Robotic Arm Workspace Domain Solution

In advanced manufacturing and robotics, automated multi-axis robotic manipulators trace complex 3D paths for precision spot welding and component assembly. Mechanical and robotics engineers must verify that the mathematical model of the robot's toolpath is valid across the operating time envelope, ensuring that physical stroke limits, pneumatic pressure lines, and safety enclosures are never breached.

An automated robotic welding tip moves through Cartesian coordinates $(x, y, z)$ in meters as a function of operating cycle time $t$ (in seconds) according to the vector-valued position function:

1. Determine the allowable operating domain for the $x$-component function $x(t) = \sqrt{2t - 6}$ based on the physical requirement that joint extension values must remain real numbers.
2. Determine the restriction on parameter $t$ for the $y$-component function $y(t) = \frac{15}{t - 8}$ where the actuator encounters a singularity.
3. Determine the allowable operating domain for the $z$-component function $z(t) = \frac{4}{\sqrt{14 - t}}$.
4. Calculate the overall valid operating cycle time domain for the entire robotic toolpath $\vec{r}(t)$ in interval notation.

Step-by-Step Solution:

1. Analyze $x$-component ($x(t) = \sqrt{2t - 6}$): The term inside the square root must be non-negative for real coordinates:

   $$2t - 6 \ge 0 \implies 2t \ge 6 \implies t \ge 3$$
   Domain of $x(t)$: $[3, \infty)$

2. Analyze $y$-component ($y(t) = \frac{15}{t - 8}$): The denominator cannot equal zero, representing an asymptotic singularity in actuator control:

   $$t - 8 \ne 0 \implies t \ne 8$$
   Domain of $y(t)$: $(-\infty, 8) \cup (8, \infty)$

3. Analyze $z$-component ($z(t) = \frac{4}{\sqrt{14 - t}}$): The radicand is located in the denominator, so it must be strictly positive:

   $$14 - t > 0 \implies 14 > t \implies t < 14$$
   Domain of $z(t)$: $(-\infty, 14)$

4. Calculate Global Domain via Set Intersection: Intersect the three individual component domains:

   $$\text{Domain}(\vec{r}) = [3, \infty) \cap \{t \ne 8\} \cap (-\infty, 14)$$
   Combining these restrictions:
   $$[3, 8) \cup (8, 14)$$

Meaning of the Answer & Real-Life Application:

The robotic toolpath is physically and mathematically valid between $t = 3\text{ s}$ and $t = 14\text{ s}$, excluding the instant $t = 8\text{ s}$. In a production line programmable logic controller (PLC), engineers program an operational safety cycle within a continuous sub-interval (e.g., $3 \le t \le 7.5\text{ s}$ or $8.5 \le t \le 13.5\text{ s}$) to prevent singularity lockup and ensure continuous, smooth kinematic execution.