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Thread Subject:
Minimizing multi-variable function

Subject: Minimizing multi-variable function

From: Matthew

Date: 25 May, 2010 00:23:05

Message: 1 of 11

Hey guys,

I need to minimize this function:
theta = acosd(sind(delta)*sind(phi)*cosd(beta)-sind(delta)*cosd(phi)*...
    sind(beta)*cosd(psi)+cosd(delta)*cosd(phi)*cosd(beta)*cosd(omega)+...
    cosd(delta)*sind(phi)*sind(beta)*cosd(psi)*cosd(omega)+cosd(delta)*...
    sind(beta)*sind(psi)*sind(omega));

to be more specific, theta(delta,phi,beta,psi,omega)

delta, phi, and omega will be constant. I just want to find phi and beta that minimizes the function as close to zero as possible.

Subject: Minimizing multi-variable function

From: Matthew

Date: 25 May, 2010 00:30:24

Message: 2 of 11

** Want to find beta and psi

Subject: Minimizing multi-variable function

From: Roger Stafford

Date: 25 May, 2010 03:00:09

Message: 3 of 11

"Matthew " <matthewnorlander@gmail.com> wrote in message <htf599$1h4$1@fred.mathworks.com>...
> Hey guys,
>
> I need to minimize this function:
> theta = acosd(sind(delta)*sind(phi)*cosd(beta)-sind(delta)*cosd(phi)*...
> sind(beta)*cosd(psi)+cosd(delta)*cosd(phi)*cosd(beta)*cosd(omega)+...
> cosd(delta)*sind(phi)*sind(beta)*cosd(psi)*cosd(omega)+cosd(delta)*...
> sind(beta)*sind(psi)*sind(omega));
>
> to be more specific, theta(delta,phi,beta,psi,omega)
>
> delta, phi, and omega will be constant. I just want to find phi and beta that minimizes the function as close to zero as possible.

> ** Want to find beta and psi
- - - - - - - - - - - -
  Do this:

 A = sind(delta)*sind(phi)+cosd(delta)*cosd(phi)*cosd(omega);
 B = -sind(delta)*cosd(phi)+cosd(delta)*sind(phi)*cosd(omega);
 C = cosd(delta)*sind(omega);
 omega = 180/pi*atan2(C,B);
 beta = 180/pi*atan2(sqrt(B^2+C^2),A);

Then the argument of acosd will be 1 and thus theta will be 0.

  The reasoning is that you can easily show that A^2+B^2+C^2 = 1 is identically true. Then the argument of acosd can be written

 A*cosd(beta) + B*sind(beta)*cosd(psi) + C*sind(beta)*sind(psi) =
 A*cosd(beta) + (B*cosd(psi) + C*sind(psi))*sind(beta)

By choosing psi = 180/pi*atan2(C,B) we make B*cosd(psi) + C*sind(psi) equal to sqrt(B^2+C^2). Thus the argument of acosd becomes

 A*cosd(beta) + sqrt(B^2+C^2)*sind(beta)

Then by choosing beta = 180/pi*atan2(sqrt(B^2+C^2),A), we make the acosd argument equal to

 sqrt(A^2+B^2+C^2)

which is just 1. Hence acosd(1) = 0 which is its minimum principal value.

Roger Stafford

Subject: Minimizing multi-variable function

From: Roger Stafford

Date: 25 May, 2010 03:49:03

Message: 4 of 11

"Roger Stafford" <ellieandrogerxyzzy@mindspring.com.invalid> wrote in message <htfefp$op0$1@fred.mathworks.com>...
> .......
> A = sind(delta)*sind(phi)+cosd(delta)*cosd(phi)*cosd(omega);
> B = -sind(delta)*cosd(phi)+cosd(delta)*sind(phi)*cosd(omega);
> C = cosd(delta)*sind(omega);
> omega = 180/pi*atan2(C,B);
> beta = 180/pi*atan2(sqrt(B^2+C^2),A);
> .......
- - - - - - - - - -
  I mistakenly changed omega instead of psi. The code should read:

 A = sind(delta)*sind(phi)+cosd(delta)*cosd(phi)*cosd(omega);
 B = -sind(delta)*cosd(phi)+cosd(delta)*sind(phi)*cosd(omega);
 C = cosd(delta)*sind(omega);
 psi = 180/pi*atan2(C,B);
 beta = 180/pi*atan2(sqrt(B^2+C^2),A);

Roger Stafford

Subject: Minimizing multi-variable function

From: Walter Roberson

Date: 25 May, 2010 03:54:28

Message: 5 of 11

Roger Stafford wrote:
> "Matthew " <matthewnorlander@gmail.com> wrote in message
> <htf599$1h4$1@fred.mathworks.com>...

>> I need to minimize this function:
>> theta = acosd(sind(delta)*sind(phi)*cosd(beta)-sind(delta)*cosd(phi)*...
>> sind(beta)*cosd(psi)+cosd(delta)*cosd(phi)*cosd(beta)*cosd(omega)+...
>> cosd(delta)*sind(phi)*sind(beta)*cosd(psi)*cosd(omega)+cosd(delta)*...
>> sind(beta)*sind(psi)*sind(omega));

> Do this:

> Then the argument of acosd will be 1 and thus theta will be 0.

There are also solutions for arbitrary phi. They are, however, more than
1000 lines long.

Subject: Minimizing multi-variable function

From: Roger Stafford

Date: 25 May, 2010 05:38:03

Message: 6 of 11

Walter Roberson <roberson@hushmail.com> wrote in message <VRHKn.21398$mi.9382@newsfe01.iad>...
> ........
> There are also solutions for arbitrary phi. They are, however, more than
> 1000 lines long.
- - - - - - - - -
  How do you mean, Walter? Which of the five angles are to be arbitrarily fixed and which variable with this 1000 line solution?

Roger Stafford

Subject: Minimizing multi-variable function

From: Walter Roberson

Date: 25 May, 2010 06:15:54

Message: 7 of 11

Roger Stafford wrote:
> Walter Roberson <roberson@hushmail.com> wrote in message
> <VRHKn.21398$mi.9382@newsfe01.iad>...
>> ........
>> There are also solutions for arbitrary phi. They are, however, more
>> than 1000 lines long.
> - - - - - - - - -
> How do you mean, Walter? Which of the five angles are to be
> arbitrarily fixed and which variable with this 1000 line solution?

In the original posting, the poster indicated that solving for phi and
beta was required, and that's what I fed into Maple. The poster followed
up indicating it was psi and beta to be solved for, but since you had
posted a much better solution, I didn't bother to retry the solution.

The solution involved converting to exp and simplify()'ing that, which
spits out a bunch of lines of the form

      - 1/8 cos(-delta - phi + beta + psi)

      - 1/16 cos(-delta + phi + beta - psi + omega)

with sign changes for the various variables.

When I do the same thing for psi and beta instead, the solution is a
list of arctans for psi and arctans for beta, with some fairly long
expressions for some of them. One of the solutions has beta as a free
parameter rather than an arctan, and the expression for psi is by far
the longest for that possibility. The major component of it is the root
of a cubic... pages and pages of it.

Subject: Minimizing multi-variable function

From: Roger Stafford

Date: 25 May, 2010 07:44:07

Message: 8 of 11

Walter Roberson <roberson@hushmail.com> wrote in message <vWJKn.12303$7d5.1931@newsfe17.iad>...
> In the original posting, the poster indicated that solving for phi and
> beta was required, and that's what I fed into Maple. The poster followed
> up indicating it was psi and beta to be solved for, but since you had
> posted a much better solution, I didn't bother to retry the solution.
>
> The solution involved converting to exp and simplify()'ing that, which
> spits out a bunch of lines of the form
>
> - 1/8 cos(-delta - phi + beta + psi)
>
> - 1/16 cos(-delta + phi + beta - psi + omega)
>
> with sign changes for the various variables.
>
> When I do the same thing for psi and beta instead, the solution is a
> list of arctans for psi and arctans for beta, with some fairly long
> expressions for some of them. One of the solutions has beta as a free
> parameter rather than an arctan, and the expression for psi is by far
> the longest for that possibility. The major component of it is the root
> of a cubic... pages and pages of it.
- - - - - - - - -
  You're right, Walter. Adjusting phi and beta is proving to be much more difficult than for psi and beta. I'll see if my brain works better tomorrow morning.

Roger Stafford

Subject: Minimizing multi-variable function

From: Roger Stafford

Date: 25 May, 2010 18:14:05

Message: 9 of 11

Walter Roberson <roberson@hushmail.com> wrote in message <vWJKn.12303$7d5.1931@newsfe17.iad>...
> Roger Stafford wrote:
> > Walter Roberson <roberson@hushmail.com> wrote in message
> > <VRHKn.21398$mi.9382@newsfe01.iad>...
> >> ........
> >> There are also solutions for arbitrary phi. They are, however, more
> >> than 1000 lines long.
> > - - - - - - - - -
> > How do you mean, Walter? Which of the five angles are to be
> > arbitrarily fixed and which variable with this 1000 line solution?
>
> In the original posting, the poster indicated that solving for phi and
> beta was required, and that's what I fed into Maple. The poster followed
> up indicating it was psi and beta to be solved for, but since you had
> posted a much better solution, I didn't bother to retry the solution.
>
> The solution involved converting to exp and simplify()'ing that, which
> spits out a bunch of lines of the form
>
> - 1/8 cos(-delta - phi + beta + psi)
>
> - 1/16 cos(-delta + phi + beta - psi + omega)
>
> with sign changes for the various variables.
>
> When I do the same thing for psi and beta instead, the solution is a
> list of arctans for psi and arctans for beta, with some fairly long
> expressions for some of them. One of the solutions has beta as a free
> parameter rather than an arctan, and the expression for psi is by far
> the longest for that possibility. The major component of it is the root
> of a cubic... pages and pages of it.
- - - - - - -
  Walter, this morning I continued to have trouble showing that a phi and beta solution could always be found, given any delta, psi, and omega, but finally I stumbled onto a counterexample! If you set delta = 20 degrees, omega = 240 degrees, and psi = 175 degrees, then no matter what phi and beta are, the argument, x, of acosd above never climbs above about .65, so that no solution is possible.

  To show this, let

 A = sind(delta);
 B = -sind(delta)*cosd(psi);
 C = cosd(delta)*cosd(omega);
 D = cosd(delta)*cosd(psi)*cosd(omega);
 E = cosd(delta)*sind(psi)*sind(omega);

For each value of phi define

 S = A*sind(phi)+C*cosd(phi);
 T = B*cosd(phi)+D*sind(phi)+E;

Then the argument x as a function of phi and beta can be expressed as

 x = S*cosd(beta)+T*sind(beta);

For each possible phi, if beta is set to 180/pi*atan2(T,S), the value of x for that fixed phi will be maximized at sqrt(S^2+T^2). This lets us plot this maximized x as a function of phi, and it is clear looking at the plot that it never gets anywhere near 1.

  I think Maple must surely have slipped a cog somewhere in its 1000 lines. I don't see any hole in the above reasoning.

Roger Stafford

Subject: Minimizing multi-variable function

From: Walter Roberson

Date: 25 May, 2010 19:14:50

Message: 10 of 11

Roger Stafford wrote:
> this morning I continued to have trouble showing
> that a phi and beta solution could always be found, given any delta,
> psi, and omega, but finally I stumbled onto a counterexample! If you
> set delta = 20 degrees, omega = 240 degrees, and psi = 175 degrees, then
> no matter what phi and beta are, the argument, x, of acosd above never
> climbs above about .65, so that no solution is possible.

> I think Maple must surely have slipped a cog somewhere in its 1000
> lines. I don't see any hole in the above reasoning.

Maple doesn't assume that solutions are to be confined to real numbers :)


I will re-run the solution a bit later and see what answer maple comes up with.

Subject: Minimizing multi-variable function

From: Walter Roberson

Date: 25 May, 2010 21:22:15

Message: 11 of 11

Roger Stafford wrote:

> - - - - - - - Walter, this morning I continued to have trouble showing
> that a phi and beta solution could always be found, given any delta,
> psi, and omega, but finally I stumbled onto a counterexample! If you
> set delta = 20 degrees, omega = 240 degrees, and psi = 175 degrees, then
> no matter what phi and beta are, the argument, x, of acosd above never
> climbs above about .65, so that no solution is possible.

Continuing this out of curiousity, knowing that it wasn't of use to the
original poster:

Solving for phi and beta, Maple finds 37 solutions, _one_ of which has beta as
a free parameter. That full expression is rather long unless you do
sub-expression elimination. I include the simplified version below, for no
good reason :)

The expression does generate a complex number for the specific case you
indicated; it appears to involve roughly -Pi/2*I . The other 36 solutions
appeared to involve +/- Pi*I and are all complex numbers for your sample angles.


Note that the below is coded for radians.

T := simplify(solve(1 = sin(delta)*sin(phi)*cos(beta) -
sin(delta)*cos(phi)*sin(beta)*cos(psi) +
cos(delta)*cos(phi)*cos(beta)*cos(omega) +
cos(delta)*sin(phi)*sin(beta)*cos(psi)*cos(omega) +
cos(delta)*sin(beta)*sin(psi)*sin(omega), [phi, beta]));

codegeneration[optimize](T);


t1 = cos(delta),
t2 = t1^2;
t3 = cos(omega);
t4 = t3^2;
t5 = t4*t2;
t6 = cos(psi);
t7 = t6^2;
t11 = RootOf(1 - 4*cos(delta)*sin(beta)*sin(psi)*sin(omega) + cos(beta)^4 -
6*cos(delta)^2*cos(psi)^2 + cos(delta)^4 +
8*cos(delta)^2*cos(beta)^2*cos(omega)^2 -
8*cos(delta)^2*cos(beta)^2*cos(psi)^2*cos(omega)^2 +
4*cos(delta)^2*cos(beta)^4*cos(psi)^2*cos(omega)^2 -
2*cos(delta)^4*cos(beta)^4*cos(psi)^2*cos(omega)^2 -
4*cos(delta)^3*sin(beta)*sin(psi)*sin(omega)*cos(beta)^2*cos(omega)^2 -
4*cos(delta)^3*sin(beta)*sin(psi)*sin(omega)*cos(beta)^2*cos(psi)^2 +
2*cos(delta)^4*cos(psi)^2*cos(omega)^2 + 6*cos(delta)^2 -
4*cos(delta)^3*sin(beta)*sin(psi)*sin(omega) - 6*cos(delta)^2*cos(beta)^2 -
2*cos(delta)^2*cos(beta)^4*cos(omega)^2 +
2*cos(delta)^4*cos(beta)^2*cos(omega)^2 +
cos(delta)^4*cos(beta)^4*cos(omega)^4 - 2*cos(beta)^2 -
2*cos(delta)^4*cos(beta)^2*cos(omega)^4 +
8*cos(delta)^2*cos(beta)^2*cos(psi)^2 - 2*cos(delta)^2*cos(beta)^4*cos(psi)^2
+ 2*cos(delta)^4*cos(beta)^2*cos(psi)^2 -
2*cos(delta)^4*cos(beta)^2*cos(psi)^4 + cos(delta)^4*cos(beta)^4*cos(psi)^4 -
2*cos(delta)^4*cos(psi)^2 + cos(delta)^4*cos(psi)^4 + (cos(beta)^4 +
cos(psi)^4 + 4*cos(delta)^2*cos(beta)^2*cos(psi)^2*cos(omega)^2 -
4*cos(delta)^2*cos(beta)^4*cos(psi)^2*cos(omega)^2 +
4*cos(delta)^4*cos(beta)^4*cos(psi)^2*cos(omega)^2 +
2*cos(delta)^2*cos(beta)^4*cos(omega)^2 -
2*cos(delta)^4*cos(beta)^4*cos(omega)^2 +
cos(delta)^4*cos(beta)^4*cos(omega)^4 - 4*cos(delta)^2*cos(beta)^2*cos(psi)^2
+ 4*cos(delta)^2*cos(beta)^4*cos(psi)^2 +
2*cos(delta)^4*cos(beta)^2*cos(psi)^2 - 2*cos(delta)^4*cos(beta)^2*cos(psi)^4
+ cos(delta)^4*cos(beta)^4*cos(psi)^4 + cos(delta)^4*cos(psi)^4 +
2*cos(beta)^2*cos(psi)^2 - 2*cos(beta)^2*cos(psi)^4 - 2*cos(beta)^4*cos(psi)^2
+ cos(beta)^4*cos(psi)^4 - 2*cos(delta)^2*cos(psi)^4 +
4*cos(delta)^2*cos(beta)^2*cos(psi)^4 - 2*cos(delta)^2*cos(beta)^4*cos(psi)^4
- 2*cos(delta)^4*cos(beta)^4*cos(psi)^2 +
2*cos(delta)^2*cos(psi)^4*cos(omega)^2 -
2*cos(delta)^4*cos(psi)^4*cos(omega)^2 + cos(delta)^4*cos(psi)^4*cos(omega)^4
- 2*cos(delta)^2*cos(beta)^4 + cos(delta)^4*cos(beta)^4 -
4*cos(delta)^2*cos(beta)^2*cos(psi)^4*cos(omega)^2 +
2*cos(delta)^2*cos(beta)^4*cos(psi)^4*cos(omega)^2 -
4*cos(delta)^4*cos(beta)^2*cos(psi)^2*cos(omega)^2 +
2*cos(delta)^4*cos(beta)^2*cos(psi)^2*cos(omega)^4 +
4*cos(delta)^4*cos(beta)^2*cos(psi)^4*cos(omega)^2 -
2*cos(delta)^4*cos(beta)^2*cos(psi)^4*cos(omega)^4 -
2*cos(delta)^4*cos(beta)^4*cos(psi)^2*cos(omega)^4 -
2*cos(delta)^4*cos(beta)^4*cos(psi)^4*cos(omega)^2 +
cos(delta)^4*cos(beta)^4*cos(psi)^4*cos(omega)^4)*_Z^4 +
(8*cos(psi)^3*sin(omega)*sin(delta)*cos(delta)^3*sin(psi)*cos(beta)^2*cos(omega)^2
+ 4*cos(psi)^3*sin(delta)*sin(beta) -
4*cos(psi)^3*sin(omega)*sin(delta)*cos(delta)*sin(psi)*cos(beta)^4 -
8*cos(psi)^3*sin(delta)*cos(delta)^3*sin(psi)*sin(omega)*cos(beta)^2 -
4*cos(psi)^3*sin(omega)*sin(delta)*cos(delta)*sin(psi) +
8*cos(psi)^3*sin(omega)*sin(delta)*cos(delta)*sin(psi)
*cos(beta)^2 +
4*cos(psi)*sin(delta)*cos(delta)*sin(psi)*sin(omega)*cos(beta)^4 -
4*cos(psi)^3*sin(delta)*sin(beta)*cos(beta)^2 -
4*cos(psi)*sin(omega)*sin(delta)*cos(delta)^3*sin(psi)*cos(beta)^4 +
4*cos(psi)^3*sin(delta)*cos(delta)^3*sin(psi)*sin(omega)*cos(beta)^4 +
4*cos(psi)*sin(delta)*sin(beta)*cos(beta)^2 +
4*cos(psi)*sin(delta)*cos(delta)^3*sin(psi)*sin(omega)*cos(beta)^2 +
4*cos(psi)*sin(delta)*cos(delta)^3*sin(psi)*sin(omega)*cos(beta)^4*cos(omega)^2
- 4*cos(psi)^3*sin(delta)*sin(beta)*cos(delta)^2 -
4*cos(psi)^3*sin(omega)*sin(delta)*cos(delta)^3*sin(psi)*cos(omega)^2 +
4*cos(psi)^3*sin(delta)*cos(delta)^3*sin(psi)*sin(omega) - 4*cos(psi)^3*
sin(delta)*sin(beta)*cos(delta)^2*cos(beta)^2*cos(omega)^2 +
4*cos(psi)^3*sin(delta)*sin(beta)*cos(delta)^2*cos(omega)^2 -
4*cos(psi)*sin(delta)*sin(beta)*cos(delta)^2*cos(beta)^2 -
4*cos(psi)*sin(delta)*cos(delta)*sin(psi)*sin(omega)*cos(beta)^2 +
4*cos(psi)^3*sin(delta)*sin(beta)*cos(delta)^2*cos(beta)^2 -
4*cos(psi)^3*sin(omega)*sin(delta)*cos(delta)^3*sin(psi)*cos(beta)^4*cos(omega)^2
+ 4*cos(psi)*sin(delta)*sin(beta)*cos(delta)^2*cos(beta)^2*cos(omega)^2 -
4*cos(psi)*sin(omega)*sin(delta)*cos(delta)^3*sin(psi)*cos(beta)^2*cos(omega)^2)*_Z^3
+ (6*cos(psi)^2 - 2*cos(beta)^4 -
4*cos(delta)^2*cos(beta)^2*cos(omega)^2 +
8*cos(delta)^2*cos(beta)^2*cos(psi)^2*cos(omega)^2 -
4*cos(delta)^2*cos(beta)^4*cos(psi)^2*cos(omega)^2 +
4*cos(delta)^4*cos(beta)^4*cos(psi)^2*cos(omega)^2 -
4*cos(delta)^3*sin(beta)*sin(psi)*sin(omega)*cos(psi)^2*cos(omega)^2 +
12*cos(delta)*sin(beta)*sin(psi)*sin(omega)*cos(beta)^2*cos(psi)^2 +
4*cos(delta)^3*sin(beta)*sin(psi)*sin(omega)*cos(beta)^2*cos(omega)^2 - 12*
cos(delta)^3*sin(beta)*sin(psi)*sin(omega)*cos(beta)^2*cos(psi)^2 +
8*cos(delta)^4*cos(psi)^2*cos(omega)^2 -
2*cos(delta)^4*cos(psi)^2*cos(omega)^4 +
4*cos(delta)^3*sin(beta)*sin(psi)*sin(omega)*cos(beta)^2*cos(psi)^2*cos(omega)^2
+ 2*cos(delta)^4*cos(beta)^4*cos(omega)^2 -
2*cos(delta)^4*cos(beta)^4*cos(omega)^4 + 2*cos(beta)^2 +
2*cos(delta)^4*cos(beta)^2*cos(omega)^4 -
4*cos(delta)^2*cos(beta)^2*cos(psi)^2 + 4*cos(delta)^2*cos(beta)^4*cos(psi)^2
+ 12*cos(delta)^4*cos(beta)^2*cos(psi)^2 - 12*cos(
delta)^4*cos(beta)^2*cos(psi)^4 + 6*cos(delta)^4*cos(beta)^4*cos(psi)^4 -
6*cos(delta)^4*cos(psi)^2 + 6*cos(delta)^4*cos(psi)^4 -
8*cos(beta)^2*cos(psi)^2 + 2*cos(beta)^4*
cos(psi)^2 - 6*cos(delta)^2*cos(psi)^4 +
12*cos(delta)^2*cos(beta)^2*cos(psi)^4 - 6*cos(delta)^2*cos(beta)^4*cos(psi)^4
- 6*cos(delta)^4*cos(beta)^4*cos(psi)^2 +
4*cos(delta)^2*cos(psi)^4*cos(omega)^2 -
6*cos(delta)^4*cos(psi)^4*cos(omega)^2 +
12*cos(delta)^3*sin(beta)*sin(psi)*sin(omega)*cos(psi)^2 -
4*cos(delta)*sin(beta)*sin(psi)*sin(omega)*cos(beta)^2 -
2*cos(delta)^4*cos(beta)^2 + 2*cos(delta)^2*cos(beta)^4 -
4*cos(delta)^2*cos(omega)^2*cos(psi)^2 -
8*cos(delta)^2*cos(beta)^2*cos(psi)^4*cos(omega)^2 +
4*cos(delta)^2*cos(beta)^4*cos(psi)^4*cos(omega)^2 -
12*cos(delta)^4*cos(beta)^2*cos(psi)^2*cos(omega)^2 +
12*cos(delta)^4*cos(beta)^2*cos(psi)^4*cos(omega)^2 +
2*cos(delta)^4*cos(beta)^4*cos(psi)^2*cos(omega)^4 -
6*cos(delta)^4*cos(beta)^4*cos(psi)^4*cos(omega)^2 +
4*cos(delta)^3*sin(beta)*sin(psi)*sin(omega)*cos(beta)^2 -
12*cos(delta)*sin(beta)*sin(psi)*sin(omega)*cos(psi)^2)*_Z^2 + ( -
12*cos(psi)^3*sin(delta)*sin(beta)*cos(delta)^2 -
4*cos(psi)*sin(delta)*cos(delta)^3*sin(psi)*sin(omega) -
4*cos(psi)*sin(delta)*cos(delta)*sin(psi)*sin(omega)*cos(beta)^4 -
8*cos(psi)^3*sin(delta)*sin(beta)*cos(delta)^2*cos(beta)^2*cos(omega)^2 +
8*cos(psi)^3*sin(delta)*sin(beta)*cos(delta)^2*cos(omega)^2 +
16*cos(psi)*sin(delta)*cos(delta)*sin(psi)*sin(omega)*cos(beta)^2 +
12*cos(psi)^3*sin(delta)*sin(beta)*cos(delta)^2*cos(beta)^2 +
4*cos(psi)*sin(delta)*sin(beta) -
8*cos(psi)*sin(delta)*sin(beta)*cos(delta)^2*cos(beta)^2 +
4*cos(psi)*sin(delta)*cos(delta)^3*sin(psi)*sin(omega)*cos(omega)^2 +
4*cos(psi)^3*sin(delta)*cos(delta)^3*sin(psi)*sin(omega)*cos(beta)^4 +
4*cos(psi)*sin(delta)*sin(beta)*cos(delta)^2*cos(beta)^2*cos(omega)^2 +
4*cos(psi)^3*sin(delta)*cos(delta)^3*sin(psi)*sin(omega) -
4*cos(psi)*sin(delta)*sin(beta)*cos(beta)^2 +
12*cos(psi)*sin(delta)*sin(beta)*cos(delta)^2 -
12*cos(psi)*sin(delta)*sin(beta)*cos(delta)^2*cos(omega)^2 -
4*cos(psi)*sin(delta)*cos(delta)^3*sin(psi)*sin(omega)*cos(beta)^4*cos(omega)^2
- 12*cos(psi)*sin(delta)*cos(delta)*sin(psi)*sin(omega) +
4*cos(psi)*sin(delta)*cos(delta)^3*sin(psi)*sin(omega)*cos(beta)^2 -
8*cos(psi)^3*sin(delta)*cos(delta)^3*sin(psi)*sin(omega)*cos(beta)^2)*_Z +
4*cos(delta)^3*sin(beta)*sin(psi)*sin(omega)*cos(omega)^2 +
4*cos(delta)^3*sin(beta)*sin(psi)*sin(omega)*cos(psi)^2 +
4*cos(delta)*sin(beta)*sin(psi)*sin(omega)*cos(beta)^2 -
6*cos(delta)^2*cos(omega)^2 + 4*cos(delta)^2*cos(omega)^2*cos(psi)^2 -
2*cos(omega)^2*cos(delta)^4 + cos(omega)^4*cos(delta)^4);
t12 = t11^2;
t13 = t7*t12;
t14 = cos(beta);
t15 = t14^2;
t19 = t11*t1*t6;
t20 = sin(delta);
t21 = sin(psi);
t22 = t21*t20;
t23 = sin(omega);
t31 = t2*t12;
t34 = t4*t7;
t36 = t7*t15;
t38 = 1 + 2*t7*t5 - t7*t2 + t13 - t15*t13 + t15*t12 - 2*t23*t22*t19 +
2*t15*t23*t22*t19 - t4*t15*t31 - t34*t31 + t36*t31;
t39 = sin(beta);
t44 = t15*t2;
t57 = - 2*t23*t21*t39*t1 - 2*t34*t44 + t2 - t5 - t15 - t7*t31
  + t4*t36*t31 + t4*t44 + t7*t44 - t15*t31 + 2*t39*t11*t20*t6;
t63 = t21*t6;
t64 = t1*t23;
t68 = t11*t20;
t77 = arctan( - 1/2/( - t6*t39 + t64*t63 - t15*t64*t63 - t15*t68 - t7*t68 +
t36*t68)/t1/t3*(t38 + t57), t11);

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