## Forward Kinematics of a Tilt Rotary Table Type 5Axis Milling Machine

When modeling a tilt rotary table type 5-axis machine, it is convenient to consider the coordinate systems illustrated in Fig. 3.3. In this figure, the machine coordinate system, Cm, is fixed to the most positive location in the work volume of the CNC machine tool. All commands sent to the machine are in terms of the machine coordinate system. All other coordinate systems are for human convenience. The programmed coordinate system, Cp, is located by the vector, m, relative to the machine coordinate system during the workpiece setup prior to machining. It is essential that this vector be set such that the center of rotation of the tilt-rotary table is coincident with the tool tip. After this vector is set, a command sent to the controller to move to position (0, 0, 0) in the programmed coordinate system will place the tool tip at the center of rotation. The tilt-rotary table coordinate systems, Ctr, and the rotating coordinate systems, CA and CC, are attached to the center of rotation of the tilt-rotary table. Note that there are many different ways to configure a tilt-rotary table, depending on the initial position of the rotary axes. The most basic configurations assume that the table is initially horizontal or vertical. This analysis assumes that the table is initially horizontal. The workpiece coordinate system, C, moves with the tilt-rotary table. The workpiece offset vector, wp, gives the position of the workpiece coordinate system relative to the tilt-rotary table coordinate systems.

When a cutter location file is post-processed, the post-processor uses the workpiece offset vector, wp, to convert the cutter location data into G-codes. Each G-code position command consists of X, Y, Z, A, and C components that describe the tool position relative to the programmed coordinate system. The tilt-rotary table coordinate systems are translated by the X, Y, and Z, commands relative to the programmed coordinate system, and the workpiece coordinate system will be rotated by the A and C commands about the x and z axes in tilt-rotary table coordinate systems. The CNC controller converts commands given in the programmed coordinate system to the machine coordinate system using the machine offset vector, m.

To model the kinematics of the CNC machine, homogeneous transformations are used to establish the relationship between the defined coordinate systems. For this exercise, a point, pwp, will be ultimately transformed from the workpiece coordinate system into the machine coordinate. The superscript on the point will refer to the coordinate system in which the point is defined and the subscripts indicate a particular component of the vector. The position of a point, pwp, in the workpiece coordinate system, Cwp, expressed in the tilt-rotary table coordinate system, Ctr, is given by:

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