9
+ 8.71730357073000E-01*h0
+ 1.02169468185000E-01*h1
+ 2.94819331087000E-01*h2
+ 2.73443411310000E-01*h3
+(-9.19189381820505E-01 - 3.93815794945351E-01*i)*zz1
+ 7.85598722235000E-01;
+ 4.75479795171427E-01*h0**2
+ 1.63173496219546E+00*h0*h3
-2.55000000000000E+00*h0*g1
-2.59807621135331E-01*h0*g2
+ 9.46521105006264E-01*h1**2
+ 2.55000000000000E+00*h1*g0
+ 1.47224318643355E+00*h1*g3
-9.37644134333082E-01*h2**2
+ 2.59807621135331E-01*h2*g0
-4.50000000000001E-01*h2*g3
-4.66602824498245E-01*h3**2
-1.47224318643355E+00*h3*g1
+ 4.50000000000001E-01*h3*g2
+(-9.34605033025402E-01 - 3.55687267474655E-01*i)*zz1;
-2.38566960690759E-01*h0**2
+ 3.26346992439092E+00*h0*h3
-3*h0*g1
-5.19615242270662E-01*h0*g2
+ 1.64559827864859E+00*h1**2
-3.26346992439092E+00*h1*h2
+ 3*h1*g0
+ 2.94448637286709E+00*h1*g3
-2.38566960690758E-01*h2**2
+ 5.19615242270662E-01*h2*g0
-3*h2*g3
+ 1.64559827864859E+00*h3**2
-2.94448637286709E+00*h3*g1
+ 3*h3*g2
+( 8.16479212436865E-01 - 5.77374831161245E-01*i)*zz1;
-3.96161138468504E-01*h0**2
+ 4.89520488658638E+00*h0*h3
-1.95000000000000E+00*h0*g1
+ 1.81865334794732E+00*h0*g2
-3.96161138468506E-01*h1**2
-4.89520488658638E+00*h1*h2
+ 1.95000000000000E+00*h1*g0
+ 1.81865334794732E+00*h1*g3
+ 2.43008672054052E+00*h2**2
-1.81865334794732E+00*h2*g0
-4.05000000000000E+00*h2*g3
+ 2.43008672054052E+00*h3**2
-1.81865334794732E+00*h3*g1
+ 4.05000000000000E+00*h3*g2
+( 3.00634681415830E-01 + 9.53739371280227E-01*i)*zz1;
+ 1.20283364898221E+00*h0**2
+ 3.26346992439092E+00*h0*h3
-1.95000000000000E+00*h0*g1
+ 2.33826859021799E+00*h0*g2
+ 2.60751029312537E-01*h1**2
+ 1.95000000000000E+00*h1*g0
-1.12583302491977E+00*h1*g3
+ 4.02908150799123E+00*h2**2
-2.33826859021799E+00*h2*g0
-4.05000000000000E+00*h2*g3
+ 3.08699888832156E+00*h3**2
+ 1.12583302491977E+00*h3*g1
+ 4.05000000000000E+00*h3*g2
+( 9.09407380757791E-01 + 4.15906498895189E-01*i)*zz1;
+ 8.84948031588529E-01*h0**2
-4.50000000000000E-01*h0*g1
+ 2.59807621135332E-01*h0*g2
+ 1.35598934142337E+00*h1**2
+ 1.63173496219546E+00*h1*h2
+ 4.50000000000000E-01*h1*g0
-1.47224318643355E+00*h1*g3
+ 2.29807196109304E+00*h2**2
-2.59807621135332E-01*h2*g0
-2.55000000000000E+00*h2*g3
+ 2.76911327092787E+00*h3**2
+ 1.47224318643355E+00*h3*g1
+ 2.55000000000000E+00*h3*g2
+(-3.69625210123089E-01 + 9.29180931811164E-01*i)*zz1;
+ 1*h0*g0
+ 1*h1*g1
+ 1*h2*g2
+ 1*h3*g3
+(-7.38370286806648E-01 - 6.74395521605139E-01*i)*zz1;
-1*h0**2
-1*h1**2
-1*h2**2
-1*h3**2
+ 1*g0**2
+ 1*g1**2
+ 1*g2**2
+ 1*g3**2
+( 1.02583165914583E-02 - 9.99947382086032E-01*i)*zz1;
+(-9.93811493219141E-01 - 1.11079772891114E-01*i)*h0
+(-2.45583317560504E-01 - 9.69375486659311E-01*i)*h1
+(-1.52133914844955E-01 - 9.88359889895350E-01*i)*h2
+( 9.97190817555422E-01 - 7.49030932815811E-02*i)*h3
+(-7.38564226845285E-01 - 6.74183122619090E-01*i)*g0
+( 7.38887324460615E-01 + 6.73829000378756E-01*i)*g1
+(-9.27254869702586E-01 - 3.74430776797048E-01*i)*g2
+(-1.81343043191790E-01 + 9.83419900493142E-01*i)*g3
+( 9.55112770805283E-01 + 2.96242459895025E-01*i)*zz1
+( 1.23362146631345E-01 + 9.92361718718788E-01*i);
TITLE : Griffis-Duffy platform, one large degree component
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 : -7.57818584116683E-01 -1.10796477190924E-01
h1 : -1.03537117898478E-01 5.56165129561824E-01
h2 : -4.99728333221264E-02 3.24943394636530E-01
h3 : -3.64514058917158E-01 -2.04934676852200E-01
zz1 : 6.69543813248381E-16 6.85639771808819E-16
g1 : -1.46166540966803E-02 -1.37976428759874E-01
g2 : 8.26626912812933E-01 1.77160093692908E-01
g0 : 2.07108151024184E-01 3.21310621958068E-02
g3 : -1.06533940644742E-01 6.59635190998907E-01
== err : 4.200E-15 = rco : 1.463E-02 = res : 1.783E-15 ==
solution 2 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -7.71792769418099E-01 1.43655574431845E-01
h1 : -3.82240082804806E-01 -1.88710968954947E-01
h2 : -5.22363474207601E-02 -2.03308764574396E-01
h3 : -2.13389616886676E-01 -1.68258111101785E-01
zz1 : 3.65811408038443E-16 -3.52186284944368E-16
g1 : -3.86068716789580E-01 7.65379952069392E-02
g2 : 5.35451637101100E-04 -3.50942745096014E-02
g0 : 3.58879634407479E-01 2.52281758455438E-01
g3 : -6.80544541929379E-01 7.81519227484477E-02
== err : 3.109E-15 = rco : 1.653E-02 = res : 8.119E-16 ==
solution 3 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -1.01708183959750E+00 2.45231556637542E-01
h1 : 2.67062076605460E-01 -5.40898791930349E-01
h2 : 8.49650317355130E-03 4.29922549468072E-01
h3 : 2.60499490213237E-01 -1.04322107307888E+00
zz1 : -5.84277241548985E-16 5.65940583335016E-15
g1 : -5.33743686761663E-01 1.94806857243626E-01
g2 : -5.45552754004731E-01 1.28749446722636E+00
g0 : 4.28783475897358E-01 -9.74756130812504E-01
g3 : 1.27364823060020E+00 4.41528951396972E-01
== err : 1.271E-14 = rco : 8.753E-03 = res : 6.883E-15 ==
solution 4 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -4.89843530631434E-01 2.14688417333045E-01
h1 : 1.53711134481107E-16 -4.23922838699398E-16
h2 : -4.29376834666086E-01 -9.79687061262863E-01
h3 : -8.48433882812560E-01 3.71851246617380E-01
zz1 : -2.77191007330907E-15 5.58382897704222E-16
g1 : -4.61473076565505E-01 2.02254226613849E-01
g2 : -5.85007126934846E+00 3.05787534198044E+00
g0 : 1.35378037271987E+00 2.52538822720596E+00
g3 : 2.74932484106801E+00 5.29704687148798E+00
== err : 2.064E-13 = rco : 1.751E-04 = res : 2.170E-14 ==
solution 5 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -2.47316147299364E+00 -2.45393851579753E-02
h1 : -1.65524842042164E-01 4.14972234525229E+00
h2 : 7.30505703821660E-02 -1.60957878897066E+00
h3 : 4.99447424957655E+00 2.63132117199930E-01
zz1 : -5.14897110559968E-14 -4.14855902556699E-14
g1 : -6.82669697168174E+00 -1.29533148874631E-01
g2 : -1.32215665580459E+00 -1.79206430978577E-01
g0 : 7.09070488081444E-02 -5.55553174014451E+00
g3 : -6.28215318506889E-02 2.49695454014366E+00
== err : 9.064E-13 = rco : 5.287E-04 = res : 6.802E-14 ==
solution 6 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -1.07123962784718E+00 -1.77037580011817E-01
h1 : -2.04425388953005E-01 1.23696097500833E+00
h2 : 9.29947302585809E-16 -1.41202425475892E-15
h3 : 6.18480487504167E-01 1.02212694476503E-01
zz1 : -1.85593275961161E-16 3.66797635453993E-16
g1 : -1.57725939733318E+00 -6.00221050870360E-01
g2 : 5.82659717882502E-01 9.62928029759197E-02
g0 : 5.67953079427884E-01 -1.65727656538952E+00
g3 : -2.16718511856424E-01 2.84031581218472E-01
== err : 2.650E-14 = rco : 5.296E-03 = res : 7.466E-15 ==
solution 7 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -5.19991618914258E-01 1.82348852322293E-01
h1 : -3.15837476924089E-01 -9.00651903469480E-01
h2 : -1.82348852322295E-01 -5.19991618914252E-01
h3 : -9.00651903469477E-01 3.15837476924088E-01
zz1 : 1.53757658237993E-16 -5.67201889467527E-16
g1 : 6.66526681423486E-02 -1.26943976428318E+00
g2 : 1.65465051614373E+00 -1.29966376610606E+00
g0 : -4.49535201390334E-01 7.69602354230569E-01
g3 : -1.76026171996759E+00 -1.46629571543667E+00
== err : 3.719E-14 = rco : 2.488E-03 = res : 4.399E-15 ==
solution 8 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -5.35797951099482E-01 7.37286764630007E-02
h1 : 2.66175534131317E-01 9.84336318791226E-01
h2 : -2.37039482096970E-01 -5.73279146195929E-01
h3 : -1.00875926355515E+00 1.52614053559576E-02
zz1 : -1.25604274851865E-15 -6.76755980970316E-16
g1 : -1.06851488614272E-01 6.74215952272880E-02
g2 : 7.42741698639127E-01 1.87570527285221E-01
g0 : 1.71441637817417E-01 2.25555054288608E-01
g3 : -2.59419987690947E-01 -6.63847835660186E-01
== err : 1.148E-14 = rco : 6.467E-03 = res : 3.296E-15 ==
solution 9 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -8.21687856252638E-01 -2.73479601727528E-01
h1 : -6.38665366460563E-02 -1.53342821819634E-02
h2 : 4.58972241754393E-02 6.58664816853119E-02
h3 : -2.79087553735311E-01 8.06559767498295E-01
zz1 : 2.32125591498932E-15 1.68518453653135E-15
g1 : -3.93798426699425E-01 -1.73975345684575E-01
g2 : 9.04756669192987E-02 -8.15613228664046E-01
g0 : 3.91724569667393E+00 -2.23304142763439E+00
g3 : -2.26602793863882E+00 -3.86414934184002E+00
== err : 1.623E-13 = rco : 2.068E-04 = res : 4.635E-15 ==
solution 10 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -1.30721832987060E+00 -7.17521130035414E-02
h1 : -6.49407076811359E-01 1.91408819867097E-01
h2 : 1.98074931440947E+00 4.88261673230089E-01
h3 : -5.98553215205546E-01 -3.69204807882787E-01
zz1 : 6.87132468915769E-15 1.03376799499886E-14
g1 : 1.19396165206360E+00 1.83301030447363E-01
g2 : 1.44994129899034E+00 2.12373718345037E-01
g0 : 9.40007442780009E-01 4.34064148912054E-01
g3 : 1.37030897348135E+00 1.62590176357119E-01
== err : 1.597E-14 = rco : 4.588E-03 = res : 9.701E-15 ==
solution 11 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -7.90744984218733E-01 -2.92359751157791E-01
h1 : -3.40173928567312E-02 4.64342662154220E-03
h2 : -4.43225441907985E-02 1.50862719804520E-01
h3 : -2.91612925097329E-01 7.67644049998241E-01
zz1 : 1.87898874667266E-15 1.30049911412686E-15
g1 : -5.01355479317071E-01 -1.80213254901840E-01
g2 : 3.55860513693219E-02 -1.71714537955376E+00
g0 : 5.45903340337069E+00 -2.06697034504043E+00
g3 : -2.14988851645916E+00 -5.23510781848022E+00
== err : 2.819E-13 = rco : 7.578E-05 = res : 2.399E-14 ==
solution 12 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -8.19542597526730E-01 -1.92730588058488E-01
h1 : -4.79872289872545E-02 -1.79609561091322E-01
h2 : -2.01067106446837E-02 -1.04694547193203E-01
h3 : -2.20696063289481E-01 7.94408221489194E-01
zz1 : 1.80744330002228E-15 1.66718226746074E-15
g1 : 7.57921853830177E-02 1.04102290326265E+00
g2 : 8.36831323944638E-01 -1.98304231895770E-01
g0 : 5.51662048281822E+00 -6.40206750726899E-01
g3 : -6.59170338313404E-01 -5.47988197303551E+00
== err : 2.700E-13 = rco : 1.448E-04 = res : 1.010E-14 ==
solution 13 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -8.72423438732191E-01 -2.33661050189763E-01
h1 : 2.02115889191233E-01 -4.83602130440217E-01
h2 : -1.43142247379857E-01 2.54764882903684E-01
h3 : -1.29083298837178E-02 6.50917826037060E-01
zz1 : 1.85885825729761E-15 -2.99674657357561E-17
g1 : -2.00123531046947E+00 -6.02181076702911E-01
g2 : 9.52483551834850E-01 -2.70741653963427E-01
g0 : -7.57814531706665E-01 2.55470183450180E+00
g3 : 1.43436551485281E+00 7.32030241992317E-01
== err : 5.913E-14 = rco : 1.747E-03 = res : 5.745E-15 ==
solution 14 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -9.19712126997014E-01 2.76298129927155E-01
h1 : 5.65520701129548E-01 -7.79383878990328E-01
h2 : 3.28741991859001E-01 -2.67068169294376E-01
h3 : -5.06708807214732E-01 -3.01676210117483E-01
zz1 : 9.09092862967815E-16 1.82257879714489E-15
g1 : -4.34906735839466E-01 1.95351467579784E-01
g2 : 6.08891575116150E-01 1.15572411687247E+00
g0 : -6.57205144986110E-01 4.75557908717305E-01
g3 : 1.33557794650129E+00 -7.00834342704109E-01
== err : 1.081E-14 = rco : 9.963E-03 = res : 4.191E-15 ==
solution 15 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -7.69755001821525E-01 6.27985614224605E-02
h1 : -1.61844786242940E-01 -1.00006721245373E+00
h2 : -4.15959644943095E-01 -4.68678643089990E-01
h3 : 8.99221534140346E-02 6.78781931641841E-01
zz1 : 4.15298494152657E-15 5.39243114715280E-16
g1 : -1.69418106960812E+00 -2.71462868191367E-01
g2 : 6.34517771492926E-01 -4.35810062081352E-01
g0 : -1.44417197566448E-02 1.78931345987120E+00
g3 : -2.44401871521921E-01 -8.67292184175702E-01
== err : 3.382E-15 = rco : 8.133E-03 = res : 4.226E-15 ==
solution 16 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -1.34612518352096E+00 2.38932640360304E-01
h1 : 1.32430380341035E-01 -2.87592222439795E-01
h2 : 8.57014040464355E-01 7.32198464235021E-01
h3 : 4.44935997561487E-01 -1.44369158881655E+00
zz1 : 4.32913114836680E-16 1.27119214133609E-14
g1 : -7.95837141827776E-01 1.01316268233063E-01
g2 : -1.97864712078321E+00 3.24787964115961E+00
g0 : 6.60313858750763E-01 -2.01438355073503E+00
g3 : 3.70126549022670E+00 2.01622908713706E+00
== err : 3.249E-14 = rco : 2.465E-03 = res : 1.298E-14 ==
solution 17 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -8.83475578411815E-01 -4.37874023682952E-01
h1 : -2.52806685444498E-01 5.10074863018522E-01
h2 : -4.37874023682951E-01 8.83475578411815E-01
h3 : 5.10074863018522E-01 2.52806685444497E-01
zz1 : 3.39298976946560E-16 -2.86242036068977E-15
g1 : -1.26812458545550E-01 -2.12854887967998E+00
g2 : -1.18071094759390E+00 -4.16308437498665E+00
g0 : 3.92491959049309E+00 -7.00178284413757E-01
g3 : -2.54106238699651E+00 9.59119445869935E-01
== err : 3.303E-14 = rco : 2.580E-03 = res : 7.272E-15 ==
solution 18 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -9.92208627373530E-01 3.27788434843091E-01
h1 : 1.89248741093904E-01 5.72851918106374E-01
h2 : -3.27788434843094E-01 -9.92208627373528E-01
h3 : 5.72851918106376E-01 -1.89248741093905E-01
zz1 : -5.19417904134113E-15 9.52220110824834E-16
g1 : -1.75798701317818E+00 3.01201752033516E-01
g2 : -3.89276638002819E-02 4.97091546600239E-01
g0 : -1.23025956252831E-02 -1.50299758103529E+00
g3 : -5.81094783379062E-01 -9.12705852350121E-01
== err : 1.757E-14 = rco : 5.923E-03 = res : 6.668E-15 ==
solution 19 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -8.83475578411820E-01 4.37874023682950E-01
h1 : -2.52806685444497E-01 -5.10074863018524E-01
h2 : -4.37874023682951E-01 -8.83475578411819E-01
h3 : 5.10074863018524E-01 -2.52806685444497E-01
zz1 : -2.57798567674619E-15 7.77189406509643E-16
g1 : 6.53037274609406E-01 3.92269134910693E-01
g2 : 1.70028412499522E-01 1.15575964089353E+00
g0 : 9.17594856399975E-01 -6.50561075679660E-01
g3 : -8.04782642227230E-01 -1.79269712714979E-01
== err : 5.357E-15 = rco : 1.163E-02 = res : 2.803E-15 ==
solution 20 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -5.81291354108615E-01 -5.32404235695645E-01
h1 : -2.77494476642597E-02 -7.32928257972158E-02
h2 : 3.42660472352470E-01 1.42969618325836E+00
h3 : -1.37892218973826E+00 1.83215791807664E-01
zz1 : -1.03913157855915E-14 6.51296808531678E-16
g1 : -5.52335562516877E-01 -2.20059862139066E-02
g2 : 4.69591682542614E-01 -2.02350257959572E-02
g0 : 5.85493227407190E-01 9.75921701198652E-01
g3 : 2.89216423933691E-01 -8.73815254890365E-02
== err : 7.451E-15 = rco : 6.498E-03 = res : 3.469E-15 ==
solution 21 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -3.04674928594827E-01 6.41613100099092E-01
h1 : -2.21263829161247E+00 2.63059136783553E+00
h2 : 1.31827220896246E+00 -1.64738003798205E+00
h3 : -2.49628392202533E+00 -1.25217958418389E+00
zz1 : 1.39326947785928E-16 -2.50473909321426E-14
g1 : -1.41308948413748E+00 -1.02487233691847E+00
g2 : 2.38576623351629E+00 3.56550941909171E-01
g0 : -5.80403660797470E-01 5.10078181548705E-02
g3 : 2.25380947320330E+00 -3.25280759358647E+00
== err : 1.729E-13 = rco : 1.064E-03 = res : 3.078E-14 ==
solution 22 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -7.36089093599414E-01 -2.57299235768000E-01
h1 : 4.21863212619846E-01 2.65986156088202E-01
h2 : -2.22345527996959E-01 -1.47651806972643E-01
h3 : -4.44250214324088E-01 8.80074222198707E-01
zz1 : 1.95104332954805E-15 2.27301079158386E-15
g1 : -1.58658827093454E+00 -6.24401011888056E-01
g2 : 1.14934528689232E+00 -4.43829685037077E-01
g0 : 1.56599729766527E+00 6.20871357107945E-01
g3 : 6.33005047806692E-01 -2.38445987925598E+00
== err : 4.761E-14 = rco : 2.443E-03 = res : 4.240E-15 ==
solution 23 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -6.64521077677107E-01 2.33584597384146E-01
h1 : -3.55679575062283E-01 7.32620862754600E-01
h2 : 2.01750211008141E-01 -4.09309276626098E-01
h3 : -8.39134481381113E-01 -5.77091912782640E-01
zz1 : 2.15224494815417E-15 -1.84884158264055E-15
g1 : -1.47240357057476E+00 4.53323892375202E-01
g2 : 1.40682781382731E+00 2.29269206604684E-01
g0 : -7.27534339547973E-01 -3.21040480955464E-01
g3 : -4.80460718955472E-02 -2.02399004589192E+00
== err : 8.908E-15 = rco : 4.110E-03 = res : 5.433E-15 ==
solution 24 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -5.79489903191199E-01 5.07650882218954E-02
h1 : 8.79277120510371E-02 1.00370595480034E+00
h2 : -5.07650882218953E-02 -5.79489903191208E-01
h3 : -1.00370595480034E+00 8.79277120510347E-02
zz1 : -1.05000674910841E-15 -4.86082504526399E-16
g1 : -5.95703197387675E-01 -6.22511089503469E-01
g2 : 9.74312024886464E-01 3.04183498891293E-01
g0 : 3.87018668201887E-01 -2.87380896101103E-02
g3 : 5.74686182203627E-01 -1.14163056345817E+00
== err : 2.546E-14 = rco : 5.480E-03 = res : 3.525E-15 ==
solution 25 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -9.92208627373520E-01 -3.27788434843099E-01
h1 : 1.89248741093907E-01 -5.72851918106368E-01
h2 : -3.27788434843098E-01 9.92208627373522E-01
h3 : 5.72851918106370E-01 1.89248741093910E-01
zz1 : -1.09629614081926E-15 -5.27935602758354E-15
g1 : -1.35557449510194E+00 1.73677932217325E-01
g2 : -7.35926590710037E-01 -1.31960728720499E+00
g0 : 8.10213144979453E-01 8.05998654125521E-01
g3 : -1.05597446762992E+00 1.31511837042634E+00
== err : 3.005E-14 = rco : 5.784E-03 = res : 4.205E-15 ==
solution 26 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -6.00725037294040E-01 -3.33777939746862E-01
h1 : 4.12090749499123E-02 4.12577396984622E-02
h2 : -3.77095412580885E-01 8.05272729579777E-01
h3 : -5.66712367317749E-01 1.80436286840595E-01
zz1 : -4.09652079836580E-16 -1.13572106886428E-15
g1 : -4.83841602091482E-01 -4.30643660462956E-01
g2 : 1.19918800167610E+00 -2.27280949930165E-02
g0 : -3.20534069561904E-01 9.92496872843385E-01
g3 : -8.75876305183831E-02 7.61410790477151E-01
== err : 1.136E-14 = rco : 1.149E-02 = res : 1.613E-15 ==
solution 27 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -5.79489903191200E-01 -5.07650882218948E-02
h1 : 8.79277120510427E-02 -1.00370595480033E+00
h2 : -5.07650882218972E-02 5.79489903191206E-01
h3 : -1.00370595480034E+00 -8.79277120510449E-02
zz1 : -1.34470797873009E-15 -1.31558434669790E-15
g1 : 4.08001759628570E-01 -8.61661579681602E-01
g2 : 3.94822697766167E-01 5.52703991186607E-01
g0 : -4.69868821875995E-01 -5.50751237510176E-01
g3 : -9.09486486981451E-01 1.37925606441923E-01
== err : 1.136E-14 = rco : 7.190E-03 = res : 3.532E-15 ==
solution 28 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -5.19991618914255E-01 -1.82348852322293E-01
h1 : -3.15837476924089E-01 9.00651903469487E-01
h2 : -1.82348852322294E-01 5.19991618914256E-01
h3 : -9.00651903469488E-01 -3.15837476924090E-01
zz1 : 1.10701198788291E-15 1.58400938731249E-15
g1 : -5.01757698136528E+00 1.18191694607091E+00
g2 : -1.28073084062136E+00 1.24913244345097E+00
g0 : -3.99003878735240E-01 -3.70498371099567E+00
g3 : -1.67273890175532E+00 -3.61793393407095E+00
== err : 1.433E-13 = rco : 5.333E-04 = res : 1.541E-14 ==
solution 29 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -6.81497710916398E-01 2.53055327927322E-01
h1 : -9.54412785707372E-02 -5.82406296466290E-01
h2 : -7.49310275427341E-02 -3.36844118451225E-01
h3 : -5.83938660274080E-01 -2.25946976807313E-01
zz1 : 1.13946309851288E-15 -5.14807796557174E-16
g1 : -3.40336091084281E-01 -2.76362347109183E-01
g2 : 8.77530550885905E-01 -1.47767662619818E-01
g0 : -4.63028367636311E-01 5.55117953738547E-01
g3 : -4.22770576606811E-01 -7.87562241897797E-01
== err : 5.283E-15 = rco : 1.116E-02 = res : 1.669E-15 ==
solution 30 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -4.89843530631430E-01 -2.14688417333046E-01
h1 : -1.11039591986822E-15 3.18538789706757E-16
h2 : -4.29376834666090E-01 9.79687061262855E-01
h3 : -8.48433882812552E-01 -3.71851246617387E-01
zz1 : -8.39092694292556E-16 -4.43422750935074E-15
g1 : -4.61473076565498E-01 -2.02254226613850E-01
g2 : 8.01188500908778E-01 7.17961847343585E-03
g0 : -1.78747107507086E-01 8.00241657922670E-01
g3 : 9.49093813194519E-02 4.63113056724228E-01
== err : 4.859E-15 = rco : 1.200E-02 = res : 2.271E-15 ==
solution 31 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -1.07123962784718E+00 1.77037580011818E-01
h1 : -2.04425388953006E-01 -1.23696097500833E+00
h2 : 2.35994077308162E-17 1.04367647007448E-15
h3 : 6.18480487504164E-01 -1.02212694476503E-01
zz1 : 8.36175702303353E-16 1.16061843795076E-16
g1 : -7.40730794406063E-01 -7.84567112881483E-03
g2 : 5.82659717882501E-01 -9.62928029759196E-02
g0 : 4.13518509806678E-02 9.32821544262327E-01
g3 : 8.73148491431638E-02 1.34232720245087E-01
== err : 4.574E-15 = rco : 1.566E-02 = res : 1.409E-15 ==
solution 32 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -5.42610268502768E-01 2.30791028348531E-01
h1 : 1.96383110475717E-01 -1.38795783503845E-01
h2 : -5.53225115618012E-01 -7.71614197787453E-02
h3 : -6.20061671042326E-01 -6.00702628791776E-01
zz1 : 2.72518433326112E-15 -3.78239835746420E-16
g1 : -9.20092666954508E-02 1.85804605589817E-01
g2 : 6.59772663100075E-01 3.79459470522897E-01
g0 : -4.93845502240083E-01 1.64563421914280E-01
g3 : -4.15895622627782E-01 -2.66127464363940E-01
== err : 3.531E-15 = rco : 1.834E-02 = res : 2.415E-15 ==
solution 33 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -7.56617862963153E-01 -1.90582750432671E-02
h1 : -3.22457791110783E-02 3.31703457590314E-01
h2 : 1.08227415205082E-01 3.94075854187767E-01
h3 : -5.65546493508479E-01 -4.88062477125627E-01
zz1 : 1.03858409578297E-15 2.74936631538159E-16
g1 : -3.27942126171374E-01 9.13255925462397E-02
g2 : 6.73649667717579E-01 4.18287214013569E-01
g0 : -1.71886199598303E-01 -6.98082864123727E-02
g3 : 2.40418778616739E-01 2.43604156480198E-01
== err : 4.042E-15 = rco : 2.548E-02 = res : 1.794E-15 ==
solution 34 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -6.77120094548120E-01 1.44748462964823E-01
h1 : -4.13909480660569E-01 1.16484781107225E+00
h2 : -2.02726987177663E+00 4.72592869008526E-01
h3 : 1.62605738599598E+00 -1.40622523086300E+00
zz1 : -9.90458679196202E-16 1.82116161641834E-14
g1 : -3.01012527427562E+00 1.02901173386818E+00
g2 : -3.21797981406217E-01 6.67632981807073E-01
g0 : -4.94540100931419E-01 -2.33026106061659E+00
g3 : -1.56792271227262E+00 1.06187646049608E+00
== err : 2.641E-14 = rco : 1.549E-03 = res : 1.191E-14 ==
solution 35 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -2.69573066406183E-01 4.93734693099919E-01
h1 : -1.53246038306504E+00 3.74797538842307E-01
h2 : -8.34486389050244E-01 2.22137075362727E-01
h3 : -5.41283166111227E-01 -1.95355480296558E+00
zz1 : 8.50647970991512E-15 1.16148848069138E-15
g1 : -6.14201499612801E-01 -2.76303962347285E+00
g2 : 5.03419564935250E-01 5.03158079138574E-01
g0 : 1.31916183896495E+00 -1.06533118793359E+00
g3 : 2.41596959509672E+00 -1.57461602761849E-01
== err : 4.335E-14 = rco : 2.720E-03 = res : 1.621E-14 ==
solution 36 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -5.91734108479863E-01 1.32068699988924E-02
h1 : 7.28257975719174E-01 2.72564839993325E-02
h2 : -7.70566495001035E-01 1.30563635582029E-02
h3 : -4.27854404915221E-01 -6.63642918235835E-02
zz1 : 1.23213210319257E-15 4.92980975244031E-16
g1 : 3.82598132494039E-01 -4.52221723513120E-02
g2 : 8.37996703892178E-01 3.65829329858449E-02
g0 : 2.79645052680764E-02 4.69039277864665E-02
g3 : -8.99074833923636E-01 -1.74647875785309E-02
== err : 2.608E-15 = rco : 2.396E-02 = res : 1.562E-15 ==
solution 37 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -8.92949308914138E-01 2.53652134715702E-02
h1 : 6.01037315764140E-01 -9.97452963027010E-01
h2 : -2.22119209132569E-01 8.67833900451083E-01
h3 : -1.13746700467722E-02 -6.43850210202357E-01
zz1 : -9.16423643540661E-16 6.58269185246993E-16
g1 : -4.58606757638361E-01 7.08911211246913E-02
g2 : -1.06188771778679E+00 7.78161953029352E-01
g0 : 6.80488249578867E-02 -7.46161203442017E-01
g3 : 9.55623846170047E-02 1.06748972247562E+00
== err : 7.767E-15 = rco : 1.028E-02 = res : 3.587E-15 ==
solution 38 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -7.50903079885540E-01 2.50401815754344E-01
h1 : -7.02929342941608E-02 -1.05520462530482E-01
h2 : -1.68268507285516E-01 -6.28005520395363E-01
h3 : -2.71438655071837E-01 -8.17490067538730E-02
zz1 : -4.96148844477132E-16 -1.36688008315450E-16
g1 : -4.76912316150929E-01 1.54353634048795E-01
g2 : -3.71588050533232E-01 3.60908215022671E-01
g0 : 3.69330650273278E-01 4.21506733792297E-01
g3 : -1.60371070551460E-01 4.36354705524193E-03
== err : 2.232E-15 = rco : 7.446E-03 = res : 1.402E-15 ==
solution 39 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -6.61174315796332E-01 -1.03460992214174E-01
h1 : -3.55897822538927E-01 4.16261373179343E-01
h2 : -1.78668208037885E-01 2.27922705933085E-01
h3 : -4.39565416571178E-01 -7.14412354894386E-02
zz1 : -2.54876675464420E-16 7.60776136047590E-16
g1 : -3.04837923682946E-01 -2.76172318592194E-02
g2 : 4.52210518805785E-01 4.15325233296978E-01
g0 : 3.50063437277406E-01 -2.93768099890693E-01
g3 : -6.78161007421020E-01 2.69045192245099E-01
== err : 1.295E-15 = rco : 2.393E-02 = res : 6.991E-16 ==
solution 40 :
t : 1.00000000000000E+00 0.00000000000000E+00
m : 1
the solution for t :
h0 : -4.37743213703473E-02 -5.62496074774880E-01
h1 : -9.77497524388686E-01 2.24436309467500E-01
h2 : -8.52634861327454E-01 1.02662916302739E+00
h3 : -1.44891286223051E+00 6.02480205423760E-01
zz1 : -1.04513356353384E-14 3.67475831200117E-16
g1 : 9.63164420845236E-01 -1.80487138543214E+00
g2 : 8.62481314019483E-01 1.63904612050788E-01
g0 : 1.65833767111446E+00 3.21418549414790E-01
g3 : -1.21905142376498E+00 7.21086428307656E-01
== err : 4.853E-14 = rco : 2.514E-03 = res : 9.884E-15 ==