Robotics and Telemetry Tracking Actuator (Computer & Robotics Engineering)

At an autonomous robotics flight testing facility, an automated optical tracking camera mount tracks an inspection drone performing a low-altitude survey. The robotic camera platform is anchored at ground level and rotates along an elevation pitch axis driven by a precision servo-actuator.

The drone flies along a straight, horizontal flight corridor at a constant altitude of $h = 300.0\text{ meters}$ directly over the camera base. Real-time GPS and optical flow telemetry indicate the drone flies at a constant horizontal speed of:

The camera elevation angle $\theta(t)$ (measured from the horizontal ground up to the drone line-of-sight) satisfies the trigonometric relation:

1. Formulate the time derivative relation connecting the camera's angular tracking velocity $\frac{d\theta}{dt}$ to the drone's horizontal flight speed $\frac{dx}{dt}$ using the Chain Rule on $\tan\theta = \frac{h}{x}$.
2. Determine the exact angular tracking velocity $\frac{d\theta}{dt}$ required by the servo-motor in radians per second ($\text{rad/s}$) and degrees per second ($^\circ/\text{s}$) at the instant the drone is at a horizontal distance of $x = 400.0\text{ meters}$ approaching the station.
3. Compute the angular velocity required when the drone is directly overhead ($x \to 0$), and explain why robotics control engineers must design pan-tilt servo actuators with sufficient angular acceleration bandwidth to prevent gimbal lock or tracking target loss.