The requested speed is never reached
A carriage makes a 20 mm move with a requested speed of 500 mm/s and acceleration of 2000 mm/s². Reaching 500 mm/s alone needs 62.5 mm of acceleration travel. Another 62.5 mm is required to decelerate at the same rate. The short move cannot contain that profile.
For an ideal symmetric triangular profile, peak velocity is sqrt(distance × acceleration). Here it is 200 mm/s. Acceleration takes 0.1 s and deceleration takes 0.1 s, so ideal travel lasts 0.2 s. Changing the requested speed from 500 to 600 mm/s changes nothing while acceleration and distance remain the limiting factors.
Build the trapezoid from distances
For speed v, acceleration a and deceleration b:
acceleration_distance = v² / (2a)
deceleration_distance = v² / (2b)
cruise_distance = travel - acceleration_distance - deceleration_distanceIf cruise distance is positive, travel includes a constant-speed segment. If it is zero or negative, calculate a lower reachable peak velocity. The profile bench handles different acceleration and deceleration in its model function; its visible trapezoid control uses equal rates to make the comparison easier.
Position is the accumulated area under the velocity curve. Acceleration is the slope of velocity. Jerk is the rate of change of acceleration, with units mm/s³. A trapezoid's abrupt acceleration changes imply unbounded instantaneous jerk in the ideal model; real mechanics cannot reproduce those changes perfectly.
Braking begins before the target
At 500 mm/s with constant deceleration of 2000 mm/s², ideal braking distance is 62.5 mm. A 10 ms delay before that deceleration begins adds 5 mm of travel. The resulting 67.5 mm still excludes jerk-limited ramping, drive response, feedback delays, compliance and uncertainty.
These are motion estimates, not validated safety distances. A protective separation calculation requires the applicable safety method, actual stop behavior and validation.
In the single-axis bench, the planner compares remaining travel with its estimated braking distance. Increasing deceleration lets it brake later. Lowering acceleration slows the start without necessarily changing the final braking segment.
What a smooth curve buys, and what it costs
A gradual acceleration change can reduce excitation of a flexible belt, screw or load. The tradeoff may be a longer move for the same velocity and acceleration limits. A fragile liquid-filled tray and a rigid empty fixture can need different dynamics even on the same axis.
The smooth teaching model uses a quintic position polynomial:
p(u) = distance × (10u³ - 15u⁴ + 6u⁵), 0 ≤ u ≤ 1Its first and second derivatives are zero at both ends. The duration is chosen to respect the entered velocity, acceleration and jerk limits. This is not the seven-segment S-curve planner of a particular drive. Keep that distinction when transferring a numerical result into a controller.
Bench investigation
Set travel to 20 mm, velocity to 500 mm/s and acceleration to 1000 mm/s². The ideal triangular peak is about 141.4 mm/s and duration is about 0.283 s. Inspect the curve and identify whether cruise exists. Increase travel until a cruise segment appears.
Switch to the smooth curve, lower jerk and watch duration grow. Scrub completion to examine position and velocity at the same fraction of travel time. A larger duration changes velocity even though the final distance remains unchanged.
Exercises
Calculate a 300 mm trapezoidal move at 100 mm/s with acceleration and deceleration of 200 mm/s². Acceleration and braking each consume 25 mm and 0.5 s. Cruise uses 250 mm and 2.5 s; total ideal duration is 3.5 s.
Then halve acceleration but leave deceleration unchanged. Recalculate each segment instead of simply doubling the whole cycle. Finally, explain why increasing peak speed may make a short machine stroke no faster.
References
PLCopen motion control defines dynamics inputs. Vendor planners define jerk conventions, zero-value defaults and override behavior; do not copy a simulator default into hardware without checking those definitions.