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

Dron lecący przez skomplikowany teren wykorzystuje ciśnienie powietrza jako dodatkową metodę do oszacowania niewielkich zmian wysokości.Pod względem wysokości ciśnienie często modeluje się jako spadek wykładniczo, a jego odwrotność daje wysokość w funkcji ciśnienia.Wzór ten jest często zbyt złożony do powtarzających się obliczeń, dlatego jest lokalnie aproksymowany jako funkcja liniowa poprzez linearyzację.Obejmuje to trzy kroki, zaczynając od początkowego punktu ciśnienia Pi , na przykład 90 kilopaskalów, gdzie przybliżenie jest wyśrodkowane. Najpierw wysokość w Pi oblicza się za pomocą oryginalnego równania, oznaczonego jako h(Pi).Po drugie, nachylenie znajduje się przez obliczenie pochodnej funkcji wysokości w Pi. Ten spadek pokazuje, jak wysokość zmienia się przy niewielkich zmianach ciśnienia.Na przykład, jeśli ciśnienie spadnie z 90 do 89,2 kilopaskala, spadek jest mnożony przez nachylenie przy początkowym ciśnieniu, aby oszacować zmianę wysokości.Na koniec wartości te są łączone w równaniu liniowego przybliżenia, aby uzyskać przybliżoną nową wysokość.Ta metoda jest dokładna przy niewielkich zmianach ciśnienia, ale staje się mniej niezawodna, gdy ciśnienie oddala się od początkowego ciśnienia.

zastosowanie linearyzacji i aproksymacji

Application of Linearization and Approximation

Dron lataj&#x105;cy w z&#x142;o&#x17C;onym terenie cz&#x119;sto wykorzystuje wi&#x119;cej ni&#x17C; jedn&#x105; metod&#x119; sensoryczn&#x105; do szacowania niewielkich zmian wysoko&#x15B;ci. Opr&#xF3;cz pomiar&#xF3;w bezpo&#x15B;rednich ci&#x15B;nienie powietrza stanowi u&#x17C;yteczny po&#x15B;redni wska&#x17A;nik ruchu pionowego. Ci&#x15B;nienie atmosferyczne maleje wraz ze wzrostem wysoko&#x15B;ci, a zale&#x17C;no&#x15B;&#x107; ta jest powszechnie opisywana za pomoc&#x105; modelu wyk&#x142;adniczego. Cho&#x107; jest to rozwi&#x105;zanie dok&#x142;adne, przeliczanie pomiar&#xF3;w ci&#x15B;nienia na warto&#x15B;ci wysoko&#x15B;ci wymaga oblicze&#x144;, kt&#xF3;re s&#x105; zbyt skomplikowane do wielokrotnego wykonywania w czasie rzeczywistym.Aby upro&#x15B;ci&#x107; te obliczenia, zale&#x17C;no&#x15B;&#x107; ci&#x15B;nienie&#x2013;wysoko&#x15B;&#x107; jest przybli&#x17C;ana lokalnie za pomoc&#x105; linearyzacji. To podej&#x15B;cie zast&#x119;puje pierwotn&#x105; zale&#x17C;no&#x15B;&#x107; nieliniow&#x105; aproksymacj&#x105; liniow&#x105;, kt&#xF3;ra jest wa&#x17C;na w pobli&#x17C;u wybranego ci&#x15B;nienia odniesienia. Proces rozpoczyna si&#x119; od wybrania pocz&#x105;tkowej warto&#x15B;ci ci&#x15B;nienia, takiej jak 90 kPa, wok&#xF3;&#x142; kt&#xF3;rej konstruowane jest przybli&#x17C;enie. Wysoko&#x15B;&#x107; odpowiadaj&#x105;ca temu ci&#x15B;nieniu odniesienia jest najpierw okre&#x15B;lana na podstawie pierwotnej zale&#x17C;no&#x15B;ci ci&#x15B;nienie&#x2013;wysoko&#x15B;&#x107; i s&#x142;u&#x17C;y jako punkt odniesienia do dalszych szacunk&#xF3;w.Nast&#x119;pnie ocenia si&#x119; czu&#x142;o&#x15B;&#x107; wysoko&#x15B;ci na niewielkie zmiany ci&#x15B;nienia przy ci&#x15B;nieniu odniesienia. Ta czu&#x142;o&#x15B;&#x107; opisuje, o ile zmienia si&#x119; wysoko&#x15B;&#x107; w odpowiedzi na niewielk&#x105; zmian&#x119; ci&#x15B;nienia. Na przyk&#x142;ad, je&#x15B;li zmierzone ci&#x15B;nienie spadnie nieznacznie poni&#x17C;ej warto&#x15B;ci odniesienia, ta niewielka r&#xF3;&#x17C;nica jest skalowana w celu oszacowania wynikaj&#x105;cej z tego zmiany wysoko&#x15B;ci.Przybli&#x17C;enie liniowe &#x142;&#x105;czy te wielko&#x15B;ci za pomoc&#x105; standardowego wzoru stycznej przedstawionego poni&#x17C;ej.\begin{equation*} L(P) = h(P_i) + h&#x27;(P_i)\,(P - P_i) \end{equation*}To przybli&#x17C;enie umo&#x17C;liwia dronowi efektywne szacowanie niewielkich zmian wysoko&#x15B;ci bez konieczno&#x15B;ci wykonywania skomplikowanych oblicze&#x144;. Metoda pozostaje dok&#x142;adna, dop&#xF3;ki zmiany ci&#x15B;nienia pozostaj&#x105; bliskie warto&#x15B;ci odniesienia, ale staje si&#x119; mniej wiarygodna, gdy ci&#x15B;nienie coraz bardziej odbiega od tej warto&#x15B;ci. Z tego powodu linearyzacja dobrze nadaje si&#x119; do monitorowania niewielkich zmian wysoko&#x15B;ci podczas lotu sta&#x142;ego i musi by&#x107; aktualizowana w przypadku wyst&#x105;pienia wi&#x119;kszych zmian.

Zastosowanie linearyzacji i aproksymacji

Dron lataj&#x105;cy w z&#x142;o&#x17C;onym terenie cz&#x119;sto wykorzystuje wi&#x119;cej ni&#x17C; jedn&#x105; metod&#x119; sensoryczn&#x105; do ...

application-of-linearization-and-approximation

Education

JoVE Core

Calculus

Unit

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

derivatives of simple functions

Slope Rules for Simple Functions

Derivatives show how fast a function changes, and they also describe slope. A derivative can tell the slope of a straight line or the slope of a tangent line to a curve at one point. In this lesson, the idea is tied to a roller coaster track, where the derivative of the horizontal position tells how steep the ride is at each spot. A flat roller coaster section can be modeled with a constant function, f(x) = c, where c is a fixed height. Its derivative is f'(x) = 0, so the slope is zero and...

the product rule

Derivative of a Changing Rectangle

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the quotient rule

Differentiating Ratios with the Quotient Rule

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derivatives of the trigonometric functions

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limits on trigonometric functions

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the chain rule

Chain Rule in a Gear System

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the chain rule: problem solving

Chain Rule in Thermal Expansion

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implicit differentiation

Tangent Lines in Circular Motion

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implicit differentiation: problem solving

Tangent Lines with Implicit Differentiation

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logarithmic differentiation

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the number e as a limit

e from Natural Log Derivatives

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derivatives of inverse trigonometric functions

derivatives-of-inverse-trigonometric-functions

Inverse Trig Derivatives in Tracking

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application of rates of change

Velocity and Acceleration in Motion Graphs

Velocity and acceleration describe how a car moves along a highway. The car’s position changes over time, so it can be written as a function of time. Rates of change help measure motion in a clear mathematical way. Velocity tells how position changes with time. Average velocity is found by dividing the change in position by the change in time. It gives the overall speed of the car during a time interval, but it does not show changes inside that interval. To find the velocity at one exact...

exponential growth

Bacterial Doubling in Exponential Growth

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related rates

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Balloon inflation is a clear example of related rates, where one changing quantity affects another through a single relationship. Related rates study how linked quantities change with respect to time. As air is added, the balloon’s volume rises, and the radius must also increase. For a hot air balloon, the inflated envelope is often modeled as a perfect sphere. This simplification makes the math easier to work with. In that model, the balloon’s volume depends directly on its radius. When air...

linearization and approximation

Tangent Line Estimates Near a Point

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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...

hyperbolic functions

Modeling Hanging Cables with Hyperbolic Curves

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Deriving the Slope of a Catenary Curve

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hyperbolic-and-inverse-hyperbolic-functions-problem-solving

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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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