Introduction

With the advent of high specification CAD packages which permit users to easily produce an accurate multidimensional path or orientation in space by the final tool element, motion control manufacturers have had to invest considerable effort and finance into research and development of complex motion algorithms to provide very smooth velocity profiles for the axes to follow whilst maintaining a very high degree of positional accuracy and further more, reduce execution cycle times to increase machine productivity.

The difficulty for the motion control designer is the part produced within the CAD package may appear to have blended intersections, however this is generally not the case when proved mathematically. In practice the CAD user cannot be expected to ensure every intersect is a perfect blend. Therefore, to attain smooth and continuous axis velocity profiles from any given data, this very complex problem is left with the control manufacturer.

Why is it necessary to produce smooth velocity profiles?

1. To prolong machine life. Motion axes being commanded to instantaneously change velocity will in a very short period of time prove to be unreliable.

2. Increase reliability of the drive chain.

3. Reduce machine maintenance.

4. Significantly improve cut quality of parts being machined.

5. Reduce cycle execution times.

6. Reduce tool wear, thus again increasing throughput.

Continuous Motion

A stream of linear (vector), arc, bezier or spline commands will generally specify a continuous multidimensional tool path. From this data the 'motion planner' needs to generate a velocity profile for each moving axis, keeping it within its acceleration, deceleration and terminal velocity limits. To simplify the generation of axis velocity profiles and make best use of modern micro-processor features, for example caching and data bursting, the input data should be converted to a common format which would be generally vectorised or splined.

To achieve this, the 'motion planner' must know what is to be executed ahead of what is currently being executed. The term for this function is 'look-ahead' and constitutes the main criteria to permit the generation of continuous smooth motion. 'Look-ahead' is the process of examining and performing some pre-calculations on programmed commands ahead of the last command to have its final velocity profile calculated and specified.

There are disadvantages associated with looking-ahead, however they are far outweighed by the advantages, however for completeness they are as follows.

1. To 'look-ahead' and perform the pre-calculations requires a significant amount of processing power which may require a multiple processor solution.

2. To store the motion commands and hold onto them until the complete velocity profile has been specified requires a significant amount of fast storage media, generally SRAM or DRAM.

The extent of the disadvantages will be determined by how far the controller is looking ahead. A number of controls on the market today do not look ahead many more than 10 commands. This significantly limits how quickly the controller executes a part program which requires high cut speeds with accurate positioning, however it does not have any detrimental effect on the overall smoothness of the velocity profile.

By 'looking-ahead' at more commands the 'motion planner' has a much better chance of achieving the programmed terminal velocity and keeping the tool speed constant when executing a series of short moves providing the controller is able to calculate quickly enough. If the 'motion planner' is looking at the next 10 moves to determine the end/final velocity of the 1st move then the end velocity of the available acceleration and deceleration distance for those 10 moves will limit that 1st move. However if the 'look-ahead' is increased to 100 then the 'motion planner' is able to accelerate the axes to a higher velocity, probably achieving the required programmed terminal velocity due to it knowing the available acceleration and deceleration distance for those 100 moves.

To recap 'look-ahead' permits the controller to accelerate, decelerate and more importantly still as far as cut quality is concerned, keep the tool speed constant across a number of consecutive move commands.

Intersection Velocity Limiting

When executing a series of vectors it is necessary to limit the velocity change of each axis at the intersect to prevent the machine from jolting at the end of one vector and the start of the next. Therefore each individual axis may need to automatically decelerate going into the intersect and then accelerate on the other side of it. With some motion controllers they leave this function to the CNC programmer. It is by far, more desirable if this complex problem is resolved by the controller as the permissible velocity changes are a function of the machine being controlled and not the part program.

The size of the velocity change will be directly proportional to the change in direction. Therefore the larger the intersect angle the larger the velocity change.

A simple, yet very effective solution to this problem is to permit the system builder to configure a set of parameters for each axis which set the maximum allowable velocity change at minimum speed and the maximum allowable velocity change at maximum speed. Using these parameters the control equations can interpolate to obtain the intermediate permitted velocity changes for any given angle change. With these parameters the step change at the intersect can now be controlled and it is left with the system builder to trade part quality against job cycle execution time. The higher the settings, the higher the throughput, however at some stage part quality will be compromised as the machine will not be able to change velocity instantaneously as being specified.

A further degree of control needs to be embedded within the motion algorithms to enhance part quality. When decelerating all axes to a stop due to a large intersect angle having been specified it is often necessary to pause at the intersect before accelerating off in the new direction. The pause is required to give the stopping axes time to settle at the intersect before changing direction which prevents against the possibility of a wavy line being machined. The length of pause/dwell should be proportional the size of angle change to minimise the overall effect on the job execution time but still sufficiently large enough to attain the required cut quality. A simple way to achieve this is to specify the minimum angle change for which a minimum pause will be required and specify the pause required for a complete change of direction i.e. 180 degrees and then interpolate to find the intermediate values.

Vector Blending

This function is required to remove the velocity step changes caused by direction changes at vector intersects. Reasonable motion smoothness can be achieved without vector blending providing the velocity changes are limited to within a range the mechanics and drive chain can cope with. To achieve the optimum velocity profile, to gain the highest cut quality and reduce job execution time vector blending is a must.

To achieve a continuous motion profile without velocity step changes the vectors must be blended to each other with other curves. The blend used is parabolic in shape and allows a smooth change of velocity, but not acceleration between successive vectors.

Whilst blending allows faster overall machine motion, the trade off is part accuracy since at the blend the tool must deviate away from the programmed path. The permitted deviation distance should be a parameter the system builder can configure to permit the trade off to be made.

The curve is a consequence of one axis accelerating and another decelerating either side of the intersect.

Generally when executing some form of arc contouring whether it be arc, ellipses, bezier, splines etc. vector blending permits the original data to be broken down to a much more coarse resolution as the vector blending algorithms will ensure a smooth velocity transition occurs at the intersect. This is quite an important realisation as breaking the original data down is mathematically intensive, therefore generally time consuming for the controller.

Before assuming it is not acceptable to deviate from the programmed path consider how accurately and repeatable the machine can position. Also consider how well the servoing function can follow the path specified for it and also bear in mind the specification of the component being machined as the tolerance may be greater than the deviation distance. In some applications it will not be acceptable to deviate away from the programmed path, however with the majority it is quite acceptable as the extra amount of productivity far out weighs the small inaccuracy of the component. In some cases the component accuracy may be improved due to the velocity profile being that much smoother.

Small circle/arc problem

It is often desirable to reduce the resultant tool speed when executing small arcs or circles to improve the quality. The CNC programmer specifying a speed change for the relevant arc has historically achieved this. However this is quite time consuming and prone to error. The ideal solution would be for the 'motion planner' to scale the programmed speed automatically based upon the angle change and distance over the next number of vectors.

The flowchart below shows how this may be achieved.

Axis Trajectory Generation

Once the final velocity profile has been specified by 'looking-ahead', limiting intersect velocities and blending where possible the next stage is to break it up into set points. Set points are either velocity or positional commands to be executed at specified time intervals. The time interval is generally termed 'trajectory time' and typically ranges from 1mS to 30mS.

Simple trapezoidal velocity profiles do not completely capture the machine characteristics and more advanced controllers allow the specification of an 'S' parameter which governs the rate of change of acceleration for each axis. This additional parameter takes into account the fact that the machine requires a finite amount of time to change its acceleration and deceleration. The effect is to smooth out discontinuities in the velocity profile and thus achieve smoother acceleration, deceleration and transitions from acceleration to constant velocity and from constant velocity to deceleration. The component accuracy can also be improved by 'S' curve ramps, as generally the axis inertia is high enough to cause quite large acceleration errors. By changing the rate of acceleration the acceleration errors can be kept to a minimum.

A major problem with generating trajectories for low-resolution axes or axes which are to move at slow cut speeds is the effect of quantisation errors.

Although the above graph showing the effect of quantisation errors is slightly exaggerated for the purpose of illustration the effect on the cut quality can be quite serious especially with very dynamic systems. Quantisation problems also result in a saw tooth occurring on the constant velocity portion of the move, again having a detrimental effect on the cut quality due to the speeding up and slowing down of the tool. With some mechanical systems the problem may be sufficiently damped by the drive chain or the inertia of the machine so as not to cause any practical problems, however if the control manufacturer is aiming to produce a system which may be fitted to any mechanical system then it is a problem which must be addressed.

The solution to what appears to be a relatively complex problem is quite simply to calculate velocity and positional set points to a higher resolution than that of the machine being controlled. This eradicates the problem created by the trajectory generator and removes the possibility of quantisation errors occurring when calculating the control signal to output to the drive amplifier from within the servoing function.

Alternatively, instead of using fixed point arithmetic floating point could be used to calculate the set points which would resolve the quantisation problem, however this leads to other problems which are outside of the scope of this paper.

Servoing

Once a smooth, accurate co-ordinated set of velocity or positional command profiles for each axis has been generated all that remains is for them to be faithfully reproduced by each axis driving motor. In the case of either AC or DC motor this objective will normally be achieved through the use of a feedback loop, usually with an optical encoder perhaps mounted on the motor shaft, providing the necessary position and velocity feedback information.

When designing the motion control if commercially available drive amplifiers are to be used which operate from a +/- 10 volt signal then the resolution of the DAC outputs must be considered as this is your control signal to the motor. Generally 12 bit DACS (11 control bits + 1 sign bit) will give acceptable control, however 16 bit DACS will give smoother control providing the servoing algorithms are able to make best use of it. Also the least significant bits should not be lost to noise problems with your DAC circuit.

The following system diagram shows how a control loop using PID strategy with feed forward and notch filtering may be implemented

If PID strategy is being deployed to make the axis follow the trajectory then the differentiation mode should be considered. Differentiating 'following error' (difference between the current axis position and the position specified by the trajectory generator) will easily give much smaller tracking errors at the expense of motion smoothness. By differentiating position the control signal will generally have a reasonable amount of high frequency oscillations, which does not bode well for motion smoothness. This is due to the dynamic nature of the error signal i.e. it is unlikely to be very stable or constant, especially during acceleration and deceleration.

Differentiating velocity is the other option. This mode permits a much smoother control signal to be calculated, as the actual velocity of the axis will be far less dynamic. By differentiating velocity the control signal and following error will be proportional to the commanded velocity as specified by the trajectory generator. Intuitively, due to the following error being relatively large at terminal velocity it is natural to assume the component accuracy is being compromised for motion smoothness. This is certainly not the case, providing the axis following errors are reasonably balanced (i.e. for 45 degree move the X:Y ratio is 1) and there is a sufficient pause to permit the axes to 'catch-up' before changing direction then the accuracy of the component is not compromised.

Feedforward Control

Feedback systems in motion controllers are able to move axes by examining the current position and looking to see where it should go next. This results in a control signal being calculated to compensate for the discrepancy. In practice, feedback configurations exhibit varying lag times between commanded and actual trajectory for each axis due to differences in dynamics, inaccuracies in acceleration and deceleration. An ideal solution to overcome the lag is to add extra intelligence into the compensation loop. This is achieved by modifying the control signal based upon the compensation loop knowing what the commanded velocity and acceleration/deceleration is, ahead of the set point currently being calculated.

The feed forward block of the compensation loop is generally configured by two system parameters, velocity feed forward (KVFF) and acceleration feed forward (KAFF). Some high performance systems, which permit independent acceleration and deceleration rates, may also incorporate a deceleration feed forward (KDFF) parameter to compensate for deceleration errors.

Notch Filtering

In the quest to achieve higher precision component manufacture it has become necessary to overcome mechanical resonance. A digital notch filter placed at the output of the compensator loop may be used to counteract the physical resonance.

We will assume that the filter is described by a set of quadratic coefficients of the form determined by one of the many approximation methods e.g. Butterworth, Chebchev, etc. There are many analogue and digital filter design software packages on the market which permit high level designs to be generated which resolves the design to produce the quadratic coefficients.

The basic quadratic block for the implementation of an infinite impulse response filter is as follows: -

Multiple order (length) filters can be implemented by cascading as shown below for a fourth order infinite impulse response filter, that is feeding the output of one quadratic block into the input of another.

We can generate the transfer function for this section by determining the expressions for the intermediate signal w(n) and the output signal y(n)

These equations can now be z-transformed to give

The above two equations can now be rewritten and combined to determine the transfer function for this section of the filter.

As can be seen in the system block diagram of the fourth order IIR filter, the input signal is first multiplied by the DC gain constant and is then processed through two quadratic factors. For every quadratic factor an intermediate signal w(n) and an output signal y(n) is calculated and every quadratic section requires the values w(n-1) and w(n-2) to be retained.

The final equations to be implemented may be written as follows: -

On a final note, filtering is really only necessary due to the control loop being unable to cope with the dynamic response characteristics of the system being controlled. In an ideal world the transfer functions would be generated for the system being controlled and suitable control equations generated from them to remove the need to have extra control sub-systems. However as we are all trying to achieve a flexible system, which may be applied to any system needing motion control, this is not a practical solution. Therefore we give the commissioning engineer the tools and training to achieve an optimal configuration for the system intuitively.