Contouring Error Bounded Control for Biaxial Systems with Structural Flexibility and Input Delay

Meng Yuan; Tianyou Chai

arxiv: 2604.11018 · v1 · submitted 2026-04-13 · 📡 eess.SY · cs.SY

Contouring Error Bounded Control for Biaxial Systems with Structural Flexibility and Input Delay

Meng Yuan , Tianyou Chai This is my paper

Pith reviewed 2026-05-10 15:29 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords biaxial contouring controlmodel predictive controlinput delay compensationstructural flexibilityrobust invariant setsprecision machiningerror bounded control

0 comments

The pith

Model predictive control guarantees bounded contouring error for arbitrary paths in biaxial systems with flexibility and delays.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper aims to show that a specific model predictive control design can maintain contouring accuracy within bounds for biaxial machines that have position-dependent flexibility and input delays. It does this by building a high-fidelity model that includes vibration dynamics and augments it with delay states, then defining robust invariant sets using switched models to guide the controller. This is important because contouring precision directly affects quality in processes like laser cutting, and the method claims to work without being tied to particular path shapes while requiring little tuning. The design keeps the system states inside safe regions that account for uncertainties. If correct, it provides a practical way to achieve reliable performance even when the machine's behavior degrades over time.

Core claim

The paper claims that augmenting the system model with input delay states and using switched linear time-invariant models to define robust control invariant sets enables model predictive control to ensure the end-effector stays on the desired contour within error bounds, independent of the curve shape, for systems with structural flexibility.

What carries the argument

High-fidelity model with augmented delay states and robust control invariant sets defined via switched LTI models, enforced by model predictive control.

If this is right

Bounded contouring error is achieved for any desired curve shape.
System constraints are satisfied despite input delays and flexibility.
The method requires low commissioning effort in degraded performance conditions.
Experimental results confirm effectiveness with discretizations and delays.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This control strategy might reduce reliance on expensive stiff mechanical designs in manufacturing.
The invariant set framework could be combined with learning methods to handle unmodeled dynamics.
Similar techniques may apply to other delayed systems like networked control or robotics with compliance.

Load-bearing premise

The high-fidelity model accurately captures the rotation dynamics with flexibility and the switched models allow invariant sets that contain all possible real system trajectories.

What would settle it

Implement the controller on a biaxial testbed with known structural modes and input delay, traverse a non-standard contour, and check if the measured contouring error stays within the prescribed bound under varying conditions.

Figures

Figures reproduced from arXiv: 2604.11018 by Meng Yuan, Tianyou Chai.

**Figure 2.** Figure 2: Block diagram of the proposed contouring error-bounded control architecture. [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Experimental test bench for contouring control. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 5.** Figure 5: Experimental tracking performance based on Tuning A: X-axis error [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: Experimental tracking performance based on Tuning B: X-axis error [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗

**Figure 7.** Figure 7: Experimental contouring error during the whole process. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗

**Figure 8.** Figure 8: Desired and actual contour based on different tuning. [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗

read the original abstract

Precision contouring control is crucial in industrial machining processes, particularly for applications such as laser and water jet cutting, where contouring accuracy directly determines product quality. This paper presents a novel control strategy for biaxial machines featuring position-dependent flexibility and input delays, ensuring that the end-effector accurately traverses the desired contour within specified contouring error bounds and system constraints. To capture the rotation dynamics for systems with mechanical vibration, we introduce a high-fidelity model and explicitly consider the input delay with augmented system states. The controller design is based on the model predictive control scheme to enforce system states staying in robust control invariant sets defined by the reference model and switched linear time-invariant control-oriented models. The proposed algorithm is not restricted to a specific shape of the curve that is being traversed. The effectiveness of the proposed control algorithm is demonstrated in an experimental environment with discretizations and input delay. The results show that a bounded contouring error can be achieved by the proposed method in a performance degradation environment with a low commissioning effort.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper offers a practical MPC scheme with invariant sets for contouring control under flexibility and delay, though the guarantees may not fully hold for the nonlinear case.

read the letter

The paper combines high-fidelity modeling of position-dependent flexibility, delay state augmentation, and MPC with robust invariant sets from switched LTI models to bound contouring errors on arbitrary curves in biaxial systems. This integration is the main contribution, along with the claim that it requires low commissioning effort even under performance degradation. The experimental demonstration adds some credibility for industrial use in machining. The soft spot is the reliance on invariant sets that must contain real trajectories. Position-dependent flexibility makes the dynamics nonlinear, so the linear approximations could fail to guarantee invariance when parameters vary or delays affect stability margins. Without equations or proofs in the provided text, it's tough to see how they handle this. Experiments are mentioned but details on tested contours and degradation levels are missing, which limits confidence in the broad claims. This is for control engineers focused on precision machine tools and manufacturing automation. A reader dealing with similar vibration and delay problems in contouring tasks would get practical ideas from the modeling and MPC setup. It deserves peer review because the problem is real and the approach is systematic, even if revisions are likely needed for the theoretical parts.

Referee Report

2 major / 2 minor

Summary. The paper proposes a novel MPC-based control strategy for biaxial contouring systems subject to position-dependent structural flexibility and input delays. A high-fidelity model is introduced to capture rotation dynamics, with state augmentation to handle input delay. The design relies on robust control invariant sets constructed from a reference model and switched LTI control-oriented models to enforce bounded contouring error for arbitrary contour shapes while satisfying system constraints. Effectiveness is shown via experiments on discretized systems with input delay, claiming bounded error achievement under performance degradation with low commissioning effort.

Significance. If the invariant-set containment holds, the approach would advance precision control for flexible biaxial machines in applications like laser cutting by providing a general, shape-independent method with explicit delay handling and practical low-tuning validation. Credit is due for the experimental demonstration under degradation conditions and the use of switched LTI models plus state augmentation to address flexibility and delay without heavy retuning.

major comments (2)

[Abstract and Controller Design] Abstract (and Controller Design section): The claim that MPC enforces states within robust control invariant sets that contain all real closed-loop trajectories for arbitrary contours rests on the switched LTI models and delay-augmented high-fidelity model accurately capturing position-dependent flexibility. However, continuous stiffness variation with position renders the true dynamics nonlinear, and linearizations plus state extension may yield sets whose invariance margin vanishes under delay-induced phase lag or high-curvature contours; the reported experiments on specific discretizations do not rule out violations for untested conditions.
[Abstract] Abstract: The assertion of 'bounded contouring error' and 'low commissioning effort' in a 'performance degradation environment' is load-bearing for the practical contribution, yet no quantitative error bounds, degradation levels, or comparison metrics are supplied to substantiate that the invariant sets deliver the claimed performance margin.

minor comments (2)

[Abstract] The abstract repeats the claim that the algorithm 'is not restricted to a specific shape of the curve' without clarifying how the switched models or invariant sets enable this generality; a brief statement on the switching logic would improve clarity.
[Experimental Results] The experimental validation is described only at a high level ('with discretizations and input delay'); specifying the discretization method, delay values, and contour types tested would aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough and constructive review of our manuscript. We address each major comment point by point below, indicating the revisions we will make to improve clarity and strengthen the claims.

read point-by-point responses

Referee: [Abstract and Controller Design] Abstract (and Controller Design section): The claim that MPC enforces states within robust control invariant sets that contain all real closed-loop trajectories for arbitrary contours rests on the switched LTI models and delay-augmented high-fidelity model accurately capturing position-dependent flexibility. However, continuous stiffness variation with position renders the true dynamics nonlinear, and linearizations plus state extension may yield sets whose invariance margin vanishes under delay-induced phase lag or high-curvature contours; the reported experiments on specific discretizations do not rule out violations for untested conditions.

Authors: We acknowledge that the true plant dynamics are nonlinear owing to continuous position-dependent stiffness variation. The switched LTI models are constructed at discrete position samples to approximate this variation, while the robust invariant sets are computed with explicit uncertainty margins that bound the linearization error between samples. State augmentation is used precisely to incorporate the known input delay and thereby limit phase-lag effects within the invariance analysis. The experiments on the discretized testbed with delay demonstrate containment for the contours examined. Nevertheless, we agree that additional verification for high-curvature trajectories would further substantiate robustness. In the revised manuscript we will add simulation results for contours with curvature radii below the minimum tested value, confirming that the invariant-set margins remain positive. This constitutes a partial revision. revision: partial
Referee: [Abstract] Abstract: The assertion of 'bounded contouring error' and 'low commissioning effort' in a 'performance degradation environment' is load-bearing for the practical contribution, yet no quantitative error bounds, degradation levels, or comparison metrics are supplied to substantiate that the invariant sets deliver the claimed performance margin.

Authors: We agree that the abstract would be strengthened by quantitative metrics. The experimental results already contain the necessary data (maximum contouring error, stiffness degradation percentages, and tuning-parameter counts relative to a baseline MPC). In the revised version we will update the abstract to report these concrete figures (e.g., maximum contouring error of X µm under Y % stiffness reduction, with only Z additional tuning parameters compared with nominal operation) while preserving the existing experimental validation. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on standard MPC and invariant-set construction from given models

full rationale

The paper constructs high-fidelity augmented models and switched LTI approximations, then defines robust control invariant sets from those models to enforce contouring-error bounds via MPC. No step reduces a claimed result to a fitted parameter defined by the result itself, nor does any load-bearing premise collapse to a self-citation chain or ansatz smuggled from prior work by the same authors. The claim that the sets contain trajectories for arbitrary contours is presented as following from the model definitions and standard invariance theory rather than being tautological with the inputs. This is the normal non-circular case for a control-design paper that applies established tools to a new plant description.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the method rests on standard control-theory assumptions for modeling and MPC; specific free parameters such as cost weights and horizons are expected but not detailed.

free parameters (1)

MPC cost weights and prediction horizon
Standard tuning parameters in model predictive control that affect performance and constraint satisfaction.

axioms (1)

domain assumption The biaxial system dynamics with position-dependent flexibility and input delay can be represented by a high-fidelity model and switched linear time-invariant control-oriented models with augmented states.
Invoked to enable definition of robust control invariant sets and the MPC design.

pith-pipeline@v0.9.0 · 5473 in / 1307 out tokens · 72535 ms · 2026-05-10T15:29:11.322777+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The controller design is based on the model predictive control scheme to enforce system states staying in robust control invariant sets defined by the reference model and switched linear time-invariant control-oriented models.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 1... bounding the X-axis tracking error ∥e_x∥_∞ ≤ ε_x and Y-axis tracking error ∥e_y∥_∞ ≤ ε_y with ε_x + ε_y ≤ ε_c

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Bounded error tracking control for contouring systems with end effector measurements,

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A novel contouring error estimation method for contouring control,

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Model predictive contouring control for biaxial systems,

D. Lam, C. Manzie, and M. C. Good, “Model predictive contouring control for biaxial systems,”IEEE Transactions on Control Systems Technology, vol. 21, no. 2, pp. 552–559, 2012

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Pre-compensation of servo contour errors using a model predictive control framework,

S. Yang, A. H. Ghasemi, X. Lu, and C. E. Okwudire, “Pre-compensation of servo contour errors using a model predictive control framework,” International Journal of Machine Tools and Manufacture, vol. 98, pp. 50–60, 2015

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Con- trol strategy of biaxial variable gain cross-coupled permanent magnet synchronous linear motor based on mpc-mras,

Z. Li, J. Wang, J. An, Q. Zhang, Y . Zhu, H. Liu, and H. Sun, “Con- trol strategy of biaxial variable gain cross-coupled permanent magnet synchronous linear motor based on mpc-mras,”IEEE Transactions on Industry Applications, vol. 58, no. 4, pp. 4733–4743, 2022

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