Making Your Point-to-Point

Typical torque/speed curves for (3A) servo and (3B) step-motor systems

The ultimate goal of any profile is to match the motion system characteristics to the desired application. Trapezoidal and S-curve profiles work well when the motion system's torque response curve is fairly flat. In other words, when the output torque does not vary that much over the range of velocities the system will be experiencing. This is true for most servo motor systems, whether DC Brush or Brushless DC.

Step motors, however, do not have flat torque/speed curves. Torque output is non-linear, sometimes having a large drop at a location called the 'mid-range instability', and generally having drop-off at higher velocities. Figure 3 gives examples of typical torque/speed curves for servo and step-motor systems.

Mid-range instability occurs at the step frequency when the motor's natural resonance frequency matches the current step rate. To address mid-range instability, the most common technique is to use a non-zero starting velocity. This means that the profile instantly 'jumps' to a programmed velocity upon initial acceleration, and while decelerating. This is shown in figure 4. While crude, this technique sometimes provides better results than a smooth ramp for zero, particularly for systems that do not use a microstepping drive technique.

To address drop-off of torque at higher velocities, a Parabolic profile, shown in figure 5 can be used. The corresponding acceleration curve has the characteristic that the acceleration is smallest when the velocity is highest. This is a good match for step-motor systems, because there is less torque available at higher speeds. But notice that starting and ending accelerations are very high, and there is no "S" phase where the acceleration smoothly transitions to zero. So if load oscillation is a problem, parabolic profiles may not work as well as an S-curve, despite the fact that a standard s-curve profile is not optimized for a step motor from the standpoint of the torque/speed curve.

Non-zero starting velocity

Parabolic Profile Equations

Parabolic profiles are closely related to S-curves because they are third-order moves. And as was the case for S-curve profiles, calculating the distance to deceleration is complicated, particularly if profile changes-on-the-fly are allowed.

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 jerk (time rate of change of acceleration)