given is
- two wheels with independent speeds from -1 to 1
- a specific distance between wheels (distance_axis )
i know those for cases case 1: ws1 = 1 ws2 = 1 radius will be infinity or zero because car drives straight forward
case 2: ws1 = 1 ws2 = 0 radius will be half the distance from one wheel to another (distance axis /2 )
case 3: ws1 = 1 ws2 = -1 radius will be zero, the car will rotate exactly around the center of the axis
is there a formula to get the radius by the ratio of the two wheel speeds ? i suspect sine/cosine or tangens something like: maybe radius = distance_axis * atan(ws1/ws2) example
Let $v_1$ and $v_2$ denote the speeds at which the wheels turn, and let $r_1$ and $r_2$ denote the radii of the circles that the wheels trace. Without loss of generality $v_1\geq v_2$ and so $r_1\geq r_2$. Then $r_1=r_2+d$, where $d$ is the length of the axis connecting the wheels.
The ratio between the speeds tells you the ratio between the arc-lengths traveled by the two wheels in any given time. This is in turn the same ase the ratio between the radii of the circles that the wheels trace. That is to say $$\frac{v_1}{v_2}=\frac{r_1}{r_2}=\frac{r_2+d}{r_2}=1+\frac{d}{r_2}.$$ A bit of algebra then shows that $$r_2=\frac{d}{\frac{v_1}{v_2}-1}=\frac{dv_2}{v_1-v_2}.$$ The radius of the circle traced by the middle of the axis is then $$r_2+\frac{d}{2}=\frac{d}{2}+\frac{dv_2}{v_1-v_2}=\frac{d}{2}\frac{v_1+v_2}{v_1-v_2}$$