Tool Axis Control 5-Axis

Theoretical Background & Mathematical Foundation

In 5-axis CNC machining, the position of the cutting tool is defined not just by the tool tip coordinates $(X,Y,Z)$ but also by the spatial orientation of the tool spindle axis, represented by a unit vector $\mathbf{T}=[i,j,k]$.

Vector Geometry of Tool Orientation

The orientation of the tool axis is calculated relative to the workpiece surface normal vector $\mathbf{N}$ and the velocity vector (direction of travel) $\mathbf{V}$ at the contact point.