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

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.

• The paper introduces an RCI-based MPC framework that guarantees contouring error bounds in dual-drive systems with input delay and structural flexibility.
• It employs switched LTI model reductions and offline robust invariant sets to ensure real-time performance and rigorous constraint satisfaction.
• Experimental validation on a HIL platform demonstrates contour errors well below prescribed limits, independent of MPC tuning parameters.

Contouring Error Bounded MPC for Biaxial Systems with Flexibility and Input Delay

Introduction

The paper "Contouring Error Bounded Control for Biaxial Systems with Structural Flexibility and Input Delay" (2604.11018) advances the state-of-the-art in precision contouring for industrial motion systems. The primary focus is on systems characterized by position-dependent flexibility and input delay, circumstances typical in modern dual-drive gantry machines. Such industrial contexts, including high-precision laser and water jet cutting, demand stringent guarantees on contouring tolerance, where contouring error—the minimal Euclidean distance from the end-effector to the prescribed path—directly impacts manufacturing quality.

Historically, contour error minimization strategies, including cross-coupled control (CCC) and coordinate frame transformations, have provided improved tracking but do not guarantee strict error bounds, particularly under model mismatch, nonlinear flexibility, or input delays. Existing model predictive control (MPC) techniques partially address operational constraints but lack formal guarantees on ultimate contouring error, especially in the presence of unmodeled flexibility and delays. The absence of real-time end-effector sensing in many industrial systems further limits performance validation and controller syntheses.

Modeling and Problem Formulation

High-Fidelity Plant Model

The authors employ a dual-drive gantry architecture featuring both lateral and rotational degrees of freedom, modeled via a Lagrangian framework. The structure includes key parameters: rigid bodies, torsional/linear flexibility at joints (spring-damper), and couplings between actuators and the beam’s center-of-mass and rotation. Full nonlinear dynamic equations are derived for three generalized coordinates: X’ translation $x_h$, Y-axis centroid $y_n$, and rotation $\theta$. Input actuation and structural disturbance terms are explicated, yielding explicit dependence on position, velocity, and higher-order coupling terms.

To enable computationally efficient control synthesis, the model is reduced axis-wise to switched linear time-invariant (LTI) approximations at several operating points—capturing local variations due to position-dependent flexibility. Notably, the unavoidable input delay present in realistic industrial networks is integrated by state augmentation, yielding fully discrete-time, delay-augmented, switched LTI models for both axes.

Control Objective

Given dynamic models with state and actuator constraints, the main control objective is as follows: for a prespecified maximal contouring error bound $\epsilon_c$ and all feasible trajectories within operational limits, synthesize control laws for each axis such that the contour error $\epsilon(k) \leq \epsilon_c$ holds for all $k$.

An important theoretical result relates contour error to axis-wise tracking errors: by guaranteeing $\|e_x\|_\infty \leq \epsilon_x$ and $\|e_y\|_\infty \leq \epsilon_y$ with $\epsilon_x + \epsilon_y \leq \epsilon_c$, path-agnostic contouring guarantee is achieved. This insight exploits conservative norm inequalities and liberates the controller from restrictions on curve geometry.

Control Synthesis

Robust Invariant Set-Based MPC

The controller structure comprises two principal innovations:

1. Path-Agnostic Robust Constraint Formulation: For each axis, an error-bounded robust model predictive control framework is employed. At the core is the offline computation of robust control invariant (RCI) sets for each delay-augmented, switched LTI subsystem, with explicit handling of worst-case bounded disturbances and rotation angle constraints. Computation is performed using iterative set-propagation algorithms with polytopic approximations to guarantee constraint satisfaction in the presence of input and modeling uncertainty.
2. Online Real-Time Fast MPC Solution: In online operation, the controller selects, based on current state feedback, the appropriate local model and associated RCI set. A constrained finite-horizon MPC problem is solved, leveraging the pre-computed robust constraint sets to enforce input, state, and tracking error bounds. The path-agnostic RCI sets ensure that tuning of cost function weights (i.e., transient/steady-state trade-offs) does not affect satisfaction of the user-prescribed ultimate contour bound.

The X-axis controller enforces stricter bounds if necessary by accommodating the influence of rotation on translational error; the Y-axis controller is synthesized at multiple linearization points using a switched controller design.

Theoretical Properties

A formal recursive feasibility result is established: if the control problem is initially feasible (i.e., the system state is within the corresponding RCI set), feasibility is preserved at all future times, implying robust constraint satisfaction for the entire operation. An explicit handling of input delay in the augmented model maintains the theoretical guarantees with communication/actuator latencies.

Experimental Validation

Hardware-in-the-Loop (HIL) Setup

Due to limited availability of direct end-effector feedback in real machines, the authors construct an HIL platform featuring back-to-back rotary motors emulating the coupled biaxial system under digital control. Realistic actuator and communication delays, as well as nonlinear disturbance terms, are injected to match practical conditions.

The model parameters are identified on a commercial laser machine; operating regions and disturbance sets are determined empirically from frequency response data.

Performance Results

The controller tracks complex reference contours, including nontrivial curves and straight lines, at a maximum speed of 0.1 m/s and acceleration 1 m/s$^2$. Experimental design parameters include:

• Prescribed contour error bound: $y_n$0 mm (5% of target radius)
• X/Y tracking bounds: $y_n$1 mm, $y_n$2 mm
• Rotation bound: $y_n$3 rad, ensuring validity of linearization assumptions
• Sampling period: $y_n$4 ms, with 1-step input delay

Across multiple controller tuning regimes (cost function weights), the actual maximum contour error measured remains under 1.7 mm for both aggressive and conservative settings, consistently below the specified 4 mm bound. The result is independent of MPC tuning parameters, reaffirming the core theoretical property of the RCI-based approach. Axis-wise maximum errors are also tightly controlled. The experimental evidence underscores that even with input delay, switched models, and practical nonlinearities, robust constraint satisfaction is achievable.

Implications and Future Directions

This work provides the first explicit, path-independent guarantee of contouring error for flexible, input-delayed biaxial machines, with experimental validation in realistic HIL environments. The introduction of RCI set-based MPC to this domain demonstrates that rigorous robustness is computationally feasible for moderate system orders even with complex delay/uncertainty structures.

From an industrial perspective, the methodology is practical: RCI set computation is an offline procedure; only standard convex QP solvers are needed at runtime. The controller generalizes to arbitrary contour geometries, multi-axis extensions, and other machine configurations where flexibility, communication delays, and uncertainty dominate.

Theoretically, the framework suggests clear research directions:

• LPV Generalization: Continuous parameter-varying dynamics (as opposed to switched LTI) present algorithmic and computational challenges in robust MPC set computation but would provide even greater model fidelity.
• Higher-Dimensional Systems: Extension to more than two axes requires scalable representations for high-dimensional invariant sets; decomposition or set-separation techniques may be employed.
• Incremental Adaptation/Identification: Online adaptation in the presence of time-varying flexibility or disturbance statistics to tighten the guaranteed error bounds with minimal conservatism.

Conclusion

RCI-based MPC for contouring underpins a robust, path-agnostic, and delay-aware approach to precision motion control in flexible industrial machines. The presented technique rigorously ensures contouring error bounds for all time, under realistic operation, independently of controller tuning, and is validated in HIL experiments. It offers significant progress towards formal, certifiable performance in advanced machining and automation environments.