求解方程组-符号值

Question

我需要找出Alpha, Beta, gamma, l1, l2, l3的值，所以有必要使用符号。我试着解决，但它解决了13个小时，没有完成。

代码如下:

import sympy as sym
from scipy.optimize import fsolve
from sympy import symbols, Eq, solve
Alpha, Beta, Gama, l1, l2, l3 = sym.symbols('Alpha, Beta, Gama, l1, l2, l3')

Eq1 = sym.Eq(-0.945592*l1 + 0.235025*l2 + 0.224988*l3 + 
0.0031173010925*sym.sin(Alpha)*sym.sin(Beta)*sym.sin(Gama) + 
0.031830140825*sym.sin(Alpha)*sym.sin(Beta)*sym.cos(Gama) - 
0.128064361336*sym.sin(Alpha)*sym.sin(Gama) + 
0.00038524122875*sym.sin(Alpha)*sym.cos(Beta) + 
0.0125420486104*sym.sin(Alpha)*sym.cos(Gama) - 
0.0125420486104*sym.sin(Beta)*sym.sin(Gama)*sym.cos(Alpha) - 
0.128064361336*sym.sin(Beta)*sym.cos(Alpha)*sym.cos(Gama) - 
0.0003687890802*sym.sin(Beta) - 0.031830140825*sym.sin(Gama)*sym.cos(Alpha) + 
0.0029841733356*sym.sin(Gama)*sym.cos(Beta) - 
0.0015499671268*sym.cos(Alpha)*sym.cos(Beta) + 
0.0031173010925*sym.cos(Alpha)*sym.cos(Gama) + 
0.030470799804*sym.cos(Beta)*sym.cos(Gama) - 0.0685469-(-0.998117*l1 + 0.0200759*l2 - 
0.0579618*l3 - 0.00035774651523*sym.sin(Alpha)*sym.sin(Beta)*sym.sin(Gama) + 
0.0025561238157*sym.sin(Alpha)*sym.sin(Beta)*sym.cos(Gama) - 
0.127083250791*sym.sin(Alpha)*sym.sin(Gama) - 
0.0006249025393*sym.sin(Alpha)*sym.cos(Beta) - 
0.0177861455049*sym.sin(Alpha)*sym.cos(Gama) + 
0.0177861455049*sym.sin(Beta)*sym.sin(Gama)*sym.cos(Alpha) - 
0.127083250791*sym.sin(Beta)*sym.cos(Alpha)*sym.cos(Gama) - 
0.0018041769486*sym.sin(Beta) - 0.0025561238157*sym.sin(Gama)*sym.cos(Alpha) + 
0.00103286188746*sym.sin(Gama)*sym.cos(Beta) + 
0.031068387859*sym.cos(Alpha)*sym.cos(Beta) - 
0.00035774651523*sym.cos(Alpha)*sym.cos(Gama) - 
0.0073798702614*sym.cos(Beta)*sym.cos(Gama) - 0.0520369),0)
Eq2 = sym.Eq(-0.998117*l1 + 0.0200759*l2 - 0.0579618*l3 - 
0.00035774651523*sym.sin(Alpha)*sym.sin(Beta)*sym.sin(Gama) + 
0.0025561238157*sym.sin(Alpha)*sym.sin(Beta)*sym.cos(Gama) - 
0.127083250791*sym.sin(Alpha)*sym.sin(Gama) - 
0.0006249025393*sym.sin(Alpha)*sym.cos(Beta) - 
0.0177861455049*sym.sin(Alpha)*sym.cos(Gama) + 
0.0177861455049*sym.sin(Beta)*sym.sin(Gama)*sym.cos(Alpha) - 
0.127083250791*sym.sin(Beta)*sym.cos(Alpha)*sym.cos(Gama) - 
0.0018041769486*sym.sin(Beta) - 0.0025561238157*sym.sin(Gama)*sym.cos(Alpha) + 
0.00103286188746*sym.sin(Gama)*sym.cos(Beta) + 
0.031068387859*sym.cos(Alpha)*sym.cos(Beta) - 
0.00035774651523*sym.cos(Alpha)*sym.cos(Gama) - 
0.0073798702614*sym.cos(Beta)*sym.cos(Gama) - 0.0520369 - (-0.999146*l1 + 0.023687*l2 - 
0.0338601*l3 - 0.00020770774995*sym.sin(Alpha)*sym.sin(Beta)*sym.sin(Gama) + 
0.002625917133*sym.sin(Alpha)*sym.sin(Beta)*sym.cos(Gama) - 
0.110764326414*sym.sin(Alpha)*sym.sin(Gama) + 
0.0003371418084*sym.sin(Alpha)*sym.cos(Beta) - 
0.0087613614021*sym.sin(Alpha)*sym.cos(Gama) + 
0.0087613614021*sym.sin(Beta)*sym.sin(Gama)*sym.cos(Alpha) - 
0.110764326414*sym.sin(Beta)*sym.cos(Alpha)*sym.cos(Gama) + 
0.00048193757532*sym.sin(Beta) - 0.002625917133*sym.sin(Gama)*sym.cos(Alpha) + 
0.000296914137885*sym.sin(Gama)*sym.cos(Beta) - 
0.0142210448472*sym.cos(Alpha)*sym.cos(Beta) - 
0.00020770774995*sym.cos(Alpha)*sym.cos(Gama) - 
0.0037536968259*sym.cos(Beta)*sym.cos(Gama) - 0.0424406),0)
Eq3 = sym.Eq(0.305065*l1 + 0.880846*l2 + 0.362004*l3 + 
0.0116832770902*sym.sin(Alpha)*sym.sin(Beta)*sym.sin(Gama) + 
0.119295616318*sym.sin(Alpha)*sym.sin(Beta)*sym.cos(Gama) + 
0.041315868145*sym.sin(Alpha)*sym.sin(Gama) + 
0.0014438387209*sym.sin(Alpha)*sym.cos(Beta) - 
0.0040462906405*sym.sin(Alpha)*sym.cos(Gama) + 
0.0040462906405*sym.sin(Beta)*sym.sin(Gama)*sym.cos(Alpha) + 
0.041315868145*sym.sin(Beta)*sym.cos(Alpha)*sym.cos(Gama) - 
0.0005933788566*sym.sin(Beta) - 0.119295616318*sym.sin(Gama)*sym.cos(Alpha) + 
0.0048015124548*sym.sin(Gama)*sym.cos(Beta) + 
0.00050004729475*sym.cos(Alpha)*sym.cos(Beta) + 
0.0116832770902*sym.cos(Alpha)*sym.cos(Gama) + 
0.049027287732*sym.cos(Beta)*sym.cos(Gama) - 0.612159 - (0.025017*l1 + 0.995996*l2 - 
0.0858224*l3 - 0.0177483499212*sym.sin(Alpha)*sym.sin(Beta)*sym.sin(Gama) + 
0.126813198708*sym.sin(Alpha)*sym.sin(Beta)*sym.cos(Gama) + 
0.003185239491*sym.sin(Alpha)*sym.sin(Gama) - 
0.031002367492*sym.sin(Alpha)*sym.cos(Beta) + 
0.0004457954349*sym.sin(Alpha)*sym.cos(Gama) - 
0.0004457954349*sym.sin(Beta)*sym.sin(Gama)*sym.cos(Alpha) + 
0.003185239491*sym.sin(Beta)*sym.cos(Alpha)*sym.cos(Gama) - 
0.0026713938448*sym.sin(Beta) - 0.126813198708*sym.sin(Gama)*sym.cos(Alpha) + 
0.00152932942128*sym.sin(Gama)*sym.cos(Beta) - 
0.000778704159*sym.cos(Alpha)*sym.cos(Beta) - 
0.0177483499212*sym.cos(Alpha)*sym.cos(Gama) - 
0.0109271654352*sym.cos(Beta)*sym.cos(Gama) - 0.574885),0)
Eq4 = sym.Eq(0.025017*l1 + 0.995996*l2 - 0.0858224*l3 - 
0.0177483499212*sym.sin(Alpha)*sym.sin(Beta)*sym.sin(Gama) + 
0.126813198708*sym.sin(Alpha)*sym.sin(Beta)*sym.cos(Gama) + 
0.003185239491*sym.sin(Alpha)*sym.sin(Gama) - 
0.031002367492*sym.sin(Alpha)*sym.cos(Beta) + 
0.0004457954349*sym.sin(Alpha)*sym.cos(Gama) - 
0.0004457954349*sym.sin(Beta)*sym.sin(Gama)*sym.cos(Alpha) + 
0.003185239491*sym.sin(Beta)*sym.cos(Alpha)*sym.cos(Gama) - 
0.0026713938448*sym.sin(Beta) - 0.126813198708*sym.sin(Gama)*sym.cos(Alpha) + 
0.00152932942128*sym.sin(Gama)*sym.cos(Beta) - 
0.000778704159*sym.cos(Alpha)*sym.cos(Beta) - 
0.0177483499212*sym.cos(Alpha)*sym.cos(Gama) - 
0.0109271654352*sym.cos(Beta)*sym.cos(Gama) - 0.574885 - (0.0313426*l1 + 0.968404*l2 - 
0.247408*l3 - 0.0084917894154*sym.sin(Alpha)*sym.sin(Beta)*sym.sin(Gama) + 
0.107356299036*sym.sin(Alpha)*sym.sin(Beta)*sym.cos(Gama) + 
0.0034746092934*sym.sin(Alpha)*sym.sin(Gama) + 
0.0137834878128*sym.sin(Alpha)*sym.cos(Beta) + 
0.00027483855801*sym.sin(Alpha)*sym.cos(Gama) - 
0.00027483855801*sym.sin(Beta)*sym.sin(Gama)*sym.cos(Alpha) + 
0.0034746092934*sym.sin(Beta)*sym.cos(Alpha)*sym.cos(Gama) + 
0.0035214075456*sym.sin(Beta) - 0.107356299036*sym.sin(Gama)*sym.cos(Alpha) + 
0.0021694836408*sym.sin(Gama)*sym.cos(Beta) + 
0.00044610549432*sym.cos(Alpha)*sym.cos(Beta) - 
0.0084917894154*sym.cos(Alpha)*sym.cos(Gama) - 
0.027427403472*sym.cos(Beta)*sym.cos(Gama) - 0.507045),0)
Eq5 = sym.Eq(-0.1131*l1 + 0.410944*l2 - 0.904618*l3 + 
0.0054506379328*sym.sin(Alpha)*sym.sin(Beta)*sym.sin(Gama) + 
0.055655378752*sym.sin(Alpha)*sym.sin(Beta)*sym.cos(Gama) - 
0.0153174723*sym.sin(Alpha)*sym.sin(Gama) + 0.0006735988576*sym.sin(Alpha)*sym.cos(Beta) 
+ 0.00150012447*sym.sin(Alpha)*sym.cos(Gama) - 
0.00150012447*sym.sin(Beta)*sym.sin(Gama)*sym.cos(Alpha) - 
0.0153174723*sym.sin(Beta)*sym.cos(Alpha)*sym.cos(Gama) + 0.0014828045947*sym.sin(Beta) 
- 0.055655378752*sym.sin(Gama)*sym.cos(Alpha) - 
0.0119985817666*sym.sin(Gama)*sym.cos(Beta) - 
0.000185387865*sym.cos(Alpha)*sym.cos(Beta) + 
0.0054506379328*sym.cos(Alpha)*sym.cos(Gama) - 
0.122515129594*sym.cos(Beta)*sym.cos(Gama) + 0.200443 - (0.0560068*l1 - 0.0871109*l2 - 
0.994623*l3 + 0.00155229010473*sym.sin(Alpha)*sym.sin(Beta)*sym.sin(Gama) - 
0.0110912211207*sym.sin(Alpha)*sym.sin(Beta)*sym.cos(Gama) + 
0.0071309537964*sym.sin(Alpha)*sym.sin(Gama) + 
0.0027115009843*sym.sin(Alpha)*sym.cos(Beta) + 
0.00099802437396*sym.sin(Alpha)*sym.cos(Gama) - 
0.00099802437396*sym.sin(Beta)*sym.sin(Gama)*sym.cos(Alpha) + 
0.0071309537964*sym.sin(Beta)*sym.cos(Alpha)*sym.cos(Gama) - 
0.030959630121*sym.sin(Beta) + 0.0110912211207*sym.sin(Gama)*sym.cos(Alpha) + 
0.0177238834731*sym.sin(Gama)*sym.cos(Beta) - 
0.0017433236636*sym.cos(Alpha)*sym.cos(Beta) + 
0.00155229010473*sym.cos(Alpha)*sym.cos(Gama) - 
0.126638384229*sym.cos(Beta)*sym.cos(Gama) + 0.207426),0)
Eq6 = sym.Eq(0.0560068*l1 - 0.0871109*l2 - 0.994623*l3 + 
0.00155229010473*sym.sin(Alpha)*sym.sin(Beta)*sym.sin(Gama) - 
0.0110912211207*sym.sin(Alpha)*sym.sin(Beta)*sym.cos(Gama) + 
0.0071309537964*sym.sin(Alpha)*sym.sin(Gama) + 
0.0027115009843*sym.sin(Alpha)*sym.cos(Beta) + 
0.00099802437396*sym.sin(Alpha)*sym.cos(Gama) - 
0.00099802437396*sym.sin(Beta)*sym.sin(Gama)*sym.cos(Alpha) + 
0.0071309537964*sym.sin(Beta)*sym.cos(Alpha)*sym.cos(Gama) - 
0.030959630121*sym.sin(Beta) + 0.0110912211207*sym.sin(Gama)*sym.cos(Alpha) + 
0.0177238834731*sym.sin(Gama)*sym.cos(Beta) - 
0.0017433236636*sym.cos(Alpha)*sym.cos(Beta) + 
0.00155229010473*sym.cos(Alpha)*sym.cos(Gama) - 
0.126638384229*sym.cos(Beta)*sym.cos(Gama) + 0.207426 - (0.0269299*l1 - 0.248258*l2 - 
0.968319*l3 + 0.0021769371633*sym.sin(Alpha)*sym.sin(Beta)*sym.sin(Gama) - 
0.027521633622*sym.sin(Alpha)*sym.sin(Beta)*sym.cos(Gama) + 
0.0029854217841*sym.sin(Alpha)*sym.sin(Gama) - 
0.0035335057656*sym.sin(Alpha)*sym.cos(Beta) + 
0.000236144253615*sym.sin(Alpha)*sym.cos(Gama) - 
0.000236144253615*sym.sin(Beta)*sym.sin(Gama)*sym.cos(Alpha) + 
0.0029854217841*sym.sin(Beta)*sym.cos(Alpha)*sym.cos(Gama) + 
0.0137822779908*sym.sin(Beta) + 0.027521633622*sym.sin(Gama)*sym.cos(Alpha) + 
0.00849104406315*sym.sin(Gama)*sym.cos(Beta) + 
0.00038329865268*sym.cos(Alpha)*sym.cos(Beta) + 
0.0021769371633*sym.cos(Alpha)*sym.cos(Gama) - 
0.107346876021*sym.cos(Beta)*sym.cos(Gama) + 0.184387),0)    

Answer 1

SymPy中的solve函数用于找到精确的解析解，但对于您的方程组来说，这些可能非常复杂。如果你只想要一个数值解，你可以使用SymPy的nsolve函数:

In [140]: from sympy import nsolve
In [141]: eqs = [Eq1, Eq2, Eq3, Eq4, Eq5, Eq6]
In [142]: nsolve(eqs, [l1, l2, l3, Alpha, Beta, Gama], [1, 1, 1, 1, 1, 1])
Out[142]: 
⎡-0.557604943463198⎤
⎢                  ⎥
⎢-0.300586875967959⎥
⎢                  ⎥
⎢0.383995113825599 ⎥
⎢                  ⎥
⎢ 3.21607797653942 ⎥
⎢                  ⎥
⎢-1.51886308101428 ⎥
⎢                  ⎥
⎣ 2.40460861693647 ⎦

请注意，可以有其他可能的解决方案的方程式，所以如果你需要找到一个不同的解决方案比一个显示，那么你应该使用不同的初始猜测([1,1,1,1,1,1]以外的东西)。