I’m trying to design a land sailer, and I wanted to make sure I had the right formula.
Using the PDF diagram attached, I was able to derive Arolin’s formula from geometry. But I think it’s only an approximation, which works best when the steering angle (angle of the wheels, not the steering wheel) is 90 degrees from straight ahead.
In the diagram, the “car” is initially the line AC, with one fixed rear wheel at A and a steerable front wheel at C. As it moves forward an infinitesimal amount dy (the distance between A and B), its front end describes an arc from C to E, determined by the steering angle θ between CD and CE, which we assume is finite and constant. The length of the “car” remains constant at s, but it moves infinitesimally to position BE.
As this happens, the front end of the “car” moves along the arc CE. By the definition of sine, the linear distance CE relates to the “tangential” distance DE (in red) that the front end of the “car” moves perpendicularly to the “straight ahead” direction AD by the sine of the steering angle θ, thus:
DE/CE = sin θ or CE = DE/sin θ
But DE is also related to the infinitesimal turning angle dψ as follows:
DE= AE sin dψ. So CE = (AE sin dψ)/sin θ
The calculations get complicated if you take the actual length of AE according to Pythagoras’ theorem. But since the changes are infinitesimal, to first order of approximation AE ≅ s and, by first-order Taylor expansion sin dψ ≅ dψ (in radians).
With these approximations, the length traced by the front end of the “car” is
CE ≈ s dψ/sin θ
Since s and θ are constants, it’s a trivial matter to integrate the length of the arc CE over the angle dψ until the finite arc equals the steering radius AC, which (by definition of radian) occurs when ψ = 1:
Steering radius = ∫ (CE)dψ = s/sin θ ∫ dψ = s/sin θ, where the integration is from 0 to 1.
This is the same as Arolin’s formula, with his a = my θ, the usual designation of angle, and his n = 1 (because we’re using the steering angle, i.e., the angle of the wheels themselves relative to straight ahead, not the angle of the steering wheel).
The formula is exact for θ = 90 degrees because then AE actually equals s. Of course, when θ = 90 degrees you burn up your tires trying to move the car forward.
I’m not sure how the formula should vary as θ → 0. At the limit it seems to work, because the steering radius becomes infinite, as it ought, when the car moves straight ahead. But the formula might not work well for small steering angles.
Anyway, wish me luck with the cactus.
