The Axis That Wouldn’t Tune Stiff: Why the Loop Gain Isn’t the Limit
A positioning axis that felt springy. When you pushed on the carriage by hand, it moved 0.3 mm and sprung back. The position loop gain was at its maximum. The customer wanted higher stiffness (less deflection under load). We couldn’t raise Kp — the axis started to ring. The problem wasn’t the gain. It was the mechanics.
What “stiff” means
Axis stiffness is how much the carriage deflects under an external force. A stiff axis deflects 0.02 mm under 10 N. A soft axis deflects 0.3 mm under the same force. The position loop tries to hold position, but it can only correct as fast as the loop bandwidth. Below the loop bandwidth, the axis is as stiff as the mechanics. Above the loop bandwidth, the loop adds stiffness.
If you can’t raise Kp without ringing, the mechanics are the weak link. The loop is already as stiff as it can be without exciting the mechanical resonance.
The mechanical stiffness chain
The total stiffness of the axis is the series stiffness of every component in the force path:
1/k_total = 1/k_belt + 1/k_coupling + 1/k_screw + 1/k_nut + 1/k_mount
The weakest link dominates. For a belt-driven axis, the belt stiffness is typically 5–20 N/μm. The coupling is 100 N·m/rad (converted to linear stiffness, about 50 N/μm). The screw is stiffer. The nut is the weakest link — a ball nut’s stiffness is about 20–50 N/μm. The belt is 5–20 N/μm. The belt dominates.
Our axis had a belt stiffness of 8 N/μm. Under 10 N, the belt stretches 1.25 mm. But we measured 0.3 mm. The belt was tensioned, and the preload reduced the effective stretch. Still, 0.3 mm is belt stiffness, not loop stiffness.
What raises stiffness
| Change | Stiffness improvement | Cost |
|---|---|---|
| Higher loop gain (Kp) | Limited by resonance | Free, but capped |
| Belt → ball screw | 5–10× stiffer | High cost, new mechanics |
| Larger belt (wider, more strands) | 2–3× stiffer | New belt, pulleys |
| Higher belt tension | 1.5–2× stiffer (up to a point) | Free, but adds bearing load |
| Bigger coupling | Slight (coupling wasn’t the limit) | New coupling |
| Linear motor (no transmission) | 10–100× stiffer | Very high cost |
What we did
We couldn’t change the belt to a screw (the machine was already built). We increased the belt tension from 5 N to 15 N. The stiffness improved from 8 N/μm to 14 N/μm. The deflection under 10 N dropped from 0.3 mm to 0.15 mm. That was close to the customer’s requirement (0.1 mm). We also raised Kp slightly (from 300 to 350 s⁻¹) because the stiffer belt allowed a higher loop gain without ringing.
For the next build, we specified a ball screw instead of a belt. The screw’s stiffness is about 50 N/μm — 6× stiffer than the belt. The deflection under 10 N is 0.02 mm. No compromise needed.
The stiffness calculation
For a ball screw, the axial stiffness is:
k_screw = (E · A) / L
Where E is Young’s modulus (200,000 N/mm²), A is the screw cross-section (π × d²/4), and L is the screw length. For Ø20 mm, 500 mm long: A = 314 mm². k = 200,000 × 314 / 500 = 125,600 N/mm = 125 N/μm. That’s 15× stiffer than the belt. The screw is the right answer for a stiff axis.
When the loop is the limit
If the mechanics are stiff (ball screw, rigid mount) and you still can’t raise Kp, the limit is the resonance frequency of the loop itself. Check the servo’s auto-tune. The auto-tune measures the mechanical resonance and sets the highest stable Kp. If you need more stiffness, you need a faster controller (higher sample rate) or a lighter load (lower inertia). But for our belt axis, the mechanics were the limit — no amount of tuning would fix the belt’s inherent flexibility.
The stiffness rule: If you can’t raise the loop gain without ringing, the mechanics are the weak link. Measure the deflection by hand. If it’s more than 0.05 mm, the belt (or coupling, or mount) is limiting. For a stiff axis, use a ball screw, not a belt. The loop gain can only compensate for so much mechanical flex.
