I am representing the orientation of a rigid body as a quaternion and the angular velocity as a vector. I am attempting to implement an angle constraint directly from the quaternion without having to store a vector alongside the quaternion in the rigid body class to represent the orientation.
C = Qa - Qb - Qi where Qi is the initial difference in orientation between the bodies
dC/dt = wa - wb which are the angular velocities of the bodies already known
So the Jacobian is [ 0 1 0 -1]
When I get to the bias calculation of bias = beta/dt * C, this will return a quaternion
So -Jv = wb - wa - bias will yield and incompatible equation as we are subtracting a quaternion from a vector. What is the preferred method without having to convert C to a vector at each frame?
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