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Abstract This article introduces a novel methodology based on conditional β-variational autoencoder (cβ-VAE) architecture to generate diverse types of planar four-bar mechanisms for a given coupler curve. Central to our contribution is the novel integration of cross- and self-attention layers within the VAE framework, facilitating an encoding and decoding process that captures the complex interdependencies of mechanism parameters and associated coupler curves. We propose a unified representation scheme for four-bar mechanisms with both revolute and prismatic joints, utilizing a consistent set of joints to describe each mechanism type. To support and validate our methodology, we have compiled an extensive dataset featuring both open and closed coupler curves of the aforementioned mechanism types. Furthermore, the article introduces three metrics aimed at quantifying the efficacy of our model, alongside an innovative algorithm designed to enhance the predictive outcomes by identifying and computing cognate mechanisms. 

Abstract Kinematic simulation of planar n-bar mechanisms has been an intense topic of study for several decades now. However, a large majority of efforts have focused on position analysis of such mechanisms with limited links and joint types. This article presents a novel, unified approach to the analysis of geometric constraints of planar n-bar mechanisms with revolute joint (R-joint), prismatic joint (P-joint), and rolling joint. This work is motivated by a need to create and program a system of constraint equations that deal with different types of joints in a unified way. A key feature of this work is that the rolling joint constraints are represented by four-point models, which enables us to use the well-established undirected graph rigidity analysis algorithms. As a result, mechanisms with an arbitrary combination of revolute-, prismatic joints, and wheel/gear/wheel-belt chains without any limitations on their actuation scheme can be analyzed and simulated efficiently for potential implementation in interactive computer software and large-scale data generation. 

Abstract The design of single-degree-of-freedom spatial mechanisms tracing a given path is challenging due to the highly non-linear relationships between coupler curves and mechanism parameters. This work introduces an innovative application of deep learning to the spatial path synthesis of one-degree-of-freedom spatial revolute-spherical-cylindrical-revolute (RSCR) mechanisms, aiming to find the non-linear mapping between coupler curve and mechanism parameters and generate diverse solutions to the path synthesis problem. Several deep learning models are explored, including multi-layer perceptron (MLP), variational autoencoder (VAE) plus MLP, and a novel model using conditional -β− VAE (c −β− VAE). We found that the c -β– VAE model withβ= 10 achieves superior performance by predicting multiple mechanisms capable of generating paths that closely approximate the desired input path. This study also builds a publicly available database of over 5 million paths and their corresponding RSCR mechanisms. The database provides a solid foundation for training deep learning models. An application in the design of human upper-limb rehabilitation mechanism is presented. Several RSCR mechanisms closely matching the wrist and elbow path collected from human movements are found using our deep learning models. This application underscores the potential of RSCR mechanisms and the effectiveness of our model in addressing complex, real-world spatial mechanism design problems. 

Abstract In recent years, there has been a strong interest in applying machine learning techniques to path synthesis of linkage mechanisms. However, progress has been stymied due to a scarcity of high-quality datasets. In this article, we present a comprehensive dataset comprising nearly three million samples of 4-, 6-, and 8-bar linkage mechanisms with open and closed coupler curves. Current machine learning approaches to path synthesis also lack standardized metrics for evaluating outcomes. To address this gap, we propose six key metrics to quantify results, providing a foundational framework for researchers to compare new models with existing ones. We also present a variational autoencoder-based model in conjunction with a k-nearest neighbor search approach to demonstrate the utility of our dataset. In the end, we provide example mechanisms that generate various curves along with a numerical evaluation of the proposed metrics. 

Abstract This paper introduces a novel matrix-based approach for the simultaneous type and dimensional synthesis of planar four-bar linkage mechanisms, accommodating various practical constraints, including position, velocity, acceleration, and joint placements. Traditional design processes segregate type synthesis, the determination of joint and link configurations, from dimensional synthesis, which involves specifying link sizes and pivot locations. This segregation often leads to complexities in addressing the complete design challenge. The novel methodology proposed in this paper departs from the conventional sequential design approach by concurrently evaluating type and dimensional parameters using a data-driven matrix formulation. The crux of the paper’s methodology involves formulating a singular design equation through a transformation matrix, parameterized by the Cartesian parameters of the mechanism’s dyads. This formulation linearly expresses a broad range of constraints, facilitating the identification of viable solutions through singular value decomposition and null space analysis. This integrated approach not only simplifies the synthesis process but also provides direct insights into the mechanism’s parameters, encompassing both type and dimensions, thereby obviating the need for further interpretative steps common to the use of quaternions and kinematic mapping. In essence, the paper presents two main contributions: the development of a unified design equation capable of encompassing a wide array of constraints within the mechanism synthesis process, and the introduction of an algorithm that effectively identifies all potential planar four-bar linkage mechanisms by accurately satisfying up to five constraints. This approach promises to enhance the design and optimization of mechanical systems by offering a more holistic and efficient pathway to mechanism synthesis. 

Abstract This paper presents a novel real-time kinematic simulation algorithm for planar N-bar linkage mechanisms, both single- and multi-degrees-of-freedom, comprising revolute and/or prismatic joints and actuators. A key feature of this algorithm is a reinterpretation technique that transforms prismatic elements into a combination of revolute joint and links. This gives rise to a unified system of geometric constraints and a general-purpose solver which adapts to the complexity of the mechanism. The solver requires only two types of methods—fast dyadic decomposition and relatively slower optimization-based—to simulate all types of planar mechanisms. From an implementation point of view, this algorithm simplifies programming without requiring handling of different types of mechanisms. This versatile algorithm can handle serial, parallel, and hybrid planar mechanisms with varying degrees-of-freedom and joint types. Additionally, this paper presents an estimation of simulation time and structural complexity, shedding light on computational demands. Demonstrative examples showcase the practicality of this method. 

Abstract This paper focuses on the representation and synthesis of coupler curves of planar mechanisms using a deep neural network. While the path synthesis of planar mechanisms is not a new problem, the effective representation of coupler curves in the context of neural networks has not been fully explored. This study compares four commonly used features or representations of four-bar coupler curves: Fourier descriptors, wavelets, point coordinates, and images. The results demonstrate that these diverse representations can be unified using a generative AI framework called variational autoencoder (VAE). This study shows that a VAE can provide a standalone representation of a coupler curve, regardless of the input representation, and that the compact latent dimensions of the VAE can be used to describe coupler curves of four-bar linkages. Additionally, a new approach that utilizes a VAE in conjunction with a fully connected neural network to generate dimensional parameters of four-bar linkage mechanisms is proposed. This research presents a novel opportunity for the automated conceptual design of mechanisms for robots and machines. 

Abstract This paper introduces a new method using deep neural networks for the interactive digital transformation and simulation of n-bar planar linkages, which consist of revolute and prismatic joints, based on hand-drawn sketches. Instead of relying solely on computer vision, our approach combines topological knowledge of linkage mechanisms with the outcomes of a convolutional deep neural network. This creates a framework for recognizing hand-drawn sketches. We generate a dataset of synthetic images that resemble hand-drawn sketches of linkage mechanisms. Next, we fine-tune a state-of-the-art deep neural network to detect discrete objects using building blocks that represent joints and links in various positions, sizes, and orientations within these sketches. We then conduct a topological analysis on the detected objects to construct a kinematic model of the sketched mechanisms. The results demonstrate the effectiveness of our algorithm in handling hand-drawn sketches and converting them into digital representations. This has practical implications for improving communication, analysis, organization, and classification of planar mechanisms. 

Abstract In this paper, we discuss the convergence of recent advances in deep neural networks (DNNs) with the design of robotic mechanisms, which entails the conceptualization of the design problem as a learning problem from the space of design specifications to a parameterization of the space of mechanisms. We identify three key inter-related problems that are at the forefront of using the versatility of DNNs in solving mechanism design problems. The first problem is that of representation of mechanisms and their design specifications, where the representation challenges arise primarily from the non-Euclidean nature of the data. The second problem is that of developing a mapping from the space of design specifications to the mechanisms where, ideally, we would like to synthesize both type and dimensions of the mechanism for a wide variety of design specifications including path synthesis, motion synthesis, constraints on pivot locations, etc. The third problem is that of designing the neural network architecture for end-to-end training and generation of multiple candidate mechanisms for a given design specification. We also present a brief overview of the state-of-the-art on each of these problems and identify questions of potential interest to the research community.