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

Karmaşık arazide uçan bir drone, küçük irtifa değişimlerini tahmin etmek için hava basıncını ek bir yöntem olarak kullanır.İrtifa açısından basınç genellikle üstel düşüş olarak modellenir ve bunun tersi yüksekliği basıncın bir fonksiyonu olarak verir.Bu formül genellikle tekrarlayan hesaplamalar için fazla karmaşıktır, bu nedenle doğrusal bir fonksiyon olarak yerel olarak lineerizasyon yoluyla yaklaştırılır.Bu, üç adımdan oluşur; başlangıçta başlanan bir basınç noktası olan Pi ile başlar, örneğin 90 kilopaskal gibi, burada yaklaşım merkezlidir. İlk olarak, Pi'deki irtifa orijinal denklem kullanılarak hesaplanır ve h(P i) olarak adlandırılır.İkinci olarak, eğim, Pi'deki irtifa fonksiyonunun türevi değerlendirilerek bulunur. Bu eğim, küçük basınç değişimleriyle irtifanın nasıl değiştiğini gösterir.Örneğin, basınç 90'dan 89,2 kilopaskal'a düşerse, düşüş başlangıç basınçtaki eğimle çarpılır ve irtifa değişimi tahmin edilir.Son olarak, bu değerler doğrusal yaklaştırma denkleminde birleştirilerek yaklaşık yeni irtifa elde edilir.Bu yöntem küçük basınç değişiklikleri için doğrudur ancak basınç başlangıç basıncından uzaklaştıkça daha az güvenilir hale gelir.

doğrusallaştırma ve yaklaştırmanın uygulanması

Karma&#x15F;&#x131;k bir arazide u&#xE7;an bir drone, irtifadaki k&#xFC;&#xE7;&#xFC;k de&#x11F;i&#x15F;imleri tahmin etmek i&#xE7;in &#xE7;o&#x11F;u zaman birden fazla alg&#x131;lama y&#xF6;ntemine ihtiya&#xE7; duyar. Do&#x11F;rudan &#xF6;l&#xE7;&#xFC;mlere ek olarak, hava bas&#x131;nc&#x131; da dikey hareketin izlenmesi i&#xE7;in yararl&#x131; bir dolayl&#x131; g&#xF6;sterge sunar. Atmosferik bas&#x131;n&#xE7;, irtifa artt&#x131;k&#xE7;a azal&#x131;r ve bu ili&#x15F;ki genellikle &#xFC;stel bir model arac&#x131;l&#x131;&#x11F;&#x131;yla tan&#x131;mlan&#x131;r. Ancak bu model y&#xFC;ksek do&#x11F;ruluk sa&#x11F;lasa da, bas&#x131;n&#xE7; &#xF6;l&#xE7;&#xFC;mlerinin irtifa de&#x11F;erlerine d&#xF6;n&#xFC;&#x15F;t&#xFC;r&#xFC;lmesi, ger&#xE7;ek zamanl&#x131; u&#xE7;u&#x15F; s&#x131;ras&#x131;nda tekrar tekrar yap&#x131;lmas&#x131; pratik olmayan karma&#x15F;&#x131;k hesaplamalar gerektirir.Bu hesaplamalar&#x131; basitle&#x15F;tirmek amac&#x131;yla, bas&#x131;n&#xE7;&#x2013;irtifa ili&#x15F;kisi yerel olarak do&#x11F;rosalla&#x15F;t&#x131;rma yoluyla yakla&#x15F;t&#x131;r&#x131;l&#x131;r. Bu y&#xF6;ntem, &#xF6;zg&#xFC;n do&#x11F;rusal olmayan ili&#x15F;kiyi, se&#xE7;ilen bir referans bas&#x131;nc&#x131; civar&#x131;nda ge&#xE7;erli olan do&#x11F;rusal bir yakla&#x15F;&#x131;mla de&#x11F;i&#x15F;tirir. S&#xFC;re&#xE7;, yakla&#x15F;t&#x131;rman&#x131;n olu&#x15F;turulaca&#x11F;&#x131;, &#xF6;rne&#x11F;in 90 kilopaskal gibi bir ba&#x15F;lang&#x131;&#xE7; bas&#x131;n&#xE7; de&#x11F;erinin se&#xE7;ilmesiyle ba&#x15F;lar. Bu referans bas&#x131;nca kar&#x15F;&#x131;l&#x131;k gelen irtifa, &#xF6;ncelikle &#xF6;zg&#xFC;n bas&#x131;n&#xE7;&#x2013;irtifa ili&#x15F;kisi kullan&#x131;larak hesaplan&#x131;r ve sonraki tahminler i&#xE7;in bir ba&#x15F;lang&#x131;&#xE7; d&#xFC;zeyi olarak kullan&#x131;l&#x131;r.Ard&#x131;ndan, referans bas&#x131;n&#xE7; de&#x11F;erinde irtifan&#x131;n k&#xFC;&#xE7;&#xFC;k bas&#x131;n&#xE7; de&#x11F;i&#x15F;imlerine kar&#x15F;&#x131; duyarl&#x131;l&#x131;&#x11F;&#x131; de&#x11F;erlendirilir. Bu duyarl&#x131;l&#x131;k, bas&#x131;n&#xE7;taki &#xE7;ok k&#xFC;&#xE7;&#xFC;k bir de&#x11F;i&#x15F;imin irtifada ne kadar bir de&#x11F;i&#x15F;ime yol a&#xE7;t&#x131;&#x11F;&#x131;n&#x131; ifade eder. &#xD6;rne&#x11F;in, &#xF6;l&#xE7;&#xFC;len bas&#x131;n&#xE7; referans de&#x11F;erin biraz alt&#x131;na d&#xFC;&#x15F;t&#xFC;&#x11F;&#xFC;nde, bu k&#xFC;&#xE7;&#xFC;k fark orant&#x131;lanarak ortaya &#xE7;&#x131;kan irtifa de&#x11F;i&#x15F;imi tahmin edilir.Do&#x11F;rusal yakla&#x15F;t&#x131;rma, bu nicelikleri a&#x15F;a&#x11F;&#x131;da g&#xF6;sterilen standart te&#x11F;et do&#x11F;ru form&#xFC;l&#xFC; kullanarak birle&#x15F;tirir.\begin{equation*} L(P) = h(P_i) + h&#x27;(P_i)\,(P - P_i) \end{equation*}Bu yakla&#x15F;t&#x131;rma, dronun karma&#x15F;&#x131;k hesaplamalara ihtiya&#xE7; duymadan k&#xFC;&#xE7;&#xFC;k irtifa de&#x11F;i&#x15F;imlerini etkin bir bi&#xE7;imde tahmin etmesine olanak tan&#x131;r. Y&#xF6;ntem, bas&#x131;n&#xE7; de&#x11F;i&#x15F;imleri referans de&#x11F;ere yak&#x131;n kald&#x131;&#x11F;&#x131; s&#xFC;rece do&#x11F;rulu&#x11F;unu korur; ancak bas&#x131;n&#xE7; bu de&#x11F;erden uzakla&#x15F;t&#x131;k&#xE7;a g&#xFC;venilirli&#x11F;i azal&#x131;r. Bu nedenle do&#x11F;rosalla&#x15F;t&#x131;rma, istikrarl&#x131; u&#xE7;u&#x15F; s&#x131;ras&#x131;nda k&#xFC;&#xE7;&#xFC;k irtifa de&#x11F;i&#x15F;imlerinin izlenmesi i&#xE7;in uygundur ve daha b&#xFC;y&#xFC;k de&#x11F;i&#x15F;imler meydana geldi&#x11F;inde g&#xFC;ncellenmesi gerekir.

Doğrusallaştırma ve Yaklaştırmanın Uygulanması

Karma&#x15F;&#x131;k bir arazide u&#xE7;an bir drone, irtifadaki k&#xFC;&#xE7;&#xFC;k de&#x11F;i&#x15F;imleri tahmin etmek i&#xE7;in &#xE7;o&#x11F;u zaman ...

application-of-linearization-and-approximation

Education

JoVE Core

Calculus

Kalkülüs

Unit

Birim

Differentiation Rules

Türev Alma Kuralları

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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 rates of change

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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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