Autonomous Drone Thrust State Estimation (Robotics & Computer Engineering)

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:

T(x, y) = \ln(x^2 + e^y)

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.

State the multivariable linear approximation formula $L(x, y)$ referencing Linear Approximations.
Compute the analytical partial derivatives $T_x(x, y)$ and $T_y(x, y)$ , and evaluate them at the hovering base state $(0, 1)$ .
Construct the explicit linearization formula $L(x, y)$ for the quadcopter's thrust controller.
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 $T_{\text{true}} = \ln(0.01 + e^{1.05}) \approx 1.0535$ .