Rotary Inverted Pendulum Model
C&DSP Lab. EE@MJU
which implies that 1 ¨ ˙ θ= −bc sin(α)α2 − cGθ ˙ ac − b2 cos2 (α) ηm ηg Kt Kg + bd sin(α) cos(α) + c V Rm 1 ˙ b2 sin(α) cos(α)α2 − b cos(α)Gθ ˙ α= ¨ 2 cos2 (α) ac − b η m η g Kt Kg + ad sin(α) + b cos(α) V Rm where a = J + mr2 , b = mLr, c = 4 mL2 , d = mgL, and 3 Fig. 1. A schematic diagram of the rotary inverted pendulum

2 ηm ηg Kt Km Kg . Rm

G=B+ Using the Lagrangian method, the equation of motion of rotary inverted pendulum can be derived as follows [1]: ¨ (J + mr )θ + mLr sin(α)α2 − mLr cos(α)¨ ˙ α ˙ = T − Bθ 4 2 ¨ ¨ 3 mL α − mLr cos(α)θ − mgL sin α = 0 2

Deﬁning x= θ α ˙ θ α ˙
T

, y= θ

α

T

, u=V

(1)

where T is the torque on the load from the motor, α is the pendulum angle, θ is the horizontal arm angle and other system parameters are given in Table I. In addition, the torque T is generated by DC motor such that [2] T = ηm ηg Kt Kg ˙ V − Kg Km θ Rm (2)

and linearizing about the upright position i.e. x = 0, yields 0 0 0 1 0 0 0 0 0 1 x + ηm ηg Kt Kg u (4) x= ˙ bd cG c R E 0 E −E 0 m η η Kt K 0 ad − bG 0 b mRg E g E E m

1 0 0 0 y= x 0 1 0 0

(5)

where E = ac − b2 . Using the system parameters of Table I, (4) becomes x = Ao x + Bo u ˙ y = Co x where 0 0 Ao = 0 0 1 Co = 0 0 0 41.68 84.05 0 0 1 0 1 0 −15.47 −14.89 0 . 0 0 0 1 , Bo = 0 , 27.13 0 0 26.12 (6)

where V is an applied armature voltage to DC motor.
TABLE I PARAMETERS OF ROTARY P ENDULUM Parameter J m r L g B Kt Kg Km Rm ηm ηg Description Moment of inertia at the load Mass of pendulum arm Rotating arm length Length to pendulum’s center of mass Gravitational constant Viscous damping coefﬁcient Motor torque constant SRV02 system gear ratio Back-EMF constant Armature resistance Motor efﬁciency Gearbox efﬁciency Value (SI) 0.0033 0.1250 0.2150 0.1675 9.81 0.0040 0.0077 70 0.0077 2.6 0.69 0.90

Now, we will design the proposed output feedback controller. Using Q =...

...Design of an LQG controller for two invertedpendulums
Objective:
The objective of this paper is to design linear quadratic controllers for a system with two invertedpendulums on 2-d Plane. To this goal, it has to be determined which control strategy delivers better performance with respect to pendulum’s angle and the cart’s position. The inverted...

...InvertedPendulum
Analysis, Design and Implementation
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Reference:
The work included in this document has been carried out in the Instrumentation and Control Lab at the Institute of Industrial Electronics Engineering, Karachi, Pakistan.
CONTENTS
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W HAT'S INSIDE T HIS REPORT
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CONTENTS IN DETAIL THE AUTHORS
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ABOUT THE AUTHOR TECHNICAL ADVISOR
PREFACE INTRODUCTION...

...Performing the real lab:
* The compound bar pendulum AB is suspended by passing a knife edge through the first hole at the end A. The pendulum is pulled aside through a small angle and released, whereupon it oscillates in a vertical plane with a small amplitude. The time for 10 oscillations is measured. From this the period T of oscillation of the pendulum is determined.
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...She isn’t a very good man! Andy Fickman’s She’s the Man did indeed hit the fan.
In this film review I will be going to talk to you about the relevance of the film to teenagers, the quality of film techniques and the quality of actor performances. I will also be talking about Shakespeare intent for the original play, and how is this achieved in the adaption of "She's the man". The social, moral of ethical message...

...One
Title: Determination of the acceleration due to gravity with a simple pendulum
Introduction and Theory: A simple pendulum performs simple harmonic motion, i.e. its periodic motion is defined by an acceleration that is proportional to its displacement and directed towards the centre of motion. It can be shown that the period T of the swinging pendulum is proportional to the square root of the length l of the pendulum: T2=...

...MODELLING AND BALANCE CONTROLLER DESIGN
FOR A ROTARYINVERTEDPENDULUM – USING LabVIEW
V.VIJAYALAKSHMI , Z.JENIFER AND ANDY SRINIVASAN
M.E. II year (C&I ), Valliammai Engineering College SRM Nagar, Kattankulathur Kancheepuram district
Asst. Professor Valliammai Engg., College SRM Nagar, kattankulathur Kancheepuram district
Prof & Head, E&I Dept.,Valliammai Engg., CollegeSRM Nagar, Kattankulathur Kancheepuram dist
Abstract
The Quanser...

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Pendulum
From Wikipedia, the free encyclopedia
This article is about pendulums. For other uses, see Pendulum (disambiguation).
"Simple gravity pendulum" model assumes no friction or air resistance. |
An animation of a pendulum showing the velocity and acceleration vectors (v and a). | |
A pendulum is a weight suspended from a pivot so...