# Determining time and distance constrained S-curve motion profile

I am attempting to determine (symbolically) the required motion profile for a servo-driven system given three critical parameters:

• dx - The total displacement of the target material

• T - The time in which the motion is to take place, start to finish

• lv - The acceleration "dwell" time, during which the system is operating at a constant velocity

• la - The acceleration "dwell" time, during which the system is operating at a constant acceleration

Based on the above, it appears trivial to determine the discrete time intervals during which segments I-VII will occur as such:

• I - Acceleration increasing to +peak

• II - Acceleration static at +peak

• III - Acceleration decreasing to 0

• IV - Acceleration static at 0

• V - Acceleration decreasing to -peak

• VI - Acceleration static at -peak

• VII - Acceleration increasing to 0

I understand that this essentially provides a motion profile that is piecewise constant for jerk, piecewise trapezoidal for acceleration, smoothed trapezoidal for velocity, and a sigmoid-type curve for position. The benefit of a profile as such is avoiding an asymptotically infinite rate of acceleration at any point in the profile, minimizing wear on mechanical components.

However, I have found significant difficulty in determining how to solve this system symbolically given the above constraints. Integration of the position curve to determine velocity tracking seems trivial, by determining the maximum velocity required during the dwell time to achieve the overall positional change required. However, moving onto integration of the velocity curve seems to be an almost-transcendental solution.

I am wondering if there is a straightforward way to approach this solution that I'm missing? My conclusion thus far is that I need to determine the symbolic expression of the acceleration profile and use this as my starting point to determine jerk by differentiation and velocity and position by integration, but this seems to be an impossible approach given that my profile definition is defined only by displacement and total time. It appears that I may have to set some concrete values for maximum acceleration and velocity, but I can't determine if this is absolutely the case.

• Can you make your question more specific? The general approach and equations for trapezoidal motion profiles are well known and can easily be found online, e.g., designworldonline.com/… Dec 17, 2022 at 5:24