Multi-Axis Synchronization Calculator — Synchronized Motion Time & Following Error Validator
Calculate synchronized motion time and maximum following error using axis stroke, velocity, acceleration, controller bandwidth, allowable error, and damping factor.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
Motion parameters
Control & accuracy
Results
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Method, application and limitations
Review the calculation method, intended application and engineering assumptions before using the result in a design decision.
Formula and calculation method
Synchronized motion time and maximum following error formula:
For each axis:
tacc = vmax / amax
sacc = 0.5 · amax · tacc2
If 2 · sacc ≥ s:
t = 2 · √(s / amax)
If 2 · sacc < s:
scruise = s − 2 · sacc
tcruise = scruise / vmax
t = 2 · tacc + tcruise
t1 = motion time for axis 1
t2 = motion time for axis 2
tsync = max(t1, t2)
Δt = |t1 − t2|
τ = 1 / (2 · π · fbw)
ademand = max(a1, a2)
edyn = (ademand · τ2) / (1 + 2 · ζ)
vglobal = max(v1, v2)
ekin = Δt · 0.5 · vglobal
efollow,max = ekin + edyn
Sync feasible = efollow,max ≤ eallow
where:
- s1 — Axis 1 stroke (length)
- s2 — Axis 2 stroke (length)
- v1 — Axis 1 max velocity (velocity)
- v2 — Axis 2 max velocity (velocity)
- a1 — Axis 1 max acceleration (acceleration)
- a2 — Axis 2 max acceleration (acceleration)
- fbw — Controller bandwidth (frequency)
- eallow — Allowable following error (length)
- ζ — Damping factor (-)
- tsync — Synchronized motion time (time)
- efollow,max — Maximum following error (length)
When to use this calculator
When to use this calculator:
- Check whether two linear axes can complete different strokes within one synchronized motion time.
- Estimate maximum following error caused by phase shift between axis motion times and controller response delay.
- Compare axis velocity and acceleration limits before defining a synchronized move profile.
- Evaluate whether controller bandwidth and damping factor keep following error below the allowable error.
- Identify whether synchronization is limited by axis 1 or axis 2 motion time.
How to interpret the result
Synchronized motion time is defined as the longer motion time required by either axis. The limiting axis is the axis with the larger calculated motion time.
Maximum following error is defined as the sum of phase-shift-related kinematic error and bandwidth-related dynamic tracking error. It depends on axis motion time mismatch, maximum axis velocity, maximum axis acceleration, controller bandwidth, and damping factor.
Increasing phase shift increases the kinematic part of the following error. Increasing the higher maximum velocity also increases the kinematic part of the following error. Increasing the higher maximum acceleration increases the dynamic tracking error. Increasing controller bandwidth decreases the control time constant and therefore decreases dynamic tracking error. Increasing damping factor decreases dynamic tracking error through the denominator 1 + 2 · ζ.
- Safe — maximum following error is less than or equal to the allowable following error, the error ratio is not above 0.6, the tracking difficulty index is not above 1, and the control margin is not below 3.
- Warning — synchronization is still feasible, but maximum following error is above 60% of the allowable error, or above 90% of the allowable error, or the tracking difficulty index is above 1, or the control margin is below 3.
- Unsafe — maximum following error is greater than the allowable following error.
- Invalid — at least one required input is zero, negative, not finite, or produces a non-finite internal result.
The result is used to decide whether the two axes can be synchronized within the specified allowable following error.
Calculation example
Example:
A user wants to synchronize two linear axes. Axis 1 has a longer stroke, while axis 2 has a shorter stroke but different velocity and acceleration limits. The user checks whether the expected following error remains below the allowed position error.
- s1 — Axis 1 stroke = 0.8 m
- s2 — Axis 2 stroke = 0.6 m
- v1 — Axis 1 max velocity = 1.2 m/s
- v2 — Axis 2 max velocity = 1.0 m/s
- a1 — Axis 1 max acceleration = 3.0 m/s²
- a2 — Axis 2 max acceleration = 2.5 m/s²
- fbw — Controller bandwidth = 20 Hz
- eallow — Allowable following error = 0.02 m
- ζ — Damping factor = 0.7
tsync = 1.067 s
efollow,max = 0.037 m
Sync feasibility = not feasible, because the maximum following error is greater than the allowable following error.
Assumptions and limitations
- Each axis motion time is calculated from stroke, maximum velocity, and maximum acceleration using either a triangular or trapezoidal motion-time model.
- Synchronized motion time is the maximum of axis 1 motion time and axis 2 motion time.
- Maximum following error is modeled as the sum of kinematic error from phase shift and dynamic tracking error from controller bandwidth and damping factor.
- Synchronization feasibility is based only on whether maximum following error is less than or equal to allowable following error.
- All required inputs must be finite and greater than zero.