Solving for rotation matrix: Align vector a with vector b, then rotate around vector b

Currently working on a quadcopter simulation. I have a desired thrust vector, t =sin(30)cos(45)a1+sin(30)sin(45)a2+cos(30)a3, and desired yaw angle, psi = 45. Because we are working with an under-actuated system, I am trying to solve for the rotation matrix that aligns the vector b3 = [0,0,1] (direction of thrust in the body-fixed frame of reference) with the direction of t. My code is included at the bottom. The problem is that it outputs an empty symbolic struct, with no warnings or errors otherwise:

What I am doing wrong? It is an issue with my theory, my implementation, or both? Thank you!

t = [sind(30) * cosd(45), sind(30) * sind(45), cosd(30)]; %desired thrust vector
psi = 45; %desired yaw angle
b3 = [0;0;1]; %body-fixed unit thrust vector

syms phi theta; %setting up to solve what's left of the rotation matrix

eqn = [[cosd(psi)*cosd(theta)-sind(phi)*sind(psi)*sind(theta), -cosd(phi)*sind(psi), cosd(psi)*sind(theta)+cosd(theta)*sind(phi)*sind(phi);
 cosd(theta)*sind(psi)+cosd(psi)*sind(phi)*sind(theta), cosd(phi)*cosd(psi), sind(psi)*sind(theta)-cosd(theta)*sind(phi)*cosd(psi);
 -cosd(phi)*sind(theta), sind(phi), cosd(phi)*cosd(theta)] * b3 == t/norm(t)]; %equation to be solved (general form of the rotation matrix multiplied by body-fixed thrust aligned with t)

S = solve(eqn,[phi theta]); %solve

Answer 1

[darova]

Here is an idea

• Use dot product to calculate angle between vectors

• Use cross product to calculate normal vector of a plane

You have normal vector and angle. COnstruct new rotation matrix

[Michael Lee]

Thank you! This pointed me in the right direction, and I didn't even need the numerical solver. For anyone else aligning vector a with vector b and then rotating around vector b, here is my code:

t = [sind(30) * cosd(45), sind(30) * sind(45), cosd(30)];
b = [0;0;1];
psi = deg2rad(45);

v = cross(b,t);
s = norm(v);
c = dot(b,t);

phiB = psi;
uB = b;
uBhat = vector2ssMat(uB);

I = eye(3);

rotA = I+vector2ssMat(v)+(vector2ssMat(v)^2)*((1-c)/s^2);
rotB = I*cos(phiB)+uB*transpose(uB)*(1-cos(phiB))+uBhat*sin(phiB);

Rdes = rotA*rotB;
disp(Rdes);

[darova]

Using ROdrigues formula

t = [sind(30) * cosd(45), sind(30) * sind(45), cosd(30)];
t = t/norm(t);
a = acos(dot(t,[0 0 1]));                       % angle to rotate (Z axis)
k = cross(t,[0 0 1]);                           % axis of rotation
k = k/norm(k);
K = zeros(3);
K([2 3 6]) = [k(3) -k(2) k(1)];
K = K - K';
ROT = @(a) eye(3) + sin(a)*K + (1-cos(a))*K^2;  % matrix of rotation

[Maaz Jamshaid]

vector2ssMat is an unrecognized function. how do i deal with this??