# Limits on Cooperative Positioning for a Robotic Swarm with Time of Flight Ranging over Two-Ray Ground Reflection Channel

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## Abstract

**:**

## 1. Introduction

## 2. Time of Flight Ranging

#### 2.1. Two-Ray Ground Reflection Channel

#### 2.2. System Model

#### 2.3. Maximum-Likelihood Estimator

#### 2.4. Lower Bound on Ranging Variance

## 3. Ranging Performance Results

#### 3.1. Ranging Variance

#### 3.2. Ranging Bias

#### 3.3. Ranging RMSE

## 4. Cooperative Positioning

#### 4.1. Lower Bound on Positioning Variance

#### 4.1.1. Non-Cooperative Positioning

#### 4.1.2. Cooperative Positioning

#### 4.2. Maximum-Likelihood Position Estimation

## 5. Non-Cooperative Positioning Performance

## 6. Cooperative Positioning Performance

## 7. Conclusions

## 8. Outlook

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

2D | two-dimensional. |

3D | three-dimensional. |

BFGS | Broyden–Fletcher–Goldfarb–Shanno. |

CDF | cumulative distribution function. |

CIR | channel impulse response. |

CRB | Cramér-Rao bound. |

DLL | delay-locked loop. |

FFT | fast Fourier transform. |

FIM | Fisher information matrix. |

FSPL | free-space path loss. |

GNSS | global navigation satellite system. |

IFFT | inverse fast Fourier transform. |

LoS | line-of-sight. |

MEE | multipath error envelope. |

ML | maximum likelihood. |

MSE | mean square error. |

OFDM | orthogonal frequency-division multiplexing. |

RMSE | root mean square error. |

RTT | round trip time. |

SNR | signal to noise ratio. |

ToF | time of flight. |

TWR | two-way ranging. |

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**Figure 1.**Lunar robotic swarm exploration scenario with multiple robots and instrument packages deployed on the lunar surface. All entities are connected with wireless links enabling communication, ranging and positioning. Infrastructure for communication and radio navigation is not available.

**Figure 4.**Ranging standard deviation ${\sigma}_{\mathrm{d}}$ over horizontal distance ${d}_{\mathrm{H}}$. Transmitter height ${h}_{\mathrm{T}}=0.8\text{}\mathrm{m}$ and receiver height ${h}_{\mathrm{R}}=0.8\text{}\mathrm{m}$.

**Figure 5.**Ranging standard deviation ${\sigma}_{\mathrm{d}}$ over horizontal distance ${d}_{\mathrm{H}}$. Transmitter height ${h}_{\mathrm{T}}=4.5\text{}\mathrm{m}$ and receiver height ${h}_{\mathrm{R}}=0.8\text{}\mathrm{m}$.

**Figure 6.**Ranging bias $\overline{d}$ over horizontal distance ${d}_{\mathrm{H}}$ for three selected carrier frequencies ${f}_{\mathrm{c}}$. Transmitter height ${h}_{\mathrm{T}}=0.8\text{}\mathrm{m}$ and receiver height ${h}_{\mathrm{R}}=0.8\text{}\mathrm{m}$.

**Figure 7.**Ranging bias $\overline{d}$ over horizontal distance ${d}_{\mathrm{H}}$ for three selected carrier frequencies ${f}_{\mathrm{c}}$. Transmitter height ${h}_{\mathrm{T}}=4.5\text{}\mathrm{m}$ and receiver height ${h}_{\mathrm{R}}=0.8\text{}\mathrm{m}$.

**Figure 8.**Ranging bias $\overline{d}$ over horizontal distance ${d}_{\mathrm{H}}$ for carrier frequencies between 1 $\mathrm{GHz}$ and 7 $\mathrm{GHz}$. Transmitter height ${h}_{\mathrm{T}}=2.5\text{}\mathrm{m}$ and receiver height ${h}_{\mathrm{R}}=2.5\text{}\mathrm{m}$.

**Figure 9.**Ranging RMSE and standard deviation over horizontal distance ${d}_{\mathrm{H}}$. Transmitter height ${h}_{\mathrm{T}}=2.5\text{}\mathrm{m}$ and receiver height ${h}_{\mathrm{R}}=2.5\text{}\mathrm{m}$.

**Figure 10.**Positioning of two swarm elements based on distance observations. Anchors have known position, and ranging links among swarm elements are used for cooperative positioning.

**Figure 11.**Positioning error of a single swarm element based on the CRB with free-space path loss model. Black triangles show the anchor node positions and contour lines are plotted for every $0.5$ $\mathrm{m}$.

**Figure 12.**Positioning error of a single swarm element based on the CRB with two-ray ground reflection model. Black triangles show the anchor node positions, and contour lines are plotted for every $0.5$ $\mathrm{m}$. The positioning error significantly increases and shows rapid spatial variations. The color coding is saturated at 2 $\mathrm{m}$, and errors of up to 13 $\mathrm{m}$ are reached.

**Figure 13.**Positioning error of a single swarm element based on ML estimation with the ranging bias and ranging variance determined by the two-ray ground reflection model. Black triangles show the anchor node positions and contour lines are plotted every $0.5$ $\mathrm{m}$. For visual clarity, we removed contour labels and the color-coding is saturated at 2 $\mathrm{m}$.

**Figure 14.**CDF of the positioning error for $Q=10$ swarm-elements. The CRB with free-space path loss provides the reference. The estimator approaches the CRB, and we see a significantly increased positioning error with the two-ray ground-reflection model and included ranging bias $\overline{d}$.

**Figure 15.**CDF of the positioning error for an increasing number of Q swarm-elements. As expected, we see an improvement for increasing Q, but it becomes small beyond five swarm-elements.

**Figure 16.**90th percentile of the positioning error over the number of swarm elements Q. An increasing number of Q results in a lower positioning error in general. However, for the two-ray ground reflection model with ranging bias $\overline{d}$, the error decrease becomes small for $Q>5$.

Parameter | Value |
---|---|

Carrier frequency ${f}_{\mathrm{c}}$ | $5.7$$\mathrm{GHz}$ |

Signal bandwidth $B={f}_{\mathrm{s}}$ | 20$\mathrm{MHz}$ |

OFDM symbol length N | 1024 |

Allocated subcarriers ${N}_{\mathrm{u}}$ | 922 |

Cyclix prefix length C | 144 |

Transmit power | 1$\mathrm{mW}$ |

Receiver temperature | 300$\mathrm{K}$ |

Polarization | vertical |

Transmitter antenna | omnidirectional |

Receiver antenna | omnidirectional |

Ground permittivity ${\u03f5}_{\mathrm{g}}$ | 3.5 |

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**MDPI and ACS Style**

Staudinger, E.; Pöhlmann, R.; Dammann, A.; Zhang, S.
Limits on Cooperative Positioning for a Robotic Swarm with Time of Flight Ranging over Two-Ray Ground Reflection Channel. *Electronics* **2023**, *12*, 2139.
https://doi.org/10.3390/electronics12092139

**AMA Style**

Staudinger E, Pöhlmann R, Dammann A, Zhang S.
Limits on Cooperative Positioning for a Robotic Swarm with Time of Flight Ranging over Two-Ray Ground Reflection Channel. *Electronics*. 2023; 12(9):2139.
https://doi.org/10.3390/electronics12092139

**Chicago/Turabian Style**

Staudinger, Emanuel, Robert Pöhlmann, Armin Dammann, and Siwei Zhang.
2023. "Limits on Cooperative Positioning for a Robotic Swarm with Time of Flight Ranging over Two-Ray Ground Reflection Channel" *Electronics* 12, no. 9: 2139.
https://doi.org/10.3390/electronics12092139