Hints for Solving a System of Nonlinear Trigonometric Equations

[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

 

[rachmaninoff]

Yikes! A little more readability can't hurt:
(https://www.physicsforums.com/showthread.php?t=8997) << read this!

$$\cos \left( t_1+t_2 \right) \cos \left( t_3+t_4 \right) \cos t_5 +\sin \left( t_1+t_2 \right) \sin t_5 =\cos \phi \cos \theta$$

$$\cos t_5 \sin \left( t_1+t_2 \right) -\cos \left( t_1+t_2 \right) \cos \left( t_3+t_4 \right) \sin t_5 =\cos \theta \sin \phi \sin \psi -\cos \psi \sin \theta$$

$$\cos \left( t_1+t_2 \right) \sin \left( t_3+t_4 \right) =\cos \psi \cos \theta \sin \phi +\sin \psi \sin \theta$$

$$l_1\cos t_1 +\cos \left( t_1+t_2 \right) (l_2+l_3 \cos t_3 +\cos \left( t_3+t_4 \right) (l_4+l_5 \cos t_5 )+d_5 \sin \left( t_3+t_4 \right) )+\sin \left( t_1+t_2 \right) (d_3+d_4+l_5 \sin t_5 )=P_x$$

$$\cos \left( t_3+t_4 \right) \cos t_5 \sin \left( t_1+t_2 \right) -\cos \left( t_1+t_2 \right) \sin t_5 =\cos \phi \sin \theta$$

$$-\cos \left( t_1+t_2 \right) \cos t_5 -\cos \left( t_3+t_4 \right) \sin \left( t_1+t_2 \right) \sin t_5 =\cos \psi \cos \theta +\sin \phi \sin \psi \sin \theta$$

$$l_1\sin t_1 +\sin \left( t_1+t_2 \right) (l_2+l_3 \cos t_3 +\cos \left( t_3+t_4 \right) (l_4+l_5 \cos t_5 )+d_5 \sin \left( t_3+t_4 \right) )-\cos \left( t_1+t_2 \right) (d_3+d_4+l_5 \sin t_5 )=P_y$$

$$\sin \left( t_1+t_2 \right) \sin \left( t_3+t_4 \right) =-\cos \theta \sin \psi +\cos \psi \sin \phi \sin \theta \cos t_5 \sin \left( t_3+t_4 \right) -\sin \phi -\sin \left( t_3+t_4 \right) \sin t_5 =\cos \phi \sin \psi$$

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

$$d_1+d_2-d_5 \cos \left( t_3+t_4 \right) +l_3 \sin t_3 +(l_4+l_5 \cos t_5 ) \sin \left( t_3+t_4 \right) =P_z$$

 

[saltydog]

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.

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.

 

[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?

 

[mikeley]

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