I need the angular velocity expressed as a quaternion for updating the quaternion every frame with the following expression in OpenCV:
q(k)=q(k-1)*qwt;
My angular velocity is
Mat w; //1x3
I would like to obtain a quaternion form of the angles
Mat qwt; //1x4
I couldn't find information about this, any ideas?
If I understand properly you want to pass from this Axis Angle form to a quaternion.
As shown in the link, first you need to calculate the module of the angular velocity (multiplied by delta(t) between frames), and then apply the formulas.
A sample function for this would be
// w is equal to angular_velocity*time_between_frames
void quatFromAngularVelocity(Mat& qwt, const Mat& w)
{
const float x = w.at<float>(0);
const float y = w.at<float>(1);
const float z = w.at<float>(2);
const float angle = sqrt(x*x + y*y + z*z); // module of angular velocity
if (angle > 0.0) // the formulas from the link
{
qwt.at<float>(0) = x*sin(angle/2.0f)/angle;
qwt.at<float>(1) = y*sin(angle/2.0f)/angle;
qwt.at<float>(2) = z*sin(angle/2.0f)/angle;
qwt.at<float>(3) = cos(angle/2.0f);
} else // to avoid illegal expressions
{
qwt.at<float>(0) = qwt.at<float>(0)=qwt.at<float>(0)=0.0f;
qwt.at<float>(3) = 1.0f;
}
}
qwt.at<float>(0)=qwt.at<float>(0)=qwt.at<float>(0)=0.0f; You set the same component over and over again.Almost every transformation regarding quaternions, 3D space, etc is gathered at this website.
You will find time derivatives for quaternions also.
I find it useful the explanation of the physical meaning of a quaternion, which can be seen as an axis angle where
a = angle of rotation
x,y,z = axis of rotation.
Then the conversion uses:
q = cos(a/2) + i ( x * sin(a/2)) + j (y * sin(a/2)) + k ( z * sin(a/2))
Here is explained thoroughly.
Hope this helped to make it clearer.
One little trick to go with this and get rid of those cos and sin functions. The time derivative of a quaternion q(t) is:
dq(t)/dt = 0.5 * x(t) * q(t)
Where, if the angular velocity is {w0, w1, w2} then x(t) is a quaternion of {0, w0, w1, w2}. See David H Eberly's book section 10.5 for proof
