# Velocity Is Key to Motion Control

02/01/1997

Motion control systems are generally designed to move a load along a specified path as fast as possible without damaging the load or the mechanism driving it. Heavy loads are particularly difficult to control since inertia tends to force the load off course during high acceleration maneuvers. Worse still, the stress of resisting the load's inertia can destroy not only the drive mechanism, but the load itself.

Rotating machinery, linear conveyors, and multiaxis robotic arms all share these problems, but a more familiar example of heavy-load motion control is the passenger elevator. An elevator must be able to move its occupants from floor to floor as fast as possible without making them sick. An elevator that lurches up the shaft and stops abruptly at the end of the trip will certainly meet the objective of high speed, but not passenger comfort. Conversely, a very slow elevator can transport its passengers comfortably, but at the expense of longer travel times.

An acceptable trade-off between speed and stress can be achieved by carefully manipulating the velocity of the load. The accompanying figure shows three techniques for accelerating a load from a stand-still up to a final velocity of V f (distance/sec for linear motion or rad/sec for circular motion). All three techniques limit inertial forces by limiting the maximum acceleration to a value of A m (distance/sec2or rad/sec2).

Technique 1 is the fastest. The velocity of the load is simply ramped up from zero to V f , then held steady thereafter. The corresponding acceleration starts at A m immediately then drops to zero when the velocity levels off. This fast and furious technique would correspond to the lurching elevator. The abrupt changes in acceleration would cause the passengers to be pinned to the floor at take-off then thrown to the ceiling once V f is achieved.

Technique 2 also starts out with maximum acceleration, but gradually reduces the acceleration to zero so that the velocity transitions smoothly to a steady state. An elevator moving this way would still lurch up the shaft at take-off, but would cause the passengers much less stress thereafter.

Technique 3 smoothes out both the initial and final changes in velocity by starting and ending with zero acceleration. The maximum acceleration A m is achieved half way through the acceleration period. Although this technique is not as fast as the first, it would be much more comfortable for an elevator's passengers.

The mathematics
A motion controller equipped with velocity feedback could be used to implement any of these three acceleration techniques. Each would require a different velocity profile v(t) for the controller to use as a reference signal. The first velocity profile would be a simple ramp...

The second velocity profile could be represented as an exponential increase...

where K = A m /V f is a constant. The corresponding acceleration would be...

There are several mathematical formulas that would produce the 'S' shape of the third velocity profile. The one shown in the accompanying figure is given by:

where the constant

is the time required to achieve the final velocity V f . Note that this interval is roughly 57 percent longer than the time required for the first technique to achieve V f using the same maximum acceleration A m . The acceleration for the third technique is given by:

Vance J. VanDoren Holds a Ph.D. in Control Engineering from Purdue University's School of Mechanical Engineering.

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