How to solve the equation to get the result in atan2 form?

Hello MATLAB Community,

I am trying to solve an equation in MATLAB, but I don't know which function or how to solve it.

syms phi theta psi

ROT = [cos(theta)*cos(phi)+sin(psi)*sin(theta)*sin(phi), -cos(theta)*sin(phi)+sin(psi)*sin(theta)*cos(phi), cos(psi)*sin(theta); ...
    cos(psi)*sin(phi), cos(psi)*cos(phi), -sin(psi); ...
    -sin(theta)*cos(phi)+sin(psi)*cos(theta)*sin(phi), sin(theta)*sin(phi)+sin(psi)*cos(theta)*cos(phi), cos(psi)*cos(theta)];

eqn = ROT(1,2) == ROT(2,1)

I tried to use functions like solve, simplify, etc. but I am not getting it in the desired form as shown below:

psi = atan2(sin(phi)*sin(theta),cos(phi)+cos(theta))

Can anyone please help me with any suggestions.

Thank you in advance!!

I really appreciate your help.

Kind regards,

Shiv

[Mohammed Hamaidi]

Can you post the Equation

[Shiv Karpoor]

Run the code which says (ROT),

This is the equation:eqn = ROT(1,2) == ROT(2,1)

[Mohammed Hamaidi]

Do you want the mathematical solution or the matlab solution?

[Shiv Karpoor]

Matlab Solution

[Mohammed Hamaidi]

Be careful. The equation has solution under some strict assumptions.

Example:

sin(x)+cos(x)==4

, has no solution,

Matlab gives:

>> solve(cos(x)+sin(x)==4,x)
 
ans =
 
 -log(- 28i^(1/2)/2 + (2 + 2i))*1i
   -log(28i^(1/2)/2 + (2 + 2i))*1i

So, you have to give the assumptions/conditions

Answer 1

[Torsten]

syms phi theta psi
ROT = [cos(theta)*cos(phi)+sin(psi)*sin(theta)*sin(phi), -cos(theta)*sin(phi)+sin(psi)*sin(theta)*cos(phi), cos(psi)*sin(theta); ...
       cos(psi)*sin(phi), cos(psi)*cos(phi), -sin(psi); ...
       -sin(theta)*cos(phi)+sin(psi)*cos(theta)*sin(phi), sin(theta)*sin(phi)+sin(psi)*cos(theta)*cos(phi), cos(psi)*cos(theta)];
eqn = ROT(1,2) == ROT(2,1);
sol = solve(eqn,psi)

[Shiv Karpoor]

Hi Torsten,

I did solve it like the way you mentioned before posting my question here,

I was able to determine the psi value, when values were given for phi & theta.

But, what I need is to solve the equation symbolically and the resulting equation should be similar to the one I mentoned in the question.

[Torsten]

For me, the solve-command from above gives a formula for psi as a function of phi and theta.

Is this not true for your MATLAB ?

[Mohammed Hamaidi]

See comment above

[Torsten]

I get two solutions:

  [       /         1         \]
  [-2*atan|-------------------|]
  [       |   /phi\    /theta\|]
  [       |tan|---|*tan|-----||]
  [       \   \ 2 /    \  2  //]
  [                            ]
  [           /    /phi\ \     ]
  [           | tan|---| |     ]
  [           |    \ 2 / |     ]
  [     2*atan|----------|     ]
  [           |   /theta\|     ]
  [           |tan|-----||     ]
  [           \   \  2  //     ]

[Shiv Karpoor]

How did you get the above solution,

I am getting the solution like this, after doing what you said:

sol =
 
 -log(((2*exp(phi*2i) + exp(phi*4i) + 2*exp(theta*2i) + exp(theta*4i) - 12*exp(phi*2i)*exp(theta*2i) + 2*exp(phi*2i)*exp(theta*4i) + 2*exp(phi*4i)*exp(theta*2i) + exp(phi*4i)*exp(theta*4i) + 16*exp(phi*2i)*exp(theta*2i)*cos(theta)^2*sin(phi)^2 + 1)^(1/2) + 4*exp(phi*1i)*exp(theta*1i)*cos(theta)*sin(phi))/(exp(phi*2i) - exp(theta*2i) - exp(phi*2i)*exp(theta*2i) + 1 - exp(theta*1i)*2i + exp(phi*2i)*exp(theta*1i)*2i))*1i
-log(-((2*exp(phi*2i) + exp(phi*4i) + 2*exp(theta*2i) + exp(theta*4i) - 12*exp(phi*2i)*exp(theta*2i) + 2*exp(phi*2i)*exp(theta*4i) + 2*exp(phi*4i)*exp(theta*2i) + exp(phi*4i)*exp(theta*4i) + 16*exp(phi*2i)*exp(theta*2i)*cos(theta)^2*sin(phi)^2 + 1)^(1/2) - 4*exp(phi*1i)*exp(theta*1i)*cos(theta)*sin(phi))/(exp(phi*2i) - exp(theta*2i) - exp(phi*2i)*exp(theta*2i) + 1 - exp(theta*1i)*2i + exp(phi*2i)*exp(theta*1i)*2i))*1i
 
>> pretty(sol)
/      / #1 + 4 exp(phi 1i) exp(theta 1i) cos(theta) sin(phi) \     \
|  -log| ---------------------------------------------------- | 1i  |
|      \                          #2                          /     |
|                                                                   |
|     /   #1 - 4 exp(phi 1i) exp(theta 1i) cos(theta) sin(phi) \    |
| -log| - ---------------------------------------------------- | 1i |
\     \                            #2                          /    /

where

   #1 == sqrt(2 #3 + exp(phi 4i) + 2 #4 + exp(theta 4i) - 12 #3 #4 + 2 #3 exp(theta 4i) + 2 exp(phi 4i) #4

                                                       2         2
      + exp(phi 4i) exp(theta 4i) + 16 #3 #4 cos(theta)  sin(phi)  + 1)

   #2 == #3 - #4 - #3 #4 + 1 - exp(theta 1i) 2i + #3 exp(theta 1i) 2i

   #3 == exp(phi 2i)

   #4 == exp(theta 2i)

[Torsten]

What do you get if you make a

sol = simplify(real(sol))

on the solution ?

I got the answer with octave, not MATLAB.

[Shiv Karpoor]

I got this:

soln =
 
angle((- exp(phi*2i)*1i + exp(theta*2i)*1i - exp(phi*2i + theta*2i)*1i + 2*(exp(phi*2i) - 2*exp(phi*2i + theta*2i) + exp(phi*2i + theta*4i))^(1/2) + 1i)/(exp(phi*2i) - exp(theta*2i) - exp(phi*2i + theta*2i) + 1 - exp(theta*1i)*2i + exp(phi*2i + theta*1i)*2i))
 angle(-(exp(phi*2i)*1i - exp(theta*2i)*1i + exp(phi*2i + theta*2i)*1i + 2*(exp(phi*2i) - 2*exp(phi*2i + theta*2i) + exp(phi*2i + theta*4i))^(1/2) - 1i)/(exp(phi*2i) - exp(theta*2i) - exp(phi*2i + theta*2i) + 1 - exp(theta*1i)*2i + exp(phi*2i + theta*1i)*2i))
 
>> pretty(soln)
/      / - exp(phi 2i) 1i + exp(theta 2i) 1i - #3 1i + #2 + 1i \ \
| angle| ----------------------------------------------------- | |
|      \                           #1                          / |
|                                                                |
|       /   exp(phi 2i) 1i - exp(theta 2i) 1i + #3 1i + #2 - i \ |
|  angle| - -------------------------------------------------- | |
\       \                           #1                         / /

where

   #1 == exp(phi 2i) - exp(theta 2i) - #3 + 1 - exp(theta 1i) 2i + exp(phi 2i + theta 1i) 2i

   #2 == sqrt(exp(phi 2i) - 2 #3 + exp(phi 2i + theta 4i)) 2

   #3 == exp(phi 2i + theta 2i)

I think, I might have to stick with written mathematical solution for now and use matlab to get the value of psi after substituting the values of theta & phi into the eqn.

Thank you so much, for suggestions

I really appreciate it

[David Goodmanson]

Hi Shiv,

how sure are you that the desired form is correct?

theta = pi/4
phi = pi/4
psi = atan2(sin(phi)*sin(theta),cos(phi)+cos(theta))   % desired form

ROT = [cos(theta)*cos(phi)+sin(psi)*sin(theta)*sin(phi), -cos(theta)*sin(phi)+sin(psi)*sin(theta)*cos(phi), cos(psi)*sin(theta); ...
    cos(psi)*sin(phi), cos(psi)*cos(phi), -sin(psi); ...
    -sin(theta)*cos(phi)+sin(psi)*cos(theta)*sin(phi), sin(theta)*sin(phi)+sin(psi)*cos(theta)*cos(phi), cos(psi)*cos(theta)];

ROT(1,2)
  ans = -0.3333
ROT(2,1)
  ans = 0.6667

whereas the expressions provided by Torsten appear to be correct.

[Shiv Karpoor]

Hi David,

I found the ROT equation from a journal paper,

I was using the above ROT equation, because the desired form was given by the journal paper.

The actual ROT that I am using is :

eROT = [(cos(phi)*cos(psi))-(cos(theta)*sin(phi)*sin(psi)) -(cos(phi)*sin(psi))-(cos(theta)*cos(psi)*sin(phi)) (sin(phi)*sin(theta));...
    (cos(psi)*sin(phi))+(cos(phi)*cos(theta)*sin(psi)) (cos(phi)*cos(theta)*cos(psi)-sin(phi)*sin(psi)) -(cos(phi)*sin(theta));...
    (sin(theta)*sin(psi)) (cos(psi)*sin(theta)) cos(theta)];

For my eROT, I am unable to find the desired form. But I guarantee you that if you take any euler angle transformations, you will get the desired form with probably different variables.

You can solve the below and see that I am able to get the psi result:

syms psi

d2r = pi/180;
r2d = 180/pi;

phi = 10*d2r;
theta = 15*d2r;


eROT = [(cos(phi)*cos(psi))-(cos(theta)*sin(phi)*sin(psi)) -(cos(phi)*sin(psi))-(cos(theta)*cos(psi)*sin(phi)) (sin(phi)*sin(theta));...
    (cos(psi)*sin(phi))+(cos(phi)*cos(theta)*sin(psi)) (cos(phi)*cos(theta)*cos(psi)-sin(phi)*sin(psi)) -(cos(phi)*sin(theta));...
    (sin(theta)*sin(psi)) (cos(psi)*sin(theta)) cos(theta)];


eqn = eROT(1,2) == eROT(2,1)
[Psi] = double(vpasolve(eqn))

For this eROT, Psi will always be opposite of phi and vice versa.

But What I need is to solve the eqn symbolically, and get the desired form without any numerical value. (like a formula).

If its possible, then Great !! or else I'll solve it by hand. (anyways I am getting the value of psi through matlab, I'll figure out the formula by hand)

[Torsten]

You didn't get David's point.

Your formula to calculate psi as

psi = atan2(sin(phi)*sin(theta),cos(phi)+cos(theta))

seems to be wrong.

But now you know the solution to write psi in terms of phi and theta in symbolic form (the expression from octave). Why don't you simply copy it and use it in your code or whereever you need it ?

[Shiv Karpoor]

Oh, Okay.

Might be wrong I can't tell, because I got it from journal paper.

I will do as suggested,

Thanks Torsten, I appreciate your help!