Application of Linearization and Approximation

A drone flying through complex terrain uses air pressure as a supplementary method to estimate small altitude changes.

In terms of altitude, pressure is often modeled as exponential decay, and its inverse gives altitude as a function of pressure.

This formula is often too complex for repeated computations, so it is approximated locally as a linear function through linearization.

This involves three steps, starting with an initial pressure point, P_i , such as 90 kilopascals, where the approximation is centered. First, the altitude at P_i is calculated using the original equation, labeled h(P_i).

Second, the slope is found by evaluating the derivative of the altitude function at P_i.This slope shows how altitude varies with small pressure changes.

For example, if pressure drops from 90 to 89.2 kilopascals, the drop is multiplied by the slope at the initial pressure to estimate altitude change.

Finally, these values are combined in the linear approximation equation to give the approximate new altitude.

This method is accurate for small pressure changes but becomes less reliable as the pressure moves farther away from the initial pressure.