Modeling of rock fragments trajectories ejected from blasting based on high-speed video analysis

This paper presents a mathematical model for the study of the 2D movement of blast bench faces based on the high-speed video analysis of three blasts at El Aljibe quarry (Spain) recorded at 2,000 frames per second. In these blasts, a set of bags of 0.75 × 0.75 m were placed in front of a selected blasthole defining a reference plane in which to study the kinematics of the movement. The burdens range from 3.86 m to 5.54 m, with a bench height of 12–13 m. An affine transformation system was calculated to convert coordinates from the video images into actual spatial coordinates. The problem is established from a multi-objective minimization approach in order to simultaneously optimize two defined objectives: trajectory fitting using Total Least Squares regression, and time–displacement regression using Ordinary Least Squares. Muti-objective minimization formulation produces Pareto plots of non-dominated solutions, i.e., optimal combinations of initial velocities and response times, (i.e. time between detonation and the onset of bench face motion). From the Pareto frontier, the most balanced solution is selected. Blast response time lasts only a few milliseconds and appears so brief at the frame rate used that it is difficult to detect visually. In this context, the bi-objective optimization-based approach proves to be a more robust tool than frame-by frame survey for quantifying such a rapid event while also maintaining a strong performance in the trajectory fitting task. Root Mean Square Error (RMSE) for the two minimizations average 0.22 m and 0.13 m, respectively. The initial velocities obtained range from 4 to 17 m/s, and the response times 0.7 ms to 21.7 ms. An analytical model is also presented for predicting fragments initial velocity and response time as functions of blast geometry and rock mass properties, explaining approximately 87% of the total variability.