20th Iranian Conference on Electrical Engineering, (ICEE2012), May 15-17, Tehran, Iran

Adaptive Locally-linear-models-based Fault Tolerant Control for Humanoid Robot with Unknown Faults

Farzad Soltanian*, Ahmad Akbari alvanagh**, and Mohammad Javad Khosrowjerdi*** * Sahand University of Technology (SUT), Tabriz, Iran, Email: Csoltanian@ sut.ac.ir

** Sahand University of Technology (SUT), Tabriz, Iran, E-mail: a.akbari@sut.ac.ir *** Sahand University of Technology (SUT), Tabriz, Iran , E-mail: khosrowjerdi@sut.ac.ir

Abstract-Today the problem of fault tolerant control

(FTC) is considered as an important and essential

counterpart of control engineering systems. Because of

importance and existence of faults that don't have a known

structure in control system, Le., fault occurred because of

tangle of complex factors, in this paper a Humanoid Robot

with unknown faults is considered and a novel FTC

architecture is presented. A neuro/fuzzy model consisting of a

few locally linear models (LLMs) with on-line updated

centers and width vectors is used to approximate the fault

model. A linear estimator is employed to estimate the states

of the system that are inputs to LLMs. The stability analysis

of system is accomplished via Lyapunov theory, from which

the parameter updating rules are derived. At the end of this

paper some numerical simulations are given to show the

effectiveness of the proposed method.

Keywords: FTC, LLMs, Humanoid Robot, neuro!fuzzy model

I. INTRODUCTION

In this paper Gymnastic humanoid configurations of the Bioloid from Korean manufacturer robotic.com with unknown faults that can affect its performance is studied. The humanoid assumes the configuration of a human gymnast hanging on a bar. The resemblance is on purpose since it is our objective to have the robot doing a hand­stand on a high bar. Consequently, it was necessary to equip this robot with two additional hand grips, in order to allow it to hang on a bar, as well as a gyroscope plugged to a specified sensor board. The humanoid Robot has 18 degrees of freedom (DOF) powered by DC servos. The "brain" of the robot is the CM5 board and it is located in the back of the humanoid [I]. Fault detection and diagnosis and fault tolerant control is considered as a main part of engineering system because of high reliability demand in control systems.Observer based fault detection is used in many studies [2, 3]. The fundamental idea behind it lies in output estimation using some nonlinear observer. The error of estimation is used as residual in this case. Then the weighted residual is used to generate fault function. A L uenberger-like observer can be used to estimate output in a deterministic framework [4, 5]. In situations where deterministic information of the output can't be measured such as chemical engineering processes [6], neural network based observers that measure system output information based on stochastic value rather than deterministic value [7]-[9] can be useful. On the other hand, since most of the faults occurring in a

dynamic system have unknown structures, model based FDD approaches, which can be considered as the major group of proposed FDD design schemes will not be effective [9]. Therefore fault approximation approaches with the ability of learning and adaptation is used [11, 12]. Therefore, in this paper an adaptive neuro/fuzzy model, with the description, detailed in section IV, is used for estimation of the fault, the inputs of this LLMs are system states, an estimator is used to estimate system states. Then stability analysis is given via L yapunov theory and parameter updating rules are derived. In this paper a neuro/fuzzy model consisting of few locally linear models (LLMs) with on-line updated centers and width vectors is used to approximate the model of the fault. This paper is organized as follows. In section II, a humanoid robot is presented. In section III, a Luenberger observer is designed to estimate system states. In section IV, LLMs based model is introduced for detection and diagnosis of fault while considering the stability analysis and obtaining parameter updating rules. In section V, the numerical fault example is presented to illustrate the effectiveness of the proposed method followed by a conclusion in section VI.

II. REPRESENTATION OF HUMANOID ROBOT

An accurate static model of the Humanoid Robot can be obtained based on the physical properties of their components. Typically, by knowing the mass, center of mass and the inertia tensor of each element of the robot it is possible to get a quite reliable model that can be further used in simulation and control. In this configuration, the humanoid was rebuilt in order to resemble a gymnast exercising on a high bar with that purpose. Designed like this, the robot can be split into 3 main blocks, arms, torso and legs (Figure la) powered by 4 servo-actuators, two for the shoulders and two for the hips (Figure 1 b). The Gymnastic humanoid robot can be seen as being compound of three main blocks. One block representing the arms, a second block representing the torso and a third block representing the legs. The joints of the robot are therefore the hands, the shoulders and the hips. Both shoulders and hips are actuated by two servos in each side. The hands are not actuated indeed, and therefore the system can be approximated by a three link under actuated pendulum. In order to obtain the humanoid robot model the following steps will be taken:

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Figure 1- Bioloid humanoid

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I. Deduce the equations of motion of the system. 2. L inearize the system dynamics about the vertical unstable equilibrium and obtain state space representation model. 3. Analyze the system poles and zeros and confirm that it is a non-minimum phase system and therefore it is impossible to use classical cancelation techniques to stabilize it. For that reason following controller is designed:

u = -KTx Where

K = [-12 .7314 -4 .5027 -1 .6007 -2.1317 -9 .7655 -3.9673 -0.8361 -1 .6433 -0 .9918 -0.7766 -0.3678] -0.2602 For simplification, it is assumed that the centers of gravity of all the blocks (arms, torso and legs) in the model of the humanoid, and therefore in its equations of motion, are aligned with the joints (e.g. case of ideal 3-link pendulum). In this way, it will be possible to test the robustness of the control strategy used. In order to describe the dynamic behavior of this multi­body robotic system, it is necessary to determine its equations of motion. These equations can be derived from the classical Euler-L agrange equations, according to Figure 2. Where:

x

Figure2- Representation of the humanoid robot seen as an under actuated 3-link inverted pendulum

A=

B=

o o o 1 0 0 o 0 0 0 1 0 o 0 0 0 0 1

74.2389 -117.8764 0.0028 0 0 0 ' -81.5857 348.777 -76.07760 0 0

7.3471 -230.9113215.85910 0 0 o 0 o 0 o 0

-176.2065 97.5517 ,C = 16x6 527.7831 -467.2635

-467.2635 697.9566

Where x E lRl.nis the state vector, u E lRl.mis the input vector, and y E lRl.P is the output vector. [(x): lRl.n --> lRl.n is the unknown vector field that represents the nonlinear fault function of the system. The matrix function of f3(t - T) is unknown matrix that represents the time profile of fault, where T denotes the fault occurrence time. Without loss of generality, it is supposed that f3(t - T) has a diagonal form as:

f3(t - T) = diag(f3i(t - T)) (2) i = 1,2, ... , n

• qi is the angle of joint i in respect to the previous Where f3i denotes the function of affecting fault on ith link. state equation.

• mi is the mass of block i. • Ii is the inertia of block i. • �i is the torque actuated on the active joint i. • Ii is the length between joint i and joint i+ l. • lei is the length between joint i and the center of

gravity of the mass i. By using given linearization of a generalized n-link pendulum system [I], the state space model IS easily obtained, where:

III. DESIGNING ESTIMATOR

To generate and estimate the system states that are inputs of fault diagnosis LLMs, nonlinear estimator such as L ipschitz [3, 13], L uenberger like [14], and High gain [15] and so on can be used. In this section a linear estimator is designed in order to estimate the system states beforehand occurring faults in system. Therefore, the system equation can be rewritten as follows: x = Ax+Bu (3)

x = Ax + B(u + f3(t - T)[(x)) y = Cx

(1) y = Cx

The observer has the following form:

1001

i = Ax + Bu + G (y - y) y = Cx

(4)

Defining e = x - x, from (3) and (4), the following state error equation is obtained:

(5)

Where A l = A - GC is designed as a Hurwitz matrix.

IV. FAULT APPROXIMATOR

A L ocally linear Neuro/Fuzzy model (LLNFM) is used in this study to model the fault. A diagram of typical LLNFM with m'LLM and n input is shown in figure 3 where each LLM has the following structure [16]:

(6)

Corresponding to each LLM there exist an activation function which specifies validity degree of that LLM for different inputs. In situation which fault structure is unknown; LLM algorithm can be modified as follow: If fault occurred, from (1), state equation has a following structure:

x = Ax + B(u + [(x)) ,t > T Y = Cx And observer equation has a following structure:

i = Ai + B(u + [(x)) + G(y - 51) y = Cx

(7)

(8)

Where the out pout of [(x) is used to approximate [(x), where

m'

[(x) = I BWiV(x)Bi(x) (9) i= l

Wi: IRl.n x IRl.n+1 --7 IRl.n(n+l)is weight function, where i denote the i th locally linear models (LLMs) are used in neuro/fuzzy model, v(x): IRl.n x IRl.n+1 --7 IRl.n(n+l)is function of input vector where vex) = [1, xv ... , XnY andBi (x) E IRl. denote validity function ofLLM s, where:

B (x) - 11; (x) (10) ; -

I�'ll1i (x) Where:

l1i (x) = (exp (_12 Cl;i:il ) 2) ... exp (�2 Cn;:

in) 2)) denote activation function of ith LLM, d E IRl.P ,where

d = [df, ... ,d�, ]and p = nm',di = [dv ... ,dn], a E IRl.P, where a = [a[, ... , an and ai = [av ... , an]. By using LLM to estimate fault function, Equation (7) can be rewritten as follow (only for theoretical considerations):

x = Ax + B(u + l:Y!'1 WiV(X)EJi(X)) + (f,t � T Y = Cx

(II)

Where a is the approximation error ofLLMs. Therefore

Fig3- Diagram of LLNFM withm' LLMs and n inputs

Observer with fault has the following structure:

£ � AH B ( U + � W'V(X)B'(X)) + G(y - y)

Y = Cx (12)

Assumptiont: Exist an ideal weight matrix W;, such asllw;l1 ::::; Wi and lIall ::::; (j for all xEAd C IRl.n, i = 1, ... ,m', where w;and (j are positive constant and Adis compact set.

Assumption2: suppose 11[(x)ll::::; f for all xEAd C IRl.n, where 1 is positive constant.

Assumption 3: system states are bounded and fault occurs after estimation of system states, because estimated states are considered as inputs of fault approximator.

The error equation is obtained as follows: e = Ale +

m' m'

B I w;v(x)ei(x,d,B) - I Wiv(x)e;(x,d,a) - 0 = ;= 1 ;= 1 Ale - 0 +

B I�'l(WiV(X)ei(X, d, B) - Wi v (x)e;(x, d, a)) (13)

The following relation is defined:

Wi = Wi- Wi V; = Vi - V; 8 = B;(x,d,a) - Bi(x,d,a)

(14)

By substituting eq. (14) in (13) the error eq. is obtained as

e = Ale + B (I�l WiV(X)e;(x, d, B) -WiV(X)ei(X, d, a)) + B I�l wiv(x)8; - 0 (15)

By expanding Taylor's series of Bi(x, d, a) at (d, B) The following formula is obtained:

1002

· 68· . 68· Where Bid. = -' Id·=d·,BUr· = -' 10"·=0'· l 8di [ l [ o(Ji l l

And d = d - d, if = 8 - (J, then by using above relation error equation can be rewritten as follow: e = Ale +

- (B (f (u\i? Bid) 1=1

+

(17) Where: ( m, m' m' )

1\= B I u\i? Bid;di + I wii? BiO';ifi - I WiVei i=l i=l i=l L emmal: if the following parameter updating equations are employed, then 1\ will be bounded:

(\I = {-YwLT eyeT (x)iY (1 + fJ - Ilwdl2) ,IIWi 112 > fJ + 1, (\IiWi > 0 , -YwLT eyeT (X)VT else (18)

d = {-YdBid,vTwTLT ey(1 + fJd -llddI2) ,lldil12 > 1 + fJd,didi > 0 ' -YdBid,vTwTLTey else

(19)

(20)

Theorml: for the nonlinear system (12), if parameter updating rules of (18), (19), (20) are used and L satisfies

PB = CT L, then error equation (17) will be bounded.

Proof: The L yapunov candidate can be selected as follows:

m' m' 1 1 I �T� 1 I -T-V = -ePe + -tr( w· w·) + -tr( d· d·) 2 2y:

1 1 2y 1 1

w i=l d i=l m'

+ � tr(""" ifT if·) 2y: L 1 1 0" i=l (21 )

By differentiation from (21), following equation is obtained:

V � -�eTQe + eTp (�(w,"e'(£.d.8l m' m'

(22)

By using assumption 3, expanding Bi(x, d, (J)at (d,8), and equation 15, the following equation obtained; for more detail consult with [3, 17].

V::; -�eTQe + eTp(-(c5 +1\)) 1

::; -ZeTQe + IleTIIIIPIIIIc5 +1\11 -1

::; TAmin(Q) IIel12 + IleTIIIIPIIIIc5 +1\11

= _II 112(�A . (Q) _ IIPIIIIc5 +1\11 e 2 mm

+ lIell

(23)

Ifllell > Am�x(P)116+i\llis designed, then by using theorem 2"Am;n(Q)

I,V < 0, also d, if are bounded.

v. SIMULATION EXAMPLE

In order to show effectiveness of this method, following example is investigated. For simulation, following parameters are designed:

10 0 0 1 0 0 o 11 0 0 1 0

G = 0 0 12 0 0 1 74.2389 -117.8764 0.0028 1 0 0' -81.5857 348.7777 -76.0776 0 5 0

7.3471 , -230.9113 215.8591 0 0 6

1003

° ° ° °

L= ° ° -88.1032 48.77584 52.7783 -46.72635

-38.9386 58.16305

Initial values that asserted on system are:

d;E[ -0.4,0.4], (J;E[0.4,0.6], i = 1,2,£2 = 4,K = 0.1, Y w = 0.012, Yd = 0.1, yO" = 0.1, m' =5,

Figure 4 shows the system states and its estimation without fault. In figure 5, applied torques to system is sketched, Figure 6 illustrates nonlinear fault and estimated fault. Figures7 and 8 sketch the system states and estimation with FTC, and applied torques to system respectively. The states and estimation states and applied torques to system after occurring fault and FTC are respectively shown in Figures 9 and 10.

10

·10

,20

·ED ·100 ·1 ED

0

Fx1l �

4 6 lime(s)

Q "' '''' xha13

4 6 lime(s)

� "" "'xha15

4 6 lime(s)

10

10

10

V -x2 ""'" xhal2

40 o

100

ED

20 o

·20 40 ·60 ·BJ

o

4 6 lime(s)

-x4 """" xhal4

4 6 lime(s)

-x6 "" "'xha16

4 6 lime(s)

10

10

10

Figure4- states and estimation states before occurring fault

1004

10 '

m�o--����--�--75 --�--��--��1O time(s)

Figure5- Applied torques

Figure6-fault and estimated fault, state error estimation, Where fault asserted on 61h state is [(x) = 2sin (O,St)

r;; • •••• �l o 5 10 15 20 25 3J 35 4J

:V ,

• '"," , • L�l l o 5 10 15 20 25 3J 35 4:1

JV : : '"," : : §j o 10 15 20 26 3J 35 4J

lime(s)

1:[ : : : ,

: : 1-", :�a�� o 5 10 15 20 25 3J 35 4J

lime(s)

:dr�:"--------:,::--------:,::--------:,::--------:,::--------:,:�I :�:�aI 5 � 10 , 5 20 25 3J 35 4J

lime(s)

�r : : : : : : E:�I� a 5 10 15 20 25 3J 35 4J time(s)

Figure7- states and estimation states after occurring fault and FTC

�t: : : : �l o 5 10 15 20 25 3J 35 4J

time(s)

if: : : : � 1 o 5 10 16 20 25 3J 35 4() tima(s)

Figure8- Applied torques after occurring fault and FTC

2<l

� . . . . . � . � ' :� o 5 10 15 20 25 3J 35 4J

time(s)

o [ � :� 10 o

o 5 10 15 20 25 3J 35 4J time(s)

·10 � 2<l � o 5 10 15 20 25 3J 35 4J

time(s)

�L : : : : : : 1-,:�a�11 o 5 10 15 20 25 3J 35 40

time(s)

. : �r- : : : : : : o 5 10 15 20 25 3J time(s)

�r : : , : : : 1-' :�aJ 10 15 20

time(s) 25 35

Figure9- states and estimation states after occurring fault without FTC

�� o 5 10 15 20 25 3J 35 4J time(s)

�� ·10 0�----�5 -----'� 0----�'5�---=20�---= 25�--���--�3�5---- -"�' time(s)

Figure 10- Applied torques after occurring fault without FTC

VI. CONCLUSION

A novel method for fault tolerant control was introduced for humanoid robot in order to resemble gymnast exercising on high bar, based on LLNFM and its stability was analyzed using Lyapunov theory and updating rules for LLN FM was extracted. The proposed method utilized unempirical controller which its parameters were invariable during FTC which in-turn caused increasing the system speed. Numerical simulations results verified the effectiveness of the proposed approach in fault tolerant situations.

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