Application of Linearization and Approximation

The original exponential pressure-altitude relationship is accurate but too complex for repeated real-time computations during drone flight. Linearization replaces this nonlinear relationship with a straight-line approximation valid near a reference pressure, enabling efficient altitude estimation without complex calculations.

First, select an initial pressure point like 90 kilopascals and calculate the altitude at that reference using the original equation. Second, find the slope by evaluating the derivative of the altitude function at that pressure point. Third, combine these values using the tangent-line formula to create the linear approximation equation.

The slope, found by evaluating the derivative at the reference pressure, shows how much altitude changes in response to small pressure variations. For example, if pressure drops from 90 to 89.2 kilopascals, multiplying this pressure drop by the slope yields the estimated altitude change at that reference point.

The linear approximation remains accurate only for small pressure changes near the reference value. As pressure deviates farther from the initial reference pressure, the approximation becomes increasingly unreliable because the straight-line model diverges from the actual exponential relationship.

The reference pressure, such as 90 kilopascals, serves as the center point around which the linear approximation is constructed. The altitude at this reference pressure becomes the baseline, and the derivative evaluated at this point determines the slope for estimating altitude changes nearby.

During steady flight with minor altitude variations, the drone measures small pressure changes and applies the linear approximation formula to efficiently estimate corresponding altitude shifts without performing complex exponential calculations. This method works well for monitoring minor changes but requires updating when larger variations occur.

Atmospheric pressure decreases predictably as altitude increases, providing an indirect but reliable indicator of vertical movement. This relationship, commonly modeled as exponential decay, allows drones to estimate altitude changes alongside direct measurements, improving navigation accuracy through complex terrain.