Research On Pythagorean Hodograph Curve And Its Interpolation

While modern manufacturing's needs and quality requirements for curved surface machining keep increasing, the traditional manner based on interpolation of beeline and circle arc approaching spline is far from sufficient to satisfy the needs; although NURBS interpolation can accomplish the direct interpolating of freeform curves and curved surfaces, its complicated calculating way and enormous amount of calculation makes it hard to realize the real-timing of interpolation process. Some simplified calculating manners can make sure its real-timing at the cost of reduction of machining precision and thus was restricted greatly in some occasions which have high requirements for precision.The Pythagorean Hodograph(PH) curve interpolation of this research which makes full use of the advantages that PH curve is fit for CNC machining, realizes direct interpolation to freeform curves and curved surfaces, thus improving the parts' machining efficiency and quality in order to adapt to the development requirements of modern CNC techniques.At the beginning of this thesis, the definition of PH curve is provided, and the expression of PH curve's arc calculation and the form of its offset is analyzed. Then the constructing methods of two common PH curves are studied which give the conditions of Bezier curve being PH curve and the computing formula of controlling points. Under the condition that the initial data can't meet direct interpolating, C~1 interpolating of PH cubic curve is accomplished by interpolating appropriate midterm data to construct PH curve of least segments. PH quintic curve's interpolating is accomplished by Bezier curve' form and the method to find its best answer is also provided. The data experiments about the constructing methods of PH cubic and quintic curves are presented then.After that, it gives the plan to realize PH curve interpolation and analyses the computing manner of PH curve according to the different feedrate forms of constant, function of time, arc and curvature, solves the problem of speed control in the process of interpolation by adopting preceding acceleration/deceleration control. Then, the actualizing flow of PH curve interpolator is provided, and the way of PH curve interpolation's CNC programming is put forward. It then probes into curved surfaces interpolation based on PH curve tentatively, discusses the process of PH curve interpolation for curved surfaces and provides the method to find tool's center locus.Finally, the application of PH curve interpolation is provided, and it's proved that the application can improve the machining quality of Abrasive Water Jet System.