Consider a robot following the dynamical system $R_B'=R_B\exp(\omega_B^{\times}), v_W'=R_B\alpha_B + g_W$, where $R_B\in SO(3), v, \alpha, \omega, g \in \mathbb{R}^3$, and $\omega^\times$ is the skew symmetric matrix associated to $\omega$. $v_B$ means a vector in the frame $B$ described by $R_B$, so that $v$ in the "world" frame is $v_W = R_B v_B$.
This robot is thus moving with angular velocity $\omega$, with non gravitational acceleration $\alpha$. $g$ is the known constant gravity vector and I am given $\alpha_B, \omega_B, R_B$.
Here is what I would like to formalize, and do: suppose the robot is turning right, and rotating the head clockwise. What should I do to have the robot going left, and turning the head counterclockwise, e.g. follow the same trajectory, but flipped around, say, the $y$ axis?
Option 1
... somehow modifying $\omega, \alpha$. It is very important that the new $\hat{\alpha}, \hat{\omega}$ have an expression in the new moving frame $C$ that is a combination of $\alpha_B, \omega_B, R_B$ (in the sense of sums, matrix products...).
For instance, call $F$ the matrix flipping the world's $y$ axis: to turn the other way around, I apply $F$ to the axis of rotation and the flip its sign, to properly tilt the axis, and reverse the sign of rotation. This is done in the world frame, so, we have $\hat{\omega}_W = -F\omega_W$, hence, $\hat{\omega}_C=-R_C^TFR_B\omega_B$
The problem already is that this way I have to know $R_C$ and this is not something I possess.
Option 2
... somehow changing the world frame into $\hat{W}$, a change of point of view, so to say. This new world frame might be a flip of the original one as before, or a rotation of it around its $x$ axis, of an angle of $\pi$. However, the robot itself is still "feeling" as it is turnign the same way as before, and this is something that is not desirable for my application. The robot should really feel a different angular velocity and acceleration. Also, the robot has to satisfy the system of the beginning, so, flipping gravity is not allowed.
Do you have any (other) idea? In italic I layed down some data/constraints that I have. Feel free to ask clarifications in case!