Context

Wally (wall-mounted lift system) moves vertically along a guide rail.

Problem: Trapezoidal velocity profiles cause abrupt acceleration changes—resulting in audible noise and mechanical vibration at transition points.

Goal: Achieve smooth, quiet lift motion without sacrificing responsiveness.

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Core Problem

Trapezoidal profiles have infinite jerk at transitions:

Trapezoidal:     S-curve:
  ___              ___
 /   \            /   \
/     \          /     \  ← smooth transitions

At each corner, acceleration changes instantaneously → mechanical shock → noise + vibration.

Key insight

Limit jerk (rate of acceleration change) to eliminate abrupt transitions. S-curve profiles provide continuous acceleration with zero jerk at endpoints.

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Approach

1) Smoothstep function for S-curve shape

Use Hermite interpolation (smoothstep):

S(t) = t² × (3 - 2t)    where t ∈ [0, 1]

Properties:

• S(0) = 0, S(1) = 1
• S’(0) = 0, S’(1) = 0 (zero derivative at endpoints)
• Continuous first and second derivatives

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2) Two-stage acceleration with symmetric jerk

Split acceleration time into two equal stages:

Stage	Duration	Jerk	Acceleration
Stage 1	0 → T/2	+J	Increasing (0 → peak)
Stage 2	T/2 → T	-J	Decreasing (peak → 0)

Jerk calculation: J = 4 × (v_target - v_current) / T²

The factor 4 comes from symmetric S-curve math—jerk magnitude is constant, only sign changes at midpoint.

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3) Integration chain: Jerk → Acceleration → Velocity

Each control loop iteration:

// Stage 1: acceleration increases
if (step < stage1) {
    accel += jerk × dt;
}
// Stage 2: acceleration decreases
else if (step < stage2) {
    accel -= jerk × dt;
}

velocity += accel × dt;

Result: Velocity follows smooth S-curve from current to target.

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4) State machine for motion phases

STOP ←→ ACCELERATING → CONSTANT_VELOCITY → DECELERATING → STOP

Each state handles:

• STOP: Zero velocity, waiting for command
• ACCELERATING: S-curve ramp up
• CONSTANT_VELOCITY: Cruising at target speed
• DECELERATING: S-curve ramp down

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Tradeoffs & Limitations

• Slightly longer motion time: S-curve trades speed for smoothness compared to trapezoidal
• Accumulation errors: Integrating jerk→accel→velocity can accumulate numerical errors over long motions
• Fixed symmetry: Equal ramp-up and ramp-down durations; cannot optimize asymmetrically

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Validation

Tested on deployed Wally units:

• A/B comparison: Trapezoidal vs S-curve with same target velocity and acceleration time
• Noise measurement: Audible clicks eliminated at motion start/stop transitions
• Vibration check: No visible oscillation on guide rail during acceleration phase
• Long-term: Continuous operation over months with no regression

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Results

Metric	Trapezoidal	S-curve
Jerk at transitions	Infinite	Bounded
Audible noise	Noticeable clicks	Smooth, quiet
Mechanical vibration	Present	Minimal

• Same algorithm, two platforms: ESP-IDF C (v1) and ROS2 C++ (v2)
• Field validated: Deployed on Wally units with no noise complaints

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Key Takeaway

Jerk-limited S-curve profiles eliminate the mechanical shock of trapezoidal motion:

• Smoothstep function: t² × (3 - 2t) provides smooth transitions
• Symmetric jerk: +J for first half, -J for second half
• Integration chain: Jerk → Acceleration → Velocity for continuous motion
• Result: Quiet, vibration-free lift operation