Autonomous Drone Thrust State Estimation (Robotics & Computer Engineering) Solution

An autonomous quadcopter drone's flight guidance computer must continuously calculate its required propulsion thrust percentage ($T$) based on forward pitch angle ($x$ in radians) and vertical climb velocity ($y$ in meters per second, $\text{m/s}$).

Aerodynamic flight dynamics establish the non-linear thrust response function:

During steady-state hovering, the drone operates at a clean nominal base state $(x_0, y_0) = (0, 1)$, yielding an exact baseline thrust of $T(0, 1) = \ln(0^2 + e^1) = \ln(e) = 1.0$ ($100\%$ nominal thrust). Suddenly, a crosswind gust perturbs the drone into a dynamic state of $(x, y) = (0.10, 1.05)$.

Because evaluating non-linear logarithmic functions on an embedded 32-bit microcontroller consumes excessive clock cycles and battery power, the flight controller executes a local linear approximation $L(x, y)$ centered at $(0, 1)$ to update thrust commands in real time.

1. State the multivariable linear approximation formula $L(x, y)$ referencing Linear Approximations.
2. Compute the analytical partial derivatives $T_x(x, y)$ and $T_y(x, y)$, and evaluate them at the hovering base state $(0, 1)$.
3. Construct the explicit linearization formula $L(x, y)$ for the quadcopter's thrust controller.
4. Estimate the required thrust $T(0.10, 1.05)$ using $L(x, y)$, and compute the absolute percentage error relative to the true numerical value $T_{\text{true}} = \ln(0.01 + e^{1.05}) \approx 1.0535$.

Step-by-Step Solution:

1. State the Linearization Formula:

   • As defined in Linear Approximations, the local linear approximation of a two-variable function at base point $(x_0, y_0)$ is:
     $$L(x, y) = T(x_0, y_0) + T_x(x_0, y_0)(x - x_0) + T_y(x_0, y_0)(y - y_0)$$

2. Compute Analytical Partial Derivatives:

   • Differentiating $T(x, y) = \ln(x^2 + e^y)$ via the Logarithmic Chain Rule:
     $$T_x(x, y) = \frac{1}{x^2 + e^y} \cdot \frac{\partial}{\partial x}(x^2 + e^y) = \frac{2x}{x^2 + e^y}$$
     $$T_y(x, y) = \frac{1}{x^2 + e^y} \cdot \frac{\partial}{\partial y}(x^2 + e^y) = \frac{e^y}{x^2 + e^y}$$
   • Evaluate at nominal base state $(x_0, y_0) = (0, 1)$:
     $$x_0^2 + e^{y_0} = 0^2 + e^1 = e$$
     $$T_x(0, 1) = \frac{2(0)}{e} = 0$$
     $$T_y(0, 1) = \frac{e^1}{e} = 1$$
   • Baseline thrust value: $T(0, 1) = \ln(e) = 1.0$.

3. Construct the Linearization Formula:

   • Substitute parameters into the formula:
     $$L(x, y) = 1.0 + 0(x - 0) + 1.0(y - 1)$$
     $$L(x, y) = 1.0 + (y - 1) = y$$
   • Physical Insight: Because $T_x(0, 1) = 0$, first-order variations in pitch angle $x$ around hover have zero linear sensitivity. In the immediate vicinity of hover, thrust commands depend almost exclusively on vertical climb rate $y$.

4. Estimate Perturbed Thrust and Compute Error:

   • Perturbed state: $(x, y) = (0.10, 1.05)$.
   • Evaluate linearization:
     $$L(0.10, 1.05) = 1.05 \quad (105\% \text{ thrust})$$
   • Compare with true value $T_{\text{true}} = \ln(0.01 + e^{1.05}) = \ln(0.01 + 2.85765) = \ln(2.86765) \approx 1.05349$:
     $$\text{Absolute Error} = |1.05349 - 1.05000| = 0.00349$$
     $$\text{Percentage Error} = \frac{0.00349}{1.05349} \times 100\% \approx 0.33\%$$
   • The microcontroller achieves a computation speedup of orders of magnitude while introducing less than $0.35\%$ error, well within actuator tolerance bands.