chapter 2shodhganga.inflibnet.ac.in › ... › 105099 › 6 › 06_chapter2.pdf · 2018-07-03 ·...

Chapter 2

Double Inverted Pendulum (A pre-robotic problem)

PP. 20-44

Mahesh A. Yeolekar 21

2.1 Introduction

At least from last five decades, the inverted pendulums have been the

most basic benchmark for researches in robotics. Balancing an

inverted pendulum in upright position is a problem of sensitive control

which is an important subset of problems in the field of robotics. The

balancing a stick on the fingertip is the simple example of invited

pendulum as shown in the Fig 2.1.

http://people.kth.se/~crro Fig 2.1 Balancing a stick

The inverted pendulum is a pendulum which has its centre of mass

above its fixed end as shown in Fig 2.2.

www.dofware.com (a) (b)

Fig 2.2(a) A inverted pendulum with pivot point set up on a cart (b) A inverted pendulum with pivot point set up on a fixed base

Modeling and Stability of Robotic Motions

22

The main objective of this problem is to stable the link of inverted

pendulum in vertically straight direction using the feedback control by

computing gain matrix. The stability of inverted pendulum can be

applicable to the following problems:

The problem of robotic arm where the centre of gravity lies

above the centre of pressure as shown in Fig 2.3(a).

The problem of missile or rocket guidance where the centre of

gravity situated above the centre of drag which indicating the

aerodynamic instability as shown in Fig 2.3(b).

The problem of self-balancing unicycle and segway where the

device keeps upright automatically by the controller as shown in

Fig 2.3(c) and (d).

http://iq.intel.com http://spaceref. com/onorbit/ (a) Robotic Arm (b) Rocket Guidance

http://www.orthoconcept.ch/de http://www.focus.de/auto/motorrad

(c) Unicycle (d) Segway Fig 2.3 Applications of inverted problem


The problem can be varied by changing the number of links of the

pendulum. The inverted pendulum with more than one link is called

multilink inverted pendulum. If the human body is separated by the

joints, then it can be considered as the multilink inverted pendulum

when it is in the standing position. The humans are capable to stand

straight which distinguish them from other animals. Moreover, it is an

important part of human bipedal locomotion, but to stand stable is not

an easy mechanism as it looks. In humans, the central nervous system

and muscles are maintaining balance, but it is not same for the

artificial human. So, the study of multilink inverted pendulum is

important to understand the biped locomotion.

In this chapter, we will consider the human body as the double

inverted pendulum (2-link inverted pendulum) where a link of fixed

end is considered as a lower part (below the hip) of body and a free

movable link is considered as an upper of body (above the hip) as

shown in Fig 2.4.

http://swaggercise.weebly.com

Fig 2.4 Human body considered as Double inverted pendulum


24

The double inverted pendulum (DIP) has two links, one above another

as shown in Fig 2.4. The inverted pendulum is a multivariable

nonlinear, highly unstable, uncontrolled system. There are two

standard ways to controlled it:

1. by moving the base

2. by applying the torque at the fixed end.

In the first method, the double inverted pendulum is placed on a

moving cart. This mechanism is inspired by the idea of balancing a

stick on a fingertip where we are balancing a stick in the upright state

by moving over hand. In the same way, the cart is moving linearly to

make stable the double inverted in the upright state as shown in Fig

2.5.

http://www.imath-asia.com/assets/index.php/quanser

Fig 2.5 Double inverted pendulum on a cart

In the second method, the base of the double inverted pendulum is

fixed at a point. To stabilize this system, the external torque is giving

by the motor joint at the fixed point. It is similar to Human posture in


quiet standing because in standing, human body is fixed at ankle and

mussels give the additional force to stand stable as shown in Fig 2.6.

http://clinicalgate.com/

(a) (b) Fig 2.6 (a) Human posture in quite standing (b) Relative positions of double inverted pendulum

In this chapter, we will discuss about the second approach to control

the double inverted pendulum because it is appropriate for

understanding of human locomotion. Various researches have been

done in this area, so a brief literature review is given in the next

section.

2.2 Literature review

The literature survey is about the existing methods to control and

stabilize the double inverted pendulum in upright unstable equilibrium

position. The system of a double inverted pendulum is a typically

nonlinear and natural characteristic of instabilities. So it is an ideal

model to test methods of advanced control theory. Kailath (1950) used


26

the inverted pendulum for educating stability of open-loop unstable

systems by linear feedback control theory. But the credit of finding

first solution of this problem goes to Roberge (1960) and then

Schaefer and Cannon (1966) got success for controlling the inverted

pendulum. Mori et. al. (1976) designed the PD controller by using

state space model for controlling the inverted pendulum. The neural

network was used to stabilize the inverted pendulum by Anderson

(1989) and fuzzy-logic used by Yamakawa in the same year. The

problem of double inverted pendulum has also got considerable

attention. The controller for the double inverted pendulum was

designed by Furuta et. al. (1980). It controlled the position of the

supporting cart on a sloping bar. Maletinsky (1981) made controller

which can work without direct measurement of upper arm angle of the

double inverted pendulum. Spong (1995) applied partial

feedback linearization techniques for controlling a double pendulum

without a cart. We noticed that the following three most popular

under-actuated mechanisms for controlling and stabilization of the

double inverted pendulum are available in the literature:

The double inverted pendulum on a cart with actuator joint with

cart explained in detail by Zhong et. al. (2001) as shown in Fig

2.6.


Pendubot, a double inverted pendulum with an actuator at the

first joint only, was described in depth by Spong (1996) and

Graichen et. al. (2005) which displayed in Fig 2.7.

http://coecsl.ece.illinois.edu/pages/pendubot.html

Fig 2.7 Pendubot

Acrobot, the double inverted pendulum with an actuator at the

second joint only, was illustrated as nonlinear controllers by

Hauser (1990), as shown fig 2.8.

http://www.thegadgetshop.co.za/products_list.php?main_cat_id=21

Fig 2.8 Acrobot

In this chapter, we have focused on the problem of controllability and

stability of double inverted pendulum in the upright unstable position

using the pole placement method. We considered the double inverted

pendulum which is pivoted at the lower end of inner arm as shown Fig

2.4. We organised this chapter in the following three sections:


28

First section contains the mathematical modelling which

described the dynamics of the system of double inverted

pendulum. The equation of motion of the double inverted

pendulum is obtained by the Euler-Lagrange formulation.

Second section includes the linearized state space model of the

nonlinear system of the double inverted pendulum. The

linearization is the key issue for controlling the nonlinear system

which is discussed by Wang et. al. (2000), Conga et. al.(2005)

and Jordan (2006).

Third section discusses about the stability and controllability

criteria which helps to control the system of the double inverted

pendulum. Considering these criteria, we used pole placement

method to control the system. In this method, the eigen-values

of the state space model are considered to be the poles for the

system in s-space. The gain matrix will be used to place the

poles at the desired position to stable a system. The numerical

and graphical illustrations are given to check the impact of

proposed pole placement method.

2.3 Mathematical Modeling

In this section, we will describe the mathematical model for the motion

of double inverted pendulum. The schematic diagram of the double

inverted pendulum is sketched in the Fig 2.9.


Fig 2.9: Schematic diagram of double inverted pendulum model

The required assumptions for modelling are listed below:

The system contains two identical rods whose masses are

concentred at the centres of their rods. So, it is considered as

two points mass system.

The lower end of a lower arm is fixed at a point which is called

pivot point of the system and its upper end is jointed with lower

end of upper arm as shown in the Fig 2.9.

It motions under the gravitational force and the actuator is

available at a pivot point to control its motion.

It is a 2D model that means the pendulum motions only in the

vertical plane.

m1g

m2g

2

1


30

We concentrated here on the simplicity of the model than its physical

realizability which is done by the above assumptions.

2.3.1 Governing Equations

Before generating the motion equation, we set up some notations: In

this model, the configuration of double inverted pendulum is described

by 1 2

T where 1 and 2 are the angles made by the lower and

upper arms with vertical line respectively as shown in Fig 2.9 and

1 2 1 2

Tx

stands for the state space vector. T and V

represent the kinetic and potential energy respectively, 1 2

T

denotes the external torque vector.

The mathematical model of DIP can be derived using the Euler-

Lagrange equation. Considering the following form of the Euler-

Lagrangian equation

d L Ldt

(2.1)

where L T V is a Lagrangian, T is kinetic energy, V is potential

energy, 1 2

T is the input generalized force vector produced by

two actuators at the lower joint (ankle) and second at joint between to

arm (knee), 1 2

T is generalized coordinate vector where 1 and

2 are angular positions of first arm, and second arm of the double


pendulum. The kinetic and potential energies in terms of generalized

coordinates can be determined as:

2 2

1 1 1 1

2 2 2 2 2

2 1 1 1 2 2 1 2 2 2

12

14 4 cos

2

m l I

m l l l l IT

(2.2)

1 1 1

2 1 1 2

cos

2 cos cos

m glV

m g l l

(2.3)

Differentiating the Lagrangian L T V by vectors of generalized

coordinate system yields Euler-Lagrange equation (2.1) as:

2 2 2 21 1 1 2 1 2 1 2 2 2 2 1 2 1 2 2 2 2 2

22 1 2 2 2 2 1 2 2 1 2

1 2 1 1 2 2 1 2 1

4 4 cos 2 cos

2 sin 4 sin2 sin sin

m l I m l m l l m l m l l m l

m l l m l lm m gl m gl

2 2 22 1 2 2 2 2 1 2 2 2 2 2 1 2 2 1

2 2 1 2 2

2 cos 2 sin

sin

m l l m l m l I m l l

m gl

The matrix form of the system is given by the following equation:

,M N G

where M is the inertia matrix, the matrix ,N contains terms of

centrifugal and coriolis forces, G includes terms of gravity as given

below:


32

2 2 2 2

1 1 1 2 1 2 1 2 2 2 2 2 1 2 2 2 22 2

2 1 2 2 2 2 2 2 2

4 4 cos 2 cos2 cos

m l I m l m l l m l m l l m lM

m l l m l m l I

2 1 2 2

2 1 2 2

2 1 2 2 1 2 1 2 1 1 2 2 1 2

2 2 1 2

0 2 sin,

2 sin 0

4 sin 2 sin sinsin

m l lN

m l l

m l l m m gl m glG

m gl

2.3.2 Linearized state space equation

In this chapter, we have used the gain scheduling method to design

the track controller of DIP. This method requires the linearized state

space system. So by considering the vertically straight position of DIP

as an equilibrium point, the system can be linearized at the equilibrium

point by taking

1 2

1 2

1 1 2 2

1 2 1 2 1 2 1 2

2 21 2

0,cos cos 1sin ; sin ;

0; cos 1; sin

0

With respect to the above values, the equations of linear system are:

2 2 2 21 1 1 2 1 2 1 2 2 2 2 1 2 2 2 2

1 2 1 1 2 2 1 2 1

2 22 1 2 2 2 1 2 2 2 2 2 2 1 2 2

4 4 212

2

m l I m l m l l m l m l l m l

m m gl m gl

m l l m l m l I m gl


The matrix form of the linear system

2 2 2 21 1 1 2 1 2 1 2 2 2 2 1 2 2 2 1

2 22 1 2 2 2 2 2 2 2

1 12 2 1 2 2 2 2

2 22 2 2 2

4 4 22

2

m l I m l m l l m l m l l m lm l l m l m l I

m m gl m gl m glm gl m gl

The state space model equation for the system is

x Ax Buy Cx Du

(2.4)

4 4 4 41 1where , , , 0A M N B M T C I D

2 21 1 1 2 1 2

2 1 2 2 222 1 2 2 2

2 22 1 2 2 2 2 2 2

1 0 0 00 1 0 0

40 0 2

40 0 2

m l I m lMm l l m l

m l l m lm l l m l m l I

1 2 1 2 2 2 2

2 2 2 2

0 0 1 00 0 0 1

2 0 00 0

Nm m gl m gl m gl

m gl m gl

0 0

0 0

1 0

0 1

T

;

1

2

1

2

x

; 1

2

u

.


34

2.4 Stability and Controllability of the system

The state space model (2.4) can also be written in the following form

x t Ax t Bu t

y t Cx t Du t

(2.5)

where , ,A B C and D are time invariant. Therefore, the system of

double inverted pendulum can be considered as continuous time

invariant linear system. Moreover, it is capable to balance itself by

calculating gain matrix without human assistant so it can be thought

as an autonomous system. However, all such continuous linear time

invariant systems are not controllable and stable. Next we discuss

criteria required for stability and controllability of the system.

2.4.1. Stability

The double inverted pendulum is said to be stable if it is stay vertically

straight and capable to manage its balance during the small

perturbation. The stability of the system can be analyzed by the

following to different approaches:

first, by the poles of the system which are the eigenvalues of a

matrix A in (2.5),

second, the system’s Lyapunov stability which does not require

the eigenvalues.


In this chapter, we have used first method for analyzing the stability of

the system of double inverted pendulum.

Stability criterion A continuous time invariant linear system (2.5) is

stable if and only if all the eigenvalues of the matrix A are inside the

unit circle.

2.4.2 Controllability

The study of controllability is an important part of any control system.

It plays a key role in various control problems, such as a problem of

optimal control or a problem of stabilization of unstable systems by

feedback control method. Although, there is no exact definition of

controllability because it varies with respect to the class of models

applied. In broader sense, we can consider that the controllability

represents the moving capability of the system in the region of its

configuration space by using only certain allowable manipulations. The

unstable system can be controlled if it satisfies the following criterion:

Controllability criterion A continuous time invariant linear system (2.5)

is controllable if and only if the controllable matrix

2 1nP B AB A B A B has rank n where n is the number of

degrees of freedom of the system.


36

The above condition of controllability shows that the initial value ( x t

in (2.5)) of a state vector can be reached to the final desired output

value ( y t in (2.5)) within some finite time interval. In other words, it

explains the capacity of an external input ( u t in (2.5)) to place the

internal state of a system from an initial state ( x t in (2.5)) to a

desired final state ( y t in (2.5)) within some finite time interval. Note

that the system is controllable, it does not mean that once it reached

the desired state place and maintained there, but it means that it can

be reached.

2.4.3 Pole placement Method

The objective of the pole placement method is to set the closed-loop

poles of linear continuous time invariant system at the desired

locations in s-plane by using feedback control. The poles of the system

are directly associated to the eigenvalues of the system and so the

placing of poles is desirable as the eigenvalues of the system must be

inside the unit circle for the stable system, in other words, the

characteristics of the response of the system are controlled by

eigenvalues. The pole placement method is applicable to the system if

it has the following properties:

The system should be state controllable.


The state variables should be measurable and accessible for

feedback.

The input for controller should be unconstrained.

The algorithm for the pole placement method is given below:

i. The output feedback control vector u t in equation (2.5) can be

constructed in the form u t Kx t where K is called the gain

matrix of the system. Note that the gain matrix K is calculated

in such a manner that poles will be located on the desired place.

ii. The state space system (2.5) is reduced to the following system

x A KB x which is called closed loop system.

iii. Gain matrix scheduling Consider the poles assigned with output

feedback as 1 2 3, , , , .n Now the problem is finding gain

matrix K for transferring the poles at the desired places. The

controllability matrix 2 1nP B AB A B A B which is an

n pn order matrix and the system is controllable, so

rank P n . That means, it has only n-linearly independent

columns among the pn-columns. Therefore, there are many ways

to construct an n n -similarity matrix which will give a multi-

input controllable canonical form. In this chapter, we use the

following technique:


38

Consider, controllable matrix in n block as follows:

1 1

1 1 1

0 1 1

n n

p p pP b b Ab Ab A b A b

Block Block Block n

Starting from the left of this matrix, check each column, keeping

count of the number of linearly independent columns we

encounter. We may stop counting when it reaches to n-linearly

independent columns. Denote this last block of nth-linearly

independent column by 1 th block. Then, the first block in

which there are no more independent columns will be the th

block. This is controllability index. Rearranging this selected n-

linearly independent column 1 11 2 1 2 1 2, , , , , , , ,pb b b Ab Ab A b A b

we will get the invertible matrix M as:

1 21 1 2 2

11 1 p

p pM b A b b A b b A b

where 1i i p are the controllability indices of ,A B .

The inverse of M is


11

1 1

21

12 2

1

m

m

m

M m

mp

mp p

such that 1 ,MM I where 11 1 10, 0,1, , 2km A b k ,

1

1

11 1 1m A b

, 11 2 10, 0,1, , 2km A b k ,

11 3 10, 0,1, , ,km A b k .

Using this inverse matrix of M , calculate transformation matrix T

as follows:


40

1 1

111 1

2 2

122 2

1

m

m A

m

Tm A

mp p

pm Ap p

Using desired poles {λ1, λ2, λ3,…, λn}, the transferred canonical

form of the system is

1 ,A T AT B TB

Using desired poles {λ1, λ2, λ3,…, λn}, the transferred canonical

form of the system is

0 1 2 1

0 1 0 0

0 0 1 0

0 0 0 1

n

A BK

or


1

2

1 2 3

1 2 3

0 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0

0 1 0 00 0 0 0 0 0 0 00 0 1 00 0 0 0 0 0 0 0

0 0 0 10 0 0 0 0 0 0 00 0 0 0 0 0 0 0

0 0 0 00 0 0 0

0 0 0 00

A BK

1 2 3

0 1 0 00 0 0 00 0 1 00 0 0 0

0 0 0 10 0 0 00 0 0 0 0 0 0 p

Solving the above matrix equation we will get gain matrix.

Block diagram of pole placement is displayed in Fig. 2.10.

Fig 2.10: Block diagram of pole placement

+ r u x y Linearized system

x Ax Bu

C

Gain matrix K


42

2.5 Simulation Results

Assumed values of parameters of the given double inverted pendulum

for the simulation are given below:

1m = mass of inner arm = 0.4 kg

2m = mass of outer arm = 0.5 kg

1l = length of inner arm = 5m

2l = length of outer arm =5m

g = gravitational acceleration = 29.8m s

So the corresponding values of state space matrices are as follows:

0 0 1 0

0 0 0 1

0.8276 1.4206 0 0

4.1012 2.1247 0 0

A

;

0 0

0 0

0.0328 0.0908

0.0908 0.1775

B

.

2.5.1 Stability of the system in absence of any external force

The eigenvalue of A of our system are: 0.0000 1.9939 ,i

0.0000 1.9939i , 1.0115 , 1.0115 which are outside the unit circle so

the system is unstable in absence any input force 1 20, . . 0, 0u i e

as shown in Fig 2.11.


Fig 2.11: Unstable system

2.5.2 Controllability of the system

With the above values of parameters, the controllable matrix

2 3P B AB A B A B has rank 4which is the degree of freedom of

the system, so the system is controllable

.


44

2.5.3 Results of pole placement method

The system is controllable so we can apply pole placement method to

control it. For the desired poles0.1, 0.1, 0.1 , 0.1i i , the calculated gain

matrix is

93.8377 24.8771 0 0

24.1188 24.3614 0 0K

By giving input force with measurement of gain matrix, the angles and

their velocities will be slow down which makes the system stable at

the desired equilibrium position as shown in Fig 2.12.

Fig 2.12 Controlled system

chapter 2shodhganga.inflibnet.ac.in › ... › 105099 › 6 › 06_chapter2.pdf · 2018-07-03 ·...

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