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Introduction
Radial Basis Function Networks
Results
Conclusion
Radial Basis Function Network (RBFN)Approximation of Finite
Element Models for
Real-Time Simulation
M. S. Narayanan1 P. Singla S. Garimella W. Waz
V. Krovi2
University at Buffalo
[email protected], 2 [email protected]
ASME Dynamic Systems and Control Conference, 2011
"filler texts"November 2, 2011
1 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
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Introduction
Radial Basis Function Networks
Results
Conclusion
Outline
1 Introduction
Objectives
Current Limitations
Background
2 Radial Basis Function NetworksRBFs
Modified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
3 Results
FE Models
RBFN Results
Post Processing
2 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
http://find/http://goback/
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3/54
Introduction
Radial Basis Function Networks
Results
Conclusion
Outline
1 Introduction
Objectives
Current Limitations
Background
2 Radial Basis Function NetworksRBFs
Modified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
3 Results
FE Models
RBFN Results
Post Processing
2 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
http://find/
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4/54
Introduction
Radial Basis Function Networks
Results
Conclusion
Outline
1 Introduction
Objectives
Current Limitations
Background
2 Radial Basis Function NetworksRBFs
Modified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
3 Results
FE Models
RBFN Results
Post Processing
2 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
http://find/http://goback/
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5/54
Introduction
Radial Basis Function Networks
Results
Conclusion
Objectives
Current Limitations
Background
Outline
1 Introduction
Objectives
Current Limitations
Background
2 Radial Basis Function NetworksRBFs
Modified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
3 Results
FE Models
RBFN Results
Post Processing
3 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
I d i
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6/54
Introduction
Radial Basis Function Networks
Results
Conclusion
Objectives
Current Limitations
Background
Goals
Nonlinear approximation methods to
parametrically capture physics in higher order models.
Radial basis function network (RBFN) method
Applicability for FE systems
Modified resource allocating network (MRAN) to estimatenetwork
parameters
Extended Kalman Filter (EKF) to optimize the parametersExploit
RBFs localization properties to model FE model
response
4 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
I t d ti
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7/54
Introduction
Radial Basis Function Networks
Results
Conclusion
Objectives
Current Limitations
Background
Goals
Nonlinear approximation methods to
parametrically capture physics in higher order models.
Radial basis function network (RBFN) method
Applicability for FE systems
Modified resource allocating network (MRAN) to estimatenetwork
parameters
Extended Kalman Filter (EKF) to optimize the parametersExploit
RBFs localization properties to model FE model
response
4 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
Introduction
http://find/http://goback/
-
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8/54
Introduction
Radial Basis Function Networks
Results
Conclusion
Objectives
Current Limitations
Background
Goals
Nonlinear approximation methods to
parametrically capture physics in higher order models.
Radial basis function network (RBFN) method
Applicability for FE systems
Modified resource allocating network (MRAN) to estimatenetwork
parameters
Extended Kalman Filter (EKF) to optimize the parametersExploit
RBFs localization properties to model FE model
response
4 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
Introduction
http://find/http://goback/
-
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9/54
Introduction
Radial Basis Function Networks
Results
Conclusion
Objectives
Current Limitations
Background
Goals
Nonlinear approximation methods to
parametrically capture physics in higher order models.
Radial basis function network (RBFN) methodApplicability for FE
systems
Modified resource allocating network (MRAN) to estimatenetwork
parameters
Extended Kalman Filter (EKF) to optimize the parametersExploit
RBFs localization properties to model FE model
response
4 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
Introduction
http://find/http://goback/
-
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10/54
Introduction
Radial Basis Function Networks
Results
Conclusion
Objectives
Current Limitations
Background
Goals
Nonlinear approximation methods to
parametrically capture physics in higher order models.
Radial basis function network (RBFN) methodApplicability for FE
systems
Modified resource allocating network (MRAN) to estimatenetwork
parameters
Extended Kalman Filter (EKF) to optimize the parametersExploit
RBFs localization properties to model FE model
response
4 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
Introduction
http://find/http://goback/
-
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11/54
Introduction
Radial Basis Function Networks
Results
Conclusion
Objectives
Current Limitations
Background
Goals
Nonlinear approximation methods to
parametrically capture physics in higher order models.
Radial basis function network (RBFN) methodApplicability for FE
systems
Modified resource allocating network (MRAN) to estimatenetwork
parameters
Extended Kalman Filter (EKF) to optimize the parametersExploit
RBFs localization properties to model FE model
response
4 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionObj ti
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12/54
Introduction
Radial Basis Function Networks
Results
Conclusion
Objectives
Current Limitations
Background
Goals
Nonlinear approximation methods to
parametrically capture physics in higher order models.
Radial basis function network (RBFN) methodApplicability for FE
systems
Modified resource allocating network (MRAN) to estimatenetwork
parameters
Extended Kalman Filter (EKF) to optimize the parametersExploit
RBFs localization properties to model FE model
response
4 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionObj ti
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13/54
Radial Basis Function Networks
Results
Conclusion
Objectives
Current Limitations
Background
Outline
1 Introduction
Objectives
Current Limitations
Background
2 Radial Basis Function NetworksRBFs
Modified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
3 Results
FE Models
RBFN Results
Post Processing
5 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionObjectives
http://find/http://goback/
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Radial Basis Function Networks
Results
Conclusion
Objectives
Current Limitations
Background
State of Art
Real-time(RT) haptic simulations used in most of thecurrent
studies are only linear FE (reliable, favorable andfeasible)
mass-spring-damper systems, surface models, hybridmethods
etc...da Vinci Skills simulator (Mimics software) uses only
linearFE
High-fidelity models (real tissue deformations) are
highlynonlinear
High haptic update rates (1000 Hz) for real-time
simulationefficient and high performance systems
(GP-GPUcomputing)Tissue property variationsNeed for high-fidelity
models for real-time simulation, forexample: simulations for
surgery
6 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionObjectives
http://find/http://goback/
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15/54
Radial Basis Function Networks
Results
Conclusion
Objectives
Current Limitations
Background
State of Art
Real-time(RT) haptic simulations used in most of thecurrent
studies are only linear FE (reliable, favorable andfeasible)
mass-spring-damper systems, surface models, hybridmethods
etc...da Vinci Skills simulator (Mimics software) uses only
linearFE
High-fidelity models (real tissue deformations) are
highlynonlinear
High haptic update rates (1000 Hz) for real-time
simulationefficient and high performance systems
(GP-GPUcomputing)Tissue property variationsNeed for high-fidelity
models for real-time simulation, forexample: simulations for
surgery
6 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionObjectives
http://find/http://goback/
-
8/3/2019 20111029 DSCC RBF Madu Template3
16/54
Radial Basis Function Networks
Results
Conclusion
Objectives
Current Limitations
Background
Outline
1 Introduction
Objectives
Current Limitations
Background
2 Radial Basis Function NetworksRBFs
Modified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
3 Results
FE Models
RBFN Results
Post Processing
7 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
Introduction
R di l B i F i N kObjectives
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Radial Basis Function Networks
Results
Conclusion
Objectives
Current Limitations
Background
Nonlinear approximation methods
multivariate polynomials, splines, tensor product methods,
local methods and global methods
RBFNs are modern ways to approximate multivariate
functions, especially in the absence of grid data.RBFN method:
Variant of artificial neuralnetworks (ANN): useful tools in
signal processing patternclassification/ clustering
dynamical system modelingfunctional approximations etc...
RBFN models can learn a systems behavior (response)
when traditional modeling is very difficult to generalize
8 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
Introduction
R di l B i F ti N t kObjectives
http://find/http://goback/
-
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18/54
Radial Basis Function Networks
Results
Conclusion
j
Current Limitations
Background
Nonlinear approximation methods
multivariate polynomials, splines, tensor product methods,
local methods and global methods
RBFNs are modern ways to approximate multivariate
functions, especially in the absence of grid data.RBFN method:
Variant of artificial neuralnetworks (ANN): useful tools in
signal processing patternclassification/ clustering
dynamical system modelingfunctional approximations etc...
RBFN models can learn a systems behavior (response)
when traditional modeling is very difficult to generalize
8 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
Introduction
Radial Basis Function NetworksObjectives
http://find/http://goback/
-
8/3/2019 20111029 DSCC RBF Madu Template3
19/54
Radial Basis Function Networks
Results
Conclusion
j
Current Limitations
Background
Nonlinear approximation methods
multivariate polynomials, splines, tensor product methods,
local methods and global methods
RBFNs are modern ways to approximate multivariate
functions, especially in the absence of grid data.RBFN method:
Variant of artificial neuralnetworks (ANN): useful tools in
signal processing patternclassification/ clustering
dynamical system modelingfunctional approximations etc...
RBFN models can learn a systems behavior (response)
when traditional modeling is very difficult to generalize
8 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
Introduction
Radial Basis Function NetworksObjectives
http://find/http://goback/
-
8/3/2019 20111029 DSCC RBF Madu Template3
20/54
Radial Basis Function Networks
Results
Conclusion
Current Limitations
Background
Nonlinear approximation methods
multivariate polynomials, splines, tensor product methods,
local methods and global methods
RBFNs are modern ways to approximate multivariate
functions, especially in the absence of grid data.RBFN method:
Variant of artificial neuralnetworks (ANN): useful tools in
signal processing patternclassification/ clustering
dynamical system modelingfunctional approximations etc...
RBFN models can learn a systems behavior (response)
when traditional modeling is very difficult to generalize
8 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Objectives
http://find/http://goback/
-
8/3/2019 20111029 DSCC RBF Madu Template3
21/54
Radial Basis Function Networks
Results
Conclusion
Current Limitations
Background
Nonlinear approximation methods
multivariate polynomials, splines, tensor product methods,
local methods and global methods
RBFNs are modern ways to approximate multivariate
functions, especially in the absence of grid data.RBFN method:
Variant of artificial neuralnetworks (ANN): useful tools in
signal processing patternclassification/ clustering
dynamical system modelingfunctional approximations etc...
RBFN models can learn a systems behavior (response)
when traditional modeling is very difficult to generalize
8 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Objectives
http://find/http://goback/
-
8/3/2019 20111029 DSCC RBF Madu Template3
22/54
Radial Basis Function Networks
Results
Conclusion
Current Limitations
Background
Nonlinear approximation methods
multivariate polynomials, splines, tensor product methods,
local methods and global methods
RBFNs are modern ways to approximate multivariate
functions, especially in the absence of grid data.RBFN method:
Variant of artificial neuralnetworks (ANN): useful tools in
signal processing patternclassification/ clustering
dynamical system modelingfunctional approximations etc...
RBFN models can learn a systems behavior (response)
when traditional modeling is very difficult to generalize
8 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Objectives
C Li i i
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23/54
Radial Basis Function Networks
Results
Conclusion
Current Limitations
Background
Nonlinear approximation methods
multivariate polynomials, splines, tensor product methods,
local methods and global methods
RBFNs are modern ways to approximate multivariate
functions, especially in the absence of grid data.RBFN method:
Variant of artificial neuralnetworks (ANN): useful tools in
signal processing patternclassification/ clustering
dynamical system modelingfunctional approximations etc...
RBFN models can learn a systems behavior (response)
when traditional modeling is very difficult to generalize
8 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
RBFsModified Resource Allocating Network (MRAN)
http://find/http://goback/
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Results
Conclusion
g ( )
Extended Kalman Filter
Final Implementation
Outline
1 IntroductionObjectives
Current Limitations
Background
2 Radial Basis Function NetworksRBFs
Modified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
3 Results
FE Models
RBFN Results
Post Processing
9 / 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
RBFsModified Resource Allocating Network (MRAN)
http://find/http://goback/
-
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Results
Conclusion
g ( )
Extended Kalman Filter
Final Implementation
Radial Basis Functions
Characteristics of RBFsRBFs are real-valued function whose value
depends only on thedistance from the origin, so that
(x) =
k (x)
k
Used in patternrecognition and
machine learning
Inherent nonlinearities
Parametric control withcompact domain support
higher order/ spectralconvergence can be achieved
Can adapt (train) with highlynonlinear models
Generally global, butextremely useful for
localizedapproximation
Parametric control withcompact domain support
fast and well conditionediterative algorithms ispossible
10/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
RBFsModified Resource Allocating Network (MRAN)
http://find/http://goback/
-
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Results
Conclusion
Extended Kalman Filter
Final Implementation
Radial Basis Functions
Characteristics of RBFsRBFs are real-valued function whose value
depends only on thedistance from the origin, so that
(x) =
k (x)
k
Used in patternrecognition andmachine learning
Inherent nonlinearities
Parametric control withcompact domain support
higher order/ spectralconvergence can be achieved
Can adapt (train) with highlynonlinear models
Generally global, butextremely useful for
localizedapproximation
Parametric control withcompact domain support
fast and well conditionediterative algorithms ispossible
10/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
RBFsModified Resource Allocating Network (MRAN)
http://find/http://goback/
-
8/3/2019 20111029 DSCC RBF Madu Template3
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Results
Conclusion
Extended Kalman Filter
Final Implementation
Radial Basis Functions
Characteristics of RBFsRBFs are real-valued function whose value
depends only on thedistance from the origin, so that
(x) =
k (x)
k
Used in patternrecognition andmachine learning
Inherent nonlinearities
Parametric control withcompact domain support
higher order/ spectralconvergence can be achieved
Can adapt (train) with highlynonlinear models
Generally global, butextremely useful for
localizedapproximation
Parametric control withcompact domain support
fast and well conditionediterative algorithms ispossible
10/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
R l
RBFsModified Resource Allocating Network (MRAN)
E d d K l Fil
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Results
Conclusion
Extended Kalman Filter
Final Implementation
Typical Gaussian RBF Units
1 D
Figure: Gaussian RBFs in 1D
(x) = exp
2
(x n)2
( )2
3
(1)
2 D
Figure: Gaussian RBF in 2D
(x) = exp(
(x
xn)T
:
R 1 : (x xn)
(2)
11/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
R lt
RBFsModified Resource Allocating Network (MRAN)
E t d d K l Filt
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Results
Conclusion
Extended Kalman Filter
Final Implementation
Radial Basis Functions Network
RBFNs
Two-layer feed-forward typenetwork in which the input
istransformed by the basis functions
at the hidden layer.
Figure: Typical (Gaussian) RBFN
Output y = w0 +
n
i=1wi
i
3
x 3 i
RBF Parametrization
mean: 3 =A n parameters
covariance matrix R 1 size:
(n
n) =A
n
2
parameters
weight (w) is a scalar =A 1parameter
12/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
RBFsModified Resource Allocating Network (MRAN)
Extended Kalman Filter
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Results
Conclusion
Extended Kalman Filter
Final Implementation
Radial Basis Functions Network
RBFNs
Two-layer feed-forward typenetwork in which the input
istransformed by the basis functions
at the hidden layer.
Figure: Typical (Gaussian) RBFN
Output y = w0 +
n
i=1wi
i
3
x 3 i
RBF Parametrization
mean: 3 =A n parameters
covariance matrix R 1 size:(n
n) =
An2 parameters
general case(symmetric)
=A rij = rji =A
n: (n+1)2
parameters
weight(
w)
is a scalar=
A 1parameter
General => number of parameters
per unit RBF = n+ n: (n+1)2
+ 1
12/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
RBFsModified Resource Allocating Network (MRAN)
Extended Kalman Filter
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Results
Conclusion
Extended Kalman Filter
Final Implementation
Radial Basis Functions Network
RBFNs
Two-layer feed-forward typenetwork in which the input
istransformed by the basis functions
at the hidden layer.
Figure: Typical (Gaussian) RBFN
Output y = w0 +
n
i=1wi
i
3
x 3 i
RBF Parametrization
mean: 3 =A n parameters
covariance matrix R 1 size:(n n) =A n2 parameters
elliptic=
A rij = rji = 0,for i T= j =A only nparameters
(diagonalelements)
weight (w) is a scalar =A 1parameter
Elliptic => number of parametersper unit RBF = n+ n+ 1
12/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
RBFsModified Resource Allocating Network (MRAN)
Extended Kalman Filter
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Results
Conclusion
Extended Kalman Filter
Final Implementation
Radial Basis Functions Network
RBFNs
Two-layer feed-forward typenetwork in which the input
istransformed by the basis functions
at the hidden layer.
Figure: Typical (Gaussian) RBFN
Output y = w0 +
n
i=1wi
i
3
x 3 i
RBF Parametrization
mean: 3 =A n parameters
covariance matrix R 1 size:(n n) =A n2 parameters
circular=
A rij = rji = 0 :V i; riis are equal : only 1parameter
weight(
w)
is a scalar=
A 1parameter
Circular => number of parametersper unit RBF = 1 + 1 + 1
12/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
RBFsModified Resource Allocating Network (MRAN)
Extended Kalman Filter
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Results
Conclusion
Extended Kalman Filter
Final Implementation
Problem Statement
Given: Input-Output response of a (FE) system (as a series
ofinput loads with displacements) To find: Parameters of the
RBFN
(i.e.) for each RBF unit (or node)
no. of nodes of RBFN(N)3
A location of origins3
A Elements of covariance matrix stacked in form of vector
w : weight factor for each RBF node
MethodsDirect Connectivity Graph
Resource Allocating Network (RAN)
Modified RAN
Modified MRAN
13/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
RBFsModified Resource Allocating Network (MRAN)
Extended Kalman Filter
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Results
Conclusion
Extended Kalman Filter
Final Implementation
Problem Statement
Given: Input-Output response of a (FE) system (as a series
ofinput loads with displacements) To find: Parameters of the
RBFN
(i.e.) for each RBF unit (or node)
no. of nodes of RBFN(N)3
A location of origins3
A Elements of covariance matrix stacked in form of vector
w : weight factor for each RBF node
MethodsDirect Connectivity Graph
Resource Allocating Network (RAN)
Modified RAN
Modified MRAN
13/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
RBFsModified Resource Allocating Network (MRAN)
Extended Kalman Filter
http://find/http://goback/
-
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Conclusion Final Implementation
Problem Statement
Given: Input-Output response of a (FE) system (as a series
ofinput loads with displacements) To find: Parameters of the
RBFN
(i.e.) for each RBF unit (or node)
no. of nodes of RBFN(N)3
A location of origins3
A Elements of covariance matrix stacked in form of vector
w : weight factor for each RBF node
MethodsDirect Connectivity Graph
Resource Allocating Network (RAN)
Modified RAN
Modified MRAN
13/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
RBFsModified Resource Allocating Network (MRAN)
Extended Kalman Filter
http://find/http://goback/
-
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36/54
Conclusion Final Implementation
Problem Statement
Given: Input-Output response of a (FE) system (as a series
ofinput loads with displacements) To find: Parameters of the
RBFN
(i.e.) for each RBF unit (or node)
no. of nodes of RBFN(N)3
A location of origins3
A Elements of covariance matrix stacked in form of vector
w : weight factor for each RBF node
MethodsDirect Connectivity Graph
Resource Allocating Network (RAN)
Modified RAN
Modified MRAN
13/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
RBFsModified Resource Allocating Network (MRAN)
Extended Kalman Filter
http://find/http://goback/
-
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37/54
Conclusion Final Implementation
Outline
1 IntroductionObjectives
Current Limitations
Background
2
Radial Basis Function NetworksRBFs
Modified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
3 ResultsFE Models
RBFN Results
Post Processing
14/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
RBFsModified Resource Allocating Network (MRAN)
Extended Kalman Filter
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Conclusion Final Implementation
Resource Allocating Network
RAN : Determining RBFN Parameters
Problem of allocating RBF nodes sequentially can be
stated as follows:
For a input data (observation)m (xn; yn), obtain posterior
information fn.Instead of an impulse function, a Gaussian RBF
will be
added which is centered at xn given by:
n(x) = exp( (x xn)T
:R 1
:(x
xn) (3)
Consider, after examining a series of input data, the RBFN
has
grown upto h nodes. Then f = h
i=1 wi (x ; i; i). So,
RBFN parameters := [N; ( i ; wi ; i) for i = 1 to h] (4)
dim( ) = h: (2n + 1)
15/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
C
RBFsModified Resource Allocating Network (MRAN)
Extended Kalman Filter
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Conclusion Final Implementation
Resource Allocating Network
RAN : Determining RBFN Parameters
For the next input data, (xm; ym), the network parameters whena
new node is inserted into RBFN is obtained as:
wh+1 = ym
f(xh+1
h+1) (5)
h+1 = xm (6)
h+1 = m (7)
Note:
For h+1, typically it can be obtained from the input data
distribution. For practical purposes: h+1 = h, where is
aconstant
16/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
C l i
RBFsModified Resource Allocating Network (MRAN)
Extended Kalman Filter
Fi l I l t ti
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Conclusion Final Implementation
Modified Resource Allocating Network
Criteria for Adding Nodes into RBFNk
xi nearest k > (8)
k yi f(xi)k > emin (9)
ermsi =
v
u
u
t
i
j=1 (Nw 1)
k
ejk
2
Nw> ermin (10)
Optimizing the Parameters
If error conditions not met, existing network parametric vector
willbe optimized. (Note: RBFN is a multi-modal nonlinear
function)
Kalman filtering technique is used to optimize the
RBFNparameters for each input data (whether the error conditions
ormet or not)
17/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
Conclusion
RBFsModified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
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Conclusion Final Implementation
Outline
1 IntroductionObjectives
Current Limitations
Background
2
Radial Basis Function NetworksRBFs
Modified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
3 ResultsFE Models
RBFN Results
Post Processing
18/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
Conclusion
RBFsModified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
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Conclusion Final Implementation
EKF Method
Parameter Estimation
RBFN is a nonlinear function =A
extended Kalman filter
is used (EKF) For a given system with measurement
model given as: y = h( k) + k (11)
withE(
k) = 0 (12)
E
l kT
= Rk (l k) (13)
Update Model
Kk = P
k HTk
Hk P
k HTk + R
1k
(14)
+k =
k + Kk
y h( k
(15)
P+k = (I KkH) P
k where, Hk =@ h( k)
@
k
= k
(16)
19/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
Conclusion
RBFsModified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
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Conclusion Final Implementation
Outline
1 IntroductionObjectives
Current Limitations
Background
2
Radial Basis Function NetworksRBFs
Modified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
3 ResultsFE Models
RBFN Results
Post Processing
20/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
Conclusion
RBFsModified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
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Conclusion Final Implementation
Training and Prediction
Fi ure: Gaussian RBFN
Input is from FE model response
(commercial software or
customized code)
Error conditions are verified for
adding a node to RBFNEKF is used to estimate the
values in real-time at every data
point (whether the error
conditions are satisfied or not)
RBFN model prediction is
discussed in the results section
21/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
Conclusion
FE ModelsRBFN Results
Post Processing
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Conclusion
Outline
1 IntroductionObjectives
Current Limitations
Background
2
Radial Basis Function NetworksRBFs
Modified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
3 ResultsFE Models
RBFN Results
Post Processing
22/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
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IntroductionRadial Basis Function Networks
Results
Conclusion
FE ModelsRBFN Results
Post Processing
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Outline
1 IntroductionObjectives
Current Limitations
Background
2 Radial Basis Function Networks
RBFs
Modified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
3 ResultsFE Models
RBFN Results
Post Processing
24/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
Conclusion
FE ModelsRBFN Results
Post Processing
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Error ComparisonCase 1
Figure: Errors in NodalDisplacements
Figure: Model Reponse
Case 2
Figure: Errors in NodalDisplacements
Figure: Model Reponse
25/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
Conclusion
FE ModelsRBFN Results
Post Processing
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Error ComparisonCase 1
Figure: Errors in NodalDisplacements
Figure: Model Reponse
Case 2
Figure: Errors in NodalDisplacements
Figure: Model Reponse
25/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
Conclusion
FE ModelsRBFN Results
Post Processing
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Outline
1 IntroductionObjectives
Current Limitations
Background
2 Radial Basis Function Networks
RBFs
Modified Resource Allocating Network (MRAN)
Extended Kalman Filter
Final Implementation
3 ResultsFE Models
RBFN Results
Post Processing
26/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
Conclusion
FE ModelsRBFN Results
Post Processing
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Pruning
Sometimes, it is essential to reduce the network size to a
minimum numberIt can be implemented by calculating the
relative
magnitude of a node for a last few 100s iterations of cycles
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FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
Conclusion
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Summary
RBFN approximation of FE models was implementedRBFN model
parameter estimationRBFN model parametric optimization
Initial studies proved promising for implementation of
real-time framework (MATLAB- Simulink at 1KHz with
VRML is possible)
Future work:
will discuss about a wide range of estimation methods and
to identify which works better for different types of
problems
apply this method for complex and highly nonlinear models
such as human tissues
compare the usage of different RBFs instead of Gaussian
functions.
28/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
IntroductionRadial Basis Function Networks
Results
Conclusion
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Thank You !!!
29/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
Appendix Bibliography
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S. Someone.
On this and that.Journal of This and That, 2(1):50100, 2000.
30/ 30 2011 ASME DSCC M S Narayanan et al. RBFN Approximation of
FE Models for Real-Time Simulation
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