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. Here is my code:

t = [sind(30) * cosd(45), sind(30) * sind(45), cosd(30)];

b3 = [0;0;1];

psi = 45;

syms phi theta; 

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), 
 -cosd(phi)*sind(theta), sind(phi), cosd(phi)*cosd(theta)] 
* b3 == t/norm(t)];

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

The problem is that it outputs an empty symbolic struct, with no warnings or errors otherwise: Output

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

  • $\begingroup$ I believe you have an error in your rotation matrix. Namely the upper right element contains sind(phi)*sind(phi). The rest of your rotation matrix looks like $Z_1\,X_2\,Y_3$ as shown here, so instead of sind(phi)*sind(phi) I believe you should use sind(phi)*sind(psi). $\endgroup$
    – fibonatic
    Mar 18, 2020 at 11:57
  • $\begingroup$ Thank you. I believe this is correct, although was able to find an analytical solution which was much faster. Will post below. $\endgroup$
    – Max Lester
    Mar 20, 2020 at 2:24

2 Answers 2


Analytic Solution:

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;

I made the same mistake, actually u need to write the thrust vector t as column vector instead of the row vector. u can simply replace t with t' at all occurrences, it will solve the issue.


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