# Trapezoidal motion/kinematics calculations - solving for slewing velocity

I am trying to program some motion control devices which have trapezoidal motion profiles to define a move. I can define the slewing(peak) velocity, accel/decel rates and distance to travel, the the hardware moves a motor the appropriate distance with the parameters given.

I need to be able to define a specific amount of time that a move will last, and calculate what the peak velocity should be when all other factors are known. Starting velocity is not always zero, but ending velocity will always zero for my purposes. The move will always be trapezoidal.

I have an equation for defining the distance traveled when all other factors are known, but I need to isolate the peak velocity to one side of the equation and I am not sharp enough at this math to be able to solve for the peak velocity. Any help would be greatly appreciated.

Assuming:
d = total distance traveled
t = total time
AC = acceleration
DC = deceleration
Vi = initial velocity
Vs = slewing (peak) velocity
Vf = final velocity

I have this equation which I believe is correct. I need to isolate Vs to one side:

d = [(Vs+Vi)/2]*[(Vs-Vi)/AC] + [(Vs+Vf)/2]*[(Vf-Fs)/DC] + Vs*[t-((Vf-Vs)/DC)-((Vs-Vi)/AC)]

I hope that's clear enough to read... If not I attached a JPG of the equation written out by hand.

#### Attachments

• KinematicEquation.jpg
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HallsofIvy
Since Vs appears in two factors of both the first and third terms, that equation is quadratic in Vs. Multiply everything out, determine the coefficients of $V_x^2$ and $V_s$, and use the quadratic formula.