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The linearization of the equations of motion of a robotics system about a given state-input trajectory, including a controlled equilibrium state, is a valuable tool for model-based planning, closed-loop control, gain tuning, and state estimation. Contrary to the case of fixed based manipulators with prismatic or revolute joints, the state space of moving-base robotic systems such as humanoids, quadruped robots, or aerial manipulators cannot be globally parametrized by a finite number of independent coordinates. This impossibility is a direct consequence of the fact that the state of these systems includes the system's global orientation, formally described as an element of the special orthogonal group SO(3). As a consequence, obtaining the linearization of the equations of motion for these systems is typically resolved, from a practical perspective, by locally parameterizing the system's attitude by means of, e.g., Euler or Cardan angles. This has the drawback, however, of introducing artificial parameterization singularities and extra derivative computations. In this contribution, we show that it is actually possible to define a notion of linearization that does not require the use of a local parameterization for the system's orientation, obtaining a mathematically elegant, recursive, and singularity-free linearization for moving-based robot systems. Recursiveness, in particular, is obtained by proposing a nontrivial modification of existing recursive algorithms to allow for computations of the geometric derivatives of the inverse dynamics and the inverse of the mass matrix of the robotic system. The correctness of the proposed algorithm is validated by means of a numerical comparison with the result obtained via geometric finite difference. [ABSTRACT FROM AUTHOR]

Abstract Three-phase flash calculation is a crucial part of the modeling and simulation of the transport phenomenon in various chemical and petroleum processes. To obtain the compositions of phases in equilibrium, utilization of Equations of State (EOSs) and thermodynamic phase equilibria rules/laws (e.g., Henry's law, Lewis/Randall rule, and modified Raoult's law) seem inevitable to model phase behaviors of liquid, vapor, and aqueous phases. This study plans to develop two thermodynamic modeling approaches (on the basis of Henry's law and EOS) to conduct three-phase flash calculations in the presence of water. It is found that both developed algorithms are capable of determining the equilibrium conditions in the three-phase region such that a desired precision is obtained. In this research study, a new stability algorithm based on a negative flash procedure is introduced to forecast the formation and/or condition of the stable phases. The modeling results reveal that the aqueous phase equilibrium calculations by using EOS lead to an inaccurate prediction of stable phases in a vicinity of phase change regions. Improving the accuracy and reliability of the phase equilibria calculations, a new analytical derivative of fugacity coefficient with respect to mole fraction is obtained so that a better convergence and less runtime (compared to numerical derivatives) are achieved. An appropriate method to guess the initial equilibrium ratio is presented in this research investigation. The best roots of the cubic EOS are then selected for all three phases by minimizing Gibbs free energy. The proposed procedure can be implemented in different multiphase systems (and simulation softwares) to increase the accuracy of multiphase equilibrium calculations with an acceptable computational cost. Graphical abstract Image 1 Highlights • This study introduces a new algorithm to obtain 3-phase equilibrium conditions. • The new technique is more accurate and reliable, compared to previous strategies. • The approach involves EOS, Henry's law, and stability criteria with low computational costs. • The new analytical derivative of fugacity coefficient is applicable within broad thermodynamic conditions. • A parametric sensitivity analysis is conducted to identify the most important input parameters. [ABSTRACT FROM AUTHOR]

We show that the pressure acting on atoms and molecular systems within the compression cavity of the eXtreme‐Pressure Polarizable Continuum method can be expressed in terms of the electron density of the systems and of the Pauli‐repulsion confining potential. The analytical expression holds for spherical cavities as well as for cavities constructed from van der Waals spheres of the constituting atoms of the molecular systems. © 2018 Wiley Periodicals, Inc. The eXtreme‐Pressure Polarizable Continuum model is a quantum chemical method aimed to describe the effects of the extreme‐high pressure (> 1 GPa) on the properties of atomic and molecular systems, by confining the systems within a var der Waals cavity of an external transmitting medium. This article explains the pressure acting on atoms and molecular systems within the cavity can be analytically expressed in terms of the electron density ρ (r) of the atomic or molecular systems and of the confining potential Z(f). [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we review the radiative transfer formalism of the matrix operator method, and present the analytic form for its differentiation with respect to aerosol optical thickness, microphysical parameters and surface parameters. This “linearization” is an exact method that allows for an accurate and speedy computation of the Jacobian matrix, which is key to most optimization-based retrieval methods. We define an aerosol in terms of its optical thickness, complex refractive index and lognormal size distribution. We consider a bimodal aerosol distribution, consisting of a fine and coarse mode, such that the two modes also differ in their respective complex refractive indices. Three types of surfaces have been considered, viz. a purely Lambertian surface, a modified Rahman–Pinty–Verstraete bidirectional reflecting surface, and a Fresnel-reflecting ocean surface. We verify our results by comparing our linearized Jacobians of normalized intensities calculated at four different wavelengths in the visible (VIS) and near-infrared (NIR) and viewing angles ranging from −75° through 0° to 75° with those computed by the method of finite differences. We demonstrate the guaranteed accuracy of the linearized approach by contrasting it with the finite difference method which can only be used as a rough estimate due to its sensitivity to step size, especially for derivatives with respect to aerosol microphysical parameters. We also report that the computational speed-up due to linearization improves with the number of parameters involved, parity being achieved with the finite difference method for just one parameter. Finally, we discuss the features of the illustrated Jacobians as a function of viewing angle and wavelengths. [Copyright &y& Elsevier]

Analytical energy gradients have been implemented in the DALTON program at the level of explicitly correlated second-order Møller-Plesset perturbation theory (MP2-R12). The implementation has been accomplished for standard approximation A using an auxiliary basis which is the same as the orbital basis for the resolution-of-the-identity, with and without frozen core. As a test calculation, the geometries of nine stationary points on the potential energy surface of the water dimer have been optimized and the results are compared with earlier studies. [ABSTRACT FROM AUTHOR]

Abstract: We derive exact expressions for the forces and torques between biaxial molecules interacting via the RE-squared potential, a recent variant of the Gay–Berne potential. Moreover, efficient routines have been provided for rigid body MD simulations, resulting in 1.6 times speedup compared to the two-point finite difference approach. It has also been shown that the time cost of a MD simulation will be almost equal to a similar MC simulation, making use of the provided routines. [Copyright &y& Elsevier]

This short note describes the relevant details of formulating and implementing general bilinear quantity–quality balances found in industrial processes when data reconciliation is applied. The modeling also allows for the straightforward generation of analytical first-order derivatives. Quantity–quality balance problems are those that involve both extensive and intensive stream variables such as flows and compositions, respectively and are related through laws of conservation of material, energy and momentum. The balance equations involve both linear and bilinear terms (multi-linear) of quantity and quality where quantity times quantity and quality times quality are not germane although they can be included easily. Two numerical examples are provided to demonstrate the new formulation technique. [Copyright &y& Elsevier]