cyclogramThe trapezoid
A trapezoidal profile accelerates at a constant rate, cruises, and decelerates. With the ramps given as fractions of the move time T, the distance is the area under the speed curve, and the peak speed follows from it:
t_a = f_a · T t_d = f_d · T t_c = T − t_a − t_d s = v · (t_a/2 + t_c + t_d/2) → v = s / (t_a/2 + t_c + t_d/2) a = v / t_a d = v / t_d200 mm in 0.5 s in equal thirds: v = 0.2 / (1/12 + 1/6 + 1/12) = 0.6 m/s, and a = 0.6 / (1/6) = 3.6 m/s², a little over a third of g.
Why a third, a third, a third
Spending the whole move on the ramps — a triangle — asks for the least acceleration but the most speed: 3.2 m/s² and 0.8 m/s for the same move. Short ramps cost acceleration, long ramps cost speed. For a load that is mostly inertia, the peak power goes with a·v and the RMS torque with a·√(ramp time), and with equal ramps of a fraction f each, both are smallest at f = 1/3:
a = s / (T² · f · (1 − f)) v = s / (T · (1 − f)) P_peak ∝ a · v ∝ 1 / (f · (1 − f)²) smallest at f = 1/3 T_rms² ∝ a² · 2f ∝ 1 / (f · (1 − f)²) smallest at f = 1/3The calculator agrees: the default ball-screw axis needs an RMS torque of 0.138 N·m in thirds and 0.150 N·m as a triangle. The triangle has a practical cost too: its 0.8 m/s turns the 10 mm screw at 4800 rpm, and the D·N climbs to 76 800 — over the usual 70 000, which step 3 flags.
The S-curve
An S-curve limits the jerk — the rate at which the acceleration changes — by ramping the acceleration up and down instead of switching it. Its ramps stay symmetric in time, so the area under each is still half the speed times the ramp time, and the peak speed is the trapezoid's. The price is a higher peak acceleration, set by the jerk fraction r, the share of each ramp spent changing the acceleration:
v = s / (t_a/2 + t_c + t_d/2) as the trapezoid a = v / (t_a · (1 − r)) j = a / (r · t_a)With r = 0.3 the default move peaks at 5.14 m/s² instead of 3.6, with a jerk of 103 m/s³. On the ball screw the peak torque rises from 0.284 to 0.391 N·m, but it arrives later in the ramp, at 2829 rpm, where the servo still has its full peak torque.
The dwell
The time the axis stands counts as well. It is part of the RMS torque, which is averaged over the whole cycle — the move out, the dwell, the move back when the axis returns, the rest at home — and it spreads the braking energy of step 5 over a longer cycle. A vertical axis holds its weight through the dwell unless a brake does.
The same move, three ways
| 200 mm in 0.5 s, 10 mm lead | Thirds | Triangle | S-curve, r = 0.3 |
|---|---|---|---|
| Peak speed | 0.6 m/s | 0.8 m/s | 0.6 m/s |
| Peak acceleration | 3.6 m/s² | 3.2 m/s² | 5.14 m/s² |
| Jerk | — | — | 103 m/s³ |
| Motor speed | 3600 rpm | 4800 rpm | 3600 rpm |
| Peak torque | 0.284 N·m | 0.257 N·m | 0.391 N·m |
| RMS torque | 0.138 N·m | 0.150 N·m | 0.152 N·m |
Questions
Trapezoid or S-curve?
A trapezoid is the usual sizing assumption and asks for the least acceleration in a given move time. An S-curve is gentler on the mechanism and the load — less vibration, less overshoot — at the cost of a higher peak acceleration and torque. Size with the profile your controller will actually run.
How should I split the move?
Equal thirds is the usual start and, for an inertia load, the cheapest in peak power and RMS torque. Shorter ramps lower the acceleration you can get away with only by raising the speed; a triangle (50 / 50) goes furthest that way.
What jerk fraction should I use?
0.5 is a pure S with no constant-acceleration part, 0.2 to 0.3 a typical smoothing, and 0 would be a trapezoid.
Does the move time include the ramps?
Yes. The move time runs from standstill to standstill — it is the time the bar takes on the cyclogram.
Related guides
- Axis sizing calculatorThe whole guided flow: motion, mechanism, motor, regeneration, and the move as a bar.
- Ball screw motor sizingSpeed, reflected inertia, torque per phase, RMS, critical speed and D·N for a ball screw axis.
- Servo motor sizingPeak torque at speed, RMS over the cycle, maximum speed and inertia ratio, each with its margin.
- Brake resistor sizingThe energy an axis returns while braking, what the drive's bus takes, and the resistor for the rest.
- Ball screw critical speedHow fast a screw may turn for its root diameter, length and supports, and the nut's D·N limit.
Every term on this page, defined: the axis and motor sizing glossary.