2D rotation dynamics/control systems as a complex number

Question

I have a dynamic system (it's a rocket in a 2D plane), that I'd like to model the orientation of using complex numbers to remove the need for trig functions in my ode.

I'm having trouble defining the angular velocity in terms of complex multiplication? I've done this before for quaternions defining rotation in 3d, and then being able to use the first order Taylor series to determine the future orientation given some constant angular velocity and initial quaternion and a small delta t. But for some reason doing this with 1 rotation axis is escaping me.

Also, if I have an angular velocity represented as a complex number $(0 + 1j)/sec$, how can I turn this into a complex number that represents a portion of this angular velocity term. I need this specifically so I can integrate the angular velocity over a small dt to get the future complex number that represents my new orientation.

I'm really looking for a function of the form $c_1 = (1/2)*omega*c_0*dt$ where c_0 and c_1 are unit complex numbers representing the rocket's orientation vector, this way I can propagate the orientation at each time step, and be able to determine the magnitude of thrust in the world frame by using the elements of the orientation (unit) complex number.

Any guidance appreciated!

Answer 1

Using quaternion with available informations will reduce to single complex number attitude representation. Assuming x-y plane is where motion is confined to, direction of omega is fixed to be z-axis. With z being axis of rotation you can set two components of quaternion to be zero (conventionally q1 and q2). This approach will also lead to two unknowns and you can easily propagate quaternion using kinematic relation (with the need of normalization after each time step).