Axis Move & Cycle Time

Q: When does a trapezoidal versus a triangular profile occur?

The threshold distance s_lim = v_max²/a decides. If the travel s is greater or equal, v_max is reached and the axis runs a trapezoidal profile with constant travel between accelerating and decelerating. If the travel is shorter, v_max is never reached and it stays a symmetric triangular profile with lower peak velocity v_peak = √(s·a). This is how Trapezoidal (auto) chooses; with the S-curve the jerk phases lengthen the ramps, so the travel needed to reach v_max is longer. "Forced" means: with Triangular (forced) the axis always runs the triangular profile, however long the travel. If s is greater than s_lim, v_peak = √(s·a) then exceeds v_max, and the calculator points this out.

Q: What is the benefit of the S-curve profile?

With the S-curve profile acceleration is ramped up and down in a jerk-limited way instead of instantly. This reduces vibration excitation and wear, but extends the motion by the jerk time a/j once v_max and a are reached. The smaller the jerk j, the smoother and slower the motion.

Q: How is the total cycle time formed?

The cycle time is the plain sum of all segment times: each move-segment time plus each dwell. Segments are executed one after another in the given order; overlapping or interpolated motions of several axes are not considered.

Q: Does the calculator cover drive sizing?

No. The calculator provides times and peak velocities of the pure motion profiles. Whether motor and gearbox deliver the required torques and speeds is checked separately by drive sizing; the accelerations and velocities determined here are its input values.

Build a sequence of move and dwell segments and calculate the total cycle time of a positioning axis. Each move segment yields its profile shape, segment time and peak velocity; dwell segments are simply added. The combined position, velocity and acceleration profile across the whole sequence is shown live as a diagram, live with every input.

Calculation

Segments

Segments in execution order. Freely mix, add, remove and reorder move and dwell segments.

1. Move
Distance s [m]v_max [m/s]Acceleration a [m/s²]Profile
2. Dwell
Duration t [s]
3. Move
Distance s [m]v_max [m/s]Acceleration a [m/s²]Profile

Model: uniformly accelerated point-to-point motions (trapezoidal/triangular) with optional jerk limiting (S-curve as jerk-limited profile, t_S = t_base + a/j when v_max and a are reached). Segments are executed sequentially and summed; multi-axis interpolation, control behaviour and drive limits are not considered. For motor and gearbox sizing use drive sizing.

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Formulas and fundamentals

Trapezoidal and triangular profile

Each move segment is a uniformly accelerated point-to-point motion over the distance s with acceleration a and velocity limit v_max. The threshold distance decides the profile shape:

s_lim = v_max²/a

With Trapezoidal (auto): if s ≥ s_lim the axis reaches v_max and runs a trapezoidal profile; if s < s_lim it stays a symmetric triangular profile. Triangular (forced) always runs the triangular profile: at s = s_lim, v_peak = v_max; for s > s_lim, v_peak exceeds v_max.

trapezoidal: t = v_max/a + s/v_max triangular: v_peak = √(s·a) t = 2·√(s/a)

The acceleration distance per ramp is s_b = v_peak²/(2·a), the acceleration time t_b = v_peak/a.

Jerk-limited S-curve profile

The jerk-limited S-curve profile ramps the acceleration up and down with the jerk j = da/dt; v_max and a remain limits. An acceleration ramp from 0 to v takes v/a + a/j as long as v ≥ a²/j. Below that it does not reach a and takes 2·√(v/j) at the peak acceleration √(v·j). If the travel suffices for two ramps to v_max, the axis runs at constant velocity in between, and if a is reached as well:

t_S = v_max/a + s/v_max + a/j = t_base + a/j

The longer ramps cover more distance themselves and shorten the constant-velocity phase; jerk limiting therefore costs a/j in total, not a/j per ramp. If the travel does not suffice, v_peak stays below v_max. The S-curve is smoother and excites fewer vibrations, at the cost of cycle time.

Dwells and total cycle time

A dwell segment contributes its dwell time unchanged. The total cycle time of the sequence is the sum of all move-segment times plus all dwells. For the diagram all segments are placed on a common time axis; dwells appear as sections with v = 0 where the cumulative position stays constant.

Worked example

A move over s = 0.5 m with v_max = 1 m/s and a = 5 m/s²: the threshold distance is s_lim = 1²/5 = 0.2 m. Since s = 0.5 m ≥ 0.2 m a trapezoidal profile results. The acceleration distance per ramp is s_b = 0.1 m at t_b = 0.2 s, the constant-velocity distance 0.3 m at t_k = 0.3 s. The segment time is t = 2·0.2 + 0.3 = 0.7 s.

The same drive over only s = 0.1 m no longer reaches v_max (0.1 m < 0.2 m): it stays a triangular profile with v_peak = √(0.1·5) = 0.707 m/s and t = 2·√(0.1/5) = 0.283 s.

A sequence of move (0.7 s), dwell (0.5 s) and another move (0.7 s) gives a total cycle time of 0.7 + 0.5 + 0.7 = 1.9 s for a total travel of 1.0 m.

Frequently asked questions

When does a trapezoidal versus a triangular profile occur?

The threshold distance s_lim = v_max²/a decides. If the travel s is greater or equal, v_max is reached and the axis runs a trapezoidal profile with constant travel between accelerating and decelerating. If the travel is shorter, v_max is never reached and it stays a symmetric triangular profile with lower peak velocity v_peak = √(s·a). This is how Trapezoidal (auto) chooses; with the S-curve the jerk phases lengthen the ramps, so the travel needed to reach v_max is longer. "Forced" means: with Triangular (forced) the axis always runs the triangular profile, however long the travel. If s is greater than s_lim, v_peak = √(s·a) then exceeds v_max, and the calculator points this out.

What is the benefit of the S-curve profile?

With the S-curve profile acceleration is ramped up and down in a jerk-limited way instead of instantly. This reduces vibration excitation and wear, but extends the motion by the jerk time a/j once v_max and a are reached. The smaller the jerk j, the smoother and slower the motion.

How is the total cycle time formed?

The cycle time is the plain sum of all segment times: each move-segment time plus each dwell. Segments are executed one after another in the given order; overlapping or interpolated motions of several axes are not considered.

Does the calculator cover drive sizing?

No. The calculator provides times and peak velocities of the pure motion profiles. Whether motor and gearbox deliver the required torques and speeds is checked separately by drive sizing; the accelerations and velocities determined here are its input values.

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