Name: estimating altitude changes with linearization
Uploaded: 2026-01-12T12:24:59.000Z
Duration: 1 min 29 s
Channel: JoVE

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, Pi , such as 90 kilopascals, where the approximation is centered. First, the altitude at Pi is calculated using the original equation, labeled h(Pi).Second, the slope is found by evaluating the derivative of the altitude function at Pi.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.

application of linearization and approximation

A drone flying through complex terrain often relies on more than one sensing method to estimate small changes in altitude. Along with direct measurements, air pressure provides a useful indirect indicator of vertical movement. Atmospheric pressure decreases as altitude increases, and this relationship is commonly described using an exponential model. Although accurate, converting pressure measurements into altitude values requires calculations that are too complex to perform repeatedly during real-time flight.To simplify these computations, the pressure&#8211;altitude relationship is approximated locally using linearization. This approach replaces the original nonlinear relationship with a straight-line approximation that is valid near a chosen reference pressure. The process begins by selecting an initial pressure value, such as ninety kilopascals, around which the approximation is constructed. The altitude corresponding to this reference pressure is first determined using the original pressure&#8211;altitude relationship and serves as the baseline for further estimation.Next, the sensitivity of altitude to small pressure changes is evaluated at the reference pressure. This sensitivity describes how much the altitude changes in response to a slight change in pressure. For example, if the measured pressure drops slightly below the reference value, this small difference is scaled to estimate the resulting change in altitude.The linear approximation combines these quantities using the standard tangent-line formula shown below.<figure class="image image-style-align-center image_resized" style="display:table;margin-left:auto;margin-right:auto;width:25%;"><img style="aspect-ratio:300/48;" src="https://cloudfront.jove.com/files/ftp_upload/19810/19810_Equation_1_1__editor_19810.svg" alt="Lagrange interpolation formula, polynomial approximation, equation in mathematical analysis diagram." data-altupdatedat="1775223722000" data-alt="19810_Equation_1_1_.svg" /></figure>This approximation enables the drone to efficiently estimate small altitude variations without requiring complex calculations. The method remains accurate as long as pressure changes remain close to the reference value, but it becomes less reliable as the pressure deviates farther from it. For this reason, linearization is well-suited for monitoring minor altitude changes during steady flight and must be updated when larger variations happen.

Application of Linearization and Approximation

A drone flying through complex terrain often relies on more than one sensing method to estimate small changes in altitude. Along with direct measurements, ...

application-of-linearization-and-approximation

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Calculus

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Differentiation Rules

KEY TERMS AND CONCEPTS

estimating altitude changes with linearization

Linearization can help a drone estimate small altitude changes from air pressure readings. Atmospheric pressure drops as altitude rises, and this relationship is often modeled with an exponential function. The model is accurate, but the math needed to convert pressure into altitude can be too complex for repeated calculations during flight.

To make the calculation easier, the pressure-altitude relationship is approximated locally with a straight line. This line is built near a chosen reference pressure, such as 90 kilopascals. First, the altitude at that reference pressure is found from the original relationship. That value becomes the starting point for the estimate.

Next, the sensitivity of altitude to pressure is measured at the reference pressure. Sensitivity means how much the altitude changes when pressure changes a little. If the measured pressure is slightly lower than the reference value, that small difference is scaled to estimate the change in altitude.

The straight-line estimate uses the standard tangent-line formula. It lets the drone quickly track small altitude changes without doing the full nonlinear calculation each time. The approximation works best when pressure stays close to the reference pressure. It becomes less reliable as the pressure moves farther away, so it must be updated when larger changes occur.

Estimating Altitude Changes with Linearization

estimating-altitude-changes-with-linearization

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Linearization can help a drone estimate small altitude changes from air pressure readings. Atmospheric pressure drops as altitude rises, and this relationship is often modeled with an exponential function. The model is accurate, but the math needed to convert pressure into altitude can be too complex for repeated calculations during flight. To make the calculation easier, the pressure-altitude relationship is approximated locally with a straight line. This line is built near a chosen reference...

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key terms and concepts

Learn how linearization turns a pressure-altitude exponential model into a quick straight-line estimate for small drone altitude changes near a reference pressure.

Estimating Altitude Changes with Linearization | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Estimating Altitude Changes with Linearization

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