The reliability of precision motion systems, such as semiconductor wafer scanners, is often influenced by nonlinear dynamics originating from components such as cable slabs. This paper introduces a data-driven framework for early fault diagnosis in these systems. Koopman operator theory is employed to derive a linear state-space model from experimental data, capturing the complex, hysteretic behavior of the cable slab. This model serves as a digital twin, and by comparing its predictions with real-time sensor measurements, operational anomalies can be detected. A systematic process for selecting observable functions yields a high-fidelity model with a tracking error of approximately ±1% across the operational range. When the proposed approach is tested against a state-of-the-art neural network model, it demonstrates a 75.4% reduction in reaction force prediction error. The framework successfully identifies an injected sensor noise fault (SNR of 20) in just 0.35 s using only force data, validating its potential to improve wafer scanner reliability.

Implementing Non-Uniform Rational B-Splines in motion controllers involves challenges like computational complexity, parameterization issues, high-fidelity interpolation needs, real-time processing requirements, hardware limitations, system integration, error management, and the demand for user-friendly design tools, all of which necessitate advanced algorithms, robust hardware, and effective interfaces for successful implementation. Expanding the standard sine profile allows for the creation of a generalized, symmetric, frequency-sensitive basis function for generating point-to-point motion trajectories, while other functions like polynomials, sigmoids, and harmonic models can also be utilized. This study employs the basis function at the jerk level to meet the motion system’s constraints, and time shifts create B-spline-like and NURBS-like profiles for CAD and motion control, eliminating the need for interpolation and curve fitting. Utilizing vector notation and algorithms reduces computational demands on both the CAD and the motion controller sides, enabling real-time implementation. Frequency-sensitive B-spline profiles enhance both repetitive and random walk motions, facilitate data collection, and help segment the workspace into regions with unique frequency characteristics. A model-free approach addresses position-dependent errors, and experimental results validate the effectiveness of these techniques. Robust statistical models based on Multivariate Analysis of Variance are developed to both characterize and assess the performance of a precision motion system under the proposed framework. For example, the mean positioning error of the motion system improved from 2.29 μm to 0.717 μm at the maximum operating frequency of 90 rad/s under the model-free approach.

A novel method for analyzing the dynamics and bend geometry of a cable slab via trained neural networks is introduced. Neural networks are trained from real-time visual feedback capture via a high-speed camera during cyclic motion to track the positions of multiple markers affixed to the cable slab through image processing techniques. Experimental parameters are systematically varied to ensure a diverse range of training patterns. Consequently, two distinct data-driven neural network models are developed: a coupled model and a decoupled model. These models accurately predict the two-dimensional positions of the markers, even during non-cyclic motion profiles. Subsequently, the marker positions are utilized as waypoints to generate a cubic spline curve with time-varying coefficients, approximating the spatiotemporal solution of the cable slab dynamics. Notably, this spline can be segmented into smaller sections tailored to specific research objectives. Experimental results validate the effectiveness of the proposed methodology.

This paper introduces a position control system for a motion stage driven by a low-energy C-core reluctance actuator. The central concept explored here is the utilization of a variable air gap to enable energy-efficient operation of the motion stage. First, we show the design and mathematical model of the reluctance-actuated motion system (RAMS). Then, by analyzing open-loop responses of the RAMS under various conditions including variable air gaps and different excitation voltages, we show that using variable air gap can reduce the required current. Finally, the paper formulates a control approach that combines a feedforward controller to linearize the RAMS’s dynamic behavior and a state feedback controller to achieve tracking performance. Experimental results demonstrate the effectiveness of this control approach in achieving tracking objectives with errors that are less than 2% for constant desired displacement and less than 10% for tracking a sinusoidal reference signal.

Stretching the definition of the standard Sine profile allows building a generalized symmetric frequency-aware basis function that can be used to generate reference motion trajectories. Other profiles such as polynomials, sigmoid, and harmonic-based models can be equally used under the proposed technique. Despite being suitable at the level of any higher-order time derivative, in this study, the generic basis function is realized at the jerk level such that the generated signals adhere to the limitations of the driven motion system. Introducing suitable time shifts, replicas of basis functions can be obtained giving rise to B-spline like frequency-aware profiles that can be used to realize the actual motion under any desired kinematical constraints, which are neatly written to reduces the computation burden at the motion controller side. Utilizing mainly the frequency-aware B-spline profiles, frequency-dependent random walk motion is presented and used to collect information about the driven motion system to help in characterizing any position-dependent errors through the statistical means, i.e. Analysis of Variance, and Design of Experiments. This allows dividing the working space in which motion takes place into several spatial regions with preferred frequency contents. The effectiveness of these proposed profiles is shown through hardware experiments using a precision motion system.

In semiconductor manufacturing, lithography machines are becoming more and more sophisticated system of systems. As an example, a TWINSCAN wafer scanner machine is composed of a wafer, and reticle handlers, reticle, optics, and two wafer chains or systems. In previous studies, we covered the interactions between the reticle, optics, and wafer chains during the step-and-scan cycle. In this study, we focus on the interaction between the additional wafer chain responsible for aligning the wafer substrate and taking its height map during the measurement cycle, and the other chains that are active during the step-and-scan cycle. Working in parallel to increase machine throughput, the inertial forces associated motion of the two cycles induce vibration that may propagate throughout the chains in the machine if no appropriate measures are taken. In this investigation, we look at the reference trajectories responsible for steering the chains throughout the two cycles, and propose two reference trajectory orchestrations that factor in the machine design, geometry, mass distribution, and functions. Theoretically, these orchestrations lead to suppressing the induced vibration without sacrificing the machine throughput while keeping the involved control loops intact.

In this paper, an immune average consensus behavior of distributed trajectory generators given in the form of a multi-agent system is presented. Starting with the well-known results of linear consensus protocols, we propose a decomposition of the invariant consensus value to enable a distributed cyber-attacks detection and mitigation mechanism among the connected agents over mainly undirected communication links. This decomposition suggests one preferred propagation of the invariant quantity along communication links of the multi-agent systems under study. Despite its simplicity, the effectiveness of this mechanism in detecting and mitigating various types of cyber-attacks is evident through a numerical simulation. Interestingly, the resulting defense mechanism will not be passive, rather it can initiate its counter-attack measures by pretending that the attack process was a success. Moreover, the trajectory generators can operate under stealth mode where the communication links get silenced or totally disconnected without affecting the intended behavior after having the consensus value locked.

Mainly, the integration of fine positioning piezo-actuated stages in precision motion systems is considered, which results in multi-stage configurations. Mostly, in such configurations, the fine stages are attached to the coarse positioning stages- that do not meet required precision- by mechanical means. Once the motion is synchronized, the fine stages enhance the overall precision of the multi-stage system. Undesirably, mechanical, and electromagnetic interference between the in-volved stages take place, which may limit the possible attainable precision. To control the fine stages, we propose the use of feedforward control based on the Prandtl-Ishlinskii model inverse in an attempt to accommodate related piezoceramics dynamic behavior and hysteresis. Targeting the semiconductor manufacturing, the needed multi-stage design steps according to the herein proposed approach are outlined. Also, the performance of a representative precision motion system comprising a planner coarse stage, and a uni-axial fine stage under step-and-scan trajectories is assessed. The results show that the proposed piezo-actuated fine stage improves the scanning accuracy of the overall motion system.

The operation time of an ideal reliable wafer scanner model is defined at the die level where the actual exposure process takes place as the time unit per die, or at the wafer substrate level as the time unit per wafer substrate. Therefore, the machine throughput is given as the reciprocal of the operation time. The involved motion profiles of a machine, namely the step-and-scan trajectories, function as the heartbeats that drive its multidisciplinary elements, which suggests that a multidisciplinary design optimization should be involved when such profiles are selected or designed. This is also true when considering the traverse motion profiles among rows and columns within the wafer substrate. The step-and-scan trajectories affect the machine throughput, performance, and die yield. The effects of tracking such profiles appear as structural vibration, tracking errors, and thermal loading at various machine elements such as the actuators, the reticle, the wafer, and the projection elements specifically when the exposure high-energy duration and frequency are not taken into consideration while designing the reference motion. From the dynamics perspective, having a reference motion with nonzero and bounded higher-order derivatives is recommended since it enhances the tracking performance of the machine, however, its ability to increase the operation time is usually overlooked. In an attempt to understand such effects, we present a case study that outlines the aforementioned aspects using three step-and-scan profiles of mainly$3^{rd}$-order. Taking the dynamics of the driven stage into consideration through input shaping, both the step-and-scan and traverse motion profiles are analyzed. We provide analytical expressions that can be used to generate both types of motion profiles on the fly without additional optimization. A simulation example of a simplified wafer scanner machine shows the usefulness of the proposed framework. Note to Practitioners—Choosing the most suitable operating conditions of a lithography machine is challenging. These conditions affect machine productivity, profit margin, and maintenance. In this paper, we reveal the relation between the selection of operating conditions based on several decision variables- and the kinodynamic step-and-scan trajectory generation based on specific machine parameters and clients’ requirements. Being chart-based, the selection process of an operating point can be less practical at some points. However, using appropriate curve fitting tools, the information provided in the optimal operating charts can be put into suboptimal closed-form expressions that facilitate the selection process. Therefore, the designed trajectories parameters can be easily saved in lookup tables for ease of evaluation and future use. This helps in accommodating changes in the operation plans and flexible manufacturing systems. Also, starting with a given set of machine parameters, it is possible to calculate the optimal machine operating point when the input shaping technique is used, as illustrated in this paper.

Adopting the ideal reliable machine model, the throughput of a lithography machine can be given as the reciprocal of the operation time. This time can be defined at the die level where the actual exposure process takes place as the time unit per die. A closer look at the motion profiles, namely step-and-scan trajectories, suggests that a multi-disciplinary design optimization should be involved when such profiles are selected or designed. Being the reference motion used, the step-and-scan trajectories not only affect the machine performance, but also affect its throughput and to an extent the die yield as well. Structural vibration, and thermal loading at the actuators due to friction and repetitive motion may build up because of following the reference motion. Moreover, since the exposure process and equipment are synchronized with the reference motion, deformation and thermal stress may affect the reticle, the wafer and the projection elements if the exposure high-energy duration and frequency are not taken into consideration while designing the reference motion. From dynamics point of view, reference motion with higher-order derivatives enhances the tracking performance of the machine, however, its operational cost is usually overlooked. In this paper, we present a case-study that outlines the aforementioned aspects using three step-and-scan profiles of the same order. We conclude by posing the following research question: what is the best combination of orders of the step and the scan trajectories that jointly meet the desired performance and operating conditions?