# Introduction

he study of humanoid robot is currently one of the most exciting research projects. Even if some of those works have already demonstrated very reliable dynamic biped walking (Yamaguchi, Soga, Inoue & Takanishi, 1999; Hirai, Hirose, Haikawa & Takenaka, 1998; Nishiwaki, Sugihara, Kagami, Kanehiro, Inaba & Inoue, 2000), we believe it is still important to understand the mathematical theoretical background of biped locomotion. In standing, it has become common to consider the body as an (single\double\triple) inverted pendulum pivoted at the ankles. Moreover, up ride of a human shoulder is also considered as a motion of an (single\double\ triple) inverted pendulum (Jadlovská, 2011;Jadlovská & Jadlovská, 2010). An inverted pendulum system is a typically nonlinear, redundancy, uncertainty, strong coupling and natural characteristics of instabilities. All these features make it the ideal model of advanced control theory and typical experiment platform of test control results. There are a number of different kinds of the inverted pendulum systems presenting a variety of control challenges. The most common types are the single inverted pendulum on a cart (Ohsumi & Izumikawa, 1995;Åström & Furuta, 2000). the double inverted pendulum on a cart (Zhong & Rock, 2001), the double inverted pendulum with an actuator at the first joint only (Pendubot) (Graichen & Zeitz, 2005), the double inverted pendulum with an actuator at the second joint only (Acrobot) (Hauser & Murray, 1990), the light weight rotary pendulum (Brockett & Hongyi, 2003).

In this paper, we have addressed the problem of stability of double inverted pendulum in the upright position using the pole placement method. For this, we have assumed that the double inverted pendulum is pivoted at the lower end of inner arm (see figure 1). The first step to achieve the objective is to understand the dynamics of the system of double inverted pendulum by developing the mathematical modelling of the system. In modelling, we have used Euler-Lagrange formulation to find equation of motion. In the second step, we linearized this non-linear system of double inverted pendulum in the up-up position and builded up its linear state space model. The linearization is one of the most important issues for control of non-linear systems. There are lots of studies in the literature regarding linearization (Jordan, 2006;Wang, Chen & Zhou, 2000;Conga, Wanga & Hill, 2005). In the next step, the stability and controllability criteria showed that the system is unstable but it is controllable.

To control this unstable system, we have employed the pole placement method. In this method, the poles are the eigen-values of linear state space model and the calculated gain matrix places the poles at desired position to stabilize a system. In the simulation part of this paper, numerical and graphical simulations for control task are given to show the effectiveness of the proposed pole placement scheme.


# II.


# Mathematical Modeling of Dip

In this section, we will describe mathematical model necessary for stability and controllability analysis. The mechanism of the double inverted pendulum is shown in Figure 1 schematically. The mathematical model of DIP can be derived using the Euler-Lagrange equation. The form of the Euler-Lagrangian equation used here is:
d L L dt ? ? ? ? ? ? = ? ? ? (1)
Where L = T -V is a Lagrangian, T is kinetic energy, V is potential energy, ? = [? 1 ? 2 ] T is Figure 1 : the input generalized force vector produced by two actuators at the lower joint (ankle) and second at joint between to arm (knee), The kinetic and potential energies in terms of generalized coordinates can be determined as:
( ) ( ) ? ? ? ? ? + = + + + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?(2)
( )
1 1 1 2 1 1 2 cos 2 cos cos m gl V m g l l ? ? ? = + ? ? ? ? ? ?(3)
Differentiating the Lagrangian L = T -V by generalized coordinate's vector yields Euler-Lagrange equation (1) as:
( ) ( ) 2 2 2 2 2 4 4 cos 2 cos 2 sin 4 sin 11 1 2 1 2 1 2 2 2 2 1 2 1 2 2 2 2 2 2 1 2 2 2 2 1 2 2 1 2 2 sin sin 1 2 1 1 2 2 1 2 1 ? ? ? ? ? ? ? ? ? ? ? + + + + ? + + ? ? ? ? ? ? ? ? ? ? + ? ? = ?? ?? ? ? ? ( ) 2 2 2 2 cos 2 sin sin 2 1 2 2 2 2 1 2 2 2 2 2 1 2 2 1 2 2 1 2 2 m l l ? ? ? ? ? ? ? ? ? ? ? ? + + + + ? = ? ? ? ? ? ? ? ? ?? ?? ?
The matrix form of the system is ( )
2 2 2 2 2 4 4 cos 2 cos 0 2 sin 11 1 2 1 2 1 2 2 2 2 2 1 2 2 2 2 2 1 2 2 1 1 2 2 2 sin 0 2 2 cos 2 1 2 2 2 2 1 2 2 2 2 2 2 2 2 4 sin 2 sin 2 1 2 2 1 2 1 2 1 1 2 2 ? ? ? ? ? ? ? ? ? + + + + ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? + ? ?? ?? ? ? ? ( ) ( ) sin 1 2 1 sin 2 2 2 1 2 m gl ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? a) Linearisation of the System
The tracking controller of DIP is designed using the Gain Scheduling method based on the linearisation of the system equations around certain equilibrium points. In this paper, we linearize the system at the vertical unstable equilibrium by taking.
( ) ( ) 0, 1 2 cos cos 1 1 2 sin ; sin ; 1 1 2 2 0; cos 1; sin 1 2 1 2 1 2 1 2 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? m 1 g m 2 g ? 2 ?1
( )
A Year So, ( ) ( ) ( ) 2 2 2 2 4 4 2 11 1 2 1 2 1 2 2 2 1 2 1 2 2 2 2 2 1 2 1 1 2 2 1 2 1 2 2 2 2 1 2 2 2 1 2 2 2 2 2 2 1 2 2 m l I m l m l l m l m l l m l m m gl m gl ? ? ? ? ? ? ? ? ? + + + + + ? ? ? ? ? ? ? ? ? ? ? + ? ? = ? ? ? ? ? ? + + + ? = ? ? ? ? ? ? ? ? ?? ?? ?? ??
The matrix of the linear system ( )
2 2 2 2 4 4 2 11 1 2 1 2 1 2 2 2 2 1 2 2 2 1 2 2 2 2 2 1 2 2 2 2 2 2 2 1 1 1 2 1 2 2 2 2 ? ? ? ? ? ? + + + + ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? + = ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ??
where ( ) 
1 0 0 0 0 1 0 0 2 2 4 1 1 1 2 1 2 0 0 2 2 1 2 2 2 2 4 2 1 2 2 2 2 2 0 0 2 2 1 2 2 2 2 2 2 0 0 1 0 0 0 0 1 2 0 0 1 2 1 2 2 2 2 0 0 2 2 2 2 m l I m l M m l l m l m l l m l m l l m l m l I N m m gl m gl m gl m gl m gl ? ? ? ? ? ? ? ? + + ? ? = ? ? ? ? + + ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? = ? ? + + ? ? ? ? ? ? ? ? ? ? 0 0 0 1 0 0 1 T = ? ? ? ? ? ? ? ? ? ? ? ? ; ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ?.A = ? ? ? ? ? ? ? ? ? ? ? ? 0 0 0 0 B = -0.0328 -0.0908 -0.0908 -0.1775 ? ? ? ? ? ? ? ? ? ? ? ? . III.

# Stability and Controllability of System


# Stability Criterion

A system (state space representation) is stable iff all the eigenvalues of the matrix A are inside the unit circle.

The eigen value of A of our system are: 0.0000 + 1.9939i, 0.0000 -1.9939i,-1.0115, 1.0115 which are outside the unit circle because the modules of eigen values are greater than 1. So the system is unstable in absence of input force (? 1 = 0, ? 2 0).


# b) Controllability criterion

A system (state space representation) is controllable iff the controllable matrix C = [B AB A 2 B ?. A n-1 B] has rank n where n is the number of degrees of freedom of the system.

In our system, the controllable matrix C = [B AB A 2 B A 3 B] has rank 4 which the degree of freedom of the system. So, the system is controllable. If the system is controllable, then all set of distinct closed loop poles are assigned arbitrarily by output feedback gain (Kimura, 1975;Kimura, 1977;Wonham, 1967).


# b) State space model for the above linear system

The state space model equation for the system is

x Ax Bu y Cx Du
= + = + ? (4) 1 2 11 4 4 4 4 A M N B M T C I D u ? ? ?? ?? ? ? ? ? ? ?? ?? ?? IV.

# Pole Placement

Block diagram of pole placement is given in Fig. 2. If the linearized system considered is completely state controllable, then poles of the closed-loop system may be placed at any desired locations by means of state feedback through an appropriate state feedback gain matrix K. In this paper, we have used the following method to calculate state feedback gain matrix:

Pole placement with output feedback is displayed in Figure 2. In this paper, the reference signal, r, is taken zero. If an output feedback control u = ?Kx is applied to (4), the closed-loop system becomes ( )
x A KB x = ? ?
The poles assigned with output feedback are ? = {? 1 , ? 2 , ? 3 ,?, ? n }. The problem considered in this paper is finding gain matrix K for transferring the poles.


# a) Gain Matrix Scheduling Consider the controllability matrix C = [B AB

A 2 B ?. A n-1 B] which is an n×pn order matrix. If the system is controllable, then the rank(C) = n. That means, it has only n linearly independent columns among the pn columns. Therefore, there will be many ways to construct an n×n similarity matrix which will give a multi-input controllable canonical form. In this paper, we use the following way: Consider controllable matrix in n block as follows:
1 1 1 1 1 0 1 1 n n p p p C b b Ab Ab A b A b Block Block Block n ? ? = ? ? ? ? ? ? ? ? ??? ?
Starting from the left in, this matrix, check each column, keeping count of the number of linearly independent columns we encounter. We may stop counting when n linearly independent columns are obtained.

Let the block in which we find the last (i.e., the nth) linearly independent column be denoted by the (µ-1) th block. Then the first block in which there are no more independent columns will be the µ th block. This µ is controllability index. 
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
Using this inverse matrix of M, calculate transformation matrix T as follows:
1 1 1 1 1 1 2 2 1 2 2 2 1 m m A m T m A m p p p m A p p µ µ µ µ µ µ µ µ µ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
Using transfer matrix T, transferring of the matrix A and B are, 
0 A BK µ µ ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1 2 3 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 p µ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? or 0 1 2 1 0 1 0 0 0 0 1 0 0 0 0 1 n A BK ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
Solving above matrix equation we will get gain matrix K.


# b) Calculation of gain matrix

Applying above method, for the desired pole 0.1, ?0.1, 0.1i, ?0.1i, the gain matrix is 93.8377 24.8771 0 0 24.1188 24.3614 0 0
K ? = ? ? ? ? ? ? ? V.

# Simulation

In absence of the input forces, the angles and their velocities increase rapidly which make the system unstable (see figure 3). By giving input force with measurement of gain matrix, the angles and their velocities will be slow down which make the system stable at the desired equilibrium place (see figurer 4) . 


# Global


# Conclusion

This study aims to understand what causes humanoids to fall, and what can be done to avoid it. Disturbances and modelling error are possible contributors to falling. For small disturbances in the walk of humanoid robot, it is simply behaving like a double inverted pendulum. So the results of this paper will be used in the development of the humanoid robot.
![? 1 and ? 2 are angular positions of first arm, and second arm of the double pendulum.](image-2.png "")
![Values of Parameters The values of parameters for the given double inverted pendulum are assumed as follows: m 1 = mass of inner arm = 0.4 kg m 2 = mass of outer arm = 0.5 kg l 1 = length of inner arm = 5 m l 2 = length of outer arm = 5 m g = gravitational acceleration = 9.8 m/s 2 So the corresponding values of state space matrices are as follows:](image-3.png "c)")
2![Figure 2 :](image-4.png "Figure 2 :")


			© 2013 Global Journals Inc. (US) Double Inverted Pendulum
			© 2013 Global Journals Inc. (US) Block Diagram of Pole Placement
			© 2013 Global Journals Inc. (US)
			© 2013 Global Journals Inc. (US)
		
		
			
			
			

				


* 
	
		Swinging up a pendulum by energy control
		
			KÅström
		
		
			KFuruta
		
	
	
		Automatica
		
			36
			295
			2000
		
	




* 
	
		A Light Weight Rotary Double Pendulum: Maximizing the Domain of Attraction
		
			RWBrockett
		
		
			HongyiLi
		
	
	
		Proceedings of the 42nd IEEE Conference on Decision and Control
				the 42nd IEEE Conference on Decision and ControlMaui, Hawaii USA
		
			2003
		
	




* 
	
		
		
			KHirai
		
		
			MHirose
		
		
			YHaikawa
		
		
			TTakenaka
		
		
			System
		
		
	




* 
	
		Transient stability and voltage regulation enhancement via coordinated control of generator excitation and SVC, Electrical Power and Energy Systems
		
			LConga
		
		
			YWanga
		
		
			DJHill
		
		
			2005
			27
			
		
	




* 
	
		Non-linear feed forward and feedback tracking control with input constraints solving the pendubot swing-up problem
		
			KGraichen
		
		
			MZeitz
		
	
	
		Preprints of 16th IFAC world congress
				Prague, CZ
		
			2005
		
	




* 
	
		Non-linear controllers for non-integrable systems: The acrobat example
		
			JHauser
		
		
			RMurray
		
	
	
		Proc. of the 1998 ICRA
				of the 1998 ICRA
		
			American Control Conference
			1990. 1998
			
		
	
	The Development of Honda Humanoid Robot




* 
	
		Modeling and Optimal Control of Inverted Pendul Systems
		
			SJadlovská
		
	
	
		Diploma (Master)
				
			2011
		
	




* 
	
		Thesis. Supervisor: prof. Ing. Ján Sarnovský, CSc
				
	




* 
	
		Inverted Pendula Simulation and Modeling -a Generalized Approach
		
			SJadlovská
		
		
			AJadlovská
		
	
	
		Proceedings of the 9 th International Scientific -Technical Conference on Process Control
				the 9 th International Scientific -Technical Conference on Process ControlCzech Republic
		
			2010. June 7-10
		
		
			University of Pardubice
		
	




* 
	
		Linearization of non-linear state equation
		
			AJJordan
		
	
	
		Bulletin of The Polish Academy of Sciences Technical Science
		
			54
			2006
		
	




* 
	
		Pole Assignment by Gain Output Feedback
		
			HKimura
		
	
	
		IEEE Transactions on Automatic Control
		
			
			1975
		
	




* 
	
		A Further Result on the Problem of Pole Assignment by Output Feedback
		
			HKimura
		
	
	
		IEEE Transactions on Automatic Control
		
			
			1977
		
	




* 
	
		Design and Development of Research Platform for Perception-Action Integration in Humanoid Robot: H6
		
			KNishiwaki
		
		
			TSugihara
		
		
			SKagami
		
		
			FKanehiro
		
		
			MInaba
		
		
			HInoue
		
	
	
		Proc. Int. Conference on Intelligent Robots and Systems
				Int. Conference on Intelligent Robots and Systems
		
			2000
			
		
	




* 
	
		Non-linear control of swing-up and stabilization of an invertedpendulum
		
			AOhsumi
		
		
			TIzumikawa
		
	
	
		Proceedings of the 34th IEEE Conference on Decision and Control
				the 34th IEEE Conference on Decision and Control
		
			1995
			4
		
	




* 
	
		The control of underactuated mechanical systems
		
			MSpong
		
	
	
		Proceedings of the First international conference on mechatronics
				the First international conference on mechatronics
		
			1996
			
		
	




* 
	
		A non-linear controller design for SVC to improve power system voltage stability
		
			YWang
		
		
			HChen
		
		
			RZhou
		
	
	
		Electrical Power and Energy Systems
		
			22
			
			2000
		
	




* 
	
		On pole assignment in multi-input controllable linear systems
		
			WMWonham
		
	
	
		IEEE Transactions on Automatic Control
		
			12
			
			1967