Robotics and Telemetry Tracking Actuator (Computer & Robotics Engineering) Solution

Problem Overview

We calculate the angular tracking velocity $\frac{d\theta}{dt}$ required of a ground-based robotic servo-camera tracking an autonomous drone flying at a fixed altitude of $h = 300.0\text{ m}$ with a constant approach speed of $\frac{dx}{dt} = -60.0\text{ m/s}$. The tracking elevation angle $\theta(t)$ is governed by:

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Step-by-Step Mathematical Solution

Step 1: Formulate the Trigonometric Time Derivative

The governing formula is the Trigonometric Tangent Ratio:

(This formula and its time derivative principles are defined in Related Rates.)

The altitude $h = 300.0\text{ m}$ is a static constant ($\frac{dh}{dt} = 0$). Rewriting the equation:

Differentiating both sides implicitly with respect to time $t$ using the Chain Rule:

Isolating the angular slew velocity $\frac{d\theta}{dt}$:

Step 2: Snapshot Calculation at $x = 400.0\text{ m}$

1. Determine the right triangle geometry at $x = 400.0\text{ m}$:
   • Altitude (opposite): $h = 300.0\text{ m}$
   • Horizontal base (adjacent): $x = 400.0\text{ m}$
   • Line-of-sight range (hypotenuse):
     $$z = \sqrt{x^2 + h^2} = \sqrt{400.0^2 + 300.0^2} = \sqrt{160000 + 90000} = \sqrt{250000} = 500.0\text{ m}$$
2. Evaluate $\cos\theta$ and $\sec^2\theta$ directly from the side lengths:
   $$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{z} = \frac{400.0}{500.0} = \frac{4}{5} = 0.80$$
   $$\sec^2\theta = \frac{1}{\cos^2\theta} = \frac{1}{(4/5)^2} = \frac{25}{16} = 1.5625$$
3. Compute the angular velocity $\frac{d\theta}{dt}$: The drone is approaching the station, so horizontal distance is decreasing: $\frac{dx}{dt} = -60.0\text{ m/s}$.
   $$\frac{d\theta}{dt} = -\frac{300.0}{(400.0)^2 (25/16)} (-60.0) = \frac{18000}{160000 \cdot (25/16)} = \frac{18000}{250000} = \frac{18}{250} = 0.072\text{ rad/s}$$
4. Convert to degrees per second:
   $$\frac{d\theta}{dt} = 0.072\text{ rad/s} \times \left(\frac{180^\circ}{\pi\text{ rad}}\right) = \frac{12.96^\circ}{\pi} \approx 4.125^\circ/\text{s}$$

Step 3: Limiting Velocity when Directly Overhead ($x \to 0$)

As the drone passes directly overhead ($x \to 0$), the line-of-sight hypotenuse approaches altitude $z \to h = 300.0\text{ m}$, and $\theta \to \frac{\pi}{2}$ ($90^\circ$). Expressing $\frac{d\theta}{dt}$ in terms of $z$:

Directly overhead ($z = 300.0\text{ m}$):

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Engineering Conclusion

In autonomous robotics and computer vision telemetry, pan-tilt servo tracking gimbals must rotate smoothly to keep dynamic targets centered inside the high-resolution optical sensor's field of view.

Notice that $\frac{d\theta}{dt} = \frac{h v}{x^2 + h^2} = \frac{h v}{z^2}$. When the drone is far away ($x = 2000\text{ m}, z \approx 2022\text{ m}$), the required angular rate is minimal ($\approx 0.0044\text{ rad/s} = 0.25^\circ/\text{s}$). As the target closes in, the required angular rate steadily climbs, reaching its absolute maximum directly overhead ($0.20\text{ rad/s} = 11.46^\circ/\text{s}$).

If the robotics engineer specifies a servo motor based only on the far-field tracking requirement, the motor will stall or lag behind the target as it passes overhead, losing lock. Control engineers use this related rates velocity profile to size servo-motor torque ratings and tune feedforward PID control loops to guarantee continuous lock throughout the flyby corridor.