M. L. Leuschen

Thesis Defense

22 October, 2001

Derivation and Application of Nonlinear Analytical

Redundancy Techniques with Applications to

Robotics

Overview

Analytical redundancy (AR): A powerful technique for fault detection» Review of standard linear AR (LAR)

techniques Derive novel nonlinear AR (NLAR)

technique using nonlinear observability NLAR application and results Conclusions and future work

Motivation

Robotic fault detection an important issue due to circumstances in which robots are used

Many robots have significant nonlinearities

Nonlinear systems degrade the efficiency of linear model-based fault detection schemes such as AR

Contribution of Thesis

Standard LAR methods inappropriate for nonlinear systems

Modern nonlinear control methods were applied to develop a accurate new nonlinear analytical redundancy (NLAR)

NLAR tested on physical and simulated systems» Compares favorably to LAR

Previous Work

Visinsky: Linear AR for electrical robots Wunnenberg/Frank: Linear observers

and AR for conventional robot dynamics Starosweicki/Comtet-Varga: Nonlinear

AR relations for certain nonlinear systems without observability based guarantees

Analytical Redundancy (AR)

First described by Chow and Willsky. (IEEE Transactions on Automatic Control, July, 1984.)

Rigorous Formal Method:» Tests based on left null-space of the

observability matrix, giving greatest possible number of independent tests

» Typical result uses time history of sensor data to test model equations, higher order dynamic response, as well as sensor redundancies

AR Data Flow

Residuals(R)

Sensor Readings

(y)

Control Inputs (u)

System Model

Residuals (R)

Controller

Physical System

Offline derivationof AR test residual

equations

Real-time evaluation of test residuals

Fault detector

AR Structure

Dynamically DerivedTest Residuals

Observability Matrix

Core Concept:» Find left null-space of canonical observability» Take product of null-space and input-output

formulation of observability to derive tests

Linear AR (LAR)

1( ) ( ) ( )

( ) ( ), 1, ,

n n q

j jj

j j

x t A x t b u t

y t c x t j m

Standard continuous-time linear control model, n states, q inputs, m sensors:

( ) ,( ) , ,( )

1, ,

T T k Tj j jLC j c c A c A

j m

Canonical observability sub-matrices for standard linear system:

1

2 0L

L L

CC O

Calculate the left null-space:

LAR Dynamically Derived Observability

Restate observability matrix in terms of known quantities:

1

( )0 0 0 ( )0

( ) 0 0 0 ( )( ) 0 0

( )( )

DDn

n n in nn

y ku t

y t CB u ty tO CAB CB

d u td CBCA BCA By t dtdt

2 ( ) ( )LDD

n

CCA

O x t O x tCA

CA

For fault-free system, dynamically derived observability is equivalent to standard linear observability times the state vector:

( ) 0 ( ) 0DD L

O O x t x t

How Many LAR Tests?

In Linear AR, the Cayley-Hamilton theorem is used to determine the number of independent tests:

Extra sensors (increases m, additional CL(j)’s) result in more tests

1

mLLAR

jN rank C j m n

LAR Example

0 1( ) ( ) ( ),0 1

x t x t u t

1 0( ) ( )0 1

y t x t

1 00 0 1 0 0 0

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

0 0 0 0 0 10

LO

1

1 1

1 1 1

2

2 2

2 2 2

( )

( ) ( )

( ) ( ) ( )

( )

( ) ( )

( ) ( ) ( )

DD

y t

y t c Bu t

y t c ABu t c Bu tO

y t

y t c Bu t

y t c ABu t c Bu t

1 1 2 1

2 1 1 1 1

3 1 2 2

4 2 2 2 2

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

DDO R

R c Bu t y t y t

R y t c ABu t c Bu t y t

R y t c Bu t y t

R y k c ABu t c Bu t y t

Sys

tem

Nul

l-Spa

ceD

D O

bser

vabi

lity

AR

Tests

R1 is the first model equation

R2 is the derivative of R1

R3 is the second model equation

R4 is the derivative of R3

Initial Approach: Nearly Nonlinear AR

(NNAR)

Nonlinear systems better modeled by using several linearizations over local regions

Taking the limit as region size goes to zero will produce nonlinear tests

Limitation: The observability used is approximate » Number of tests and coverage of tests is ad hoc

NLAR: a Nonlinear Observability Based

Approach

AR adapted using nonlinear control theory to develop accurate Nonlinear Analytical Redundancy method (NLAR)

Superior to previous methods » Follows nonlinear model, unlike LAR » Uses full observability space, unlike NNAR

Nontrivial to derive» Nonlinear systems lack useful principles such as

superposition

1

( ) ( ( )) ( ( )) ( )

( ) ( ) , 1, ,

q

j jj

j j

x t f x t g x t u t

y t h x t j m

Standard continuous nonlinear linear control model; n states, q inputs, m sensors:

1 2,

lj jk k kspan h L L L h

Nonlinear observation space for this system is:

1

1, ,1,2,

, , , qi

k

j ml

k span f g g

L

Observability for Nonlinear Systems (Isidori)

If rank(grad( )) = n (the system rank), the system is ‘locally observable’

is the Lie derivative

A Reformulated Nonlinear Observability for AR

Normal nonlinear not suitable for NLAR- Unable to construct a dynamically derived observability!

Triangular Nonlinear Observability

( ) ( ) ( )( , , , ) ( )mj l k m k l k ju u uL j l m L Cx t

0

, 0( )

, 1

1

j

f jk j

g j

u

xCx h tt

0

00

0 00

( )

( )

( , )

( , , )

q

jq q

jlq q q

m jl

C x t

L j

L j l

L j l m

Where:

Nonlinear Dynamically Derived Observability

Express in an input-output format:( ) 0( ) 0

( ) ( )

( ) ( ) ( )

( ) ( ) ( )( )

( ) ( )

i i i

i ji

i j

g

i i ig xg g f

i i j g gfg

i j g g

DD

y ty t

y t u t L

u t L u t L u t L

u t L u t u t Ly t

u t u t L

Repeated derivatives of y, similar to LAR

Number of NLAR Tests

Cayley-Hamilton does not apply to nonlinear systems

However, we have shown that the number of independent NLAR tests is:

1

m

NLNLARj

N rank C j m n

- number of states - number of sensors

- j th sub-matrix of ( )NL

nm

C j

LARNLARNN

Conceptual NLARM

OD

EL

-BA

SE

D O

BS

ER

VA

BIL

ITY

SP

AC

E DY

NA

MIC

AL

LY

DE

RIV

ED

O

BS

ER

VA

BIL

ITY

SP

AC

E

DERIVE OBSERVABILITY NULL-SPACE:

0

00

0 00

( )

( )

( , )

( , , )

q

jq q

jlq q q

m jl

C x t

L j

L j l

L j l m

NEGATIVE AR TEST

Fault Free

Faulty

0

CONTROL MODEL

REAL TIME CONTROL INPUTS & SENSOR DATA

0DD

R

( ) 0

( ) 0

( ) ( )

( )

( )

( )( )

( )

( ) ( )

( ) ( )

i

i

i

i

i j

i j

g

i g

i xg

i g f

i fg

i j g g

i j g g

y t

y t

y t u t L

u t L

u t L

u t Ly t

u t L

u t u t L

u t u t L

DD

POSITIVE AR TEST

0DD

R

The Rosie Mobile Worksystem

L.C. Bares, L.S. Conley, and B.R. Thompson. Rosie: A Mobile Worksystem for D&D: Overview of System Capabilities and CP-5 Reactor Application. In Proceedings of the ANS 7th Topical Meeting on Robotics and Remote Systems, pages 471-477

L. Conley, W.R. Hamel, and B.R. Thompson. Rosie: A Mobile Worksystem for Decontamination and Dismantlement Operations. In Proceedings of the ANS 6th Topical Meeting on Robotics and Remote Systems, pages 231-238

The Rosie Hydraulic Testbed

Physical testbed based on the Rosie mobile worksystem, a decommissioning robot built for DOE

Motorp

(k)

p (k)

Control ValveS

l

m

xv

.

Rosie NLAR

2

0 1 00

0 0

4( )4 4

0

m m

t t ve f

l l s le m e tmt

t t

B d

J Jx uK

p p p pd Cv

v v

0 1 0, ,

0 0 1 l

y C x C yp

1 1 2 1( ) ( ) ( )m m

t t

B dR y t y t y t

J J

22

22 1 22

2 12

44( ) ( )

4( ( )) ( )

mm m m e m tme

t tt t tt

e m fs

t t

B d d CdBR y t y t

J vJ v JJ

d Kp y t u y t

J v

3 1 2 22244 4

( ) ( ) ( )( ( ))e fe m e tm

st t t

Kd CR y t y t u y tp y t

v v v

22

4 12 22

2 222 2 22

2 22 22

2

22222

84 16 2( )

( ( ))

1684 16 2( )

( ( ))

16 4( ( ))( ( ))

e m fm e m e m tm

t t st t

e fe f tme m e tm

t t s tt t

e f tm e fss

tt

d KB d d CR y tu

J v p y tv v

KK Cd Cy t uu

J v p y t vv v

K C Ku p y t u yp y t

vv

2 ( )t

Sys

tem

NLA

R T

ests

R1 is the first model equation, R2 is the derivative of R1,R3 is the second model equation, R4 is the derivative of R3

Rosie NLAR Results: Servovalve Winding Fault

0 10 20 30 40 50-4

-2

0

2

4

6R1

0 10 20 30 40 50-20

0

20

40

60

80R2

0 10 20 30 40 50-1

0

1

2

3x 105 R3

0 10 20 30 40 50-6

-4

-2

0

2

4x 104 R4

RC0303

Faulty Run (BF0193)

Speed: 5 RPM

Load: 1500 PSI

The Integrated Motion Inc. Robot

W.E. Dixon, I.D. Walker, Darren M. Dawson, J.P. Hartranft. Fault Detection for Robot Manipulators with Parametric Uncertainty: A Prediction-Error-Based Approach. IEEE Transactions on Robotics and Automation, 16(6):689-699, 2001.

A two joint planar manipulator at Clemson University» Canonical robot example» Many interesting nonlinearities

Extensively modeled for dynamic control» Control model used to derive NLAR tests» Control model converted to simulation to

generate copious data quickly

NLAR for the IMI Robot

IMI Robot Model Equations

2 cos( ) cos( )1 3 2 2 3 21 1cos( )2 2 3 2 2 2

sin( ) sin( ) 1 2 13 2 2 3 2sin( ) 0 23 2 1

0 0 ( )1 11 10 02 22

p p q p p qu q

u p p q p q

q q qp q q p q

qp q q

f fq sign qd sf q f ssd

( )2ign q

pi are inertias, fi are frictions, qi are joint positions, and ui are control input torques

IMI Robot NLAR Tests

R1: Second model equation checked against shoulder resolver. Tests the acceleration of thesystem.R2: Derivative of second model equation checked against shoulder resolver. Tests the jerk of thesystem.R3: Second derivative of second model equation checked against shoulder resolver. Tests thederivative of the jerk.R4: Sensor comparison of shoulder tachometer with derivative of shoulder resolver.R5: Derivative of R4.R6: Second derivative of R4.R7: Fourth model equation checked against elbow resolver. Tests the acceleration of the system.R8: Derivative of fourth model equation checked against elbow resolver. Tests the jerk of thesystem.R9: Sensor comparison of elbow tachometer and derivative of elbow resolver.R10: Derivative of R9.R11: Second derivative of fourth model equation checked against elbow tachometer. Tests thederivative of the jerk.

NLAR test residuals complex, stated here in terms of the model equations

IMI Robot NLAR Results: Interpretation

0 time (s) 10-100

0

100R1

5.5 time 6.5

-5

0

5

10

15R1 Fault Detail

1 time 4-2

0

2R1 Noise Detail

Fault free signal shows low-

magnitude noise

Fault promptly generates large spike signal

IMI Robot NLAR Results: Frozen Elbow Sensor Fault

0 10-1

0

1x 104 R2

time (s)

0 10-100

0

100R1

time (s)

0 10-2

-1

0

1x 106 R3

time (s)

0 10-10

0

10

20R4

time (s)

0 10-100

0

100R5

time (s)

4

0 10-1

0

1x 10 R6

time (s)

0 10-200

0

200

400R7

time (s)

0 10-5000

0

5000

10000R8

time (s)

0 10-0.5

0

0.5R9

time (s)

0 10-20

-10

0

10R10

time (s)

0 10-5

0

5x 10 R115

time (s)

Elbow resolver freezes at t=6s This fault is subtle, yet is still detected in one or two

steps by most NLAR tests

IMI Robot NLAR Results: Shoulder Motor Fault

0 10-100

-50

0

50R1

time (s)

0 10-500

0

500

1000R2

time (s)

0 10-5

0

5x 10

4 R3

time (s)

0 10-0.1

-0.05

0

0.05

0.1R4

time (s)

0 10-2

0

2

4R5

time (s)

0 10-10

-5

0

5R10

time (s)

0 10-400

-200

0

200

400R6

time (s)

0 10-100

0

100

200R7

time (s)

0 10-2000

-1000

0

1000R8

time (s)

0 10-0.05

0

0.05R9

time (s)

0 10-2

-1

0

1x 10

5 R11

time (s)

Shoulder motor limp at t=6s Fault signal very clear on most tests

Comparison of NLAR vs. LAR: IMI Robot Motor Fault

0 10-100

-50

0

50R1

time (s) 0 10-500

0

500

1000R2

time (s)

0 10-100

0

100

200R7

time (s) 0 10-2000

-1000

0

1000R8

time (s)

Four tests where NLAR (thick, solid) and LAR (thin, dotted) comparable

NLAR clearly outperforms LAR on these tests

Summary and Conclusions

Our new nonlinear analytical redundancy technique (NLAR) is a powerful model-based fault detection method for nonlinear systems

NLAR a clear improvement over LAR NLAR shows excellent results on our

example nonlinear systems

Contributions

New nonlinear analytical redundancy (NLAR) technique for fault detection in nonlinear systems» Reformulated nonlinear observability for NLAR

null matrix determination» Nonlinear dynamically derived observability » Calculation of number of valid NLAR tests

Application of NLAR to test robots» Automatic calculation of NLAR tests

Directions for Future Work

Further testing on data from physical testbeds and systems

Analysis of test residuals» Thresholding and sensitivity» Fault classification

Non smooth nonlinearities Automated NLAR software package

The End

Appendix: Additional Information

Rosie Details

Foster-Miller testbed is based on the Rosie mobile worksystem, a decommissioning robot built for DOE

Wheeled platform with heavy-duty robotic manipulator

Central hydraulic power source Wheel actuators of special interest to

prevent failures that trap robot

Physical Testbed

Collaborating with Foster-Miller Technologies Incorporated (FM)» Considerable experience in evaluating the

reliability of hydraulic systems» Contracted by DOE to examine hydraulic fault

issues for hazardous environments FM constructed physical testbed to acquire

fault data» Faults simulated by modifications to test rig

Hydraulic Test Rig

Dynamic Model

e

ltlemimmmlsfv

pvpccdppKxQ

4

)()(2

e

ltlemimmmlcvq

pvpccdpkxkQ

4)(

lmmmtmlg TBJdpT

Linearized Flow:

Flow Equation:

Torque Equation:

Linearization Consequences

The linearization of the flow equation causes significant error

However, for standard AR, we must use linear equation

0 500 1000 1500 2000 2500 30000

0.2

0.4

Pressure (PSI)

Control Valve Position (in.)

Linear Approximation Error

50%25%

0%-25% -50%

Error between original and linearized flow equations

,0)()(42

)2()(42

12

,0)()1()(1

2

2

2

22

2

22

kxkpvJ

tMd

J

td

J

tBd

kkvJ

td

J

tB

J

tBV

kpJ

tdkk

J

JtBV

tt

me

t

m

t

mm

tt

m

t

m

t

m

t

m

t

tm

,0)(4

)1()(4

)(4

3

kxv

tkkpkp

v

vtMk

v

tdV

t

qe

t

te

t

me

.0)1(4

)(164

)(8164

1

)2()(1684

4

2

22

2

22222

2

222

kxv

tkkx

v

tMktkv

kpv

tM

v

tM

vJ

td

kpkv

tMd

v

td

vJ

tdBV

t

qe

t

qeqte

t

e

t

e

tt

me

t

me

t

me

tt

mme

Linear System AR Tests

Linear AR Test Characteristics

V1 and V3 are the original discrete-time model equations

V2 and V4 are related to the first derivatives of V1 and V3 respectively, but use both model equations

More AR tests than model equations! However, linearization of flow equation

severely degrades performance

Idea: Piecewise Linear Division of Workspace

-0.5 in. 0.5 in.

3000 PSI

-3000 PSI

Pre

ssu

re

Valve Position

Linearization Points

Piecewise linear division of workspace

TransitionRegions

Comparison of Linear AR to NNAR

(RC0303)

0 10 20 30 40-10

-8

-6

-4

-2

0

x 105 V4 and NV4

Nonlinear

Linear

NV

4 is

zer

o m

ean

0 10 20 30 40

NV4 and translated V4(both magnified)

Nonlinear

Linear

NV

4 ha

s sm

alle

r va

rian

ce

Fault Free Run

(5 RPM, 1500 PSI load, steady state input)

Ideal AR test should be zero mean and low variance

time time

0 10 20 30 40-4

-3

-2

-1

0

1x 10

6 V4

0 10 20 30 40-2

-1

0

1

2x 10

5 NV4

Comparison of Linear AR to NNAR

Control Valve Open Winding Fault

(BF0193)

(5RPM, 1500PSI load, control valve winding open from 11s until 31s)

V4 drift hides fault signal NV4 fault signal prominent

timetime

Why go Beyond NNAR?

Nonlinear extension works better than linear, but it is not rigorous» Linear model used to derive basic tests» Essentially looking at AR for system

linearized about current point of operation Want to examine nonlinear equivalent to

the observability space complement idea in AR

IMI Robot System Model

2 111 11

2 122 2 1 2 1 221

132 23

144 2 1 2 1 24 2 3 2

2

0( )

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

( ) 0( , , )( ) cos( )

( )

p gq qf xq gf q q q q qf x

f x g xgq qf xgf q q q q qf x p p q

q

2 3 2 21

2 222

23

241 3 2

2

0cos( )

( ), ( )

02 cos( )

( )

p p q gq g

g xggp p q

q

2 3 23 2 1 2 2 1 22 3 22 1 2 2 1 2

2 2

1 3 2 2 3 23 2 1 2

2 1 22

2 3 23 22 1 2

cos( )sin( ) sin( )( , , ) , ( , , ) ,

( ) ( )

2 cos( ) cos( )sin( )( , , ) ,

( )

cos( )sin( )( , , )

p p qp q q p q q qp p qq q q q q q

q q

p p q p p qp q q qq q q

q

qp p qp qq q q

1 2 2 3 22 1 2 2 3 2

2

, ( ) cos ( )( )

qq p p p p q

q

IMI Fault Run Details

0.5sin 0.310.5cos 0.50.42

ttdttd

1: 2 1 12 2 2 12

cos( )sin( ) 2 3 23 2 1 2 2 2 11

cos( )sin( ) 1 2 2 3 22 3 2 22

cos( )2 3 22 1 2

1 2

R f u g u g yx x x

p p qp q q p q p f qd

q q p p qp p q f qd

p p qp u u

p p

2 3 2cos ( )2 3 2p p q

4: 02 1

R y y

Input signal:

Example Test Residuals: