5.25 5.31 3.84 7.11 5.63 3.78 7.09 5 7.68 4.65 6.33 7.74 3.78 7.85 7.69 3.71 8.16 6 9.63 4.65 5.62 10.19 4.06 7.43 10.16 4.02 7.49 7 11.35 4.65 4.89 12.28 4.43 6.57 12.27 4.47 6.67 8 12.98 4.65 3.80 14.11 4.67 5.48 13.79 4.59 5.66 9 14.72 4.65 2.86 15.09 4.81 5.05 15.05 4.85 4.92 10 16.86 4.65 3.19 16.15 4.80 5.15 16.32 4.80 5.14 11 18.32 4.65 3.87 17.38 4.68 5.79 17.31 4.70 5.65 12 20.00 4.65 5.36 18.63 4.50 6.81 18.63 4.50 6.81 -
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