Freeform Surfaces Research Articles

Results of round robin form measurements of optical aspheres and freeform surfaces

High-quality aspherical and freeform surfaces are in high demand, and the high-accuracy form measurement of such surfaces is a challenging task. To explore the current status of form measurement systems for complex surfaces such as aspheres and freeforms, interlaboratory comparison measurements are performed. This study presents the pseudonymized results obtained using three different surfaces (metal asphere, glass asphere, toroidal surface) in a total of six different round robins. These results were taken from a total of 13 different measurement instruments based on 9 different measurement principles and operated at 12 different laboratories. They were analyzed using a sophisticated procedure that was first developed in 2018 and then refined and tested on simulated data in 2022 to address the challenges of such a comparison at this level of accuracy. In the current study, we applied these refined methods todata acquired from tactile and optical point measurements as well as from optical areal measurements. As there are no absolutely measured and very well characterized reference standard aspherical and freeform surfaces available at the accuracy level of a few tens of nanometers root-mean-square, the approximated true forms of the surfaces were derived from the measurements and indicate the manufacturing accuracy of the surface forms. Then, the measurement’s differences to the approximated true forms were analyzed, which directly indicate the systematic measurement errors of the instruments. By also comparing the approximated true forms from the two different round robins for each surface, additional insights into the reliability and stability of these so-called virtual reference topographies were gained.

Concept for improving the form measurement results of aspheres and freeform surfaces in a tilted-wave interferometer

Abstract. Accurate and flexible form measurements for aspherical and freeform surfaces are in high demand, and non-null-test interferometric methods such as tilted-wave interferometry have gained attention as a promising response to this need. Interferometric methods, however, display ambiguities between the measurement of certain form errors and the misalignment of the measured specimen. Therefore, improved knowledge of the absolute measurement position of the specimen in relation to the interferometer setup may improve the form measurement result. In this work, we propose a concept that uses a white light interferometer to measure the absolute distance between a transparent specimen's surface and the interferometer's objective and present preparatory data to qualify the white light interferometer for the improvement of tilted-wave interferometer measurements.

Vortex phase-based adaptive moiré technique accommodating micro displacement measurement of freeform surfaces.

The traditional displacement measurement interferometer (DMI) provides elegant performance by straight interference fringe movement counting to convert a phase calculation into an image motion calculation. However, it cannot be applied to a curve surface displacement measurement. The counting of the movement of irregular fringes is not achievable. We provide an adaptive moiré technique with a vortex phase to realize micro displacement measurement of a freeform surface with any continuous shape. The technique produces straight moiré fringes that rotate in a circle regardless of the shape of interference fringes and tested surface shapes. The vortex phase is used to record only one interferogram before the measurement for subsequent data processing, and then it no longer participates in the displacement measurement process. Therefore, this technology can be employed to remold traditional DMIs. Simulations and experiments validating the method are presented.

Compact off-axis reflective optical system design combining freeform mirror and freeform detector

Minimizing system size and weight without sacrificing the desired imaging performance is a crucial and challenging goal in the design of off-axis optical systems. In this paper, we propose a design concept that combines freeform mirror and freeform detector to achieve more compact off-axis optical systems. Initially, by applying nodal aberration theory, the optimal focal surface shape for off-axis systems is demonstrated to be an off-axis paraboloid or even a freeform surface, rather than the commonly assumed spherical shape. Subsequently, we show the ability of the design concept to reduce off-axis system volume through a design example of a fast and wide-field off-axis three-mirror system. Results indicate that, under the same design parameters and imaging quality, the design with a freeform detector is 80% smaller than that with a flat detector in terms of volume. Finally, the bending capability of the freeform detector presented in this paper is ensured through finite element analysis, and its surface accuracy is achievable based on the current precision level of curved detector fabrication. The proposed design concept holds promise for application in planetary science, where the imaging performance of the instrument is highly required and the size and weight are strictly limited.

Bayesian uncertainty evaluation applied to the tilted-wave interferometer

The tilted-wave interferometer is a promising technique for the development of a reference measurement system for the highly accurate form measurement of aspheres and freeform surfaces. The technique combines interferometric measurements, acquired with a special setup, and sophisticated mathematical evaluation procedures. To determine the form of the surface under test, a computational model is required that closely mimics the measurement process of the physical measurement instruments. The parameters of the computational model, comprising the surface under test sought, are then tuned by solving an inverse problem. Due to this embedded structure of the real experiment and computational model and the overall complexity, a thorough uncertainty evaluation is challenging. In this work, a Bayesian approach is proposed to tackle the inverse problem, based on a statistical model derived from the computational model of the tilted-wave interferometer. Such a procedure naturally allows for uncertainty quantification to be made. We present an approximate inference scheme to efficiently sample quantities of the posterior using Monte Carlo sampling involving the statistical model. In particular, the methodology derived is applied to the tilted-wave interferometer to obtain an estimate and corresponding uncertainty of the pixel-by-pixel form of the surface under test for two typical surfaces taking into account a number of key influencing factors. A statistical analysis using experimental design is employed to identify main influencing factors and a subsequent analysis confirms the efficacy of the method derived.

Freeform surface profiling by iterative learning-extremum seeking control

Freeform surface profiling by iterative learning-extremum seeking control

Influence of Tool Inclination and Effective Cutting Speed on Roughness Parameters of Machined Shaped Surfaces

Free-form surfaces in the automotive or aviation industry where the future shape of the product will contain complex surfaces raises the question of how to achieve the necessary shape of the required quality in the milling process. One of the methods of their production is the use of 5-axis milling, in which it is necessary to consider not only the input data of the process itself, but also the methodology for evaluating the desired results. Correctly answered questions can thus facilitate the choice of the inclination of the tool when machining parts of the surfaces defined in the experiment. The primary goal of the paper was to monitor the influence of tool inclination on the quality of the machined surface and effective cutting speed by evaluating surface roughness and surface topography. The experiment was designed to show the effect of different tool positions while the feed per tooth fz for the finishing operation remained constant. The best result in terms of surface quality was achieved with a tool inclination of 15° in the cutting process. The most unfavorable result was obtained with a tool axis inclination of zero degrees due to unfavorable cutting conditions.

Improving Machined Accuracy Under a Constant Feed Speed Vector at the End-Milling Point by Estimating Machining Force in Tool Approach

A five-axis machining center (5MC) is capable of synchronous control, which makes it a feasible tool for quickly and accurately machining complicated three-dimensional surfaces, such as propellers and hypoid gears. Recently, the necessity of improving both the machined shape accuracy and the machined surface roughness of free-form surfaces is growing. Therefore, in our previous study, we aimed to maintain the feed speed vector at the end-milling point by controlling two linear axes and a rotary axis of the 5MC to improve the quality of the machined surface. Additionally, we developed a method for maintaining the feed speed vector at the end-milling point by controlling the three axes of the 5MC to reduce the shape error of the machined workpieces (referred to as the shape error herein), considering the approach path of the tool determined via calculation. However, a high machining force at the start of the workpiece cutting was observed and the factor contributing to this phenomenon was not determined, although this phenomenon leads to a shape error to a certain degree according to the machining condition. In this study, the main objective is to suggest a method to reduce the machining force at the start of the workpiece cutting and shape error. Hence, we develop a theoretical method to estimate the machining force by using an instantaneous cutting force model, which considers the synchronized motion of two linear axes and a rotary axis of the 5MC. Subsequently, we determine the most suitable approach path of the tool considering the prediction of the machining force. The results of this study indicate that the machining force can be estimated by applying an instantaneous cutting force using the feed per tooth and machining angle, and that both a high machining force at the start of the workpiece cutting and shape error reduction can be realized by using the proposed approach path of the tool.

A shape-performance synergistic strategy for design and additive manufacturing of continuous fiber reinforced transfemoral prosthetic socket

A shape-performance synergistic design and additive manufacturing strategy was developed for the transfemoral prosthetic sockets (TPS) to fulfill the demands of customized geometry and long-term load bearing. The 3D printing path of the continuous fiber reinforced polymer composites (CFRPC) was generated by reversing the milling path of subtractive manufacturing. The mechanical properties of the 3D printed CFRPC samples were investigated as a foundation, and the customized fiber trajectories were designed based on the analyzed stress pattern of the TPS in daily gait. The CFRPC TPS with weight of 28 % less than those made by pure resin had a safety factor larger than 3 and passed the fatigue testing of 3 million cycles. The design and additive manufacturing strategy developed in this study was also applicable for other parts with freeform surfaces.

A novel pose calibration method for laser displacement sensor in the five-axis on-machine measurement system

The five-axis on-machine measurement system, equipped with a laser displacement sensor, offers an effective approach for conducting on-site inspections of freeform surface parts. It is crucial to address the calibration error pertaining to the sensor’s pose, encompassing both the beam direction and the zero position, as it directly impacts the measurement accuracy. This paper presents a new approach for the calibration of a laser displacement sensor’s pose, which is mounted on the spindle of a five-axis gantry machine tool. The proposed method utilizes measurements taken from multiple angles of a standard ball. Through the utilization of spherical constraints and coordinate translation transformation, the Levenberg-Marquardt iteration method is employed to derive the sensor’s beam direction, while the least square method is used to determine the sensor’s zero position. Applying this proposed method, the maximum error in the distance between the centers of the standard double ball is below 0.008 mm, which can meet the precision requirements of on-site inspection of freeform surface parts.

Optical design of low-location illumination based on asymmetric double freeform surfaces.

The development of optical optics for low-location road lighting is a challenging problem in providing high luminance and uniformity of illumination and meeting many other specific requirements. This study proposes an optical design method of low-location illumination based on an asymmetric double freeform surface lens. The ray emitted from the light source is refracted and reflected through the different surface types to the corresponding area of the receiving surface. In the design example, the road has dual-side mounted luminaires and a width of 6m, and a height of 0.8m. Simulation results indicate that, compared with conventional high-pole streetlights, the luminance uniformity had increased from 0.60 to 0.66, the illuminance uniformity had improved from 0.75 to 0.86, and the glare had been reduced.

Exploiting image quality measure for automatic trajectory generation in robot-aided visual quality inspection

AbstractCurrently, the standard method of programming industrial robots is to perform it manually, which is cumbersome and time-consuming. Thus, it can be a burden for the flexibility of inspection systems when a new component with a different design needs to be inspected. Therefore, developing a way to automate the task of generating a robotic trajectory offers a substantial improvement in the field of automated manufacturing and quality inspection. This paper proposes and evaluates a methodology for automatizing the process of scanning a 3D surface for the purpose of quality inspection using only visual feedback. The paper is divided into three sub-tasks in the same general setting: (1) autonomously finding the optimal distance of the camera on the robot’s end-effector from the surface, (2) autonomously generating a trajectory to scan an unknown surface, and (3) autonomous localization and scan of a surface with a known shape, but with an unknown position. The novelty of this work lies in the application that only uses visual feedback, through the image focus measure, for determination and optimization of the motion. This reduces the complexity and the cost of such a setup. The methods developed have been tested in simulation and in real-world experiments and it was possible to obtain a precision in the optimal pose of the robot under 1 mm in translational, and 0.1$$^\circ $$ ∘ in angular directions. It took less than 50 iterations to generate a trajectory for scanning an unknown free-form surface. Finally, with less than 30 iterations during the experiments it was possible to localize the position of the surface. Overall, the results of the proposed methodologies show that they can bring substantial improvement to the task of automatic motion generation for visual quality inspection.

Initial design with confocal conic mirrors for freeform optical imaging systems

Freeform surfaces that can offer high degrees of freedom for aberration correction have been widely used in various imaging applications. Initial design is particularly critical in the optical design of freeform imaging systems due to the significantly expanded solution space. Here, we present a method to find an initial point for the optical design of four-mirror freeform imaging systems. The method is computationally simple, easily accessible, and theoretically supported. The effectiveness of the method is demonstrated by designing a four-mirror confocal system without field-constant aberration and linear astigmatism. We also generalize the proposed method to four-mirror confocal systems with “double-pass surface.” The proposed method can yield a promising starting point for the design of freeform off-axis imaging systems.

An integrated climate-based daylight performance evaluation framework for indoor arenas' roof system

Daylight environment plays a crucial role in indoor arenas' comfort and energy performance, characterized by their large spans and large spaces. Design parameters of roof systems in these arenas are essential for enhancing the daylight within competition halls. However, due to the complexity of spatial archetypes in these halls and the unique characteristics of freeform surfaces, more research is needed on the interplay between roof system design parameters and daylight performance. Rapid evaluation of daylight performance in competition halls during the design phase remains a challenge. This study proposes a climate-based evaluation framework for assessing daylight performance in indoor arenas and integrates an efficient practical and research workflow using Rhino + Grasshopper in conjunction with Python.Further, based on this framework and workflow, controlled variable simulation experiments were conducted to explore the relationships between external design conditions, such as climate (represented by three typical climatic zones) and seating capacity, and roof system design parameters, including roof form, redundant height, orientation, skylight form, distribution, and skylight-to-floor ratio (SFR), with time-series, climate-based daylight performance indicators. To integrate the critical factors affecting indoor lighting environments in arenas - illuminance potential, glare evaluation, and uniformity assessment - this study selected Continuous Daylight Autonomy (cDA), Useful Daylight Illuminance (UDI), and Glare Autonomy (GA) as performance indicators, supplemented by two time-series climate indicators specifically for evaluating illuminance uniformity in indoor arenas: “Uniformity Autonomy (UA) “and “Uniformity Variance Autonomy (UVA) ". Statistical analysis of five performance indicators across 860 experimental models in seven groups shows that different design conditions and parameters affect the optimal range of SFR. The significance of the impact of various design parameters varies across different performance indicators. There is a strong correlation between cDA, UDI, and GA, while UA and UVA effectively supplement the evaluation of the first three indicators.

Interfacial Self-Assembly of Oriented Semiconductor Monolayer for Chemiresistive Sensing.

Semiconductor nanofilm fabrication with advanced technology is of great importance for next-generation electronics/optoelectronics. Fabrication of high-quality and perfectly oriented semiconductor thin films and integration into high-performance electronic devices with low cost and high efficiency are huge challenges. Here we exquisitely utilized the Marangoni effect to perfectly guide tin disulfide (SnS2) nanocoins into an ordered assembly in milliseconds, resulting in an uniaxial-oriented monolayer semiconductor film. Further exploration revealed that the formed "crumple zone" at the interface caused by the Marangoni force endows the nanofilm with a rapid healable capability, which can be easily transferred to arbitrary substrates. As a proof of concept, the nanocoin-monolayer was transferred onto a micro-interdigitated electrode substrate to form a high-performance chemiresistive sensor that can effectively monitor the trace amounts of toxic gases. In addition, the assembled monolayer nanofilms can be conformally printed on freeform surfaces: both flat and nonflat substrates. This efficient and low-cost Marangoni force-assisted surface self-assembly (MFA-SSA) strategy is promising for advanced microelectronics and real industrial applications.

Surface reconstruction and thickness error calculation of optical components with a complex curved surface

The increasing demand for free-form irregular optical components in both military and civilian sectors has made the inspection of such unique shapes a central challenge that hinders their production and use. In particular, the shape and thickness errors of low- and medium-precision components thermally pressed from flat optical materials are greater than those of hard brittle optical components fabricated by subtractive manufacturing, and the resulting impact on human vision is more severe. Reasonable, convenient, efficient, and accurate 3D scanning and data processing for surface reconstruction that combines application scenarios and batch manufacturing needs are urgently needed. Based on the principles of optical ray tracing and triangulation processing, the sampling and calculation of optical path thickness proposed in this paper effectively establish a theoretical model for macroscopic distortion, providing a reasonable solution for distortion correction, batch manufacturing of free-form surface pressing formed components, and defect repair.

Fabrication of the freeform Fresnel lens by swinging-rotating diamond ruling

Freeform optics are widely applied across various industries, encompassing both imaging and non-imagining optical systems. To compact the freeform surface, the freeform Fresnel lens has been devised by optical designers. However, conventional slow slide servo techniques cannot effectively fabricate the intricate lens, due to its asymmetry and discontinuity. In this paper, a novel swinging-rotating diamond ruling method is proposed to fabricate a biconical freeform Fresnel lens, with the synchronized motions of X-, Z-, C- and A-axes. The morphed Archimedean driving spiral based on contour lines is constructed to avoid the tool-microstructure interference. The tool path generation algorithm is proposed to generate smooth and accurate tool path, encompassing an adaptive projection principle, a transition tool path interpolation method and a postprocessing method. Finally, the feasibility and superiority of the proposed method is validated through two case studies. It is demonstrated that the precise and efficient fabrication of the freeform Fresnel lens can be realized by swinging-rotating diamond ruling method.

Clustering and optimization of nodes, beams and panels for cost-effective fabrication of free-form surfaces

Free-form surfaces are increasingly used in contemporary architectural designs for their unique and elegant shapes. However, fabricating these doubly curved surfaces using panel and frame systems presents challenges due to the shape variability of nodes, beams and panels. In this study, we propose a mesh-based computational design framework that clusters and optimizes these components together, reducing the shape variety of elements for free-form surfaces. Our method employs a vertex-based similarity metric to partition panels into user-defined groups and clusters beams based on edge lengths. A box-constrained optimization is introduced to achieve congruent faces and matching beam lengths while considering various functional constraints. Additionally, connection holes on node surfaces are clustered and optimized to allow their use at multiple locations. The practicality of our approach is demonstrated through the design and construction of a full-scale pavilion, resulting in a significant reduction in the shape variety of building elements.

Free-form lens design of LED extended light source based on improved weighted superposition optimization

Free-form surface has a wide range of applications in imaging and non-imaging optics, which can simplify the optical components, support structures required by the system and make the overall optical space more compact, but because of the large freedom of design, the design difficulty is greatly increased, how to meet the design requirements while improving efficiency, is the focus and difficulty of current research. Therefore, this paper proposed an improved optimization method for designing of free-form lens for light-emitting diode (LED) light distribution. Based on the lens matching curve obtained by weighted superposition, feedback optimization method was used to further improve the optimization possibility of lens. The weight factors of each surface are optimized by particle swarm optimization algorithm to minimize the evaluation function based on illuminance uniformity of the target surface. The illuminance uniformity on the target plane is optimized from 72.8% to 93.2% start form initial structure. The results show that the proposed algorithm has the advantages of high optimization efficiency and good optimization effect.

Trajectory optimization for automated tape placement on triangular mesh surfaces considering gap requirements

Abstract Automated tape placement (ATP) is an important automated process adopted for the fabrication of large composite components. Trajectory planning is the key link of ATP, which directly affects the precision and efficiency of the layup process, and the quality of final products. Presently, most existing trajectory optimization methods for ATP focus on smooth surfaces. Nevertheless, as commercial CAD/CAM software generally uses NURBS (Non-uniform Rational B-Splines) for modelling, the difficulty of finding the solution and the low efficiency associated with the calculation process are inevitable. The discrete methods provide alternatives for designing layup trajectories, whereas their accuracy is seldom analyzed. Furthermore, a path optimization algorithm for eliminating gap problems while preventing wrinkles on discrete models is rarely reported. In this paper, the adjustment of layup trajectories for ATP is considered on triangular meshes. Firstly, the triangular mesh is reconstructed as a Nagata patch to recover the original geometry with good accuracy. Then, a numerical method for tracing desired paths on the Nagata patch set is provided, and the computation efficiency is validated. Next, two optimization methodologies are proposed to improve the layup of composite tapes while avoiding wrinkles. Finally, the presented two strategies are examined on a discrete hyperbolic surface and a discrete freeform surface, and some of the results are delivered.