Nonlinear, Phase-Based Oscillator to Generate and Assist Periodic Motions

2017 ◽  
Vol 9 (2) ◽  
Author(s):  
Juan De la Fuente ◽  
Thomas G. Sugar ◽  
Sangram Redkar
Keyword(s):  
Limit Cycle ◽  
Phase Angle ◽  
Trajectory Planning ◽  
Nonlinear Oscillator ◽  
Oscillatory Behavior ◽  
Periodic Motions ◽  
Nonlinear Phase ◽  
Forcing Function ◽  
Line Trajectory ◽  
Bendixson Criterion

Oscillatory behavior is important for tasks, such as walking and running. We are developing methods for wearable robotics to add energy to enhance or vary the oscillatory behavior based on the system's phase angle. We define a nonlinear oscillator using a forcing function based on the sine and cosine of the system's phase angle that can modulate the amplitude and frequency of oscillation. This method is based on the state of the system and does not use off-line trajectory planning. The behavior of a limit cycle is shown using the Poincaré–Bendixson criterion. Linear and rotational models are simulated using our phase controller. The method is implemented and tested to control a pendulum.

Nonlinear Phase-Based Oscillator to Generate and Assist Periodic Motions

2016 ◽  
Author(s):  
Juan De la Fuente ◽  
Thomas G. Sugar ◽  
Sangram Redkar ◽  
Andrew R. Bates
Keyword(s):  
Phase Angle ◽  
Nonlinear Oscillator ◽  
Mechanical Systems ◽  
Oscillatory Behavior ◽  
Periodic Motions ◽  
Nonlinear Phase ◽  
Forcing Function ◽  
Hip Motion ◽  
The Stability ◽  
Bendixson Criterion

Oscillatory behavior is important for tasks such as walking and running. We are developing methods to add energy to enhance or vary the oscillatory behavior based on the system’s phase angle. We define a nonlinear oscillator using a forcing function based on the sine and cosine of the system’s phase angle that can modulate the amplitude and frequency of oscillation. The stability of the system is proved using the Poincaré-Bendixson criterion. Linear and rotational mechanical systems are simulated using our phase controller. The method is implemented and tested to control a pendulum. Lastly, we propose how to assist hip motion during walking using the phase-based forcing function.

Bifurcation Trees of Periodic Motions to Chaos in a Parametric, Quadratic Nonlinear Oscillator

2014 ◽  
Vol 24 (05) ◽  
pp. 1450075 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Fourier Series ◽  
Nonlinear Systems ◽  
Analytical Solutions ◽  
Nonlinear Oscillator ◽  
Series Solution ◽  
Parametric Oscillator ◽  
Periodic Motions ◽  
Parametric Oscillators ◽  
Harmonic Term ◽  
Nonlinear Behaviors

In this paper, bifurcation trees of periodic motions to chaos in a parametric oscillator with quadratic nonlinearity are investigated analytically as one of the simplest parametric oscillators. The analytical solutions of periodic motions in such a parametric oscillator are determined through the finite Fourier series, and the corresponding stability and bifurcation analyses for periodic motions are completed. Nonlinear behaviors of such periodic motions are characterized through frequency–amplitude curves of each harmonic term in the finite Fourier series solution. From bifurcation analysis of the analytical solutions, the bifurcation trees of periodic motion to chaos are obtained analytically, and numerical illustrations of periodic motions are presented through phase trajectories and analytical spectrum. This investigation shows period-1 motions exist in parametric nonlinear systems and the corresponding bifurcation trees to chaos exist as well.

Galloping Vibration of Nonlinear Cables Through a Two Degree-of-Freedom Oscillator

2016 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Transmission Line ◽  
Analytical Solutions ◽  
Transmission Lines ◽  
Nonlinear Oscillator ◽  
Numerical Solutions ◽  
Degree Of Freedom ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Two Degree Of Freedom

In this paper, galloping vibrations of a lightly iced transmission line are investigated through a two-degree-of-freedom (2-DOF) nonlinear oscillator. The 2-DOF nonlinear oscillator is used to describe the transverse and torsional motions of the galloping cables. The analytical solutions of periodic motions of galloping cables are presented through generalized harmonic balanced method. The analytical solutions of periodic motions for the galloping cable are compared with the numerical solutions, and the corresponding stability and bifurcation of periodic motions are analyzed by the eigenvalues analysis. To demonstrate the accuracy of the analytical solutions of periodic motions, the harmonic amplitudes are presented. This investigation will help one better understand galloping mechanism of iced transmission lines.

Poincaré Bifurcation of Some Nonlinear Oscillator of Generalized Liénard Type Using Symbolic Computation Method

2018 ◽  
Vol 28 (08) ◽  
pp. 1850096 ◽  
Author(s):  
Hongying Zhu ◽  
Bin Qin ◽  
Sumin Yang ◽  
Minzhi Wei
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Nonlinear Oscillator ◽  
Melnikov Function ◽  
Computation Method ◽  
Rigorous Proof ◽  
Heteroclinic Loop ◽  
Period Annulus ◽  
Domain Iii ◽  
Weak Damping

In this paper, we study the Poincaré bifurcation of a nonlinear oscillator of generalized Liénard type by using the Melnikov function. The oscillator has weak damping terms. When the damping terms vanish, the oscillator has a heteroclinic loop connecting a nilpotent cusp to a hyperbolic saddle. Our results reveal that: (i) the oscillator can have at most four limit cycles bifurcating from the corresponding period annulus. (ii) There are some parameters such that three limit cycles emerge in the original periodic orbit domain. (iii) Especially, we give a rigorous proof that [Formula: see text] limit cycle(s) can emerge near the original singular loop and [Formula: see text] limit cycle(s) can emerge near the original elementary center with [Formula: see text].

Bifurcation Trees of Period-1 Motions to Chaos in a Two-Degree-of-Freedom, Nonlinear Oscillator

2015 ◽  
Vol 25 (13) ◽  
pp. 1550179 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Dynamical System ◽  
Fourier Series ◽  
Analytical Solutions ◽  
Nonlinear Oscillator ◽  
Degree Of Freedom ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Series Transformation ◽  
Two Degree Of Freedom ◽  
Analytical Bifurcation Trees

In this paper, analytical solutions for period-[Formula: see text] motions in a two-degree-of-freedom (2-DOF) nonlinear oscillator are developed through the finite Fourier series. From the finite Fourier series transformation, the dynamical system of coefficients of the finite Fourier series is developed. From such a dynamical system, the solutions of period-[Formula: see text] motions are obtained and the corresponding stability and bifurcation analyses of period-[Formula: see text] motions are carried out. Analytical bifurcation trees of period-1 motions to chaos are presented. Displacements, velocities and trajectories of periodic motions in the 2-DOF nonlinear oscillator are used to illustrate motion complexity, and harmonic amplitude spectrums give harmonic effects on periodic motions of the 2-DOF nonlinear oscillator.

Adding and Subtracting Energy to Body Motion: Phase Oscillator

2014 ◽  
Author(s):  
Jason Kerestes ◽  
Thomas G. Sugar ◽  
Matthew Holgate
Keyword(s):  
Phase Angle ◽  
Metabolic Cost ◽  
Oscillatory Behavior ◽  
Body Motion ◽  
Phase Oscillator ◽  
Motion Energy

We are developing methods to add a bounded amount of energy to assist body motion. Energy is added based on the phase angle of the limb to create a “phase oscillator.” The energy is added assisting motion creating an oscillatory behavior. An anti-phase angle can be used to subtract energy from body motion as well. Using a “phase oscillator” controller, a powered hip exoskeleton assisted a runner and demonstrated a reduction in metabolic cost.

Sign in / Sign up

Export Citation Format

Share Document