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An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is unstable and falls over without additional help. It can be suspended stably in this inverted position by using a control system to monitor the angle of the pole and move the pivot point horizontally back under the center of mass when it starts to fall ...
A classic example along these lines is the inverted pendulum. [42] Other examples are vibrations of a string with periodically varying tension [41] stability of railroad rails as trains drive over them; seasonally forced population dynamics; the phenomenon of parametric resonance in forced oscillators; motion of ions in a quadrupole ion trap [43]
The Gudermannian function relates the area of a circular sector to the area of a hyperbolic sector, via a common stereographic projection.If twice the area of the blue hyperbolic sector is ψ, then twice the area of the red circular sector is ϕ = gd ψ.
The techniques have been extensively used to solve a wide range of problems such as those arising in UAV trajectory generation, missile guidance, control of robotic arms, vibration damping, lunar guidance, magnetic control, swing-up and stabilization of an inverted pendulum, orbit transfers, tether libration control, ascent guidance and quantum ...
After determining the correct path, Christiaan Huygens attempted to create pendulum clocks that used a string to suspend the bob and curb cheeks near the top of the string to change the path to the tautochrone curve. These attempts proved unhelpful for a number of reasons. First, the bending of the string causes friction, changing the timing.
The mathematical MIMO-model which is needed in order to simulate a ballbot and to design a sufficient controller which stabilizes the system, is very similar to an inverted pendulum on a cart. The LEGO NXT Ballbot, [ 38 ] Adelaide Ballbot, [ 40 ] Rezero [ 32 ] and Kugle [ 51 ] include actuator models in their robot models, whereas CMU Ballbot ...
Verlet integration (French pronunciation:) is a numerical method used to integrate Newton's equations of motion. [1] It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics.
The Van der Pol oscillator was originally proposed by the Dutch electrical engineer and physicist Balthasar van der Pol while he was working at Philips. [2] Van der Pol found stable oscillations, [3] which he subsequently called relaxation-oscillations [4] and are now known as a type of limit cycle, in electrical circuits employing vacuum tubes.