The problem is to find the trajectory of the center of mass of a car when it is rotating and translating (after an off center collision, for instance)

What i know as input is the trajectory of the wheels (skidd marks deposited on the surface of the road) I also am given the geometry of the vehicle, so the distance of all four wheels to the center of mass is know, as are the Wheelbase ($WB$) and Track lenghts ($T$).

Let $y = f_i(x), i=1,..,4$ be the function describing the trajectory of the $i$th wheel, and $r_i$ be the distance of the $i$th wheel to the center of mass.

How would i go about finding a function $y = f_{CM}(x)$ that describes the trajectory of the center of mass? Assume planar motion.

  • $\begingroup$ Is this a homework problem or an exam question? $\endgroup$
    – Solar Mike
    May 1, 2021 at 17:13
  • $\begingroup$ Shouldn't the center of mass be used as the reference point for the equation of Y with respect to the displacement in X at time t? $\endgroup$
    – r13
    May 1, 2021 at 17:14
  • $\begingroup$ You don't have enough information unless you know (how?) that all 4 skid marks stop and start at the same time. In real skids typically one locks up one axle first. Anyway, you may have to do some polite massaging of the data, but basically you set up a geometrical model with the properties at the centre of mass of x,y, and theta. Then best fit to your known xy trajectories of each wheel. $\endgroup$ May 23, 2023 at 2:26

1 Answer 1


This requires a few assumptions, and its not the most optimal/efficient algorithm, but its relatively easy to program.


  • you have(or you can create) a list of points for the coordinates of each tire $FR(i),FL(i),RR(i),RL(i)$ (Front Right, Front Left, Rear Right, Rear Left). So for $FR(1)$ you would get something like (3,2.5), meaning 3 m on the axis and 2.5 m on the y-axis
  • the thickness of the tires is really small (so orientation does not have an effect.
  • The car and the tires do not deform

The main idea is that you will start with one track -lets assume FR()- and you find which points from the other curves correspond to that point, based on geometric constraints.


  1. You start by the first point in the list FR(i=1).
  2. With center that point you draw 3 circles with radius ($WB$, $T$ and $\sqrt{WB^2+T^2}$ (if you want you can generate circles with $\pm$ a few cm)
  3. if the circles
  • coincide with only one point on each other curve, then you find the index of that curve.
  • intersect more than one points on the other curves then you use the orientation of the vehicle to select the best fit.
  1. Assuming you arrive at indexes $FR(i), FL(k),RR(l),RL(m)$, you know that the center of geometry is exactly between the coordinate $FR(i)$ and $RL(m)$.
  2. you proceed to the next point ($FR(i+1)$) and go back step 2. The main difference, now is that assuming points $FR(i)$ and $FR(i+1)$ are close you have a good starting point.

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