# System of nonlinear equations

1. Aug 11, 2005

### mikeley

Hello,

I have a system of trigonometric equations from which I should find theta1,..., theta5. Is it possible you can give me a hint on how to proceed. Thanks.

theta, phi, psi, Px, Py, Pz, l1, l2, l3, l4, l5, d1, d2, d3, d4, d5 are all constants.

Cos[t1+t2] Cos[t3+t4] Cos[t5]+Sin[t1+t2] Sin[t5]=Cos[phi] Cos[theta]

Cos[t5] Sin[t1+t2]-Cos[t1+t2] Cos[t3+t4] Sin[t5]=Cos[theta] Sin[phi] Sin[psi]-Cos[psi] Sin[theta]

Cos[t1+t2] Sin[t3+t4]=Cos[psi] Cos[theta] Sin[phi]+Sin[psi] Sin[theta]

l1Cos[t1]+Cos[t1+t2] (l2+l3 Cos[t3]+Cos[t3+t4] (l4+l5 Cos[t5])+d5 Sin[t3+t4])+Sin[t1+t2] (d3+d4+l5 Sin[t5])=Px

Cos[t3+t4] Cos[t5] Sin[t1+t2]-Cos[t1+t2] Sin[t5]=Cos[phi] Sin[theta]

-Cos[t1+t2] Cos[t5]-Cos[t3+t4] Sin[t1+t2] Sin[t5]=Cos[psi] Cos[theta]+Sin[phi] Sin[psi] Sin[theta]

l1Sin[t1]+Sin[t1+t2] (l2+l3 Cos[t3]+Cos[t3+t4] (l4+l5 Cos[t5])+d5 Sin[t3+t4])-Cos[t1+t2] (d3+d4+l5 Sin[t5])=Py

Sin[t1+t2] Sin[t3+t4]=-Cos[theta] Sin[psi]+Cos[psi] Sin[phi] Sin[theta] Cos[t5] Sin[t3+t4]-Sin[phi]-Sin[t3+t4] Sin[t5]=Cos[phi] Sin[psi]

-Cos[t3+t4]=Cos[phi]Cos[psi]

d1+d2-d5 Cos[t3+t4]+l3 Sin[t3]+(l4+l5 Cos[t5]) Sin[t3+t4]=Pz

2. Aug 11, 2005

### rachmaninoff

Yikes! A little more readability can't hurt:

Last edited by a moderator: Aug 11, 2005
3. Aug 12, 2005

### saltydog

Feel like posting the values for all these constants?

Then me anyway, in some desperate attempt at approaching it, I would then convert each to:

$$t1=f(t1,t2,t3,t4,t5; constants)$$

$$t2=g(t1,t2,t3,t4,t5;constants)$$

and so on and then use iteration of some sort to analyze if it converges to a solution. There is a sufficiency condition for this sort of iteration to converge to a solution and involves the partials of each function above.

Oh yea, I'd rely heavily on Mathematica too.

Edit: I just noticed you have 10 equation and one in particular:

$$-\cos \left( t_3+t_4 \right) =\cos \phi \cos \psi$$

You can immediately start cleaning them up by substituting this one and it's Sin equivalent.

Last edited: Aug 12, 2005
4. Aug 12, 2005

### saltydog

I change my mind. I can do that. There're 9 equations in 9 unknowns. For example:

$$Cos[t1+t2]=u1$$

What are the rest?

5. Aug 12, 2005

### mikeley

Thanks a lot. I managed to get the reduction you mentioned.