I have been repeatedly told when dealing with some data for Navigation purposes that I can assume it follows a "bicycle model". I am not very familiar with vehicle dynamics so I consulted online. I saw many different models varying on the level of sophistication but most of them seems to consist in planar Dynamics similar to the following:
\begin{equation} \begin{bmatrix} x_{k+1} \\ y_{k+1} \\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} x_{k} \\ y_{k} \\ \theta_{k} \end{bmatrix} + \begin{bmatrix} v\cdot\Delta t\cdot\cos{\theta_k} \\ v\cdot\Delta t\cdot\sin{\theta_k} \\ f(\gamma,v) \end{bmatrix} \end{equation}
where $\gamma$, and $v$ are the steering angle and the velocity, both inputs from the vehicle sensors, and $\theta$ is the angle respect to the origin of a fixed frame. $f(\gamma,v)$ is some function relating both angles.
My doubt arises from the fact that the data I have been granted contains only the following fields: $x$,$y$,$\theta$. My assumption is that, if we know it must follow a bicycle model it should hold a correlation like the following:
\begin{equation} \begin{bmatrix} x_{k+1} \\ y_{k+1} \\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} x_{k} \\ y_{k} \\ \theta_{k} \end{bmatrix} + \begin{bmatrix} d_k\cdot\cos{\theta_k} \\ d_k\cdot\sin{\theta_k} \\ \Delta\theta_k \end{bmatrix} \end{equation}
Where $d_k$ and $\Delta\theta_k$ are known at every time step and are the distance traveled and the change in the angle respectively.
If I generate data this way, starting from an initial point $(x_0,y_0)$ and building up, it doesn't seem to correlate well with the actual data I have been granted. My question is if the deduction exposed has any conceptual mistake, so I can clear out if the problem is that the data given is very noisy and deviates after a while or that what I think is a bicycle model is not
x,y,theta
then you don't need to solve the system you are showing here. Its not really clear what you are trying to do. $\endgroup$