# Path curvature in kinematic bicycle model

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

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 $$\theta_{dot}$$

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 :)

• 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. – Chuck Feb 14 at 15:10