Lidar Safety Envelopes (Robotics & Computer Science) Solution

In autonomous vehicle design, a lidar sensor scans the surroundings by emitting light pulses across a specific field of view. The domain of the range-finding function is restricted by the sensor's physical limits: the horizontal scanning angle $\theta$ is restricted to $\left[-\frac{\pi}{3}, \frac{\pi}{3}\right]$ radians, and the radial distance $d$ is restricted to $[0.1, 150]$ meters.

To determine how the vehicle should respond to obstacles, software engineers program a piecewise control function $A(d)$ to evaluate safety zones:

Determine the vehicle's response and safety state if an obstacle is detected at a distance of:

1. $d = 2\text{ meters}$
2. $d = 10\text{ meters}$
3. $d = 80\text{ meters}$

Step-by-Step Solution:

1. For $d = 2\text{ m}$, since $0.1 \le 2 < 3$, we use the first piecewise condition:
   $$A(2) = \text{Full Emergency Braking}$$
2. For $d = 10\text{ m}$, since $3 \le 10 < 15$, we use the second piecewise condition:
   $$A(10) = \text{Proportional Deceleration}$$
3. For $d = 80\text{ m}$, since $15 \le 80 \le 150$, we use the third piecewise condition:
   $$A(80) = \text{Maintain Cruise Speed}$$

Meaning of the Answer & Real-Life Application:

Identifying domain constraints and evaluating piecewise functions in real-time allows the vehicle's onboard computer to calculate safe velocity profiles without executing undefined operations or misinterpreting out-of-range sensor noise. For example, if an obstacle is detected outside the sensor's domain ($d > 150$ or $d < 0.1$), the system ignores the signal or triggers a sensor fault handler.