Autonomous Drone LiDAR Obstacle Clearance (Computer Engineering) Solution

Problem Overview

An autonomous quadcopter following the trajectory $y = 0.5x^2$ on $x \in [-4, 4]$ detects a stationary transmission pylon at $(0, 4)$ (coordinates in decameters, $1\text{ dam} = 10\text{ m}$ ). We determine the closest point of approach (CPA) and minimum clearance distance using the Squared-Distance Optimization Technique and the Second Derivative Test, verifying compliance with the $2.0\text{-dam}$ ( $20\text{-m}$ ) collision-avoidance safety margin.

Step-by-Step Mathematical Solution

Step 1: Formulate the Euclidean Distance Function

The Euclidean distance between the pylon at $(x_0, y_0) = (0, 4)$ and any point $(x, y)$ along the flight path is:

d(x) = \sqrt{(x - x_0)^2 + (y - y_0)^2} = \sqrt{(x - 0)^2 + (0.5x^2 - 4)^2}

(Governing equations defined in Applied Optimization Modeling Framework.)

Step 2: Formulate the Squared-Distance Objective Function

Applying the Squared-Distance Optimization Technique, we differentiate the squared distance $D(x) = [d(x)]^2$ to eliminate the radical chain rule:

D(x) = x^2 + (0.5x^2 - 4)^2

Expand the quadratic term:

D(x) = x^2 + \left( 0.25x^4 - 4x^2 + 16 \right) = 0.25x^4 - 3x^2 + 16

Feasible domain:

x \in [-4, 4]\text{ dam}

Step 3: Differentiate and Find Critical Numbers

Differentiate $D(x)$ with respect to $x$ :

D'(x) = \frac{d}{dx}\left[ 0.25x^4 - 3x^2 + 16 \right] = x^3 - 6x

Set

D'(x) = 0

to identify stationary points:

x(x^2 - 6) = 0 \implies x = 0 \quad \text{or} \quad x = \pm\sqrt{6} \approx \pm 2.449\text{ dam}

All three critical numbers lie strictly within the operational flight corridor

[-4, 4]

Step 4: Classify Critical Points via the Second Derivative Test

Compute the second derivative:

D''(x) = 3x^2 - 6

Evaluate

D''(x)

at each critical number:

At $x = 0$ : $D''(0) = 3(0)^2 - 6 = -6 < 0$ Because $D''(0) < 0$ , the flight path directly beneath the pylon at $(0, 0)$ represents a local maximum of the distance function among the stationary points ( $d = \sqrt{16} = 4\text{ dam}$ ).
At $x = \pm\sqrt{6}$ : $D''(\pm\sqrt{6}) = 3(6) - 6 = 18 - 6 = 12 > 0$ Because $D''(\pm\sqrt{6}) > 0$ , both symmetrical points represent local minima.

Step 5: Evaluate Coordinates, Endpoints, and Safety Margin

Evaluate the trajectory altitude $y = 0.5x^2$ at the critical numbers:

y = 0.5(\pm\sqrt{6})^2 = 0.5(6) = 3\text{ dam}

The two closest points of approach on the drone trajectory are:

P_1\left(-\sqrt{6}, 3\right) \approx (-2.45, 3.00)\text{ dam} \quad \text{and} \quad P_2\left(\sqrt{6}, 3\right) \approx (2.45, 3.00)\text{ dam}

Calculate the minimum squared distance:

D(\pm\sqrt{6}) = 0.25(6)^2 - 3(6) + 16 = 0.25(36) - 18 + 16 = 9 - 18 + 16 = 7\text{ dam}^2

Taking the square root gives the minimum clearance:

d_{\min} = \sqrt{7} \approx 2.646\text{ dam} = 26.46\text{ meters}

Evaluate the domain boundary endpoints ( $x = \pm 4$ ):

D(\pm 4) = 0.25(4)^4 - 3(4)^2 + 16 = 0.25(256) - 48 + 16 = 64 - 48 + 16 = 32\text{ dam}^2

d(\pm 4) = \sqrt{32} \approx 5.657\text{ dam} = 56.57\text{ meters}

Comparing outputs:

Endpoints: $d(\pm 4) \approx 26.46\text{ m} < 56.57\text{ m}$ .
Central stationary point: $d(0) = 40.0\text{ m}$ .
Closest points: $d(\pm\sqrt{6}) = 26.46\text{ m}$ .

Engineering Significance & Conclusion

Safety Clearance: The minimum physical distance between the UAV and the high-voltage transmission pylon is $26.46\text{ meters}$ ( $2.646\text{ dam}$ ). Because this clearance exceeds the required minimum safety buffer of $20.0\text{ meters}$ ( $2.0\text{ dam}$ ), the navigation flight computer does not trigger an emergency trajectory abort.
LiDAR Firmware Optimization: Computing $D(x) = [d(x)]^2$ rather than $d(x)$ allows embedded flight microcontrollers to execute real-time collision checks in simple polynomial arithmetic, saving processor cycles and reducing latency in obstacle avoidance pipelines.

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