Robotic Manufacturing & CNC Toolpaths

In precision manufacturing and aerospace robotics, automated laser cutters and multi-axis CNC milling tools must trace intricate 3D contoured profiles along metal workpieces. To prevent localized heat buildup, material burning, or excessive tool wear, the cutting head must maintain a perfectly uniform physical feed rate across spatial coordinates, regardless of the mathematical parameterization of the motor drives.

An automated robotic laser cutter at a UNLV advanced manufacturing laboratory traces a cutting path along an aerospace titanium bracket according to the vector position function:

where coordinates $(x, y, z)$ are measured in millimeters and $t \ge 0$ represents an internal motor cycle parameter.

1. State the formula for the accumulated arc length function $s(t)$ and compute $s(t)$ starting from initial cycle $t = 0$.
2. Invert the arc length function to solve for the motor cycle parameter $t$ in terms of physical traversal distance $s$.
3. Formulate the intrinsic arc length parametrization $\vec{r}(s)$ and determine the exact coordinate location of the laser cutter after it has traversed an arc length of $s = \frac{7}{3}\text{ mm}$.