High-speed Machining with 10 Series Numerical Controls

With the advent of extremely rigid machines, the machining of profiles as a sequence of linear micro-interpolations is no longer ideal. There are two possible ways of solving the problem, by regulating either the dynamics of the movement or the geometry of the trajectories.

Today mechanical technology has evolved to such a degree that what used to be considered insuperable limits in terms of machining speeds and accelerations have almost become the norm in the world of machine tools.

Machines capable of traverse speeds in the order of 80 - 100 m/minute and accelerations of the order of one g, if not higher, are now quite common. The typical field of application of the machines described above is high-speed milling of surfaces generated by CAD systems.

The solution normally offered by numerical controls for the machining of profiles as a series of linear micro-interpolations is no longer acceptable, because the mechanics are capable of following the continual changes in direction, resulting in undesirable effects on the surface finish. Many numerical controls have been designed to tackle this problem, sometimes in widely differing ways.

First of all, let's analyse the meaning of programming as input to the numerical control. Normally the programmed points are understood to be obligatory points, through which the machine must pass at all costs and "G01" programming is understood to be a strictly linear interpolation between these points.

This means that, to obtain a sufficiently accurate point curve, the points have to be programmed so close together that the programmed segmented line cannot be distinguished from the desired curve.

This type of approach is still normally used with machine tools with high inertia or the capability to mechanically smooth the segmented line.

With the advent of extremely rigid machines together with adequately sized axis drives capable of providing high torque values (brush-less motors) the solution described above has proved to be no longer acceptable. This is because all the sudden variations in direction are recognised by the machine tool and cause, at best, poor quality surface finish.

Possible Solutions: Dynamic Approach and Geometrical Approach

There are two possible ways of solving the problem, by managing either the dynamics of the movement or the geometry of the trajectories. It is either a question of filtering the actuation of the commands, creating a "soft" output to the servomotors, without changing the programming concept based on microscopic straight lines, or revolutionising the programming method, by interpreting the programmed values as a set of points to be approximated in the best possible way.

OSAI have chosen to pursue both options, tackling the problem from both the dynamic and the geometric points of view.

Geometrical Approach

Before defining how the geometrical data provided as input to the NC is processed, it is essential to verify what the data represents, as programming may relate to three different types of co-ordinates, that is:

1. Part co-ordinates, which refer to the actual cutting point
   
2. Tool co-ordinates, which refer to the point normally indicated as the centre of the tool
   
3. Machine co-ordinates, which refer to an arbitrary point integral to the machining axes

Part co-ordinates are linked with tool co-ordinates by the geometry and orientation of the tool.

Machine co-ordinates are linked with tool co-ordinates by the geometry of the machine tool. In the case of three-axis machining, the co-ordinates are simply translated, while in the case of machining with five co-ordinated axes, rototranslation matrices are applied that take into account the geometrical transformations due to the movement of rotary axes.

Typically, CAD/CAM systems are capable of providing machine co-ordinates directly, by processing the data that describes the surface to be milled. This means that the programs normally generated by CAD are closely dependent upon the geometry of the tool and in the case of machining with 5 co-ordinated axes, the kinematics of the machine.

Figure 1 shows what is meant by part co-ordinates, tool co-ordinates and machine co-ordinates.

The OSAI 10 Series control is capable of handling all three types of co-ordinates, so that it can be used with different types of CAD CAM systems and satisfy various programming requirements.

In all possible types of programming, the geometry used by the control is based on the generation of polynomial curves that approximate the set of data points within a configurable tolerance range.

The feed rate may refer either to the cutting point or the centre of the tool, if programming with part co-ordinates; to the centre of the tool if programming with tool co-ordinates; or to the programmed co-ordinates, if programming with machine co-ordinates.

Let's now consider the most complex case (part co-ordinates) and analyse the sequence of the geometrical transformations involved in its processing.

The input format requires both the three-dimensional co-ordinates of the part to be machined (X Y and Z) and the programming of the two versors m, n, o and i, j, k indicated in figure 1, which represent the versor orthogonal to the machined surface and orientation of the tool, respectively. All the data is expressed in part co-ordinates where the part to be machined is understood to be stationary and the tool is moved with respect to it, irrespective of the location of the rotary axes in the machine. In other words, if there is a rotary table, programming is carried out as though the tool rotated about the part and not vice versa, as happens in reality.

A third versor (called p, q, d) which determines the shape of the tool may also be programmed to identify the characteristic of the cut as accurately as possible. This versor is not shown in the figure and to avoid complicating the concepts will not be mentioned further in this paper.

Starting from the programmed values, using exclusive algorithms, the trajectory is split into sets of fifth-power polynomial functions (hereinafter called splines) consisting of a number of elements that corresponds to the number of programmed elements; each set represents a section of the program.

Each spline generated is a polynomial function which may be represented on a Cartesian plane, with the variable represented on the Y axis and an arbitrary parameter called s on the X axis. The choice of the parameter s is particularly important, in that it represents the only link between the various splines.

Each value of the parameter s corresponds to a point on all the functions generated, with a precise position of the axes with a specific tool orientation.

In our case, the parameter s corresponds to the length of the programmed trajectories and has its field of definition (domain of the polynomial function), defined for each set of spline segments.

The trajectory will therefore be split into a set of fifth-power functions X(s), Y(s), Z(s), m(s), n(s), o(s), i(s),j(s),k(s) where the parameter s is between S₀ and S₁.

The values S₀ and S₁ therefore correspond to the limits of the field of definition of the spline segments generated. If a constant speed of the centre of the tool has been requested the new splines have to be reparameterized by generating a new parameter s, which corresponds to the length of the trajectory of the centre of the tool.

Figure 2 shows an example of how the trajectory may be modified by switching from the part co-ordinates to the tool co-ordinates.

At this point in the process, the information about the tool has already been used and the output of this phase is represented by the co-ordinates of the centre of the tool and its orientation.

The tool co-ordinates still have to be transformed into machine co-ordinates, so that information needed for moving the machine tool axes may finally be obtained.

This transformation is performed by taking the kinematic chain consisting of the mechanics of the machining head into account. In the case of three-axis machining, the co-ordinates will simply be translated to take into account the length of the tool. In the case of the most complex machining process with five axes, rototranslation matrices will be applied to the splines calculated previously, so as to pass from the set of splines X(s), Y(s), Z(s), i(s), j(s), k(s) to X(s), Y(s), Z(s), A(s), C(s) where A and C are the rotary axes of the machine.

Note that the splines have not yet been reparameterized, so the five polynomial curves calculated are dependent upon the parameter used originally (length of the part trajectory or length of the tool trajectory). We will see later why this information is important and what implications it has on the dynamic part of the algorithms.

Figure 3 provides a graphical representation of all the geometrical transformations performed, from the programmed points to the definition of the spline curves for the machine axes.

The operations performed so far are exclusively geometrical and tend to generate a rigorously continuous trajectory of the single axes with respect to the programmed trajectory.

The use of fifth-power polynomials enables the generation of sufficiently long splines to copy very twisty profiles by minimising the number of segments used for modelling the input points.

The solution of the geometrical problem, that movements of the single axes are always gradual, continuous even in their derivatives, enables smooth and accurate trajectories to be generated but does not completely solve the motion problem. It does not consider the dynamics of the motion, that is the relationship between the geometrical shape and time.

Dynamic Approach

To move the axes of a machine tool correctly, the limits of the system under control must be considered. At the beginning, the parameters to be used are maximum speed and maximum acceleration supported by the kinematic mechanism connected to the axis.

Considering only the limits laid down by the speed and acceleration, one may think that the best way of driving a NC axis is to remain within these limits, keeping both the speed and the torque supplied by the motors under control.

This solution leads to a movement that uses constant acceleration ramps (trapezoidal speed diagrams). Figure 4 shows space, speed and acceleration in time.

It can be seen that acceleration is discontinuous in the time domain and causes abrupt changes in the torque provided by the motors. This kind of motion, if used at the maximum acceleration is certain to cause the machine to oscillate, thus resulting in a poor finish.

A better approach to the movement of the machines is represented by trapezoidal acceleration ramps, which are capable of generating a continuous acceleration diagram.

Figure 5 shows space, speed, acceleration and a further parameter called jerk, which represents the derivative of the acceleration, in relation to time.

The latter parameter, which is easy to define analytically but not physically, is extremely important in keeping stress on the mechanics to a minimum. With trapezoidal acceleration, jerking is limited, but still presents some discontinuities that correspond to variations in speed.

With a view to "looking after" the mechanics of the machines, we chose to use an acceleration diagram that guarantees the continuity of the derivative before acceleration. Experience has shown that the best results are obtained with a continuity of the second derivative of jerking.

Figure 6 shows space, speed, acceleration and jerk in relation to time for the type of axis drive used in the OSAI 10 Series control.

In this figure it can be seen that acceleration is particularly gradual and smooths any oscillatory transients generated by the machine tool which correspond to changes in speed.

By analysing the type of drive chosen to control the axes, it can also be seen to be referable to a polynomial function in the time domain.

The aim of handling the dynamic part is to determine a new spline, in the time domain, which enables parameters to be calculated with respect to the characteristics of each axis.

The polynomial interpolation algorithms implemented in the 10 Series CNC, described simply, enable the maximum speed to be calculated for each spline segment calculated in the geometrical phase (depending on the programmed speed and the characteristics of each axis involved in the motion). These constant speed sections are then linked with polynomial ramps.

This type of approach, which is correct in the case of the programming of movements referred to long spline sections that handle all the acceleration and deceleration ramps, becomes particularly complex when the spline segments do not enable the maximum speed to be reached and a single acceleration ramp corresponds to a large number of geometrical splines.

Figure 7 shows a simple example of how this function is performed, by generating a single ramp (Ramp 2) instead of the Ramp 1 and Ramp 3 sequence which, despite respecting the dynamic characteristics set (acceleration and jerk), would generate an undulating trajectory that fails to respect the dynamics of the machine as a whole.

Examples

Example are provided by figures 8, 9, and 10 which show the results of the same machining process, that is, exactly the same part program executed on the 10 Series NC with normal linear interpolation and linear acceleration ramps.

The diagrams shown in the figures represent an enlargement of the same area of the program and show the speed of the X and Y axes during the course of five-axis interpolation.

The undulations in speed that can be seen in figure 8 may be attributed to the changes in speed carried out by the NC in passing from one block to another, generated by the fact that the geometrical outline is not consistent with the algorithms used for regulating the speeds.

In figure 9, the undulations have almost completely disappeared in that, due to the algorithms used and the powerful look ahead, the speed diagram on the profile has become particularly fluid. There are still some irregularities, due to the fact that the trajectory was generated as a broken line and the angles between the program blocks, though certainly small and kept under control by the CAD, cause disturbances to the dynamics of the trajectory. The microvariations in direction caused by these angles are the cause of disturbances in the speed diagram.

In figure 10, the speed diagram is very similar to the one shown in figure 10, in that the algorithm for generating the acceleration ramps is the same but, as a result of the different geometry applied, the disturbances are significantly smoother. It should be pointed out that, in this case, in order to use the same program, it was not possible to parameterise the splines with respect to the centre of the tool, but only the machine co-ordinates, thus losing some of the benefits deriving from the full use of polynomial interpolation.

This difference, which may seem marginal, is in reality highly significant in that it does not permit control of the trajectory of the tool on the part and may generate dynamically inaccurate movements on the work point.

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