# A Factor-Graph-Based Approach to Vehicle Sideslip Angle Estimation

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## Abstract

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## 1. Introduction

## 2. Vehicle Dynamic Model

## 3. Factor Graph for Vehicle Lateral Dynamics

#### 3.1. The Estimation Problem

#### 3.2. Implementation

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Appendix A

- For priors, the actual value is the guess and the function is the difference between these two values; hence:$${b}_{1}=\left({\beta}_{0}-\overline{){\beta}_{0}}+\overline{){\beta}_{0}}\phantom{\rule{1.em}{0ex}}\right)/{\sigma}_{\beta}$$
- For dynamic factors, the actual value is zero, since it is the difference between the forward value and the same obtained by integrating the differential equation; for instance:$${b}_{5}=\left[0-{\beta}_{1}+\left(1-dt{\displaystyle \frac{{C}_{f}+{C}_{r}}{m{u}_{1}}}\right){\beta}_{0}-dt\left({\displaystyle \frac{{C}_{f}{l}_{f}-{C}_{r}{l}_{r}}{m{u}_{1}^{2}}}+1\right){r}_{0}+dt{\displaystyle \frac{{C}_{f}{\delta}_{1}}{m{u}_{1}}}\right]/{\sigma}_{r}$$
- Finally, measures follow this rationale (only yaw rate is taken for the sake of brevity):$${b}_{3}=\left({\dot{\phi}}_{1}+{r}_{0}\right)/{\sigma}_{\dot{\phi}}$$

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**Figure 3.**FG relative to the first three steps of the estimation problem: the first step with priors is shown in the black dashed window, and the generic step is shown in the gray dashed-dotted window, with the involved factors.

**Figure 4.**Vehicle sideslip angle $\beta $ estimate at the top and yaw rate r at the bottom obtained by using a KF-based observer applied to the linear bicycle model.

**Figure 5.**Vehicle sideslip angle $\beta $ estimate at the top and yaw rate r at the bottom obtained by using FG applied to the linear bicycle model.

**Figure 6.**Estimation of vehicle sideslip angle $\beta $ by considering both KF and FG estimators with a window of 5 samples.

**Figure 7.**Estimation of vehicle sideslip angle $\beta $ by considering both KF and FG estimators for small values of the sideslip angle.

**Figure 8.**Estimation of vehicle sideslip angle $\beta $ by considering both KF and FG estimators with a window of 5 samples, for another set of real data.

**Figure 9.**Estimation of vehicle sideslip angle $\beta $ by considering both KF and FG batch estimators.

**Figure 10.**Absolute error in the estimation of sideslip angle $\beta $ by considering KF against the FG batch estimator.

**Figure 11.**Path followed by the vehicle during a single lap: black is ground truth, red represents KF, and green indicates FG.

Factor | ${\Delta}_{1}$ | ${\Delta}_{2}$ | ${\Delta}_{3}$ | ${\Delta}_{4}$ | ${\Delta}_{5}$ | ${\Delta}_{6}$ | b |
---|---|---|---|---|---|---|---|

p1 | ${A}_{1,1}$ | ${b}_{1}$ | |||||

p2 | ${A}_{2,2}$ | ${b}_{2}$ | |||||

m1 | ${A}_{3,2}$ | ${b}_{3}$ | |||||

m2 | ${A}_{4,1}$ | ${A}_{4,2}$ | ${b}_{4}$ | ||||

d1 | ${A}_{5,1}$ | ${A}_{5,2}$ | ${A}_{5,3}$ | ${b}_{5}$ | |||

d2 | ${A}_{6,1}$ | ${A}_{6,2}$ | ${A}_{6,4}$ | ${b}_{6}$ | |||

m3 | ${A}_{7,4}$ | ${b}_{7}$ | |||||

m4 | ${A}_{8,3}$ | ${A}_{8,4}$ | ${b}_{8}$ | ||||

d3 | ${A}_{9,3}$ | ${A}_{9,4}$ | ${A}_{9,5}$ | ${b}_{9}$ | |||

d4 | ${A}_{10,3}$ | ${A}_{10,4}$ | ${A}_{10,6}$ | ${b}_{10}$ | |||

m5 | ${A}_{11,6}$ | ${b}_{11}$ | |||||

m6 | ${A}_{12,5}$ | ${A}_{12,6}$ | ${b}_{12}$ |

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**MDPI and ACS Style**

Leanza, A.; Reina, G.; Blanco-Claraco, J.-L.
A Factor-Graph-Based Approach to Vehicle Sideslip Angle Estimation. *Sensors* **2021**, *21*, 5409.
https://doi.org/10.3390/s21165409

**AMA Style**

Leanza A, Reina G, Blanco-Claraco J-L.
A Factor-Graph-Based Approach to Vehicle Sideslip Angle Estimation. *Sensors*. 2021; 21(16):5409.
https://doi.org/10.3390/s21165409

**Chicago/Turabian Style**

Leanza, Antonio, Giulio Reina, and José-Luis Blanco-Claraco.
2021. "A Factor-Graph-Based Approach to Vehicle Sideslip Angle Estimation" *Sensors* 21, no. 16: 5409.
https://doi.org/10.3390/s21165409