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

Um drone voando por terrenos complexos usa pressão atmosférica como método suplementar para estimar pequenas mudanças de altitude.Em termos de altitude, a pressão é frequentemente modelada como decaimento exponencial, e seu inverso dá altitude em função da pressão.Essa fórmula é frequentemente complexa demais para cálculos repetidos, por isso é aproximada localmente como uma função linear por meio da linearização.Isso envolve três etapas, começando com um ponto de pressão inicial, P i , como 90 quilopascales, onde a aproximação está centralizada. Primeiro, a altitude em Pi é calculada usando a equação original, rotulada h(Pi).Segundo, a inclinação é encontrada avaliando a derivada da função altitude em Pi. Essa inclinação mostra como a altitude varia com pequenas variações de pressão.Por exemplo, se a pressão cair de 90 para 89,2 quilopascales, a queda é multiplicada pela inclinação na pressão inicial para estimar a mudança de altitude.Finalmente, esses valores são combinados na equação de aproximação linear para fornecer a nova altitude aproximada.Esse método é preciso para pequenas variações de pressão, mas se torna menos confiável à medida que a pressão se afasta da pressão inicial.

aplicação de linearização e aproximação

Um drone voando por terrenos complexos frequentemente depende de mais de um m&#xE9;todo para estimar pequenas varia&#xE7;&#xF5;es de altitude. Al&#xE9;m das medi&#xE7;&#xF5;es diretas, a press&#xE3;o do ar fornece um indicador indireto &#xFA;til do deslocamento vertical. A press&#xE3;o atmosf&#xE9;rica diminui com o aumento da altitude, e essa rela&#xE7;&#xE3;o &#xE9; comumente descrita por um modelo exponencial. Embora preciso, a convers&#xE3;o das medi&#xE7;&#xF5;es de press&#xE3;o em valores de altitude requer c&#xE1;lculos complexos demais para serem realizados repetidamente durante o voo em tempo real.Para simplificar esses c&#xE1;lculos, a rela&#xE7;&#xE3;o entre press&#xE3;o e altitude &#xE9; aproximada localmente por meio de lineariza&#xE7;&#xE3;o. Essa abordagem substitui a rela&#xE7;&#xE3;o n&#xE3;o linear original por uma aproxima&#xE7;&#xE3;o linear v&#xE1;lida pr&#xF3;ximo a uma press&#xE3;o de refer&#xEA;ncia. O processo come&#xE7;a com a escolha de um valor de press&#xE3;o inicial, como noventa quilopascais, em torno do qual se constr&#xF3;i a aproxima&#xE7;&#xE3;o. A altitude correspondente a essa press&#xE3;o de refer&#xEA;ncia &#xE9; determinada inicialmente com base na rela&#xE7;&#xE3;o entre press&#xE3;o e altitude original e serve como base para estimativas posteriores.Em seguida, a sensibilidade da altitude a pequenas varia&#xE7;&#xF5;es de press&#xE3;o &#xE9; avaliada na press&#xE3;o de refer&#xEA;ncia. Essa sensibilidade descreve quanto varia a altitude em resposta a uma pequena mudan&#xE7;a na press&#xE3;o. Por exemplo, se a press&#xE3;o medida cair ligeiramente abaixo do valor de refer&#xEA;ncia, essa pequena diferen&#xE7;a &#xE9; ajustada para estimar a varia&#xE7;&#xE3;o resultante na altitude.A aproxima&#xE7;&#xE3;o linear combina essas quantidades por meio da f&#xF3;rmula padr&#xE3;o da reta tangente, mostrada abaixo.\begin{equation*} L(P) = h(P_i) + h&#x27;(P_i)\,(P - P_i) \end{equation*}Essa aproxima&#xE7;&#xE3;o permite que o drone estime eficientemente pequenas varia&#xE7;&#xF5;es de altitude sem exigir c&#xE1;lculos complexos. O m&#xE9;todo permanece preciso enquanto as varia&#xE7;&#xF5;es de press&#xE3;o se mantiverem pr&#xF3;ximas ao valor de refer&#xEA;ncia, mas torna-se menos confi&#xE1;vel conforme a press&#xE3;o se desvia desse valor. Por esse motivo, a lineariza&#xE7;&#xE3;o &#xE9; adequada para monitorar pequenas varia&#xE7;&#xF5;es de altitude durante voo est&#xE1;vel e precisa ser atualizada quando ocorrem varia&#xE7;&#xF5;es maiores.

Aplicação de Linearização e Aproximação

Um drone voando por terrenos complexos frequentemente depende de mais de um m&#xE9;todo para estimar pequenas varia&#xE7;&#xF5;es de altitude. Al&#xE9;m ...

application-of-linearization-and-approximation

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Calculus

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Differentiation Rules

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KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

Application of Linearization and Approximation

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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application of linearization and approximation

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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