How to update quaternion based on 3d gyro data?

Question

I've got an initial quaternion , q0. And I get angular velocity measurments, I integrate the the velocities so I got 3 angles at 50Hz or so. How can make a quaternion based on the 3 angles? I can't just make 3 quaternions, can I?

So to make it clear.

Q.new=Q.new*Q.update(alfa,beta,gamma)

Q.new represents my current orientation in a quaternion, I want to update it by multiplying with a Q.update quaternion. How can I make the Q.update with the angles?

Thanks!

Answer 1

Forgive me thread necromancing, but the answers all seem to complicated and some, like me, may prefer this more 'convenient' approach:

Say omega=(alpha,beta,gamma) is the measured angular velocity vector of the gyros. Then we rotate

theta = ||omega||*dt; //length of angular velocity vector

many units (deg or rad dependent on gyro) around

v = omega / ( ||omega|| ); // normalized orientation of angular velocity vector

Thus, we can construct the rotation quaternion as:

Q.update = (cos(theta/2),v_x * sin(theta/2), v_y * sin(theta/2), v_z * sin(theta/2));

All that is left now is to rotate our current rotation by Q.update. This is trivial:

Q.new = multiply_quaternions(Q.update,Q.new); 
// note that Q.update * Q.new != Q.new * Q.update for quaternions

Done. Quaternions are beautiful, aren't they?

Some slides on gyros and quaternions that may be useful: http://stanford.edu/class/ee267/lectures/lecture10.pdf