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.

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.

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?