Structural Design, Simulation and Experiment of Quadruped Robot

This paper carried out a series of designs, simulations and implementations by using the physical-like mechanism of a bionic quadruped robot dog as a vehicle. Through an investigation of the walking mechanisms of quadrupeds, a bionic structure is proposed that is capable of omnidirectional movements and smooth motions. Furthermore, the kinematic and inverse kinematic solutions based on the DH method are explored to lay the foundation for the gait algorithm. Afterward, a classical compound pendulum equation is applied as the foot-end trajectory and inverse kinematic solutions are combined to complete the gait planning. With appropriate foot–ground contact modeling, MATLAB and ADAMS are used to simulate the dynamic behavior and the diagonal trot gait of the quadruped robot. Finally, the physical prototype is constructed, designed and debugged, and its performance is measured through real-world experiments. Results show that the quadruped robot is able to balance itself during trot motion, for both its pitch and roll attitude. The goal of this work is to provide an affordable yet comprehensive platform for novice researchers in the field to study the dynamics, contact modeling, gait planning and attitude control of quadruped robots.

Approximate Solutions to the Forced Damped Parametric Driven Pendulum Oscillator: Chebyshev Collocation Numerical Solution

In this work, novel semi-analytical and numerical solutions to the forced damped driven nonlinear (FDDN) pendulum equation on the pivot vertically for arbitrary angles are obtained for the first time. The semi-analytical solution is derived in terms of the Jacobi elliptic functions with arbitrary elliptic modulus. For the numerical analysis, the Chebyshev collocation numerical method is introduced for analyzing tthe forced damped parametric driven pendulum equation. Moreover, the semi-analytical solution and Chebyshev collocation numerical solution are compared with the Runge-Kutta (RK) numerical solution. Also, the maximum distance error to the obtained approximate solutions is estimated with respect to the RK numerical solution. The obtained results help many authors to understand the mechanism of many phenomena related to the plasma physics, classical mechanics, quantum mechanics, optical fiber, and electronic circuits.

On the Analytical Solutions of the Forced Damping Duffing Equation in the Form of Weierstrass Elliptic Function and its Applications

In this study, a novel analytical solution to the integrable undamping Duffing equation with constant forced term is obtained. Also, a new approximate analytical (semianalytical) solution for the nonintegrable linear damping Duffing oscillator with constant forced term is reported. The analytical solution is given in terms of the Weierstrass elliptic function with arbitrary initial conditions. With respect to it, the semianalytical solution is constructed depending on a new ansatz and the exact solution of the standard Duffing equation (in the absence of both damping and forced terms). A comparison between the obtained solutions and the Runge–Kutta fourth-order (RK4) is carried out. Moreover, some complicated oscillator equations such as the constant forced damping pendulum equation, forced damping cubic-quintic Duffing equation, and constant forced damping Helmholtz–Duffing equation are reduced to the forced damping Duffing oscillator, in which its solution is known. As a practical application, the proposed techniques are applied to investigate the characteristics behavior of the signal oscillations arising in the RLC circuit with externally applied voltage.

Limit Cycles in the Equation of Whirling Pendulum With Piecewise Smooth Perturbations

This paper deals with the problem of limit cycles for the whirling pendulum equation ẋ = y, ẏ = sin x(cos x-r) under piecewise smooth perturbations of polynomials of cos x, sin x and y of degree n with the switching line x = 0. The upper bounds of the number of limit cycles in both the oscillatory and the rotary regions are obtained by using the Picard-Fuchs equations which the generating functions of the associated first order Melnikov functions satisfy. Further, the exact bound of a special case is given by using the Chebyshev system.

Generalized Bessel Functions and Their Use in Bremsstrahlung and Multi-Photon Processes

The theory of Generalized Bessel Functions is reviewed and their application to various problems in the study of electro-magnetic processes is presented. We consider the cases of emission of bremsstrahlung radiation by ultra-relativistic electrons in linearly polarized undulators, including also exotic configurations, aimed at enhancing the harmonic content of the emitted radiation. The analysis is eventually extended to the generalization of the FEL pendulum equation to treat Free Electron Laser operating with multi-harmonic undulators. The paper aims at picking out those elements supporting the usefulness of the Generalized Bessel Functions in the elaboration of the theory underlying the study of the spectral properties of the bremsstrahlung radiation emitted by relativistic charges, along with the relevant flexibility in accounting for a large variety of apparently uncorrelated phenomenolgies, like multi-photon processes, including non linear Compton scattering.

Chaos Suppression in a Pendulum Equation through Parametric Excitation with Phase Shift for Ultra-Subharmonic Resonance

Under ultra-subharmonic resonance, we investigate the chaos suppression of pendulum equation by using Melnikov methods, and get the conditions of suppressing chaos for homoclinic and heteroclinic orbits, respectively. At the same time, we give some numerical simulations including the bifurcation diagrams of system and corresponding phase diagrams, and observe that the chaos behaviors of system may be suppressed to period-n(n ∈ Z+) orbits by adjusting the value of Ψ. Although our results are only necessary, not sufficient. Numerical simulations show that our method is effect in suppressing chaos for this case.

On the existence of periodic oscillations for pendulum-type equations

We provide new sufficient conditions for the existence of T-periodic solutions for the ϕ-laplacian pendulum equation (ϕ(x′))′ + k x′ + a sin x = e(t), where e ∈ C͠T. Our main tool is a continuation theorem due to Capietto, Mawhin and Zanolin and we improve or complement previous results in the literature obtained in the framework of the classical, the relativistic and the curvature pendulum equations.

Angle-action variables for orbits trapped at a Lindblad resonance

ABSTRACT The conventional approach to orbit trapping at Lindblad resonances via a pendulum equation fails when the parent of the trapped orbits is too circular. The problem is explained and resolved in the context of the Torus Mapper and a realistic Galaxy model. Tori are computed for orbits trapped at both the inner and outer Lindblad resonances of our Galaxy. At the outer Lindblad resonance, orbits are quasi-periodic and can be accurately fitted by torus mapping. At the inner Lindblad resonance, orbits are significantly chaotic although far from ergodic, and each orbit explores a small range of tori obtained by torus mapping.

Roles of magnetospheric convection on nonlinear drift resonance between electrons and ULF waves

<p>In the Earth's inner magnetosphere, charged particles can be accelerated and transported by ultralow frequency (ULF) waves via drift resonance. We investigate the effects of magnetospheric convection on the nonlinear drift resonance process, which provides an inhomogeneity factor S to externally drive the pendulum equation that describes the particle motion in the ULF wave  field. The S factor, defined as the ratio of the driving amplitude to the square of the pendulum trapping frequency, is found to vary with magnetic local time and as a consequence, oscillates quasi-periodically at the particle drift frequency. To better understand the particle behavior governed by the driven pendulum equation, we carry out simulations to obtain the evolution of electron distribution functions in energy and L-shell phase space. We find that resonant electrons can remain trapped by the low-m ULF waves under strong convection electric  field, whereas for high-m ULF waves, the electrons trajectories can be significantly modified. More interestingly, the electron drift frequency is close to the nonlinear trapping frequency for intermediate-m ULF waves, which corresponds to chaotic motion of resonant electrons. These  findings shed new light on the nature of particle coherent and diffusive transport in the inner magnetosphere.</p>