基于弧长参数的Akima刀位轨迹拟合算法研究

在数控加工中，通常用小线段表达刀位轨迹，往往会导致刀位点庞大且轨迹不平滑。基于Akima曲线具有光顺连接且端点连接处保证G1以上连续等特点，将刀位点拟合成Akima样条曲线，提出了基于弧长参数的保凸Akima拟合刀位轨迹算法。该算法分为刀位点搜索和拟合两个阶段：首先利用拟合刀位轨迹的误差测试（双弦误差测试和弦切误差测试）约束，获得该段的首末刀位点；然后在该段内根据首末刀位点计算切线矢量，用弧长信息对刀位点参数化，生成一段Akima样条。利用UG软件生成了内含75个刀位点的鞋底模型，通过MATLAB编程和仿真实验，设置不同测试阈值，对比了弧长参数化Akima曲线与节点参数化Akima曲线、NURBS曲线之间的拟合效果，验证了算法的有效性。

Study on Akima Curve-fitting Algorithm with Tool-path Based on Arc-length Parameters

In NC machining, tool-path was usually expressed by micro-line shocks, which might result in the magnitude of the cutter location(CL)points whose trajectory was unsmooth. The CL points were fitted into Akima spline curve with its advantages of smooth junction and the node points for over G1-continuity,thus the Akima curve-fitting algorithm preserving convexity for tool-path was proposed based on arc-length parameters. The algorithm was divided into two stages including CL points' searching and curve fitting.  Firstly the features of CL points, the error testing constraints including bi-chord error tests and tangent error tests for fitting the tool trajectory were used to judge the adjacent CL points by the threshold of the testing constraints, if they were satisfied with the threshold continue to search the next tool location point（TLP）, otherwise, CL points' searching might be suspended and the first and end points of the segment were attained, then the tangent vectors were calculated according to the first and end points in this segment to ensure the concavity and convexity of the fitted curves, and the arc-length was used to parameterize the CL points ,so that a section of Akima spline between the adjoining CL points which had paused to searching were generated. When this segment of Akima curve fitting was accomplished, then the end point of the section was regarded as the first point of the next section, as the circle to the end. The model of the shoe sole(consists of 75 CL points) were generated by ultilizing UG software. The different testing thresholds were set up to compare the Akima curves whose arc-length was parameterized with the Akima curves whose knots were parameterized and NURBS curves based on MATLAB programming and simulation experiments were used to validate effectiveness of the algorithm.