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ramp_cubic_quadratic.md

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Ramp generator concept for cubic/quadratic ramp combination

Piecewise ramp:

  • ramp up with s = 1/6 * j * t^3 and v = 1/2 * j * t^2 so t = sqrt(2v/j) and s = 1/6 * j * (2v/j)^(3/2) and (6s/j)^(2/3) = 2v / j and v = j/2 * (6/j)^(2/3) * s^(2/3) = 6^(2/3)/2 * j^(1/3) * s^(2/3)
  • define hand over point s_h
  • ramp up to max speed with s = 1/2 * a * t'^2 = 1/2 * v^2 / a

Conditions at hand over point s_h:

  • s_h = 1/6 * j * t_h^3 = 1/2 * a * t'_h^2 + ds
  • v_h = 1/2 * j * t_h^2 = a * t'_h
  • a_h = j * t_h = a

Replacing j = a / t_h:

  • s_h = 1/6 * a * t_h^2 = 1/2 * a * t'_h^2 + ds
  • v_h = 1/2 * a * t_h = a * t'_h

With t'_h = t_h - dt:

  • s_h = 1/6 * a * t_h^2 = 1/2 * a * (t_h^2 - 2t_hdt + dt^2) + ds
  • v_h = 1/2 * a * t_h = a * t_h - a * dt

Consequently from v_h: dt = th/2

And for s:

  • s_h = 1/6 * a * t_h^2 = 1/8 * a * t_h^2 + ds

So ds: ds = (8-6)/48 * a * t_h^2 = 1/24 * a * t_h^2 = s_h/4

So we have clear relation from s to the ramp speed:

if s <  s_h, then cubic ramp:      v(s) = 6^(2/3)/2 * j^(1/3) * s^(2/3)
if s >= s_h, then quadratic ramp:  v(s) = sqrt(2 * a * (s-s_h/4))

Still need to reduce j and s_h to one parameter of freedom. If choose s_h as user defined parameter, then:

t_h = sqrt(6 * s_h / a)

and so:

j = a / t_h = a / sqrt(6 * s_h / a) = sqrt(a^3 / (6 * s_h))

For the speed calculation cubic ramp, this yields:

v(s) = [6^(2/3)/2 * (a^3/6/s_h)^(1/6)] * s^(2/3)
     = [6^(4/6)/2 * (a^3)^(1/6) * 6^(-1/6) * s_h^(-1/6)] * s^(2/3)
     = [6^(3/6)/2 * sqrt(a)                * s_h^(-1/6)] * s^(2/3)
     = [sqrt(6)/2 * sqrt(a)                * s_h^(-1/6)] * s^(2/3)
     = [sqrt(3/2) * sqrt(a) / s_h^(1/6)                ] * s^(2/3)

Check cubic: v(s_h) = [sqrt(3/2) * sqrt(a) / s_h^(1/6) ] * s_h^(2/3) = [sqrt(3/2) * sqrt(a) ] * s_h^(1/2)

and quadratic: v(s_h) = sqrt(2 * a * s_h * 3 / 4) = sqrt(3/2 * a * s_h) equals cubic

With s_h = 0 there will be no cubic ramp start.