9
+ 3.42336889130000E-01*h0
+ 7.73122708067000E-01*h1
+ 6.38635548911000E-01*h2
+ 9.05029265582000E-01*h3
+(-4.48538843751698E-02 + 9.98993558065546E-01*i)*zz1
+ 9.34003271419000E-01;
+ 2.25000000000000E+00*h0**2
+ 2.59807621135332E+00*h0*h3
-3*h0*g1
+ 3*h1**2
+ 3*h1*g0
+ 1.73205080756888E+00*h1*g3
+ 7.50000000000000E-01*h3**2
-1.73205080756888E+00*h3*g1
+( 9.94654043308859E-01 + 1.03263421061569E-01*i)*zz1;
+ 2.25000000000000E+00*h0**2
+ 5.19615242270663E+00*h0*h3
-3*h0*g1
+ 5.25000000000000E+00*h1**2
-5.19615242270663E+00*h1*h2
+ 3*h1*g0
+ 3.46410161513775E+00*h1*g3
+ 2.25000000000000E+00*h2**2
-3*h2*g3
+ 5.25000000000000E+00*h3**2
-3.46410161513775E+00*h3*g1
+ 3*h3*g2
+( 7.39016515969049E-01 - 6.73687308122224E-01*i)*zz1;
+ 2.25000000000000E+00*h0**2
+ 7.79422863405995E+00*h0*h3
-1.50000000000000E+00*h0*g1
+ 2.59807621135332E+00*h0*g2
+ 2.25000000000000E+00*h1**2
-7.79422863405995E+00*h1*h2
+ 1.50000000000000E+00*h1*g0
+ 2.59807621135332E+00*h1*g3
+ 6.75000000000000E+00*h2**2
-2.59807621135332E+00*h2*g0
-4.50000000000000E+00*h2*g3
+ 6.75000000000000E+00*h3**2
-2.59807621135332E+00*h3*g1
+ 4.50000000000000E+00*h3*g2
+(-8.09276140434650E-01 - 5.87428402891107E-01*i)*zz1;
+ 2.25000000000000E+00*h0**2
+ 5.19615242270663E+00*h0*h3
-1.50000000000000E+00*h0*g1
+ 2.59807621135332E+00*h0*g2
+ 7.50000000000000E-01*h1**2
+ 1.50000000000000E+00*h1*g0
-8.66025403784439E-01*h1*g3
+ 6.75000000000000E+00*h2**2
-2.59807621135332E+00*h2*g0
-4.50000000000000E+00*h2*g3
+ 5.25000000000000E+00*h3**2
+ 8.66025403784439E-01*h3*g1
+ 4.50000000000000E+00*h3*g2
+(-2.91752263234989E-01 + 9.56493918902395E-01*i)*zz1;
+ 7.50000000000000E-01*h1**2
+ 2.59807621135332E+00*h1*h2
-1.73205080756888E+00*h1*g3
+ 2.25000000000000E+00*h2**2
-3*h2*g3
+ 3*h3**2
+ 1.73205080756888E+00*h3*g1
+ 3*h3*g2
+(-4.69789445909537E-01 - 8.82778497989167E-01*i)*zz1;
+ 1*h0*g0
+ 1*h1*g1
+ 1*h2*g2
+ 1*h3*g3
+(-4.19390965637989E-01 - 9.07805715966382E-01*i)*zz1;
-1*h0**2
-1*h1**2
-1*h2**2
-1*h3**2
+ 1*g0**2
+ 1*g1**2
+ 1*g3**2
+ 1*g2**2
+(-1.54218246879440E-01 - 9.88036807173413E-01*i)*zz1;
+(-7.14591623411357E-01 + 6.99541858469042E-01*i)*h0
+(-3.05477026118637E-01 - 9.52199446814434E-01*i)*h1
+(-6.77010516461900E-01 + 7.35973342316141E-01*i)*h2
+(-4.28248318706532E-01 + 9.03661096609248E-01*i)*h3
+(-9.11272245030594E-01 - 4.11804438340460E-01*i)*g0
+( 7.21464646265866E-01 - 6.92451272067911E-01*i)*g1
+( 9.77858291825326E-01 - 2.09268155982833E-01*i)*g3
+( 4.32748124060744E-01 + 9.01514870161279E-01*i)*g2
+(-4.13027206873742E-01 - 9.10718686742550E-01*i)*zz1
+(-5.30779024078570E-01 + 8.47510252208314E-01*i);
TITLE : Griffis-Duffy platform, components of low degree
REFERENCES :
M. Griffis and J. Duffy:
"Method and apparatus for controlling geometrically
simple parallel mechanisms with distinctive connections".
US Patent 5,179,525, 1993.
M.L. Husty and A. Karger:
"Self-motions of Griffis-Duffy type parallel manipulators".
2000 IEEE Int. Conf. Robotics and Automation, CDROM,
San Francisco, CA, April 24--28, 2000.
A.J. Sommese, J. Verschelde, and C.W. Wampler:
"Advances in polynomial continuation for solving problems in kinematics".
ASME Journal of Mechanical Design 126(2):262-268, 2004.
NOTE :
The polynomial system above is an "embedded system" and jointly with
the 40 solutions below, we have a witness set for the solution curve.
The original system can be obtained by removing the last hyperplane
and removing all terms which have the slack variable zz1 in them.
The following Maple code was used to generate the system:
# This worksheet gnerates the equations for a moving Stewart-Gough
# platform with Soma coordinates. This type of platform was proposed by
# Griffis and Duffy, whence the name Griffis-Duffy platform. Running
# this worksheet creates two files: gdplat1 and gdplat2.
Digits := 16:
with(linalg):
# 1. The Maple Procedures
# There are two Maple procedures: one to generate the equations of the
# Griffis-Duffy platform and another one to export the polynomials to a
# file in a format digestable by PHCpack (notice that we replaced the
# original e0, e1, e2, e3 variables by h0, h1, h2, h3).
platform := proc(a1,a2,a3,a4,a5,b1,b2,b3,b4,b5,
L0,L1,L2,L3,L4,L5::numeric)
local g, e, p0,p1,p2,p3,p4,p5,p6,p7,i,i3,i4,lam,
cp,r3,r4,amb,bma,sp:
g := vector([g0,g1,g2,g3]):
e := vector([h0,h1,h2,h3]):
p6 := innerprod(g,e);
p7 := innerprod(g,g) - innerprod(e,e);
i3 := matrix(3,3,[1,0,0,0,1,0,0,0,1]):
i4 := matrix(4,4,[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1]):
for i from 1 to 5 do
cp := crossprod(b||i,a||i):
r3 := evalm(evalm(b||i &* transpose(a||i))
+ evalm(a||i &* transpose(b||i))
- scalarmul(i3,innerprod(a||i,b||i))):
r4 := matrix(4,4,
[ innerprod(a||i,b||i), cp[1], cp[2], cp[3],
cp[1], r3[1,1], r3[1,2], r3[1,3],
cp[2], r3[2,1], r3[2,2], r3[2,3],
cp[3], r3[3,1], r3[3,2], r3[3,3] ]):
A||i := scalarmul(evalm(scalarmul(i4,innerprod(b||i,b||i)
+ innerprod(a||i,a||i) - (L||i)^2 + (L0)^2)
- scalarmul(r4,2)),(1/L0));
lam := matrix(3,3,[0,-a||i[3]-b||i[3],a||i[2]+b||i[2],
a||i[3]+b||i[3],0,-a||i[1]-b||i[1],
-a||i[2]-b||i[2],a||i[1]+b||i[1],0]):
amb := evalm(a||i - b||i): bma := evalm(b||i - a||i):
B||i := matrix(4,4,[0,amb[1],amb[2],amb[3],
bma[1],-lam[1,1],-lam[1,2],-lam[1,3],
bma[2],-lam[2,1],-lam[2,2],-lam[2,3],
bma[3],-lam[3,1],-lam[3,2],-lam[3,3]]):
p||i := innerprod(e,evalm(A||i &* e))
+ 2*innerprod(g,evalm(B||i &* e)):
end do:
p0 := sum('rand()/1.0e+12*h||i','i'=0..3) + rand()/1.0e+12:
sp := [p0,seq(p||i,i=1..5),p6,p7];
return sp;
end proc:
# The second procedure is usually used shortly after the
# "writeto(filename)" command:
write_in_phc_format := proc(sys)
description `writes the system in a PHC format`:
local i,sp:
printf(` %d\n`,nops(sys));
for i from 1 to nops(sys) do
sp := convert(sys[i],string):
sp := ` `||sp||`;`:
printf(`%s\n`,sp):
end do:
end proc:
# 2. Case 1 : several irreducible components of small degree
# In the first case we generate points on base and end plate, for which
# the system decomposes in several irreducible components of small
# degree. In this case, the end plate and base plate have the same size
# and the lengths are all equal to one.
t := eval(Pi/3):
a0 := [0,0,0]:
a2 := map(evalf,[1-cos(2*t),sin(2*t),0]):
a4 := map(evalf,[1-cos(4*t),sin(4*t),0]):
a6 := a0:
a1 := 0.5*(a0+a2):
a3 := 0.5*(a2+a4):
a5 := 0.5*(a4+a6):
b1 := a0-a5:
b2 := a1-a5:
b3 := a2-a5:
b4 := a3-a5:
b5 := a4-a5:
L0 := 1:
for i from 1 to 5 do
L||i := 1:
end do:
sys := platform(a1,a2,a3,a4,a5,b1,b2,b3,b4,b5,L0,L1,L2,L3,L4,L5):
writeto(gdplat1);
write_in_phc_format(fnormal(sys));
writeto(terminal);
# 3. Case 2 : one large irreducible component
# In this case the end plate has the same shape as the base plate, but
# it is scaled by a factor.
s:= 0.7:
t := evalf(Pi/3):
a0 := [0,0,0]:
a2 := [1-cos(2*t),sin(2*t),0]:
a4 := [1-cos(4*t),sin(4*t),0]:
a6 := a0:
a1 := 0.5*(a0+a2):
a3 := 0.5*(a2+a4):
a5 := 0.5*(a4+a6):
b0 := [0,0,0]:
b1 := s*(a0-a5):
b2 := s*(a1-a5):
b3 := s*(a2-a5):
b4 := s*(a3-a5):
b5 := s*(a4-a5):
# Quaternion elementary matrices :
Q0 := matrix(4,4,[ 1,0,0,0, 0, 1,0,0, 0,0, 1,0, 0,0,0, 1]):
Qi := matrix(4,4,[0,0,0, 1, 0,0,-1,0, 0,1, 0,0, -1,0,0,0]):
Qj := matrix(4,4,[0,0, 1,0, 0,0,0, 1, -1,0,0,0, 0,-1,0,0]):
Qk := matrix(4,4,[0,-1,0,0, 1,0,0,0, 0,0,0, 1, 0,0,-1,0]):
# Make a unit-length 4-vector (manually randomized)
ee := [.456,-.337,.628,.193]:
ee := ee/sqrt(innerprod(ee,ee)):
# Form the quaternion for rotation
E := evalm(scalarmul(Q0,ee[1]) + scalarmul(Qi,ee[2])
+ scalarmul(Qj,ee[3]) + scalarmul(Qk,ee[4])):
# Choose a position vector ( manually randomized)
posn := [.476,-.381,.933]:
# Compute compatible leg lenghts:
for i from 0 to 5 do
# b||i as quaternion in end-effector coordinates
bquat1 := evalm(scalarmul(Qi,b||i[1]) + scalarmul(Qj,b||i[2])
+ scalarmul(Qk,b||i[3])):
# transform to base coordinates
bquat0 := evalm( E &* evalm( bquat1 &* transpose(E) ) ):
# extract vector
bvec := [bquat0[1,4],bquat0[2,4],bquat0[3,4]]:
lvec := -a||i + posn + bvec:
L||i := sqrt( innerprod( lvec,lvec ) ):
end do:
# Generate the equations for the system:
sys := platform(a1,a2,a3,a4,a5,b1,b2,b3,b4,b5,L0,L1,L2,L3,L4,L5):
# Prepare to write to file:
writeto(gdplat2);
write_in_phc_format(fnormal(sys));
writeto(terminal);
THE SOLUTIONS :
40 9
===========================================================
solution 1 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -1.13801036929069E+00 6.78785181811673E-01
h1 : 5.57283616327522E-16 1.41802311330974E-15
h2 : 1.94618603616506E-01 7.39034394757603E-01
h3 : -7.38882957687825E-01 -7.78258638339318E-01
zz1 : -9.15917184510493E-16 -4.14111939977075E-15
g1 : -1.17345348285854E+00 1.72093010600492E-01
g0 : -3.77616033311447E+00 3.51114515169338E-01
g3 : -9.02244287390915E-02 -2.73728268678052E+00
g2 : 4.36007665463969E-01 2.81473352139762E+00
== err : 6.055E-14 = rco : 4.361E-04 = res : 9.833E-15 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -8.80903827010905E-01 5.53629704613943E-01
h1 : -1.34125143965878E-01 -3.29886079034908E-01
h2 : -1.54866275891304E+00 5.55553989063721E-01
h3 : 5.08590061654918E-01 -3.19638258990232E-01
zz1 : 8.14870949559562E-15 -2.85243047760367E-16
g1 : -6.15657817313033E-01 -1.06686186554749E+00
g0 : -1.66439347656213E+00 7.13049288955357E-01
g3 : -1.24590033467826E+00 7.07341509470806E-01
g2 : 3.17090982279578E-01 -1.73859760089071E-01
== err : 2.707E-14 = rco : 2.423E-03 = res : 4.746E-15 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -2.35987197109235E-01 2.44368273833226E-01
h1 : -4.23258266037085E-01 -4.08741815328982E-01
h2 : -2.44368273833245E-01 -2.35987197109242E-01
h3 : -4.08741815328994E-01 4.23258266037089E-01
zz1 : 1.53246226262449E-15 1.21059716535803E-15
g1 : 2.60135467046062E-01 7.86324550424426E-01
g0 : 8.77242956895040E-01 2.58552533404194E-01
g3 : 5.32197289247188E-02 -9.68097058373796E-01
g2 : 9.67672912582785E-01 -3.92531841216207E-01
== err : 1.279E-13 = rco : 5.266E-04 = res : 1.777E-15 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -8.36991382810080E-01 1.70547971068197E+00
h1 : -3.66284795512152E-01 -6.95291388384885E-01
h2 : -2.11474625289007E-01 -4.01426670249241E-01
h3 : -2.53286769262042E-01 2.32105515639479E-01
zz1 : -3.20624816992483E-16 2.16153112893375E-15
g1 : -4.49656434786568E-01 1.47000030910934E+00
g0 : 3.16351751301293E-01 3.52034158283954E-01
g3 : -3.10962313953065E-01 -7.33849685178659E-01
g2 : 8.45176690488384E-01 -9.76441954375624E-01
== err : 8.073E-15 = rco : 4.228E-03 = res : 4.045E-15 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -4.06432763374973E+00 -2.91601317712484E-01
h1 : 1.65151330281963E+00 2.03063122738495E-01
h2 : -9.53501649953167E-01 -1.17238548575556E-01
h3 : -2.32607262948370E-01 1.95638772227842E-02
zz1 : -1.82182601993628E-14 2.59297955411755E-14
g1 : -2.61911405582411E+00 -1.24293062664487E-01
g0 : -1.30653545799503E+00 -1.44395442906467E-01
g3 : -2.93572935998568E+00 -3.48293429581451E-01
g2 : 1.74475346811675E+00 5.21967559649562E-02
== err : 3.059E-13 = rco : 1.587E-03 = res : 1.799E-14 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 1.72691629511631E-01 2.51823279446906E+00
h1 : -3.93030211111348E-16 9.50296538755971E-16
h2 : -2.00411313263416E-15 -4.65297510323633E-17
h3 : -1.09733687562706E+00 -9.52548181311347E-01
zz1 : -6.40616601577835E-15 -6.05811261662410E-15
g1 : -3.45642083267520E-01 1.47620913417965E+00
g0 : -1.15912887758085E+00 -8.83828224908878E-01
g3 : -3.33016566844307E-01 -2.51005122036023E+00
g2 : 1.29689342544483E+00 1.00258440312542E-01
== err : 6.334E-14 = rco : 2.424E-03 = res : 6.587E-15 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -2.03756001625813E+01 1.12024958939925E+01
h1 : -6.32345513398884E+00 -1.45848370976406E+01
h2 : 3.65084852381703E+00 8.42055962440970E+00
h3 : 9.50086559178317E+00 2.27967483416650E+00
zz1 : -1.31027436457969E-12 -1.78019710289524E-12
g1 : -4.11715971073986E+01 1.94932592738428E+01
g0 : 1.88161483839981E+01 4.23136343613962E+01
g3 : 1.70691872372984E+01 6.78913344809027E+00
g2 : 1.42695670811403E+01 -1.35341133233028E+01
== err : 1.092E-10 = rco : 1.469E-05 = res : 1.402E-12 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -1.24311125178841E+00 -3.19070216891305E-01
h1 : -7.22594221984349E-18 2.67571145089335E-16
h2 : 3.05142796440822E-16 1.66736348777937E-15
h3 : -5.61794465632326E-01 1.20691683262154E-01
zz1 : 3.07684872802217E-16 -3.40107840428351E-16
g1 : -1.17559757831286E+00 -1.87041630803214E-01
g0 : 2.17842378748830E-01 3.99275006246010E-01
g3 : -3.71647012739505E-02 -1.01520355415135E+00
g2 : 1.24052604392992E+00 -1.27031473682515E-02
== err : 4.534E-15 = rco : 5.233E-03 = res : 1.491E-15 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 1.39119162012519E-01 8.37730921765757E-01
h1 : -4.83664173189936E-01 8.03204856373637E-02
h2 : -8.37730921765740E-01 1.39119162012511E-01
h3 : -8.03204856373625E-02 -4.83664173189934E-01
zz1 : 4.70682680960085E-16 -4.76917408870687E-15
g1 : -1.18537602883798E-01 -9.54689163774785E-01
g0 : -9.28073877308476E-01 8.48324223461175E-02
g3 : -3.01907218873820E-01 9.01411401349660E-02
g2 : 3.56483061099134E-02 -2.02577617523594E-01
== err : 3.740E-15 = rco : 3.762E-03 = res : 1.825E-15 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 2.21299191280052E+00 -7.33562210268627E-01
h1 : 1.03412560518194E-15 -1.13375088616159E-15
h2 : 1.59481967839090E-15 -1.89200439670828E-15
h3 : -1.86910203111404E+00 2.77477662432496E-01
zz1 : -4.82040730134002E-16 1.65994182156888E-14
g1 : 8.50399013995463E-01 -4.30020305376837E-01
g0 : 1.57347203480280E+00 -2.92658805856477E-01
g3 : 1.85044578426153E+00 -6.89333060524882E-01
g2 : 1.37812393146517E+00 -2.92053233661718E-02
== err : 1.062E-14 = rco : 2.658E-03 = res : 7.985E-15 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -8.69630710554809E-01 -3.30626672217738E-01
h1 : -4.43705720868673E-01 6.84456361367076E-01
h2 : 2.56173617384507E-01 -3.95171064483834E-01
h3 : -5.04800482041897E-01 -1.80781733547355E-01
zz1 : 6.22729292733532E-15 -3.30951943870831E-16
g1 : -8.26007040897522E-01 -1.13559896840341E+00
g0 : 9.13232854034393E-01 -1.44135136104066E-01
g3 : 1.29313248452747E+00 -1.27878757409813E+00
g2 : 9.81695869456604E-01 8.36420103646528E-01
== err : 9.551E-15 = rco : 3.940E-03 = res : 3.969E-15 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 2.02900577966358E-01 1.00528609700378E+00
h1 : -1.16080439743545E+00 2.34289406615215E-01
h2 : -3.91256855977406E-16 -9.67801483839745E-17
h3 : -1.17144703307608E-01 -5.80402198717724E-01
zz1 : -1.06427111644967E-15 2.02355646942349E-15
g1 : -1.15548392828321E+00 3.62427424555290E-01
g0 : 7.49173005731341E-01 9.12819908077190E-01
g3 : 5.72750860475193E-01 -7.29917397616374E-01
g2 : -1.75717054961411E-01 -8.70603298076585E-01
== err : 2.054E-14 = rco : 6.640E-03 = res : 3.164E-15 ==
solution 13 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -3.14132346963160E-01 4.75257290110047E-01
h1 : 6.77340546233622E-16 -9.93156670344324E-16
h2 : -5.21170829922528E-16 3.17053922424095E-15
h3 : -9.13190559040153E-01 -1.79771095167845E-01
zz1 : -1.73887020467879E-15 9.06443673033780E-16
g1 : -6.31022371534812E-01 2.78599799941782E-01
g0 : -5.83929420427169E-02 1.30040855184273E+00
g3 : -7.22728745665152E-01 -3.35446222749563E-01
g2 : 1.27751149511045E+00 1.89214256752838E-02
== err : 5.117E-15 = rco : 5.363E-03 = res : 2.259E-15 ==
solution 14 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -1.77278940042740E+00 -6.56515876583770E-01
h1 : -4.29898685052401E-01 1.02391298169449E+00
h2 : -2.48202121539269E-01 5.91156435608067E-01
h3 : 1.80947817872945E-01 -1.04349585702953E+00
zz1 : 4.86720675476170E-15 1.63400736524611E-15
g1 : 1.21434102115571E+00 -1.36996561536454E+00
g0 : 1.54867124894819E+00 2.62649789206704E+00
g3 : -2.57309303654138E+00 1.80890734640581E+00
g2 : 1.96495904497274E+00 1.34285319584239E+00
== err : 1.239E-13 = rco : 1.095E-03 = res : 1.171E-14 ==
solution 15 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -2.35987197109250E-01 -2.44368273833246E-01
h1 : -4.23258266037088E-01 4.08741815328988E-01
h2 : -2.44368273833246E-01 2.35987197109246E-01
h3 : -4.08741815328987E-01 -4.23258266037089E-01
zz1 : -1.68414293267635E-15 3.50389092699316E-16
g1 : 7.61224239632747E-02 -1.95720504317387E+00
g0 : 1.55325112457282E+00 -3.64792513362430E-01
g3 : 1.22410022167413E+00 7.84084015291021E-01
g2 : 8.61432932624538E-01 -2.83476326461544E-01
== err : 3.563E-14 = rco : 1.453E-03 = res : 3.713E-15 ==
solution 16 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -4.31055092444107E-01 -1.58089184096799E-01
h1 : -2.73818498982767E-01 7.46609320974469E-01
h2 : 1.58089184096801E-01 -4.31055092444094E-01
h3 : -7.46609320974465E-01 -2.73818498982763E-01
zz1 : 3.59637649226034E-15 -1.64736305932727E-15
g1 : -1.03651823780659E+00 -1.25973019621046E+00
g0 : 5.90396318397039E-01 -2.25129423130364E-01
g3 : 1.49686397235566E+00 -1.68310087647273E+00
g2 : 1.34504340459206E+00 1.00112406687117E+00
== err : 2.656E-14 = rco : 1.975E-03 = res : 4.831E-15 ==
solution 17 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 1.63901832883802E-01 -6.99000596717361E-01
h1 : -1.60829362542201E-01 -9.00887870641937E-01
h2 : -9.28548757573324E-02 -5.20127854624788E-01
h3 : -8.91100087950041E-01 1.40101844896965E+00
zz1 : 9.11890402208833E-15 2.55327376755724E-15
g1 : -4.98142768819242E-01 -2.27898073781234E-01
g0 : 1.68149812199673E-01 1.05472257512087E+00
g3 : -2.05986742751521E-01 -7.68701234633815E-02
g2 : 9.07104119248203E-01 -1.62775241368215E+00
== err : 2.548E-14 = rco : 4.152E-03 = res : 4.163E-15 ==
solution 18 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 1.39119162012524E-01 -8.37730921765757E-01
h1 : -4.83664173189923E-01 -8.03204856373601E-02
h2 : -8.37730921765751E-01 -1.39119162012521E-01
h3 : -8.03204856373662E-02 4.83664173189933E-01
zz1 : 3.15492490713230E-15 -3.69627856938746E-15
g1 : 1.68551385990269E-01 -9.20308954416843E-02
g0 : 8.84898446555933E-01 4.12420292677338E-01
g3 : -1.34862727809031E+00 -3.77230129009052E-01
g2 : 5.32901021133389E-01 -1.61039470634084E+00
== err : 5.616E-15 = rco : 3.159E-03 = res : 2.415E-15 ==
solution 19 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -3.37906425214687E-01 2.25980190654332E-01
h1 : -9.65043502199908E-17 5.27766874438662E-17
h2 : -4.51960381308665E-01 -6.75812850429374E-01
h3 : -5.85271096675811E-01 3.91409171717406E-01
zz1 : -3.46267393486496E-16 -2.78728073242349E-16
g1 : -5.06859637822031E-01 3.38970285981499E-01
g0 : 1.70953139234556E-02 7.71271225933199E-01
g3 : 7.07550524249772E-01 2.34960022537630E+00
g2 : -2.42044909688011E+00 6.21304385423029E-01
== err : 1.192E-13 = rco : 3.830E-04 = res : 4.988E-15 ==
solution 20 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -2.06028035001627E+00 1.83864460318905E-01
h1 : -2.09284707915540E+00 2.05273544398899E-01
h2 : 4.89792487247280E-01 -1.96626342497308E-01
h3 : 1.18950341468799E+00 -1.06154195659531E-01
zz1 : -3.54408758455560E-15 3.25124439280727E-14
g1 : -7.14794375783201E-01 -4.81919361412117E-02
g0 : 1.24475621778122E+00 -3.36368472246772E-02
g3 : 1.82848713024411E+00 -4.53291375120419E-01
g2 : -2.22857651427593E+00 8.73811418776237E-02
== err : 3.897E-14 = rco : 2.469E-03 = res : 7.043E-15 ==
solution 21 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 2.73975091814179E-01 5.71392820223175E-01
h1 : -5.64189211777552E-01 3.28135093152718E-01
h2 : -3.25734793293651E-01 1.89448884362285E-01
h3 : -4.23833523236351E-01 -6.30130258014378E-01
zz1 : -2.10315091234391E-15 -3.51012369522961E-15
g1 : -5.01242680134387E-01 -8.54490154855783E-01
g0 : -1.71823363839185E-01 -6.39074172877738E-02
g3 : 6.44531367485480E-01 -4.00013363458700E-01
g2 : 1.09088299034357E-01 -4.29854317576059E-02
== err : 2.060E-14 = rco : 3.719E-03 = res : 2.246E-15 ==
solution 22 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 2.42373353722436E+00 2.58637590477849E+00
h1 : 1.49324482484942E+00 -1.39934321016040E+00
h2 : -2.58637590477850E+00 2.42373353722436E+00
h3 : -1.39934321016041E+00 -1.49324482484943E+00
zz1 : 4.04431804084220E-14 2.32041484307073E-14
g1 : 4.05520763258130E+00 3.81178788853364E+00
g0 : -2.12247581606767E+00 2.82579602439069E+00
g3 : -3.94733982580185E+00 3.21599297909178E+00
g2 : 1.37223360609053E+00 2.35725865848061E+00
== err : 4.058E-13 = rco : 4.965E-04 = res : 6.482E-14 ==
solution 23 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -8.64847471027710E-01 -7.85769407289120E-01
h1 : 2.03978910233738E-17 4.68591629871914E-17
h2 : -6.29826240102804E-01 5.74045159211031E-01
h3 : -2.60438707357430E-01 -1.07850424952601E-01
zz1 : 6.45162692093599E-16 4.60421646257768E-16
g1 : -7.61408871620941E-01 -6.36027659375790E-01
g0 : 2.63270149420604E-01 2.50820604274859E-01
g3 : -2.67217063306311E-01 2.29970748754977E-01
g2 : 4.14884260777212E-01 -3.44067442620553E-01
== err : 4.697E-15 = rco : 5.121E-03 = res : 1.996E-15 ==
solution 24 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -3.19728998010567E-01 6.78547212130632E-01
h1 : -1.84198386688375E-01 3.36136749393069E-01
h2 : 1.06346988138830E-01 -1.94068642746612E-01
h3 : -8.28765743834873E-01 -4.06868126277165E-01
zz1 : 3.62374470075326E-16 6.22296258997842E-16
g1 : -4.73979919418055E-01 1.76214949480899E+00
g0 : 4.13955077117146E-01 2.77727024090183E+00
g3 : -3.10704812279314E+00 -1.27544743648258E-01
g2 : 1.10241817790137E+00 -6.10509358903165E-01
== err : 5.852E-14 = rco : 1.060E-03 = res : 8.049E-15 ==
solution 25 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -2.94523218093560E-01 1.93068846241079E-01
h1 : -4.02358866865686E-01 -4.14714032743256E-01
h2 : -2.32302000095736E-01 -2.39435258441033E-01
h3 : -4.12967879894705E-01 4.50197724046949E-01
zz1 : 1.66900746325557E-15 1.04955228386258E-15
g1 : 9.41489283795918E-02 8.56226545760087E-01
g0 : 9.69503750571993E-01 3.85953705484773E-01
g3 : 1.66301047323038E-01 -9.12191533002485E-01
g2 : 9.28873317859295E-01 -3.48156484912661E-01
== err : 5.778E-14 = rco : 5.569E-04 = res : 1.631E-15 ==
solution 26 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 4.30832795003530E-01 5.79303117781669E+00
h1 : 8.04243900150323E-15 6.18764796934547E-15
h2 : 4.90380306490201E-01 7.61171012427278E+00
h3 : -1.54101925699561E+00 -7.56249240295218E+00
zz1 : 2.20034982807776E-14 -2.42769423264577E-13
g1 : -3.44156315886953E-01 1.07011811492081E+00
g0 : 5.65276236555265E-01 7.27176963401987E+00
g3 : 9.01843031758311E-01 9.14463572148276E+00
g2 : 1.67951882175280E+00 3.50880873607056E+00
== err : 6.217E-13 = rco : 1.766E-04 = res : 1.441E-13 ==
solution 27 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -4.31055092444093E-01 1.58089184096792E-01
h1 : -2.73818498982760E-01 -7.46609320974474E-01
h2 : 1.58089184096796E-01 4.31055092444097E-01
h3 : -7.46609320974475E-01 2.73818498982765E-01
zz1 : -1.96403673348944E-15 -3.60609098512452E-15
g1 : 8.97361195411155E-01 2.29686818071412E-01
g0 : -4.29950324853856E-03 -8.91396388218185E-01
g3 : 4.66820594216598E-01 -2.50778556745012E-01
g2 : 2.28517593243520E-01 -4.06428245225603E-01
== err : 1.956E-14 = rco : 2.563E-03 = res : 1.193E-15 ==
solution 28 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -1.06332077418059E+00 9.21436937220716E-01
h1 : -7.34575855987566E-17 1.41023413854326E-17
h2 : -6.27887258219428E-01 -7.32035691622785E-01
h3 : -1.86732327957206E-01 1.68019054087226E-01
zz1 : 7.24045214726147E-16 5.68879260364521E-16
g1 : -8.78348050494813E-01 7.63832087495222E-01
g0 : 3.91202308027711E-02 7.16813238439555E-02
g3 : -3.14932067680705E-01 -3.69336750800315E-01
g2 : -1.00620731473851E-02 -2.66848786166397E-02
== err : 1.080E-14 = rco : 5.001E-03 = res : 2.466E-15 ==
solution 29 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 6.00483942438715E-01 9.07457503899008E-01
h1 : 6.81651273625422E-01 1.23231812852715E+00
h2 : -3.93551546321086E-01 -7.11479203232408E-01
h3 : -1.56374517474744E+00 -8.93908547381683E-01
zz1 : -2.07753926504334E-14 -1.33133983267589E-14
g1 : -4.17396540620349E-01 -3.48663887951388E-01
g0 : -1.86153637320987E-01 -4.82968501781324E-01
g3 : -5.45631961680177E-01 -1.55925503829706E+00
g2 : 1.80472917983342E+00 1.09520973694712E+00
== err : 2.561E-14 = rco : 3.286E-03 = res : 8.299E-15 ==
solution 30 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 3.55062168897418E-01 -6.58793817483642E-01
h1 : 8.65122688235075E-16 5.13233621632621E-17
h2 : -6.99881452025659E-01 -5.57604591211007E-01
h3 : -6.72447839671453E-01 6.42670422226320E-01
zz1 : 2.47162951652330E-15 1.03523277295336E-15
g1 : -2.48818292646590E-02 -2.15810907158299E-01
g0 : 3.13295497458149E-01 -2.31908447187466E-01
g3 : -1.78128951327466E+00 -1.46926219146222E+00
g2 : 1.52568419760988E+00 -1.85209575083376E+00
== err : 2.763E-14 = rco : 4.005E-03 = res : 5.778E-15 ==
solution 31 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -3.97212465974088E-01 8.10757746037958E-01
h1 : -5.32489817125910E-01 -2.41113073210387E-01
h2 : -3.07433139258378E-01 -1.39206697723154E-01
h3 : -2.09943587065549E-01 -2.47499918354666E-03
zz1 : 1.20731861807830E-15 -5.52415134162948E-16
g1 : -3.17017817986829E-01 -2.56542973910398E-01
g0 : -5.31045817320590E-01 2.10317546770931E-01
g3 : -5.62135395713953E-01 -5.03215531393190E-01
g2 : 4.75881144279140E-01 -8.46539460654437E-01
== err : 3.149E-15 = rco : 5.451E-03 = res : 1.846E-15 ==
solution 32 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 1.20606214953554E+00 2.66803629825001E-01
h1 : -4.57957460139961E-16 -4.11131736722730E-16
h2 : -1.32423546111093E+00 -1.90947378668478E-01
h3 : -5.53771037306898E-01 3.38210713291072E-02
zz1 : -7.95316821349576E-16 1.27203046906612E-15
g1 : 6.64756719057740E-01 2.14747675845858E-01
g0 : -1.11585798851694E+00 -1.21712261606686E-01
g3 : -1.26976900196572E+00 -1.23082702034050E-01
g2 : -4.93025369305623E-01 -2.45538602184865E-01
== err : 1.134E-14 = rco : 4.679E-03 = res : 2.346E-15 ==
solution 33 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -3.88165977092905E-01 1.41746547719912E+00
h1 : -5.05251073281063E-01 -5.36210537489197E-01
h2 : -9.60364196606838E-01 1.04904637704470E+00
h3 : 2.24107731364845E-01 -8.18374074827915E-01
zz1 : 6.90068272148205E-15 -1.15935883560961E-14
g1 : -2.98699968147054E-01 -9.88921702777758E-01
g0 : -1.15569369394966E+00 9.81321211648346E-01
g3 : -3.35745964856912E-01 1.47137544710295E+00
g2 : 2.99858265236615E-01 -4.58340665075808E-01
== err : 5.981E-14 = rco : 2.074E-03 = res : 4.885E-15 ==
solution 34 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 2.02900577966356E-01 -1.00528609700379E+00
h1 : -1.16080439743545E+00 -2.34289406615213E-01
h2 : -3.46788258958458E-16 9.03139660731560E-18
h3 : -1.17144703307606E-01 5.80402198717725E-01
zz1 : -1.16701064404116E-15 6.51980355925717E-18
g1 : 1.16646524718559E+00 -1.86690230033454E-01
g0 : 5.96980130885627E-01 1.09804706417512E+00
g3 : 6.60619457736112E-01 -4.31057190118026E-01
g2 : -1.75717054961411E-01 8.70603298076588E-01
== err : 1.840E-14 = rco : 7.164E-03 = res : 3.164E-15 ==
solution 35 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -3.37906425214687E-01 -2.25980190654332E-01
h1 : -7.48668676369706E-18 -8.05348069324338E-17
h2 : -4.51960381308664E-01 6.75812850429374E-01
h3 : -5.85271096675811E-01 -3.91409171717405E-01
zz1 : -4.07327333630646E-16 4.53289037533107E-16
g1 : -5.06859637822031E-01 -3.38970285981498E-01
g0 : -5.06423071941813E-01 6.26830668879195E-01
g3 : -1.99209918765331E-01 7.19832905970868E-02
g2 : 3.75754692744673E-01 4.25732386307518E-01
== err : 7.637E-15 = rco : 3.999E-03 = res : 1.947E-15 ==
solution 36 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 2.42373353722443E+00 -2.58637590477853E+00
h1 : 1.49324482484944E+00 1.39934321016045E+00
h2 : -2.58637590477854E+00 -2.42373353722443E+00
h3 : -1.39934321016045E+00 1.49324482484945E+00
zz1 : -3.78662419912045E-14 8.73314561458121E-16
g1 : 1.06325573447402E+01 -7.28327548952220E+00
g0 : -8.13526871882521E+00 -1.42180999049983E+01
g3 : -4.75852224813389E-01 3.36135673306692E+00
g2 : -1.00200702745170E+01 3.65553424427689E+00
== err : 1.830E-12 = rco : 1.150E-04 = res : 1.061E-13 ==
solution 37 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 8.87333448795487E-01 5.71131808118920E-02
h1 : -1.17050106202412E-01 1.97961584736694E-01
h2 : -1.07045037057375E+00 -2.23535676219990E-01
h3 : -5.12302205523034E-01 -3.29743103160224E-02
zz1 : -2.19662431638435E-15 -3.47786251569689E-16
g1 : 8.65930793472035E-01 -4.46008216856838E-01
g0 : -8.96735449816699E-01 -4.83411688297159E-01
g3 : -9.86576737030125E-01 -2.27645407317612E-01
g2 : -2.73615270138913E-01 -4.31768984950880E-02
== err : 2.654E-14 = rco : 3.640E-03 = res : 3.459E-15 ==
solution 38 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 9.02253209186252E-01 -3.09498839434083E-02
h1 : -2.12697243146744E-01 -1.08622313370825E-01
h2 : -9.50451967814528E-01 1.22764486230802E-01
h3 : -5.20916133200887E-01 1.78689238261145E-02
zz1 : 1.06974236703126E-15 -7.10524942997707E-16
g1 : 9.53616313018668E-01 1.52885125140607E-01
g0 : -5.75516377780198E-01 2.86844473782494E-01
g3 : -9.09202200207360E-01 1.12606677989292E-01
g2 : -2.41617603714046E-01 3.78231594945250E-02
== err : 2.465E-14 = rco : 6.200E-03 = res : 2.518E-15 ==
solution 39 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 4.39557070154697E-01 -7.97626816828049E-01
h1 : -4.67874836750222E-01 8.88351689198215E-02
h2 : -7.72080945056142E-01 -3.32581677639246E-01
h3 : -2.53778392778018E-01 4.60510057408538E-01
zz1 : 3.68339630597356E-15 -3.91010446618247E-15
g1 : 2.63754427448688E-01 -2.16229449603230E-01
g0 : 1.03125264783036E+00 2.07591256584139E-01
g3 : -1.34832391699439E+00 -6.95658897363183E-01
g2 : 7.91450152452166E-01 -1.70228643792234E+00
== err : 1.817E-14 = rco : 2.620E-03 = res : 3.039E-15 ==
solution 40 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : 4.19784586126185E-01 -5.66201222792871E-02
h1 : -1.64180600767541E-01 -7.34743826307057E-01
h2 : 9.47897140488540E-02 4.24204545903796E-01
h3 : -1.11743939072849E+00 3.49732547075056E-01
zz1 : 5.49246555751842E-16 -1.19462227266188E-15
g1 : 6.09557926988737E-01 1.18028064527871E-01
g0 : -4.26024191016892E-02 4.36010649527860E-01
g3 : 1.97436823019562E-01 6.47630819739270E-02
g2 : 7.65510957494871E-01 -4.17876081902153E-01
== err : 5.398E-15 = rco : 5.788E-03 = res : 2.954E-15 ==