Curvilinear trajectory estimation of a supersonic bullet using ballistic shock wave arrivals at asynchronous acoustic sensor nodes

The trajectory of a supersonic bullet, which is subjected to drag and gravity, is curvilinear and the supersonic flight of the bullet generates a ballistic shock wave (SW). A model for the differential time of arrival (DTOA) of the SW at a pair of acoustic sensors is derived for a given bullet trajectory, which is fully described by seven parameters including the drag coefficient exponent and ballistic constant of the bullet. Assuming that the drag coefficient exponent is 0.5, the DTOA model is used to develop a nonlinear least-squares (NLS) method to estimate the other six trajectory parameters using DTOA of SW measurements from each node (which comprises a small acoustic sensor array) of an asynchronous sensor network. The position of the shooter and the muzzle speed of the bullet are then determined by tracing the estimated bullet trajectory back to topographic or man-made obstructions on a digital map. The effectiveness of the NLS method is verified using simulated data for different types of real bullets, and the error standard deviations in the parameter estimates are close to the Cramer-Rao lower bounds.