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# Copyright INRIA
if SearchText(`Release 3`,interface(version))<>0 then
matadd:=add;
elif SearchText(`Release 4`,interface(version))<>0 then
1;
else
ERROR(`Unknown Release`)
fi;
read(`Euler.map`);
#----------------------------------------------------------------------------
# AND NOW THE N-LINK PNDULUM
#-----------------------------------------------------------------------------
#----------------------------------------------------------------------------
# TeX Notations for my problem ( x= `x` is in fact useless ut just help
# to remind the state names )
#-----------------------------------------------------------------------------
addnotations( { x = `x` , th = `\\theta`, y =`y` , t = `\\eta`} );
#----------------------------------------------------------------------------
# Lagrangian for My problem : the N-link pendulum
# l[i] : length of links r[i]:=l[i]/2;
#---------------------------------------------------------------------------
# number of link in the pendulum
n:=10;
# [x,y,theta] is of size three
mm:=3:
LLi:=proc(i)
(1/2)*m[i]*( ( xd[i-1]- r[i] *sin(th[i])*thd[i])**2 +
( yd[i-1]+ r[i]*cos(th[i])*thd[i])**2 ) + 1/2*J[i]*(thd[i])**2
-m[i]*g*(y[i-1]+r[i]*sin(th[i])):
end:
# The point zero is fixed
x[0]:=0:xd[0]:=0:
y[0]:=0:yd[0]:=0:
L:=sum( LLi(j),j=1..n):
sorties(`systeme.tex`,`Lagrangian`,L):
# Lagrangian variables :
q := [ seq (op([x[i],y[i],th[i]]),i=1..n)]:
qd := [ seq (op([xd[i],yd[i],thd[i]]),i=1..n)]:
qdd:= [ seq (op([xdd[i],ydd[i],thdd[i]]),i=1..n)]:
# Lhs of Euler equations
EL:=euler_equations(L,q,qd,qdd):
EE:=map((i)->rhs(i),EL):
sortiesM(`systeme.tex`,`Lhs of Euler Equations`,EL):
#-----------------------------------------------------
# Rewritting the Euler equations to have a canonical form
# for TeX output
# .. . .
# El= ME(q) q + CE(q) q^2 + RE(q,q)
# x^2 means transpose( x[1]^2 ,....,x[n]^2)
# Computation of ME,CE,RE
#-----------------------------------------------------
XX:=CEuler(EE,q,qd,qdd):
# simplifying notations for output
ME1:=XX[1]:
ME1:=subs(seq(m[k]*r[k]*cos(th[k])=mrc[k],k=1..n),
seq(m[k]*r[k]*sin(th[k])=mrs[k],k=1..n),eval(ME1)):
sortiesI(`systeme.tex`,`$mrc_k=m_k r_k \\cos(\\theta_k)$`);
sortiesI(`systeme.tex`,`$mrs_k=m_k r_k \\sin(\\theta_k)$`);
sorties(`systeme.tex`,`$M(q)\\ddot{q}+C(q)\\dot{q}^2 +R(q,\\dot{q})$,M:`,
eval(ME1)):
sorties(`systeme.tex`,`C~:`,XX[2]):
sorties(`systeme.tex`,`R~:`,XX[3]):
#------------------------------------------------
# Constraints on the N-link Pendulum
#------------------------------------------------
ncont:=2*n;
cont:=[ seq(op([ x[i]-x[i-1] -2*r[i]*cos(th[i]),
y[i]-y[i-1] -2*r[i]*sin(th[i])]),i=1..n)]:
sortiesM(`systeme.tex`,`Constraints `,cont);
# time derivative of constraints;
cont1:=map ( (exp)->( time_diff(exp,q,qd,qdd)),cont):
# Derivatives of constraints are of type Aprim qd = 0
Aprim:=genmatrix(cont1,qd):
sorties(`systeme.tex`,`Time derivative of constraints : $A'(q)\\dot{q}$`,Aprim);
#---------
# Total Energy Ec + Ep
#---------
EEi:=proc(i)
(1/2)*m[i]*( ( xd[i-1]- r[i] *sin(th[i])*thd[i])**2 +
( yd[i-1]+ r[i]*cos(th[i])*thd[i])**2 ) + 1/2*J[i]*(thd[i])**2
+m[i]*g*(y[i-1]+r[i]*sin(th[i])):
end:
ET:=sum( EEi(j),j=1..n):
Scont1:=solve({op(cont1)},{ seq (op([xd[i],yd[i]]),i=1..n)}):
Scont2:=solve({op(cont)},{ seq (op([x[i],y[i]]),i=1..n)}):
ET:=subs(Scont1,ET):
ET:=subs(Scont2,ET):
sorties(`systeme.tex`,`N-pendulum energy~:`,ET);
#----------------------------------------------
# Computing SS:=S(q);
# S(q) will solve Aprim S(q) = 0
# in The Euler equations Equ= A(q)' lambda + u
# the term A(q)lambda can be eliminated
# if we left-multiply euler equations by S(q)'
#----------------------------------------------
SS:=linsolve(Aprim,matrix(ncont,1,0)):
#------------------------------------------------------------------------
# The constraints are now dotq=S(q)eta
# can be used to see that eta[i]=thd[i] ( eta[i] is the maple variable ti )
# but the indices can be mixed and linsolve doesn't
# always return the same result
# We have to check the correspondance between t.sig(i)=thd[i]
# and to change SS to have a good corespondance
#-------------------------------------------------------------------------
permut:={seq(SS[mm*i,1]=t_s[i],i=1..n)}:
SS:=subs(permut,eval(SS)):
permut:={seq(t_s[i]=t[i],i=1..n)}:
SS:=subs(permut,eval(SS)):
S:= genmatrix(convert(convert(SS,vector),list),[seq( t[i],i=1..n)]):
sorties(`systeme.tex`,`$\\dot{q}=S(q)\\eta$ Kernel of $A(q)'$~:`,S):
#-----------------------------------------------------
# this multiplication eliminates the term A(q) lambda
# in the Euler equations
#-----------------------------------------------------
E1:=multiply(transpose(S),EE):
# sortiesM(`systeme.tex`,`$S(q)^T E$`,E1);
#-----------------------------------------------------
# since Aprim(q) dotq=0
# .
# q= S(q) eta ; here eta = [t1,t2]
# ..
# we use this equation to compute q
# Warning : t1 and t2 are time dependent
#-----------------------------------------------------
qt := [ seq (t[i] ,i=1..n)]:
qtd := [ seq (td[i] ,i=1..n)]:
qtdd:= [ seq (tdd[i],i=1..n)]:
qqdd:=map((x,y,z,t)-> time_diff(x,y,z,t),eval(SS),
[op(q),op(qt)],[op(qd),op(qtd)],[op(qdd),op(qtdd)]):
#-----------------------------------------------------
# .. .
# using q= d/dt [ S(q) eta] and q= S(q) eta
# we can subsitute these expressions in E1
#-----------------------------------------------------
E2:=subs(seq(qdd[i]=qqdd[i,1],i=1..nops(qdd)),eval(E1)):
E3:=subs(seq(qd[i]=SS[i,1],i=1..nops(qd)),eval(E2)):
#-----------------------------------------------------
# The global system is now
# .
# E3 = 0 and q= S(q) eta
#-----------------------------------------------------
E3:=map((x)-> simplify(x),E3):
sortiesM(`systeme.tex`,
`$S(q)^T E$ simplified with $\\dot{q}=S(q)\\eta $`,E3);
#------------------------------------------------------------------
# Trying to find canonical representation
# for the simplified euler equations
# .
# El= ME(q) t + CE(q) t^2 + RE
# we use CEuler with a little trick in the parameter call qt,qt,qtd
#------------------------------------------------------------------
XX1:=CEuler(E3,qt,qt,qtd):
MM3:=map((i)->factor(combine(i,trig)),XX1[1]):
CC3:=map((i)->factor(combine(i,trig)),XX1[2]):
RR3:=map((i)->factor(combine(i,trig)),XX1[3]):
sorties(`systeme.tex`,`a cononical form $M(q)\\dot{t}+C(q)t^2 +R$,M:`,MM3);
sorties(`systeme.tex`,`C:`,CC3);
sorties(`systeme.tex`,`R:`,RR3);
sortiesI(`systeme.tex`,`Since $M,C,R$ only depends on $\\theta$ `);
sortiesI(`systeme.tex`,`,we can forget $x_i,y_i$ in the dynamical system `);
sortiesI(`systeme.tex`,` and just keep the $\\dot{\theta}=\\eta$ in the constraints`);
#-----------------------------------------------------
# FORTRAN GENERATION
#-----------------------------------------------------
#-----------------------------------------------------
# First routine npend(neq,t,th,ydot)
# .
# we have MM3(theta) eta + CC3(theta)*eta^2 +RR3(theta) = 0
# .
# theta = eta
# .
# ==> th=[ theta,eta] ydot= th
#-----------------------------------------------------
flist:=[subroutinem,`npend`,[`neq`,`t`,`th`,`ydot`],
[
[ parameterf,[`n=`.n]],
[ declaref,doubleprecision,[`t,th(2*n),ydot(2*n),r(n),j(n),m(n)`]],
[ declaref,doubleprecision,[`me3s(n,n),cc3s(n,n),const(n,1)`]],
[ declaref,doubleprecision,[`w(3*n),rcond `]],
[ declaref,`implicit doubleprecision`,[`(t)`] ],
[ declaref,integer,[`i,k,neq,ierr`]],
[ declaref,`data g`,[`/ 9.81/`]],
[ declaref,`data r`,[`/ n*1.0/`] ],
[ declaref,`data m`,[`/ n*1.0/`] ],
[ declaref,`data j`,[`/ n*0.3/`] ],
[ matrixm,`me3s`,MM3 ] ,
[ matrixm,`cc3s`,CC3 ] ,
[ matrixm,`const`,RR3 ] ,
[ dom , `i ` ,1,`n `,1,[ equalf,`ydot(i)`,`th(i+n)`]],
[ dom , `i ` ,1,`n `,1,[
[ dom , `k ` ,1,`n `,1,
[[ equalf,`Const(i,1)`,
` Const(i,1)+CC3S(i,k)*(th(k+n)**2)`]]],
[ equalf,` Const(i,1)`,`-Const(i,1)`]]],
[commentf,` we must solve M z =const `],
[commentf,` which gives ydot((n+1)..2*n) `],
[ callf , `dlslv`,[`me3s,n,n,Const,n,1,w, rcond,ierr,1`]],
[ if_then_m,ierr<>0,[
[writef,6,ff_w,[]],
[formatf,ff_w,[`'Matrice mal conditionnee'`]]]],
[ dom , `i ` ,1,`n `,1,[ equalf,`ydot(n+i)`,`const(i,1)`]],
[returnf]]]:
Gener(`npend.f`,flist):
# New ener.f
ET:=matrix(1,1,[ET]):
flist:=[subroutinem,`ener`,[`th`,`e`],
[
[ parameterf,[`n=`.n]],
[ declaref,doubleprecision,[`th(2*n),thd(n),et(1,1),r(n),j(n),m(n)`]],
[ declaref,`implicit doubleprecision`,[`(t)`] ],
[ declaref,integer,[`i `]],
[ declaref,`data g`,[`/ 9.81/`]],
[ declaref,`data r`,[`/ n*1.0/`] ],
[ declaref,`data m`,[`/ n*1.0/`] ],
[ declaref,`data j`,[`/ n*0.3/`] ],
[ dom , `i ` ,1,`n `,1,[ equalf,`thd(i)`,`th(i+n)`]],
[ matrixm,`et`,ET ] ,
[ equalf,`e`,`et(1,1)`],
[returnf]]]:
Gener(`ener.f`,flist):
# New np.f
flist:=[subroutinem,`np`,[`i `],
[
[ declaref,integer,[`i `]],
[ equalf,`i `,n],
[returnf]]]:
Gener(`np.f`,flist):
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