# The State Space Subdivision Filter for Estimation on SE(2)

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## Abstract

**:**

## 1. Introduction

## 2. Density Representation and Approximation

- (A1)
- The function value in each area is identical to the corresponding grid value, i.e., $\forall \alpha \in {A}_{i}:{f}^{\omega}\left(\alpha \right)={\gamma}_{i}$.
- (A2)
- The conditional density ${f}^{\mathrm{c}}\left({\underline{x}}^{\tau}\right|\alpha )$ is the same for all $\alpha \in {A}_{i}$ (we denote it by ${f}_{i}^{\mathrm{c}}\left({\underline{x}}^{\tau}\right)$).
- (A3)
- The conditional density ${f}^{\mathrm{c}}\left({\underline{x}}^{\tau}\right|\alpha )$ is Gaussian for every considered $\alpha $.

#### 2.1. Deriving Parameters for a Given Density

#### 2.2. Normalization

#### 2.3. Providing Continuous Densities

## 3. Filter Derivation

#### 3.1. Update Step

#### 3.2. Prediction Step

- (B1)
- ${f}^{\mathrm{T},\omega}\phantom{\rule{-1.00374pt}{0ex}}\left({x}_{t+1}^{\omega}\right|{x}_{t}^{\omega})={f}^{\mathrm{T},\omega ,\mathrm{full}}\phantom{\rule{-1.00374pt}{0ex}}\left({x}_{t+1}^{\omega}\right|{\underline{x}}_{t}^{\tau},{x}_{t}^{\omega})$ holds for all ${\underline{x}}_{t}^{\tau}\in {\mathbb{R}}^{2}$.
- (B2)
- The function value of ${f}^{\mathrm{T},\omega}\phantom{\rule{-1.00374pt}{0ex}}\left({x}_{t+1}^{\omega}\right|{x}_{t}^{\omega})$ in each area in $\mathbb{S}\times \mathbb{S}$ is identical to the grid value for this area, i.e., ${f}^{\mathrm{T},\omega}\phantom{\rule{-1.00374pt}{0ex}}\left({\alpha}_{1}\right|{\alpha}_{2})={f}^{\mathrm{T},\omega}\phantom{\rule{-1.00374pt}{0ex}}\left({\beta}_{i}\right|{\beta}_{j})={\gamma}_{i,j}^{\mathrm{T}}$ for all ${\alpha}_{1}\in {A}_{i}$, ${\alpha}_{2}\in {A}_{j}$.
- (B3)
- The conditional density ${f}^{\mathrm{T},\mathrm{c},\mathrm{full}}\left({\underline{x}}_{t+1}^{\tau}\right|{\alpha}_{1},{\underline{x}}_{t}^{\tau},{\alpha}_{2})$ is the same function for all ${\alpha}_{1}\in {A}_{i}$, ${\alpha}_{2}\in {A}_{j}$ (we denote it by ${f}_{i,j}^{\mathrm{T},\mathrm{c}}\left({\underline{x}}_{t+1}^{\tau}\right|{\underline{x}}_{t}^{\tau})$).
- (B4)
- Each conditional density can be written as$${f}_{i,j}^{\mathrm{T},\mathrm{c}}\left({\underline{x}}_{t+1}^{\tau}\right|{\underline{x}}_{t}^{\tau})={f}_{\mathrm{N}}\left(\right)open="("\; close=")">{\underline{x}}_{t+1}^{\tau};{\mathbf{F}}_{t,i,j}{\underline{x}}_{t}^{\tau}+{\underline{\widehat{u}}}_{t,i,j},{\mathbf{C}}_{t,i,j}^{\underline{w}}$$

## 4. Evaluation

#### 4.1. Scenario Description

#### 4.2. Models Used by the Filters

#### 4.3. Evaluation Metrics

#### 4.4. Evaluation Results

#### 4.4.1. Evaluation Considering All Time Steps

#### 4.4.2. Evaluation of the Last Estimates and Run Times

## 5. Discussion, Summary, and Outlook

#### 5.1. Discussion

#### 5.2. Summary

#### 5.3. Outlook

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Naïve approximation of the density shown in Figure 1 based on a subdivision of the state space. 10 areas are used in this example.

**Figure 4.**Continuous density obtained by the use of a trigonometric polynomial for the periodic part and distance-based mixtures of Gaussians for the linear part.

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Pfaff, F.; Li, K.; Hanebeck, U.D.
The State Space Subdivision Filter for Estimation on SE(2). *Sensors* **2021**, *21*, 6314.
https://doi.org/10.3390/s21186314

**AMA Style**

Pfaff F, Li K, Hanebeck UD.
The State Space Subdivision Filter for Estimation on SE(2). *Sensors*. 2021; 21(18):6314.
https://doi.org/10.3390/s21186314

**Chicago/Turabian Style**

Pfaff, Florian, Kailai Li, and Uwe D. Hanebeck.
2021. "The State Space Subdivision Filter for Estimation on SE(2)" *Sensors* 21, no. 18: 6314.
https://doi.org/10.3390/s21186314