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Technical Briefs

# Friction and the Inverted Pendulum Stabilization Problem

[+] Author and Article Information
Sue Ann Campbell, Stephanie Crawford, Kirsten Morris

Department of Applied Mathematics, University of Waterloo, Waterloo, ON, N2L 3G1, Canada

J. Dyn. Sys., Meas., Control 130(5), 054502 (Aug 04, 2008) (7 pages) doi:10.1115/1.2957631 History: Received July 25, 2007; Revised April 03, 2008; Published August 04, 2008

## Abstract

We consider an experimental system consisting of a pendulum, which is free to rotate $360deg$, 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 aims to stabilize the pendulum in the upright position. We show that a controller designed using a simple viscous friction model has poor performance—small amplitude oscillations occur when the controller is implemented. We consider various models for stick slip friction between the cart and the track and measure the friction parameters experimentally. We give strong evidence that stick slip friction is the source of the small amplitude oscillations. A controller designed using a stick slip friction model stabilizes the system, and the small amplitude oscillations are eliminated.

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## Figures

Figure 1

Inverted pendulum system

Figure 2

Comparison of experiment with control gain K=Kvis and simulation of the pendulum cart system (Eq. 1) with the same controller and the exponential friction model

Figure 5

Implementation of the controller based on the dynamic friction model in the experiment (solid line) and in a numerical simulation of the system with initial condition (x,θ,ẋ,θ̇)=(0,0.02rad,0,0) (dashed line)

Figure 3

Simulations of the pendulum cart system (Eq. 1) with the dynamic friction model (Eqs. 5,6) and the control gain K=Kvis. (a) The Initial angle is chosen for comparison with experiment. (b) Two initial angles very close to zero are chosen to illustrate the bistability between the equilibrium point and the periodic solution.

Figure 4

Implementation of the controller based on the exponential friction model in the experiment (solid line) and in a numerical simulation of the system with initial condition (x,θ,ẋ,θ̇)=(0,0.06rad,0,0) (dashed line).

Figure 6

Results of the numerical simulations of Eq. 1 with the exponential friction model and corresponding feedback gain K=Kexp. Dashed lines correspond to using the friction model (Eqs. 3,4). Solid lines correspond to replacing sgn(ẋ) with tanh(100ẋ). Initial conditions used are as given in Eq. 12 with various values of θ0 for each time delay, τ. The bottom graph shows a zoom in of the top graph.

Figure 7

Results of the numerical simulations of the nonlinear system (Eq. 1) with the dynamic friction model and corresponding feedback gain. Initial conditions used are as given in Eq. 12 with various values of θ0 for each time delay, τ. The bottom graph shows a zoom in of the top graph.

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