how to extrapolate a rotation matrix values?

Question

I am using chai3d api, which uses a 3x3 floating pt matrix for storing objects orientation in my virtual world.

I want to predict these orientations on client side, after periodic updates from server, so that I have a consistent virtual graphical world.

I predict the objects (e.g. opengl cube) position by sending a position and velocity value.

Is angular velocity for orientation same as velocity for position?

if yes how do I calculate the angular velocity from this 3x3 matrix and use it for extrapolation?

Answer 1

A transformation matrix is essentially a representation of a new coordinate system within another coordinate system. If you add a column you can even put the translation into it. If you remember your calculus and physics, then you may remember

r = 1/2 a² t + v0 t + r0

v = d/dt r = a + v0

a = d/dt v

To get from velocity 'v' to position 'r' you have to integrate. In the case of scalars you multiply v with time. But scalar multiplication with a matrix will just scale it, not rotate it. So you must do something else. The keyword, if you want to do this using matrices is matrix powers, i.e. calculating the powers of a matrix.

Say you have a differential rotation, d/dt R, then you would integrate this, by multiplying the corresponding rotation matrix infinitesimaly often with itself, i.e. take a power.

But there's also a mathematically much nicer way to do this. Something very close to just multiplying with a factor. And that is: Using quaternions instead of matrices to represent orientations. It turns out that simply scaling a quaternions is the same as just multiplying on the rotation it desscribes.

The keywords you should Google for (because StackOverflow is the wrong place for introducing one into the whole theory of quaternions) are:

• quaternion
• angular velocity
• angular interpolation
• SLERP http://en.wikipedia.org/wiki/Slerp