Uncertainty in Radius Determined by Multi-Point Curve Fits for Use in the Critical Curve Speed Formula 2019-01-0428

The critical curve speed formula used for estimating vehicle speed from yaw marks depends on the tire-to-road friction and the mark’s radius of curvature. This paper quantifies uncertainty in the radius when it is determined by fitting a circular arc to three or more points. A Monte Carlo analysis was used to generate points on a circular arc given three parameters: number of points n, arc angle θ, and point measurement error σ. For each iteration, circular fits were performed using three techniques. The results show that uncertainty in radius is reduced for increasing arc length, decreasing point measurement error, and increasing number of points used in the curve fit. Radius uncertainty is linear if the ratio of the standard deviation in point measurement error (σ) to the curve’s middle ordinate (m) is less than 0.1. The ratio σ/m should be less than 0.018 for a radius found using a 3-point circular fit to be within 5% of the actual value 95% of the time. Increasing the number of points used for the fit reduces uncertainty in radius: a 15-point fit reduces the 95% prediction interval width by up to 42%, or allows a shorter yaw mark arc angle with the same uncertainty in radius.