Motion Control of Three Link Underactuated Manipulator by Bifurcation Control

2014 ◽  
Author(s):  
Hiroshi Yabuno ◽  
Kazuya Endo
Keyword(s):  
Feedback Control ◽  
Motion Control ◽  
Multiple Scales ◽  
Equilibrium Points ◽  
Pitchfork Bifurcation ◽  
Upright Position ◽  
State Feedback Control ◽  
Vertical Excitation ◽  
Underactuated Manipulator ◽  
Pitchfork Bifurcations

Motion control of a three link underactuated manipulator, whose first joint has an actuator and a sensor and second and third joints do not have actuator or sensor, is theoretically proposed without feedback control with respect to the motion of the free links. By using high-frequency vertical excitation, so called Kapitza pendulum is stabilized at the upright position without state feedback control. The phenomenon can be regarded as a subcritical pitchfork bifurcation. On the other hand, it is known that the horizontal excitation causes supercritical pitchfork bifurcation in a pendulum. Also, the inclination of excitation from the horizontal and vertical directions produces the perturbation of the complete supercritical and subcritical pitchfork bifurcations, respectively. In this paper, we apply the method of multiple scales to obtain the averaged equations governing the motion of the free links. We perform the bifurcation analysis of the free links and clarify the equilibrium points in the free links and their stability. Then, we propose a strategy to swing up the free links and to stabilize them at the upright position by actuating the perturbation of the pitchfork bifurcations based on the change of the inclination of excitation.

Swing-Up Control of a Three-Link Underactuated Manipulator by High-Frequency Horizontal Excitation

2012 ◽  
Vol 8 (1) ◽  
Author(s):  
Kazuya Endo ◽  
Hiroshi Yabuno
Keyword(s):  
High Frequency ◽  
Multiple Scales ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Averaged Equations ◽  
Underactuated Manipulator ◽  
High Frequency Excitation ◽  
Pitchfork Bifurcations ◽  
The Stability

In the present paper, we consider a three-link underactuated manipulator, the first joint of which is active and the second and third joints of which exhibit passive motion, on a plane inclined at slight angle from horizontal the plane. We analytically investigate changes in the stability of equilibrium points of the free links connected to the passive joints using high-frequency horizontal excitation of the first link. We derive autonomous averaged equations from the dimensionless equations of motion using the method of multiple scales. We clarify that the two free links can be swung up through pitchfork bifurcations and stabilized at some configurations by producing nontrivial and stable equilibrium points due to the high-frequency excitation. Furthermore, it is experimentally verified that increasing the excitation frequency multiplies stable and nontrivial equilibrium points.

scholarly journals PATTERNS OF OSCILLATION IN A RING OF IDENTICAL CELLS WITH DELAYED COUPLING

2007 ◽  
Vol 17 (09) ◽  
pp. 3109-3125 ◽  
Author(s):  
SHARENE D. BUNGAY ◽  
SUE ANN CAMPBELL
Keyword(s):  
Periodic Solutions ◽  
Nearest Neighbor ◽  
Equilibrium Points ◽  
Pitchfork Bifurcation ◽  
Codimension Two ◽  
Bifurcation Points ◽  
Pitchfork Bifurcations ◽  
Numerical Bifurcation ◽  
Secondary Bifurcations ◽  
Codimension Two Bifurcation

We investigate the behavior of a neural network model consisting of three neurons with delayed self and nearest-neighbor connections. We give analytical results on the existence, stability and bifurcation of nontrivial equilibria of the system. We show the existence of codimension two bifurcation points involving both standard and D3-equivariant, Hopf and pitchfork bifurcation points. We use numerical simulation and numerical bifurcation analysis to investigate the dynamics near the pitchfork–Hopf interaction points. Our numerical investigations reveal that multiple secondary Hopf bifurcations and pitchfork bifurcations of limit cycles may emanate from the pitchfork–Hopf points. Further, these secondary bifurcations give rise to ten different types of periodic solutions. In addition, the secondary bifurcations can lead to multistability between equilibrium points and periodic solutions in some regions of parameter space. We conclude by generalizing our results into conjectures about the secondary bifurcations that emanate from codimension two pitchfork–Hopf bifurcation points in systems with Dn symmetry.

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