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 invertedpendulums represents a challenging control problem, since it continually moves toward an uncontrolled state.
Introduction:
Invertedpendulum has been the subject of numerous studies in automatic control since the forties. The invertedpendulum is a typical representative of a class of high-order nonlinear and non-minimum phase systems [1]. Since the system is inherently nonlinear, it is useful to illustrate some of the ideas in nonlinear control.
Many diﬀerent control methods are proposed for the invertedpendulum problem. The Proportional-Integral-Derivative (PID) and Proportional-Derivative (PD) controllers, Model Predictive Control (MPC), and fuzzy control to mention a few. However one of the obstacles by using the PID and PD controllers are that they alone cannot eﬀectively control all of the pendulum state variables (modes) since they are of lower order...

...InvertedPendulum
Analysis, Design and Implementation
IIEE Visionaries
Document Version 1.0
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
4
W HAT'S INSIDE T HIS REPORT
(CONTENTS IN DETAIL)
CONTENTS IN DETAIL THE AUTHORS
ü ü
4
… … 3 6
ABOUT THE AUTHOR TECHNICAL ADVISOR
PREFACE INTRODUCTION
ü ü
… …
9 12
INTRODUCTION TO INVERTEDPENDULUM APPLICATIONS OF INVERTEDPENDULUM o SIMULATION OF DYNAMICS OF A ROCKET VEHICLE o MODEL OF A HUMAN STANDING STILL ü PROBLEM DESCRIPTION
MATHEMATICAL WORK
ü
…
19
MATHEMATICAL ANALYSIS o SETUP DESCRIPTION o INVERTEDPENDULUM SYSTEM EQUATIONS o ACTUATION MECHANISM o TRANSFER FUNCTION OF THE W HOLE SYSTEM ü SYSTEM P ARAMETERS
ANALYSIS OF UNCOMPENSATED SYSTEM
ü ü ü ü ü ü POLE ZERO MAP OF UNCOMPENSATED OPEN LOOP S YSTEM IMPULSE RESPONSE OF UNCOMPENSATED OPEN LOOP S YSTEM ROOT LOCUS OF THE UNCOMPENSATED S YSTEM STEP RESPONSE OF UNCOMPENSATED OPEN LOOP S YSTEM SIMULINK M ODEL FOR THE OPEN LOOP IMPULSE RESPONSE SIMULINK M ODEL FOR THE OPEN LOOP STEP RESPONSE
…
26
COMPENSATION DESIGN
ü HOW CAN THE COMPENSATION BE DESIGNED? (POSSIBLE OPTIONS)
…
34
ROOT LOCUS SYSTEM DESIGN
ü W HY COMPENSATION IS REQUIRED? ü...

...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.
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
* In a similar manner, periods of oscillation are determined by suspending the pendulum through the remaining holes on the same side of the centre of mass G of the bar. The bar is then inverted and periods of oscillation are determined by suspending the pendulum through all the holes on the opposite side of G. The distances d of the top edges of different holes from the end A of the bar are measured for each hole.The position of the centre of mass of the bar is found by balancing the bar horizontally on a knife edge. The mass M of the pendulum is determined by weighing the bar with an accurate scale or balance.
* A graph is drawn with the distance d of the various holes from the end A along the X-axis and the period T of the pendulum at these holes along the Y-axis. The graph has two branches, which...

...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 conveyed in the film and its value to teenagers.
Shakespeare's original play had the theme of love, appearance vs reality, madness, dramatic irony, through the portrayal of a female losing her brother and who is trying to make her way in life as a man because she knows that women could not make their way because the society back then was sexist, rude and extremely cruel. Andy Fickman's ‘She's the Man’ had the same relevancy as Shakespeare's play but did not quite reach the standard of Shakespeare's work , it seems like he wanted to get as close to Shakespeare’s original intent but just went in the complete opposite direction. He failed miserably and made a horrible movie.
She's the Man is about Viola Hastings, a girl whose passion is Soccer. She then finds out that the Girl's Soccer Program gets cancelled at her school. She then tries out for the guy's team, but they tell her that girls aren't good enough. So when Viola finds out that her fraternal twin brother, Sebastian is sneaking...

...Science
Vol.1,Issue.1/Oct. 2013
ISSN : 2347-5420
Research Papers
REAL TIME 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 RotaryInvertedPendulum is highly non-linear, open loop and unstable system.
Solving the operation of the Quanser rotaryinvertedpendulum is one of the classical and fundamental
problems in the area of control theory. The practical controller design and implementation for such a
system is a challenging task. The main objective is to design a stabilizing controller that balances the
invertedpendulum in the up-right position. This paper describes modern control technique that include
Full State Feedback (FSF) controller design to control the RotaryInvertedPendulum using LabVIEW
Interface.
KEY WORDS:
FSF (Full state-feedback), LabVIEW (Laboratory Virtual Instrumentation Engineering
Workbench), Quanser RotaryInvertedPendulum,...

...Report : Experiment 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 thependulum: T2= (4π2l)/g
with T the period in seconds, l the length in meters and g the gravitational acceleration in m/s2. Our raw
data should give us a square-root relationship between the period and the length. Furthermore, to find an accurate value for ‘g’, we will also graph T2 versus the length of the pendulum. This way, we will be
able to obtain a straight-line graph, with a gradient equal to 4π2g–1.
Procedure: Refer to lab manual.
Measurement / Data:
Length of Pendulum ( l +/- 0.1 cm) | Time for 20 Oscillations (s) | Time for 1 Oscillation (Periodic Time) T (s) | T^2 ( s^2) |
| 1 | 2 | Mean | | |
35 | 24.00 | 23.87 | 23.94 | 1.20 | 1.43 |
45 | 26.50 | 26.75 | 26.63 | 1.33 | 1.77 |
55 | 29.94 | 29.81 | 29.88 | 1.49 | 2.23 |
65 | 32.44 | 32.31 | 32.38 | 1.62 | 2.62 |
75 | 35.06 | 35.00 | 35.03 | 1.75 | 3.07 |
85 | 37.06 | 36.87 | 36.97 | 1.85 | 3.42 |
95 | 39.25 | 39.19 | 39.22 | 1.96 | 3.85 |
Length of...

...-------------------------------------------------
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 that it can swing freely.[1] When a pendulum is displaced from its restingequilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force combined with the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period. A pendulum swings with a specific period which depends (mainly) on its length.
From its discovery around 1602 by Galileo Galilei the regular motion of pendulums was used for timekeeping, and was the world's most accurate timekeeping technology until the 1930s.[2] Pendulums are used to regulate pendulum clocks, and are used in scientific instruments such as accelerometers and seismometers. Historically they were used as gravimeters to measure the acceleration of gravity in geophysical...