64 lines
2.0 KiB
Markdown
64 lines
2.0 KiB
Markdown
# Ramp generator concept for cubic/quadratic ramp combination
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Piecewise ramp:
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- ramp up with s = 1/6 * j * t^3 and v = 1/2 * j * t^2
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so t = sqrt(2*v/j) and s = 1/6 * j * (2*v/j)^(3/2)
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and (6*s/j)^(2/3) = 2*v / j
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and v = j/2 * (6/j)^(2/3) * s^(2/3) = 6^(2/3)/2 * j^(1/3) * s^(2/3)
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- define hand over point s_h
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- ramp up to max speed with s = 1/2 * a * t'^2 = 1/2 * v^2 / a
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Conditions at hand over point s_h:
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- s_h = 1/6 * j * t_h^3 = 1/2 * a * t'_h^2 + ds
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- v_h = 1/2 * j * t_h^2 = a * t'_h
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- a_h = j * t_h = a
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Replacing j = a / t_h:
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- s_h = 1/6 * a * t_h^2 = 1/2 * a * t'_h^2 + ds
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- v_h = 1/2 * a * t_h = a * t'_h
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With t'_h = t_h - dt:
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- s_h = 1/6 * a * t_h^2 = 1/2 * a * (t_h^2 - 2*t_h*dt + dt^2) + ds
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- v_h = 1/2 * a * t_h = a * t_h - a * dt
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Consequently from v_h:
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dt = th/2
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And for s:
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- s_h = 1/6 * a * t_h^2 = 1/8 * a * t_h^2 + ds
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So ds:
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ds = (8-6)/48 * a * t_h^2 = 1/24 * a * t_h^2 = s_h/4
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So we have clear relation from s to the ramp speed:
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if s < s_h, then cubic ramp: v(s) = 6^(2/3)/2 * j^(1/3) * s^(2/3)
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if s >= s_h, then quadratic ramp: v(s) = sqrt(2 * a * (s-s_h/4))
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Still need to reduce j and s_h to one parameter of freedom.
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If choose s_h as user defined parameter, then:
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t_h = sqrt(6 * s_h / a)
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and so:
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j = a / t_h = a / sqrt(6 * s_h / a) = sqrt(a^3 / (6 * s_h))
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For the speed calculation cubic ramp, this yields:
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v(s) = [6^(2/3)/2 * (a^3/6/s_h)^(1/6)] * s^(2/3)
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= [6^(4/6)/2 * (a^3)^(1/6) * 6^(-1/6) * s_h^(-1/6)] * s^(2/3)
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= [6^(3/6)/2 * sqrt(a) * s_h^(-1/6)] * s^(2/3)
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= [sqrt(6)/2 * sqrt(a) * s_h^(-1/6)] * s^(2/3)
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= [sqrt(3/2) * sqrt(a) / s_h^(1/6) ] * s^(2/3)
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Check cubic:
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v(s_h) = [sqrt(3/2) * sqrt(a) / s_h^(1/6) ] * s_h^(2/3)
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= [sqrt(3/2) * sqrt(a) ] * s_h^(1/2)
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and quadratic:
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v(s_h) = sqrt(2 * a * s_h * 3 / 4) = sqrt(3/2 * a * s_h) equals cubic
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With s_h = 0 there will be no cubic ramp start.
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