Solving nonlinear system of equations for variables
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I would like to solve the following system of equations for $α_1$ and $α_2$:
$$
begin{bmatrix}
frac{alpha_1 sin(alpha_1 +alpha_2)-sin(alpha_1) alpha_1 +sin(alpha_1) alpha_2}{alpha_1 alpha_2}\
frac{-alpha_1 cos(alpha_1+alpha_2)+cos(alpha_1)alpha_1+alpha_2-alpha_2 cos(alpha_1)}{alpha_1 alpha_2}
end{bmatrix}
=
begin{bmatrix}
x\
y
end{bmatrix}
$$
Obviously, this is not straight forward. The equations describe a Cartesian robotic end-effector position depending on the two angles $α_1$ and $α_2$. I would like to get your feedback how to proceed.
1.) Is there an analytic solution to compute $α_1$ and $α_2$ for a given $x$ and $y$?
2.) If not, would you try to fit an approximation around the working point?
3.) Is there any other approach I don't have in mind?
Every feedback is appreciated. Thank you very much!
trigonometry systems-of-equations nonlinear-system
asked Nov 22 at 16:33
Markus
1
add a comment |
up vote
0
down vote
favorite
I would like to solve the following system of equations for $α_1$ and $α_2$:
$$
begin{bmatrix}
frac{alpha_1 sin(alpha_1 +alpha_2)-sin(alpha_1) alpha_1 +sin(alpha_1) alpha_2}{alpha_1 alpha_2}\
frac{-alpha_1 cos(alpha_1+alpha_2)+cos(alpha_1)alpha_1+alpha_2-alpha_2 cos(alpha_1)}{alpha_1 alpha_2}
end{bmatrix}
=
begin{bmatrix}
x\
y
end{bmatrix}
$$
Obviously, this is not straight forward. The equations describe a Cartesian robotic end-effector position depending on the two angles $α_1$ and $α_2$. I would like to get your feedback how to proceed.
1.) Is there an analytic solution to compute $α_1$ and $α_2$ for a given $x$ and $y$?
2.) If not, would you try to fit an approximation around the working point?
3.) Is there any other approach I don't have in mind?
Every feedback is appreciated. Thank you very much!
trigonometry systems-of-equations nonlinear-system
asked Nov 22 at 16:33
Markus
1
I doubt there is a closed form solution. Did you consider Newton's Method? The danger is that there may be periodic roots and you might have to deal with that once roots are found.
– Moo
Nov 22 at 16:39
Usually, when you have both angle and a trigonometric function of the angle, the only way to solve it is numerically.
– Andrei
Nov 22 at 16:40
@Moo - Yes, I thought of Newton's Method and yes there are periodic roots. I hoped for any more elegant solution. Thanks, for your feedback.
– Markus
Nov 22 at 16:59
@Andrei - that's what I guessed - I just hoped someone might know a better solution. Thanks though!!
– Markus
Nov 22 at 17:00
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I would like to solve the following system of equations for $α_1$ and $α_2$:
$$
begin{bmatrix}
frac{alpha_1 sin(alpha_1 +alpha_2)-sin(alpha_1) alpha_1 +sin(alpha_1) alpha_2}{alpha_1 alpha_2}\
frac{-alpha_1 cos(alpha_1+alpha_2)+cos(alpha_1)alpha_1+alpha_2-alpha_2 cos(alpha_1)}{alpha_1 alpha_2}
end{bmatrix}
=
begin{bmatrix}
x\
y
end{bmatrix}
$$
Obviously, this is not straight forward. The equations describe a Cartesian robotic end-effector position depending on the two angles $α_1$ and $α_2$. I would like to get your feedback how to proceed.
1.) Is there an analytic solution to compute $α_1$ and $α_2$ for a given $x$ and $y$?
2.) If not, would you try to fit an approximation around the working point?
3.) Is there any other approach I don't have in mind?
Every feedback is appreciated. Thank you very much!
trigonometry systems-of-equations nonlinear-system
asked Nov 22 at 16:33
Markus
1
I would like to solve the following system of equations for $α_1$ and $α_2$:
$$
begin{bmatrix}
frac{alpha_1 sin(alpha_1 +alpha_2)-sin(alpha_1) alpha_1 +sin(alpha_1) alpha_2}{alpha_1 alpha_2}\
frac{-alpha_1 cos(alpha_1+alpha_2)+cos(alpha_1)alpha_1+alpha_2-alpha_2 cos(alpha_1)}{alpha_1 alpha_2}
end{bmatrix}
=
begin{bmatrix}
x\
y
end{bmatrix}
$$
Obviously, this is not straight forward. The equations describe a Cartesian robotic end-effector position depending on the two angles $α_1$ and $α_2$. I would like to get your feedback how to proceed.
1.) Is there an analytic solution to compute $α_1$ and $α_2$ for a given $x$ and $y$?
2.) If not, would you try to fit an approximation around the working point?
3.) Is there any other approach I don't have in mind?
Every feedback is appreciated. Thank you very much!
trigonometry systems-of-equations nonlinear-system
trigonometry systems-of-equations nonlinear-system
asked Nov 22 at 16:33
Markus
1
asked Nov 22 at 16:33
Markus
1
asked Nov 22 at 16:33
Markus
1
asked Nov 22 at 16:33
Markus
1
asked Nov 22 at 16:33
Markus
1
1
I doubt there is a closed form solution. Did you consider Newton's Method? The danger is that there may be periodic roots and you might have to deal with that once roots are found.
– Moo
Nov 22 at 16:39
Usually, when you have both angle and a trigonometric function of the angle, the only way to solve it is numerically.
– Andrei
Nov 22 at 16:40
@Moo - Yes, I thought of Newton's Method and yes there are periodic roots. I hoped for any more elegant solution. Thanks, for your feedback.
– Markus
Nov 22 at 16:59
@Andrei - that's what I guessed - I just hoped someone might know a better solution. Thanks though!!
– Markus
Nov 22 at 17:00
add a comment |
I doubt there is a closed form solution. Did you consider Newton's Method? The danger is that there may be periodic roots and you might have to deal with that once roots are found.
– Moo
Nov 22 at 16:39
Usually, when you have both angle and a trigonometric function of the angle, the only way to solve it is numerically.
– Andrei
Nov 22 at 16:40
@Moo - Yes, I thought of Newton's Method and yes there are periodic roots. I hoped for any more elegant solution. Thanks, for your feedback.
– Markus
Nov 22 at 16:59
@Andrei - that's what I guessed - I just hoped someone might know a better solution. Thanks though!!
– Markus
Nov 22 at 17:00
I doubt there is a closed form solution. Did you consider Newton's Method? The danger is that there may be periodic roots and you might have to deal with that once roots are found.
– Moo
Nov 22 at 16:39
I doubt there is a closed form solution. Did you consider Newton's Method? The danger is that there may be periodic roots and you might have to deal with that once roots are found.
– Moo
Nov 22 at 16:39
Usually, when you have both angle and a trigonometric function of the angle, the only way to solve it is numerically.
– Andrei
Nov 22 at 16:40
Usually, when you have both angle and a trigonometric function of the angle, the only way to solve it is numerically.
– Andrei
Nov 22 at 16:40
@Moo - Yes, I thought of Newton's Method and yes there are periodic roots. I hoped for any more elegant solution. Thanks, for your feedback.
– Markus
Nov 22 at 16:59
@Moo - Yes, I thought of Newton's Method and yes there are periodic roots. I hoped for any more elegant solution. Thanks, for your feedback.
– Markus
Nov 22 at 16:59
@Andrei - that's what I guessed - I just hoped someone might know a better solution. Thanks though!!
– Markus
Nov 22 at 17:00
@Andrei - that's what I guessed - I just hoped someone might know a better solution. Thanks though!!
– Markus
Nov 22 at 17:00
add a comment |
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I doubt there is a closed form solution. Did you consider Newton's Method? The danger is that there may be periodic roots and you might have to deal with that once roots are found.
– Moo
Nov 22 at 16:39
Usually, when you have both angle and a trigonometric function of the angle, the only way to solve it is numerically.
– Andrei
Nov 22 at 16:40
@Moo - Yes, I thought of Newton's Method and yes there are periodic roots. I hoped for any more elegant solution. Thanks, for your feedback.
– Markus
Nov 22 at 16:59
@Andrei - that's what I guessed - I just hoped someone might know a better solution. Thanks though!!
– Markus
Nov 22 at 17:00