| The ability of manufacturing industry reflects the comprehensive national strength of a country, and the accuracy molding tools industry is an important part of it. Molding tools always have"black gold"of call, and this industry is also called"the prime mover of coming into wealthy society because of its huge production value and profits. However, as the last and most important working producer of molding's process, grinding and polishing haven't used CNC in large scale for some special reasons of themselves. According to the Stat., the handicraft man-hour of free curved surface take up 37%~40% of the total amount in some developed country like American, Japanese and Germany, but it's above 50% in our country. Therefore, the demands of high accuracy and automatic polishing machine tools for free curved surface is very urgent.In order to carry out the intelligence process of grinding and polishing for large free curved face, our intelligent accuracy manufacturing group of JiLin University developed the hybrid virtual axis polishing machine tool as fig.1 shows. Based on the key item"the embedded system for apery free curved surface polishing machine tool", this paper designs the inverse kinematics control arithmetic of series-parallel machine tool according the theory of our group member. This paper also design a kind of interpolation method about the grinding and polishing based on the B-Spline theory.In order to analyze the kinematics of series-parallel mechanism, it needs to be simplified. The fig.2 and fig.3 are the simplification of parallel and series mechanism respectively. The structure of parallel mechanism is 3-PTT, linking it there is the series mechanism which is making up of two poles and the joints connecting them. A simplification of Denavit-Hartenberg method called"moving coordinates method"is bring forward in this paper, which can be used analyzing the position and pose between parts. This kind of method uses the multiplication of six base position-posture-array to replace the complicated relations among the parts of a mechanism. The result can be gained quickly by this way. Asking for help from simulation experiment of the parallel mechanism, it could be proved that it only have three translation DOFs along three axis. This conclusion provides the base of theory about kinematics of the parallel parts.The analysis of inverse kinematics about parallel and series mechanisms can be discussed respectively. The parallel one has one and only result of inverse kinematics, its corresponding expression is: There isn't one and only inverse kinematics result about the series mechanism , which depends the technical and motion restricting conditions. Appropriate location of the polishing tool bearing rod axes should on the semi-cone in fig.4.The work-space of parallel mechanism is a column space which is made up of three circular-arc surfaces, and its top and bottom are both the complex curved surface consists of four ball-arc surfaces. Fig.5 shows the lattice of this work-space. Considering the influence of series mechanism, the vertical cutaway view of whole work-space is like fig.6.The control-program arithmetic of inverse kinematics should be designed as following step: first of all, make a judgment whether the processing spot is included by the main work-space; second, make sure the right flexible-workspace and select the function to get result. The process should be stopped if processing spot doesn't locate on the main work-space.In the field of free curved surfaces interpolation, more and more CAD/CAM systems and top grade CNC began using some surface description methods such as cubic-spline, B-spline and NURBS. Considering the specific processing situation, this paper designs a new interpolation control method based on the B-spline theory. Comparing with NURBS, the B-spline theory is more mature and there are many good arithmetic about it. The expression of B-spline is simple and recursive, so it's easy to be used in program with less calculation.Because the grinding and polishing have different principle, they could use the different interpolation control method. Considering the B-spline have this character:"when inserted unlimited nodes, the control polygon will converge to the B-spline", we can use it as the basic theory of interpolation for grinding. Fig.7 shows that the control polygon(linked by stars) is closer to the B-spline than before. It could be realized by Boehm arithmetic.Because the polishing demands a high accuracy but a relaxed requirements of time, we can evaluation the points on B-spline and converge to the spline by the line linked them. The de Boor method is a simple way to evaluation the points on B-spline. Fig.8 shows its theory.The interpolation of grinding and polishing just use the control points of B-spline and avoid the complicated calculation of basis function, so the calculation is compact. Both the interpolation control arithmetic of grinding and polishing are recursive, so it is easy to be used in program. The arithmetic of grinding interpolation can be thought as the one rank recursive of polishing, who can use the result in its calculation.The outside normal vector ( n 0) and the velocity of polishing tool ( v 0) are used in the inverse kinematics arithmetic. They can be solved through B-spline theory. The derivative vector of B-spline is just the vector v 0. When the curve is concave, the host normal vector is n 0, which has a opposite direction when the curve is convex.No matter which arithmetic, the interpolator in CNC system should generate the new node vector u or its increment ?u by Real-time. The expansion of Taylor formula can be used in designing a efficient way to generate the nodes or their increment, its expression is:(2)...
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