# Robust Multipath-Assisted SLAM with Unknown Process Noise and Clutter Intensity

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Environment Map and Signal Model

**J**PTs. The UE position is unknown and time-varying, which is denoted by ${\mathit{u}}_{n}\in {\mathbb{R}}^{2}$, while the PT positions ${\mathit{a}}_{1}^{(j)}\in {\mathbb{R}}^{2},j\in \left\{1,\dots ,J\right\}$ may be unknown, where the number of PT

**J**is supposed to be known. Assume that there are ${L}_{n}^{(j)}-1$ VTs associated with the

**j**th PT and that their positions are unknown and represented by ${\mathit{a}}_{l}^{(j)}\in {\mathbb{R}}^{2},l\in \left\{2,\dots ,{L}_{n}^{(j)}\right\}$. PTs and VTs are features of the environmental map, and their number is unknown and varies with the UE position. At each discrete time n, the radio signal $s(t)$ is transmitted from the

**j**th PT to the mobile UE, and the received signal of the UE can be expressed as [17]

**j**th PT ($l=1$) or the associated VTs ($l\in \left\{2,\dots ,{L}_{n}^{(j)}\right\}$); thus, we have ${\tau}_{l,n}^{(j)}=\Vert {\mathit{u}}_{n}-{\mathit{a}}_{l}^{(j)}\Vert /c$, where

**c**is the speed of light. Furthermore, ${g}_{n}^{(j)}(t)$ in (1) denotes the diffuse MPCs, which is regarded as interference. Finally, ${\omega}_{\mathrm{GWN}}(t)$ represents additive white Gaussian noise.

**m**th MPC of PT

**j**are obtained by multiplying the delay estimation ${\widehat{\tau}}_{m,n}^{(j)}$ with the speed of light

**c**, which are the measurements of the proposed algorithm used in this paper. For convenience, we define the stacked vectors of ${z}_{m,n}^{(j)}$, ${\mathit{z}}_{n}^{(j)}\triangleq {\left[{z}_{1,n}^{(j)}\cdots {z}_{{M}_{n}^{(j)},n}^{(j)}\right]}^{\mathrm{T}}$, ${\mathit{z}}_{n}\triangleq {\left[{\mathit{z}}_{n}^{(1)\mathrm{T}}\cdots {\mathit{z}}_{n}^{(J)\mathrm{T}}\right]}^{\mathrm{T}}$ and $\mathit{z}\triangleq {\left[{\mathit{z}}_{1}^{\mathrm{T}}\cdots {\mathit{z}}_{n}^{\mathrm{T}}\right]}^{\mathrm{T}}$ and the stacked vectors of ${M}_{n}^{(j)}$, ${\mathit{m}}_{n}\triangleq {\left[{M}_{n}^{(1)}\cdots {M}_{n}^{(J)}\right]}^{\mathrm{T}}$ and $\mathit{m}={\left[{\mathit{m}}_{1}^{\mathrm{T}}\cdots {\mathit{m}}_{n}^{\mathrm{T}}\right]}^{\mathrm{T}}$.

## 3. System Model with Unknown Process Noise and Clutter Intensity

#### 3.1. UE State and PF States

#### 3.2. Markov Chain Modeling of the Process Noise and the Clutter Intensity

#### 3.3. State Evolution with Unknown Process Noise

#### 3.4. Prior Distributions with Unknown Clutter Intensity

#### 3.5. Measurement Model and Likelihood Function

## 4. The Proposed Algorithm

#### 4.1. Detection and Estimation

#### 4.2. Joint Posterior Distribution and Factor Graph

#### 4.3. BP Message Passing Algorithm

- Prediction.

- 2.
- Measurement evaluation of legacy PFs.

- 3.
- Measurement evaluation of new PFs.

- 4.
- Iterative DA

- 5.
- Measurement update for legacy PFs

- 6.
- Measurement update for new PFs

- 7.
- Calculation of beliefs

## 5. Simulation Results and Discussions

#### 5.1. Simulation Parameters

#### 5.2. First Scenario: Unknown Process Noise Only

#### 5.3. Second Scenario: Unknown Clutter Intensity Only

#### 5.4. Third Scenario: Unknown Process Noise and Clutter Intensity

#### 5.5. Calculation Complexity Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Factor graph representing the factorization of the joint posterior probability density function (PDF) in (30). All factor nodes, variable nodes and messages related to the user equipment (UE) augmented state are represented in red color, those related to legacy potential features (PF) states are represented in purple color, those related to new PF states are represented in blue color, and the section of the clutter intensity and the iterative data association (DA) are represented in orange color and green color, respectively. The abbreviations are defined as $K\triangleq {K}_{n}^{(j)}$, $M\triangleq {M}_{n}^{(j)}$, $\mathit{x}\triangleq {\mathit{x}}_{n}$, ${\underset{\_}{\mathit{y}}}_{k}\triangleq {\underset{\_}{\mathit{y}}}_{k,n}^{(j)}$, ${h}_{k}\triangleq {h}_{k,n}^{(j)}$, ${q}_{m}\triangleq {q}_{m,n}^{(j)}$, ${\lambda}_{\mathrm{CL}}\triangleq {\lambda}_{\mathrm{CL},n}^{(j)}$, ${\phi}_{k,m}\triangleq \phi ({h}_{k,n}^{(j)},{q}_{m,n}^{(j)})$, ${c}_{m}\triangleq c({\mathit{x}}_{n},{\overline{\mathit{a}}}_{m,n}^{(j)},{\overline{e}}_{m,n}^{(j)},{q}_{m,n}^{(j)},{\lambda}_{\mathrm{CL},n}^{(j)};{\mathit{z}}_{n}^{(j)})$, ${b}_{k}\triangleq b({\mathit{x}}_{n},{\underset{\_}{\mathit{a}}}_{k,n}^{(j)},{\underset{\_}{e}}_{k,n}^{(j)},{h}_{k,n}^{(j)},{\lambda}_{\mathrm{CL},n}^{(j)};{\mathit{z}}_{n}^{(j)})$, $f\triangleq f({\mathit{x}}_{n}|{\mathit{x}}_{n-1})$, ${f}_{k}\triangleq f({\underset{\_}{\mathit{y}}}_{k,n}^{(j)}|{\mathit{y}}_{k,n-1}^{(j)})$, $p\triangleq p({\lambda}_{\mathrm{CL},n}^{(j)}|{\lambda}_{\mathrm{CL},n-1}^{(j)})$, $\alpha \triangleq \alpha ({\mathit{x}}_{n})$, ${\alpha}_{k}\triangleq {\alpha}_{k}({\underset{\_}{\mathit{a}}}_{k,n}^{(j)},{\underset{\_}{e}}_{k,n}^{(j)})$, ${\stackrel{\u2322}{f}}^{-}\triangleq \stackrel{\u2322}{f}({\mathit{x}}_{n-1})$, $\stackrel{\u2322}{f}\triangleq \stackrel{\u2322}{f}({\mathit{x}}_{n})$, ${\stackrel{\u2322}{\underset{\_}{f}}}_{k}^{-}\triangleq \underset{\_}{\stackrel{\u2322}{f}}({\mathit{a}}_{k,n-1}^{(j)},{e}_{k,n-1}^{(j)})$, ${\stackrel{\u2322}{\underset{\_}{f}}}_{k}\triangleq \underset{\_}{\stackrel{\u2322}{f}}({\underset{\_}{\mathit{a}}}_{k,n}^{(j)},{\underset{\_}{e}}_{k,n}^{(j)})$, ${\stackrel{\u2322}{\overline{f}}}_{m}\triangleq \stackrel{\u2322}{\overline{f}}({\overline{\mathit{a}}}_{m,n}^{(j)},{\overline{e}}_{m,n}^{(j)})$, ${\stackrel{\u2322}{p}}^{-}\triangleq \stackrel{\u2322}{p}({\lambda}_{\mathrm{CL},n-1}^{(j)})$, ${\stackrel{\u2322}{p}}^{-}\triangleq \stackrel{\u2322}{p}({\lambda}_{\mathrm{CL},n}^{(j)})$, ${\beta}_{k}\triangleq \beta ({\lambda}_{\mathrm{CL},n}^{(j)})$, $\chi \triangleq \chi ({\lambda}_{\mathrm{CL},n}^{(j)})$, ${\epsilon}_{k}\triangleq \epsilon ({h}_{k,n}^{(j)})$, ${\xi}_{m}\triangleq \xi ({q}_{m,n}^{(j)})$, ${\gamma}_{k}\triangleq \gamma ({h}_{k,n}^{(j)})$, ${\nu}_{m}\triangleq \nu ({q}_{m,n}^{(j)})$, ${\delta}_{m,k}\triangleq {\delta}_{m\to k}^{(t)}({h}_{k,n}^{(j)})$, ${\varsigma}_{k,m}\triangleq {\varsigma}_{k\to m}^{(t)}({q}_{m,n}^{(j)})$, ${\eta}_{k}^{(j)}\triangleq {\eta}_{k}^{(j)}({\mathit{x}}_{n})$, ${\eta}_{k}\triangleq \eta ({\underset{\_}{\mathit{a}}}_{k,n}^{(j)},{\underset{\_}{e}}_{k,n}^{(j)})$, and ${\mu}_{m}\triangleq \mu ({\overline{\mathit{a}}}_{m,n}^{(j)},{\overline{e}}_{m,n}^{(j)})$.

**Figure 3.**Results for the first scenario: (

**a**) mean optimal subpattern assignment (MOSPA) error for physical transmitter (PT) 1 and the associated virtual transmitters (VTs); (

**b**) MOSPA error for PT2 and the associated VTs; (

**c**) average number of detected PFs associated with PT1 every 60 s; (

**d**) average number of detected PFs associated with PT2 every 60 s; and (

**e**) user equipment (UE) position root mean square errors (RMSEs).

**Figure 4.**Results for the second scenario: (

**a**) MOSPA error for PT1 and the associated VTs; (

**b**) MOSPA error for PT2 and the associated VTs; (

**c**) average number of detected PFs associated with PT1 every 60 s; (

**d**) average number of detected PFs associated with PT2 every 60 s; and (

**e**) UE position RMSE.

**Figure 5.**Results for the third scenario: (

**a**) MOSPA error for PT1 and the associated VTs; (

**b**) MOSPA error for PT2 and the associated VTs; (

**c**) average number of detected PFs associated with PT1 every 60 s; (

**d**) average number of detected PFs associated with PT2 every 60 s; and (

**e**) UE position RMSE.

${\sigma}_{a}$ | ${\sigma}_{b,1}$ | ${\lambda}_{new,1}^{(j)}$ | ${v}_{m,n}^{(j)}$ | ${P}_{d}$ | ${\lambda}_{b}^{(j)}$ | ${P}_{s}$ | ${P}_{th}$ | ${P}_{pr}$ | ${N}_{par}$ | Simulation Runs |
---|---|---|---|---|---|---|---|---|---|---|

10^{−4} m | 10^{−3} m | 6 | 0.1 m | 0.95 | 10^{−4} | 0.999 | 0.5 | 10^{−4} | 10,000 | 100 |

Algorithm | UE RMSE | MOSPA | No. of PFs |
---|---|---|---|

BP-SLAM A | 0.2891 m | 1.1143 m | 0.1090 |

BP-SLAM B | 0.1984 m | 1.0442 m | 0.2564 |

Proposed algorithm | 0.1605 m | 0.8759 m | 0.0005 |

BP-SLAM (known params) | 0.0949 m | 0.7739 m | 0 (True) |

Algorithm | UE RMSE | MOSPA | No. of PFs |
---|---|---|---|

BP-SLAM A | 0.3768 m | 1.2199 m | 8.0151 |

BP-SLAM B | 0.1083 m | 1.0006 m | 0.8329 |

Proposed algorithm | 0.0741 m | 0.7659 m | 0.0678 |

BP-SLAM (known params) | 0.0714 m | 0.7376 m | 0 (True) |

Algorithm | UE RMSE | MOSPA | No. of PFs |
---|---|---|---|

BP-SLAM A | 1.4636 m | 2.2396 m | 8.4696 |

BP-SLAM B | 0.2452 m | 1.0713 m | 0.8099 |

Proposed algorithm | 0.1121 m | 0.7659 m | 0.0628 |

BP-SLAM (known params) | 0.0699 m | 0.6738 m | 0 (True) |

Algorithm | Complexity Order | CPU Run Time |
---|---|---|

BP-SLAM (unknown params) | $\mathcal{O}({N}_{\mathrm{par}})$ | 0.0645 s |

Proposed algorithm | $\mathcal{O}({N}_{\mathrm{par}}+{N}_{b}^{2}+{N}_{g}^{2})$ | 0.0761 s |

BP-SLAM (known params) | $\mathcal{O}({N}_{\mathrm{par}})$ | 0.0652 s |

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**MDPI and ACS Style**

Dan, Z.; Lian, B.; Tang, C.
Robust Multipath-Assisted SLAM with Unknown Process Noise and Clutter Intensity. *Remote Sens.* **2021**, *13*, 1625.
https://doi.org/10.3390/rs13091625

**AMA Style**

Dan Z, Lian B, Tang C.
Robust Multipath-Assisted SLAM with Unknown Process Noise and Clutter Intensity. *Remote Sensing*. 2021; 13(9):1625.
https://doi.org/10.3390/rs13091625

**Chicago/Turabian Style**

Dan, Zesheng, Baowang Lian, and Chengkai Tang.
2021. "Robust Multipath-Assisted SLAM with Unknown Process Noise and Clutter Intensity" *Remote Sensing* 13, no. 9: 1625.
https://doi.org/10.3390/rs13091625