S-Curve Acceleration and Jerk Limiting for Precision Motion

Key Takeaway

S-curve acceleration limits jerk so that a machine axis can start and stop without exciting its structural resonance. On a ball screw axis, trapezoidal velocity causes a step in acceleration that the machine feels as a bang; an S-curve ramps acceleration up and down, so the machine moves smoothly and settles faster.

Figure 1: Motion profiles: trapezoid versus S-curve – parameters and where it matters

1. What Jerk Is and Why It Matters
2. Trapezoidal Velocity and Its Acceleration Step
3. S-Curve Velocity and Its Seven Phases
4. Acceleration Limit, Jerk Limit and Time Budget
5. Where S-Curve Matters: Resonance and Settling
6. What S-Curve Costs: Time and Torque
Related Reading
Sources

1. What Jerk Is and Why It Matters

Jerk is the rate of change of acceleration. In SI units it is metres per second cubed (m/s^3). It is the third derivative of position, after velocity and acceleration, and it is the quantity that connects a motion command to the way a machine actually moves.

Every machine axis has structure: a screw, a coupling, a guideway, a frame. Each element has stiffness and mass, and each has a natural frequency. When acceleration changes suddenly, the machine feels it as a step input, and the structure rings at its natural frequency. That ringing is vibration, and vibration is settling time, and settling time is cycle time.

The three derivatives

Derivative	Symbol	Unit	Physical meaning
Position	x	m	Where the axis is
Velocity	v = dx/dt	m/s	How fast it moves
Acceleration	a = dv/dt	m/s^2	How fast velocity changes
Jerk	j = da/dt	m/s^3	How fast acceleration changes

The force a machine structure feels is mass times acceleration (F = ma). When acceleration changes suddenly, the force changes suddenly, and the structure responds to that sudden force. A smooth force produces a smooth response; a step force produces a step response, complete with overshoot and ringing.

2. Trapezoidal Velocity and Its Acceleration Step

Trapezoidal velocity is the simplest move profile: accelerate at a constant rate until cruise speed, hold cruise speed, then decelerate at a constant rate to stop. It is easy to plan and easy to tune, and it is the default on most servo drives.

The acceleration step

At the start of a trapezoidal move, acceleration jumps from zero to the acceleration limit in zero time. That step in acceleration is a step in force, and the machine structure feels it as an impact. The impact excites every natural frequency below the move bandwidth, and the axis rings.

j = da/dt = infinity at the step

A step in acceleration has theoretically infinite jerk. In practice the jerk is limited by the drive’s current loop bandwidth and the electrical time constant, but it is still large enough to excite the structure.

What the step does to the machine

Ball screw whip.A long screw under sudden acceleration whips laterally, and the whip shows up as position error at the nut.
Coupling wind-up.The coupling between motor and screw twists under the sudden torque and then unwinds, producing an oscillation that the position loop sees as error.
Guideway stick-slip.The carriage breaks static friction suddenly and then slides, producing a jump in position that the loop must correct.
Frame ringing.The machine frame, especially a tall column or a long gantry beam, rings at its bending mode.

3. S-Curve Velocity and Its Seven Phases

An S-curve velocity profile replaces the acceleration step with a ramp. Acceleration starts at zero, ramps up to the limit, holds, ramps down to zero, cruises, then reverses the process to stop. The result is seven phases instead of three, and the acceleration changes smoothly instead of stepping.

The seven phases

Jerk ramp up.Acceleration increases from zero to the limit at constant jerk.
Constant acceleration.Acceleration holds at the limit.
Jerk ramp down.Acceleration decreases from the limit to zero at constant jerk.
Constant velocity.Acceleration is zero; the axis cruises.
Jerk ramp up (negative).Acceleration becomes negative at constant jerk.
Constant deceleration.Deceleration holds at the limit.
Jerk ramp down (negative).Deceleration returns to zero at constant jerk.

The velocity curve is literally S-shaped: flat at the start, steep in the middle, flat at the end. The acceleration curve is trapezoidal instead of rectangular, and the jerk curve is a pair of rectangular pulses instead of impulses.

What smooth jerk does

No structure ring.With no step in force, the structure has nothing to ring at.
Lower peak torque.The drive does not have to deliver the full acceleration torque instantly; it builds up over the jerk ramp.
Less wear.Bearings, screws and couplings see smooth torque instead of shock torque.
Faster settling.Without ring, the axis reaches position on the first attempt.

4. Acceleration Limit, Jerk Limit and Time Budget

Choosing the acceleration limit

The acceleration limit is set by the machine dynamics, not by the drive. It is the largest acceleration the structure can survive without ring, and it is usually found empirically: start low, increase until the axis rings, then back off 20 percent.

For a ball screw axis, the acceleration limit is also limited by the screw’s critical speed and by the motor’s torque. A 63 mm ball screw over 2 m can accelerate roughly 2 to 5 m/s^2 depending on the end support, and the motor must produce enough torque to move the carriage at that rate.

Choosing the jerk limit

The jerk limit is the S-curve parameter that replaces the acceleration step. It is set by the structure’s natural frequency:

j_limit = a_limit / t_ramp

where t_ramp is the time over which acceleration builds up. A longer ramp means lower jerk and less ring, but it also means the axis takes longer to reach cruise speed.

A practical starting point: set the jerk ramp time to 50 to 100 ms. For a 2 m/s^2 acceleration limit, that is a jerk limit of 20 to 40 m/s^3.

The time budget

S-curve takes longer than trapezoidal because of the two jerk ramps. The extra time is approximately:

t_extra = 2 x t_ramp

For a 100 ms ramp, that is 200 ms of extra move time. On a pick-and-place axis making 10 moves per minute, that is 2 seconds per minute, or about 3 percent of cycle time. The trade is 3 percent cycle time for zero ring and full settling on the first attempt.

Position budget

The S-curve covers the same distance as the trapezoid, so the position at the end of the move is identical. The difference is how the axis gets there, not where it ends.

5. Where S-Curve Matters: Resonance and Settling

The resonance connection

Every machine axis has natural frequencies: the first bending mode of a long screw, the torsional mode of a shaft and coupling, the bending mode of a gantry beam, the mode of a cross slide with a heavy carriage. When acceleration changes suddenly, the move contains energy at all frequencies, and whichever natural frequency is closest to the move bandwidth gets excited.

S-curve removes the sudden change, so the move no longer contains the high-frequency content that excites the structure. The axis moves smoothly, and the position loop does not have to fight ring.

Settling time comparison

Profile	Move time	Ring after move	Settling time
Trapezoidal, 2 m/s^2	1.00 s	0.30 s	1.30 s
S-curve, 2 m/s^2, 100 ms ramp	1.20 s	0.02 s	1.22 s

The S-curve takes 20 percent longer to execute but settles 15 times faster. On a machine making hundreds of moves per hour, the S-curve is faster overall because every move settles immediately.

Where S-curve is essential

Glass and optical stages.Sub-micron positioning cannot tolerate ring, and a glass plate on a moving stage is a classic ring source.
Chip breaking in machining.A CNC machine breaking a chip needs a smooth acceleration so the chip does not chatter and the tool does not deflect.
Camera and inspection axes.A camera moving over a part must not blur the image with vibration, and S-curve keeps the image sharp.
Z axis on a vertical machine.A vertical axis lifting a load must not drop or bounce when it starts, and S-curve gives a controlled lift.

6. What S-Curve Costs: Time and Torque

The time cost

S-curve always takes longer than trapezoidal for the same distance and the same acceleration limit, because of the jerk ramps. The extra time scales with the ramp time and inversely with the distance: short moves pay a higher percentage penalty than long moves.

For a 100 mm move at 1 m/s cruise with a 2 m/s^2 acceleration limit and a 100 ms jerk ramp, the trapezoid takes about 0.20 s and the S-curve takes about 0.24 s. The penalty is 20 percent on a short move, but on a 1 m move it falls to about 4 percent.

The torque benefit

S-curve does not require more peak torque; it spreads the same torque over a longer time. The motor sees a gentler demand, which is easier on the drive and the power supply.

When not to use S-curve

Moves so short that the jerk ramp consumes most of the move time.
Machines with no structural resonance (rare).
Cycles where every millisecond matters and the machine can tolerate ring.

For almost every precision axis, S-curve is worth the time. The jerk limit is the knob that trades time for smoothness, and it is the one parameter that turns a ringing axis into a quiet one.

Sources

Key Takeaways

Jerk is the rate of change of acceleration, and a step in acceleration excites every structural natural frequency below the move bandwidth.
Trapezoidal velocity has an acceleration step and therefore theoretically infinite jerk; S-curve ramps acceleration up and down and eliminates the step.
S-curve has seven phases instead of three, and it takes about 2 x jerk_ramp_time longer than trapezoidal for the same move.
The jerk limit is chosen from the structure’s natural frequency: start with a 50 to 100 ms ramp and adjust until the axis no longer rings.
On precision axes (glass, optics, inspection, vertical Z), S-curve is essential because it removes ring and settles on the first attempt.