A simple qualitative physical explanation is suggested for the phenomenon of subharmonic resonances of a rigid planar pendulum whose axis is forced to oscillate with a high frequency in the vertical direction. An approximate quantitative theory based on the suggested approach is developed. The spectral composition of the subharmonic resonances is investigated quantitatively, and the boundaries ...

This article describes two methods for controlling the surface of a liquid in an open container as it is being carried by a robot arm. Both methods make use of the fundamental mode of oscillation and damping of the liquid in the container as predicted from a boundary element model of the fluid. The first method uses an infinite impulse response filter to alter the acceleration profile so that t...

The harmonic oscillator is a paradigm of predictability. The resonance response of a harmonic oscillator can be exploited to make clocks accurate to 1 part in 10. Harmonic oscillations are normally observed in systems operating near equilibrium and governed by linear or “Hooke’s law” restoring forces. Nonetheless, for small enough excursions from equilibrium, systems governed by nonlinear resto...

Abstract We consider an experimental system consisting of a pendulum, which is free to rotate 360 degrees, attached to a cart. The cart can move in one dimension. We study the effect of friction on the design and performance of a feedback controller, a linear quadratic regulator, that aim to stabilize the pendulum in the upright position. We show that a controller designed using a simple viscou...

The Kinect sensor was introduced in November 2010 by Microsoft for the Xbox 360 video game system. The sensor unit is connected to a base with a motorized pivot. It was designed to be positioned above or below a video display and to track player body and hand movements in 3D space, which allows users to interact with the Xbox 360. The device contains a RGB camera, depth sensor, IR light source,...

The inverted pendulum is a canonical problem in both Nonlinear Dynamics and Control Theory. In this article, the phenomenon of dynamic stabilization of the vertically driven inverted pendulum is investigated experimentally and numerically. We resolve the first stabilizing boundary in driving parameter space, as well as investigate the effects of frictional damping on the dynamics of the pendulu...

Over the course of a few decades, the pendulum problem has become a prime example in the instruction of many concepts in linear and nonlinear dynamics. Particularly, the inverted pendulum has been illuminating in the areas of dynamical stability and control. Our intent with this dynamics problem was to explore the stability of the inverted state while vertically driving the pivot point. Forcing...

We address the safety-critical control problem for systems whose dynamics evolve on manifolds using the concepts of control Lyapunov functions (CLFs) and control Barrier functions (CBFs). We first extend the concepts of CLFs and CBFs from Cartesian spaces to manifolds, resulting in geometric CLFs and CBFs. We then formulate a state-dependent online quadratic program (QP) that imposes the constr...