I Wanna Go Fast

Since trapezoidal profiles spend their time at full acceleration or full deceleration, they are, from the standpoint of profile execution, faster than S-curve profiles. But if this 'all on'/'all off' approach causes an increase in settling time, the advantage is lost. Often, only a small amount of "S" (transition between acceleration and no acceleration) can substantially reduce induced vibration. And so to optimize throughput the S-curve profile must be 'tuned' for each a given load and given desired transfer speed.

What s-curve form is right for a given system? On an application by application basis, the specific choice of the form of the s-curve will depend on the mechanical nature of the system, and the desired performance specifications. For example in medical applications which involve liquid transfers that should not be jostled, it would be appropriate to choose a profile with no phase II & VI segment at all, instead spreading the acceleration transitions out as far as possible, thereby maximizing smoothness.

In other applications involving high speed pick and place, overall transfer speed is most important, so a good choice might be an s-curve with transition phases (phases I, III, V, and VII) that are 5-15 % of phase II & VI. In this case the s-curve profile will add a small amount of time to the overall transfer time, but because of reduced load oscillation at the end of the move, the total effective transfer time can be considerably decreased. Trial and error using a motion measurement system is generally the best way to determine the right amount of "S", because modelling the response to vibrations is complicated, and not always accurate.

Trapezoidal Profile Equations

The basic math required to execute trapezoidal profiles is straghtforward. There, however, two forms that can be used; the continuous form, that will be familiar from High School Physics, and the discrete time form, which is used in most motion systems that utilize microprocessors or DSPs (Digital Signal Processor) to generate a new set of motion parameters at each tick of the motion 'clock'.

Continuous form
PT = P0 + V0T + 1/2AT2
VT = V0 + AT
and
Pdecel = V2/2A
Discrete time form
PT = PT + VT +1/2A
VT = VT+A
where:
P0 and V0, are the starting position and velocities
PT and VT, are the position and velocity at time T
A is the profile acceleration

S-Curve Profile Equations

Because they are third versus second-order curves, and because there are seven versus three separate motion segments, point-to-point S-curves are more complicated then Trapezoids. In particular it is not simple to calculate the stopping distance for a given set of profile values. Accordingly, many s-curve profiling systems restrict changes-on-the-fly, or do not allow asymmetric profiles. These restrictions allow information about how long, and over what distance, the profile previously took to accelerate to determine when to start decelerating.

Continuous form
PT = P0 + V0T + 1/2A0T2 + 1/6JT3
VT = V0 + A0T + 1/2 JT2
AT = A0 + JT
Discrete time form
PT = PT + VT +1/2AT + 1/6J
VT = VT+AT + 1/2JT
AT = AT+JT
where
P0, V0, and A0 are the starting position, velocity, and accelerations
PT , VT, and AT are the position, velocity, and acceleration at time T

J is the profile jerk (time rate of change of acceleration)