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

複雑な地形を飛行するドローンは、気圧を補助的に利用して小さな高度変化を推定します。高度の観点から見ると、圧力はしばしば指数関数的減衰としてモデル化され、その逆数は圧力の関数としての高度を示します。この式は繰り返し計算には複雑すぎることが多いため、線形化によって局所的に線形関数として近似されます。これは、90キロパスカルのような初期圧力点Pから始まり、近似が中心となる3ステップから始まります。まず、Pi の高度は元の式 h(Pi) を用いて計算します。次に、傾斜はPiにおける高度関数の微分を評価することで求められます。この傾斜は、わずかな気圧変化によって高度がどのように変化するかを示しています。例えば、気圧が90から89.2キロパスカルに低下した場合、その圧力の低下に初期圧力の傾斜を掛けて高度変化を推定します。最後に、これらの値を線形近似方程式で組み合わせて、新しい近似高度を算出します。この方法は小さな圧力変化に対して正確ですが、圧力が初期圧力から遠ざかるほど信頼性は低下します。

線形化と近似の適用

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P_i) \end{equation*}&#x3053;&#x306E;&#x8FD1;&#x4F3C;&#x306B;&#x3088;&#x308A;&#x3001;&#x30C9;&#x30ED;&#x30FC;&#x30F3;&#x306F;&#x8907;&#x96D1;&#x306A;&#x8A08;&#x7B97;&#x3092;&#x5FC5;&#x8981;&#x3068;&#x305B;&#x305A;&#x306B;&#x3001;&#x5C0F;&#x3055;&#x306A;&#x9AD8;&#x5EA6;&#x5909;&#x52D5;&#x3092;&#x52B9;&#x7387;&#x7684;&#x306B;&#x63A8;&#x5B9A;&#x3067;&#x304D;&#x307E;&#x3059;&#x3002;&#x3053;&#x306E;&#x65B9;&#x6CD5;&#x306F;&#x3001;&#x5727;&#x529B;&#x5909;&#x5316;&#x304C;&#x57FA;&#x6E96;&#x5024;&#x306B;&#x8FD1;&#x3044;&#x9650;&#x308A;&#x7CBE;&#x5EA6;&#x3092;&#x4FDD;&#x3061;&#x307E;&#x3059;&#x304C;&#x3001;&#x5727;&#x529B;&#x304C;&#x57FA;&#x6E96;&#x5024;&#x304B;&#x3089;&#x5927;&#x304D;&#x304F;&#x96E2;&#x308C;&#x308B;&#x307B;&#x3069;&#x4FE1;&#x983C;&#x6027;&#x306F;&#x4F4E;&#x4E0B;&#x3057;&#x307E;&#x3059;&#x3002;&#x305D;&#x306E;&#x305F;&#x3081;&#x3001;&#x7DDA;&#x5F62;&#x5316;&#x306F;&#x5B9A;&#x5E38;&#x98DB;&#x884C;&#x4E2D;&#x306E;&#x5C0F;&#x3055;&#x306A;&#x9AD8;&#x5EA6;&#x5909;&#x52D5;&#x306E;&#x76E3;&#x8996;&#x306B;&#x9069;&#x3057;&#x3066;&#x304A;&#x308A;&#x3001;&#x5927;&#x304D;&#x306A;&#x5909;&#x52D5;&#x304C;&#x767A;&#x751F;&#x3057;&#x305F;&#x5834;&#x5408;&#x306F;&#x66F4;&#x65B0;&#x3059;&#x308B;&#x5FC5;&#x8981;&#x304C;&#x3042;&#x308A;&#x307E;&#x3059;&#x3002;

application-of-linearization-and-approximation

Education

JoVE Core

Calculus

微積分学

Unit

単位

Differentiation Rules

微分法（導関数の公式）

KEY TERMS AND CONCEPTS

重要用語と概念

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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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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