scholarly journals Oscillatory Behavior of the Rate of Escape through an Unstable Limit Cycle

1996 ◽  
Vol 77 (24) ◽  
pp. 4860-4863 ◽  
Author(s):  
Robert S. Maier ◽  
D. L. Stein
Keyword(s):  
Limit Cycle ◽  
Oscillatory Behavior ◽  
Unstable Limit Cycle ◽  
Rate Of Escape

scholarly journals Design of active disturbance rejection controller for compass-like biped walking

2018 ◽  
Vol 15 (3) ◽  
pp. 172988141877684 ◽  
Author(s):  
Sumian Song ◽  
Chong Tang ◽  
Zidong Wang ◽  
Yinan Wang ◽  
Gangfeng Yan
Keyword(s):  
Limit Cycle ◽  
Disturbance Rejection ◽  
Controller Design ◽  
Linear Time ◽  
Biped Robot ◽  
Transverse Coordinate ◽  
Linear Time Invariant ◽  
Active Disturbance Rejection ◽  
Unstable Limit Cycle ◽  
Coordinate Control

This article proposes an active disturbance rejection controller design scheme to stabilize the unstable limit cycle of a compass-like biped robot. The idea of transverse coordinate transformation is applied to form the control system based on angular momentum. With the linearization approximation, the limit cycle stabilization problem is simplified into the stabilization of an linear time-invariant system, which is known as transverse coordinate control. In order to solve the problem of poor adaptability caused by linearization approximation, we design an active disturbance rejection controller in the form of a serial system. With the active disturbance rejection controller, the system error can be estimated by extended state observer and compensated by nonlinear state error feedback, and the unstable limit cycle can be stabilized. The numerical simulations show that the control law enhances the performance of transverse coordinate control.

Steinkamp's Toy Can Hop 100 Times But Can't Stand Up

2017 ◽  
Vol 9 (1) ◽  
Author(s):  
Gregg Stiesberg ◽  
Tim van Oijen ◽  
Andy Ruina
Keyword(s):  
Stability Analysis ◽  
Experimental Observation ◽  
Limit Cycle ◽  
Periodic Motion ◽  
Largest Eigenvalue ◽  
Unstable Limit Cycle ◽  
Stable Attractor ◽  
Linearized Stability ◽  
The Stability ◽  
Basic Motion

We have experimented with and simulated Steinkamp's passive-dynamic hopper. This hopper cannot stand up (it is statically unstable), yet it can hop the length of a 5 m 0.079 rad sloped ramp, with n≈100 hops. Because, for an unstable periodic motion, a perturbation Δx0 grows exponentially with the number of steps (Δxn≈Δx0×λn), where λ is the system eigenvalue with largest magnitude, one expects that if λ>1 that the amplification after 100 steps, λ100, would be large enough to cause robot failure. So, the experiments seem to indicate that the largest eigenvalue magnitude of the linearized return map is less than one, and the hopper is dynamically stable. However, two independent simulations show more subtlety. Both simulations correctly predict the period of the basic motion, the kinematic details, and the existence of the experimentally observed period ∼11 solutions. However, both simulations also predict that the hopper is slightly unstable (|λ|max>1). This theoretically predicted instability superficially contradicts the experimental observation of 100 hops. Nor do the simulations suggest a stable attractor near the periodic motion. Instead, the conflict between the linearized stability analysis and the experiments seems to be resolved by the details of the launch: a simulation of the hand-holding during launch suggests that experienced launchers use the stability of the loosely held hopper to find a motion that is almost on the barely unstable limit cycle of the free device.

The Effects of Friction in Axial Splines on Rotor System Stability

1993 ◽  
Vol 115 (2) ◽  
pp. 272-278
Author(s):  
A. F. Artiles
Keyword(s):  
Friction Coefficient ◽  
Limit Cycle ◽  
Stable Limit Cycle ◽  
Threshold Value ◽  
System Stability ◽  
Stable Limit ◽  
Spin Speed ◽  
Side Load ◽  
Unstable Limit Cycle ◽  
Rotor Bearing

An in-depth parametric evaluation of the effects of Coulomb friction in an axial spline joint on the stability of the rotor-bearing system was conducted through time transient integration of the equations of motion. The effects of spin speed, friction coefficient, spline torque, external damping, imbalance, and side load as well as asymmetric bearing stiffnesses were investigated. A subsynchronous instability is present at the bending critical speed when the spin speed is above this critical. The limit cycle orbit is circular, proportional to the product of the friction coefficient and spline torque (μT), inversely proportional to the external damping, and independent of spin speed. When imbalance is applied to the rotor, beating between the subsynchronous natural frequency and the synchronous (spin speed) frequency occurs. The subsynchronous component of the orbit is proportional to μT, while the synchronous component is proportional to the imbalance. When a static side load is applied, the unstable node at the center of the orbitally stable limit cycle grows into an ellitpical orbitally unstable limit cycle, separating stable from unstable regions of the phase plane. Below a threshold value of side load, the transient motion approaches one of two asymptotic solutions depending on the initial conditions: the larger stable limit cycle or a point at the center of the smaller unstable limit cycle. Beyond the threshold value of side load, the rotor-bearing is stable and all motions decay to a point. Asymmetry in the bearing stiffnesses reduces the size of the subsynchronous whirl orbit.

scholarly journals The Effects of Friction in Axial Splines on Rotor System Stability

1991 ◽  
Author(s):  
Antonio F. Artiles
Keyword(s):  
Friction Coefficient ◽  
Limit Cycle ◽  
Stable Limit Cycle ◽  
Threshold Value ◽  
Stable Limit ◽  
Spin Speed ◽  
Side Load ◽  
Unstable Limit Cycle ◽  
Rotor Bearing ◽  
Bearing System

An in depth parametric evaluation of the effects of Coulomb friction in an axial spline joint on the stability of the rotor-bearing system was conducted through time transient integration of the equations of motion. Effects of: spin speed, friction coefficient, spline torque, external damping, imbalance and side load as well as asymmetric bearing stiffnesses were investigated. A subsynchronous instability is present at the bending critical speed when the spin speed is above this critical. The limit cycle orbit is circular, is proportional to the product of the friction coefficient and spline torque (μT), is inversely proportional to the external damping and is independent of spin speed. When imbalance is applied to the rotor, beating between the subsynchronous natural frequency and the synchronous (spin speed) frequency occurs. The subsynchronous component of the orbit is proportional to μT, while the synchronous component is proportional to the imbalance. When a static side load is applied, the unstable node at the center of the orbitally-stable limit cycle grows into an elliptical orbitally-unstable limit cycle, separating stable - from unstable regions of the phase plane. Below a threshold value of side load, the transient motion approaches one of two asymptotic solutions depending on the initial conditions: the larger stable limit cycle or a point at the center of the smaller unstable limit cycle. Beyond the threshold value of side load the rotor-bearing system is stable and all motions decay to a point. Asymmetry in the bearing stiffnesses reduces the size of the subsynchronous whirl orbit.

Two disc dynamos with viscous friction and time delay

1972 ◽  
Vol 71 (1) ◽  
pp. 135-153 ◽  
Author(s):  
A. E. Cook
Keyword(s):  
Time Delay ◽  
Numerical Integration ◽  
Limit Cycle ◽  
Direct Method ◽  
Viscous Friction ◽  
Singular Points ◽  
Unstable Limit Cycle

AbstractTwo extensions of the Rikitake model are examined; the first includes viscous friction and the second includes time delay. By use of Liapounov's direct method the viscous system is shown to have an unstable limit cycle associated with two of the three singular points. By numerical integration true ‘reversals of field’ are shown to occur for both the viscous and time delay cases.

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.

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