3D Coordinate Mapping Solution

In robotic systems and autonomous navigation, depth cameras (like RGB-D sensors) represent the surroundings as a 3D point cloud. A robot arm is situated at the origin $(0,0,0)$, and its software needs to calculate distances between detected objects to prevent collisions.

A depth sensor detects a cylindrical obstacle with a safety envelope modeled as a sphere centered at $C(3, -4, 2)\text{ meters}$ with a radius of $r = 1.5\text{ meters}$. A point obstacle is detected at $P(4, -2.5, 3)\text{ meters}$.

1. Find the distance between the robot arm at the origin $O(0,0,0)$ and the center of the spherical safety envelope $C$.
2. Determine the standard equation of the spherical safety envelope.
3. Calculate the distance from the point obstacle $P$ to the center of the safety envelope $C$. Determine if the point obstacle has breached the safety envelope (i.e., is it inside the sphere?).

Step-by-Step Solution:

1. Find the Distance from the Origin to $C$: Using the 3D distance formula between $O(0,0,0)$ and $C(3, -4, 2)$:

   $$d_{OC} = \sqrt{(3-0)^2 + (-4-0)^2 + (2-0)^2} = \sqrt{3^2 + (-4)^2 + 2^2}$$
   $$d_{OC} = \sqrt{9 + 16 + 4} = \sqrt{29} \approx 5.39\text{ meters}$$

2. Determine the Equation of the Sphere: The standard equation of a sphere is $(x-h)^2 + (y-k)^2 + (z-l)^2 = r^2$. Substituting the center $(h,k,l) = (3, -4, 2)$ and radius $r = 1.5$:

   $$(x-3)^2 + (y+4)^2 + (z-2)^2 = (1.5)^2$$
   $$(x-3)^2 + (y+4)^2 + (z-2)^2 = 2.25$$

3. Check if the Safety Envelope is Breached: Calculate the distance from the point obstacle $P(4, -2.5, 3)$ to the center $C(3, -4, 2)$:

   $$d_{PC} = \sqrt{(4-3)^2 + (-2.5 - (-4))^2 + (3-2)^2}$$
   $$d_{PC} = \sqrt{1^2 + (1.5)^2 + 1^2} = \sqrt{1 + 2.25 + 1} = \sqrt{4.25} \approx 2.06\text{ meters}$$
   Compare the distance $d_{PC}$ to the radius $r$: Since $d_{PC} \approx 2.06\text{ m} > 1.5\text{ m}$, the point obstacle is outside the safety envelope. The safety envelope has not been breached.

Meaning of the Answer & Real-Life Application:

Robotic collision-avoidance algorithms use these simple geometric checks in real time. By comparing distance vectors between spatial coordinates and spherical safety envelopes, robots can instantly halt or reroute their paths when a breach is detected, avoiding mechanical damage or injury.