Re: Solving equations
- To: mathgroup at smc.vnet.net
- Subject: [mg17223] Re: Solving equations
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Mon, 26 Apr 1999 01:20:54 -0400
- Organization: University of Western Australia
- References: <7f9gh9$539@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Marcos Duarte wrote:
>
> I'm trying to solve the following equations in Mathematica to find a
> solution for a2 and a1:
> x=d1*Cos[a1]+d2*Cos[a1+a2]
> y=d1*Sin[a1]+d2*Sin[a1+a2]
> These equations express the end-point position (x,y) of a two-link arm for a
> planar case.
>
> The answer (which are the angles (a1,a2) for a given (x,y)) is
> straightforward:
> a2=ArcCos[(x^2+y^2-d1^2-d2^2)/(2*d1*d2)]
> a1=ArcTan[y/x]-ArcTan[(d2*Sin[a2])/(d1+d2*Cos[a2])]
>
> But M. gets lost with all possible values that x,y,d1, and d2 can assume and
> because of the use of inverse functions.
> I also imposed the constraint that all the 4 parameters are different from
> zero, but still I don't get anything close to the answer given above.
Here is one possible approach to simplify the solution (which mimics the
solution-by-hand). First use TrigExpand:
In[1]:= eqs = {x == TrigExpand[d1*Cos[a1] + d2*Cos[a1 + a2]],
y == TrigExpand[d1*Sin[a1] + d2*Sin[a1 + a2]]}
Out[1]= {x == d1 Cos[a1] + d2 Cos[a1] Cos[a2] - d2 Sin[a1] Sin[a2],
y == d1 Sin[a1] + d2 Cos[a2] Sin[a1] + d2 Cos[a1] Sin[a2]}
Then, eliminate Cos[a1]:
In[2]:= Simplify[Eliminate[eqs, {Cos[a1]}]]
Out[2]=
2 2
(d1 + d2 + 2 d1 d2 Cos[a2]) Sin[a1] + d2 x Sin[a2] == y (d1 + d2
Cos[a2])
which leads to a simple equation for Sin[a1]:
In[3]:= Solve[%, Sin[a1]]
Out[3]=
-(d1 y) - d2 y Cos[a2] + d2 x Sin[a2]
{{Sin[a1] -> -(-------------------------------------)}}
2 2
d1 + d2 + 2 d1 d2 Cos[a2]
The solutions for a2 can be obtained directly:
In[4]:= a2 /. Solve[eqs, {a1, a2}]
Out[4]=
2 2 2 2
-d1 - d2 + x + y
{-ArcCos[--------------------],
2 d1 d2
2 2 2 2
-d1 - d2 + x + y
-ArcCos[--------------------],
2 d1 d2
2 2 2 2 2 2 2 2
-d1 - d2 + x + y -d1 - d2 + x + y
ArcCos[--------------------], ArcCos[--------------------]}
2 d1 d2 2 d1 d2
Cheers,
Paul
____________________________________________________________________
Paul Abbott Phone: +61-8-9380-2734
Department of Physics Fax: +61-8-9380-1014
The University of Western Australia
Nedlands WA 6907 mailto:paul at physics.uwa.edu.au
AUSTRALIA http://www.physics.uwa.edu.au/~paul
God IS a weakly left-handed dice player
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