A Discrete-Time Extended Kalman Filter Approach Tailored for Multibody Models: State-Input Estimation
Abstract
:1. Introduction
2. Multi-Body Model and Time-Disretization
2.1. The Multibody Equations of Motion
2.2. The Differential-Algebraic form of the EOMs
2.3. EOMs: The Discrete Index-3 Form
3. An Explicit Linearized Approximation for Use of the Multibody Model in State-Estimation
4. State-Input Estimation for MB Models
4.1. Model and Measurement Equations with Uncertainty
4.2. The Augmented Constraint Measurement Equations
4.3. An Efficient Strategy for the Measurement Sensitivities Computation
4.4. Augmented Discrete Extended Kalman Filter
4.5. The Adopted Extended Kalman Filter Scheme
- A-priori step: Assuming that the augmented states at the previous filter step and the input are known, the a-priori state prediction and generalized accelerations can be computed solving the ID-DAEs of Equation (17):Knowing the estimated state covariance matrix for the previous timestep, the a-priori covariance at the current time () step can be approximated from Equation (47) asThe predicted measurement can then be evaluated from Equation (30) as:The Kalman filter gain allows achieving a desireable trade-off between the confidence in the model and the available measurements, and can be evaluated as:
- A-posteriori step: When the real measurement becomes available together with the predicted measurement , the a posteriori state vector is obtained as:The inclusion of the actual measurements also affects the posterior covariance matrix and can be evaluated as:
5. Validation: Joint State-Input Estimation
5.1. The Slider-Crank System
5.2. Results
5.3. Kalman Filter Tuning
6. Discussion and Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
TPA | Transfer Path Analysis |
KF | Kalman Filter |
ADE-KF | Augmented Discrete Extended Kalman Filter |
MB | MultiBody |
FE | Finite Element |
EOM | Equation Of Motion |
(I-), (E-) DAE | (Implicit), (Explicit) Differential Algebraic Equation |
(I-), (E-) ODE | (Implicit), (Explicit) Ordinary Differential Equation |
MBRC | MultiBody Research Code |
FNCF | Flexible Natural Coordinates Formulation |
BDF | Backward Differentiation Formula |
VS | Virtual Sensor |
MEMS | Micro Electro-Mechanical Systems |
PID | Proportional Integrative Derivative |
integer numbers set | |
real numbers set | |
scalar | |
column vector | |
vertical vector concatenation | |
matrix | |
identity matrix | |
zero vector | |
zero matrix | |
transpose operator | |
inverse matrix operator | |
a priori prediction | |
a posteriori prediction | |
th time step | |
natural coordinates | |
, | time derivatives |
total derivative | |
partial derivative | |
second partial derivative | |
2-norm operator |
Appendix A. Influence of the Forward Differentiation Scheme to the Linearization of the EOMs
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Adduci, R.; Vermaut, M.; Naets, F.; Croes, J.; Desmet, W. A Discrete-Time Extended Kalman Filter Approach Tailored for Multibody Models: State-Input Estimation. Sensors 2021, 21, 4495. https://doi.org/10.3390/s21134495
Adduci R, Vermaut M, Naets F, Croes J, Desmet W. A Discrete-Time Extended Kalman Filter Approach Tailored for Multibody Models: State-Input Estimation. Sensors. 2021; 21(13):4495. https://doi.org/10.3390/s21134495
Chicago/Turabian StyleAdduci, Rocco, Martijn Vermaut, Frank Naets, Jan Croes, and Wim Desmet. 2021. "A Discrete-Time Extended Kalman Filter Approach Tailored for Multibody Models: State-Input Estimation" Sensors 21, no. 13: 4495. https://doi.org/10.3390/s21134495