The Problem
Stick-slip is the dominant source of positioning error and surface-finish degradation in servo-driven industrial machinery operating at low feed rates. When a ball-screw axis traverses the Stribeck velocity zone, the closed-loop system destabilizes into a limit cycle — periodic velocity oscillation, position error, and visible surface patterns in precision machining.
Every existing MPC approach treats the velocity weight as a tuning parameter, selected by cross-validation or heuristic. This means: no principled answer to "why this weight?", no proof of optimality, and retuning required every time the system changes.
What FERRUM Does
FERRUM is an LTV-MPC architecture running at 500 Hz where the velocity weight is derived from a proprietary theoretical framework. The derivation is not empirical — it identifies a specific mathematical property of the stick-slip process and proves that a unique weight ratio follows from that property.
The resulting weight is not one option among many. The paper proves it is the only ratio that sits on the optimal Pareto frontier at the point of maximum roughness reduction per unit of position error tolerance. Any other ratio is provably suboptimal.
The method is not public. The results are.
Simulation Results
Four-condition comparison on an industrial cruise scenario (constant velocity in the Stribeck zone, 3.08 s cruise phase):
| Metric | PD Baseline | MPC (standard) | MPC + FERRUM | Δ vs PD |
|---|---|---|---|---|
| RMS position error | 3426 μm | 537 μm | 763 μm | −77.7% |
| Velocity roughness | 0.0440 | 0.0123 | 0.0086 | −80.4% |
| Stick-slip events | 6 | 0 | 0 | −100% |
FERRUM achieves 30.1% lower velocity roughness than standard MPC while sitting at the provably optimal trade-off point. The position error increase (763 vs 537 μm) is the exact cost of operating at that optimum — and the paper proves no other weight achieves a better roughness-per-position ratio.
By the Numbers
- −80.4% velocity roughness vs PD baseline
- −77.7% RMS position error vs PD baseline
- 0 stick-slip events (complete elimination)
- 30.1% roughness improvement over standard MPC
- 500 Hz control loop, < 0.5 ms solve time
- Provably Pareto-optimal — not just good, formally proven
Architecture
The MPC architecture itself uses standard components — the novelty is entirely in how the weights are determined:
LTV-MPC at 500 Hz — Linear Time-Varying formulation with re-linearization at each control step. 15-step horizon (30 ms prediction window).
LuGre friction feedforward — full contact dynamics modeling: bristle stiffness, bristle damping, viscous friction, Stribeck velocity-dependent friction capacity. Three-state plant: position, velocity, bristle deflection.
DARE terminal cost with asymptotic stability proven via Lyapunov argument. Closed-loop spectral radius 0.9983.
Computational footprint — the online solve is a 15×15 linear system. No iterative QP solver needed. Sub-millisecond execution on any modern DSP.
Honest Limitations
The stability guarantee is local — computed at the nominal linearization point. The LuGre model omits temperature dependence and lubrication film dynamics. The detailed balance assumption holds at steady cyclic operation; transient phases require separate analysis. All performance figures are simulation results. Hardware validation on a physical ball-screw axis is the next milestone.
Industrial Path
500 Hz matches standard servo drives (Beckhoff EtherCAT, Siemens S120, Allen-Bradley Kinetix). LuGre feedforward is implementable as an IEC 61131-3 function block. The DARE terminal cost is computed offline; the online MPC runs on commodity real-time hardware. Directly deployable on CNC machines, precision grinding stages, EDM systems, and coordinate measuring machines.
Application Fit
For surface-finish-critical applications (precision grinding, EDM, laser cutting) where velocity smoothness dominates — FERRUM is the correct operating point.
For absolute-positioning-critical applications (pick-and-place, assembly) where position accuracy dominates — a lower velocity weight is appropriate, and FERRUM's framework can derive that weight too.
The choice is not intuition. It is derivable.