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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