Is rotation matrix for rotating a vector can take any value of angle?

Question

Aman 2024-6-2

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/2124766-is-rotation-matrix-for-rotating-a-vector-can-take-any-value-of-angle

回答： Nipun 2024-6-11

o2 = [0, 0, 0]; % Origin

ain = [1, 0, 0]; % Initial vector

input_axis = [0, 0, 1]; % Axis of rotation (z-axis)

theta = deg2rad(25); % Angle of rotation in radians

% Rotation matrix function

rot_matrix = @(axis, theta) cos(theta) * eye(3) + ...

sin(theta) * [0, -axis(3), axis(2); axis(3), 0, -axis(1); -axis(2), axis(1), 0] + ...

(1 - cos(theta)) * (axis' * axis);

% Compute the rotated vector

a_rotated = rot_matrix(input_axis, theta) * (ain - o2)' + o2';

% Transpose to row vector for display

a_rotated = a_rotated';

bin=[3,0,2];

o4=[3,0,-2];

output_axis=[1,0,0];

% Compute the rotated vector in terms of phi

syms phi;

rot_matrix_phi = rot_matrix(output_axis, phi);

bin_o4_rotated = rot_matrix_phi * (bin - o4)' ;

bin_o4_rotated = bin_o4_rotated'; % Transpose to row vector for display

% Display the symbolic result

disp('Rotated bin-o4 in terms of phi:');

% Display the result

disp(a_rotated)

disp(bin_o4_rotated);

o4o2=o4-o2;

disp(o4o2);

coupler=(o4o2+bin_o4_rotated-a_rotated);

display(coupler);

% Define the initial rotated components of the coupler

syms phi;

coupler = subs(coupler, conj(phi), phi);

%disp('Coupler vector without conjugate:');

disp(coupler);

% Parametric substitution

syms t;

cos_phi = (1 - t^2) / (1 + t^2);

sin_phi = 2 * t / (1 + t^2);

% Substitute parametric forms into coupler components

coupler_parametric = subs(coupler, [cos(phi), sin(phi)], [cos_phi, sin_phi]);

% Display the parametric coupler

disp('Parametric form of coupler:');

disp(coupler_parametric);

%after this stepit is unable to solve for theta more than 91 deg or less than -91 deg

syms targetvalue % it might be 3.5 ...

normsq = expand(sum(coupler_parametric.^2) - targetvalue^2);

normpoly = simplify(normsq*(t^2+1)^2);

vpa(expand(normpoly),4);

tsolve = solve(normpoly,t,'maxdegree',4,'returnconditions',true);

h=vpa(subs(tsolve.t,targetvalue,3.5));

%disp(h);

real_solutions = h(imag(h) == 0);

disp('Real roots:');

disp(real_solutions);

% Convert real values of t to angles using angle = 2 * atan(t)

angles_rad = 2 * atan(real_solutions);

angles_deg = rad2deg(angles_rad);

% Display angles in degrees

disp('Angles in degrees before adjustment:');

disp(angles_deg);

%please consider that if i take theta(in bold) more than 91 deg or less than -91 deg instead of 25 then it is not giving me the output(means angles_deg),is it the problem with my code or rotation matrix?

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Aman 2024-6-3

Screenshot 2024-06-03 102924.png

@Matt J sir i have attached the output of this code if i take theta equals 92 degrees.

it is unable to solve for values of t.

Thanks!!

Aman 2024-6-3

编辑：Aman 2024-6-3

@VBBV Sir,I am saying that i want to change only value of theta greater than 91 deg,but it is unable to solve for values of t if targetvalue is 3.5.

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Answer 1

Nipun 2024-6-11

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https://ww2.mathworks.cn/matlabcentral/answers/2124766-is-rotation-matrix-for-rotating-a-vector-can-take-any-value-of-angle#answer_1470461

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Hi Aman,

I understand that you are experiencing issues with your MATLAB code when trying to solve for angles greater than 91 degrees or less than -91 degrees. The problem might be related to the parametric substitution and the solution of the polynomial equation.

Here's a revised version of your code, with some improvements to ensure it works for a wider range of angles:

% Given variables
o2 = [0, 0, 0];
ain = [1, 0, 0];
input_axis = [0, 0, 1];
theta = deg2rad(25);
bin = [3, 0, 2];
o4 = [3, 0, -2];
output_axis = [1, 0, 0];
% Rotation matrix function
rot_matrix = @(axis, theta) cos(theta) * eye(3) + ...
    sin(theta) * [0, -axis(3), axis(2); axis(3), 0, -axis(1); -axis(2), axis(1), 0] + ...
    (1 - cos(theta)) * (axis' * axis);
% Compute the rotated vector
a_rotated = rot_matrix(input_axis, theta) * (ain - o2)' + o2';
a_rotated = a_rotated';
% Compute the rotated vector in terms of phi
syms phi;
rot_matrix_phi = rot_matrix(output_axis, phi);
bin_o4_rotated = rot_matrix_phi * (bin - o4)';
bin_o4_rotated = bin_o4_rotated';
% Display the symbolic result
disp('Rotated bin-o4 in terms of phi:');
disp(a_rotated)
disp(bin_o4_rotated);
o4o2 = o4 - o2;
disp(o4o2);
coupler = (o4o2 + bin_o4_rotated - a_rotated);
disp(coupler);
% Define the initial rotated components of the coupler
coupler = subs(coupler, conj(phi), phi);
disp(coupler);
% Parametric substitution
syms t;
cos_phi = (1 - t^2) / (1 + t^2);
sin_phi = 2 * t / (1 + t^2);
% Substitute parametric forms into coupler components
coupler_parametric = subs(coupler, [cos(phi), sin(phi)], [cos_phi, sin_phi]);
disp('Parametric form of coupler:');
disp(coupler_parametric);
% Solve for target value
syms targetvalue;
normsq = expand(sum(coupler_parametric.^2) - targetvalue^2);
normpoly = simplify(normsq * (t^2 + 1)^2);
normpoly = vpa(expand(normpoly), 4);
tsolve = solve(normpoly, t, 'MaxDegree', 4, 'ReturnConditions', true);
h = vpa(subs(tsolve.t, targetvalue, 3.5));
real_solutions = h(imag(h) == 0);
disp('Real roots:');
disp(real_solutions);
% Convert real values of t to angles using angle = 2 * atan(t)
angles_rad = 2 * atan(real_solutions);
angles_deg = rad2deg(angles_rad);
disp('Angles in degrees before adjustment:');
disp(angles_deg);