Autonomous Drone LiDAR Obstacle Clearance (Computer Engineering)

In autonomous robotics and unmanned aerial vehicle (UAV) guidance systems, real-time path planning algorithms must compute the minimum clearance distance between a drone's programmed flight trajectory and physical obstacles detected by onboard LiDAR point clouds.

A computer engineering team programs an autonomous surveillance drone executing an inspection survey over a power substation. In a local coordinate system (scaled in decameters, where $1\text{ dam} = 10\text{ meters}$), the drone follows a pre-computed quadratic ascent path modeled by:

The drone's forward-looking solid-state LiDAR sensor detects an unmapped high-voltage transmission pylon located at coordinate point $P(0, 4)$. The drone's collision-avoidance firmware requires calculating the closest point of approach (CPA) and the absolute minimum clearance distance between the flight path and the pylon to determine if a trajectory replanning interrupt must be triggered (threshold clearance: $d_{\min} \ge 2.0\text{ dam}$).

                          y ^
                            |      P(0, 4) [Pylon]
                            |        o
                            |       / \
                            |  d   /   \
                            |     /     \
                            |    v       v
                            |   *         * [Closest Points]
                            |  /           \
                            | /  y = 0.5x^2 \
                            +-------------------> x

1. Formulate the Euclidean distance function $d(x)$ between the pylon at $(0, 4)$ and an arbitrary point $(x, y)$ on the flight trajectory.
2. Apply the Squared-Distance Optimization Technique to define the simplified single-variable objective function $D(x) = [d(x)]^2$.
3. Differentiate $D(x)$ to locate all critical numbers of the distance function along the flight corridor $x \in [-4, 4]$.
4. Classify each critical point using the Second Derivative Test, identifying whether each represents a local maximum or local minimum clearance.
5. Compute the coordinates of the closest approach points and the absolute minimum clearance distance $d_{\min}$ (in decameters and meters). Does the drone satisfy the $2.0\text{-dam}$ safety margin?