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//
// Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
// Copyright (C) ????-2008 - INRIA
//
// This file is distributed under the same license as the Scilab package.
//
function [xdot]=car(t,x)
//
//
xdot=zeros(1,4) ;
// car length for the control computation
LCpct = bigL * 1.;
// calcul de u1 et u2
//
if (t<0)
u1=0 ; u2=0 ;
elseif (t < bigT)
// motion 1
tau = t/bigT ;
phi=tau*tau*(3-2*tau) ;
dphi = 6*tau*(1-tau) ;
a = (1-phi)*a1 + phi*a0 ;
da = ((a0-a1)/bigT) * dphi ;
f = p(1).*(a-a0)^3 + p(2).*(a-a0)^4 + p(3).*(a-a0)^5 ;
df = 3*p(1).*(a-a0)^2 + 4*p(2).*(a-a0)^3 + 5*p(3).*(a-a0)^4 ;
ddf = 6*p(1).*(a-a0) + 12*p(2).*(a-a0)^2 + 20*p(3).*(a-a0)^3 ;
dddf= 6*p(1).*1 + 24*p(2).*(a-a0) + 60*p(3).*(a-a0)^2 ;
k = ddf / ((1+df*df)^(3/2)) ;
dk = ( dddf - 3*df*ddf*ddf/(1+df*df)) / ((1+df*df)^(3/2)) ;
u1 = ((1+df*df)^(1/2)) * da ;
u2 = (bigL*dk / (1+(bigL*k)^2) ) * da ;
elseif (t < (2*bigT) )
// motion 2
tau = t/bigT -1 ;
phi=tau*tau*(3-2*tau) ;
dphi = 6*tau*(1-tau) ;
a = (1-phi)*a0 + phi*a1 ;
da = ((a1-a0)/bigT) * dphi ;
f = p(1).*(a-a0)^3 + p(2).*(a-a0)^4 + p(3).*(a-a0)^5 ;
df = 3*p(1).*(a-a0)^2 + 4*p(2).*(a-a0)^3 + 5*p(3).*(a-a0)^4 ;
ddf = 6*p(1).*(a-a0) + 12*p(2).*(a-a0)^2 + 20*p(3).*(a-a0)^3 ;
dddf= 6*p(1).*1 + 24*p(2).*(a-a0) + 60*p(3).*(a-a0)^2 ;
k = ddf / ((1+df*df)^(3/2)) ;
dk = ( dddf - 3*df*ddf*ddf/(1+df*df)) / ((1+df*df)^(3/2)) ;
u1 = ((1+df*df)^(1/2)) * da ;
u2 = (LC*dk / (1+(LC*k)^2) ) * da ;
else
u1=0; u2=0;
end
xdot(1)= u1 * cos(x(1,3)) ;
xdot(2)= u1 * sin(x(1,3)) ;
xdot(3)= u1 * tan(x(1,4)) / bigL ;
xdot(4)= u2 ;
function [xdot]=car2T(t,x)
//
xdot=zeros(1,6) ;
// calcul de u1 et u2
//
if (t<0)
u1=0 ; u2=0 ;
elseif (t < bigT)
// motion
tau = t/bigT ;
phi=tau*tau*(3-2*tau) ;
dphi = 6*tau*(1-tau) ;
a = (1-phi)*a1 + phi*a0 ;
da = ((a0-a1)/bigT) * dphi ;
[ff df d2f d3f d4f d5f] = cr2Tfjt(a) ;
[k2 k1 k0 dk0]=cr2Tfk(df,d2f,d3f,d4f,d5f) ;
u1 = ( (1+(d2*k2)^2)*(1+(d1*k1)^2)*(1+df^2) )^(1/2) * da ;
u2 = da * bigL * dk0 / (1+(bigL*k0)^2) ;
else
u1=0; u2=0;
end
xdot(1)= u1 * cos(x(1,3)) ;
xdot(2)= u1 * sin(x(1,3)) ;
xdot(3)= u1 * tan(x(1,6)) / bigL ;
xdot(4)= u1 * sin(x(1,3)-x(1,4)) / d1 ;
xdot(5)= u1 * sin(x(1,4)-x(1,5)) * cos(x(1,3)-x(1,4)) / d2 ;
xdot(6)= u2 ;
function [ff,df,d2f,d3f,d4f,d5f]=cr2Tfjt(a)
//
//
M= [
(a-a0)^5 (a-a0)^6 (a-a0)^7 (a-a0)^8 (a-a0)^9
5*(a-a0)^4 6*(a-a0)^5 7*(a-a0)^6 8*(a-a0)^7 9*(a-a0)^8
20*(a-a0)^3 30*(a-a0)^4 42*(a-a0)^5 56*(a-a0)^6 72*(a-a0)^7
60*(a-a0)^2 120*(a-a0)^3 210*(a-a0)^4 336*(a-a0)^5 504*(a-a0)^6
120*(a-a0)^1 360*(a-a0)^2 840*(a-a0)^3 1680*(a-a0)^4 3024*(a-a0)^5
120 720*(a-a0)^1 2520*(a-a0)^2 6720*(a-a0)^3 15120*(a-a0)^4
]*p ;
ff = M(1) + b0 ;
df = M(2) ;
d2f = M(3) ;
d3f = M(4) ;
d4f = M(5) ;
d5f = M(6) ;
function [k2,k1,k0,dk0]=cr2Tfk(df,d2f,d3f,d4f,d5f)
//
//
// computation of curvatures from derivatives of b=f(a)
//
g = (1+df^2)^(-1/2);
dg = - df*d2f*g^3 ;
d2g = - g*g*(d2f^2*g+df*d3f*g+3*df*d2f*dg) ;
d3g = ....
- 2*g*dg*(d2f^2*g+df*d3f*g+3*df*d2f*dg) ....
- g^2*(3*d2f*d3f*g+df*d4f*g ....
+ 4*d2f^2*dg+4*df*d3f*dg+3*df*d2f*d2g) ;
//
k2 = d2f * g^3 ;
dk2 = d3f*g^3 + 3*d2f*g^2*dg ;
d2k2 = g^2*(d4f*g+6*d3f*dg+3*d2f*d2g) + 6*g*dg^2*d2f ;
d3k2 = 2*g*dg*(d4f*g+6*d3f*dg+3*d2f*d2g) ....
+ g^2*(d5f*g+7*d4f*dg+9*d3f*d2g+3*d2f*d3g) ....
+6*dg^3*d2f+12*g*dg*d2g*d2f+6*g*dg^2*d3f ;
//
g2 = (1+(d2*k2)^2)^(-1/2) ;
dg2 = -d2^2*k2*dk2*g2^3 ;
d2g2 = -d2^2*g2^2*(dk2^2*g2+k2*d2k2*g2+3*k2*dk2*dg2) ;
//
h2 = g2^3*g ;
dh2 = g2^2*(3*dg2*g+g2*dg);
d2h2 = 2*g2*dg2*(3*dg2*g+g2*dg) ....
+ g2^2*(3*d2g2*g+4*dg2*dg+g2*d2g) ;
//
k1 = g2*k2 + d2*h2*dk2 ;
dk1 = dg2*k2 + g2*dk2 + d2 * (dh2*dk2+h2*d2k2) ;
d2k1 = d2g2*k2 + 2*dg2*dk2 + g2*d2k2 ....
+ d2 * (d2h2*dk2+2*dh2*d2k2+h2*d3k2) ;
//
g1 = (1+(d1*k1)^2)^(-1/2) ;
dg1 = - d1^2*k1*dk1*g1^3 ;
//
k0 = g1*k1 + d1*g1^3*g2*g*dk1 ;
dk0 = dg1*k1 + g1*dk1 ....
+ d1*g1^2*(3*dg1*g2*g*dk1+g1*dg2*g*dk1 ....
+ g1*g2*dg*dk1+g1*g2*g*d2k1) ;
function coef=cr2Tkf(b,theta2,theta12,theta01,phi)
//
//
M = [
1*(a1-a0)^5 1*(a1-a0)^6 1*(a1-a0)^7 1*(a1-a0)^8 1*(a1-a0)^9
5*(a1-a0)^4 6*(a1-a0)^5 7*(a1-a0)^6 8*(a1-a0)^7 9*(a1-a0)^8
20*(a1-a0)^3 30*(a1-a0)^4 42*(a1-a0)^5 56*(a1-a0)^6 72*(a1-a0)^7
60*(a1-a0)^2 120*(a1-a0)^3 210*(a1-a0)^4 336*(a1-a0)^5 504*(a1-a0)^6
120*(a1-a0)^1 360*(a1-a0)^2 840*(a1-a0)^3 1680*(a1-a0)^4 3024*(a1-a0)^5
] ;
//
//
df = tan(theta2) ;
//
// curvatures
k2=tan(theta12)/d2;k1=tan(theta01)/d1;k0=tan(phi)/bigL;
//
ddf = k2*((1+df*df)^(3/2)) ;
//
// derivative of k2
dk2ds2 = ( (1+(d2*k2)^2)/d2 )*( (1+(d2*k2)^2)^(1/2)*k1 - k2 ) ;
dk2 = (1+df*df)^(1/2) * dk2ds2 ;
//
dddf = dk2 * (1+df*df)^(3/2) + 3*k2*df*ddf*(1+df*df)^(1/2) ;
//
// second derivative of k2
dk1ds1 = ( (1+(d1*k1)^2)/d1 )*( (1+(d1*k1)^2)^(1/2)*k0 - k1 ) ;
dk1ds2 = dk1ds1 * (1+(d2*k2)^2)^(1/2) ;
//
ddk2ds2 = ....
3*d2*k2*dk2ds2*(1+(d2*k2)^2)^(1/2)*k1 ....
+ (1+(d2*k2)^2)^(3/2)*dk1ds2/d2 ...
- (1/d2+3*d2*k2*k2)*dk2ds2 ;
//
ddk2 = ....
df*ddf*((1+df*df)^(-1/2))*dk2ds2 ....
+ (1+df*df)*ddk2ds2 ;
//
ddddf = ....
ddk2 * (1+df*df)^(3/2) ....
+ 6*dk2*df*ddf*(1+df*df)^(1/2) ....
+ 3*k2*ddf*ddf*(1+df*df)^(1/2) ....
+ 3*k2*df*dddf*(1+df*df)^(1/2) ....
+ 3*k2*(df*ddf)^2*(1+df*df)^(-1/2) ;
//
//
//
//
coef =inv(M)* [b;df;ddf;dddf;ddddf] ;
function []=ptcr(state)
//
x=state(1);y=state(2);theta=state(3);phi=state(4);
//
// plot the car
//
// rear wheels
wheel1 = [ -1/8 1/8 ; 1/6 1/6 ] ;
wheel2 = [ -1/8 1/8 ; -1/6 -1/6 ] ;
// front wheels
wheel3 = [(1-cos(phi)/8) (1+cos(phi)/8);
(1/6-sin(phi)/8) (1/6+sin(phi)/8) ] ;
wheel4 = [(1-cos(phi)/8) (1+cos(phi)/8);
(-1/6-sin(phi)/8) (-1/6+sin(phi)/8) ] ;
//
// car + wheels
xy_car = [ [ -1/3 7/6 4/3 4/3 7/6 -1/3 -1/3;
-1/3 -1/3 -1/6 1/6 1/3 1/3 -1/3;
] wheel1 wheel2 wheel3 wheel4 ] ;
xy_car = ....
[x x x x x x x x x x x x x x x;
y y y y y y y y y y y y y y y ] + ....
bigL*[cos(theta) -sin(theta);sin(theta) cos(theta)]*xy_car ;
//
xpoly(xy_car(1,1:7),xy_car(2,1:7),"lines") ;
// The 4 wheels
xpolys(matrix(xy_car(1,8:15),2,4),matrix(xy_car(2,8:15),2,4),[1 1 1 1])
function ptcr2T(state)
// plot the car with two trailers
//
x0=state(1);y0=state(2);theta0=state(3);theta1=state(4);
theta2=state(5);phi=state(6);
theta = theta0 ; x=x0 ; y=y0 ;
// rear wheels
wheel1 = [ -1/8 1/8 ; 1/6 1/6 ] ;
wheel2 = [ -1/8 1/8 ; -1/6 -1/6 ] ;
// front wheels
wheel3 = [
(1-cos(phi)/8) (1+cos(phi)/8)
(1/6-sin(phi)/8) (1/6+sin(phi)/8)
] ;
wheel4 = [
(1-cos(phi)/8) (1+cos(phi)/8)
(-1/6-sin(phi)/8) (-1/6+sin(phi)/8)
] ;
//
// car + wheels
xy_car = [ [ -1/3 7/6 4/3 4/3 7/6 -1/3 -1/3
-1/3 -1/3 -1/6 1/6 1/3 1/3 -1/3 ] wheel1 wheel2 wheel3 wheel4 ] ;
xy_car = diag([x,y])*ones(2,15) + ...
bigL*[cos(theta0) -sin(theta0);sin(theta0) cos(theta0)]*xy_car ;
//
//
// Trailer 1
x = x - d1*cos(theta1) ;
y = y - d1*sin(theta1) ;
//
// shift + rotation
xy_T1 =diag([x,y])*ones(2,11) + ...
[cos(theta1) -sin(theta1);sin(theta1) cos(theta1)]*xy_T1 ;
//
// Trailer 2
x = x - d2*cos(theta2) ;
y = y - d2*sin(theta2) ;
//
// shift + rotation
xy_T2 = diag([x,y])*ones(2,11)+ ...
[cos(theta2) -sin(theta2);sin(theta2) cos(theta2)]*xy_T2 ;
//
// plots
//
// Car
xpoly(xy_car(1,1:7),xy_car(2,1:7),"lines") ;
// The 4 wheels
xpolys(matrix(xy_car(1,8:15),2,4),matrix(xy_car(2,8:15),2,4),[1 1 1 1])
// trailer 1
xpoly(xy_T1(1,1:5),xy_T1(2,1:5),"lines") ;
// hitch and wheels
xpolys(matrix(xy_T1(1,6:11),2,3),matrix(xy_T1(2,6:11),2,3),[1 1 1 1]);
//
// trailer 2
xpoly(xy_T2(1,1:5),xy_T2(2,1:5),"lines") ;
// hitch and wheels
xpolys(matrix(xy_T2(1,6:11),2,3),matrix(xy_T2(2,6:11),2,3),[1 1 1 1]);
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