A continuous switching contact model for virtual environment based teleoperation

Virtual environment (VE), as the proxy of slave contact environment, is the most promising technology to solve the time-delay problems in teleoperation. The accuracy of the predicted force depends not only on the reliability of the contact model but also on the estimation algorithm’s adaptability. A new contact model is proposed to be applicable in various materials, which includes both the Kelvin–Voigt model (KVM) and Hunt–Crossley model (HCM). An extra parameter is set in the model to express the capacity of continuous switching between KVM and HCM, whose rationality is proved based on the energy loss. The energy loss is proportional to a power of impact velocity, and the exponent is bounded at [2,3], which exactly lies between KVM and HCM. Furthermore, to estimate the parameters with a single-stage method, the nonlinear model is linearized approximatively with logarithm function and polynomials. Then, the recursive least squares (RLS) algorithm combining forgetting factor and self-perturbing action is designed to identify the four parameters online. Finally, the model’s continuous switching is verified with ideal simulation, and the model parameters are continuously changed without jumpy switch error. In the experiment, sponge, foam, and human hand represent the complex contact materials of the slave environment where the predicted force is shown to follow the real contact force with enough accuracy. Therefore, the virtual model can be considered the substitution of slave contact environment so that the feedback force in master can be calculated in real-time.

Students’ Development of a Logarithm Function in Python Using Taylor Expansions: A Teaching Design Case Study

We present here the lessons learned by iteratively designing a tutorial for first-year university students using computer programming to work with mathematical models. Alternating between design and implementation, we used video-taped task interviews and classroom observations to ensure that the design promoted student understanding. The final version of the tutorial we present here has students make their own logarithm function from scratch, using Taylor polynomials. To ensure that the resulting function is accurate and reasonably fast, the students have to understand and apply concepts from both computing and mathematics. We identify four categories of such concepts and identify three design features that students attended to when demonstrating such understandings. Additionally, we describe seven important take-aways from a teaching design point of view that resulted from this iterative design process.

A Modified Whale Optimization Algorithm with Single-Dimensional Swimming for Global Optimization Problems

As a novel meta-heuristic algorithm, the Whale Optimization Algorithm (WOA) has well performance in solving optimization problems. However, WOA usually tends to trap in local optimal and it suffers slow convergence speed for large-scale and high-dimension optimization problems. A modified whale optimization algorithm with single-dimensional swimming (abbreviated as SWWOA) is proposed in order to overcome the shortcoming. First, tent map is applied to generate the initialize population for maximize search ability. Second, quasi-opposition learning is adopted after every iteration for further improving the search ability. Third, a novel nonlinearly control parameter factor that is based on logarithm function is presented in order to balance exploration and exploitation. Additionally, the last, single-dimensional swimming is proposed in order to replace the prey behaviour in standard WOA for tuning. The simulation experiments were conducted on 20 well-known benchmark functions. The results show that the proposed SWWOA has better performance in solution precision and higher convergence speed than the comparison methods.

A Note on a New Type of Degenerate poly-Euler Numbers and Polynomials

Kim-Kim [12] introduced the new type of degenerate Bernoulli numbers and polynomials arising from the degenerate logarithm function. In this paper, we introduce a new type of degenerate poly-Euler polynomials and numbers, are called degenerate poly-Euler polynomials and numbers, by using the degenerate polylogarithm function and derive several properties on the degenerate poly-Euler polynomials and numbers. In the last section, we also consider the degenerate unipoly-Euler polynomials attached to an arithmetic function, by using the degenerate polylogarithm function and investigate some identities of those polynomials. In particular, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

On an integral involving the logarithm function

We use the Ramanujan's master theorem to evaluate the integral $$\int_{0}^{\infty}\frac{x^{l-1}}{(1+x)^{m+1}}\log^{n}(1+x)\, dx $$ in terms of the digamma function, the gamma function, and the Hurwitz zeta function.

Entropic Divergence and Entropy Related to Nonlinear Master Equations

We reverse engineer entropy formulas from entropic divergence, optimized to given classes of probability distribution function (PDF) evolution dynamical equation. For linear dynamics of the distribution function, the traditional Kullback–Leibler formula follows from using the logarithm function in the Csiszár’s f-divergence construction, while for nonlinear master equations more general formulas emerge. As applications, we review a local growth and global reset (LGGR) model for citation distributions, income distribution models and hadron number fluctuations in high energy collisions.

Causal homogenization of metamaterials

We propose a homogenization scheme for metamaterials that utilizes causality to determine their effective parameters. By requiring the Kramers-Kronig causality condition in the homogenization of metamaterials, we show that the effective parameters can be chosen uniquely, in contrast to the conventional parameter retrieval method which has unavoidable phase ambiguity arising from the multivalued logarithm function. We demonstrate that the effective thickness of metamaterials can also be determined to a specific value by saturating the minimum-error condition for the causality restriction. Our causal homogenization provides a robust and accurate characterization method for metamaterials.