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Robotics & Automated Manufacturing (CNC Laser Alignment) Solution

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Problem Statement

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

Step-by-Step Mathematical Solution

1. Direction Vector and Parametric Equations

Compute direction vector v⃗=CD⃗\vec{v} = \vec{CD}:

v⃗=⟨80−20,−20−10,15−45⟩=⟨60,−30,−30⟩ mm\vec{v} = \langle 80 - 20, -20 - 10, 15 - 45 \rangle = \langle 60, -30, -30 \rangle\text{ mm}

Using point C(20,10,45)C(20, 10, 45) as initial position r⃗0\vec{r}_0:

  • x(t)=20+60tx(t) = 20 + 60t
  • y(t)=10−30ty(t) = 10 - 30t
  • z(t)=45−30tz(t) = 45 - 30t

2. Symmetric Form

Solve each parametric equation for tt:

t=x−2060=y−10−30=z−45−30t = \frac{x - 20}{60} = \frac{y - 10}{-30} = \frac{z - 45}{-30}

Equating the terms yields the symmetric equation:

x−2060=y−10−30=z−45−30\frac{x - 20}{60} = \frac{y - 10}{-30} = \frac{z - 45}{-30}

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

Substitute x=50x = 50 into the symmetric relation:

t=50−2060=3060=0.5t = \frac{50 - 20}{60} = \frac{30}{60} = 0.5

Now solve for the corresponding yy and zz values:

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

The laser beam crosses the x=50x = 50 plane at the coordinate (50,−5,30) mm(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)(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)(50, -5, 30) against the safety envelope of the physical clamp, engineers avoid catastrophic tool collisions and costly equipment damage.