Problem: I have a semi-autonomous vehicle that needs it's steering adjusted. I can attempt to send it in a "straight line" of distance $(l)$ -- but unless it's steering is perfect, it will veer from this imaginary straight line. At the end of $(l)$, the vehicle ends up offset from the intended straight path by $(x)$ units. The path length $(l)$, and length of vehicle from axle to axle $(b)$ are both variables, and I am assuming that the steering angle $\theta$ and also arc radius $(r)$ remain constant during transit. Photo of problem
What I've tried: Started with the radius of the turn from the wheel angle and vehicle length
$$r = \frac{b}{\sin\theta}$$
and subbed this into an equation of a circle, offset to one side by (r),
$$\left(x-\left(\frac{b}{\sin\theta}\right)\right)^2 + y^2 = \left(\frac{b}{\sin\theta}\right)^2$$
and then solved for $\theta$:
$$\theta = \operatorname{arcsin}\left(\frac{2bx}{x^2+y^2}\right)$$ and as far as I can tell, this is correct given a {x,y} endpoint, however, I have {x,l}. $y$ will only be the same as $l$ if vehicle drives perfectly straight -- which kind of makes this equation useless to me as it stands.
How can I solve for the angle of the drive wheel using arc length instead of $y$?