| In 1994 on the Chicago international exhibition of machine tool (IMTS'94), the parallel machine tools of Hexapod and Variax are displayed by the Ingersoll and Giddings & Lewis company of America for the first time, and it is paid close attention by the whole world. Based on the three dimensional parallel mechanism, the parallel machine tool founds a new principle of the construction and the kinematics for the machine tool, and is known as " the biggest transformation and innovation of the mechanism for machine tool in this century "," the machine tool of the 21 century ". In after years, the manufacturers of Italy, Japan, Russia, Norway, Switzerland, Sweden and Danish etc. develop parallel machine tool competitively. In the end of 1997, the first parallel machine tool VAMT1Y of our country was developed by the Tsinghua University and Tianjin University jointly.Compared with the traditional series machine tool, the parallel machine tool has many merits on the aspect of the structure and the kinematics characteristic such as higher proportion of the rigidity and weight, less quality of the moving parts, rapider respond speed, less error accumulation, easier realization for 6 axles uniting to move, easier calculation for inverse kinematics result that is expedient for real time control, simpler structure of machine tool, higher technical additional value and so on. However, the parallel machine tool has also the disadvantages such as smaller work space, more difficult calculation for forward kinematics result, deeper nonlinear coupling among each axis and more complex control for interpolation, more difficult kinematics marking, weaker rigidity and movement precision of the machine tool. It is against the shortcomings of the parallel machine tool that the hybrid machine tool with series-parallel conformation has been developed.The series-parallel hybrid machine tool has inherited all advantages of the parallel virtual axis machine tool nearly, and its workspace to be increased, the difficulty to calculate the forward kinematics result and to control the machine tool to be reduced. For these reasons, theseries-parallel hybrid machine tool has shown powerful vitality though its rigidity and movement precision is still lower than the series machine tool. Observing the conformation of the machine tools to be made recently, mostly of them are series-parallel one.Studying the polishing for many years, the group knows that the polishing machining belongs to flexible machining. In the polishing process, the requirements for the rigidity and the movement precision of the machine tool are both lower because the active force of the polishing process is smaller and the displacement-force obedience controller is installed in the bearing rod of the polishing tool. For these reasons, the series-parallel hybrid machine tool can be used as a better polishing machine tool.Based on the polishing machine tool made by the group, these questions such as the conception design, the kinematics analysis, the interpolation control and the polishing experiment of the series-parallel hybrid polishing machine tool have been discussed deeply in the paper. Fig. 1 is the photo of the machine tool.Fig. 1 The photo of the machine tool1. Column 2. Drive bolt 3. Sliding block 4. Bearing rod 5. U-joint 6 Moving platform 7. Series mechanism S. Polishing tool 9. Fixed platform Fig. 2 The mechanism of the machine toolAfter analyzing the structure characteristic, the kinematics characteristic and the control characteristic among the series conformation, the parallel conformation and the series-parallel conformation, select the conformation connected 2 series rotation mechanisms on the 3 DOFS translation moving platform as the conformation of the polishing machine tool. Fig. 2 is the mechanisms of the machine tool.The chain structure of 3 DOFS translation platform is analyzed. Through the method ofhomogeneous transformation and the analysis of dimensional geometry connection about the position-posture of the parallel platform, the closed position connection of the parallel platform is given out, and based on it, the conclusion is revealed that the most simple mechanism of the 3 DOFS translation platform is 3 single-link chains mechanism which is symmetrical in space and jointed by Hook joints. The structure parameters are discussed under the condition of the biggest ratio between the workspace and the structure space of the platform.Based on the closed position connection of the parallel platform, the forward kinematics result of the machine tool is given out, and the workspace of the machine tool is discussed in form and shape. The workspace of the series-parallel hybrid virtual axis polishing machine tool is a column space that is made up of 3 circular-arc surfaces, and its top and bottom are both the complex curved surface consists of 4 ball-arc surfaces. The concept of tool flexibility was introduced firstly in the analysis of machine tool workspace, and according to it, the workspace of the machine tool is divided into four kinds of flexible workspace, the full flexible-workspace, the most flexible-workspace, the fraction flexible-workspace and unhandy workspace. The main influence factor on the full flexible-workspace of the machine tool is the bearing rod length of the polishing tool. The workspace of the machine tool is shown as Fig. 3.Based on the closed position connection of the parallel platform and the requirement of position-posture on polishing tool, the inverse kinematics result of the machine tool is analyzed in form and calculating method. Being the polishing posture angle 8X, the axes of the polishing tool bearing rod can form a cone with half-angle 5X (shown as Fig. 4). That makes the inverse kinematics result of the machine tool have the characteristic of more values and uncertainty. In order to make the inverse kinematics result uniquely, the technical restricting conditions and the motion restricting conditions should be appended. The relations among the active force, the polishing velocity and the polishing posture angle Sx are analyzed, and the technical restricting conditions, the motion restricting conditions are put forward. The method of calculating the polishing posture in four kinds of flexible workspace is introduced. After that, the inverse kinematics result of the machine tool is given out. Fig. 5 is the semi-cone that the axes of the polishing tool bearing rod should be located on. ,F. F.Semi-coneBearing rod^ Polishing toolPolishing surfaceFig. 4 The posture cone of the polishing tool Fig. 5 Semi-cone the axes of the polishing toolbearing rod located onAccording to the property of the variable definition region of the NURBS curved surface, the polishing path has been planned. For the reason of its excellent property, online direct interpolation control is chosen as the interpolation control method of the new developing system.Based on the analysis about the row space of the polishing trace and its influence factor, the interpolation step size and its influence factor, the methods how to determine the row space of the polishing trace and the interpolation step size are given out. The relation between row space and radius of the curved surface is shown in Fig. 6, and the relation between interpolation step size and radius of the curved surface is shown in Fig. 7.
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