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Numerical Minimization of Large Expression
Combined numerical minimization and maximizationusing array elements in minimization$L_infty$ norm minimizationNumerical Instability?Simple minimization not evaluatingMinimization and integrationIssue with numerical minimizationMinimization over the integersProducing a smooth curve from numerical minimizationoptimization problem with NDSolve and Import cannot get any results, even without error message
1
$begingroup$
I want to minimize a function which contains ArcTan functions. The function is also very complicated.
I have the following code.
LenFact = 1.3745/0.31571;
xC = 0.157392 LenFact; yC = 0.035495 LenFact;
q=x1[t],y1[t],x2[t],y2[t],x3[t],y3[t],x4[t],y4[t],x5[t],y5[t],x6[t],y6[t];
CE =
-0.609155 + x1[t]^2 + y1[t]^2,
-0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + y2[t])^2,
-1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + y2[t])^2,
-1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + y6[t])^2,
-0.428366 + ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]],
-0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + y4[t])^2,
-0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + y5[t])^2,
-0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]],
-0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + y6[t])^2,
-0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + y3[t])^2,
-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]]
;
y6Start = 0.15 LenFact;
VPenalty=3.90351 + 1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 +
3006.52 (2.92648 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 +
1082.47 (2.78921 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] +
ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 +
6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] +
ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 +
239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] +
4.93484 y5[t] + 97.0584 y6[t] +
1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 +
ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 +
ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 +
x1[t]^2 +
y1[t]^2)^2 + (-1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 +
y2[t])^2)^2 + (-0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] +
y2[t])^2)^2 + (-0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] +
y3[t])^2)^2 + (-0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] +
y4[t])^2)^2 + (-0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] +
y5[t])^2)^2 + (-1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] +
y6[t])^2)^2 + (-0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] +
y6[t])^2)^2);
[Theta]1 = ArcTan[x1[t], y1[t]];
[Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
[Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
[Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
[Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
[Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
[Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];
eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0,
90 [Degree] < [Theta]1 < 180 [Degree],
20 [Degree] < [Theta]21 < 95 [Degree],
145 [Degree] < [Theta]3 < 180 [Degree],
80 [Degree] < [Theta]5 < 100 [Degree], 0 < y6[t] < y6Start,
q, Reals, MaxIterations -> 10^4]
It is running for several hours with no result. Can any body suggest any specific way of minimizing such large scale expression? I also could not give initial points. It is always showing syntax error. Is it not possible to give one initial point only for each variable?
mathematical-optimization
asked 4 hours ago
Soumyajit RoySoumyajit Roy
1809
$endgroup$
add a comment |
1
$begingroup$
I want to minimize a function which contains ArcTan functions. The function is also very complicated.
I have the following code.
LenFact = 1.3745/0.31571;
xC = 0.157392 LenFact; yC = 0.035495 LenFact;
q=x1[t],y1[t],x2[t],y2[t],x3[t],y3[t],x4[t],y4[t],x5[t],y5[t],x6[t],y6[t];
CE =
-0.609155 + x1[t]^2 + y1[t]^2,
-0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + y2[t])^2,
-1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + y2[t])^2,
-1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + y6[t])^2,
-0.428366 + ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]],
-0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + y4[t])^2,
-0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + y5[t])^2,
-0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]],
-0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + y6[t])^2,
-0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + y3[t])^2,
-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]]
;
y6Start = 0.15 LenFact;
VPenalty=3.90351 + 1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 +
3006.52 (2.92648 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 +
1082.47 (2.78921 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] +
ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 +
6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] +
ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 +
239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] +
4.93484 y5[t] + 97.0584 y6[t] +
1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 +
ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 +
ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 +
x1[t]^2 +
y1[t]^2)^2 + (-1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 +
y2[t])^2)^2 + (-0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] +
y2[t])^2)^2 + (-0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] +
y3[t])^2)^2 + (-0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] +
y4[t])^2)^2 + (-0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] +
y5[t])^2)^2 + (-1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] +
y6[t])^2)^2 + (-0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] +
y6[t])^2)^2);
[Theta]1 = ArcTan[x1[t], y1[t]];
[Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
[Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
[Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
[Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
[Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
[Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];
eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0,
90 [Degree] < [Theta]1 < 180 [Degree],
20 [Degree] < [Theta]21 < 95 [Degree],
145 [Degree] < [Theta]3 < 180 [Degree],
80 [Degree] < [Theta]5 < 100 [Degree], 0 < y6[t] < y6Start,
q, Reals, MaxIterations -> 10^4]
It is running for several hours with no result. Can any body suggest any specific way of minimizing such large scale expression? I also could not give initial points. It is always showing syntax error. Is it not possible to give one initial point only for each variable?
mathematical-optimization
asked 4 hours ago
Soumyajit RoySoumyajit Roy
1809
$endgroup$
$begingroup$
I am sorry. I have incorporated CE in the code now.
$endgroup$
– Soumyajit Roy
4 hours ago
$begingroup$
And what abouty6Start?
$endgroup$
– Henrik Schumacher
4 hours ago
1
$begingroup$
And after all: What is the context? This looks a bit like discrete bending energy minimization with some angle constraints... Maybe the problem can be rephrased into another one that is simpler to solve.
$endgroup$
– Henrik Schumacher
4 hours ago
$begingroup$
y6Start = 0.15 LenFact; This is about finding the static equilibrium of a mechanism. Static equilibrium states can be found by minimizing the potential energy function.
$endgroup$
– Soumyajit Roy
4 hours ago
add a comment |
1
1
1
$begingroup$
I want to minimize a function which contains ArcTan functions. The function is also very complicated.
I have the following code.
LenFact = 1.3745/0.31571;
xC = 0.157392 LenFact; yC = 0.035495 LenFact;
q=x1[t],y1[t],x2[t],y2[t],x3[t],y3[t],x4[t],y4[t],x5[t],y5[t],x6[t],y6[t];
CE =
-0.609155 + x1[t]^2 + y1[t]^2,
-0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + y2[t])^2,
-1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + y2[t])^2,
-1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + y6[t])^2,
-0.428366 + ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]],
-0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + y4[t])^2,
-0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + y5[t])^2,
-0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]],
-0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + y6[t])^2,
-0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + y3[t])^2,
-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]]
;
y6Start = 0.15 LenFact;
VPenalty=3.90351 + 1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 +
3006.52 (2.92648 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 +
1082.47 (2.78921 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] +
ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 +
6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] +
ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 +
239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] +
4.93484 y5[t] + 97.0584 y6[t] +
1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 +
ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 +
ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 +
x1[t]^2 +
y1[t]^2)^2 + (-1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 +
y2[t])^2)^2 + (-0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] +
y2[t])^2)^2 + (-0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] +
y3[t])^2)^2 + (-0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] +
y4[t])^2)^2 + (-0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] +
y5[t])^2)^2 + (-1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] +
y6[t])^2)^2 + (-0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] +
y6[t])^2)^2);
[Theta]1 = ArcTan[x1[t], y1[t]];
[Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
[Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
[Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
[Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
[Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
[Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];
eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0,
90 [Degree] < [Theta]1 < 180 [Degree],
20 [Degree] < [Theta]21 < 95 [Degree],
145 [Degree] < [Theta]3 < 180 [Degree],
80 [Degree] < [Theta]5 < 100 [Degree], 0 < y6[t] < y6Start,
q, Reals, MaxIterations -> 10^4]
It is running for several hours with no result. Can any body suggest any specific way of minimizing such large scale expression? I also could not give initial points. It is always showing syntax error. Is it not possible to give one initial point only for each variable?
mathematical-optimization
asked 4 hours ago
Soumyajit RoySoumyajit Roy
1809
$endgroup$
I want to minimize a function which contains ArcTan functions. The function is also very complicated.
I have the following code.
LenFact = 1.3745/0.31571;
xC = 0.157392 LenFact; yC = 0.035495 LenFact;
q=x1[t],y1[t],x2[t],y2[t],x3[t],y3[t],x4[t],y4[t],x5[t],y5[t],x6[t],y6[t];
CE =
-0.609155 + x1[t]^2 + y1[t]^2,
-0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] + y2[t])^2,
-1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 + y2[t])^2,
-1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] + y6[t])^2,
-0.428366 + ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]],
-0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] + y4[t])^2,
-0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] + y5[t])^2,
-0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]],
-0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] + y6[t])^2,
-0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] + y3[t])^2,
-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]]
;
y6Start = 0.15 LenFact;
VPenalty=3.90351 + 1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 +
3006.52 (2.92648 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 +
1082.47 (2.78921 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] +
ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 +
6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] +
ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 +
239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] +
4.93484 y5[t] + 97.0584 y6[t] +
1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 +
ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 +
ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 +
x1[t]^2 +
y1[t]^2)^2 + (-1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 +
y2[t])^2)^2 + (-0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] +
y2[t])^2)^2 + (-0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] +
y3[t])^2)^2 + (-0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] +
y4[t])^2)^2 + (-0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] +
y5[t])^2)^2 + (-1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] +
y6[t])^2)^2 + (-0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] +
y6[t])^2)^2);
[Theta]1 = ArcTan[x1[t], y1[t]];
[Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
[Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
[Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
[Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
[Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
[Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];
eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0,
90 [Degree] < [Theta]1 < 180 [Degree],
20 [Degree] < [Theta]21 < 95 [Degree],
145 [Degree] < [Theta]3 < 180 [Degree],
80 [Degree] < [Theta]5 < 100 [Degree], 0 < y6[t] < y6Start,
q, Reals, MaxIterations -> 10^4]
It is running for several hours with no result. Can any body suggest any specific way of minimizing such large scale expression? I also could not give initial points. It is always showing syntax error. Is it not possible to give one initial point only for each variable?
mathematical-optimization
mathematical-optimization
asked 4 hours ago
Soumyajit RoySoumyajit Roy
1809
asked 4 hours ago
Soumyajit RoySoumyajit Roy
1809
edited 4 hours ago
Soumyajit Roy
asked 4 hours ago
Soumyajit RoySoumyajit Roy
1809
asked 4 hours ago
Soumyajit RoySoumyajit Roy
1809
asked 4 hours ago
Soumyajit RoySoumyajit Roy
1809
1809
$begingroup$
I am sorry. I have incorporated CE in the code now.
$endgroup$
– Soumyajit Roy
4 hours ago
$begingroup$
And what abouty6Start?
$endgroup$
– Henrik Schumacher
4 hours ago
1
$begingroup$
And after all: What is the context? This looks a bit like discrete bending energy minimization with some angle constraints... Maybe the problem can be rephrased into another one that is simpler to solve.
$endgroup$
– Henrik Schumacher
4 hours ago
$begingroup$
y6Start = 0.15 LenFact; This is about finding the static equilibrium of a mechanism. Static equilibrium states can be found by minimizing the potential energy function.
$endgroup$
– Soumyajit Roy
4 hours ago
add a comment |
$begingroup$
I am sorry. I have incorporated CE in the code now.
$endgroup$
– Soumyajit Roy
4 hours ago
$begingroup$
And what abouty6Start?
$endgroup$
– Henrik Schumacher
4 hours ago
1
$begingroup$
And after all: What is the context? This looks a bit like discrete bending energy minimization with some angle constraints... Maybe the problem can be rephrased into another one that is simpler to solve.
$endgroup$
– Henrik Schumacher
4 hours ago
$begingroup$
y6Start = 0.15 LenFact; This is about finding the static equilibrium of a mechanism. Static equilibrium states can be found by minimizing the potential energy function.
$endgroup$
– Soumyajit Roy
4 hours ago
$begingroup$
I am sorry. I have incorporated CE in the code now.
$endgroup$
– Soumyajit Roy
4 hours ago
$begingroup$
I am sorry. I have incorporated CE in the code now.
$endgroup$
– Soumyajit Roy
4 hours ago
$begingroup$
And what about
y6Start?$endgroup$
– Henrik Schumacher
4 hours ago
$begingroup$
And what about
y6Start?$endgroup$
– Henrik Schumacher
4 hours ago
1
1
$begingroup$
And after all: What is the context? This looks a bit like discrete bending energy minimization with some angle constraints... Maybe the problem can be rephrased into another one that is simpler to solve.
$endgroup$
– Henrik Schumacher
4 hours ago
$begingroup$
And after all: What is the context? This looks a bit like discrete bending energy minimization with some angle constraints... Maybe the problem can be rephrased into another one that is simpler to solve.
$endgroup$
– Henrik Schumacher
4 hours ago
$begingroup$
y6Start = 0.15 LenFact; This is about finding the static equilibrium of a mechanism. Static equilibrium states can be found by minimizing the potential energy function.
$endgroup$
– Soumyajit Roy
4 hours ago
$begingroup$
y6Start = 0.15 LenFact; This is about finding the static equilibrium of a mechanism. Static equilibrium states can be found by minimizing the potential energy function.
$endgroup$
– Soumyajit Roy
4 hours ago
add a comment |
2 Answers
2
active
oldest
votes
2
$begingroup$
I corrected typos. The code is fast and gives an answer, albeit with messages.
LenFact = 1.3745/0.31571;
xC = 0.157392 LenFact; yC = 0.035495 LenFact;
q = x1[t], y1[t], x2[t], y2[t], x3[t], y3[t], x4[t], y4[t], x5[t],
y5[t], x6[t],
y6[t]; CE = -0.609155 + x1[t]^2 +
y1[t]^2, -0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] +
y2[t])^2, -1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 +
y2[t])^2, -1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] +
y6[t])^2, -0.428366 +
ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] +
y6[t]], -0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] +
y4[t])^2, -0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] +
y5[t])^2, -0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] +
y5[t]], -0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] +
y6[t])^2, -0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] +
y3[t])^2, -ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]];
VPenalty = 3.90351 +
1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 +
3006.52 (2.92648 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 +
1082.47 (2.78921 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] +
ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 +
6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] +
ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 +
239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] +
4.93484 y5[t] + 97.0584 y6[t] +
1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 +
ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 +
ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 +
x1[t]^2 +
y1[t]^2)^2 + (-1.64323 + (-0.685234 +
x2[t])^2 + (-0.154534 +
y2[t])^2)^2 + (-0.0383062 + (-x1[t] +
x2[t])^2 + (-y1[t] +
y2[t])^2)^2 + (-0.0138516 + (-x2[t] +
x3[t])^2 + (-y2[t] +
y3[t])^2)^2 + (-0.783885 + (-x3[t] +
x4[t])^2 + (-y3[t] +
y4[t])^2)^2 + (-0.0353049 + (-x4[t] +
x5[t])^2 + (-y4[t] + y5[t])^2)^2 + (-1.88925 + (-x1[t] +
x6[t])^2 + (-y1[t] +
y6[t])^2)^2 + (-0.0228637 + (-x5[t] +
x6[t])^2 + (-y5[t] + y6[t])^2)^2);
[Theta]1 = ArcTan[x1[t], y1[t]];
[Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
[Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
[Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
[Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
[Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
[Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];
With[y6Start = 0.15 LenFact,
eqbr = NMinimize[VPenalty[[1, 1]],
Flatten[CE] == Table[0, Length[CE]], Pi/2 < [Theta]1 < Pi,
20*Pi/180 < [Theta]21 < 95*Pi/180, 145*Pi/180 < [Theta]3 < Pi,
80*Pi/180 < [Theta]5 < 100*Pi/180, 0 < y6[t] < y6Start, q,
Reals]]
(*1652.07, x1[t] -> -0.729157, y1[t] -> 0.278358, x2[t] -> -0.573507,
y2[t] -> 0.397027, x3[t] -> -0.457944, y3[t] -> 0.374765,
x4[t] -> 0.423225, y4[t] -> 0.460944, x5[t] -> 0.60875,
y5[t] -> 0.431194, x6[t] -> 0.611298, y6[t] -> 0.582379*)
answered 4 hours ago
Alex TrounevAlex Trounev
7,8831521
$endgroup$
$begingroup$
Thanks Alex. At least the constraints are matching to a reasonable degree. Although I am still not sure about the obtained value of Vtotal to be the global one.
$endgroup$
– Soumyajit Roy
4 hours ago
$begingroup$
HereVPenalty[]is a non-linear function, so the min can be local.
$endgroup$
– Alex Trounev
3 hours ago
add a comment |
2
$begingroup$
The method "RandomSearch" usually gives a result reasonably quickly (other than using FindMinimum). It's less robust, though, than the other methods. The following takes about 30 sec.
eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0,
90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °,
145 ° < θ3 < 180 °, 80 ° < θ5 < 100 °, 0 < y6[t] < y6Start, q,
Method -> "RandomSearch"]
NMinimize::incst: NMinimize was unable to generate any initial points satisfying the inequality constraints...
(*
161.292, x1[t] -> -0.718754, y1[t] -> 0.30422, x2[t] -> -0.567146,
y2[t] -> 0.427983, x3[t] -> -0.452162, y3[t] -> 0.402877,
x4[t] -> 0.424915, y4[t] -> 0.523793, x5[t] -> 0.611471,
y5[t] -> 0.501395, x6[t] -> 0.610863, y6[t] -> 0.652602
*)
While NMinimize did not find good starting points, it found a solution within the constraints:
90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °, 145 ° < θ3 < 180 °,
80 ° < θ5 < 100 °, 0 < y6[t] < y6Start /.
Last@eqbr
(* True, True, True, True, True *)
answered 3 hours ago
Michael E2Michael E2
149k12200479
$endgroup$
add a comment |
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2 Answers
2
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votes
2 Answers
2
active
oldest
votes
active
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active
oldest
votes
2
$begingroup$
I corrected typos. The code is fast and gives an answer, albeit with messages.
LenFact = 1.3745/0.31571;
xC = 0.157392 LenFact; yC = 0.035495 LenFact;
q = x1[t], y1[t], x2[t], y2[t], x3[t], y3[t], x4[t], y4[t], x5[t],
y5[t], x6[t],
y6[t]; CE = -0.609155 + x1[t]^2 +
y1[t]^2, -0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] +
y2[t])^2, -1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 +
y2[t])^2, -1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] +
y6[t])^2, -0.428366 +
ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] +
y6[t]], -0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] +
y4[t])^2, -0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] +
y5[t])^2, -0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] +
y5[t]], -0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] +
y6[t])^2, -0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] +
y3[t])^2, -ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]];
VPenalty = 3.90351 +
1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 +
3006.52 (2.92648 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 +
1082.47 (2.78921 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] +
ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 +
6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] +
ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 +
239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] +
4.93484 y5[t] + 97.0584 y6[t] +
1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 +
ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 +
ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 +
x1[t]^2 +
y1[t]^2)^2 + (-1.64323 + (-0.685234 +
x2[t])^2 + (-0.154534 +
y2[t])^2)^2 + (-0.0383062 + (-x1[t] +
x2[t])^2 + (-y1[t] +
y2[t])^2)^2 + (-0.0138516 + (-x2[t] +
x3[t])^2 + (-y2[t] +
y3[t])^2)^2 + (-0.783885 + (-x3[t] +
x4[t])^2 + (-y3[t] +
y4[t])^2)^2 + (-0.0353049 + (-x4[t] +
x5[t])^2 + (-y4[t] + y5[t])^2)^2 + (-1.88925 + (-x1[t] +
x6[t])^2 + (-y1[t] +
y6[t])^2)^2 + (-0.0228637 + (-x5[t] +
x6[t])^2 + (-y5[t] + y6[t])^2)^2);
[Theta]1 = ArcTan[x1[t], y1[t]];
[Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
[Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
[Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
[Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
[Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
[Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];
With[y6Start = 0.15 LenFact,
eqbr = NMinimize[VPenalty[[1, 1]],
Flatten[CE] == Table[0, Length[CE]], Pi/2 < [Theta]1 < Pi,
20*Pi/180 < [Theta]21 < 95*Pi/180, 145*Pi/180 < [Theta]3 < Pi,
80*Pi/180 < [Theta]5 < 100*Pi/180, 0 < y6[t] < y6Start, q,
Reals]]
(*1652.07, x1[t] -> -0.729157, y1[t] -> 0.278358, x2[t] -> -0.573507,
y2[t] -> 0.397027, x3[t] -> -0.457944, y3[t] -> 0.374765,
x4[t] -> 0.423225, y4[t] -> 0.460944, x5[t] -> 0.60875,
y5[t] -> 0.431194, x6[t] -> 0.611298, y6[t] -> 0.582379*)
answered 4 hours ago
Alex TrounevAlex Trounev
7,8831521
$endgroup$
$begingroup$
Thanks Alex. At least the constraints are matching to a reasonable degree. Although I am still not sure about the obtained value of Vtotal to be the global one.
$endgroup$
– Soumyajit Roy
4 hours ago
$begingroup$
HereVPenalty[]is a non-linear function, so the min can be local.
$endgroup$
– Alex Trounev
3 hours ago
add a comment |
2
$begingroup$
I corrected typos. The code is fast and gives an answer, albeit with messages.
LenFact = 1.3745/0.31571;
xC = 0.157392 LenFact; yC = 0.035495 LenFact;
q = x1[t], y1[t], x2[t], y2[t], x3[t], y3[t], x4[t], y4[t], x5[t],
y5[t], x6[t],
y6[t]; CE = -0.609155 + x1[t]^2 +
y1[t]^2, -0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] +
y2[t])^2, -1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 +
y2[t])^2, -1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] +
y6[t])^2, -0.428366 +
ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] +
y6[t]], -0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] +
y4[t])^2, -0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] +
y5[t])^2, -0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] +
y5[t]], -0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] +
y6[t])^2, -0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] +
y3[t])^2, -ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]];
VPenalty = 3.90351 +
1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 +
3006.52 (2.92648 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 +
1082.47 (2.78921 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] +
ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 +
6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] +
ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 +
239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] +
4.93484 y5[t] + 97.0584 y6[t] +
1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 +
ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 +
ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 +
x1[t]^2 +
y1[t]^2)^2 + (-1.64323 + (-0.685234 +
x2[t])^2 + (-0.154534 +
y2[t])^2)^2 + (-0.0383062 + (-x1[t] +
x2[t])^2 + (-y1[t] +
y2[t])^2)^2 + (-0.0138516 + (-x2[t] +
x3[t])^2 + (-y2[t] +
y3[t])^2)^2 + (-0.783885 + (-x3[t] +
x4[t])^2 + (-y3[t] +
y4[t])^2)^2 + (-0.0353049 + (-x4[t] +
x5[t])^2 + (-y4[t] + y5[t])^2)^2 + (-1.88925 + (-x1[t] +
x6[t])^2 + (-y1[t] +
y6[t])^2)^2 + (-0.0228637 + (-x5[t] +
x6[t])^2 + (-y5[t] + y6[t])^2)^2);
[Theta]1 = ArcTan[x1[t], y1[t]];
[Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
[Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
[Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
[Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
[Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
[Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];
With[y6Start = 0.15 LenFact,
eqbr = NMinimize[VPenalty[[1, 1]],
Flatten[CE] == Table[0, Length[CE]], Pi/2 < [Theta]1 < Pi,
20*Pi/180 < [Theta]21 < 95*Pi/180, 145*Pi/180 < [Theta]3 < Pi,
80*Pi/180 < [Theta]5 < 100*Pi/180, 0 < y6[t] < y6Start, q,
Reals]]
(*1652.07, x1[t] -> -0.729157, y1[t] -> 0.278358, x2[t] -> -0.573507,
y2[t] -> 0.397027, x3[t] -> -0.457944, y3[t] -> 0.374765,
x4[t] -> 0.423225, y4[t] -> 0.460944, x5[t] -> 0.60875,
y5[t] -> 0.431194, x6[t] -> 0.611298, y6[t] -> 0.582379*)
answered 4 hours ago
Alex TrounevAlex Trounev
7,8831521
$endgroup$
$begingroup$
Thanks Alex. At least the constraints are matching to a reasonable degree. Although I am still not sure about the obtained value of Vtotal to be the global one.
$endgroup$
– Soumyajit Roy
4 hours ago
$begingroup$
HereVPenalty[]is a non-linear function, so the min can be local.
$endgroup$
– Alex Trounev
3 hours ago
add a comment |
2
2
2
$begingroup$
I corrected typos. The code is fast and gives an answer, albeit with messages.
LenFact = 1.3745/0.31571;
xC = 0.157392 LenFact; yC = 0.035495 LenFact;
q = x1[t], y1[t], x2[t], y2[t], x3[t], y3[t], x4[t], y4[t], x5[t],
y5[t], x6[t],
y6[t]; CE = -0.609155 + x1[t]^2 +
y1[t]^2, -0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] +
y2[t])^2, -1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 +
y2[t])^2, -1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] +
y6[t])^2, -0.428366 +
ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] +
y6[t]], -0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] +
y4[t])^2, -0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] +
y5[t])^2, -0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] +
y5[t]], -0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] +
y6[t])^2, -0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] +
y3[t])^2, -ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]];
VPenalty = 3.90351 +
1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 +
3006.52 (2.92648 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 +
1082.47 (2.78921 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] +
ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 +
6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] +
ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 +
239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] +
4.93484 y5[t] + 97.0584 y6[t] +
1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 +
ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 +
ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 +
x1[t]^2 +
y1[t]^2)^2 + (-1.64323 + (-0.685234 +
x2[t])^2 + (-0.154534 +
y2[t])^2)^2 + (-0.0383062 + (-x1[t] +
x2[t])^2 + (-y1[t] +
y2[t])^2)^2 + (-0.0138516 + (-x2[t] +
x3[t])^2 + (-y2[t] +
y3[t])^2)^2 + (-0.783885 + (-x3[t] +
x4[t])^2 + (-y3[t] +
y4[t])^2)^2 + (-0.0353049 + (-x4[t] +
x5[t])^2 + (-y4[t] + y5[t])^2)^2 + (-1.88925 + (-x1[t] +
x6[t])^2 + (-y1[t] +
y6[t])^2)^2 + (-0.0228637 + (-x5[t] +
x6[t])^2 + (-y5[t] + y6[t])^2)^2);
[Theta]1 = ArcTan[x1[t], y1[t]];
[Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
[Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
[Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
[Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
[Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
[Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];
With[y6Start = 0.15 LenFact,
eqbr = NMinimize[VPenalty[[1, 1]],
Flatten[CE] == Table[0, Length[CE]], Pi/2 < [Theta]1 < Pi,
20*Pi/180 < [Theta]21 < 95*Pi/180, 145*Pi/180 < [Theta]3 < Pi,
80*Pi/180 < [Theta]5 < 100*Pi/180, 0 < y6[t] < y6Start, q,
Reals]]
(*1652.07, x1[t] -> -0.729157, y1[t] -> 0.278358, x2[t] -> -0.573507,
y2[t] -> 0.397027, x3[t] -> -0.457944, y3[t] -> 0.374765,
x4[t] -> 0.423225, y4[t] -> 0.460944, x5[t] -> 0.60875,
y5[t] -> 0.431194, x6[t] -> 0.611298, y6[t] -> 0.582379*)
answered 4 hours ago
Alex TrounevAlex Trounev
7,8831521
$endgroup$
I corrected typos. The code is fast and gives an answer, albeit with messages.
LenFact = 1.3745/0.31571;
xC = 0.157392 LenFact; yC = 0.035495 LenFact;
q = x1[t], y1[t], x2[t], y2[t], x3[t], y3[t], x4[t], y4[t], x5[t],
y5[t], x6[t],
y6[t]; CE = -0.609155 + x1[t]^2 +
y1[t]^2, -0.0383062 + (-x1[t] + x2[t])^2 + (-y1[t] +
y2[t])^2, -1.64323 + (-0.685234 + x2[t])^2 + (-0.154534 +
y2[t])^2, -1.88925 + (-x1[t] + x6[t])^2 + (-y1[t] +
y6[t])^2, -0.428366 +
ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] +
y6[t]], -0.783885 + (-x3[t] + x4[t])^2 + (-y3[t] +
y4[t])^2, -0.0353049 + (-x4[t] + x5[t])^2 + (-y4[t] +
y5[t])^2, -0.25649 + ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] +
y5[t]], -0.0228637 + (-x5[t] + x6[t])^2 + (-y5[t] +
y6[t])^2, -0.0138516 + (-x2[t] + x3[t])^2 + (-y2[t] +
y3[t])^2, -ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]];
VPenalty = 3.90351 +
1.13817*10^6 (2.74102 - 1. ArcTan[x1[t], y1[t]])^2 +
3006.52 (2.92648 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]])^2 +
1082.47 (2.78921 -
1. ArcTan[-0.685234 + x3[t], -0.154534 + y3[t]] +
ArcTan[-1. x3[t] + x4[t], -1. y3[t] + y4[t]])^2 +
6825.76 (2.48452 - 1. ArcTan[x1[t], y1[t]] +
ArcTan[-1. x1[t] + x6[t], -1. y1[t] + y6[t]])^2 +
239.418 y1[t] + 29.1166 y2[t] + 6.00899 y3[t] + 7.28423 y4[t] +
4.93484 y5[t] + 97.0584 y6[t] +
1.90484*10^8 ((-ArcTan[0.685234 - x2[t], 0.154534 - y2[t]] +
ArcTan[-x2[t] + x3[t], -y2[t] + y3[t]])^2 + (-0.25649 +
ArcTan[-x3[t] + x4[t], -y3[t] + y4[t]] -
ArcTan[-x4[t] + x5[t], -y4[t] + y5[t]])^2 + (-0.428366 +
ArcTan[-x1[t] + x2[t], -y1[t] + y2[t]] -
ArcTan[-x1[t] + x6[t], -y1[t] + y6[t]])^2 + (-0.609155 +
x1[t]^2 +
y1[t]^2)^2 + (-1.64323 + (-0.685234 +
x2[t])^2 + (-0.154534 +
y2[t])^2)^2 + (-0.0383062 + (-x1[t] +
x2[t])^2 + (-y1[t] +
y2[t])^2)^2 + (-0.0138516 + (-x2[t] +
x3[t])^2 + (-y2[t] +
y3[t])^2)^2 + (-0.783885 + (-x3[t] +
x4[t])^2 + (-y3[t] +
y4[t])^2)^2 + (-0.0353049 + (-x4[t] +
x5[t])^2 + (-y4[t] + y5[t])^2)^2 + (-1.88925 + (-x1[t] +
x6[t])^2 + (-y1[t] +
y6[t])^2)^2 + (-0.0228637 + (-x5[t] +
x6[t])^2 + (-y5[t] + y6[t])^2)^2);
[Theta]1 = ArcTan[x1[t], y1[t]];
[Theta]21 = ArcTan[x2[t] - x1[t], y2[t] - y1[t]];
[Theta]3 = ArcTan[x3[t] - xC, y3[t] - yC];
[Theta]22 = ArcTan[x6[t] - x1[t], y6[t] - y1[t]];
[Theta]41 = ArcTan[x4[t] - x3[t], y4[t] - y3[t]];
[Theta]42 = ArcTan[x5[t] - x4[t], y5[t] - y4[t]];
[Theta]5 = ArcTan[x6[t] - x5[t], y6[t] - y5[t]];
With[y6Start = 0.15 LenFact,
eqbr = NMinimize[VPenalty[[1, 1]],
Flatten[CE] == Table[0, Length[CE]], Pi/2 < [Theta]1 < Pi,
20*Pi/180 < [Theta]21 < 95*Pi/180, 145*Pi/180 < [Theta]3 < Pi,
80*Pi/180 < [Theta]5 < 100*Pi/180, 0 < y6[t] < y6Start, q,
Reals]]
(*1652.07, x1[t] -> -0.729157, y1[t] -> 0.278358, x2[t] -> -0.573507,
y2[t] -> 0.397027, x3[t] -> -0.457944, y3[t] -> 0.374765,
x4[t] -> 0.423225, y4[t] -> 0.460944, x5[t] -> 0.60875,
y5[t] -> 0.431194, x6[t] -> 0.611298, y6[t] -> 0.582379*)
answered 4 hours ago
Alex TrounevAlex Trounev
7,8831521
answered 4 hours ago
Alex TrounevAlex Trounev
7,8831521
answered 4 hours ago
Alex TrounevAlex Trounev
7,8831521
answered 4 hours ago
Alex TrounevAlex Trounev
7,8831521
7,8831521
$begingroup$
Thanks Alex. At least the constraints are matching to a reasonable degree. Although I am still not sure about the obtained value of Vtotal to be the global one.
$endgroup$
– Soumyajit Roy
4 hours ago
$begingroup$
HereVPenalty[]is a non-linear function, so the min can be local.
$endgroup$
– Alex Trounev
3 hours ago
add a comment |
$begingroup$
Thanks Alex. At least the constraints are matching to a reasonable degree. Although I am still not sure about the obtained value of Vtotal to be the global one.
$endgroup$
– Soumyajit Roy
4 hours ago
$begingroup$
HereVPenalty[]is a non-linear function, so the min can be local.
$endgroup$
– Alex Trounev
3 hours ago
$begingroup$
Thanks Alex. At least the constraints are matching to a reasonable degree. Although I am still not sure about the obtained value of Vtotal to be the global one.
$endgroup$
– Soumyajit Roy
4 hours ago
$begingroup$
Thanks Alex. At least the constraints are matching to a reasonable degree. Although I am still not sure about the obtained value of Vtotal to be the global one.
$endgroup$
– Soumyajit Roy
4 hours ago
$begingroup$
Here
VPenalty[] is a non-linear function, so the min can be local.$endgroup$
– Alex Trounev
3 hours ago
$begingroup$
Here
VPenalty[] is a non-linear function, so the min can be local.$endgroup$
– Alex Trounev
3 hours ago
add a comment |
2
$begingroup$
The method "RandomSearch" usually gives a result reasonably quickly (other than using FindMinimum). It's less robust, though, than the other methods. The following takes about 30 sec.
eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0,
90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °,
145 ° < θ3 < 180 °, 80 ° < θ5 < 100 °, 0 < y6[t] < y6Start, q,
Method -> "RandomSearch"]
NMinimize::incst: NMinimize was unable to generate any initial points satisfying the inequality constraints...
(*
161.292, x1[t] -> -0.718754, y1[t] -> 0.30422, x2[t] -> -0.567146,
y2[t] -> 0.427983, x3[t] -> -0.452162, y3[t] -> 0.402877,
x4[t] -> 0.424915, y4[t] -> 0.523793, x5[t] -> 0.611471,
y5[t] -> 0.501395, x6[t] -> 0.610863, y6[t] -> 0.652602
*)
While NMinimize did not find good starting points, it found a solution within the constraints:
90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °, 145 ° < θ3 < 180 °,
80 ° < θ5 < 100 °, 0 < y6[t] < y6Start /.
Last@eqbr
(* True, True, True, True, True *)
answered 3 hours ago
Michael E2Michael E2
149k12200479
$endgroup$
add a comment |
2
$begingroup$
The method "RandomSearch" usually gives a result reasonably quickly (other than using FindMinimum). It's less robust, though, than the other methods. The following takes about 30 sec.
eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0,
90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °,
145 ° < θ3 < 180 °, 80 ° < θ5 < 100 °, 0 < y6[t] < y6Start, q,
Method -> "RandomSearch"]
NMinimize::incst: NMinimize was unable to generate any initial points satisfying the inequality constraints...
(*
161.292, x1[t] -> -0.718754, y1[t] -> 0.30422, x2[t] -> -0.567146,
y2[t] -> 0.427983, x3[t] -> -0.452162, y3[t] -> 0.402877,
x4[t] -> 0.424915, y4[t] -> 0.523793, x5[t] -> 0.611471,
y5[t] -> 0.501395, x6[t] -> 0.610863, y6[t] -> 0.652602
*)
While NMinimize did not find good starting points, it found a solution within the constraints:
90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °, 145 ° < θ3 < 180 °,
80 ° < θ5 < 100 °, 0 < y6[t] < y6Start /.
Last@eqbr
(* True, True, True, True, True *)
answered 3 hours ago
Michael E2Michael E2
149k12200479
$endgroup$
add a comment |
2
2
2
$begingroup$
The method "RandomSearch" usually gives a result reasonably quickly (other than using FindMinimum). It's less robust, though, than the other methods. The following takes about 30 sec.
eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0,
90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °,
145 ° < θ3 < 180 °, 80 ° < θ5 < 100 °, 0 < y6[t] < y6Start, q,
Method -> "RandomSearch"]
NMinimize::incst: NMinimize was unable to generate any initial points satisfying the inequality constraints...
(*
161.292, x1[t] -> -0.718754, y1[t] -> 0.30422, x2[t] -> -0.567146,
y2[t] -> 0.427983, x3[t] -> -0.452162, y3[t] -> 0.402877,
x4[t] -> 0.424915, y4[t] -> 0.523793, x5[t] -> 0.611471,
y5[t] -> 0.501395, x6[t] -> 0.610863, y6[t] -> 0.652602
*)
While NMinimize did not find good starting points, it found a solution within the constraints:
90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °, 145 ° < θ3 < 180 °,
80 ° < θ5 < 100 °, 0 < y6[t] < y6Start /.
Last@eqbr
(* True, True, True, True, True *)
answered 3 hours ago
Michael E2Michael E2
149k12200479
$endgroup$
The method "RandomSearch" usually gives a result reasonably quickly (other than using FindMinimum). It's less robust, though, than the other methods. The following takes about 30 sec.
eqbr = NMinimize[VPenalty[[1, 1]], Flatten[CE] == 0,
90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °,
145 ° < θ3 < 180 °, 80 ° < θ5 < 100 °, 0 < y6[t] < y6Start, q,
Method -> "RandomSearch"]
NMinimize::incst: NMinimize was unable to generate any initial points satisfying the inequality constraints...
(*
161.292, x1[t] -> -0.718754, y1[t] -> 0.30422, x2[t] -> -0.567146,
y2[t] -> 0.427983, x3[t] -> -0.452162, y3[t] -> 0.402877,
x4[t] -> 0.424915, y4[t] -> 0.523793, x5[t] -> 0.611471,
y5[t] -> 0.501395, x6[t] -> 0.610863, y6[t] -> 0.652602
*)
While NMinimize did not find good starting points, it found a solution within the constraints:
90 ° < θ1 < 180 °, 20 ° < θ21 < 95 °, 145 ° < θ3 < 180 °,
80 ° < θ5 < 100 °, 0 < y6[t] < y6Start /.
Last@eqbr
(* True, True, True, True, True *)
answered 3 hours ago
Michael E2Michael E2
149k12200479
answered 3 hours ago
Michael E2Michael E2
149k12200479
answered 3 hours ago
Michael E2Michael E2
149k12200479
answered 3 hours ago
Michael E2Michael E2
149k12200479
149k12200479
add a comment |
add a comment |
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$begingroup$
I am sorry. I have incorporated CE in the code now.
$endgroup$
– Soumyajit Roy
4 hours ago
$begingroup$
And what about
y6Start?$endgroup$
– Henrik Schumacher
4 hours ago
1
$begingroup$
And after all: What is the context? This looks a bit like discrete bending energy minimization with some angle constraints... Maybe the problem can be rephrased into another one that is simpler to solve.
$endgroup$
– Henrik Schumacher
4 hours ago
$begingroup$
y6Start = 0.15 LenFact; This is about finding the static equilibrium of a mechanism. Static equilibrium states can be found by minimizing the potential energy function.
$endgroup$
– Soumyajit Roy
4 hours ago