Quadratic Travel Curve: How a CNC System Moves

• 1
  Key takeaways
• 2
  What the Quadratic Travel Curve Actually Describes
• 3
  How the Curve Shows Up on the Machine
• 4
  When the Quadratic Travel Curve Stops Being the Right Answer
• 5
  Practical Settings When You Cannot Change the Profile
• 6
  Step by Step: Isolating a Travel-Curve Problem
• 7
  Quadratic vs Cubic vs S-Curve Travel Profiles

A quadratic travel curve is the position profile you get when acceleration is held constant over a move. Position along the axis follows x(t) = a t² + b t + c, where t is time and a, b, c set the shape. Velocity is the first derivative, a straight ramp. Acceleration is the second derivative, a flat constant. That is the whole idea, and everything else follows from it.

Most CNC controllers do not name it that way. They call it linear acceleration, trapezoidal velocity, or simply a linear accel/decel ramp. When a control is described as a CNC system analyzing the quadratic travel curve, it means the interpolator is fitting parabolic segments between programmed points instead of stepping velocity directly.

The coefficient a is half the commanded acceleration. The coefficient b is the initial velocity at the start of the segment. The coefficient c is the starting position. On a real machine these are not free variables: a is capped by the drive current limit and by the servo error budget, b is set by the previous block, and c must match the endpoint of the last move or the axis jumps.

That last point matters more than the algebra. Parabolic blending only works when consecutive segments share position, velocity, and acceleration at the join. If the CAM post outputs points that do not, the controller has to insert its own transition, and the result is no longer a clean quadratic curve.

Constant acceleration means the axis pushes steadily. On a 4,000 mm travel machine doing a long roughing pass, that is exactly what you want: the servo stays loaded, the tool load stays even, and there is no hunting. Chip load is predictable, which matters on 6061 and 7075 aluminum where a sudden load change will chatter.

The cost appears at the joints. Acceleration is constant within a segment and then must change to whatever the next segment demands. That change is a jerk step, and a jerk step is a force step delivered to the ballscrew, the linear guide, and the part. On a light finishing pass in Ra 0.8–1.6 μm territory, that step shows up as a witness mark at the block boundary.

You can hear it. Run a part with many short G01 moves and listen at the transition points. A clean quadratic ramp sounds like a smooth ramp up and down. A poorly blended one clicks or rings at each corner. The ringing frequency tells you which structural mode is being excited, and it usually sits between 200 Hz and 900 Hz on a mid-size vertical mill.

Servo tuning changes how much of that jerk step reaches the part. A stiffer position loop with higher feedforward will track the parabola closely and pass more of the step through. A softer loop smooths it but adds following error. Neither setting removes the step; it only moves the energy somewhere else.

Short moves are the first failure case. If a segment lasts less than about three servo time constants, the axis never reaches the commanded acceleration before it has to reverse. The parabola is truncated, the velocity profile becomes triangular, and the jerk step frequency climbs with the block rate. On a 1 kHz interpolation loop, 1 mm segments at 6,000 mm/min produce a step every few milliseconds.

Thin-wall parts are the second. A jerk step is a broadband excitation. A 1.5 mm aluminum wall has a low first bending mode, and a repeated step at that frequency will sing the wall. The fix is not more clamping; it is a cubic or S-curve profile that spreads the acceleration change over time.

Large machines amplify the problem. On a 4,000 × 400 × 150 mm travel, the moving mass is high and the structural damping is low. A jerk step that is invisible on a compact 500 × 500 × 450 mm machine becomes a visible mark on the large one. If you are cutting a long aerospace rib, plan the toolpath so that direction changes land on support features.

Tool life is the quiet cost. Every jerk step is a small impact at the cutting edge. On hardened steel above 45 HRC, or on Inconel, that impact adds up. Quadratic profiles are fine on 6061 and most plastics; they are a poor default on difficult alloys.

Sometimes the control only offers linear acceleration. In that case, tune the surrounding parameters. Lower the commanded acceleration to about 60–70% of the drive limit. This keeps the parabola below the point where the servo saturates and shortens the jerk step slightly, because the acceleration difference between segments shrinks.

Increase the block join tolerance. Most controllers blend consecutive blocks within a tolerance band. A tolerance of 0.01–0.02 mm on a roughing pass lets the interpolator round the corner and remove the hardest part of the step. On a finishing pass, drop it to 0.002–0.005 mm and accept a slower cycle.

Use exact stop only where geometry demands it. G61 exact stop forces the axis to reach zero velocity at every block, which turns one jerk step into two. Reserve it for datum features and sharp internal corners. Everywhere else, use continuous mode and let the control blend.

Check the post processor. Many CAM posts output points at a fixed chord tolerance, which produces uneven segment lengths. Uneven segments mean uneven parabola lengths, which means the jerk step lands at irregular intervals and excites a wider frequency band. A post that adapts point spacing to curvature gives the controller a much easier job.