Drone Flight Path Linearization (Robotics & Computer Engineering) Solution

Problem Overview

We linearize the quadcopter flight path $y(x) = 4x - x^2$ at waypoint $x = 3$ and evaluate local path estimation accuracy.

Step-by-Step Mathematical Solution

Step 1: Compute General Derivative $y'(x)$

Set up the difference quotient limit:

Step 2: Evaluate Point of Tangency and Slope at $x = 3$

• Altitude coordinate:
  $$y(3) = 4(3) - (3)^2 = 12 - 9 = 3\text{ m} \implies (3, 3)$$
• Instantaneous tangent slope:
  $$m = y'(3) = 4 - 2(3) = 4 - 6 = -2$$

Step 3: Construct the Tangent Line (Linearization $L(x)$)

Using point-slope form $y - y_1 = m(x - x_1)$:

Step 4: Compare Linear Approximation with Exact Nonlinear Value at $x = 3.1$

• Linear Tangent Approximation:
  $$L(3.1) = -2(3.1) + 9 = -6.2 + 9 = 2.80\text{ m}$$
• Exact Trajectory Value:
  $$y(3.1) = 4(3.1) - (3.1)^2 = 12.4 - 9.61 = 2.79\text{ m}$$
• Approximation Error:
  $$\text{Error} = |2.79 - 2.80| = 0.01\text{ m} \quad (\approx 0.36\% \text{ relative error})$$

Engineering Conclusion

Within a narrow window near $x = 3$, the tangent line linearization $L(x) = -2x + 9$ predicts the vehicle position to within $1\text{ centimeter}$ of accuracy. In robotics and embedded flight control, substituting tangent line models for nonlinear curves allows low-power flight microprocessors to execute guidance loop corrections at over $500\text{ Hz}$ without computational latency.