R = [
  1.0000000e-01,  0.0000000e+00,  0.0000000e+00,  
  0.0000000e+00,  1.0000000e-01,  0.0000000e+00,  
  0.0000000e+00,  0.0000000e+00,  1.0000000e-01,  
];

M = [
  0.0000000e+00,  1.0000000e+00,  1.0000000e+00,  
  1.0000000e+00,  0.0000000e+00,  1.0000000e+00,  
  1.0000000e+00,  1.0000000e+00,  0.0000000e+00,  
];

C = [
  4.0800000e+02,  0.0000000e+00,  0.0000000e+00,  
  0.0000000e+00,  4.0800000e+02,  0.0000000e+00,  
  0.0000000e+00,  0.0000000e+00,  4.0800000e+02,  
];

Np = [
  2.0000000e+00,  
];

Vmax = [
  6.0000000e+02,  
];


% data for inverse flux characteristic for fixed theta,
% ie, F = f_1^{-1}(psi, theta) 

psi_vec = [
  0.0000000e+00,  4.7899357e-04,  9.5798713e-04,  1.4369807e-03,  1.9159743e-03,  2.3949678e-03,  2.8739614e-03,  3.3529550e-03,  3.8319485e-03,  4.3109421e-03,  
];

F_mat = [
  0.0000000e+00,  4.4007822e+02,  9.0419641e+02,  1.3801058e+03,  1.9123314e+03,  2.0273319e+03,  3.4046301e+02,  -4.5508225e+03,  -1.4049072e+04,  -2.9556832e+04,  
  0.0000000e+00,  4.2517851e+02,  8.4624724e+02,  1.2937893e+03,  1.8352913e+03,  2.2753368e+03,  1.4116750e+03,  -2.0160241e+03,  -9.2680906e+03,  -2.1604855e+04,  
  0.0000000e+00,  3.0308410e+02,  6.4062747e+02,  1.0418604e+03,  1.4533311e+03,  -8.9628239e+02,  -1.9749976e+04,  -7.1594354e+04,  -1.7291967e+05,  -3.4021617e+05,  
  0.0000000e+00,  1.7444030e+02,  3.7638116e+02,  6.6325394e+02,  1.0436851e+03,  1.4726391e+03,  2.0900968e+03,  2.1283471e+03,  3.0949317e+02,  -4.6443620e+03,  
  0.0000000e+00,  1.1748755e+02,  2.3818636e+02,  3.9708086e+02,  6.3017490e+02,  9.6920977e+02,  1.4701870e+03,  7.8783632e+02,  -4.7371124e+03,  -1.8788238e+04,  
  0.0000000e+00,  8.9529633e+01,  1.8167793e+02,  2.8172015e+02,  4.0888342e+02,  5.9250626e+02,  9.0163761e+02,  1.3321668e+03,  -3.6748371e+04,  -4.5764816e+05,  
  0.0000000e+00,  7.0585099e+01,  1.4617116e+02,  2.2443220e+02,  3.1518185e+02,  4.2964619e+02,  5.8263245e+02,  8.7504215e+02,  1.3921557e+03,  -1.3481517e+04,  
  0.0000000e+00,  5.6427758e+01,  1.2173607e+02,  1.8357302e+02,  2.5617049e+02,  3.4408913e+02,  4.5229072e+02,  6.1902197e+02,  1.0229172e+03,  1.6118756e+03,  
  0.0000000e+00,  5.7301740e+01,  1.0635094e+02,  1.6604688e+02,  2.2500887e+02,  3.0156504e+02,  4.0692108e+02,  5.5201709e+02,  9.0334330e+02,  1.5090426e+03,  
  0.0000000e+00,  5.6427758e+01,  1.2173607e+02,  1.8357302e+02,  2.5617049e+02,  3.4408913e+02,  4.5229072e+02,  6.1902197e+02,  1.0229172e+03,  1.6118756e+03,  
  0.0000000e+00,  7.0585099e+01,  1.4617116e+02,  2.2443220e+02,  3.1518185e+02,  4.2964619e+02,  5.8263245e+02,  8.7504215e+02,  1.3921557e+03,  -1.3481517e+04,  
  0.0000000e+00,  8.9529633e+01,  1.8167793e+02,  2.8172015e+02,  4.0888342e+02,  5.9250626e+02,  9.0163761e+02,  1.3321668e+03,  -3.6748371e+04,  -4.5764816e+05,  
  0.0000000e+00,  1.1748755e+02,  2.3818636e+02,  3.9708086e+02,  6.3017490e+02,  9.6920977e+02,  1.4701870e+03,  7.8783632e+02,  -4.7371124e+03,  -1.8788238e+04,  
  0.0000000e+00,  1.7444030e+02,  3.7638116e+02,  6.6325394e+02,  1.0436851e+03,  1.4726391e+03,  2.0900968e+03,  2.1283471e+03,  3.0949317e+02,  -4.6443620e+03,  
  0.0000000e+00,  3.0308410e+02,  6.4062747e+02,  1.0418604e+03,  1.4533311e+03,  -8.9628239e+02,  -1.9749976e+04,  -7.1594354e+04,  -1.7291967e+05,  -3.4021617e+05,  
  0.0000000e+00,  4.2517851e+02,  8.4624724e+02,  1.2937893e+03,  1.8352913e+03,  2.2753368e+03,  1.4116750e+03,  -2.0160241e+03,  -9.2680906e+03,  -2.1604855e+04,  
  0.0000000e+00,  4.4007822e+02,  9.0419641e+02,  1.3801058e+03,  1.9123314e+03,  2.0273319e+03,  3.4046301e+02,  -4.5508225e+03,  -1.4049072e+04,  -2.9556832e+04,  
];

theta_vec = [
  0.0000000e+00,  1.9634954e-01,  3.9269908e-01,  5.8904862e-01,  7.8539816e-01,  9.8174770e-01,  1.1780972e+00,  1.3744468e+00,  1.5707963e+00,  1.7671459e+00,  1.9634954e+00,  2.1598449e+00,  2.3561945e+00,  2.5525440e+00,  2.7488936e+00,  2.9452431e+00,  3.1415927e+00,  
];


% data for flux characteristic function, psi = f_1(F, theta)

F_vec = [
  -6.1200000e+03,  -5.9951020e+03,  -5.8702041e+03,  -5.7453061e+03,  -5.6204082e+03,  -5.4955102e+03,  -5.3706122e+03,  -5.2457143e+03,  -5.1208163e+03,  -4.9959184e+03,  -4.8710204e+03,  -4.7461224e+03,  -4.6212245e+03,  -4.4963265e+03,  -4.3714286e+03,  -4.2465306e+03,  -4.1216327e+03,  -3.9967347e+03,  -3.8718367e+03,  -3.7469388e+03,  -3.6220408e+03,  -3.4971429e+03,  -3.3722449e+03,  -3.2473469e+03,  -3.1224490e+03,  -2.9975510e+03,  -2.8726531e+03,  -2.7477551e+03,  -2.6228571e+03,  -2.4979592e+03,  -2.3730612e+03,  -2.2481633e+03,  -2.1232653e+03,  -1.9983673e+03,  -1.8734694e+03,  -1.7485714e+03,  -1.6236735e+03,  -1.4987755e+03,  -1.3738776e+03,  -1.2489796e+03,  -1.1240816e+03,  -9.9918367e+02,  -8.7428571e+02,  -7.4938776e+02,  -6.2448980e+02,  -4.9959184e+02,  -3.7469388e+02,  -2.4979592e+02,  -1.2489796e+02,  0.0000000e+00,  1.2489796e+02,  2.4979592e+02,  3.7469388e+02,  4.9959184e+02,  6.2448980e+02,  7.4938776e+02,  8.7428571e+02,  9.9918367e+02,  1.1240816e+03,  1.2489796e+03,  1.3738776e+03,  1.4987755e+03,  1.6236735e+03,  1.7485714e+03,  1.8734694e+03,  1.9983673e+03,  2.1232653e+03,  2.2481633e+03,  2.3730612e+03,  2.4979592e+03,  2.6228571e+03,  2.7477551e+03,  2.8726531e+03,  2.9975510e+03,  3.1224490e+03,  3.2473469e+03,  3.3722449e+03,  3.4971429e+03,  3.6220408e+03,  3.7469388e+03,  3.8718367e+03,  3.9967347e+03,  4.1216327e+03,  4.2465306e+03,  4.3714286e+03,  4.4963265e+03,  4.6212245e+03,  4.7461224e+03,  4.8710204e+03,  4.9959184e+03,  5.1208163e+03,  5.2457143e+03,  5.3706122e+03,  5.4955102e+03,  5.6204082e+03,  5.7453061e+03,  5.8702041e+03,  5.9951020e+03,  6.1200000e+03,  
];

psi_mat = [
  -1.5983192e-03,  -1.5686817e-03,  -1.5350885e-03,  -1.4981301e-03,  -1.4694098e-03,  -1.4272047e-03,  -1.3876380e-03,  -1.3562909e-03,  -1.3273596e-03,  -1.2973860e-03,  -1.2478885e-03,  -1.2191453e-03,  -1.1863360e-03,  -1.1559684e-03,  -1.1248880e-03,  -1.0924705e-03,  -1.0630595e-03,  -1.0371100e-03,  -1.0162568e-03,  -9.8553393e-04,  -9.5900649e-04,  -9.3410077e-04,  -9.0998148e-04,  -8.8581333e-04,  -8.6076102e-04,  -8.3400026e-04,  -8.0543947e-04,  -7.5817398e-04,  -7.2081514e-04,  -6.9288740e-04,  -6.6210136e-04,  -6.1114084e-04,  -5.8102200e-04,  -5.4254725e-04,  -5.0665943e-04,  -4.7626745e-04,  -4.4580096e-04,  -4.1298581e-04,  -3.8361176e-04,  -3.5089721e-04,  -3.1959504e-04,  -2.8732249e-04,  -2.4255500e-04,  -2.0545779e-04,  -1.7494001e-04,  -1.4475445e-04,  -1.1057026e-04,  -7.4911470e-05,  -3.8121959e-05,  0.0000000e+00,  3.8121959e-05,  7.4911470e-05,  1.1057026e-04,  1.4475445e-04,  1.7494001e-04,  2.0545779e-04,  2.4255500e-04,  2.8732249e-04,  3.1959504e-04,  3.5089721e-04,  3.8361176e-04,  4.1298581e-04,  4.4580096e-04,  4.7626745e-04,  5.0665943e-04,  5.4254725e-04,  5.8102200e-04,  6.1114084e-04,  6.6210136e-04,  6.9288740e-04,  7.2081514e-04,  7.5817398e-04,  8.0543947e-04,  8.3400026e-04,  8.6076102e-04,  8.8581333e-04,  9.0998148e-04,  9.3410077e-04,  9.5900649e-04,  9.8553393e-04,  1.0162568e-03,  1.0371100e-03,  1.0630595e-03,  1.0924705e-03,  1.1248880e-03,  1.1559684e-03,  1.1863360e-03,  1.2191453e-03,  1.2478885e-03,  1.2973860e-03,  1.3273596e-03,  1.3562909e-03,  1.3876380e-03,  1.4272047e-03,  1.4694098e-03,  1.4981301e-03,  1.5350885e-03,  1.5686817e-03,  1.5983192e-03,  
  -1.6847983e-03,  -1.6569041e-03,  -1.6262104e-03,  -1.5931627e-03,  -1.5641794e-03,  -1.5252363e-03,  -1.4884573e-03,  -1.4588862e-03,  -1.4184528e-03,  -1.3804910e-03,  -1.3513909e-03,  -1.3197567e-03,  -1.2726839e-03,  -1.2325246e-03,  -1.2016725e-03,  -1.1729330e-03,  -1.1479719e-03,  -1.1157669e-03,  -1.0792570e-03,  -1.0463195e-03,  -1.0149489e-03,  -9.8486762e-04,  -9.5510063e-04,  -9.2467003e-04,  -8.9259784e-04,  -8.5774939e-04,  -8.1902331e-04,  -7.7822764e-04,  -7.3760325e-04,  -6.9938246e-04,  -6.6388453e-04,  -6.2979639e-04,  -5.9634412e-04,  -5.6275378e-04,  -5.2825489e-04,  -4.9274324e-04,  -4.5676218e-04,  -4.2071335e-04,  -3.8494812e-04,  -3.4926100e-04,  -3.1360259e-04,  -2.7802856e-04,  -2.4259456e-04,  -2.0735623e-04,  -1.7235488e-04,  -1.3753933e-04,  -1.0289689e-04,  -6.8426740e-05,  -3.4128052e-05,  0.0000000e+00,  3.4128052e-05,  6.8426740e-05,  1.0289689e-04,  1.3753933e-04,  1.7235488e-04,  2.0735623e-04,  2.4259456e-04,  2.7802856e-04,  3.1360259e-04,  3.4926100e-04,  3.8494812e-04,  4.2071335e-04,  4.5676218e-04,  4.9274324e-04,  5.2825489e-04,  5.6275378e-04,  5.9634412e-04,  6.2979639e-04,  6.6388453e-04,  6.9938246e-04,  7.3760325e-04,  7.7822764e-04,  8.1902331e-04,  8.5774939e-04,  8.9259784e-04,  9.2467003e-04,  9.5510063e-04,  9.8486762e-04,  1.0149489e-03,  1.0463195e-03,  1.0792570e-03,  1.1157669e-03,  1.1479719e-03,  1.1729330e-03,  1.2016725e-03,  1.2325246e-03,  1.2726839e-03,  1.3197567e-03,  1.3513909e-03,  1.3804910e-03,  1.4184528e-03,  1.4588862e-03,  1.4884573e-03,  1.5252363e-03,  1.5641794e-03,  1.5931627e-03,  1.6262104e-03,  1.6569041e-03,  1.6847983e-03,  
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  -2.0330581e-03,  -1.9733051e-03,  -1.9294029e-03,  -1.9007906e-03,  -1.8725403e-03,  -1.8285258e-03,  -1.7860881e-03,  -1.7558724e-03,  -1.7249841e-03,  -1.6691749e-03,  -1.6395490e-03,  -1.6115999e-03,  -1.5720407e-03,  -1.5244892e-03,  -1.4903905e-03,  -1.4609703e-03,  -1.4215413e-03,  -1.3821742e-03,  -1.3495645e-03,  -1.3041229e-03,  -1.2702954e-03,  -1.2406302e-03,  -1.2048555e-03,  -1.1593283e-03,  -1.1251280e-03,  -1.1033568e-03,  -1.0544822e-03,  -1.0218459e-03,  -9.8357430e-04,  -9.3310319e-04,  -8.9023063e-04,  -8.6181552e-04,  -8.1598462e-04,  -7.7063250e-04,  -7.2729318e-04,  -6.7901884e-04,  -6.3580035e-04,  -6.0648281e-04,  -5.3895791e-04,  -4.9065801e-04,  -4.5595966e-04,  -4.1055166e-04,  -3.5815467e-04,  -3.1377485e-04,  -2.6169120e-04,  -1.9601523e-04,  -1.3981914e-04,  -8.7451206e-05,  -4.0361479e-05,  0.0000000e+00,  4.0361479e-05,  8.7451206e-05,  1.3981914e-04,  1.9601523e-04,  2.6169120e-04,  3.1377485e-04,  3.5815467e-04,  4.1055166e-04,  4.5595966e-04,  4.9065801e-04,  5.3895791e-04,  6.0648281e-04,  6.3580035e-04,  6.7901884e-04,  7.2729318e-04,  7.7063250e-04,  8.1598462e-04,  8.6181552e-04,  8.9023063e-04,  9.3310319e-04,  9.8357430e-04,  1.0218459e-03,  1.0544822e-03,  1.1033568e-03,  1.1251280e-03,  1.1593283e-03,  1.2048555e-03,  1.2406302e-03,  1.2702954e-03,  1.3041229e-03,  1.3495645e-03,  1.3821742e-03,  1.4215413e-03,  1.4609703e-03,  1.4903905e-03,  1.5244892e-03,  1.5720407e-03,  1.6115999e-03,  1.6395490e-03,  1.6691749e-03,  1.7249841e-03,  1.7558724e-03,  1.7860881e-03,  1.8285258e-03,  1.8725403e-03,  1.9007906e-03,  1.9294029e-03,  1.9733051e-03,  2.0330581e-03,  
  -1.6847983e-03,  -1.6569041e-03,  -1.6262104e-03,  -1.5931627e-03,  -1.5641794e-03,  -1.5252363e-03,  -1.4884573e-03,  -1.4588862e-03,  -1.4184528e-03,  -1.3804910e-03,  -1.3513909e-03,  -1.3197567e-03,  -1.2726839e-03,  -1.2325246e-03,  -1.2016725e-03,  -1.1729330e-03,  -1.1479719e-03,  -1.1157669e-03,  -1.0792570e-03,  -1.0463195e-03,  -1.0149489e-03,  -9.8486762e-04,  -9.5510063e-04,  -9.2467003e-04,  -8.9259784e-04,  -8.5774939e-04,  -8.1902331e-04,  -7.7822764e-04,  -7.3760325e-04,  -6.9938246e-04,  -6.6388453e-04,  -6.2979639e-04,  -5.9634412e-04,  -5.6275378e-04,  -5.2825489e-04,  -4.9274324e-04,  -4.5676218e-04,  -4.2071335e-04,  -3.8494812e-04,  -3.4926100e-04,  -3.1360259e-04,  -2.7802856e-04,  -2.4259456e-04,  -2.0735623e-04,  -1.7235488e-04,  -1.3753933e-04,  -1.0289689e-04,  -6.8426740e-05,  -3.4128052e-05,  0.0000000e+00,  3.4128052e-05,  6.8426740e-05,  1.0289689e-04,  1.3753933e-04,  1.7235488e-04,  2.0735623e-04,  2.4259456e-04,  2.7802856e-04,  3.1360259e-04,  3.4926100e-04,  3.8494812e-04,  4.2071335e-04,  4.5676218e-04,  4.9274324e-04,  5.2825489e-04,  5.6275378e-04,  5.9634412e-04,  6.2979639e-04,  6.6388453e-04,  6.9938246e-04,  7.3760325e-04,  7.7822764e-04,  8.1902331e-04,  8.5774939e-04,  8.9259784e-04,  9.2467003e-04,  9.5510063e-04,  9.8486762e-04,  1.0149489e-03,  1.0463195e-03,  1.0792570e-03,  1.1157669e-03,  1.1479719e-03,  1.1729330e-03,  1.2016725e-03,  1.2325246e-03,  1.2726839e-03,  1.3197567e-03,  1.3513909e-03,  1.3804910e-03,  1.4184528e-03,  1.4588862e-03,  1.4884573e-03,  1.5252363e-03,  1.5641794e-03,  1.5931627e-03,  1.6262104e-03,  1.6569041e-03,  1.6847983e-03,  
  -1.5983192e-03,  -1.5686817e-03,  -1.5350885e-03,  -1.4981301e-03,  -1.4694098e-03,  -1.4272047e-03,  -1.3876380e-03,  -1.3562909e-03,  -1.3273596e-03,  -1.2973860e-03,  -1.2478885e-03,  -1.2191453e-03,  -1.1863360e-03,  -1.1559684e-03,  -1.1248880e-03,  -1.0924705e-03,  -1.0630595e-03,  -1.0371100e-03,  -1.0162568e-03,  -9.8553393e-04,  -9.5900649e-04,  -9.3410077e-04,  -9.0998148e-04,  -8.8581333e-04,  -8.6076102e-04,  -8.3400026e-04,  -8.0543947e-04,  -7.5817398e-04,  -7.2081514e-04,  -6.9288740e-04,  -6.6210136e-04,  -6.1114084e-04,  -5.8102200e-04,  -5.4254725e-04,  -5.0665943e-04,  -4.7626745e-04,  -4.4580096e-04,  -4.1298581e-04,  -3.8361176e-04,  -3.5089721e-04,  -3.1959504e-04,  -2.8732249e-04,  -2.4255500e-04,  -2.0545779e-04,  -1.7494001e-04,  -1.4475445e-04,  -1.1057026e-04,  -7.4911470e-05,  -3.8121959e-05,  0.0000000e+00,  3.8121959e-05,  7.4911470e-05,  1.1057026e-04,  1.4475445e-04,  1.7494001e-04,  2.0545779e-04,  2.4255500e-04,  2.8732249e-04,  3.1959504e-04,  3.5089721e-04,  3.8361176e-04,  4.1298581e-04,  4.4580096e-04,  4.7626745e-04,  5.0665943e-04,  5.4254725e-04,  5.8102200e-04,  6.1114084e-04,  6.6210136e-04,  6.9288740e-04,  7.2081514e-04,  7.5817398e-04,  8.0543947e-04,  8.3400026e-04,  8.6076102e-04,  8.8581333e-04,  9.0998148e-04,  9.3410077e-04,  9.5900649e-04,  9.8553393e-04,  1.0162568e-03,  1.0371100e-03,  1.0630595e-03,  1.0924705e-03,  1.1248880e-03,  1.1559684e-03,  1.1863360e-03,  1.2191453e-03,  1.2478885e-03,  1.2973860e-03,  1.3273596e-03,  1.3562909e-03,  1.3876380e-03,  1.4272047e-03,  1.4694098e-03,  1.4981301e-03,  1.5350885e-03,  1.5686817e-03,  1.5983192e-03,  
];


% data for torque function

F_vec = [
  -6.1200000e+03,  -5.9951020e+03,  -5.8702041e+03,  -5.7453061e+03,  -5.6204082e+03,  -5.4955102e+03,  -5.3706122e+03,  -5.2457143e+03,  -5.1208163e+03,  -4.9959184e+03,  -4.8710204e+03,  -4.7461224e+03,  -4.6212245e+03,  -4.4963265e+03,  -4.3714286e+03,  -4.2465306e+03,  -4.1216327e+03,  -3.9967347e+03,  -3.8718367e+03,  -3.7469388e+03,  -3.6220408e+03,  -3.4971429e+03,  -3.3722449e+03,  -3.2473469e+03,  -3.1224490e+03,  -2.9975510e+03,  -2.8726531e+03,  -2.7477551e+03,  -2.6228571e+03,  -2.4979592e+03,  -2.3730612e+03,  -2.2481633e+03,  -2.1232653e+03,  -1.9983673e+03,  -1.8734694e+03,  -1.7485714e+03,  -1.6236735e+03,  -1.4987755e+03,  -1.3738776e+03,  -1.2489796e+03,  -1.1240816e+03,  -9.9918367e+02,  -8.7428571e+02,  -7.4938776e+02,  -6.2448980e+02,  -4.9959184e+02,  -3.7469388e+02,  -2.4979592e+02,  -1.2489796e+02,  0.0000000e+00,  1.2489796e+02,  2.4979592e+02,  3.7469388e+02,  4.9959184e+02,  6.2448980e+02,  7.4938776e+02,  8.7428571e+02,  9.9918367e+02,  1.1240816e+03,  1.2489796e+03,  1.3738776e+03,  1.4987755e+03,  1.6236735e+03,  1.7485714e+03,  1.8734694e+03,  1.9983673e+03,  2.1232653e+03,  2.2481633e+03,  2.3730612e+03,  2.4979592e+03,  2.6228571e+03,  2.7477551e+03,  2.8726531e+03,  2.9975510e+03,  3.1224490e+03,  3.2473469e+03,  3.3722449e+03,  3.4971429e+03,  3.6220408e+03,  3.7469388e+03,  3.8718367e+03,  3.9967347e+03,  4.1216327e+03,  4.2465306e+03,  4.3714286e+03,  4.4963265e+03,  4.6212245e+03,  4.7461224e+03,  4.8710204e+03,  4.9959184e+03,  5.1208163e+03,  5.2457143e+03,  5.3706122e+03,  5.4955102e+03,  5.6204082e+03,  5.7453061e+03,  5.8702041e+03,  5.9951020e+03,  6.1200000e+03,  
];

tau_mat = [
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