This lidar-measured coordinate gives us the first important ingredient in generating a lidar-validated home run distance projection. In fact, we give this late-trajectory impact coordinate heavy control over the trajectory solution we will eventually find. The ingredients we need to generate a trajectory are shown in Table 1:

Table 1. List of ingredients and sources to create a lidar-validated trajectory solution.

Flight time or time from impact (bat) to impact (banner) is harvested from the broadcast video down to the closest frame. The flight time and impact coordinate together make the high-confidence late-trajectory coordinate. Next, we pull in local environmental conditions that will affect our eventual trajectory solution. In this case, these aren’t that interesting since Tropicana Field has a dome. In general, air density and wind conditions help to model the ball’s flight path more precisely. Finally, we use the initial conditions of the actual trajectory measured with high confidence: exit velocity, launch angle, and spray direction.

We’ve collected all the ingredients, now we need to use them to create this trajectory solution. Thankfully, esteemed physicist Alan Nathan has done some of this work already by parameterizing the equations of motion for a baseball trajectory. Our treatment closely follows Dr. Nathan’s method, including drag and lift coefficient ranges from Robert Adair’s The Physics of Baseball.

The trajectory solution is calculated using a nonlinear-optimization or solver algorithm. In essence, we will tether the solution to parameters in which we have high confidence (launch velo, launch angle, temperature, etc) and allow the optimization algorithm to vary other parameters within physically realistic constraints. The goal of the optimizer is to minimize the sum of the squares of the differences (△x²+ △y²+ △z²) between the calculated-trajectory’s coordinates at the time of impact and the lidar-measured impact coordinates. We consider a solution convergent when the residual (△x²+ △y²+ △z²) is essentially zero.

The result of all of this is a closed-form trajectory solution which matches our high-confidence observations of what happened. You will note that this trajectory is calculated independently from in-flight tracking data and the derivative distance projections are independent of our tracking vendor’s projected distances.

The lidar-validated trajectory solution is plotted in Figure 6. The dashed lines represent the projected trajectories back to ground level that would have occurred had this HR not impacted the banner. The 459-foot projected range at the z=0 root is called out in the plot.