Developments and Applications of Adaptive Cerebellar Model ... · The Cerebellar Cortex A model of the cerebellar cortex (1969 Marr ) 1. Introduction of Cerebellar Model Articulation

Developments and Applications of Adaptive Cerebellar Model Articulation

Controller

Chih-Min Lin林志民

Yuan Ze University, IEEE Fellow , IET Fellow

2

Content1. Introduction of Cerebellar Model Articulation Controller (CMAC)2. Missile Guidance Law Design Using CMAC3. Linear Piezoelectric Ceramic Motor (LPCM) Control Using Adaptive CMAC4. Recurrent CMAC Control for Unknown Nonlinear Systems

Referred Papers

5. RCMAC Fault Accommodation Control of a Biped Robot

3

The Cerebellar Cortex

A model of the cerebellar cortex (1969 Marr )

1. Introduction of Cerebellar Model Articulation Controller (CMAC)

GranuleCell

Layer

Mossy FiberFeedback from Limbs

Mossy Fiber Inputfrom Higher Centers

Selection of ActiveParallel Fibers

-

-

+

+

+

+

+

+

+

PurkinjeCells

StellateCells

BasketCells

AdjustWeights

ClimbingFiber Input

Output

AdjustableWeight Synapses

Summation ofSynaptic Influence

1. Information is stored in overlap layers

2. Quickly recall of the stored information

4

1w

2w

3w4w

Rnw

kw

Rna

ka

1a2a

3a4a

5a 5w

LearningAlgorithm

oy

dy

Input Space Q

Output

Weight MemorySpace W

Association Memory Space A

Commands fromHigher Level

Feedback fromSensors

AaAbBaBbCc

Ef

Hh

Sum ofSelected Weights

Referenceof Output

Receptive-FieldSpace T

Original Cerebellar Model Articulation Controller

The basic concept of an original CMAC (Albus 1971).

5

Mapping : Transforms the input vector into an association memory selection vector .

AQ qAqa )(

Tnk Raaaa )](,),(,),(),([)( 21 qqqqqa

Mapping : Each location of A corresponds to a receptive-field (binary receptive-field).

TA

Mapping :WT T

nk Rwwww ],,,,,[ 21 w

Output computation :oy

k

n

kk

To way

R

1

)()()( qwqaq

(1.1)

(1.2)

(1.3)

6

1 Variable q

A B

HG

FE

DC

a

h

g

f

e

d

c

b

1

1

2

2 3

3

4

5

5

2 Variable q

Layer 1

Layer 1 Layer 2Layer 2Layer 3

State (2,2)

Layer 3

Layer 4

Layer 4

4

Bb

Cc

Ee

Gg

Layer 1

Layer 2

Layer 3

Layer 4

The schematic representation of a 2-D CMAC. (binary receptive-fields )

1 Variable q

A B

HG

FE

DC

a

h

g

f

e

d

c

b

1

1

2

2 3

3

4

4

5

5

2 Variable q

Layer 1


State (2,2)

Layer 3

Layer 4

Layer 4

Bb

Cc Ee Gg

Non-differentiable receptive-fieldsNon-smooth function approximationNo stability analysis

7

A B

IHG

FED

C

a

i

h

g

f

e

d

c

b

71

1

2

2 3

3

4

4

6

5

65

Layer 1


Gg

State (3,3)7

Layer 3

J K L

l

k

j

8

9

8 9Layer 4

Layer 4

Jj 1 Variable q

2 Variable q

Bb

Ee

The schematic diagram of a general 2-D CMAC. General Cerebellar Model Articulation Controller

Differentiable receptive-fieldsSmooth function approximationStability analysis

8

Gaussian receptive-field basis function :

2

2)(exp)(ik

ikiiik v

mqq

Multidimensional receptive-field function:

n

i ik

ikin

iiikkkk v

mqqb1

2

2

1

)(exp)(),,( vmq

(1.4)

(1.5)

9

The output of CMAC

The receptive-field Bc with including a Gaussian receptive-field basis function.

Rn

kkkkk

To bwy

1),,(),,( vmqvmqw

Ca

A

B

C

cb

aBc

Ba

Aa

AcAb

Bb

Cb

Cc

1 Variable q

2 Variable q

(1.6)

10

1q

nq

Input Space Q

Outputoy

Receptive-Field Space T

Weight Memory Space W


k1

nk

kb kw

CMAC Neural Network

Layer 1

Layer 2 Layer 3 Layer 4

1x

2x

2jy 3ky

4oy

Good generalization capabilityLess computationBetter approximation ability

11

Formulation of Missile-Target Engagement

IZ

IX

IY

MXMYMZ

my

mx

m

ty

tx t

tm

mm

Target

Missile

Ground tracker

mRtR

xaycazca

The 3-D missile-target pursuit diagram.

2. Missile Guidance Law Design Using CMAC

12

The motion of the missile in the inertial frame.

(2.1)mmvmmczcmmcycmmmmczcmmmcycm

vmmczcmmcycmxm

mmcmmmczc

mmcmmmcycmmxm

mmcmmmczc

mmcmmmcycmmxm

vgvava

vava

gaaaza

aaya

aax

/cos/cos/sin

)cos/(sin)cos/(cos

coscoscossinsin)cossinsinsin(cos

)coscossinsin(sinsincos)sinsincossin(cos

)sincoscossin(sincoscos

Target

Missile

LYLZ

LX

LOSGround tracker

mR

pR1e

2e

P

13

m

m

m

tmtmtmtmtm

tmtm

zyx

ee

)cos()sin()sin()cos()sin(0)cos()sin(

2

1

The missile position in the LOS frame.

(2.2)

14

The tracking error dynamic equation

)(),(),(

)(),(),(),(),(

),(),(

2221

1211

2

1

2

1

ttt

ttGtGtGtG

tFtF

ee

uxGxF

uxxxx

xx

(2.4)

Define

Tzcyc

T

Tmmmmmmmm

T

aauu

zyxzyx

xxxxxxxx

],[],[

],,,,,,,[

],,,,,,,[

21

87654321

u

x

(2.3)

15

The CMAC control system.The control law:

CCMACuuu (2.5)

CMACe

dtd

AdaptationLaws

21 ˆ,ˆ ww

21,

CMACu

CompensationController

CuuTarget

Maneuvertt , Calculation of the

Tracking Errormm ,

MissileManeuver

Limiter

Adaptive CMAC Control System

16

A feedback linearization control law

]),()[,(1 eKeKxFxGu pvT tt

Substituting (2.6) into (2.4), yields 0eKeKe pv

(2.6)

(2.7)

The minimum approximation error TjjjCMACjj uu ),(

*wq

CMAC control system )()(ˆ)()ˆ,( quqwquwquu C

TCCMACT

Error equation])[,( TCCMACpv t uuuxGeKeKe

(2.9)

(2.10)

(2.8)

17

),(),(00

),(),(00

),(),,(),(

2221

121121

tGtG

tGtGttt mmm

xx

xxxGxGxG

whereTeeee ],,,[ 2211 E

22

11

001000000010

vp

vp

kk

kk

The error dynamics in the state-space form

])[,(

])[,(

])[,(

2222

1111

TCCMACm

TCCMACm

TCCMACm

uuut

uuut

t

xG

xGE

uuuxGEE

(2.11)

18

Theorem 2.1: If the control law is designed as (2.5), in which the adaptation law of CMAC

and the compensation control is designed as

jmjT

wjjj PGEww ~ˆ

then the stability of the guidance system can be guaranteed.

)sgn( mjT

jCju PGE

(2.12)

(2.13)

19

222

111

~~2

1~~2

121)( wwwwPEE T

w

T

w

TtV

The Lyapunov equation RPP T

Taking the derivative of the Lyapunov function.

222222

111111

22

2

11

1

~][

~][

~~1~~121)(

T

m

T

Cm

T

T

m

T

Cm

T

T

w

T

w

T

uu

tV

wPGEPGEwPGEPGE

wwwwREE

Proof: A Lyapunov function is defined as

(2.14)

(2.15)

(2.16)

20

Choosing (2.12), then (2.16) can be simplified

||

||21)(

2222

1111

m

T

Cm

T

m

T

Cm

TT

u

utV

PGEPGE

PGEPGEREE

Setting (2.13), then (2.17) can be rewritten as

021

)|(|)|(|21)(

222111

REE

PGEPGEREE

T

m

T

m

TTtV

(2.17)

(2.18)

21

Numerical SimulationsThe target motion model.

ttvtzt

tttyt

vttzt

tttzttyt

tttzttyt

vga

vagaz

aay

aax

/)cos(

)cos/(cos

sinsincos

cossinsin

For scenarios 1 and 2:for the first 2.5 sec until interception

vty ga 5 vtz ga vty ga 5 vtz ga 5

For scenarios 3: for the first 2.5 sec until interception

vty ga 0vty ga 5.0

vtz ga

vtz ga

(2.19)

22

The feedback linearization guidance law ]),()[,(1 eKeKxFxGu pvT tt

where ,140014

vK

490049

pK

Adaptive CMAC-based guidance law

The design parameters are set as follows:

,

16830083588000016830083588

R ,1521 ww 01.021

(2. 20)

(2.21)

23

Engagement scenario 1 with feedback linearization guidance law.

0 1 2 3 4 5 6 7-2

0

2

4

0 1 2 3 4 5 6 7-4

-2

0

2

1e

2e

(sec) Time

(sec) Time

0 1 2 3 4 5 6 7-300

-200

-100

0

100

200

0 1 2 3 4 5 6 7-100

0

100

200

300

)(m

/sec

2

yca)

(m/s

ec

2zca

(sec) Time

(sec) Time

0

1000

2000

3000

0

2000

4000

60000

200

400

600

800

1000

1200

(m)

z

(m)x(m)

y

MD=4.4539m

y trajectorMissile

jectoryTarget trapointIntercept

24


0 1 2 3 4 5 6 7-2

0

2

4

6

0 1 2 3 4 5 6 7-0.5

0

0.5

1

1.5

2

1e

2e

(sec) Time

(sec) Time

0 1 2 3 4 5 6 7-400

-200

0

200

0 1 2 3 4 5 6 7-400

-200

0

200

400

)(m

/sec

2

yca)

(m/s

ec

2zca

(sec) Time

(sec) Time

0

2000

4000

6000

0

100

200300

4000

1000

2000

3000

4000

(m)

z

(m)x(m)

y

MD=3.758m

y trajectorMissile

jectoryTarget tra

pointIntercept

25


0 1 2 3 4 5 6 7 8 9-2

0

2

4

6

0 1 2 3 4 5 6 7 8 9-20

-10

0

10

20

1e

2e

(sec) Time

(sec) Time

0 1 2 3 4 5 6 7 8 9-400

-200

0

200

400

0 1 2 3 4 5 6 7 8 9-400

-200

0

200

400

)(m

/sec

2

yca)

(m/s

ec

2zca

(sec) Time

(sec) Time

0

2000

4000

6000

0

2000

4000

60000

2000

4000

6000

8000

(m)

z

(m)x(m)

y

MD=1.8434m

y trajectorMissile

jectoryTarget tra

pointIntercept

26

Engagement scenario 1 with adaptive CMAC-based guidance law.

0 1 2 3 4 5 6 7-2

0

2

4

0 1 2 3 4 5 6 7-4

-2

0

2

1e

2e

(sec) Time

(sec) Time

0 1 2 3 4 5 6 7-200

-100

0

100

200

0 1 2 3 4 5 6 7-200

-100

0

100

200

)(m

/sec

2

yca)

(m/s

ec

2zca

(sec) Time

(sec) Time

0

1000

2000

3000

0

2000

4000

60000

200

400

600

800

1000

1200

(m)

z

(m)x(m)

y

MD=0.5737m

y trajectorMissile

jectoryTarget trapointIntercept

27


0 1 2 3 4 5 6 7-4

-2

0

2

4

6

0 1 2 3 4 5 6 7-2

0

2

4

1e

2e

(sec) Time

(sec) Time

0 1 2 3 4 5 6 7-400

-200

0

200

400

0 1 2 3 4 5 6 7-400

-200

0

200

400

)(m

/sec

2

yca)

(m/s

ec

2zca

(sec) Time

(sec) Time

0

2000

4000

6000

0

100

200300

4000

1000

2000

3000

4000

(m)

z

(m)x(m)

y

MD=1.5612m

y trajectorMissile

jectoryTarget tra

pointIntercept

28


0 1 2 3 4 5 6 7 8 9-4

-2

0

2

4

6

0 1 2 3 4 5 6 7 8 9-15

-10

-5

0

5

1e

2e

(sec) Time

(sec) Time

0 1 2 3 4 5 6 7 8 9-400

-200

0

200

400

0 1 2 3 4 5 6 7 8 9-400

-200

0

200

400

)(m

/sec

2

yca)

(m/s

ec

2zca

(sec) Time

(sec) Time

0

2000

4000

6000

0

2000

4000

60000

2000

4000

6000

8000

(m)

z

(m)x(m)

y

y trajectorMissile

jectoryTarget tra

pointIntercept

MD=0.2781m

29

0.27811.56120.5737Adaptive CMAC-based Guidance

Law

1.84343.7584.4539Feedback

LinearizationGuidance Law

Scenario 3Scenario 2Scenario 1Scenario

GuidanceLaw

Comparison of Miss-distance

30

Summary

CMAC guidance law can achieve satisfactory performance for different engagement scenarios.

CMAC guidance law performs better than the feedback linearization guidance law.

31

3. Linear Piezoelectric Ceramic Motor (LPCM) Control Using Adaptive CMAC

Structure of LPCM

Unknown dynamic equation);()();();()( txdtutxgtxftx (3.1)

32

Adaptive CMAC Control System

The adaptive CMAC control law

CCMAC uuu (3.2)

CMAC

dtd

AdaptationLaws

ikik vm ˆ,ˆ,ŵ

CMACu

CompensatedControl

Cu

u x

Linear Piezoelectic Ceramic Motor Drive System

*x

dx+

cvmw ,,,

+ +e

LC ResonantInverter

Linear PiezoelectricCeramic Motor

PerformaceIndex

ReferenceModel

r

33

Tracking errorxxe d

Performance index dekeketr t 012 )()(

Ideal control law]);();([);( 12

1 ekektxdtxfxtxgu dT

Substituting (3.5) into (3.1), then0)( 12 ekeketr

(3.3)

(3.4)

(3.5)

(3.6)

34

CT

CCMAC uuuu wvmwq ˆ)ˆ,ˆ,ˆ,(

Adaptive CMAC control

Error equation])ˆ,ˆ,ˆ,()[;()( CCMACT uuutxgtr vmwq

(3.7)

(3.8)A minimum approximation error

TTCMACT uuu***** ),,,( wvmwq (3.9)

According to (3.9), (3.8) can be rewritten as

]~)[;(

])ˆ()[;()( *

CT

CT

utxg

utxgtr

w

ww

(3.10)

35

Theorem 3.1: The adaptive CMAC control law is designed as (3.8), in which the adaptation law

then the stability of the control system can be guaranteed.

with bound estimation algorithm given in

and the compensated control is designed as);()(ˆ txgtrww

)];()(sgn[ˆ txgtruC

|);()(|ˆ txgtrc

(3.11)

(3.12)

(3.13)

36

Proof: A Lyapunov function is chosen as 22 ~

21~~

21)(

21)(

cT

wtrtV ww

ˆ~1ˆ~1]~)[;()()(c

T

wC

T utxgtrtV www

Substituting (3.11)-(3.13) into (3.15), gives

0|);()(||)|(

ˆ]ˆ[1);()(|||);()(|

~~1);()();()()(

txgtr

utxgtrtxgtr

utxgtrtxgtrtV

cC

cC

(3.14)

(3.15)

(3.16)

37

According to the gradient descent method,

kCMAC

wk

wkwk buV

wVbtxgtrw

ˆ

);()(̂

The adaptation laws of means and variances

21

1

)()(2ˆ);()(

ˆ

ik

ikik

n

kkm

ik

ik

ik

kn

k k

CMAC

CMACmik

vmqbwtxgtr

mb

bu

uVm

R

R

(3.17)

(3.18)

(3.19)

R

R

n

k ik

ikikkv

n

k ik

ik

ik

k

k

CMAC

CMACvik

vmqbwtxgtr

vb

bu

uVv

13

21

)()(2ˆ);()(

ˆ

38

)];(sgn[)(ˆ txgtrww

)];(sgn[)](sgn[ˆ txgtruC

|)(|ˆ trc

21 )(

)(2ˆ)];(sgn[)(ˆik

ikik

n

kkmik v

mqbwtxgtrmR

Rn

k ik

ikikkvik v

mqbwtxgtrv1

3

2

)()(2ˆ)];(sgn[)(ˆ

The adaptation laws can be reconstructed as (3.20)

(3.21)

(3.22)

(3.23)

(3.24)

The in the tuning algorithms can be reorganized as in practical applications.

);( txg)];(sgn[|);(| txgtxg

39

dx

ParallelI/O

EncorderInterface

and Timer

D/AConverter

Servo Control Card

Personal Computer

DigitalOscilloscope

CW,CCW

u

x

LinearScale

Linear PiezoelectricCeramic Motor Moving Table

LC ResonantDrivingCircuit

x

Control Computer andServo Control Card

LC ResonantDriving System

DigitalOscilloscope

Linear Scale

Linear PiezoelectricCeramic Motor

MovingTable

The PC-based experimental control system.Experimental Results

40

Block diagram of PI position control system.

eV

u x

dx LC ResonantInverter+

Linear Piezoelectric CeramicMotor Drive System

PK

dtd

sKI

SK Linear Piezoelectric

Ceramic MotorReference

Model

*x

+ ++

x

The parameters of the PI position control system are chosen as follows:

,20SK ,1PK 25IK

36.7313.17s36.73

2 2222

sss nnn

The reference model for the periodic step command:

(3.25)

41

Experimental results of PI position control for LPCM due to periodic step command.

Reference Model

Table Position

Tracking Error

Control Effort

Start

0 cm

4.5 cm

2 sec

Reference Model

Table Position

Start

0 cm

4.5 cm

2 sec

Start

0 cm

2 sec

0.5 cm Tracking Error

Start

0 cm

2 sec

0.5 cm

Start

0 V

2 sec

5 V Control Effort

Start

0 V

2 sec

5 V

(a)

(b)

(c)

(d)

(e)

(f)

No load caseLoad case

42

Reference Model

Table Position

Start

0 cm

4.5 cm

2 sec

-4.5 cm

Reference Model

Table Position

Start

0 cm

4.5 cm

2 sec

-4.5 cm

Tracking Error

Start

0 cm

2 sec


Start

0 cm

2 sec

0.5 cm

Control Effort

Start

0 V

2 sec

5 V Control Effort

Start

0 V

2 sec

5 V

(a)

(b)

(c)

(d)

(e)

(f)

Experimental results of PI position control for LPCM due to sinusoidal command.

43

The adaptive CMAC control Parameters

,04.0w 01.0c,02.0 vm

20ikv]35,25,15,5,5,15,25,35[

],,,,, ,,[ 87654321

iiiiiiii mmmmmmmm

The initial values of the parameters are chosen as

,251 k ,102 k

,4 ,5En 422 Rn

]/)(exp[)( 22 ikikiiik vmqq

The receptive-field basis functions are chosen as

,42Bn

44

Reference Model

Table Position

Start

0 cm

4.5 cm

2 sec

Reference Model

Table Position

Start

0 cm

4.5 cm

2 sec

Tracking Error

Start

0 cm

2 sec


Start

0 cm

2 sec

0.5 cm

Control Effort

Start

0 V

2 sec

5 V Control Effort

Start

0 V

2 sec

5 V

(a)

(b)

(c)

(d)

(e)

(f)

Experimental results of robust CMAC control for LPCM due to periodic step command.

45

Reference Model

Table Position

Start

0 cm

4.5 cm

2 sec

-4.5 cm

Reference Model

Table Position

Start

0 cm

4.5 cm

2 sec

-4.5 cm

Tracking Error

Start

0 cm

2 sec


Start

0 cm

2 sec

0.5 cm

Control Effort

Start

0 V

2 sec

5 V Control Effort

Start

0 V

2 sec

5 V

(a)

(b)

(c)

(d)

(e)

(f)

Experimental results of robust CMAC control for LPCM due to sinusoidal command.

46

Summary

The successful development of adaptive CMAC control system.

The successful application of adaptive CMAC control for an LPCM.

47

4. Recurrent CMAC Control for Unknown Uncertain Nonlinear Systems

Problem FormulationThe nth-order nonlinear dynamic system

xytdtugfx n )()()()()( xx

The tracking error vector is defined asTneee ],,,[ )1( E

(4.1)

(4.2)The ideal control law

])()([)(

1 )( EKxx

TndI xtdfg

u (4.3)

48

The error dynamics0)1(1

)( ekeke nnn

Recurrent CMACArchitecture of a recurrent CMAC

(4.4)

1q

nq

Input Space Q

Receptive-Field Space T

Output

oy

Weight Memory Space W


1z

1z

1rw

rnw

kwkb

1rq

nrq

k1

nk

49

The inputs of every block are represented as

nTnrrrr qqq ],,,[ 21 q

)1( Nyorr wqq (4.5)The receptive-field basis function

2

2)(exp)(ik

ikririik v

mqq (4.6)

The RCMAC is utilized to estimate the perfect control law, so that

TorrRCMAC yu wwvmwq ),,,,( (4.7)

50

Recurrent CMAC Control SystemControl law

CRCMAC uuu

Recurrent CMAC

riikikk wvmw ˆ,ˆ,ˆ,ˆ

RCMACu

CompensatedController

u x +

E

AdaptiveEstimation Law

̂

Cu

dx

e

Adaptive Recurrent CMAC

Plantxytdtugfx n ),()()()()( xx

Adaptive Laws

TrackingError Vector

E

++

(4.8)

51

Theorem 4.1: The adaptive law of the recurrent CMAC is designed as

and the compensated controller is designed as

with the adaptive estimation law given in

where and are positive constants, then the stability of the control system can be guaranteed.

mT

w PBEw ̂

)sgn(ˆ mT

Cu PBE

||ˆ mT

c PBE

w c

(4.9)

(4.10)

(4.11)

52

On-line parameter training algorithm

),,,(

ˆ),,,(ˆ

rkkrkRCMAC

w

k

RCMAC

RCMACwrkkrkm

Twk

bu

Vw

uu

Vbw

wvmq

wvmqPBE

(4.11)

53

The adaptive laws of means, variances and recurrent weights:

2)(2ˆˆ

ik

ikrikkm

Tmik v

mqbwm PBE

3

2)(2ˆˆik

ikrikkm

Tvik v

mqbwv PBE

)1()(2ˆ

ˆˆ

2

Nuv

qmbw

wq

qb

bu

uVw

RCMACik

riikkkm

Tr

ri

ri

ri

ik

ik

k

k

RCMAC

RCMACrri

PBE

(4.13)

(4.14)

(4.12)

54

where and .

If is unknown:

rT

w PBEw ̂

)sgn(ˆ rT

Cu PBE

||ˆ rT

c PBE

2)(2ˆˆ

ik

ikrikkr

Tmik v

mqbwm PBE

3

2)(2ˆˆik

ikrikkr

Tvik v

mqbwv PBE

)1()(2ˆˆ 2

Nuv

qmbww RCMACik

riikkkr

Trri PBE

jj nT

r ]1,,0,0[ B

(4.16)

(4.17)

(4.15)

(4.19)

(4.20)

(4.18)

)(xg

|);(| txg

55

Illustrative Examples

Example 4.1: Consider the Duffing forced oscillation system

)cos(121.0 3 tuxxx

)(10

0010

2

1

2

1 dugfxx

xx

It can be rewritten as

(4.21)

(4.22)

56

Time (sec)

State response

Time (sec)

State response

Phase-plane portrait

2x 2x2x 2x

1x 1x

1x(a)

(b)

(c)

Simulated results of the Duffing forced oscillation system (without control)

57


1x

1dx

2x

2dx

Time (sec)

Time (sec)

Control effort u

Tacking error e

Time (sec)

Time (sec)

State response

State response

1x2x 2x 2x 2x

1x 1x

u

e

(a)

(b)

(c)

(e)

(d)

Simulated results of RCMAC control for the Duffingforced oscillation system.

58

Example 4.2: The delta wing of the wing rock motion control

uqcqqcqqcqcqccq 3543210 ||||

)(10

0010

2

1

2

1 dugfxx

xx

The state equation

The aerodynamic parameters of the delta wing for a 25-deg angle of attack are chosen as

,00 c ,01859521.01 c ,015162375.02 c,06245153.03 c ,00954708.04 c

02145291.05 c

(4.23)

(4.24)

59

Z

80

x

Wing

Axis of rotation

d

d

Axis of rotation

Wing

U

(a)

(b)

(c)

Two initial conditions:

a small initial condition

a large initial condition

deg6)0(1 xsecdeg/3)0(2 x

deg30)0(1 x

secdeg/10)0(2 x

60

The reference model is defined as

,25.6w

,]6,9[ TK

05.0c,75.0 vm ,01.0r

,29990

R ,

15530

P

The parameters are selected as

2

1

2

16.164.0

10

d

d

d

dxx

xx

(4.25)

61


c)(d

egre

e/se

2xc)

(deg

ree/

se2x

c)(d

egre

e/se

2xc)

(deg

ree/

se2x

(degree)1x (degree)1x (degree)1x (degree)1x

State response

Time (sec)

State response

Time (sec) Time (sec)

Time (sec)

State response

State response

(deg

ree)

1x(d

egre

e)1x

c)(d

egre

e/se

2xc)

(deg

ree/

se2x

c)(d

egre

e/se

2xc)

(deg

ree/

se2x

(deg

ree)

1x(d

egre

e)1x


(a)

(b)

(c) (f)

(e)

(d)

Simulated results of the wing rock motion system (without control)

62

Initial condition

Intelligent hybrid control

RNN adaptive control





1x

1dx

(RNN adaptive control)

1x (Intelligent hybrid control)

2x

2dx



(degree)1x (degree)1x

c)(d

egre

e/se

2xc)

(deg

ree/

se2x

)c

(deg

ree/

se2

u)

c(d

egre

e/se

2u


State response

Time (sec)

State response

Time (sec)

(deg

ree)

1x(d

egre

e)1x

c)(d

egre

e/se

2xc)

(deg

ree/

se2x

Time (sec)

Control effort u

Tacking error e

(deg

ree)

e(de

gree

)e

(a)

(b)

(c)

(e)

(d)Time (sec)

Simulated results of RCMAC control and RNN control for small initial condition.

63

Initial condition







State response

1x

1dx



2x

2dx



c)(d

egre

e/se

2xc)

(deg

ree/

se2x

(degree)1x (degree)1x

)c

(deg

ree/

se2

u)

c(d

egre

e/se

2u


Time (sec)

State response

Time (sec)

(deg

ree)

1x(d

egre

e)1x

c)(d

egre

e/se

2xc)

(deg

ree/

se2x

Time (sec)

Control effort u

Tacking error e

(deg

ree)

e(de

gree

)e

(a)

(b)

(c)

(e)

(d)Time (sec)

Simulated results of RCMAC control and RNN control for large initial condition.

64

Summary

An RCMAC control scheme has been proposed for a class of nonlinear dynamical system.

RCMAC is introduced which has both the merits of RNN and conventional CMAC.

65

Input Space

Receptive -FieldSpace

Weight MemorySpace

Association MemorySpace

Recurrent Unit

k1

nk

Output Space

kow

-

kpw anp

1p k1

kna

k

ikr Tikr T

Ono

1o

sI

sA

sR

sW

sO Input Space

Receptive -FieldSpace

Weight MemorySpace

Association MemorySpace

Recurrent Unit

k1

nk

Output Space

kow

-

kpw anp

1p k1

kna

k

ikr Tikr Tikr TTikr T

Ono

1o

sI

sA

sR

sW

sO

Structures of MIMO RCMAC)()()( Ttrtptp ikikirik

dn

kkkp

Tpp wo

1Φw

(5.1)

(5.2)

6. RCMAC Fault Accommodation Control of a Biped Robot

66

4l

1

4m

x

1a 1l

1m1q

2

2q 2a

2l2m

3a3

3m3q

5a

4a 4q

5q

5l 5

6

b

6q

6m

6a6l

4

5m

4l

1

4m

x

1a 1l

1m1q

2

2q 2a

2l2m

3a3

3m3q

5a

4a 4q

5q

5l 5

6

b

6q

6m

6a6l

4

5m

yy

),() ()(),()( t01 qqfqgqqqCτqMq tt

• The unknown fault-occurrence time

,1 ,0

) (0

00 ttif

ttiftt

• A biped robot is subjected to nonlinear faults

(5.3)

(5.4)

67

In the absence of a fault )(),()(1 qgqqqCτqMq

Computed torque controller

)(),(2)( 20 qgqqqCeKeKqqMτ dqqe dwhere the tracking error vector

Error dynamics 0 eKeKe 22

Fault occurs:

Robust fault-accommodation controller

(5.5)

(5.6)

(5.7)

68

RCMAC-based fault accommodation control

RCMAC-Based Fault-Tolerant Control System

_

Nonlinear Estimation Model

Computed TorqueController

rvcw ,,, Adaptive Laws

RCMAC

Biped Robot

dq],[ qq

],[ dd qq 0τ

tf̂rτ

+

_

+_

qq

ζ

+],[ ee

tf̂

τ

ω

)(qM

rvcW ˆ,ˆ,ˆ,ˆ

RCMAC-Based Fault-Tolerant Control System

_

Nonlinear Estimation Model

Computed TorqueController

rvcw ,,, Adaptive Laws

RCMAC

Biped Robot

dq],[ qq

],[ dd qq 0τ

tf̂rτ

+

_

+_

q

q

ζ

+],[ ee

tf̂

τ

ω

)(qM

rvcW ˆ,ˆ,ˆ,ˆ

Accommodation controller

),(ˆ)( t qqfqMτ r

Fault-accommodation control law

rτττ 0

t1 ˆ)(),()()( fqgqqqCτqMωqω c

RCMACTo estimate the nonlinear fault

(5.8)

(5.9)

(5.10)

69

Simulation Results The joint angle of each link

CMAC RCMAC

70

Simulation Results The fault function and the output

CMAC RCMAC

71

RCMAC can achieve favorable accommodation control for the faults of a biped robot.

Summary

72

Chih-Min Lin and Ya-Fu Peng, “Adaptive CMAC-based supervisory control for uncertain nonlinear systems,” IEEE Trans. System, Man, and Cybernetics Part B, Vol. 34, No. 2, pp. 1248-1260, 2004.

Chih-Min Lin, Ya-Fu Peng, and Chun-Fei Hsu, “Robust cerebellarmodel articulation controller design for unknown nonlinear systems,” IEEE Transactions on Circuits and Systems-II, Vol. 51, No. 7, pp. 354-358, 2004.

Chih-Min Lin and Ya-Fu Peng, “Missile guidance law design using adaptive cerebellar model articulation controller,” IEEE Trans. Neural Networks, Vol. 16, No. 3, pp. 636-644, 2005.

Chih-Min Lin and Chiu-Hsiung Chen, “Robust fault-tolerant control for biped robot using recurrent cerebellar model articulation controller,” IEEE Trans. Systems, Man, and Cybernetics, Part B, Vol. 37, No. 1, pp. 110-123, 2007.

Referred Papers

73

Thank You for Attention

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