Ambiguity in DCM to Quaternion conversion using the default Simulink library block

Question

I am simulating a system where I need Direction Cosine Matrix to quaternion conversion. I use the default DCM to Quaternion conversion block available in simulink. However at some points of the simulation, the output quaternion components reverse sign.

Unfortunately I cannot attach the plot image.

Though this is mathematically correct I desire a smooth change. Any idea on how to avoid this and have a smooth curve for the quaternion?

Update 1: http://tinypic.com/view.php?pic=33dayap&s=6

Above is the simulated plot. The first plot is of the output quaternion. Second plot is of the Direction Cosine Matrix. As you see that even though the dcm components change smoothly, the quaternion changes sign abruptly.

Answer 1

The problem arises because of the double covering property of quaternions: Two unit quaternions correspond to every rotation. At some point, according to some rule, the Matlab implementation switched from one quaternion to the other. There is not much you can do about it.

A messy workaround would be to write your own rotation matrix to quaternion conversion, and pick that representation of the two possibilities that is closer to the previous one, hence avoiding the sudden jumps. It's messy.

Plotting the quaternions is typically not needed in practical applications. Most likely you are rotating an object / vector. If you plot that object / vector (or some projections of it) you won't get any sudden jumps even if there are jumps in the representation of the rotation. Another benefit of plotting the projections of the rotated object is that it is usually much easier to interpret these plots than the quaternions. I don't know whether it makes sense in your application; it worked beautifully in mine.