A Framework for Collaborative Multi-Robot Mapping using Spectral Graph WaveletsJournal Title

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A Framework for Collaborative

Multi-Robot Mapping using Spectral

Graph Wavelets

Journal Title

XX(X):1–14

©The Author(s) 2016

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DOI: 10.1177/ToBeAssigned

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SAGE

Lukas Bernreiter1, Shehryar Khattak2, Lionel Ott1, Roland Siegwart1, Marco Hutter2and

Cesar Cadena1

Abstract

The exploration of large-scale unknown environments can beneﬁt from the deployment of multiple robots for collaborative

mapping. Each robot explores a section of the environment and communicates onboard pose estimates and maps to a

central server to build an optimized global multi-robot map. Naturally, inconsistencies can arise between onboard and

server estimates due to onboard odometry drift, failures, or degeneracies. The mapping server can correct and overcome

such failure cases using computationally expensive operations such as inter-robot loop closure detection and multi-modal

mapping. However, the individual robots do not beneﬁt from the collaborative map if the mapping server provides no

feedback. Although server updates from the multi-robot map can greatly alleviate the robotic mission strategically, most

existing work lacks them, due to their associated computational and bandwidth-related costs. Motivated by this challenge,

this paper proposes a novel collaborative mapping framework that enables global mapping consistency among robots

and the mapping server. In particular, we propose graph spectral analysis, at different spatial scales, to detect structural

differences between robot and server graphs, and to generate necessary constraints for the individual robot pose graphs.

Our approach speciﬁcally ﬁnds the nodes that correspond to the drift’s origin rather than the nodes where the error

becomes too large. We thoroughly analyze and validate our proposed framework using several real-world multi-robot ﬁeld

deployments where we show improvements of the onboard system up to 90% and can recover the onboard estimation

from localization failures and even from the degeneracies within its estimation.

Keywords

Multi-Robot Mapping, Spectral Graph Theory

Introduction

Over recent years, an abundance of localization and mapping

frameworks have been proposed and successfully deployed

in various robotic scenarios. As part of this research, many

traditional SLAM challenges have been fully or partially

addressed. Despite this development, new challenges readily

arise with the need for more robotic autonomy and the

deployment of heterogeneous robotic teams in large-scale

environments. In particular, an increase in the number of

deployed robots and autonomy requires a higher degree of

robustness and efﬁciency. At the same time, scalability and

persistence across all systems become a pertaining issue.

While it is difﬁcult to maintain a consistent estimate of the

environment across all employed systems, it is an essential

prerequisite for operating robotic teams in applications like

disaster response or search and rescue.

With the recent advent of high-bandwidth mobile networks

such as 5G networks, collaborative robotic approaches have

received increased attention in the robotics community due to

their improved practical feasibility. A promising research

direction is to employ a centralized mapping approach.

Centralized servers running in the local network or a remote

cloud environment have more computational capacity than

individual robots. Therefore, they can perform expensive

operations such as global optimizations, loop closing, and

exploitation of all available sensor data to improve accuracy

and overcome onboard failures.

Most collaborative mapping approaches focus on building

accurate maps on the server and ignore the use of global

multi-robot information to provide localization corrections to

individual robots. Especially in centralized settings without

feedback, pose estimation discrepancies may arise between

robots during large missions leading to severe drift between

robot and server maps resulting in increased optimization

time at the server for collaborative mapping. Therefore, it is

desirable to provide additional constraints to improve onboard

estimation and collaborative mapping performance for large-

scale multi-robot missions.

Furthermore, multi-robot missions often deploy a

heterogeneous set of robots, e.g., aerial and ground robots,

which additionally might rely on heterogeneous sensory

systems. Carrying a diverse set of multi-modal sensors

onboard and effectively utilizing different algorithms for,

e.g., localization and mapping, can be highly beneﬁcial for

the deployment as it becomes more ﬂexible and robust.

However, no common layer sharing data to improve pose

1Autonomous Systems Lab, ETH Zurich, Switzerland

2Robotics Systems Lab, ETH Zurich, Switzerland

Corresponding author:

Lukas Bernreiter, Autonomous Systems Lab, ETH Zurich, Zurich,

Switzerland 8092.

Email: berlukas@ethz.ch

Prepared using sagej.cls [Version: 2017/01/17 v1.20]

arXiv:2210.13856v2 [cs.RO] 2 Nov 2022

2Journal Title XX(X)

estimation and mapping estimates among the employed robots

is readily available with diverse sensory systems. Hence, a

sensor modality-invariant approach that can incorporate and

communicate relevant consistency information among robots

while maintaining low network bandwidth requirements is

essential for large-scale multi-robot ﬁeld deployments.

This paper proposes a novel multi-robot pose graph

consistency approach independent of the underlying robot

pose estimation processes. Our proposed approach relies

only on a sparse abstraction of the estimated poses in

SE(3)

. Moreover, the framework operates in the graph

spectral domain of the pose graphs to identify structural

anomalies in the individual robot pose graphs using a multi-

scale analysis. By examining the structural components of

the pose graphs at different scales, our system identiﬁes

discrepancies in the local and coarser neighborhoods and

adds corresponding constraints to improve the pose estimation

accuracy of individual robots and make the individual robot

and collaborative server maps consistent with each other. The

key contributions of this paper are:

•

Graph spectral analysis of pose graphs to identify

discrepancies between onboard and server pose graphs.

•

Automatic adaptive inference of multi-scale constraints

to correct onboard estimation failures.

•

Comparison against current state-of-the-art approaches

on datasets and a thorough quantitative analysis on

large-scale multi-robot ﬁeld deployments are presented

to validate the proposed approach.

Related Work

In this section, we review the state-of-the-art collaborative

multi-robot localization and mapping approaches as well

as the current applications of graph signal processing and

degeneracy and failure detection.

Collaborative Multi-Robot Mapping

Collaborative multi-robot approaches can be distinguished

into centralized (Deutsch et al. 2016;Schmuck and Chli 2019;

Karrer et al. 2018) and distributed solutions (Cunningham

et al. 2013;Dong et al. 2015). Deutsch et al. (2016) proposed

a vision-based centralized multi-robot SLAM approach where

a mapping server performs loop closures and replaces robot

pose graphs with corrected graphs. A similar approach was

proposed by Schmuck and Chli (2019) in which robots

send local maps to a mapping server which then returns

optimized keyframes and landmarks to each robot to include

in their onboard optimizations, thus increasing the bandwidth

requirements for real-world robot deployments. The work of

Van Opdenbosch and Steinbach (2019) proposes an encoding

and decoding of visual features during the transmission of

the maps to reduce the required bandwidth. CoSLAM (Zou

and Tan 2013) proposes to make use of GPU computing

to circumvent the need for large computational processes

and improve the speed of onboard optimizing tasks, hence

requiring a GPU onboard individual robots.

Different from vision-only approaches, LAMP (Ebadi et al.

2020;Chang et al. 2022) proposes a large-scale collaborative

multi-modal SLAM framework. However, their proposed

approach does not provide any pose corrections from the

centralized server to the individual robots.

In contrast to centralized approaches, distributed

approaches require each robot to run a full onboard

SLAM solution (Dong et al. 2015) and share marginalized

information with other robots (Cunningham et al. 2013), thus

making full information available to each robot. Additionally,

they have the advantage of scaling well to large swarms of

robotic systems (Ziegler et al. 2021) but typically increase

the onboard compute requirements signiﬁcantly.

A crucial aspect of multi-robot SLAM is the ability to

incorporate inter-robot loop closures. Kim et al. (2010)

aims to achieve consistent maps across multiple robots

independently of the employed sensing modalities by

detecting loop closures between robots and connecting

their pose graphs. In the same direction, Mangelson et al.

(2018,2019) aim to robustly select inter-robot loop closure

candidates by maintaining pair-wise consistent measurements.

More recently Lajoie et al. (2020) proposed a distributed

system with distributed loop closure detection.

The more robots are deployed for a speciﬁc task,

the more information needs to be processed, potentially

leading to delays or longer processing times, especially for

components such as the factor graph optimization. Recently,

COVINS (Schmuck et al. 2021) demonstrated a collaborative

deployment of 12 individual agents while maintaining a

reasonable collaborative trajectory error. Although their

system propagates optimized poses from the centralized

server back to individual agents, the poses are only used

for drift quantization by comparing the optimized to the

onboard estimate. Thus, the onboard pose estimations are

not corrected.

Concluding, many existing approaches are limited to

a single modality only (Lajoie et al. 2020;Karrer et al.

2018;Deutsch et al. 2016) often incorporated in tightly

coupled multi-robot frameworks, exchanging large data

structures such as descriptors (Tian et al. 2022), partial or

complete (Schmuck and Chli 2019) factor graphs. As a

consequence, the systems become less ﬂexible and maintain

little versatility for the application of different robotic tasks.

Conversely, this paper proposes to detect discrepancies

between the robot graphs using spectral analysis and a sparse

abstraction of the server graph to generate an individual set of

constraints for each robot. Hence, the proposed approach

achieves high accuracy and mapping consistency while

maintaining low network and compute requirements.

Failure and Degeneracy Detection

Pose estimation from onboard sensors is subject to drift

(accumulation of small errors) and to degeneracies (errors due

to speciﬁc sensor modality’s deﬁciency). Recognizing such

errors enables corrective actions to avoid possible catastrophic

losses (e.g., platform crashes and wrong decision making).

However, evaluating the quality of poses or maps is not

trivial when no ground truth is available for comparison.

In Schwertfeger and Birk (2013), a metric to assess the quality

of the maps was proposed by matching topological graphs

from the robot with a ground truth map. Some research

also approaches the problem using redundant estimation

systems (Sundvall and Jensfelt 2006) to ﬁnd inconsistencies.

Moreover, the recent work of Nobili et al. (2018) learns a

Prepared using sagej.cls

Bernreiter et al. 3

Odometry

Robot 1

Base Station

Centralized Mapping Server

Map merging

Intra-/Inter-robot Loop closure

Global multi-robot optimization

Graph Monitor

Graph creation

Kron reduction

Signal generation

Onboard Graph

Onboard Map 2

Onboard Map 1

Sparse Graph

Graph Evaluation

Graph wavelets

Spectral comparison

Constraint generation

Legend: Onboard Estimation Graph Processing Global Mapping Onboard Mapping

Robot 2

Robot 3

Onboard Map 3

Multi-Robot

Map

Figure 1.

Overview of our approach. We consider multiple robots simultaneously exploring an environment and sending incremental

maps to a centralized mapping server. The server accumulates all robot maps and jointly optimizes them. A relaxation of the

collaborative multi-robot map is sent back to the robots, where a multi-scale graph spectral analysis is performed to identify

discrepancies onboard and server maps and to generate necessary constraints for making them consistent.

model to predict failures for pointcloud alignments. Akai et al.

(2019) infers a failure type based on the distribution of the

residual error. The work of Zhang et al. (2016), proposes to

analyze the structure of the constraints using the eigenvalues

to derive a degeneracy factor.

We approach the problem differently by taking into

account the underlying graph structure, precisely its spectral

properties. Thus, making our approach independent of the

employed sensor system and enabling us to evaluate the

discrepancies at multiple scales to be more precise when

resolving the spurious estimations.

Graph Signal Processing

Spectral graph theory is an active research area and has

gained popularity in the past years in the context of robotics.

Spectral graph theory approaches have been proposed for

robotic mapping (Brunskill et al. 2007), planning (Indelman

2018), and more recently, in combination with graph neural

networks for various robotic tasks (Chandra et al. 2020;Moon

and Lee 2020). In general, graphs are irregular structures

and are capable of modeling large, complex, and distributed

problems (Mateos et al. 2019), e.g. Egilmez and Ortega

(2014) proposes an anomaly detection for spatial proximity

of graph nodes using spectral graph ﬁltering. Furthermore,

graph signal processing aims at applying signal processing

techniques on graph structures, thus allowing the use of

existing concepts such as the Laplacian operator (Sandryhaila

and Moura 2014) and multi-scale analysis (Hammond et al.

2011,2019). Similarly, Donnat et al. (2018) aims to learn

a multi-scale structural embedding using graph wavelets by

treating the wavelet coefﬁcients as a probability distribution.

A good introduction and overview of graph signal processing

are presented in Ortega et al. (2018).

Our approach also performs a structural analysis of graph

signals to detect discrepancies between the onboard and server

graphs. Using localized graph wavelets in the graph domain,

our approach directly compares the trajectories at different

scales to estimate the severity of the inconsistency of the

individual pose graphs.

Preliminaries

This section introduces the fundamental and necessary

concepts for analyzing and comparing graph structures in

the graph spectral domain. We ﬁrst introduce the underlying

methods to use graphs for modeling complex problems. Next,

the analysis of harmonic signals in the Euclidean and graph

domain are covered.

Fundamental Graph Theory Review

In this work, we exploit the graph structure that serves

as the primary foundation for most modern SLAM

backends (Cadena et al. 2016). In particular, we extract the

pose information of the factor graphs, i.e., disregarding any

other sort of constraints to visual landmarks, GPS sensors, etc.

Thus, we consider in this work, weighted undirected graphs

G= (E,V, w)

consisting of a set of nodes

with cardinality

, edges

and weights

w:E 7→ R+

denoting how strong

two nodes are connected with each other.

A graph

is uniquely described by

and

in the

form of a weighted symmetric adjacency matrix

A∈RN×N

with

An,m >0

if two nodes

and

are connected. The

weight can be chosen freely, such as, for instance, the spatial

proximity, or the number of co-observed landmarks between

nodes, but ought to measure the relationship between the

nodes. Another fundamental construct in graph theory, is the

degree matrix

, deﬁned as a diagonal matrix with entries

Dn,n =Pn0An,n0where n0are all incident nodes of n.

Finally, signals in traditional signal processing are often

expressed as functions over time, such as

x(t) : R→R

mapping a scalar value to each discrete point in time

. In a

similar vein, signals on graphs are deﬁned as

x(n) : V → R

associating a scalar value to each node

in the graph. While a

traditional signal

x(t)

changes over time, a signal deﬁned

on a graph

x(n)

alters between the nodes in the graph,

leading to certain variations within the signal. Analyzing

these signal variations and, consequently, their trends can lead

to a more fundamental understanding of the signal’s nature

and is generally termed spectral or frequency analysis.

Prepared using sagej.cls

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