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1. A Real-Time Algorithm for Computing the Tension Force in a Suspended Elastic Sagging Cable

   Baskar, Aravind; Plecnik, Mark; Hauenstein, Jonathan D; Wampler, Charles W (May 2024, Springer)

   Larochelle, Pierre; McCarthy, J Michael; Lusk, Craig P (Ed.)

   An algorithm is presented for computing the tension in an elastic cable subject to sagging under its own weight, a problem highly relevant in tethered systems such as cable-driven parallel robots. This requires solving the two coupled equations of the Irvine cable model, which give the endpoint position as a function of vertical and horizontal components of tension. Via a change of variables, we reformulate this system as a pair of uncoupled equations, which are shown to have a unique solution. We develop an efficient numerical procedure to solve one of these, after which closed-form formulas provide the solution of the second equation and ultimately the tension components. 

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2. Geolocation with FDOA measurements via polynomial systems and RANSAC

   https://doi.org/10.1109/RADAR.2018.8378640  

   Cameron, Karleigh J.; Bates, Daniel J. (April 2018, 2018 IEEE Radar Conference (RadarConf18))

   The problem of geolocation of a radiofrequency transmitter via time difference of arrival (TDOA) and frequency difference of arrival (FDOA) is given as a system of polynomial equations. This allows for the use of homotopy continuation-based methods from numerical algebraic geometry. A novel geolocation algorithm employs numerical algebraic geometry techniques in conjunction with the random sample consensus (RANSAC) method. This is all developed and demonstrated in the setting of only FDOA measurements, without loss of generality. Additionally, the problem formulation as polynomial systems immediately provides lower bounds on the number of receivers or measurements required for the solution set to consist of only isolated points. 

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3. Global, Unified Representation of Heterogenous Robot Dynamics Using Composition Operators: A Koopman Direct Encoding Method

   https://doi.org/10.1109/TMECH.2023.3253599  

   Asada, H. Harry (May 2023, IEEE/ASME Transactions on Mechatronics)

   The dynamic complexity of robots and mechatronic systems often pertains to the hybrid nature of dynamics, where governing equations consist of heterogenous equations that are switched depending on the state of the system. Legged robots and manipulator robots experience contact-noncontact discrete transitions, causing switching of governing equations. Analysis of these systems have been a challenge due to the lack of a global, unified model that is amenable to analysis of the global behaviors. Composition operator theory has the potential to provide a global, unified representation by converting them to linear dynamical systems in a lifted space. The current work presents a method for encoding nonlinear heterogenous dynamics into a high dimensional space of observables in the form of Koopman operator. First, a new formula is established for representing the Koopman operator in a Hilbert space by using inner products of observable functions and their composition with the governing state transition function. This formula, called Direct Encoding, allows for converting a class of heterogenous systems directly to a global, unified linear model. Unlike prevalent data-driven methods, where results can vary depending on numerical data, the proposed method is globally valid, not requiring numerical simulation of the original dynamics. A simple example validates the theoretical results, and the method is applied to a multi-cable suspension system. 

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4. Improved quantitative unique continuation for complex-valued drift equations in the plane

   https://doi.org/10.1515/forum-2022-0114  

   Davey, Blair; Kenig, Carlos; Wang, Jenn-Nan (June 2022, Forum Mathematicum)

   Abstract In this article, we investigate the quantitative unique continuation properties of complex-valued solutions to drift equations in the plane.We consider equations of the form Δ ⁢ u + W ⋅ ∇ ⁡ u = 0 {\Delta u+W\cdot\nabla u=0} in ℝ 2 {\mathbb{R}^{2}} ,where W = W 1 + i ⁢ W 2 {W=W_{1}+iW_{2}} with each W j {W_{j}} being real-valued.Under the assumptions that W j ∈ L q j {W_{j}\in L^{q_{j}}} for some q 1 ∈ [ 2 , ∞ ] {q_{1}\in[2,\infty]} , q 2 ∈ ( 2 , ∞ ] {q_{2}\in(2,\infty]} and that W 2 {W_{2}} exhibits rapid decay at infinity,we prove new global unique continuation estimates.This improvement is accomplished by reducing our equations to vector-valued Beltrami systems.Our results rely on a novel order of vanishing estimate combined with a finite iteration scheme. 

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5. Numerical study of non-uniqueness for 2D compressible isentropic Euler equations.

   https://doi.org/10.1016/j.jcp.2021.110588  

   Bressan, Alberto; Jiang, Yi; and Liu, Hailiang (July 2021, Journal of computational physics)

   In this paper, we numerically study a class of solutions with spiraling singularities in vorticity for two-dimensional, inviscid, compressible Euler systems, where the initial data have an algebraic singularity in vorticity at the origin. These are different from the multi- dimensional Riemann problems widely studied in the literature. Our computations provide numerical evidence of the existence of initial value problems with multiple solutions, thus revealing a fundamental obstruction toward the well-posedness of the governing equations. The compressible Euler equations are solved using the positivity-preserving discontinuous Galerkin method. 

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