Self-Calibration of Two-Dimensional Precision Metrology Systems

In modern industry, there usually exist high-precision stages, such as reticle stages and wafer stages in VLSI lithography tools, metrology stages in coordinate measurement machines, motion stages in CNC machine tools, etc. These stages need high-precision measurement/metrology systems for monitoring its XY movement. As the metrology systems are quite accurate, we often cannot find a standard tool with better accuracy to implement a traditional calibration process for systematic measurement error (i.e., stage error) determination and measurement accuracy compensation. Subsequently, self-calibration technology is developed to meet this challenge and to solve the calibration problem. In this chapter, we study the self-calibration of two-dimensional precision metrology systems and present a holistic self-calibration strategy. This strategy utilizes three measurement views of an artifact plate with mark positions not precisely known on the un-calibrated two-dimensional metrology stage and constructs relevant symmetry, transitivity, and redundancy of the stage error of the metrology stage. The misalignment errors of all measurement views, especially including those of the translation view, are totally determined by detailed mathematical manipulations. Then, a redundant equation group is synthesized, and a least-square–based robust estimation law is employed to calculate out the stage error even under the existence of random measurement noise. Furthermore, as the determination of the misalignment error components of the measurement views is rather complicated but important in previous and the proposed methods, this chapter also significantly analyzes the necessity of this costly computation. The proposed approach is investigated by simulation computation, and the simulation results prove that the proposed determination scheme can calculate out the stage error rather exactly without random measurement noise. Furthermore, when there exist various random measurement noises, the calibration accuracy of the proposed strategy is also investigated, and the results illustrate that the standard deviations of the calibration error are consistently with the same level of those of the random measurement noises. All these results verify that the proposed scheme can realize the stage error rather accurately even under the existence of random measurement noise. The practical procedure for performing a standard self-calibration is also introduced for engineers to facilitate actual implementation

Learning Predictive Safety Filter via Decomposition of Robust Invariant Set

Ensuring safety of nonlinear systems under model uncertainty and external disturbances is crucial, especially for real-world control tasks. Predictive methods such as robust model predictive control (RMPC) require solving nonconvex optimization problems online, which leads to high computational burden and poor scalability. Reinforcement learning (RL) works well with complex systems, but pays the price of losing rigorous safety guarantee. This paper presents a theoretical framework that bridges the advantages of both RMPC and RL to synthesize safety filters for nonlinear systems with state- and action-dependent uncertainty. We decompose the robust invariant set (RIS) into two parts: a target set that aligns with terminal region design of RMPC, and a reach-avoid set that accounts for the rest of RIS. We propose a policy iteration approach for robust reach-avoid problems and establish its monotone convergence. This method sets the stage for an adversarial actor-critic deep RL algorithm, which simultaneously synthesizes a reach-avoid policy network, a disturbance policy network, and a reach-avoid value network. The learned reach-avoid policy network is utilized to generate nominal trajectories for online verification, which filters potentially unsafe actions that may drive the system into unsafe regions when worst-case disturbances are applied. We formulate a second-order cone programming (SOCP) approach for online verification using system level synthesis, which optimizes for the worst-case reach-avoid value of any possible trajectories. The proposed safety filter requires much lower computational complexity than RMPC and still enjoys persistent robust safety guarantee. The effectiveness of our method is illustrated through a numerical example

Self-Calibration of Precision XY<em>θ_z</em> Metrology Stages

This chapter studies the on-axis calibration for precision XYθz metrology stages and presents a holistic XYθz self-calibration approach. The proposed approach uses an artifact plate, specially designed with XY grid mark lines and angular mark lines, as a tool to be measured by the XYθz metrology stages. In detail, the artifact plate is placed on the uncalibrated XYθz metrology stages in four measurement postures or views. Then, the measurement error can be modeled as the construction of XYθz systematic measurement error (i.e. stage error), artifact error, misalignment error, and random measurement noise. With a new property proposed, redundance of the XYθz stage error is obtained, while the misalignment errors of all measurement views are determined by rigid mathematical processing. Resultantly, a least square-based XYθz self-calibration law is synthesized for final determination of the stage error. Computer simulation is conducted, and the calculation results validate that the proposed scheme can accurately realize the stage error even under the existence of various random measurement noise. Finally, the designed artifact plate is developed and illustrated for explanation of a standard XYθz self-calibration procedure to meet practical industrial requirements

Safe Reinforcement Learning with Dual Robustness

Reinforcement learning (RL) agents are vulnerable to adversarial disturbances, which can deteriorate task performance or compromise safety specifications. Existing methods either address safety requirements under the assumption of no adversary (e.g., safe RL) or only focus on robustness against performance adversaries (e.g., robust RL). Learning one policy that is both safe and robust remains a challenging open problem. The difficulty is how to tackle two intertwined aspects in the worst cases: feasibility and optimality. Optimality is only valid inside a feasible region, while identification of maximal feasible region must rely on learning the optimal policy. To address this issue, we propose a systematic framework to unify safe RL and robust RL, including problem formulation, iteration scheme, convergence analysis and practical algorithm design. This unification is built upon constrained two-player zero-sum Markov games. A dual policy iteration scheme is proposed, which simultaneously optimizes a task policy and a safety policy. The convergence of this iteration scheme is proved. Furthermore, we design a deep RL algorithm for practical implementation, called dually robust actor-critic (DRAC). The evaluations with safety-critical benchmarks demonstrate that DRAC achieves high performance and persistent safety under all scenarios (no adversary, safety adversary, performance adversary), outperforming all baselines significantly

Time-Optimal Control for High-Order Chain-of-Integrators Systems with Full State Constraints and Arbitrary Terminal States

Time-optimal control for high-order chain-of-integrators systems with full state constraints and arbitrary given terminal states remains a challenging problem in the optimal control theory domain, yet to be resolved. To enhance further comprehension of the problem, this paper establishes a novel notation system and theoretical framework, successfully providing the switching manifold for high-order problems in the form of switching law. Through deriving properties of switching laws on signs and dimension, this paper proposes a definite condition for time-optimal control. Guided by the developed theory, a trajectory planning method named the manifold-intercept method (MIM) is developed. The proposed MIM can plan time-optimal jerk-limited trajectories with full state constraints, and can also plan near-optimal higher-order trajectories with negligible extra motion time. Numerical results indicate that the proposed MIM outperforms all baselines in computational time, computational accuracy, and trajectory quality by a large gap

Coordinated Adaptive Robust Contouring Control of an Industrial Biaxial Precision Gantry With Cogging Force Compensations

Cogging force is an important source of disturbances for linear-motor-driven systems. To obtain a higher level of contouring motion control performance for multiaxis mechanical systems subject to significant nonlinear cogging forces, both the coordinated control of multiaxis motions and the effective compensation of cogging forces are necessary. In addition, the effect of unavoidable velocity measurement noises needs to be sufficiently attenuated. This paper presents a discontinuous-projection-based desired compensation adaptive robust contouring controller to address these control issues all at once. Specifically, the presented approach explicitly takes into account the specific characteristics of cogging forces in the controller designs and employs the task coordinate formulation for coordinated motion controls. Theoretically, the resulting controller achieves a guaranteed transient performance and a steady-state contouring accuracy even in the presence of both parametric uncertainties and uncertain nonlinearities. In addition, the controller also achieves asymptotic output tracking when there are parametric uncertainties only. Comparative experimental results obtained on a high-speed Anorad industrial biaxial precision gantry are presented to verify the excellent contouring performance of the proposed control scheme and the effectiveness of the cogging force compensations