# “Bicycle model” in vehicle dynamics

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:

$$\begin{bmatrix} x_{k+1} \\ y_{k+1} \\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} x_{k} \\ y_{k} \\ \theta_{k} \end{bmatrix} + \begin{bmatrix} v\cdot\Delta t\cdot\cos{\theta_k} \\ v\cdot\Delta t\cdot\sin{\theta_k} \\ f(\gamma,v) \end{bmatrix}$$

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:

$$\begin{bmatrix} x_{k+1} \\ y_{k+1} \\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} x_{k} \\ y_{k} \\ \theta_{k} \end{bmatrix} + \begin{bmatrix} d_k\cdot\cos{\theta_k} \\ d_k\cdot\sin{\theta_k} \\ \Delta\theta_k \end{bmatrix}$$

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

• if you are given 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. – agentp Mar 27 '17 at 20:54
• Hi @agentp. I understand it may sound confusing. The data given has errors, so what I want to do is to compare the data values from the ones you would retrieve expanding the time series form the beginning using the "perfect" dynamics without noise. However since I don't know the Dynamics beforehand, only know they are from a "bicycle model", I am trying to find what this generally means in this context – A. Frenzy Mar 28 '17 at 0:19
• "bicycle model" simply means you are not concerning yourself with the details of steering geometry, accounting for the slight differences between inside and outside wheel angle and so on. – agentp Mar 28 '17 at 0:29
• Thank you @agentp. Therefore an agent that moves using the above Dynamics would be following a bicycle model? As a contextualization, the real agent is supposed to be a pedestrian, whose movement has been simplified to the "bicycle model" – A. Frenzy Mar 28 '17 at 1:56