Mechatronics & Robotics

Motion Profile Calculator — Minimum Move Time & Feasibility Check

Check whether a point-to-point move can meet its target time using symmetric trapezoidal or jerk-limited S-curve motion limits, with minimum move time, peak velocity, time margin and active kinematic constraint.

Unit-aware inputs Deterministic calculation Engineering interpretation
Calculation workspace

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Deterministic calculation
01
Parameters

Input parameters

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Motion profile

Motion parameters

02
Output

Results

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Engineering reference

Method, application and limitations

Review the calculation method, intended application and engineering assumptions before using the result in a design decision.

01
Method

Formula and calculation method

Motion profile feasibility calculation

This calculator evaluates a symmetric rest-to-rest point-to-point move. It first calculates the minimum achievable motion time from the entered travel distance and kinematic limits, then compares that minimum with the target motion time.

Symbols

  • s — travel distance [m]
  • t — target motion time [s]
  • vmax — maximum allowed velocity [m/s]
  • amax — maximum acceleration/deceleration magnitude [m/s²]
  • jmax — maximum jerk magnitude for S-curve motion [m/s³]
  • tmin — minimum achievable motion time [s]
  • vpeak — peak velocity reached during the minimum-time move [m/s]
  • apeak — peak acceleration magnitude [m/s²]

Trapezoidal motion profile

For a symmetric trapezoidal velocity profile, the acceleration time required to reach the velocity limit is:

tacc = vmax / amax

The total distance required for acceleration and deceleration at vmax is:

sramps = vmax² / amax

Profile reaches vmax

If s ≥ vmax² / amax, the move contains acceleration, constant-velocity and deceleration phases:

vpeak = vmax

sacc = vmax² / (2amax)

sconst = s − 2sacc

tconst = sconst / vmax

tmin = 2tacc + tconst

Equivalently:

tmin = s / vmax + vmax / amax

Triangular velocity profile

If the travel distance is too short to reach vmax, the constant-velocity phase disappears and the velocity profile becomes triangular:

vpeak = √(s · amax)

tacc = vpeak / amax

tmin = 2tacc = 2√(s / amax)

Jerk-limited S-curve motion profile

The S-curve mode uses a symmetric jerk-limited motion profile in which acceleration rises and falls at the entered jerk limit. Depending on distance and limits, the constant-acceleration and constant-velocity phases may disappear.

The time required to ramp acceleration from zero to amax is:

tj,lim = amax / jmax

The velocity reached at the acceleration-limit boundary is:

va = amax² / jmax

Acceleration side when vmax is reached

If vmax ≤ va, the acceleration limit is not reached:

tj = √(vmax / jmax)

ta = 0

tacc = 2tj

apeak = jmax · tj

If vmax > va, the profile reaches amax and contains a constant-acceleration interval:

tj = amax / jmax

ta = vmax / amax − tj

tacc = 2tj + ta

apeak = amax

The combined acceleration and deceleration distance required to reach vmax is:

sramps = vmax · tacc

S-curve with constant-velocity phase

If s ≥ sramps:

vpeak = vmax

sacc = 0.5 · vpeak · tacc

sconst = s − 2sacc

tconst = sconst / vpeak

tmin = 2tacc + tconst

Short S-curve move limited by jerk

The distance boundary for reaching amax without a constant-velocity phase is:

sa = 2amax³ / jmax²

If s < sa, the move is jerk-limited and never reaches amax:

tj = ∛(s / (2jmax))

apeak = jmax · tj

vpeak = jmax · tj²

tmin = 4tj

Short S-curve move limited by acceleration

If the move reaches amax but does not reach vmax:

tj = amax / jmax

ta = (√(tj² + 4s / amax) − 3tj) / 2

vpeak = amax(tj + ta)

tacc = 2tj + ta

tmin = 2tacc

Target-time feasibility

After calculating the minimum motion time, the entered target time is evaluated using:

Feasibility ratio = t / tmin

Δt = t − tmin

Time margin [%] = (t / tmin − 1) · 100

A target is classified as feasible when the target motion time is not below the calculated minimum time, subject to the calculator's numerical tolerance.

02
Application

When to use this calculator

When to use this calculator

Use this motion profile calculator when a point-to-point axis must travel a specified distance within a target time and you already know the maximum velocity and acceleration available to the motion system. For jerk-limited motion, also enter the maximum jerk.

The calculator is useful for:

  • checking whether a required move time is kinematically achievable;
  • calculating the minimum move time allowed by distance, velocity, acceleration and jerk constraints;
  • determining whether a trapezoidal profile reaches its maximum velocity or becomes triangular;
  • evaluating a symmetric jerk-limited S-curve motion profile;
  • finding the peak velocity and peak acceleration actually reached during short moves;
  • identifying whether velocity, acceleration or jerk is the active kinematic limitation;
  • estimating acceleration, cruise and jerk-ramp durations before servo or actuator sizing;
  • checking the available time margin between a required motion time and the theoretical minimum.

Typical engineering applications

The calculation is applicable to simplified point-to-point motion of servo axes, linear actuators, positioning stages, automated handling equipment, pick-and-place mechanisms and other machine motions that can be represented by a symmetric rest-to-rest profile.

What this calculator does not calculate

The feasibility result is a kinematic check only. It does not calculate motor torque, motor power, reflected inertia, gearbox loading, structural vibration, following error, controller bandwidth, settling time or thermal duty. It also does not generate sampled position commands or a controller-ready trajectory.

A motion can therefore satisfy the calculated minimum-time criterion while still requiring additional verification of actuator torque, motor speed, drive limits, mechanical loads and control-system performance.

03
Decision support

How to interpret the result

How to interpret the results

Minimum motion time — tmin

This is the shortest motion time calculated for the selected profile while respecting the entered travel distance and kinematic limits. Reducing the target time below tmin makes the requested move infeasible within this model.

Feasibility ratio — t / tmin

  • Ratio > 1: the target time is longer than the calculated minimum and has positive time margin.
  • Ratio ≈ 1: the target lies close to the minimum-time boundary.
  • Ratio < 1: the requested target time is shorter than the calculated minimum.

FEASIBLE / INFEASIBLE status

FEASIBLE means the target motion time is not below the calculated minimum time. INFEASIBLE means the target is too short for the entered distance and motion limits.

This status evaluates only the kinematic timing criterion. It does not confirm that the motor, drive, transmission or mechanical structure can produce the required motion.

Peak velocity — vpeak

Peak velocity is the highest velocity actually reached by the minimum-time profile. It equals vmax when the move is long enough to reach the entered velocity limit. For shorter moves, vpeak can be lower than vmax.

Peak acceleration — apeak

Peak acceleration is the largest acceleration magnitude reached by the calculated profile. In trapezoidal mode it equals amax. In S-curve mode, a sufficiently short jerk-limited move may have apeak below amax.

Target-time margin — Δt

The time margin is:

Δt = t − tmin

A positive value indicates available time beyond the theoretical minimum. A negative value indicates how much the requested target time falls below the calculated minimum.

Time margin percentage

The percentage margin expresses the same difference relative to tmin. A small positive margin means the target operates close to the calculated kinematic boundary and leaves little timing allowance for effects that are outside this model.

Acceleration-side duration — tacc

This is the time from the beginning of the move until peak velocity is reached. The deceleration side uses the same duration because the implemented profiles are symmetric.

Constant-velocity duration — tconst

A positive tconst means the move reaches its peak velocity and remains there for part of the travel. A zero value means the acceleration and deceleration portions meet without a cruise phase.

Acceleration and constant-velocity distances

sacc is the distance covered on one acceleration side of the motion. sconst is the distance covered during the constant-velocity phase. For a symmetric move, the deceleration-side distance equals sacc.

S-curve jerk and constant-acceleration durations

For S-curve motion, tj is the duration of each jerk ramp. ta is the duration of the constant-acceleration portion when that phase exists. Either the constant-acceleration phase or the cruise phase can collapse to zero on shorter moves.

Active profile regime

The profile-mode result identifies which phases are present in the calculated minimum-time motion, such as a triangular trapezoidal profile, an S-curve with cruise, an acceleration-limited S-curve without cruise or a purely jerk-limited short move.

Active kinematic limit

The limiting-factor result identifies the constraint that governs the minimum-time solution, such as maximum velocity, maximum acceleration or maximum jerk. This helps indicate which entered limit is controlling the achievable motion time.

04
Worked case

Calculation example

Worked example — jerk-limited S-curve feasibility check

A positioning axis must move 1.00 m from rest to rest. The target motion time is 1.80 s. The available kinematic limits are:

  • s = 1.00 m
  • t = 1.80 s
  • vmax = 1.00 m/s
  • amax = 2.00 m/s²
  • jmax = 10.0 m/s³
  • Profile = S-curve

1. Jerk-ramp time

tj = amax / jmax

tj = 2.00 / 10.0 = 0.20 s

2. Check whether the acceleration limit is reached before vmax

va = amax² / jmax

va = 2.00² / 10.0 = 0.40 m/s

Because vmax = 1.00 m/s is greater than 0.40 m/s, the acceleration limit is reached and a constant-acceleration phase is present.

3. Constant-acceleration duration

ta = vmax / amax − tj

ta = 1.00 / 2.00 − 0.20 = 0.30 s

The acceleration-side duration is:

tacc = 2tj + ta = 2(0.20) + 0.30 = 0.70 s

4. Check whether the move reaches maximum velocity

The acceleration and deceleration ramps together require:

sramps = vmax · tacc

sramps = 1.00 · 0.70 = 0.70 m

Because the total travel is 1.00 m, the axis reaches vmax and has a constant-velocity phase.

5. Constant-velocity phase

The acceleration-side distance is:

sacc = 0.5 · 1.00 · 0.70 = 0.35 m

The constant-velocity distance is:

sconst = 1.00 − 2(0.35) = 0.30 m

Therefore:

tconst = 0.30 / 1.00 = 0.30 s

6. Minimum motion time

tmin = 2(0.70) + 0.30 = 1.70 s

7. Feasibility ratio and time margin

Feasibility ratio = 1.80 / 1.70 = 1.0588

Δt = 1.80 − 1.70 = 0.10 s

Time margin = (1.0588 − 1) · 100 ≈ 5.9%

Result

The 1.80 s target is FEASIBLE within the calculator's kinematic model because it is 0.10 s longer than the calculated minimum motion time of 1.70 s.

The minimum-time profile reaches the entered 1.00 m/s velocity limit and 2.00 m/s² acceleration limit. Its active regime contains jerk ramps, a constant-acceleration phase and a constant-velocity phase.

This result does not verify motor torque, drive current, structural dynamics, positioning accuracy or servo settling performance. Those checks remain separate from the kinematic time-feasibility calculation.

05
Model boundaries

Assumptions and limitations

Assumptions and limitations

  • The calculation represents a one-dimensional point-to-point move from rest to rest.
  • Acceleration and deceleration are symmetric and use the same magnitude limits.
  • The entered travel distance, target time, maximum velocity and maximum acceleration must be positive.
  • Maximum jerk must also be positive when S-curve motion is selected.
  • The trapezoidal mode assumes constant acceleration and deceleration segments and does not impose a finite jerk limit.
  • The S-curve mode uses a symmetric jerk-limited profile family in which jerk, acceleration and velocity phases may collapse when the move is too short to reach the corresponding limits.
  • The entered vmax, amax and jmax values are treated as kinematic limits. The calculator does not determine whether the actual actuator, motor or drive can deliver them.
  • The target motion time is used as a feasibility criterion. It does not reshape the profile or solve for new required velocity, acceleration or jerk limits.
  • Initial and final velocities other than zero are not included.
  • Different acceleration and deceleration limits are not supported by this calculation.
  • Motor torque, motor power, reflected inertia, gearbox efficiency, friction, gravity and external forces are not included.
  • Servo following error, feedback bandwidth, controller tuning, encoder resolution and settling time are outside the calculation.
  • Structural flexibility, backlash, compliance, vibration, resonance and other machine dynamics are not modeled.
  • The calculation does not generate sampled position, velocity, acceleration or jerk setpoints for direct controller use.
  • Multi-axis synchronization and path interpolation are not included.
  • A FEASIBLE result confirms only the evaluated kinematic time criterion. Final machine design may require separate motor sizing, drive verification, mechanical analysis and control-system validation.
06
Questions

Frequently asked questions

What does a motion profile feasibility calculator calculate?
It calculates the minimum time required to complete a specified point-to-point move while respecting the entered velocity and acceleration limits and, for S-curve motion, the jerk limit. It then compares the required target time with that minimum and reports whether the target is kinematically feasible.
How is minimum move time calculated?
Minimum move time is obtained by constructing the fastest symmetric rest-to-rest profile allowed by the entered limits. A long move may reach maximum velocity and include a constant-velocity phase. A shorter move may reach only a lower peak velocity, causing one or more profile phases to disappear.
What is the difference between a trapezoidal and an S-curve motion profile?
A trapezoidal velocity profile uses constant acceleration and deceleration segments with instantaneous changes in acceleration at their boundaries. The S-curve mode limits the rate of change of acceleration using a maximum jerk value, so acceleration is ramped rather than changed instantaneously. This generally increases the minimum move time for the same finite velocity and acceleration limits.
When does a trapezoidal motion profile become triangular?
The profile becomes triangular when the travel distance is too short to reach the entered maximum velocity. For the symmetric model used here, this occurs when s is less than v_max² / a_max. The constant-velocity phase then disappears and the calculated peak velocity is √(s · a_max).
What does the feasibility ratio mean?
The feasibility ratio is the target motion time divided by the calculated minimum motion time. A ratio of 1 means the target lies at the minimum-time boundary. A value above 1 indicates positive time margin, while a value below 1 means the requested target time is shorter than the minimum allowed by the entered kinematic limits.
What happens if the target motion time is shorter than the minimum motion time?
The calculator reports the target as INFEASIBLE. Meeting that time would require changing at least one assumption or available kinematic limit, such as increasing allowable velocity, acceleration or, for an S-curve profile, jerk. The calculator does not automatically increase those limits.
Is jerk required for a trapezoidal motion profile calculation?
No. The trapezoidal mode uses travel distance, target time, maximum velocity and maximum acceleration. Maximum jerk is required only when the S-curve profile is selected.
Why can peak velocity be lower than the entered maximum velocity?
Maximum velocity is a limit rather than a value that every move must reach. On a short move, there may not be enough distance to accelerate to v_max and decelerate back to rest. The calculator therefore determines the lower peak velocity that satisfies the distance while using the applicable acceleration and jerk limits.
Does the S-curve mode always reach the maximum acceleration?
No. A sufficiently short move can be limited entirely by jerk, so acceleration begins to ramp down before a_max is reached. Longer S-curve moves can reach the acceleration limit, and still longer moves may also reach the velocity limit and include a constant-velocity phase.
Does a FEASIBLE result mean the servo motor and drive are correctly sized?
No. FEASIBLE means only that the requested motion time is not below the calculated kinematic minimum for the entered profile limits. Motor torque, motor speed, reflected inertia, drive current, thermal loading, following error, mechanical vibration and settling performance require separate verification.
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Engineering notice. Verify input data, assumptions, applicable standards and required safety factors before using results for design release or safety-critical decisions.