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1. Synthesizing the Transmission Properties of a Five-bar Linkage by Shaping Workspace Bounds

   Ramesh, Shashank; Plecnik, Mark (July 2024, Springer)

   Lenarčič, Jadran; Husty, Manfred (Ed.)

   The multidirectional transmission characteristics of a five-bar linkage can be visualized by plotting Jacobian-defined velocity ellipses inside its workspace. The orientation, size, and aspect ratio of these ellipses indicate directional force and velocity multiplication from the actuators to the end-effector. Our broader goal is approximate dimensional synthesis to achieve desired ellipses. On a workspace bound, the minor axis of a velocity ellipse collapses while the major axis aligns tangential to the bound. Interior to the workspace, ellipses vary with continuity. Therefore, the shape of a workspace bound influences the interior ellipses. The workspace bounds of a five-bar linkage are formed from segments of four-bar coupler curves (the locus of endpoint positions while the five-bar is held in output singularity conditions) and circular segments. Therefore, interior ellipses can be influenced by the path synthesis of four-bar linkages that represent the five-bar situated with certain links held colinear (the output singularity conditions). This paper details the synthesis of these four-bar coupler curves for forming the workspace bounds of a five-bar in order to influence its interior ellipses. Our approach employs saddle graphs that detail the connectivity of critical points over an optimization function. 

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2. An Invariant Representation of Coupler Curves Using a Variational AutoEncoder: Application to Path Synthesis of Four-Bar Mechanisms

   https://doi.org/10.1115/1.4063726  

   Nurizada, Anar; Purwar, Anurag (January 2024, Journal of Computing and Information Science in Engineering)

   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. 

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3. Design a Four-Bar Mechanism for Specific Upper Limb Muscle Strength Rehabilitation Using Genetic Algorithm

   https://doi.org/10.1142/S0219843623500056  

   Quarnstrom, Joel; Zaman, Rahid; Xiang, Yujiang (August 2023, International Journal of Humanoid Robotics)

   In this study, a novel human-in-the-loop design method using a genetic algorithm (GA) is presented to design a low-cost and easy-to-use four-bar linkage medical device for upper limb muscle rehabilitation. The four-bar linkage can generate a variety of coupler point trajectories by using different link lengths. For this medical device, patients grab the coupler point handle and rotate the arm along the designed coupler point trajectory to exercise upper limb muscles. The design procedures include three basic steps: First, for a set of link lengths, a complete coupler point trajectory is generated from four-bar linkage kinematics; second, optimization-based motion prediction is utilized to predict arm motion (joint angle profiles) subjected to hand grasping and joint angle limit constraints; third, the predicted joint angles and given hand forces are imported into an OpenSim musculoskeletal arm model to calculate the muscle forces and activations by using the OpenSim static optimization. In the GA optimization formulation, the design variables are the four-bar link lengths. The objective function is to maximize a specific muscle’s exertion for a complete arm rotation. Finally, different four-bar configurations are designed for different muscle strength exercises. The proposed human-in-the-loop design approach successfully integrates GA with linkage kinematics, arm motion prediction, and OpenSim static optimization for four-bar linkage design for upper limb muscle strength rehabilitation. 

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4. A Dataset of 3M Single-DOF Planar 4-, 6-, and 8-Bar Linkage Mechanisms With Open and Closed Coupler Curves for Machine Learning-Driven Path Synthesis

   https://doi.org/10.1115/1.4067014  

   Nurizada, Anar; Dhaipule, Rohit; Lyu, Zhijie; Purwar, Anurag (April 2025, Journal of Mechanical Design)

   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. 

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5. Finding straight line generators through the approximate synthesis of symmetric four-bar coupler curves

   https://doi.org/10.1016/j.mechmachtheory.2023.105310  

   Baskar, Aravind; Plecnik, Mark; Hauenstein, Jonathan D. (October 2023, Mechanism and Machine Theory)

   The approximate path synthesis of four-bar linkages with symmetric coupler curves is presented. This includes the formulation of a polynomial optimization problem, a characterization of the maximum number of critical points, a complete numerical solution by homotopy continuation, and application to the design of straight line generators. Our approach specifies a desired curve and finds several optimal four-bar linkages with a coupler trace that approximates it. The objective posed simultaneously enforces kinematic accuracy, loop closure, and leads to polynomial first order necessary conditions with a structure that remains the same for any desired trace leading to a generalized result. Ground pivot locations are set as chosen parameters, and it is shown that the objective has a maximum of 73 critical points. The theoretical work is applied to the design of straight line paths. Parameter homotopy runs are executed for 1440 different choices of ground pivots for a thorough exploration. These computations found the expected linkages, namely, Watt, Evans, Roberts, Chebyshev, and other previously unreported linkages which are organized into a 2D atlas using the UMAP algorithm. 

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