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 , and point measurement error . 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 ( to the curve’s middle ordinate (m) is less than 0.1. The ratio 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.