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.