When a motion profile is designed with the desired acceleration, velocity, and position, total motion control results. There is no other way to achieve motion control. Either of two methods can be used to build a motion profile: the mathematical approach or the geometric approach. This month, we'll examine the mathematical approach, and next month we'll study the geometric approach.
The mathematical approach
Countless ways exist to specify a profile, so no treatment can be exhaustive. The methods presented here will be limited to the so-called "constant acceleration profiles." They are characterized by flattopped acceleration waves, which result in velocities that are trapezoidal in nature and position profiles with parabolic and ramp intervals.
Velocity is the derivative of position, and acceleration is the derivative of velocity, making the acceleration wave the second derivative of the position wave. For simplicity of discussion, and without spending an entire page of derivation, we'll assume constant velocity and acceleration. Therefore:
X = 1/2 a t 2 + v0t + X0
The first term is parabolic and exists whenever acceleration is non-zero. The second term is a ramp and is present any time there is an initial velocity, V0. X0is merely the position when t = 0. The process is reversed by differentiating the equation to show that the velocity and acceleration portions are reconstructed.The point at which t = 0 is arbitrary. It is usually shifted from interval to interval for simple trapezoidal-profiles. That is, a new t = 0 is made at the beginning of the Δ t1 interval, again at the beginning of Δ t2 and so on.
Acceleration, velocity and position are not independent of one another. An important concept that comes from this statement is that certain mathematical relationships bind the acceleration, velocity, and position waves together. Specifically, it must be understood that:
When one of the three wave shapes (acceleration, velocity, or position) is given values at all points in time, then the other two wave shapes follow from it, and have NO arbitrary features whatsoever.
|To go from:||and arrive at:|
| — || Integrate |
Table 1. Mathematical operations necessary to make one wave shape from another.
The practical interpretation is that, as we design a motion profile, one of the three wave shapes is arbitrary. We can give it any value at anytime, provided that it is not double valued. However, once that one wave shape is quantified, the shapes of the other two waves are determined by it. Furthermore, at profile design time, we can select any one of the three as a starting point, and let the others follow from it. The table above should be committed to memory and fully understood if motion control is to be mastered.
Profile design is complicated by the fact that its specifications may be incomplete. It is the exception that a particular motion profile will be specified completely by whomever is designing the machine. For example, it may be known only that a slide is to extend and retract some specified distance and must complete a given number of cycles in a minute, with no acceleration times or limits.
In other instances, the acceleration may be required not to exceed some critical value. The acceleration may even be limited so severely that the required cycle time cannot be met, meaning the profile specifications cannot physically be accomplished as stated. The ability to synthesize profiles (create them from stated specifications) and to analyze them is crucial. The integral and derivative relationships can make the process confusing, even to those with experience in such matters. The variety of ways in which profiles can be described adds to the confusion.