Robotics & Automated Manufacturing (CNC Laser Alignment) Solution

Problem Statement

A robotic laser cutter trims along a straight line from approach point $C(20, 10, 45)$ to destination point $D(80, -20, 15)$ (dimensions in millimeters).

Step-by-Step Mathematical Solution

1. Direction Vector and Parametric Equations

Compute direction vector $\vec{v} = \vec{CD}$:

Using point $C(20, 10, 45)$ as initial position $\vec{r}_0$:

• $x(t) = 20 + 60t$
• $y(t) = 10 - 30t$
• $z(t) = 45 - 30t$

2. Symmetric Form

Solve each parametric equation for $t$:

Equating the terms yields the symmetric equation:

3. Coordinates at $x = 50\text{ mm}$ Plane Crossing

Substitute $x = 50$ into the symmetric relation:

Now solve for the corresponding $y$ and $z$ values:

• $\frac{y - 10}{-30} = 0.5 \implies y - 10 = -15 \implies y = -5\text{ mm}$
• $\frac{z - 45}{-30} = 0.5 \implies z - 45 = -15 \implies z = 30\text{ mm}$

The laser beam crosses the $x = 50$ plane at the coordinate $(50, -5, 30)\text{ mm}$.

Real-Life Engineering Meaning & Application

• Physical Interpretation: The symmetric equations represent the spatial geometry of the laser line independent of the feed rate. The coordinate $(50, -5, 30)$ is the exact spatial point where the beam pierces the critical clamping plane.
• Engineering Application: In Computer-Aided Manufacturing (CAM) and automated fabrication cells, collision-detection routines perform plane-intersection checks before running G-code scripts. By verifying the beam coordinate $(50, -5, 30)$ against the safety envelope of the physical clamp, engineers avoid catastrophic tool collisions and costly equipment damage.