I'm trying to develop an understanding of the control law for longitudinal and lateral control for self driving cars. While doing so, I came across the kinematic bicycle model which is used to derive relations for these.

While studying about path tracking, the curvature was defined as

$k(s)$ = $e_{ra}$*$\theta_p(s)$/$ds$

Where $k(s)$: Curvature; $e_{ra}$: Orthagonal distance between rear axle centre and path; $\theta_p(s)$: Angle between path tangent and global x at point s

I'm attaching a picture for reference below kinematic bicycle model in path coordinates

I have a few doubts on this,

I'm unable to understand how the curvature formula was derived

Furthermore, two more relations were given for $s_{dot}$ and $\theta_{dot}$ s_dot,theta_dot

I'm unable to understand how these relations were derived, I've been sitting on this for a long time, I sincerely can use some help. Thank you so much for taking the time to read and help with this :)

  • $\begingroup$ You've given us the results and asked how they're derived, but you haven't linked the original paper or given any context. I think you haven't gotten any answers because of this. For example, you ask how $\dot{e}_{ra}$ is derived but haven't given an expression for $e_{ra}$. Please link the source or give more context. $\endgroup$ – Chuck Feb 14 at 15:10

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