Robotic Safety Zone Boundary (Robotics & Computer Engineering)

A robotic manufacturing arm operates inside an automated assembly cell adjacent to an inclined conveyor belt. To prevent the robotic end-effector from colliding with the moving conveyor structure, a safety automation engineer must define the conveyor's flat face as a virtual planar boundary barrier in the robot's trajectory-planning controller.

Metrology calibration instruments establish that the conveyor's surface is perpendicular to an anchor calibration normal vector $\vec{n} = \langle 1, -2, 2 \rangle = \hat{i} - 2\hat{j} + 2\hat{k}$ and passes through a fixed structural mounting pivot located at Cartesian coordinate $P_0(3, 1, 4)$ meters relative to the robot's base origin.

1. State the point-normal vector formula for a plane in 3D space, identifying the components of the normal vector $\vec{n}$ and base point $P_0$.
2. Formulate the geometric equation of the safety boundary plane in point-normal form.
3. Simplify the equation into standard general linear form ($ax + by + cz = d$).
4. Explain how the robot's real-time motion controller evaluates the scalar expression $f(x, y, z) = x - 2y + 2z - 9$ to trigger emergency collision-avoidance braking for a tool tip located at $(x_t, y_t, z_t)$.