Nevermind...I suspect you just let b be called c in the first case and then
let c = a+b in the second case. If you can solve for a and c you can solve
for a and a+b, hence a and b in the second case.
/c
On Thursday, December 18, 2025 at 12:13:12 AM UTC-6 Chris Smith wrote:
> What substitution were you thinking? I would expand the 2nd set and
> rewrite in terms of exponentials and solve for them.
>
> a,b,x,y,l1,l2=symbols('a b x y l1 l2')
> eq = [l1*cos(a)+l2*cos(a+b)-x,l1*sin(a)+l2*sin(a+b)-y]
> eq=[i.expand(trig=1).rewrite(exp).subs(exp(I*a),var('A')).subs(exp(I*b),var("B")).as_numer_denom()[0]
>
> for i in eq]
> g = groebner(eq,(A,B))
> solve(g,(A,B),simplify=0,verify=0)
>
> which gives 3 solutions (but the one with A=0 is not valid, so just 2)
>
> /c
>
>
>
>
> On Tuesday, November 11, 2025 at 4:02:36 AM UTC-6 [email protected]
> wrote:
>
>> this works (A)(although takes awfully long)
>> a,b,x,y,l1,l2=symbols('a b x y l1 l2')
>> eq = [l1*cos(a)+l2*cos(b)-x,l1*sin(a)+l2*sin(b)-y]
>> res = solve(eq,[a,b])
>>
>> this does not (B):
>> a,b,x,y,l1,l2=symbols('a b x y l1 l2')
>> eq = [l1*cos(a)+l2*cos(a+b)-x,l1*sin(a)+l2*sin(a+b)-y]
>> res = solve(eq,[a,b])
>>
>> solution from (A) could be used to find solution for (B) by a simple
>> substitution, so, analytic solution exists, sympy, however, produces error
>>
>> python 3.11.13
>> sympy 1.14.0
>>
>>
>> ---------------------------------------------------------------------------
>> TypeError Traceback (most recent call last) Cell In[11], line 1 ----> 1
>> res2 = solve(eq,[a,b]) File
>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\solvers\solvers.py:1170
>>
>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/solvers/solvers.py#line=1169>,
>>
>> in solve(f, *symbols, **flags) 1168 solution = _solve(f[0], *symbols,
>> **flags)
>> 1169 else: -> 1170 linear, solution = _solve_system(f, symbols, **flags)
>> 1171 assert type(solution) is list 1172 assert not solution or type
>> (solution[0]) is dict, solution File
>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\solvers\solvers.py:1963
>>
>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/solvers/solvers.py#line=1962>,
>>
>> in _solve_system(exprs, symbols, **flags) 1961 for s in ok_syms: 1962 try:
>> -> 1963 soln = _vsolve(eq2, s, **flags) 1964 except NotImplementedError:
>> 1965 continue File
>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\solvers\solvers.py:2638
>>
>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/solvers/solvers.py#line=2637>,
>>
>> in _vsolve(e, s, **flags) 2636 def _vsolve(e, s, **flags): 2637 """return
>> list of scalar values for the solution of e for symbol s""" -> 2638
>> return [i[s] for i in _solve(e, s, **flags)] File
>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\solvers\solvers.py:1743
>>
>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/solvers/solvers.py#line=1742>,
>>
>> in _solve(f, *symbols, **flags) 1738 result = [r for r in result if 1739
>> not any(checksol(d, r, **flags) 1740 for d in dens)] 1741 if check: 1742 #
>> keep only results if the check is not False -> 1743 result = [r for r in
>> result if 1744 checksol(f_num, r, **flags) is not False] 1745 return
>> result File
>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\solvers\solvers.py:1744
>>
>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/solvers/solvers.py#line=1743>,
>>
>> in <listcomp>(.0) 1738 result = [r for r in result if 1739 not
>> any(checksol(d,
>> r, **flags) 1740 for d in dens)] 1741 if check: 1742 # keep only results
>> if the check is not False 1743 result = [r for r in result if -> 1744
>> checksol(f_num, r, **flags) is not False] 1745 return result File
>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\solvers\solvers.py:354
>>
>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/solvers/solvers.py#line=353>,
>>
>> in checksol(f, symbol, sol, **flags) 352 return val == 0 353 if
>> numerical and val.is_number: --> 354 return (abs(val.n(18).n(12, chop=
>> True)) < 1e-9) is S.true 356 if flags.get('warn', False): 357 warnings.
>> warn("\n\tWarning: could not verify solution %s." % sol) File
>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\core\decorators.py:248
>>
>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/core/decorators.py#line=247>,
>>
>> in _SympifyWrapper.make_wrapped.<locals>._func(self, other) 246 if not
>> isinstance(other, expectedcls): 247 return retval --> 248 return func(
>> self, other) File
>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\core\expr.py:400
>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/core/expr.py#line=399>,
>>
>> in Expr.__lt__(self, other) 397 @sympify_return([('other', 'Expr')],
>> NotImplemented) 398 def __lt__(self, other): 399 from .relational import
>> StrictLessThan --> 400 return StrictLessThan(self, other) File
>> ~\Documents\FH\Python\venv\control\Lib\site-packages\sympy\core\relational.py:850
>>
>> <http://localhost:8888/lab/tree/Teaching/RobReg/R2D/~/Documents/FH/Python/venv/control/Lib/site-packages/sympy/core/relational.py#line=849>,
>>
>> in _Inequality.__new__(cls, lhs, rhs, **options) 848 raise
>> TypeError("Invalid
>> comparison of non-real %s" % me) 849 if me is S.NaN: --> 850 raise
>> TypeError("Invalid NaN comparison") 851 # First we invoke the
>> appropriate inequality method of `lhs` 852 # (e.g., `lhs.__lt__`). That
>> method will try to reduce to 853 # boolean or raise an exception. It may
>> keep calling (...) 857 # exception). In that case, it must call us with
>> 858 # `evaluate=False` to prevent infinite recursion. 859 return
>> cls._eval_relation(lhs,
>> rhs, **options) TypeError: Invalid NaN comparison
>>
>
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