# Distributed Relative Localization Algorithms for Multi-Robot Networks: A Survey

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## Abstract

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## 1. Introduction

- This work summarizes the distributed multi-robot network relative localization methods and classifies various methods according to their design methodologies and types of measurements. A detailed introduction is presented for each type of distributed relative localization algorithm. The advantages and limitations of each type of distributed relative localization algorithm are analyzed and compared.
- Key problems in distributed localization algorithms, namely local subnetwork organization, communication efficiency, and the robustness of distributed localization, are investigated.
- The popular simulation experiment platforms are investigated, and the characteristics of each platform are analyzed from different aspects, which can be a reference for researchers conducting simulation experiments.

## 2. Related Reviews

## 3. Distance-Based Distributed Relative Localization Methods

- Barycentric coordinate-based methods, which present a linear representation of the localization problem and derive naturally distributed iterative algorithms running at distributed nodes that can guarantee converging to the correct states under specific conditions.
- Distributed algorithms developed from their centralized versions, which include: (i) distributed multi-dimensional scaling (MDS), (ii) distributed semi-definite programming (SDP), and (iii) distributed nonlinear optimization.

#### 3.1. Barycentric Coordinate-Based Algorithms

#### 3.1.1. Formulation of Barycentric Distributed Localization

#### 3.1.2. Dealing with Convex Hull Constraints and Noises

#### 3.1.3. Communication Problems for Barycentric Distributed Localization

#### 3.2. Distributed Algorithms Transformed from Centralized Algorithms

#### 3.2.1. Distributed Multi-Dimensional Scaling

- Step 1: Divide the network into sub-networks (clusters). In the cluster, pairwise distances are needed.In distributed systems, each agent in the network runs the division algorithm individually and asynchronously without centralized control and scheduling. Generally, we can divide the network according to the characteristics of the graph nodes. The simplest way is to divide the nodes according to their degree centrality [9,33]. First, for each node i in the network, construct the set ${S}_{i}$ consisting of all the direct neighbors of node i. Then, update ${S}_{i}$ by ${S}_{i}={S}_{i}\cup {S}_{j}$ for $\forall j\in {S}_{i}$. Search the largest cardinality group in the network and set it as a cluster. Repeat the process in the last part of the network until the network is divided.
- Step 2: Perform the MDS-based algorithm in each cluster to calculate the relative coordinates. This step constructs the accurate local graph.We can conduct the classic MDS method to calculate the relative coordinates in the cluster. While to reduce the computational complexity, the improved MDS, MDS-MAP [34], is deployed in many papers [26,33,35]. The MDS-MAP algorithm uses the Floyd algorithm in each cluster to fill in the missing range measurements, thus, reducing the range error caused by too many hops.However, the method does not address the problem of errors in the measurement itself. When the measurement error is significant, the matrix in the MDS algorithm may not be able to perform the eigenvalue decomposition. The operations on linear equations and matrix decomposition in MDS are susceptible to measurement errors. In paper [26], the authors use singular value decomposition instead of eigenvalue decomposition to deal with the connection matrix and compute the relative coordinates matrix Z. Then, a more accurate matrix Z is obtained by minimizing $\parallel B-{Z}^{T}{Z\parallel}_{2}^{2}$, where B can be derived from the matrix composed of the square of the distances.
- Step 3: Merge all the local graphs into a global network based on the common nodes in each local graph. This is a key step in the algorithm, especially for large networks [36,37].The traditional method uses the overlapping edges between subgraphs for merging [38], but this method is prone to ambiguity. Kim et al. [35] add a distance constraint to the splicing process, i.e., they choose the network that does not generate new edges as the result.In fact, the uniqueness of subgraph merging can be determined: Two individual maps can be uniquely merged in m-dimensional space if and only if at least $m+1$ common nodes exist. Dan et al. [33] propose that merging remains unique when the number of common nodes is less than m+1; however, some specific edges exist. Based on these conditions, they design corresponding merging strategies that guarantee uniqueness.

#### 3.3. Distributed Semi Definite Programming

#### 3.4. Distributed Nonlinear Optimization

## 4. Bearing-Based Distributed Relative Localization Methods

#### 4.1. Formulation of Bearing-Based Localization

#### 4.2. Distributed Bearing-Based Localization

## 5. Multiple Measurement Fusion

#### 5.1. Filter-Based Multi-Sensor Fusion and Localization

#### 5.1.1. Particle Filter and Relative Localization

#### 5.1.2. Formulation of Kalman-Filter-Based Localization

#### 5.1.3. Distributed Kalman-Filter-Based Relative Localization

**Distributed Kalman Filter:**Kalman filtering or its modifications are also applied in multi-robot network localization systems. The paper [66] transformed the centralized extended Kalman filter into a decentralized form. A “small” Kalman filter runs distributively on each robot. When no observation occurs between robots, each robot estimates its pose as a single robot. When two robots observe each other, they communicate to exchange states and measurements. The robots calculate the Kalman gain based on the synergy information and update their state in the update phase.

**Distributed MSCKF and Distributed VIO:**Due to the accuracy, efficiency, and robustness of MSCKF and high-quality open-source vision odometry, such as open-VINS [67], the MSCKF framework has been used in recent years for filter-based cooperative multi-robot localization. In the distributed MSCKF, co-viewed environmental feature points can be used as observations between two robots, which is applied to the update of states and covariances [68]. In the algorithm proposed by Zhu et al. [69], IMU pre-integration is used for the prediction process of filtering.

**Distributed MSCKF Incorporating Ranging Information**: In addition to fuse vision and IMU data, the MSCKF can further utilize the ranging information between robots [70]. There may be distance measurements between robots apart from the direct target detection and common visual feature point matching. The distance measurement model is added to the observation equation in the method proposed by Chenchana et al. [71] to achieve better localization precision. However, sensors for ranging, such as UWB, can be unstable in complex environments. Particularly in the presence of occlusion, the measurement error is significant, and thus the raw range information needs to be filtered. The noise point criterion is that the covariance of the data exceeds a certain threshold.

#### 5.2. Distributed Graph-Optimization-Based Multi-Sensor Fusion and Localization

#### 5.2.1. The Formulation of Distributed Graph Optimization

**The distance edge:**The sensors and model of distance measurements were introduced in Section 3. The formulation of distance edge is:

**The re-projection edge:**When vision-detection algorithms are available, we can detect in images landmarks or other robots whose positions are known or can be estimated. The difference between the detected pixel coordinates and the pixel coordinates obtained by reprojection is the reprojection error:

**The ego-motion edge:**When a single robot has an odometry system to estimate the ego-motion, we can build an ego-motion edge (also called an odometry edge):

**The loop closure edge:**When the robot arrives at a scene that it has experienced before (intra-robot loop closure) or that other robots have experienced (inter-robot loop closure), the robot can obtain a new constraint by matching it with the observations in the historical data:

#### 5.2.2. Algorithms for Distributed Graph Optimization

#### 5.3. Other Solvers for Distributed Graph Optimization

- (1)
- Gauss–Seidel Algorithm

- (2)
- Non-Convex Riemannian Optimization

- (3)
- Factor Graph Optimization

## 6. Distributed Simultaneous Localization and Mapping

#### 6.1. Distributed SLAM Algorithms

#### 6.1.1. Distributed Loop Closures

#### 6.1.2. Relocalization and Global Localization

#### 6.1.3. Data Association

#### 6.2. Distributed Mapping-Oriented Algorithms

## 7. Local Subnetwork Organization and Communication

#### 7.1. Local Subnetwork Organization

#### 7.2. Communication Efficiency

## 8. Robust Distributed Localization

#### 8.1. Initialization

#### 8.2. Measurement Noises

## 9. Challenges and Future Study on Distributed Localization

#### 9.1. Dynamic Topology

#### 9.2. Scalability

#### 9.3. Distributed Active Localization

## 10. Simulation Platforms and Datasets

- Open source. The open source experimental platform allows researchers to change the details of the simulation according to their experimental needs, such as the model of the robot, the type and parameters of the sensors, and the communication mode. The open source platform has better scalability.
- Supported OS. Windows, Linux, and Mac are the most popular operating systems on the market. Robots or other edge devices may run on any of these operating systems. Therefore, a simulation platform adapted to these operating system is closer to the real world.
- ROS supported. Robot Operating System (ROS) [130] is one of the most popular robot software frameworks. The simulation platform can provide data sources and operational feedback for ROS to simulate real-world experiments.
- Model. The model of the simulation object is the most critical part of the experiment. The model here includes not only the physical appearance but also sensors, communication modules, and computing power.
- Scene quality. The robot’s ability to perceive and operate on the environment is basic. Therefore, the quality of the environment should be as close to reality as possible and also simulate the noise in the environment.
- Distributed capabilities. Simulation experiments are usually executed on a single machine; however, the multi-robot distributed localization algorithm is designed to be executed on multiple robots in a distributed manner. This is also important for the implementation of distributed simulation.

## 11. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Sequential iteration (

**top**) and parallel iteration (

**bottom**). ${X}_{i}^{p}$ denotes the state of the i-th node after $p-th$ iteration.

**Figure 3.**Distributed graph optimization for multi-robot localization. The left is the global pose graph, and the right figure is the local pose graph of Robot 2.

**Figure 4.**Estimated trajectories of Kimera-Multi [119] compared to the ground truth (GT). The absolute trajectory error (ATE) in that experiment was 1.46 m.

Survey | Date | Focus | Multi-Robot Localization | Distributed Methods | Challenges & Trend | Experiment Platform |
---|---|---|---|---|---|---|

[15] | 2014 | Collaboration systems | Yes | Few | No | No |

[16] | 2016 | Localization | Yes | Few | Yes | No |

[17] | 2019 | Localization, IoT | Yes | Medium | Yes | No |

[18] | 2020 | Localization, UWB | Yes | Few | No | No |

[14] | 2021 | Localization, single robot | No | No | Yes | No |

[21] | 2021 | Localization, single robot | No | No | Yes | No |

[20] | 2022 | SLAM | Yes | Few | Yes | No |

Our work | 2022 | Distributed relative localization | Yes | Most | Yes | Yes |

**Table 2.**Comparison of distributed location algorithms based on distance measurement. Note that s is a parameter set in advance, which may be different in different papers.

Algorithm Category | Paper | Anchor Number | Characteristics |
---|---|---|---|

Barycentric Coordinate-Based | [24,27,29] | ≥3 | Slow convergence speed; Sensitive to noise; Topology may be limited to convex hull. |

Distributed MDS | [9,33,50] | ≥0 | Dense network; Sensitive to noise. |

Distributed SDP | [41,43,49] | ≥0 | Dense network; Sensitive to noise; High convergence speed. |

Nonlinear Optimization | [11,46] | ≥s | Tolerate to sparsity in some extend; High convergence speed; General model. |

Solution | Year | Method | Sensors | Characteristics |
---|---|---|---|---|

Graph optimization | 2020 | Xu et al. [78] | Camera, IMU, UWB | Involing range measurement. Decentralized scheme. |

2022 | Omni-swarm [80] | Fisheye-camera, IMU, UWB | Omni-directional. | |

2022 | Nguyen et al. [83] | Camera, IMU, UWB | Range-focused fusion. | |

Filter | 2018 | Chenchana et al. [71] | Camera, IMU, UWB | Involving range measurement |

2021 | DISC-VIO [69] | Camera, IMU | Using MSCKF. Only simulation experiment. | |

Factor graph | 2013 | DDF-sam [10] | Camera, IMU, GPS | Factor graph. Anti-factor. |

Gauss–Seidel | 2016 | Choudhary et al. [87] | LIDAR, IMU, wheel odometry | Two-stage distributed Gauss–Seidel approach. |

Year | Method | Sensors (Frontend) | Backend | Characteristics |
---|---|---|---|---|

2017 | Choudhary et al. [118] | LIDAR, IMU, wheel odometry | Gauss–Seidel | Object-based model |

2018 | Cieslewski et al. [106] | Camera | Gauss–Seidel | Data efficient. |

2020 | DOOR-SLAM [108] | Camera | Graph optimization | Outlier rejection. Communication Efficiency. |

2021 | Karrer et al. [121] | Camera, IMU, UWB | Filter | Invole range measurement |

2022 | Kimera-multi [122] | Camera, IMU | Graph optimization | Semantic map. Outlier Rejection. |

Platform | Open Source | Supported OS | ROS Supported | Model | Scene Quality | Distributed Capabilities |
---|---|---|---|---|---|---|

AirSim [131] | ✓ | Win, Linux | ✓ | Car, Quadrotor | High | |

Gazebo [132] | ✓ | Linux, MacOS | ✓ | Robot, Quadrotor | Middle | |

RflySim [133] | ✓ | Win | ✓ | Car, Quadrotor, Fixed-wing | High | ✓ |

Jmavsim [134] | ✓ | Win, Linux, MacOS | ✓ | Quadrotor | Low | |

Morse [135] | ✓ | Linux | ✓ | Robot, Quadrotor | Middle | |

FightGear [136] | ✓ | Win, Linux, MacOS | Quadrotor, Fixed-wing | Low | ||

XPlane [137] | ✓ | Win, Linux | Fixed-wing | High | ||

HackFlightSim [138] | ✓ | Win, Linux | ✓ | Fixed-wing, Multirotor | Middle |

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Wang, S.; Wang, Y.; Li, D.; Zhao, Q.
Distributed Relative Localization Algorithms for Multi-Robot Networks: A Survey. *Sensors* **2023**, *23*, 2399.
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**AMA Style**

Wang S, Wang Y, Li D, Zhao Q.
Distributed Relative Localization Algorithms for Multi-Robot Networks: A Survey. *Sensors*. 2023; 23(5):2399.
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**Chicago/Turabian Style**

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https://doi.org/10.3390/s23052399