# Autonomous Vehicles: Vehicle Parameter Estimation Using Variational Bayes and Kinematics

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## Abstract

**:**

## 1. Introduction

**H0:**

## 2. State of the Art

#### 2.1. Bayesian State Estimation Approaches

#### 2.2. Advances in Gaussian-Process Models

## 3. SGP Motion Model Implementation

- If the derived kinematic model is in accordance with real measurements, the machine learning model will be able to represent the corresponding physical behavior.
- However, if the kinematic model does not rely on data acquired from real-life measurements, the machine learning model might identify incoherences on the sample data that would prevent it from making the appropriate decisions.

#### 3.1. SGP Model Training

#### 3.2. Vehicle Parameter Estimation

#### 3.3. Comparison of the Presented Approach with Blackbox and Whitebox Models

## 4. Results

#### 4.1. Ground Truth Data Generation

#### 4.2. Motion Model and Vehicle Parameter Estimation

**Motion Model:**We need to fit the sGP motion model to the data. In doing so, we need to optimize parameters and to ensure that the model represents the simulated movement.**Kinematic Parameter Estimation:**The kinematic parameter estimation is based on the result of the motion model and simplified vehicle kinematics.

#### 4.3. Comparison of Whitebox and Blackbox Models

## 5. Conclusion and Future Work

**H1:**

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CAD | computer-aided design |

EKF | Extended Kalman Filter |

EKFDR | Extended Kalman Filter Dead Reckoning |

GNSS | Global navigation satellite system |

GP | Gaussian process |

ICC | instantaneous center of curvature |

ICR | Instantaneous center of rotation |

IMU | Inertial measurement unit |

KF | Kalman Filter |

LIDAR | light detection and ranging |

MCMC | Makrov chain monte carlo |

M m | Map Matching |

PF | Particle Filter |

RMS | Root mean square |

RTK | Real time kinematic |

sGP | Sparsed Gaussian processes |

UAV | autonomous unmanned air vehicles |

VB | Variational Bayes |

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**Figure 1.**Graphical probabilistic model describing the Bayes filter. Each node describes a random variable. ${x}_{j}$ describes the vehicle’s state at time j, ${z}_{j}$ describes a measurement and ${u}_{j}$ a control state. Note that due to readability, we skipped the vector arrow in the figure.

**Figure 2.**Proposed approach of this study for vehicle parameter estimation. The on-board sensory systems measure the wheel movement ($\Delta {\varphi}_{r}$ and $\Delta {\varphi}_{l}$). Based on simulated global vehicle poses (gray arrows), we derive hidden vehicle parameters ($\Theta =({r}_{r},{r}_{l},B)$).

**Figure 3.**Graphical model including the vehicle parameters (

**a**). Since we perform simulation from a known pose using the movement model, the connections to ${x}_{t}$ disappear (

**b**).

**Figure 4.**Visualization of the kinematic models used for the performed mobile robot (

**a**) and car experiments (

**b**). For both models, we assumed a rigid frame. For the car model, we used a simplified version of Ackermann steering using a virtual wheel. The doted lines are the zero-motion lines of the wheels and thus represent kinematic constrains of the wheels.

**Figure 5.**Overview of the setup of the simulation environments, namely the test factory (

**a**) and the car environment (

**b**).

**Figure 6.**Visualization of implemented sparsed Gaussian processe (sGP) models. We select 100 auxiliary points for the mobile robot (

**a**). After analyzing the ${R}^{2}$ value of the Ackermann motion models, we chose 20 auxiliary points for the car simulation (

**b**).

**Figure 8.**Baseline estimation for both kinematic models using kernel density estimation and maximum (red line). The used bin sizes are 0.002 m for mobile robot baseline (

**a**) and 0.001 m for Ackermann steering (

**b**).

**Figure 9.**Estimation of T using the instantaneous center of rotation (ICR). The used bin size is 0.05 m.

**Table 1.**Summarized results of derived vehicle parameters from the sGP motion models (M) and simulated ground truth (GT).

${\overline{\mathit{r}}}_{\left(\mathit{b}\right)}$ | B | ${\overline{\mathit{r}}}_{\mathit{v}}$ | T | |||||
---|---|---|---|---|---|---|---|---|

M | GT | M | GT | M | GT | M | GT | |

Mobile Robot | 3.329 cm | 3.300 cm | 15.96 cm | 16 cm | - | - | - | - |

Ackermann Steering | 30.79 cm | 31.26 cm | 1.561 m | 1.586 m | 30.797 cm | 31.265 cm | 2.757 m | 2.86 m |

**Table 2.**Summarized results additional simulation results for Ackermann (AM) steering. AM 1 and AM 2 are based on the standard friction of $0.9$ for dry concrete. For AM 3, we changed the friction coefficient to $0.45$. The histograms for AM 1–AM 3 can be found in the Supplementary Materials.

${\overline{\mathit{r}}}_{\left(\mathit{b}\right)}$ | B | ${\overline{\mathit{r}}}_{\mathit{v}}$ | T | |||||
---|---|---|---|---|---|---|---|---|

M | GT | M | GT | M | GT | M | GT | |

AM 1 | 26.890 cm | 26.265 cm | 1.629 m | 1.586 m | 35.428 cm | 36.270 cm | 2.522 m | 2.860 m |

AM 2 | 29.105 cm | 30.000 cm | 1.534 m | 1.586 m | 23.142 cm | 25.000 cm | 2.437 m | 2.860 m |

AM 3 | 29.872 cm | 30.000 cm | 1.581 m | 1.586 m | 26.890 cm | 25.000 cm | 2.443 m | 2.860 m |

**Table 3.**Summarized results based on the blackbox model Bayesian neuronal networks of vehicle parameters from the sGP motion models (M) and simulated ground truth (GT). For mobile robot experiments, 15 hidden neurons were used in the hidden layer. For Ackermann steering experiments, 40 hidden neurons were used.

${\overline{\mathit{r}}}_{\left(\mathit{b}\right)}$ | B | ${\overline{\mathit{r}}}_{\mathit{v}}$ | T | |||||
---|---|---|---|---|---|---|---|---|

M | GT | M | GT | M | GT | M | GT | |

Mobile Robot | 3.683 cm | 3.300 cm | 17.866 cm | 16 cm | - | - | - | - |

Ackermann Steering | 28.429 cm | 31.26 cm | 1.436 m | 1.586 m | 28.429 cm | 31.265 cm | 2.582 m | 2.86 m |

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**MDPI and ACS Style**

Wöber, W.; Novotny, G.; Mehnen, L.; Olaverri-Monreal, C.
Autonomous Vehicles: Vehicle Parameter Estimation Using Variational Bayes and Kinematics. *Appl. Sci.* **2020**, *10*, 6317.
https://doi.org/10.3390/app10186317

**AMA Style**

Wöber W, Novotny G, Mehnen L, Olaverri-Monreal C.
Autonomous Vehicles: Vehicle Parameter Estimation Using Variational Bayes and Kinematics. *Applied Sciences*. 2020; 10(18):6317.
https://doi.org/10.3390/app10186317

**Chicago/Turabian Style**

Wöber, Wilfried, Georg Novotny, Lars Mehnen, and Cristina Olaverri-Monreal.
2020. "Autonomous Vehicles: Vehicle Parameter Estimation Using Variational Bayes and Kinematics" *Applied Sciences* 10, no. 18: 6317.
https://doi.org/10.3390/app10186317