File:Chronogrammes loi trapezoidale en acceleration.svg

Summary[edit]

DescriptionChronogrammes loi trapezoidale en acceleration.svg	English: Chronograms for a movement with a trapezoidal acceleration law, in order to control the jerk. A body is moved from the position x = 0 to x = xf, the duration of the movement is T. There are three parts which last T/3: Acceleration (a > 0). Constant velocity (a = 0). Braking (a < 0). In the firts and third parts, a has a trapezoid shape (each subpart lasts T/9), with a maximum value of a0. Français : Chronogrammes pour un mouvement suivant une loi trapézoïdale en accélération, afin de maîtriser l'à-coup (jerk). Il y a trois partie d'une durée de T/3 : Accélération (a > 0). Mouvement uniforme (a = 0). Freinage (a < 0). Dans la première et la troisième partie, a a une forme trapézoïdale (chaque sous-partie durant T/9), avec une accélération maximale valant a0.
Date	13 February 2014
Source	Own work
Author	Christophe Dang Ngoc Chan (Cdang )
SVG development InfoField	The source code of this SVG is invalid due to an error.   This W3C-invalid geometry was created with Batik, Scilab.   This geometry uses embedded text.

1.

${\displaystyle a={\frac {9a_{0}t}{\mathrm {T} }}}$

${\displaystyle v={\frac {9a_{0}t^{2}}{2\mathrm {T} }}}$

${\displaystyle x={\frac {3a_{0}t^{3}}{2\mathrm {T} }}}$

${\displaystyle a_{1}=a_{0}}$

${\displaystyle v_{1}={\frac {a_{0}\mathrm {T} }{18}}={\frac {a_{0}\mathrm {T} }{2\times 3^{2}}}\simeq 0,055\,6a_{0}\mathrm {T} }$

${\displaystyle x_{1}={\frac {a_{0}\mathrm {T} ^{2}}{486}}={\frac {a_{0}\mathrm {T} ^{2}}{2\times 3^{5}}}\simeq 0,002\,06a_{0}\mathrm {T} ^{2}}$

2.

${\displaystyle a=a_{0}}$

${\displaystyle v=v_{1}+a_{0}\left(t-{\frac {\mathrm {T} }{9}}\right)=a_{0}\left(t-{\frac {\mathrm {T} }{18}}\right)}$

${\displaystyle x=a_{0}\left({\frac {t^{2}}{2}}-{\frac {\mathrm {T} t}{18}}+{\frac {\mathrm {T} ^{2}}{486}}\right)}$

${\displaystyle a_{2}=a_{0}}$

${\displaystyle v_{2}={\frac {a_{0}\mathrm {T} }{6}}={\frac {a_{0}\mathrm {T} }{2\times 3}}\simeq 0,167a_{0}\mathrm {T} }$

${\displaystyle x_{2}={\frac {7a_{0}\mathrm {T} ^{2}}{486}}={\frac {7a_{0}\mathrm {T} ^{2}}{2\times 3^{5}}}\simeq 0,014\,4a_{0}\mathrm {T} ^{2}}$

3.

${\displaystyle a=a_{0}\left(-{\frac {9t}{\mathrm {T} }}+3\right)}$

${\displaystyle v=a_{0}\left(-{\frac {9t^{2}}{2\mathrm {T} }}+3t-{\frac {\mathrm {5T} }{18}}\right)}$

${\displaystyle x=a_{0}\left({\frac {-3t^{3}}{2\mathrm {T} }}+{\frac {3t^{2}}{2}}-{\frac {5\mathrm {T} t}{18}}+{\frac {\mathrm {T} ^{2}}{54}}\right)}$

${\displaystyle a_{3}=0}$

${\displaystyle v_{3}={\frac {2a_{0}\mathrm {T} }{9}}={\frac {2a_{0}\mathrm {T} }{3^{2}}}\simeq 0,222a_{0}\mathrm {T} }$

${\displaystyle x_{3}={\frac {a_{0}\mathrm {T} ^{2}}{27}}={\frac {a_{0}\mathrm {T} ^{2}}{3^{3}}}\simeq 0,037\,0a_{0}\mathrm {T} ^{2}}$

4.

${\displaystyle a=0}$

${\displaystyle v=v_{3}}$

${\displaystyle x=x_{3}+v_{3}\left(t-{\frac {\mathrm {T} }{3}}\right)={\frac {a_{0}\mathrm {T} }{9}}\left(2t-{\frac {\mathrm {T} }{3}}\right)}$

${\displaystyle a_{4}=0}$

${\displaystyle v_{4}=v_{3}}$

${\displaystyle x_{4}={\frac {a_{0}\mathrm {T} ^{2}}{9}}={\frac {a_{0}\mathrm {T} ^{2}}{3^{2}}}\simeq 0,111a_{0}\mathrm {T} ^{2}}$

5.

${\displaystyle a=-{\frac {9a_{0}}{\mathrm {T} }}\left(t-{\frac {2\mathrm {T} }{3}}\right)=a_{0}\left({\frac {-9t}{\mathrm {T} }}+6\right)}$

${\displaystyle v=a_{0}\left({\frac {-9t^{2}}{2\mathrm {T} }}+6t-{\frac {16\mathrm {T} }{9}}\right)}$

${\displaystyle x=a_{0}\left(-{\frac {3t^{3}}{2\mathrm {T} }}+3t^{2}-{\frac {16\mathrm {T} t}{9}}+{\frac {11\mathrm {T} ^{2}}{27}}\right)}$

${\displaystyle a_{5}=-a_{0}}$

${\displaystyle v_{5}=v_{2}}$

${\displaystyle x_{5}={\frac {65a_{0}\mathrm {T} ^{2}}{486}}={\frac {13\times 5a_{0}\mathrm {T} ^{2}}{2\times 3^{5}}}\simeq 0,134a_{0}\mathrm {T} ^{2}}$

6.

${\displaystyle a=-a_{0}}$

${\displaystyle v=a_{0}\left(-t+{\frac {17\mathrm {T} }{18}}\right)}$

${\displaystyle x=a_{0}\left({\frac {-t^{2}}{2}}+{\frac {17\mathrm {T} t}{18}}-{\frac {145\mathrm {T} ^{2}}{486}}\right)}$

${\displaystyle a_{6}=-a_{0}}$

${\displaystyle v_{6}=v_{1}}$

${\displaystyle x_{6}={\frac {71a_{0}\mathrm {T} ^{2}}{486}}={\frac {71a_{0}\mathrm {T} ^{2}}{2\times 3^{5}}}\simeq 0,146a_{0}\mathrm {T} ^{2}}$

7.

${\displaystyle a=a_{0}\left({\frac {9t}{\mathrm {T} }}-9\right)}$

${\displaystyle v=a_{0}\left({\frac {9t^{2}}{2\mathrm {T} }}-9t+{\frac {9\mathrm {T} }{2}}\right)}$

${\displaystyle x=a_{0}\left({\frac {3t^{3}}{2\mathrm {T} }}-{\frac {9t^{2}}{2}}+{\frac {9\mathrm {T} t}{2}}-{\frac {73\mathrm {T} ^{2}}{54}}\right)}$

${\displaystyle a_{7}=0}$

${\displaystyle v_{7}=0}$

${\displaystyle x_{\mathrm {f} }={\frac {4a_{0}\mathrm {T} ^{2}}{27}}={\frac {2^{2}\times a_{0}\mathrm {T} ^{2}}{3^{3}}}\simeq 0,148a_{0}\mathrm {T} ^{2}}$

Max.

${\displaystyle v_{\max }={\frac {2a_{0}\mathrm {T} }{9}}={\frac {3x_{\mathrm {f} }}{2\mathrm {T} }}}$

${\displaystyle a_{\max }=a_{0}={\frac {27x_{\mathrm {f} }}{4\mathrm {T} ^{2}}}}$

${\displaystyle j_{\max }={\frac {9a_{0}}{\mathrm {T} }}={\frac {243x_{\mathrm {f} }}{4\mathrm {T} ^{3}}}}$

Scilab source

This media was created with Scilab, a free open-source software. Here is a listing of the Scilab source used to create this file.

function [t1, t2, t3, t4, t5, t6, t7] = indice(t)
    t1 = (t < 1/9);
    t2 = (t >= 1/9) & (t < 2/9);
    t3 = (t >= 2/9) & (t < 1/3);
    t4 = (t >= 1/3) & (t <= 2/3);
    t5 = (t > 2/3) & (t <= 7/9);
    t6 = (t > 7/9) & (t <=8/9);
    t7 = (t > 8/9);
endfunction

function [j]=jerk(t)
    [t1, t2, t3, t4, t5, t6, t7] = indice(t);
    j = zeros(t);
    j(t1) = 9;
    j(t2) = 0;
    j(t3) = -9;
    j(t4) = 0;
    j(t5) = -9;
    j(t6) = 0;
    j(t7) = 9;
endfunction

function [a]=acceleration(t)
    [t1, t2, t3, t4, t5, t6, t7] = indice(t);
    a = zeros(t);
    a(t1) = 9*t(t1);
    a(t2) = 1;
    a(t3) = (3 - 9*t(t3));
    a(t4) = 0;
    a(t5) = 6 - 9*t(t5);
    a(t6) = -1;
    a(t7) = 9*t(t7) - 9;
endfunction

function [v] = vitesse(t)
    [t1, t2, t3, t4, t5, t6, t7] = indice(t);
    v = zeros(t);
    v(t1) = 9/2*t(t1).^2;
    v(t2) = t(t2) - 1/18;
    v(t3) = (-9/2*t(t3) + 3).*t(t3) - 5/18;
    v(t4) = 2/9;
    v(t5) = (-9/2*t(t5) + 6).*t(t5) - 16/9;
    v(t6) = -t(t6) + 17/18;
    v(t7) = (9/2*t(t7) - 9).*t(t7) + 9/2;
endfunction

function [x] = position(t)
    [t1, t2, t3, t4, t5, t6, t7] = indice(t);
    x = zeros(t);
    x(t1) = 3/2*t(t1).^3;
    x(t2) = (0.5*t(t2) - 1/18).*t(t2) + 1/486;
    x(t3) = ((-3/2*t(t3) + 3/2).*t(t3) - 5/18).*t(t3) + 1/54;
    x(t4) = 1/9*(2*t(t4) - 1/3);
    x(t5) = ((-3/2*t(t5) + 3).*t(t5) - 16/9).*t(t5) + 11/27;
    x(t6) = (-0.5*t(t6) + 17/18).*t(t6) - 145/486;
    x(t7) = ((3/2*t(t7) - 9/2).*t(t7) + 9/2).*t(t7) - 73/54;
endfunction

t = 0:0.005:1;

j = jerk(t);
a = acceleration(t);
v = vitesse(t);
x = position(t);

scf(0);
clf;

subplot(4, 1, 1)
plot(t, x)
xtitle("position", "t/T", "$x/(a_0 \mathrm{T}^2)$")

subplot(4, 1, 2)
plot(t, v)
xtitle("vitesse", "t/T", "$v/(a_0 \mathrm{T})$")

subplot(4, 1, 3)
plot(t, a)
xtitle("accélération", "t/T", "$a/a_0$")

subplot(4, 1, 4)
xtitle("à-coup", "t/T", "$j \times \mathrm{T}/a_0$")
plot(t, j)

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