Indirect Estimation of Tire Pressure on Several Road Pavements via Interacting Multiple Model Approach
Abstract
1. Introduction
2. Vehicle Dynamics and Road Profile Synthesis Modelling
2.1. Road Surface Profile Synthesis
3. Interacting Multiple Model Filter
- 1.
- Mode Probability Prediction:
- 2.
- Mixing of the previous estimates:
- 3.
- Mode dependent UKFs, whose equations can be found in [34].
- 4.
- Mode Probability Correction:
- 5.
- IMMUF’s corrected estimates:
4. Tire Inflation Pressure Estimation
5. Algorithm Validation
5.1. Simulation Platform
5.2. 2DOF QC Parameters Identification
5.3. Monte Carlo Simulation
- Mean estimation error from all the Monte Carlo samples (the red line in Figure 6). For each -th time step, the mean estimation error is calculated as
- Standard deviation of the estimation errors (the dashed blue line in Figure 6). For each -th time step, the standard deviation of the errors is calculated as
- Covariance bound computed by the filter (the dashed black line in Figure 6). For each -th time step, the -bounds is calculated by the square root of the diagonal elements of the covariance matrix P, as
- Comparison of sample estimation error (the green line in Figure 6) with the -bounds.
5.4. Tire Inflation Pressure Estimation
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
NHTSA | National Highway Traffic Safety Administration |
TPMS | Tire Pressure Monitoring System |
dTPMS | direct Tire Pressure Monitoring System |
iTPMS | indirect Tire Pressure Monitoring System |
KF | Kalman Filter |
EKF | Extended Kalman Filter |
UKF | Unscented Kalman Filter |
MM | Multiple Model |
IMM | Interactive Multiple Model |
IMMUF | Interacting Multiple Model Unscented Filter |
2DOF | Two Degree Of Freedom |
QC | Quarter Car |
PSD | Power Spectral Density |
MTM | Markov Transition Matrix |
RMSE | root Mean Square Error |
ASM | Automotive Simulation Model |
Nomenclature
Symbol | Description | Symbol | Description |
Sprung mass | Unsprung mass | ||
vehicle chassis mass | loading mass | ||
total sprung mass of the vehicle | l | wheelbase | |
rear semiwheel base | inflation pressure increment | ||
Linear damping coefficient | Non-linear square damping coefficient | ||
Linear spring stiffness coefficient | Non-linear cube spring stiffness coefficient | ||
Tire stiffness coefficient | vertical stiffness at the nominal inflation pressure | ||
sprung mass vertical displacement | unsprung mass vertical displacement | ||
Symbol | Description | Symbol | Description |
road profile | nominal pressure | ||
effective pressure | pressure effect on vertical stiffness | ||
angular frequency | PSD | ||
road roughness variance | v | vehicle longitudinal velocity | |
linear shape filter parameter | probability vector | ||
reference spacial frequency | lower spatial frequency | ||
upper spatial frequency | Probability transition matrix | ||
Q | process noise covariance matrix | state vector | |
measurement vector | process function | ||
measurement function | P | covariance matrix | |
noise covariance matrix | sprung mass acceleration variance | ||
time step | noise |
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Quantity | Value | Quantity | Value |
---|---|---|---|
Mass of vehicle chassis (vehicle body) | 1788 kg | Height of the center of mass | 0.6 m |
Front semi-wheelbase | 1.347 m | Rear semi-wheelbase | 1.471 m |
Front track width | 1.606 m | Rear track width | 1.6364 m |
Yaw moment of inertia | 3230 kg m2 | Frontal area | 2.75 m2 |
Front wheel mass | 61.14 kg | Rear wheel mass | 52.75 kg |
Nominal pressure of inflated tire | 250 kPa | vertical stiffness at | 264,700 N/m |
Parameter | Description |
---|---|
Linear damping coefficient | |
Non-linear square damping coefficient | |
Linear spring stiffness coefficient | |
Non-linear cube spring stiffness coefficient |
Parameter | Road Class A | Road Class B | Road Class C | Road Class D |
---|---|---|---|---|
[Ns/m] | 6576 | 4585 | 14,819 | 14,708 |
[Ns/m2] | 4319 | 6016 | 5839 | 4555 |
[N/m] | 113,086 | 173,415 | 94,544 | 118,114 |
[N/m3] | 130,098 | 151,886 | 62,836 | 121,723 |
Parameter | Range |
---|---|
(130–230) [kPa] | |
v | 40–80 [km/h] |
a | 0–3 [-] |
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Brancati, R.; Tufano, F. Indirect Estimation of Tire Pressure on Several Road Pavements via Interacting Multiple Model Approach. Machines 2022, 10, 1221. https://doi.org/10.3390/machines10121221
Brancati R, Tufano F. Indirect Estimation of Tire Pressure on Several Road Pavements via Interacting Multiple Model Approach. Machines. 2022; 10(12):1221. https://doi.org/10.3390/machines10121221
Chicago/Turabian StyleBrancati, Renato, and Francesco Tufano. 2022. "Indirect Estimation of Tire Pressure on Several Road Pavements via Interacting Multiple Model Approach" Machines 10, no. 12: 1221. https://doi.org/10.3390/machines10121221
APA StyleBrancati, R., & Tufano, F. (2022). Indirect Estimation of Tire Pressure on Several Road Pavements via Interacting Multiple Model Approach. Machines, 10(12), 1221. https://doi.org/10.3390/machines10121221