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1. TMTDyn : A Matlab package for modeling and control of hybrid rigid–continuum robots based on discretized lumped systems and reduced-order models

   https://doi.org/10.1177/0278364919881685  

   Sadati, S.M. Hadi; Naghibi, S. Elnaz; Shiva, Ali; Michael, Brendan; Renson, Ludovic; Howard, Matthew; Rucker, Caleb D.; Althoefer, Kaspar; Nanayakkara, Thrishantha; Zschaler, Steffen; et al (January 2021, The International Journal of Robotics Research)

   null (Ed.)

   A reliable, accurate, and yet simple dynamic model is important to analyzing, designing, and controlling hybrid rigid–continuum robots. Such models should be fast, as simple as possible, and user-friendly to be widely accepted by the ever-growing robotics research community. In this study, we introduce two new modeling methods for continuum manipulators: a general reduced-order model (ROM) and a discretized model with absolute states and Euler–Bernoulli beam segments (EBA). In addition, a new formulation is presented for a recently introduced discretized model based on Euler–Bernoulli beam segments and relative states (EBR). We implement these models in a Matlab software package, named TMTDyn, to develop a modeling tool for hybrid rigid–continuum systems. The package features a new high-level language (HLL) text-based interface, a CAD-file import module, automatic formation of the system equation of motion (EOM) for different modeling and control tasks, implementing Matlab C-mex functionality for improved performance, and modules for static and linear modal analysis of a hybrid system. The underlying theory and software package are validated for modeling experimental results for (i) dynamics of a continuum appendage, and (ii) general deformation of a fabric sleeve worn by a rigid link pendulum. A comparison shows higher simulation accuracy (8–14% normalized error) and numerical robustness of the ROM model for a system with a small number of states, and computational efficiency of the EBA model with near real-time performances that makes it suitable for large systems. The challenges and necessary modules to further automate the design and analysis of hybrid systems with a large number of states are briefly discussed. 

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2. Stabilization of Nonholonomic Pendulum Skate by Controlled Lagrangians

   https://doi.org/10.1109/CDC51059.2022.9993264  

   Garcia, Jorge Silva; Ohsawa, Tomoki (January 2022, 2022 IEEE 61st Conference on Decision and Control (CDC))

   We consider the problem of stabilizing what we call a pendulum skate, a simple model of a figure skater developed by Gzenda and Putkaradze. By exploiting the symmetry of the system as well as taking care of the part of the symmetry broken by the gravity, the equations of motion are given as nonholonomic Euler–Poincaré equation with advected parameters. Our main interest is the stability of the sliding and spinning equilibria of the system. We show that the former is unstable and the latter is stable only under certain conditions. We use the method of Controlled Lagrangians to find a control to stabilize the sliding equilibrium. 

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3. On the approximation of vorticity fronts by the Burgers–Hilbert equation

   https://doi.org/10.3233/asy-211724  

   Hunter, John K.; Moreno-Vasquez, Ryan C.; Shu, Jingyang; Zhang, Qingtian (August 2022, Asymptotic Analysis)

   This paper proves that the motion of small-slope vorticity fronts in the two-dimensional incompressible Euler equations is approximated on cubically nonlinear timescales by a Burgers–Hilbert equation derived by Biello and Hunter (2010) using formal asymptotic expansions. The proof uses a modified energy method to show that the contour dynamics equations for vorticity fronts in the Euler equations and the Burgers–Hilbert equation are both approximated by the same cubically nonlinear asymptotic equation. The contour dynamics equations for Euler vorticity fronts are also derived. 

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4. High-precision quantum algorithms for partial differential equations

   https://doi.org/10.22331/q-2021-11-10-574  

   Childs, Andrew M.; Liu, Jin-Peng; Ostrander, Aaron (November 2021, Quantum)

   Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity p o l y ( 1 / ϵ ) , where ϵ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be p o l y ( d , log ⁡ ( 1 / ϵ ) ) , where d is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations. 

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5. Optimal error estimates of ultra-weak discontinuous Galerkin methods with generalized numerical fluxes for multi-dimensional convection-diffusion and biharmonic equations

   https://doi.org/10.1090/mcom/3927  

   Chen, Yuan; Xing, Yulong (September 2024, Mathematics of Computation)

   In this paper, we study ultra-weak discontinuous Galerkin methods with generalized numerical fluxes for multi-dimensional high order partial differential equations on both unstructured simplex and Cartesian meshes. The equations we consider as examples are the nonlinear convection-diffusion equation and the biharmonic equation. Optimal error estimates are obtained for both equations under certain conditions, and the key step is to carefully design global projections to eliminate numerical errors on the cell interface terms of ultra-weak schemes on general dimensions. The well-posedness and approximation capability of these global projections are obtained for arbitrary order polynomial space based on a wide class of generalized numerical fluxes on regular meshes. These projections can serve as general analytical tools to be naturally applied to a wide class of high order equations. Numerical experiments are conducted to demonstrate these theoretical results. 

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