Kinematics Of Machine Tools Pdf
Vanya Lamunyon
Coordinated multi-axis CNC machine tools controlled with LinuxCNC, require a special kinematics component for each type of machine. This chapter describes some of the most popular 5-axis machine configurations and then develops the forward (from work to joint coordinates) and inverse (from joint to work) transformations in a general mathematical process for two types of machine.
CAD/CAM systems are typically used to generate the 3D CAD models of the workpiece as well as the CAM data for input to the CNC 5-axis machine. The tool or cutter location (CL) data, is composed of the cutter tip position and the cutter orientation relative to the workpiece coordinate system. Two vectors, as generated by most CAM systems and shown in Fig. 1, contain this information:
The K vector is equivalent to the 3rd vector from the pose matrix E6 that was used in the 6-axis robot kinematics [3] and the Q vector is equivalent to the 4th vector of E6. In MASTERCAM for example this information is contained in the intermediate output ".nci" file.
Homogeneous transformations provide a simple way to describe the mathematics ofmulti-axis machine kinematics. A transformation of the space H is a 4x4 matrix and can represent translation and rotation transformations. Given a point x,y,x described by a vector u = x,y,z,1T, then its transformation v is represented by the matrix product
The matrix T(a,b,c) implies a translation in the X, Y, Z coordinate directions by the amounts a, b, c respectively. The R matrices imply rotations of the angle theta about the X, Y and Z coordinate axes respectively. The C and S symbols refer to cosine and sine functions respectively.
A rotary table which rotates about the vertical Z-axes (C-rotation, secondary) mounted on a tilting table which rotates about the X- or Y-axis (A- or B-rotation, primary). The workpiece is mounted on the rotary table.
A tilting table which rotates about the X- or Y-axis (A- or B-rotation, secondary) is mounted on a rotary table which rotates about the Z-axis (C-rotation, primary), with the workpiece on the tilting table.
A multi-axis machine can be considered to consist of a series of links connected by joints. By embedding a coordinate frame in each link of the machine and using homogeneous transformations, we can describe the relative position and orientation between these coordinate frames
We need to describe a relationship between the workpiece coordinate system and the tool coordinate system. This can be defined by a transformation matrix wAt, which can be found by subsequent transformations between the different structural elements or links of the machine, each with its own defined coordinate system. In general such a transformation may look as follows:
Matrix multiplication is a simple process in which the elements of each row of the lefthand matrix A is multiplied by the elements of each column of the righthand matrix B and summed to obtain an element in the result matrix C, ie.
In Fig. 2 a generic configuration with coordinate systems is shown [4]. It includes table rotary/tilting axes as well as spindle rotary/tilting axes. Only two of the rotary axes are actually used in a machine tool.
First we will develop the transformations for the first type of configuration mentioned above, ie. a table tilting/rotary (trt) type with no rotating axis offsets. We may give it the name xyzac-trt configuration.
We deal here with a simplified configuration in which the tilting axis and rotary axis intersects at a point called the pivot point as shown in Fig. 4. therefore the two coordinate systems Ows and Owp of Fig. 2 are coincident.
In these equations Lx, Ly, Lz defines the offsets of the pivot point of the two rotary axes A and C relative to the workpiece coordinate system origin. Furthermore, Px, Py, Pz are the relative distances of the pivot point to the cutter tip position, which can also be called the "joint coordinates" of the pivot point. The pivot point is at the intersection of the two rotary axes. The signs of the SA and SC terms are different to those in [2,3] since there the table rotations are negative relative to the workpiece coordinate axes (note that sin(-theta) = -sin(theta), cos(-theta) = cos(theta)).
We deal here with a extended configuration in which the tilting axis and rotary axis do not intersect at a point but have an offset Dy.Furthermore, there is also an z-offset between the two coordinate systems Ows and Owp of Fig. 2, called Dz.A vismach model is shown in Fig. 5 and the offsets are shown in Fig. 6 (positive offsets in this example).To simplify the configuration, the offsets Lx, Ly, Lz of the previous case are not included.They are probably not necessary if one uses the G54 offsets in LinuxCNC by means of the "touch of" facility.
In these equations Dy, Dz defines the offsets of the pivot point of the rotary axes A relative to the workpiece coordinate system origin. Furthermore, Px, Py, Pz are the relative distances of the pivot point to the cutter tip position, which can also be called the "joint coordinates" of the pivot point. The pivot point is on the A rotary axis.
We deal here again with a extended configuration in which the tilting axis (about the y-axis) and rotary axis do not intersect at a point but have an offset Dx.Furthermore, there is also an z-offset between the two coordinate systems Ows and Owp of Fig. 2, called Dz.A vismach model is shown in Fig. 7 (negative offsets in this example) and the positive offsets are shown in Fig. 8.
In these equations Dx, Dz defines the offsets of the pivot point of the rotary axes B relative to the workpiece coordinate system origin.Furthermore, Px, Py, Pz are the relative distances of the pivot point to the cutter tip position, which can also be called the "joint coordinates" of the pivot point.The pivot point is on the B rotary axis.
LinuxCNC includes kinematics modules for the xyzac-trt and xyzbc-trt topologiesdescribed in the mathematics detailed above. For interested users, the source codeis available in the git tree in the src/emc/kinematics/ directory.
The example configurations include the required INI files and an examples subdirectorywith G-code (NGC) files. These sim configurations invoke a realistic 3-dimensional modelusing the LinuxCNC vismach facility.
Vismach is a library of python routines to display a dynamic simulation of a CNC machine on the PC screen. The python script for a particular machine is loaded in HAL and data passed by HAL pin connections. The non-realtime vismach model is loaded by a HAL command like:
In order to use tools from a tool table sequentially with tool-length compensation applied automatically, a further Z-offset is required. For a tool that is longer than the "master" tool, which typically has a tool length of zero, LinuxCNC has a variable called "motion.tooloffset.z". If this variable is passed on to the kinematic component (and vismach python script), then the necessary additional Z-offset for a new tool can be accounted for by adding the component statement, for example:
LinuxCNC implements kinematics using a HAL component that is loadedat startup of LinuxCNC. The most common kinematics module, trivkins,implements identity (trivial) kinematics where there is a one-to-onecorrespondence between an axis coordinate letter and a motor joint.Additional kinematics modules for more complex systems (including xyzac-trtand xyzbc-trt described above) are available.
The kinematics modules provided by LinuxCNC are typically written in theC-language. Since a standard structure is used, creation of a customkinematics module is facilitated by copying an existing source file to a userfile with a new name, modifying it, and then installing.
where "kinsname" is the name you give to your component. The sudo prefix isrequired to install it and you will be asked for your root password. See thehalcompile man page for more information (\$ man halcompile)
where "kinsname" is the name of your kins program.Additional HAL pins may be created by the module for variable configuration itemssuch as the Dx, Dy, Dz, tool-offset used in the xyzac-trt kinematics module.These pins can be connected to a signal for dynamic control or set once with HAL connections like:
With small moving masses, parallel kinematic machine tools achieve very high dynamics of motion. Nevertheless, such machine tools are still rare today. The reason for this is the necessary very high rigidity of the axes, which must be designed as direct drives. At the same time, it is difficult to achieve a perfect surface quality. Conventional servo drives fail to meet these requirements.
The solution with Triamec Servo Drives together with a Beckhoff TwinCAT CNC controller leads to success for a well-known Swiss machine tool manufacturer: The target data of the CNC controllers are converted into the joint coordinates using inverse kinematics. At the same time, a workspace monitoring system is running to protect the mechanics from damage. The manipulated variables are then interpolated and sent to the Triamec servo drives in 10kHz cycles via the Tria-Link field bus.
Parallel kinematics is a branch of mechanics that focusses on manipulators formed by closed kinematic chains, i.e., mechanisms that have an endeffector joined to the fixed frame by several limbs. Such a kinematic structure provides some advantages regarding stiffness, acceleration and weight, but has some drawbacks due to mechanical complexity and limited workspaces. In the field of machining, there have been several applications of such mechanisms to machine tools. Earlier designs based on hexapods did not fulfil expectations but new topologies are promising. In this chapter, there is first a description of the evolution of parallel kinematics in the manufacturing industry. Second, the authors expose a design methodology giving some hints on the main problems to overcome. Third, there is a study on calibration processes that can be applied to these machines. And, at the end, there is a description of control issues.
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