CNC Motion Interpolation

The interpolator is the mathematical heart of a CNC controller. Without it, a machine could only move from one absolute point to another in a disjointed, ‘point-to-point’ fashion. The interpolator enables ‘contouring’, or continuous path control, which is essential for all modern machining. When the controller reads a G-code block like ‘G01 X10 Y20’, it knows the current position (e.g., X0 Y0) and the target position. The interpolator’s job is to break down this single vector into a series of very small, discrete step commands for each axis motor (e.g., X and Y). It calculates the required velocity for each axis so that they start and stop simultaneously, resulting in a perfectly straight line between the two points. The algorithm used is often a variation of a Digital Differential Analyzer (DDA) or Bresenham’s line algorithm.

For circular interpolation (G02/G03), the calculation is more complex. The G-code provides the start point (current position), the end point, and the center of the circle (or the radius). The interpolator must then calculate a series of intermediate points that lie on the specified arc. It does this by solving the circle equation incrementally, generating coordinated velocity commands for the X and Y axes that maintain the correct tangential speed and radial distance. Advanced CNC controllers feature higher-order interpolation, such as helical (combining circular motion with linear motion in a third axis), spline, or NURBS (Non-Uniform Rational B-Spline) interpolation. NURBS interpolation is particularly powerful as it allows the machine to follow complex, free-form curves defined by a single mathematical equation, resulting in smoother motion and better surface finishes than approximating the curve with many small linear segments.