m. l. leuschen thesis defense 22 october, 2001 derivation and application of nonlinear analytical...
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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:
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