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1. A General Method for Constructing Planar Cognate Mechanisms

   https://doi.org/10.1115/1.4050293  

   Sherman, Samantha N.; Hauenstein, Jonathan D.; Wampler, Charles W. (June 2021, Journal of Mechanisms and Robotics)

   null (Ed.)

   Abstract Cognate linkages are mechanisms that share the same motion, a property that can be useful in mechanical design. This article treats planar curve cognates, that is, planar mechanisms with rotational joints whose coupler points draw the same curve, as well as coupler cognates and timed curve cognates. The purpose of this article is to develop a straightforward method based solely on kinematic equations to construct cognates. The approach computes cognates that arise from permuting link rotations and is shown to reproduce all of the known results for cognates of four-bar and six-bar linkages. This approach is then used to construct a cognate of an eight-bar and a ten-bar linkage. 

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2. 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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3. Eclipse Timing the Milky Way’s Gravitational Potential

   https://doi.org/10.3847/2041-8213/ac5c43  

   Chakrabarti, Sukanya; Stevens, Daniel J.; Wright, Jason; Rafikov, Roman R.; Chang, Philip; Beatty, Thomas; Huber, Daniel (April 2022, The Astrophysical Journal Letters)

   Abstract We show that a small but measurable shift in the eclipse midpoint time of eclipsing binary (EBs) stars of ∼0.1 s over a decade baseline can be used to directly measure the Galactic acceleration of stars in the Milky Way at ∼kiloparsec distances from the Sun. We consider contributions to the period drift rate from dynamical mechanisms other than the Galaxy’s gravitational field and show that the Galactic acceleration can be reliably measured using a sample of Kepler EBs with orbital and stellar parameters from the literature. The contribution from tidal decay we estimate here is an upper limit assuming the stars are not tidally synchronized. We find there are about 200 detached EBs that have estimated timing precision better than 0.5 s, and for which other dynamical effects are subdominant to the Galactic signal. We illustrate the method with a prototypical, precisely timed EB using an archival Kepler light curve and a modern synthetic HST light curve (which provides a decade baseline). This novel method establishes a realistic possibility to constrain dark matter substructure and the Galactic potential using eclipse timing to measure Galactic accelerations, along with other emerging new methods, including pulsar timing and extreme-precision radial velocity observations. This acceleration signal grows quadratically with time. Therefore, given baselines established in the near future for distant EBs, we can expect to measure the period drift in the future with space missions like JWST and the Roman Space Telescope. 

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4. Path Generative Model Based on Conditional β -Variational Auto Encoder for Four-Bar Mechanism Design

   https://doi.org/10.1115/1.4067169  

   Nurizada, Anar; Lyu, Zhijie; Purwar, Anurag (June 2025, Journal of Mechanisms and Robotics)

   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. 

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5. GeoLatent: A Geometric Approach to Latent Space Design for Deformable Shape Generators

   https://doi.org/10.1145/3618371  

   Yang, Haitao; Sun, Bo; Chen, Liyan; Pavel, Amy; Huang, Qixing (December 2023, ACM Transactions on Graphics)

   We study how to optimize the latent space of neural shape generators that map latent codes to 3D deformable shapes. The key focus is to look at a deformable shape generator from a differential geometry perspective. We define a Riemannian metric based on as-rigid-as-possible and as-conformal-as-possible deformation energies. Under this metric, we study two desired properties of the latent space: 1) straight-line interpolations in latent codes follow geodesic curves; 2) latent codes disentangle pose and shape variations at different scales. Strictly enforcing the geometric interpolation property, however, only applies if the metric matrix is a constant. We show how to achieve this property approximately by enforcing that geodesic interpolations are axis-aligned, i.e., interpolations along coordinate axis follow geodesic curves. In addition, we introduce a novel approach that decouples pose and shape variations via generalized eigendecomposition. We also study efficient regularization terms for learning deformable shape generators, e.g., that promote smooth interpolations. Experimental results on benchmark datasets show that our approach leads to interpretable latent codes, improves the generalizability of synthetic shapes, and enhances performance in geodesic interpolation and geodesic shooting. 

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