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University of Southern California Dissertations and Theses
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Remote exploration with robotic networks: queue-aware autonomy and collaborative localization
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Remote exploration with robotic networks: queue-aware autonomy and collaborative localization
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REMOTE EXPLORATION WITH ROBOTIC NETWORKS: QUEUE-AWARE AUTONOMY AND COLLABORATIVE LOCALIZATION by Lillian Clark A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) May 2023 Copyright 2023 Lillian Clark Acknowledgements I am so fortunate to have an amazing support system of mentors, friends, and family, and I would like to express my sincere gratitude to everyone who has helped me during this PhD journey. I would like to thank my advisor and committee chair Professor Bhaskar Krishnamachari, who taught me what it means to be a researcher. Aside from providing countless hours of technical mentoring and guidance, he has given me unforgettable life advice. During the pandemic, he encouraged me to “move where you have momentum" even if that is in unexpected directions. I appreciate this bit of wisdom and would like to pass it along. I would also like to thank my advisor Professor Kostas Psounis, who taught me to be confident, resilient to criticism from reviewers, and to appreciate the importance of being a well- rounded individual. For the last five years, Bhaskar and Kostas have been the people I turn to for answers, advice, and inspiration in research and in life. I am grateful for my research collaborator Dr. Joseph Galante, whose spacecraft domain expertise and valuable feedback have shaped my research direction and career trajectory. I would also like to thank Professor Gaurav Sukhatme for his time and effort in serving on my committee, as well as his time and effort in voluntarily teaching an extracurricular robotics seminar in 2018, which also shaped my research direction. I am grateful for Professor Michael Neely, whose course on stochastic network optimization planted the seed for a significant portion of this research. His insightful feedback during my qualifying exam demonstrated a level of investment in my success that I will always appreciate. ii I’d like to extend my sincere thanks to the entire Autonomous Networks Research Group and my other colleagues and friends on the fourth floor of Ronald Tutor Hall. In particular, I’d like to recognize Sampad Mohanty who has a wealth of knowledge I’ve been lucky to tap into, David Millard who is a constant source of intellectually stimulating conversation, and Kegan Strawn who motivates me to not just work hard but be kind. I’d also like to thank some of the people who improved my graduate school experience outside of classes and research, including my fellow members of the USC PhD Women in Computing Club and our advisor Professor Leana Golubchik. I’m grateful for Diane Demetras, who has provided ample help and guidance. I would also like to thank Sara Gazarek, whose vocal jazz class gave me a creative outlet and supportive community outside Viterbi. I would not have pursued a PhD without the encouragement and support of my community at MIT, including my advisor Professor Eytan Modiano, my research mentors Dr. Igor Kadota and Dr. Tiago Stegun Vaquero, Professors Brian Williams and Nick Roy, and my dear friend Rachel Morgan. I am also grateful for the meaningful connections I’ve made in industry during graduate school, including Ali-akbar Agha- mohammadi, Jeffrey Edlund, Maira Saboia, Matthew Clark, Peter Kairouz, and Anthony Clark. And for the support system offered to me by the Brooke Owens Fellowship (especially Lori Garver, Will Pomerantz, and Dr. Laurie Leshin) and Matthew Isakowitz Fellowship (especially Steve Isakowitz), through which I’ve developed a wide network and lasting friendships. Over the years, I have been fortunate to receive funding from many sources, including the Annenberg Graduate Fellowship, USC Women in Science and Engineering, Qualcomm, and the Ming Hsieh Institute. I would like to express my sincere gratitude to NASA for inspiring me from a young age and supporting me through a Space Technology Research Fellowship (Grant No. 80NSSC19K1189). It is a blessing to make a living thinking about the things you love. iii My loving family, Michael, Sarah, and Amelia Clark, taught me everything I know and taught me how to learn in the first place. My parents have unwaveringly supported my education, ambitions, and dreams and this privilege is not lost on me. I am lucky and proud to be their daughter and to have Amelia as a sister. My family taught me to be clever, driven, creative, and resourceful and I would not be here without them. I would also like to thank my dear friends Isabel Rayas and Justin Gong, who have been like family to me for many years, for being by my side and making Los Angeles feel like home. Countless others have shaped my life over the last twenty-six years, and while there are too many to name I’m grateful for each of them. iv TableofContents Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Chapter 2: Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1 Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.4 Mathematical and Computational Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.2 Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Chapter 3: Queue-Stabilizing Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3.1 Information-Theoretic Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3.2 Stabilizing the Delay Queue via Network Reliability . . . . . . . . . . . . . . . . . 28 3.3.3 Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.3.4 Localizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3.5 Lyapunov Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.3.6 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.7 Local Optima and Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.4.1 Performance: Exploration Coverage . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.4.2 Performance: Localizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4.3 Role of Virtual Queues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.4.4 Additional Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.4.5 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 v 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 Chapter 4: Accurate Signal Strength Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.3 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3.1 PropEM: Propagation Environment Modeling . . . . . . . . . . . . . . . . . . . . . 48 4.3.2 PropEM-L: Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.4 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4.1 Performance: Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.4.2 Performance: Generalizability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Chapter 5: Robust Network Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.3 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.3.1 Sparse Matrix Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.3.2 Linear Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.3.3 Comparison with Graph Convolutional Networks . . . . . . . . . . . . . . . . . . . 71 5.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.4.1 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 5.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Chapter 6: Collaborative Localization and Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.4.1 Qualitative Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 6.4.2 Quantitative Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.5.1 Performance: Resulting Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.5.2 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Chapter 7: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.1 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 7.2 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 vi ListofTables 3.1 Summary of results: maximum coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.1 Correlation between PropEM output features and RSS. . . . . . . . . . . . . . . . . . . . . 52 4.2 Mean absolute error (dB) for each model (offline) . . . . . . . . . . . . . . . . . . . . . . . . 56 4.3 Mean absolute error (dB) for each NN model trained online. . . . . . . . . . . . . . . . . . 59 5.1 SMILE parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 6.1 Relevant parameters for simulation experiments . . . . . . . . . . . . . . . . . . . . . . . . 94 6.2 Additional relevant parameters for testbed experiments . . . . . . . . . . . . . . . . . . . . 98 6.3 Runtime comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 vii ListofFigures 1.1 Mobile robots exploring remote, subterranean environments with applications in disaster mitigation. Credit: Caltech/NASA Jet Propulsion Lab. . . . . . . . . . . . . . . . . . . . . . 2 1.2 We consider a network of four or more mobile robots with sensors for mapping and radios for communication and ranging. All exploration data needs to be transferred to a stationary data sink. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1 Comparison of occupancy grid and topological map, adapted from [34]. (a) occupancy grid map, (b) segmented map, where each color corresponds to a different location, (c) topological map, where colored graph nodes correspond to rooms. . . . . . . . . . . . . . . 9 2.2 The objective of network localization is to accurately determine the location of nodes (grey) based on the known locations of anchors (yellow). LOS links are depicted in green while NLOS links are depicted in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Radio propagation is a multi-scale process. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 The two-ray model is a third order propagation model which captures interference from the ground-reflected signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1 Given an upper bound onq(t) and minimum rate of data transfer, the maximum size of the delay queue is achieved by the controller which results in the largest area under the q(t) curve (top). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Simulated environments with obstacles (purple), robots (yellow), and data sink (red). Left: 4 robots in 30x30 space with 20 obstacles. Right: With 75 obstacles. Additional experiments considered 8 robots and 50x50 space, and results are summarized in Table 3.1. 38 3.3 Map cells known at data sink over time, averaged over six independent trials, with the following parameters: QS (our novel approach): Solid - k q = 0.005,k Q = 0,k Z = 0,k Y = 100, Dashed - k q = 1000,k Q = 0,k Z = 0,k Y = 100, TP: Solid - ρ = 0.35, Dashed -ρ =0.99,MO:w 1 =10,w 2 =0.1,w 3 =0.1,w 4 =100. . . . . . . . . . . . . . . 39 viii 3.4 Average percent of time spent with CRB below θ CRB per robot against number of map cells known at data sink for 4 robots in 30x30 space with 20 obstacles. Each point is the average performance of 6 trials, and ideal is at the top right. We show more points for the more parameterized approaches (MO and our novel QS), while SC and UN are captured by a single point because they do not depend on parameters. . . . . . . . . . . . . . . . . . 40 3.5 QS connectivity and localizability performance with and without stabilizing the relevant virtual queues. Left: Averageλ 2D against coverage. Right: Average percent of time spent with CRB belowθ CRB per robot against coverage. Notation¬k Q indicatesk Q = 0, and in both plots ideal is at the top right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.6 Average percent of time spent with CRB below θ CRB per robot against number of map cells known at data sink for 8 robots in 30x30 space with 20 obstacles. Each point is the average performance of 6 trials using 8-hop-connectivity, and ideal is at the top right. . . . 42 3.7 Team of mobile robots during field-testing for the DARPA Subterranean Challenge [8]. . . 43 4.1 Left: Network of exploring robots, stationary relay radios, and human supervisor base station. The network provides communication during exploration, disaster mitigation, or search and rescue and is visualized over a ground truth map of the 2021 DARPA Subterranean Challenge course. Right: 3D map of connectivity predicted by PropEM-L, learned from sparse point cloud data, which can be used to enable communication-aware exploration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2 PropEM-L is a framework for predicting the signal strength between two radios, given their positions, which learns from the geometry of their environment. PropEM acts as an encoding layer from raw sensor data to specific features which are relevant to propagation modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.3 3D occupancy grid constructed for predicting line of sight. . . . . . . . . . . . . . . . . . . 49 4.4 Left: Occupied (red), maybe occupied (yellow) and free (green) voxels intersected during ray tracing. Right: Voxels which comprise the first Fresnel zone between two radios (black), where colors help illustrate perspective. . . . . . . . . . . . . . . . . . . . . . . . . 49 4.5 Signal attenuation plotted against distance. Measurements at each distance are averaged. Line-of-sight (LOS) links see less degradation than non-line-of-sight (NLOS) links. . . . . . 52 4.6 Example performance of the learning models visualized against ground-truth attenuation (PL). Top: conventional data-driven propagation models. Bottom: neural-network based prediction. These plots depict the link between a single exploring robot and the base station. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.7 Error probability density functions for each NN model. . . . . . . . . . . . . . . . . . . . . 56 ix 4.8 Spatial plot of signal strength from a static transmitter (red) in the environment depicted in Fig. 4.1. A simple path loss model predicts the signal will attenuate with distance (circles around transmitter). However, branch 2 sees more degradation than predicted due to barriers between branches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.9 Performance of NN-vox trained online in the competition environment on day 3. By the final minute of exploration, mean absolute error is 7.54 dB. . . . . . . . . . . . . . . . . . . 59 5.1 An overview of Sparse Matrix Inference and Linear Embedding (SMILE). . . . . . . . . . . 65 5.2 The final value of the sparse matrix inference objective function f (Eq. 5.3) when we assume different levels of sparsity ˆ β , plotted alongside the true sparsityβ (dotted line) for various simulated matricesE∈R 500× 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.3 Localization with GCN (Yan et al. [13]) and novel SMILE for a 500 node network with 50 anchors. GCN achieves RMSE 0.11 and SMILE achieves RMSE 0.06. . . . . . . . . . . . . . 74 5.4 Error density for GCN and SMILE on the data from Fig. 5.3, where SMILE results in a more desirable distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.5 RMSE (solid line) and runtime (dashed line) trade-off as we vary the number of neighbors k. Each point is the average of 10 trials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.6 Performance of SMILE and GCN as the probability of NLOS links, standard deviation of Gaussian noise, and threshold for distances which can be measured are varied (T =2). . . 76 5.7 Performance of SMILE and GCN in an ideal setting (p NLOS =0,σ =0) and a noisy setting (p NLOS = 1 10 ,σ = 0.5). SMILE is robust to noise without sacrifice performance in ideal settings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.8 Compute time for different sizes of networks for GCN and SMILE. Previous work has shown that learned approaches (GCN, multi-layer perception, neural tangent kernel) scale better than optimization approaches (LS, expectation conditional maximization, SDP) [13]. SMILE outperforms GCN for very large networks. . . . . . . . . . . . . . . . . . . . . 78 5.9 Ablation study withσ = 0.3,θ = 5. Top row, left to right: classical MDS achieves RMSE 1.49, de-noising via PSVD improves RMSE to 0.25, and sparse matrix inference further improves RMSE to 0.19. Bottom row, left to right: LLE with direct position recovery achieves RMSE 1.16, de-noising via PSVD improves RMSE to 0.19, and sparse matrix inference further improves RMSE to 0.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.10 Measurement error for 11 node wireless sensor network outdoors, comparing distance from RSS (Eq. 5.9) with ground truth. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.11 Localization accuracy of GCN (RMSE of 1.6) and SMILE (RMSE of 1.63) on the sensor network. Edges in the left graph represent observed distances less thanθ GCN . Edges in the right graph connect each nodei to its nearest neighborsNN(i). . . . . . . . . . . . . . . . 82 x 5.12 Measurement error for 18 node robotic network in GPS-denied environment. . . . . . . . . 83 5.13 Localization accuracy of GCN (RMSE 45.8) and SMILE (RMSE 34.22) on the robotic network. Edges indicate error in distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.1 TEAM leverages collaborative localization and coordinated mobility. Top left: robots initiate a shared coordinate frame. Top right: the first robot moves and maps while using its neighbors to trilaterate. Bottom left and right: subsequent robots take turns moving and mapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.2 Robot 3 trilateration, using the received position estimates of robots 0, 1, and 2 as anchors. 91 6.3 Time-division multiple access scheme for 4 robots performing TEAM. Each robot initializes two-way ranging (TWR) during its TDMA window, and this cycle repeats. . . . 91 6.4 The top image resulted from SLAM with 50 particles, the bottom image resulted from TEAM, each after 25 simulated minutes. The true environment dimensions are overlaid in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.5 The left image resulted from SLAM with 50 particles, the right image resulted from TEAM. The true environment dimensions are overlaid in red. . . . . . . . . . . . . . . . . . . . . . 94 6.6 The left image resulted from SLAM with 50 particles and scans throttled to the low frequency of 0.1Hz. The right image resulted from TEAM with the same low LiDAR sample rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.7 The left image resulted from SLAM with 1 particle. The right image resulted from TEAM, which is computationally comparable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6.8 These graphs show the localization error over time for SLAM and TEAM for the environment depicted in Fig. 6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.9 This graph shows the map error as a function of time for the environment depicted in Fig. 6.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.10 Turtlebot3 Burger with Pozyx UWB anchor. . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.11 SLAM in the hallway: merged map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.12 SLAM in the hallway: map created by a single robot . . . . . . . . . . . . . . . . . . . . . . 99 6.13 TEAM in the hallway: merged map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 6.14 TEAM in the hallway: map created by a single robot . . . . . . . . . . . . . . . . . . . . . . 99 xi Abstract Lunar and planetary exploration puts stringent requirements on a robotic system, including high reliability, accurate localization and mapping, and the ability to operate and communicate findings with a remote base station despite the lack of existing infrastructure. Robotic networks are well-suited for operation in these harsh, remote environments because the system is robust to the failure of a single robot and agents can communicate and collaborate. In this dissertation, we identify four key problems in the field of networked robotic exploration and provide the necessary solutions to meet the end goal of enabling exploration and mapping in harsh, GPS-denied, communication-restricted environments with a team of mobile robots. We focus on two subdomains in robotic network research: communication and localization. First, we consider that while connectivity is necessary for communicating exploration data, strictly maintaining connectivity can limit exploration. Thus, we propose a queue-aware distributed controller which more flexibly approaches connectivity by focusing on time-average constraints. This improves exploration efficiency without sacrificing timely data transfer. Second, we note that predicting connectivity, which is a key component of queue-aware exploration, is challenging in unknown environments with obstacles that prevent line-of-sight and significantly attenuate signal strength. Thus, we propose a data-driven approach to signal strength prediction which combines the strengths of well-known models of signal propagation phenomena (e.g. shadowing, reflection, diffraction) and machine learning, and can adapt online to new environments. This leads to accurate signal strength prediction which enables communication-aware autonomy for the network of robots. xii Third, we observe that accurate signal strength models can also enable collaborative localization; given the position of a few robots, we can leverage pairwise inter-robot signal strength measurements to deter- mine the positions of all robots. However, the accuracy of this network localization is again stressed by obstacles which prevent line-of-sight. Thus, we propose a centralized algorithm which first infers and extracts the non-line-of-sight component of attenuation. This leads to accurate collaborative localization which is robust to obstacles in unknown environments. Finally, we acknowledge that a centralized approach to collaborative localization has certain disadvan- tages, namely communication overhead and synchronization. Thus, we propose a distributed approach to localization for a team of robots with coordinated mobility. Our trilateration-based approach reduces the computational complexity of localization and mapping. This distributed algorithm leads to accurate localization and mapping suitable for resource-constrained robots. We validate these algorithms and models in simulated environments which offer fine control of simu- lated failures and signal noise. We also validate our proposed methods on robotic systems in real-world en- vironments. We test our low-complexity localization algorithm on a network of four resource-constrained wheeled robots with ultra-wideband positioning devices in an indoor environment. We test our signal strength predictive model and our robust network localization algorithm on a network of three wheeled and three quadruped mobile robots in large-scale subterranean environments. We evaluate our solutions with respect to high-level performance metrics including localization accuracy, mapping accuracy, and exploration efficiency. Further, we analyze implementation-focused metrics including complexity, robust- ness to noise and failures, and scalability to large networks. Our findings support that (1) queue-aware exploration can improve coverage by 12% compared to the state-of-the-art approach to exploration with intermittent connectivity, (2) data-driven models of the prop- agation environment can improve signal strength prediction accuracy up to 44% compared to a distance- based model, (3) careful non-line-of-sight inference and matrix manipulation can reduce localization error xiii by 45% compared to the state-of-the-art graph-based learning approach, and (4) trilateration-based local- ization can reduce complexity by an order of magnitude compared to a well-known simultaneous local- ization and mapping approach. Together, the four proposed solutions in this dissertation enable a team of mobile robots to efficiently (in terms of time and complexity) explore and map remote environments, e.g. the lunar subsurface, while allowing timely data transfer. Timely data transfer mitigates the risk of losing valuable data due to unexpected failures in harsh environments, and thus we advance the field of remote exploration with robotic networks. xiv Chapter1 Introduction The study of multi-robot systems is an active research domain which has emerged over the last few decades [1, 2]. Within this broader area,networkedrobots refer to multiple robots operating together in coordination or cooperatively, where communication between entities is fundamental [3]. While this communication can improve task performance or efficiency through collaboration, modeling it in order to better maintain and/or improve connectivity is challenging. This is especially true when considering the realistic dynamics of a wireless channel and the specifics of networking and data transfer protocols, which are often abstracted or simplified. Sensor networks are a related research area in which network nodes are static, and communication between nodes has been studied extensively [4]. One of the challenges facing sensor networks is that nodes are typically equipped with low-cost sensors with limited computational capabilities, meaning they require low-complexity solutions which are robust to sensor noise. In this work we are interested in the intersection of these two research domains, often referred to as roboticsensornetworks,roboticwirelesssensornetworks, or simplyroboticnetworks. We investigate modeling and controlling connectivity for teams of robots, and low-complexity algorithms for localization which are robust to noise. 1 Figure 1.1: Mobile robots exploring remote, subterranean environments with applications in disaster mit- igation. Credit: Caltech/NASA Jet Propulsion Lab. Motivating applications for this work come from two domains. Scientists believe tunnels formed by cooled flowing lava exist below the surface of the moon and may be a favorable environment for human habitation [5]. It is not yet safe or practical to send humans to explore lunar lava tubes, and robotic networks have been proposed for lunar exploration because multiple small autonomous robots can cover more ground than a single rover [6]. Networks of resource-constrained robots can also assist larger, more capable planetary rovers or human scientists in investigating geologic and biologic diversity in lava tubes, caves, and other rocky bodies [7]. This research also has application in disaster response, especially in complex underground settings which are too high-risk for military and civilian first responders, as illustrated in Fig. 1.1 [8, 9]. The capability of a team of robots to rapidly map, navigate, and search underground environments makes them ideal for time-sensitive combat operations or disaster response scenarios. This was the focus of the 2021 DARPA Subterranean Challenge, which brought together multidisciplinary teams from around the world to address the autonomy, perception, networking, and mobility technologies necessary to map subsurface networks in unpredictable conditions [8]. Motivated by these space exploration and disaster response applications, we consider remote explo- ration with multiple resource-limited, networked robots in GPS-denied, harsh environments which lack communication infrastructure. A key consideration is that exploration data must be shared with a station- ary base station, or data sink, as illustrated in Fig. 1.2. The presence of a data sink is what differentiates 2 remote exploration. Examples of data sinks include the base station where a human can supervise au- tonomous robotic first response from a safe distance, or a planetary lander which has powerful enough equipment to communicate directly to Earth. In these settings, data from the robots needs to be transferred to allow situational awareness and the possibility of human intervention, or to enable scientific analysis and discovery. Because these environments are difficult to traverse, we seek solutions which enable exploration even when one or more robots fail or get stuck on obstacles in the environment. Timeliness of data transfer is a key component of our proposed algorithms, because it mitigates the risk that valuable data is lost when a robot fails. To clarify the relevant terminology, we define infrastructure to be anything which exists in the envi- ronment for the sake of communication or localization: global navigation satellite systems (e.g. Global Position Systems (GPS)), wireless access points, localization reference points (e.g. fiducial markers), bea- cons, cell towers, etc. We define anchors as network nodes at known locations. Anchors are distinguished from infrastructure in this dissertation because we assume they are either instantaneous anchors which are temporarily static robots at known locations or deployed anchors which are static but placed in the environment during exploration. For instance, these could be static radios autonomously deployed by the robots. The deployed radios can assist in both localization and communication, and are capable of relaying data. As shown in Fig. 1.2, we assume the network is made up of four or more robots which (1) can com- municate peer-to-peer and relay data, (2) can use radios for inter-robot distance measurements, (3) fail probabilistically, (4) have onboard sensors for mapping the environment (nominally LiDAR), and (5) have controlled mobility. We nominally assume holonomic ground robots in two-dimensional space. 3 Figure 1.2: We consider a network of four or more mobile robots with sensors for mapping and radios for communication and ranging. All exploration data needs to be transferred to a stationary data sink. 1.1 Contribution In this dissertation, we address the following four questions 1. How should a team of robots balance the competing goals of exploring a large-scale environment and maintaining a connectivity with a stationary base station? 2. How can robots leverage partial maps available via their onboard sensors to improve signal strength prediction in unknown environments? 3. How can a centralized server estimate the positions of all robots based on inter-robot signal strength measurements in a way that is robust to non-line-of-sight attenuation? 4. How can a robot in a team of exploring robots use its neighbors as localization reference points to reduce localization complexity? We address the first question in Chapter 3, where we present a queue-stabilizing framework for net- worked multi-robot exploration which manages time-average constraints on queuing delay to encourage timely data transfer. The framework uses Lyapunov-based stochastic optimization to maximize new infor- mation while using virtual queues to constrain time-average expectations of metrics of interest [10]. These include queuing delay, network connectivity, and localization uncertainty. The result is a distributed online controller which autonomously and strategically breaks and restores connectivity as needed. We explic- itly account for obstacle avoidance, limited sensing ranges, and noisy communication/ranging links with 4 line-of-sight occlusions. We demonstrate in simulation that our queue-stabilizing controller can reduce localization uncertainty and achieve better coverage than two state-of-the-art approaches. We address the second question in Chapter 4, where we present Propagation Environment Modeling and Learning (PropEM-L). PropEM-L leverages real-time sensor-derived 3D geometric representations of an environment to extract information about line of sight between radios and attenuating walls/obstacles in order to accurately predict received signal strength (RSS). Our data-driven approach combines the strengths of well-known models of signal propagation phenomena (e.g. shadowing, reflection, diffrac- tion) and machine learning, and can adapt online to new environments. We demonstrate the performance of PropEM-L on a six-robot team in a communication-restricted environment with subway-like, mine-like, and cave-like characteristics, constructed for the 2021 DARPA Subterranean Challenge [8]. Our findings indicate that PropEM-L can improve signal strength prediction accuracy by up to 44% over a log-distance path loss model. In Chapter 5, to address the third question, we present Sparse Matrix Inference and Linear Embed- ding (SMILE), a novel approach to network localization which draws on both the well-known Locally Linear Embedding (LLE) algorithm [11] and recent advances in sparse plus low-rank matrix decomposi- tion [12]. We demonstrate that our approach is robust to noisy signal propagation, severe attenuation due to non-line-of-sight, and missing pairwise measurements. Our experiments include simulated large-scale networks, an 11-node sensor network, and an 18-node network of mobile robots and static anchor radios in a GPS-denied limestone mine. Our findings indicate that SMILE outperforms classical multidimensional scaling (MDS) which ignores the effect of non-line of sight (NLOS), as well as outperforming state-of- the-art robust network localization algorithms that do account for NLOS attenuation including a graph convolutional network-based approach [13]. We demonstrate that this improved accuracy is not at the cost of complexity, as SMILE sees reduced computation time for very large networks which is important for position estimation updates in a dynamic setting, e.g for mobile robots. 5 Finally, Chapter 6 addresses the fourth question through the design of an approach to mapping un- known environments called Trilateration for Exploration And Mapping (TEAM). TEAM is designed to leverage the capability of commercially-available ultra-wideband (UWB) radios on board the robots to provide range estimates with centimeter accuracy and perform anchorless localization in a shared, sta- tionary frame. It is well-suited for feature-deprived environments, where feature-based localization ap- proaches suffer. We provide experimental results in varied Gazebo simulation environments [14] as well as on a testbed of Turtlebot3 Burgers with Pozyx UWB radios. We compare TEAM to the popular Rao- Blackwellized Particle Filter for Simultaneous Localization and Mapping (SLAM) [15]. We demonstrate that TEAM requires an order of magnitude less computational complexity and reduces the necessary sam- ple rate of LiDAR measurements by an order of magnitude. These advantages do not require sacrificing performance, as TEAM reduces the maximum localization error by 50% and achieves up to a 28% increase in map accuracy in feature-deprived environments and comparable map accuracy in other settings. 6 Chapter2 Background Robotic exploration is defined as the act of moving through an unknown environment while building a map that can be used for subsequent navigation [16]. Exploration in remote environments requires robust and reliable map generation, precise localization, safe navigation, and efficient path planning; each of these are research domains with rich bodies of literature. Multi-robot exploration of communication-restricted environments introduces important challenges which have only been addressed in the last decade [17]. In this dissertation, we focus on challenges specific to connectivity and localization. This section provides a brief overview of exploration (Sec. 2.1), localization (Sec. 2.2), and connectivity (Sec. 2.3). We also provide background for certain mathematical and computation tools used throughout this work (Sec. 2.4). ∗ 2.1 Exploration An exploration strategy decides, given what is known about the world, where a robot should move to gain new information. A seminal work on robotic exploration introduces the concept of frontiers, regions on the border between explored and unexplored space, and describes methods for detecting and navigating to these frontiers [16]. Frontier-based exploration extends easily to multiple robots who share their map representations [19]. ∗ Parts of this chapter are adapted from Azpurua et al. [18]. 7 Frontier-based exploration relies on building a feasible trajectory between the robot’s current position and the frontier. Path planning algorithms which address this challenge are well-researched [20, 21, 22]. Given a graph-based representation of the environment, perhaps the most well-known algorithm is Dijk- stra’s algorithm which returns the shortest path [23]. More computationally efficient approaches such as A* [24] or D* [25] use heuristics to find the best path. Other popular search algorithms, such as RRT [26] and RRT* [27], use probabilistic sampling to create a tree-based representation of the environment. To re- duce complexity, a popular approach to navigation involves a hierarchical approach: global path planning offers a trajectory between point A and point B, and local path planning addresses obstacle avoidance and low-level motion planning [8]. In the last decade, advancements in approximations of mutual information for ranging sensors [28] have led to the development of exploration approaches that directly maximize mutual information [29, 30]. These information-theoretic exploration strategies decide, given a probabilistic sensor model, where to move to minimize uncertainty in the map. Maps are used for both navigation and environment reconstruction, and can typically be classified as either topological maps or metric/geometric maps. A topological map is an abstract description of the structural characteristics of an environment. For example, topological maps can represent the connectivity of different places such as rooms, floors, or buildings connected by a sequence of robot actions. Generally, these types of maps are modeled as graphs, where nodes/vertices are locations and edges are ways to reach them [31, 32]. On the other hand, metric maps capture an environment’s geometric properties, similar to a floor plan (see Fig. 2.1). In practice, metric maps are finer-grained than topological maps, but this comes with an extra computational cost [33]. Occupancy grids were one of the first probabilistic metric map representations used for mobile robots [35]. These maps discretize an environment into small portions called grid cells, and every cell has infor- mation about the area it covers, such as the probability of cell occupation. Occupancy grids have great 8 Figure 2.1: Comparison of occupancy grid and topological map, adapted from [34]. (a) occupancy grid map, (b) segmented map, where each color corresponds to a different location, (c) topological map, where colored graph nodes correspond to rooms. representation capability because no prior information of the map or model of the environment is needed. Traditional occupancy grids are capable of multiple map resolutions but focus on only two dimensions. Other less popular 2D map representations are feature-maps [36] and point maps [37]. Information gathering, a closely related problem to exploration, can be viewed as sequential decision problems in which actions are chosen to maximize an objective function, and the decision is typically solved myopically by maximizing over a limited time horizon [38] or greedily by optimizing over a single time step. Rather than selecting a frontier and navigating towards it using a path-planning algorithm, this approach chooses the next finite number of positions which will be the most informative. From a decision-theoretic standpoint, the sequential decision problem of exploration is an instance of a partially observable Markov decision process (POMDP). The difficulty in expressing model parame- ters for the exploration problem as a continuous POMDP, much less scaling algorithms which solve the problem [39], makes their use challenging for real-world applications [40]. Reinforcement learning is a popular approach to solving large sequential decision-making problems by letting the learner optimize its performance from experience, and has been successfully applied to robotic exploration [40]. Reinforce- ment learning with multiple robots introduces new challenges for scalability, as the number of states can grow exponentially with the number of robots [41]. When exploring with multiple robots, it is possible to improve exploration efficiency through coordi- nation [1]. Research in this area is often concerned with how robot exploration can be distributed [30, 42] 9 and what information they should communicate with each other [38]. Intuitively, efficiency is improved when robots divide-and-conquer, moving in opposite directions and spreading over the space. However, this behavior can be at odds with other objectives. 2.2 Localization While exploring, each robot needs to address two critical problems: mapping the environment and find- ing its location within the map. Simultaneous localization and mapping (SLAM) algorithms solve these problem iteratively; a robot localizes within a previously unexplored environment while constructing a consistent and incremental map. The interdependence of localization and mapping raises the complexity of the problem, but is well studied [43, 44, 45]. Less well-studied is the interdependence of exploration and localization [46], although the choice of movement does affect how well the robot can localize with respect to the map. For multi-robot SLAM, two key questions are how to represent and share local maps [47], and how to merge multiple maps with the appropriate transformations [48]. Localization accuracy can be improved via inter-robot detection [49, 50] and inter-robot ranging [51]. In contrast with map-based localization, landmark-based localization is common when fixed elements (e.g. visual markers, motion capture cameras, wireless beacons) exist at known locations. Recently, ultra- wideband (UWB) wireless radios have emerged as a popular solution for highly accurate indoor local- ization, with commercially-available products able to offer centimeter accuracy [52]. For robotics appli- cations, UWB radios offer the opportunity to trilaterate a robot’s position given the presence of three reference radios [53]. A recent survey by Shule et al. on UWB localization for collaborative multi-robot systems highlights several trends [54], including fusing UWB ranging data with other sensors [54, 55, 56]. Recently, UWB radios onboard robots have been demonstrated for relative positioning via inter-robot ranging with applications in autonomous docking [57], formation flying [58], and leader-follower [59, 60]. 10 Figure 2.2: The objective of network localization is to accurately determine the location of nodes (grey) based on the known locations of anchors (yellow). LOS links are depicted in green while NLOS links are depicted in red. Commercial, anchor-based systems mainly use Time Difference of Arrival (TDOA), a technique which requires nanosecond clock synchronization between anchors, limiting scalability and making distributed operation challenging [61, 62]. Two-way ranging (TWR) is a more flexible ranging solution, but introduces the key challenge of access control to a shared medium. Time Division Multiple Access (TDMA) is a simple mechanism which allows high channel utilization. Several prior works consider self-localization in UWB anchor networks. Di Franco et al. presented a method for calibration-free, infrastructure-free localization in sensor networks based on inter-node UWB ranging [63]. Hamer and D’Andrea presented a self-localizing network of UWB anchors which then allow multiple robots to localize based on received signals [64]. Subramanian and Lim presented a scalable distributed localization scheme and introduce the possibility of anchor node mobility [65]. There is also a significant body of research on localization in sensor networks, where resources are often more limited [66, 67, 68]. In the network localization problem, nodes determine their locations by measuring the distance to their neighbors [69]. Given a graph where graph nodes represent network nodes and edges represent pairwise measurements between them, a network localization algorithm seeks the positions of all nodes which is consistent with the measurements (see Fig. 2.2). It is an instance of the 11 more general problem of mappingn points to a low-dimensional space consistent with measurements be- tween them, which has applications in information visualization (graph embedding [70]) and data analysis (manifold learning [71]). Network localization is well-researched, and is commonly formulated as a least squares (LS) problem [72, 73, 74]. Multidimensional scaling (MDS) and its extensions are popular methods for solving this prob- lem [75, 76, 77]. Classical MDS uses the distance matrix to compute a matrix of scalar products, typically called the Gram matrix, that captures pairwise correlation of the position vectors [78]. The principal com- ponents from eigen-decomposition of this matrix are then used to recover relative node positions [68]. Rather than compute over the entire distance matrix, Locally Linear Embedding (LLE) [79, 11] applies principal component analysis to small neighborhoods, which improves performance when the reduction from the noisy (high-dimensional) data to the low-dimensional true positions is non-linear, and has shown promise in sensor network localization [80]. If the data is well-described by a particular statistical model (e.g. Gaussian or log-normal), we can in- stead form the maximum likelihood estimation problem [81], and solve using semi-definite programming (SDP) methods [82, 83]. Recent works extend SDP methods to consider non-Gaussian noise [84] and additional attenuation caused by walls and obstacles which prevent line-of-sight measurements [85, 86, 87]. Jin et al. demonstrate that when the NLOS attenuation has a certain structure, namely non-negative and sparse, a sparsity-promoting term in the objective function can improve the performance of SDP ap- proaches [87]. However, SDP methods suffer with respect to complexity and are intractable for very large networks [13]. The network localization problem seeks a low-dimensional node embedding, and whether there is exactly one solution is closely related to the rigidity of the graph. Rigidity is a property of a graph which means that the edges present are enough to constrain the shape of the graph. For multi-robot systems, the accuracy of localization, exploration, and mapping can benefit from rigid formations [88]. From rigidity 12 theory, it is possible to derive controllers which improve or constrain localizability [88, 89, 90]. However, even rigid graphs can experience ambiguities which lead to challenges in robotic network localization [91, 92]. The Cramer-Rao bound is a useful metric which provides a lower bound on the covariance of any unbiased location estimator and has been shown to accurately predict tracking performance [93]. The geometric dilution of precision (GDOP), commonly used to describe how localization error depends on GPS satellite positions, is the ratio of location estimate variance to signal variance and is closely related to the Cramer-Rao bound [73]. 2.3 Connectivity Many of the approaches to multi-robot exploration we have discussed rely on communication between robots, and therefore connectivity. Connectivity-aware planning has been studied in single robot and multi-robot settings [94, 95], with significant attention paid to control strategies which maintain connec- tivity [96, 97, 98, 2]. These works have created a strong precedent for the use of graph theoretic concepts when discussing communication. In particular, we consider three specific graph-theoretic metrics: two-terminal network reliability, k- hop-connectivity, and algebraic connectivity. Consider a communication graph where vertices represent robots and edges represent the probability of successful communication. The two-terminal network reli- ability is the probability that at least one successful path exists between two vertices [99]. This is known to be NP-hard to compute for a general graph [100], but methods exist to compute it exactly [101, 102]. The closely related k-hop-connectivity of two vertices is the weighted number of paths ofk hops or fewer which connect the first to the second, where weights indicate the probability of success along the path [97]. Both of these metrics will increase as the number or reliability of the paths which connect two robots increases. For a given communication graph, it is possible to describe the adjacency matrix whose entries 13 Received power (Pr/Pt) dB log(Distance) Path loss (free space) Shadowing (obstacles) Multipath (reflections) Figure 2.3: Radio propagation is a multi-scale process. represent pairwise connections. From this we can compute the weighted Laplacian, whose second-smallest eigenvalue is known as the Fiedler value and quantifies the algebraic connectivity of the graph [103]. Maintaining connectivity at all times can limit the network of robots in certain applications [2]. Strate- gies for application where intermittent connectivity is sufficient or even desirable are an interesting emerg- ing research area. A recent survey by Amigoni et al. provides an overview of communication-restricted multi-robot exploration [17]. Some of these works take a periodic approach [104], others are rendezvous- based [45], and others are deployment-based, allowing robots to lost connectivity en route to a prede- termined configuration [9]. Avoiding the challenges of selecting optimal rendezvous locations or time periods, we are also interested in role-based approaches which assign connectivity and exploration as distinct tasks [105, 106, 107] and multi-objective approaches which allow robots to trade between connec- tivity and other metrics [108]. For each of these, modeling connectivity requires accurate models of signal propagation over point-to-point links. Radio signal propagation is a multi-scale process where received signal strength is a function of dis- tance between the transmitter and receiver, shadowing due to obstacles, and multi-path phenomena that 14 d LOS d reflected h tx h rx d Figure 2.4: The two-ray model is a third order propagation model which captures interference from the ground-reflected signal. result from reflections and diffraction (see Fig. 2.3). Free space path loss is a first-order model which quan- tifies the expected attenuation in an obstacle-free environment. It is typically modeled as a logarithmic function of distanced given by PL dB =PL(d 0 ) dB +η 10log 10 (d/d 0 ). (2.1) PL(d 0 ) dB is the reference path loss in dB at a known distanced 0 andη is the path loss exponent which captures how quickly the signal falls off. Note that the path loss exponent η takes on different values in different environments: in free space η = 2, in indoor areas with line-of-sight 1.6 ≤ η ≤ 1.8, in urban outdoor areas2.7≤ η ≤ 3.5, and in shadowed urban outdoor areas3≤ η ≤ 5 [109]. Especially for long- ranges, the effect of shadowing is often captured in this model by the addition of a log-normal random variable with zero mean. Shadowing is complex to model as it depends on the environment itself, and second-order models (capturing path loss and shadowing) are usually determined by fitting samples taken within a specific environment. Third-order models capture additional information about reflected and diffracted signals. The two-ray model is commonly used to capture the constructive or destructive interference caused by signals which 15 bounce off the ground, and depends on the height of the antennas at the transmitter and receiver ( h tx and h rx ) (see Fig. 2.4). The two-ray model [110] is given by PL= G tx G rx λ 2 (4πd ) 2 1+Γexp 2iπ (l ′ − l) λ 2 (2.2) l ′ = p (d 2 +(h tx +h rx ) 2 ) (2.3) l = p (d 2 +(h tx − h rx ) 2 ) (2.4) whereG tx ,G rx are the transmitter and receiver antenna gains andλ is the wavelength of the radio signal. Γ is the ground reflection coefficient, and we assume perfect reflection ( Γ = -1) for simplicity. GtxGrxλ 2 (4πd ) 2 also models the free space path loss, therefore the isolated reflection loss can be modeled by RL= 1− exp 2iπ (l ′ − l) λ 2 (2.5) where negative values indicate destructive interference and positive values indicate constructive interfer- ence. This model assumes a flat ground surface and a large distance between the transmitter and receiver, while we discuss more complex geometries in the next paragraph. Note that there exist six-ray and ten-ray models which account for additional reflections, but we limit our attention to the two-ray model to avoid the additional complexity [110]. Given knowledge of the three-dimensional space surrounding the transmitter and receiver, we can also model the Fresnel zone which is a series of ellipsoids defining the area between the transmitter and receiver where obstructions may cause interference. The radius of the first Fresnel zone at a given distance d 1 from the transmitter andd 2 to the receiver can be approximated by r = r d 1 d 2 λ d 1 +d 2 (2.6) 16 making the simplifying assumption that the distance between the radios is much larger than the radius [110]. Even given a clear line-of-sight path, obstacles within the first Fresnel zone can have a significant impact on signal strength [111]. For a single object, like a hill or boulder, the attenuation caused by diffraction can be estimated by treating the obstruction as a diffracting knife edge where the additional diffraction loss is given by DL dB =20log 10 (|F(v)|). (2.7) F(v) is the Fresnel integral, a function of the Fresnel-Kirchoff diffraction parameter v, defined as v =h s 2(d 1 +d 2 ) λd 1 d 2 (2.8) where h is the relative height of the obstruction (i.e. 0 if exactly in the line of sight). An approximate solution for Eq. 2.7 is given in [112]. As demonstrated by Filiposka and Trajanov, the combination of the two-ray model and the knife-edge model can predict values within a few dB of the ground truth and can identify isolated weak-reception areas well [112]. These accurate models of signal strength are used by controllers which maintain or control connec- tivity. When robots do not have constant connectivity, it makes sense to model their store of local data as a queue. As they generate or collect data, this information waits to be transferred. In particular, let Q(t) represent the content of a queue over time slotst∈{0,1,2...}. AssumeQ(0) is a non-negative real valued random variable. Future states are driven by stochastic arrival and server processesa(t) andb(t) according to the following dynamics: Q(t+1)=max[Q(t)− b(t),0]+a(t) for t∈0,1,2,.... (2.9) 17 This discrete time processQ(t) is mean rate stable if: lim t→∞ E[Q(t)] t =0 (2.10) To analyze stability, we employ mathematical tools introduced in the next section. 2.4 MathematicalandComputationalTools This section introduces several mathematical and computation tools which will be referred to throughout this dissertation. 2.4.1 Matrices In linear algebra, the rank of a matrixA is the dimension of the vector space generated or spanned by its columns [113]. This corresponds to the maximal number of linearly independent columns. The rank ofA is also equal to the number of non-zero singular values, or the number of non-zero diagonal elements in Σ in the singular value decompositionA=UΣV T (assumingV is a real matrix). A sparse matrix is a matrix in which most of the elements are zero, and the sparsity of the matrix is the number of zero-valued elements divided by the total number of elements. In recent years, sparse and low-rank matrix recovery has drawn attention due to its relevance in signal processing, statistics, and machine learning [114]. Bertsimasetal. recently proposed a discrete optimization approach to sparse and low-rank recovery which uses alternating minimization [12]. We will see that this method can serve as an important component of a robust network localization algorithm. Euclidean distance matrices (EDMs) are matrices of the squared distances between points [115]. For many applications, it is desirable to have an analytic expression for the EDM in terms of the point matrix it resulted from. Consider a collection of n points in a d-dimensional Euclidean space, ascribed by the 18 columns of matrixX∈R d× n ,X = [x 1 ,x 2 ,...,x n ],x i ∈R n . The squared distance betweenx i andx j is given by d ij =||x i − x j || 2 =(x i − x j ) T (x i − x j ) (2.11) =x i T x i − 2x i T x j +x j T x j . Therefore, the EDMD=[d ij ] is given by D=1diag(X T X) T − 2X T X+ diag(X T X)1 T . (2.12) Because the rank ofX is at mostd, the rank ofX T X is also at mostd. The remaining two summands in Eq. (2.12) have rank one. The rank of a sum of matrices cannot exceed the sum of the ranks of the summands. Thus, the rank of an EDM corresponding to points inR d is at most d+2. The dimension of the affine subspace that contains the points is also important: for example if the points lie on a plane (but not a line) inR 3 , the rank of the EDM will be 4 and not 5. However, if the nodes are in general position, meaning a general case situation as opposed to more special or coincidental cases (e.g. n points in 3D which lie on a line), the rank is exactly min(n,d+2). Please refer to [115] for additional details on EDMs. 2.4.2 MachineLearning Neural networks (NNs) are a machine learning tool where computations happen in layers which are each made up of units or neurons. Each neuron computes a weighted linear combination of its inputs from the 19 previous layer and then applies a non-linear activation function before passing the output to the subsequent layer [116]. For example, a model with one hidden layer between the input and output looks like Y =W (2) σ (W (1) X) (2.13) whereW (1) andW (2) are the weights between layers andσ (x) is the activation function e.g. the sigmoid function σ (x) = 1/(1+e − βx ). In this equation, β dictates the steepness of the curve near the origin. Training the neural network involves passing data to the input layer, and comparing the result of the output layer with the ground truth. The difference (or loss) between the computed value and the ground truth is then used to update the NN’s weights through back-propagation [116]. With sufficiently rich data, NN models perform well on many tasks. LetG = (V,E) be a simple undirected graph, whereV ={1,...,n} is the set ofn∈Z vertices andE is the set of edges. LetA = (a ij )∈R n× n be the adjacency matrix ofG wherea ij = 1 if an edge exists between thei th andj th vertices [117]. The degree matrix ofG is given byD= diag(d(1),...d(n))∈R n× n , whered(i) = P j∈V a ij is the degree of thei th vertex. The augmented adjacency matrix adds self loops: ˜ A = A+I and an associated degree matrix ˜ D. The Laplacian ofG is given byL = D− A ∈ R n× n . Graph signal processing considers data associated with the vertices (called the graph signal or feature vectors) as well as the graph structure specified by its adjacency matrix. Previous work has posited that observations of the graph signal will be a combination of some noise and the underlying true signal which is low frequency [118]. Graph neural networks (GNNs) are a class of neural networks which can learn from graph-structured data, and have recently become a popular method for approaching optimization problems in wireless net- works [119]. Compared to multi-layer perceptrons like the neural network described previously, GNNs can 20 exploit extra information from the edges. Instead of using the graph signal exclusively, an additional step combines the feature vector for a single network node with the feature vectors of its adjacent neighbors. Graph convolutional networks (GCNs) are a representative type of GNNs, where each layer carries out three actions: feature propagation, linear transformation, and an element-wise nonlinear activation [13]. During the feature propagation step, at each layer, the output at that layer for each network node is averaged with its neighbors. To clarify, a GCN modifies Eq. (2.13) as follows: Y = ˆ Aσ ( ˆ AXW (1) )W (2) (2.14) where ˆ A = ˜ D − 1 2 ˜ A ˜ D − 1 2 is the augmented normalized adjacency matrix. Multiplying by the augmented adjacency matrix encourages similar predictions for connected network nodes. Interestingly, multiplying by ˆ A repeatedly is also equivalent to passing a “low-pass" filter in the context of graph signal processing. In Chapter 5 we will comment on the parallels between low-pass filtering of a graph signal and low- rank decomposition of a Euclidean distance matrix, as they relate to the network localization problem. 2.4.3 Optimization A set is convex if the line segment formed by any two points in the set is also in the set. A functionf(x) defined over a convex set X is a convex function if for any two pointsa,b∈X and any two probabilities p,q >=0 such thatp+q =1,f(pa+qb)≤ pf(a)+qf(b) [10]. Let x= lim t→∞ P t− 1 τ =0 x(τ ) t (2.15) denote the time average of a functionx(t). Consider a stochastic network which operates in discrete time t∈{0,1,2,...}, where the network is described by a collection of queue backlogs(Q 1 (t),Q 2 (t),...). Every 21 timet, a control action is taken which affects arrivals and departures of the queues, and attributes of the system x(t)={x 1 (t),...x M (t)} (2.16) y(t)={y 0 (t),y 1 (t),...y L (t)} (2.17) e(t)={e 1 (t),...e J (t)} (2.18) for some non-negative integersM,L,J. More generally, these attributes may be a function of a random event observed at this time ω(t), and the control action taken α (t). Note that α (t) is chosen from the action setA ω(t) which may depend on the random event. The objective is to optimize convex functions of time averages; to find a solution to the following problem: Minimize: y 0 +f(x) (2.19) Subject to: y l +g l (x)≤ 0 ∀l∈1,...,L e j =0 ∀j∈1,...,J x∈X α (t)∈A ω(t) ∀t Stability of all network queues wheref(x) is a convex function fromR M toR,X is a closed and convex subset ofR M , andx indicates a vector of time averages. A solution to this stochastic program is an algorithm for choosing control actions over time in reaction to the existing network state, such that all of the constraints are satisfied and the quantity to be minimized is as small as possible [10]. 22 We solve the above problem by constructing virtual queues and defining a function L(t) as the sum of squares of backlog in all virtual and actual queues [10]. This is called a Lyapunov function and is a scalar measure of network congestion. TheLyapunovdrift is defined as ∆( t)=L(t+1)− L(t). Control decisions which greedily minimize the Lyapunov drift intuitively maintain network stability. To additionally min- imize the objective functiony 0 (t), we instead minimize the drift-plus-penalty expression,∆( t)+Vy 0 (t) whereV is a non-negative tunable parameter which allows a smooth tradeoff between backlog reduction and objective function minimization. The problem described by Eq. (2.19) has many applications, and we will see in the next chapter that the drift-plus-penalty method leads to promising emergent behavior for queue-aware autonomy. 23 Chapter3 Queue-StabilizingExploration This chapter presents a distributed controller which achieves improved exploration coverage at a remote base. It is important to note that during exploration, the configuration of robots affects both network connectivity (i.e. whether data is transferred or accumulates in queues) and the accuracy of relative local- ization. In this chapter, we will use Lyapunov-based stochastic optimization to maximize new information while using virtual queues to constrain time-average expectations of metrics of interest: queuing delay, network connectivity, and localization uncertainty. We will explicitly account for obstacle avoidance, lim- ited sensing ranges, and noisy communication/ranging links with line-of-sight occlusions. ∗ 3.1 ProblemFormulation Our objective is to design a distributed controller for a multi-robot system where each robotr in the set of robotsR can collect information about its environment and store this information in a local mapm r . We assume the stationary data sinkD is at positionx D . The goal of the system is to create a complete map at the data sinkm D . Each robot r can communicate with any robot i for which link quality f c r,i is above some threshold θ c . Map information is shared over these communication links such that m r ← m r ∪m i for each pair ∗ This chapter is adapted from Clark et al. [120, 121]. 24 (r,i) for whichf c r,i >θ c . Robots also share their time-stamped location estimate, which times out aftert to time steps. This means a robot has an absolute location estimate for any robot it has communicated with (through one or more hops) in the lastt to time steps, and we call this set of robotsC r . Each robotr can also collect a range measurement between itself and any roboti for which link qualityf l r,i is above some threshold θ l . These measurements containing range estimates and robot identifiers can be obtained, for example, via ultra-wideband radio and used for relative localization [122, 123]. Note that since communi- cation and localization may use different technologies, f c andf l may not have the same dependence on distance and may have different levels of noise. The data sink D is equipped with the same communication and ranging capabilities as the robots. We assume time is discretized,t∈{0,1,...}. The stateS r (t) captures what is known by robotr at time t: the position of the robotx r (t), the robot’s mapm r (t), and the estimated position of all robotsx i (t) in the setC r (t). At each timet, the controller deployed on a single robot observes the current state and makes a myopic decision α r (t) choosing from the discrete finite set of locations which can be reached in the next time step,A r (S r (t)). We assume perfect obstacle detection, therefore we limit the decision space to obstacle-free locations.S r (t+1) depends on the previous stateS r (t) and decisionα r (t). It also depends on unknowns in the environment and decisionsα i (t)∀i∈R; therefore, the transition probabilities between states are unknown. Our distributed controller makes online decisionsα r (t) based solely on the observable state S r (t) and memory stored in queues q r (t), D r (t), Q r (t), and Z r (t). We will define these queues in Sec. 3.3.2, 3.3.3, and 3.3.4. Wear and tear or unforeseen hazards may cause failures. Robot failure is typically modeled by a con- stant hazard rate, γ , and a probability of failure given by Pr[Robotr fails at timet] = γe − γt [124]. To consider robots which are constrained by a given resource (e.g. fuel), we could define the probability of failure as a function of the remaining quantity of that resource, rather than time. For example, in [105] the authors propose an exploration strategy which considers remaining battery life. 25 3.2 RelatedWork We will consider two state-of-the-art approaches to intermittent connectivity which have been explored in the literature. Benavides et al. [108] derive a multi-objective function which weighs discovering new information with maintaining connectivity and adapts to a human operator’s input on the importance of each objective. This approach is advantageous in that it does not require specifying roles or deciding ren- dezvous locations. But this approach assigns a utility to connectivity which does not depend on whether or not there is new information to share. Spirin et al. [107] introduces a constraint on the ratio of infor- mation at the data sink and a crude approximation of total information across all mobile robots. Based on this constraint, robots choose a role-based controller (explore orreturntothedatasink). The authors show desirable emergent behavior, but their controller does not consider the location of neighboring robots so they are limited to relaying opportunistically. As we will see in the following section, our approach has the advantages of both balancing competing objectives with weights proportional to the specified impor- tance (as in [108]), and adapting to the amount of new, untransferred data resulting in desirable emergent behavior (as in [107]). 3.3 Approach The crux of our approach is that a backlog in the data queue puts pressure on the robot to transmit this data and therefore move to recover a communication path to the data sink. To achieve this, we introduce a Lya- punov function which depends on the amount of untransferred data, the time this data has been waiting in the queue, and metrics on connectivity and localizability, which are functions of the network configu- ration and can be estimated locally. From this we derive a distributed, online controller which balances maximizing new information and stabilizing this Lyapunov function. One advantage of this approach is 26 that it is flexible enough to consider constraints on the time-average expectation of any bounded metric. Another is that computations are over a single time step, minimizing complexity. In this section we describe the four objectives of our controller: (1) the task-specific goal of maximizing new information, (2) keeping the average queuing delay low, to prevent data loss upon failure by increasing the network reliability, (3) maintaining overall network connectivity, which encourages the robots to act as relays, and (4) maintaining network localizability, which keeps location uncertainty low to ensure new information is useful. We then formulate the constrained optimization problem and use Lyapunov drift minimization to derive our multi-objective queue-stabilizing controller. 3.3.1 Information-TheoreticExploration The objective of information-theoretic exploration is to minimize entropy in the map by maximizing the mutual information gained with new sensor measurements [30]. We represent the environment as an occupancy grid where cells are occupied or free with some prob- ability. The random variable X i describes the probability that the i th cell is occupied and the random variable Z i is associated with the observation of the i th cell, where cells take on values in{0,1}. Our sensor model is therefore the conditional probabilityPr(Z i |X i ). Prior to making an observation, each cell is associated with an entropy which depends on the map thus far,H(X i |m r ). The entropy after taking ob- servationz i isH(X i |m r ,Z i =z i ). The value of observing thei th cell is given by the mutual information or reduction in entropy, I(Z i ;X i |m r )=H i (X i |m r )− H i (X i |m r ;Z i ). (3.1) We assumeX i are independent (as in [30]). We letZ(x r ) describe the set of map cells which are within the sensor coverage of the robot, and we define the expected information gain associated with a robot position x r given mapm r asI(x r ;m r )= P i∈Z(xr) I(Z i ;X i |m r ). 27 A simple information-theoretic controller maximizesI(α r (t);m r (t)) subject toα r (t)∈A r (t) where A r (t) is the set of map cells accessible within one time step [46]. This directs the robot towards increasing information gain and constantly adapts to new information, but suffers the risk of getting stuck in plateaus, areas where all location decisionsα r (t)∈A r (t) result in the same value ofI(α r (t);m r (t)). To avoid this, we define the frontier goal, x f r (t), as the closest point which offers positive information gain, breaking ties according to maximum information gain and subsequent ties arbitrarily. Now our information-theoretic objective is to minimize the distance to the frontier goal, Y(α r (t))= α r (t)− x f r (t) mr(t) (3.2) where notation∥x 1 − x 2 ∥ mr refers to the length of the shortest path fromx 1 tox 2 in the mapm r , found via Dijkstra’s algorithm. If no frontiers remain, the map has been fully explored and the robot returns to the data sink. 3.3.2 StabilizingtheDelayQueueviaNetworkReliability We define the communication graph at robot r at timet,G c r (t), to have vertices representingr, data sink D, and other robots in the setC r (t). An edge in this graph e ij represents the probability of successful communication between nodesi andj. Assuming link qualityf c ij falls off exponentially with distance, the probability of successful communication has the shape of a sigmoid function [125] and can be modeled as e ij = 0 if non-line-of-sight 1 1+e η ( ∥ x i − x j∥ − d θ c ) otherwise (3.3) 28 where d θ c is the distance at whichE[f c ij ] falls below θ c , η defines the steepness of the sigmoid, and we assume radio-frequency-impermeable obstacles such that the probability is zero if an obstacle obstructs the line of sight path. We assume that links fail independently. We define p ij to be the probability that at least one successful path exists between vertices i and j, which is referred to as the two-terminal network reliability and is known to be NP-hard to compute for a general graph [99, 100]. After enumerating the set of all simple paths fromi toj we can exactly computep ij by constructing a sum of disjoint products [101]. Letπ k ij be the event pathk exists which connectsi→j. The probabilityPr[π k ij ] is the product of the edge probabilities which comprise the path. Let ϕ ij be the event any path i → j exists, which is the Boolean sum over all simple paths betweeni andj. To calculatep ij = Pr[ϕ ij ] we employ the approach presented in [102] to transform a set of simple paths into a set of disjoint events. We then sum the probabilities of these disjoint events. This calculation is intractable as the number of robots increases, and we discuss an approximate heuristic for larger networks in Sec. 3.4.4. We now introduce a queueq r (t) of untransferred data stored at robotr with the following dynamics: q r (t+1)=max[q r (t)+I(α r (t);m r (t))− b1(α r (t)),0] (3.4) where1(α r (t)) indicates that a successful communication path exists betweenα r (t) and data sinkD, and b is the (constant) finite capacity or quantity of information which can be transmitted in a single time step. Assuming each robot has finite memory, the queue size is bounded by q r (t)≤ q max , and any additional new information will not be stored. We assume that signal propagation time is negligible, so the robot transfers its data and receives an acknowledgement within one time step. 29 Intuitively, our objective is to minimize the time that data waits in the queue to be transferred. Follow- ing [126], we introduce a threshold on the average delayθ d , and a virtual delay queue with the following dynamics: D r (t+1)=max[D r (t)− θ d b1(α r (t)),0]+q r (t) (3.5) where we have carefully ensured that ifq r (t)>0, the delay queue is positive. Note that in this chapter we use capital letters to differentiate virtual queues from real queues of data. The delay queue grows with each time step that the data queue is non-empty, and decreases when1(α r (t))=1. The goal of our controller is to achieve mean rate stability, defined as lim t→∞ E[Dr(t)] t =0 with probability 1. 3.3.3 Connectivity Intuitively, we expect that if each robot maximizesp rD , the likelihood it can communicate with the data sink, the result will be a network with high connectivity. Additionally, our framework allows us to also control network connectivity explicitly. Closely following [97], we introduce a weighted Laplacian matrix, L r . This matrix is directly con- structed from the adjacency matrix, whose elements are given by f c i,j . The second-smallest eigenvalue of this matrix, λ 2 , is known as the Fiedler value and quantifies the algebraic connectivity of the graph [103]. If and only if λ 2 > 0, the graph is connected. A controller designed to increase λ 2 will result in well-connected graphs with several paths from all robots to the data sink. We use notationλ 2r (α r (t)) to denote the Fiedler value of the Laplacian if robotr makes location decisionα r (t). Note thatλ 2r indicates only the connectivity ofG c r (t), the graph available to robotr. Since eachG c r (t) contains the data sink, if each graph is connected, then the overall network is connected. Thus we can take actions which increase connectivity in a distributed fashion. 30 ConsiderQ r , a virtual queue with the following dynamics: Q r (t+1)=max[Q r (t)+(θ λ − λ 2r (α r (t))),0]. (3.6) Stabilizing Q r (t) will result in a time-average expectation of λ 2r which is greater than or equal to the constraintθ λ . Formally,λ 2r =lim t→∞ 1 t P t− 1 τ =0 E[λ 2r (τ )]≥ θ λ . 3.3.4 Localizability Creating accurate maps requires accurate localization, but a controller which results in efficient exploration may not result in high localizability. The trade between minimizing map uncertainty and minimizing localization uncertainty was previously explored by Bourgault et al. for a single robot [46], and [88, 89, 90] for robotic networks. We consider the Cramer-Rao bound (CRB) which provides a lower bound on the covariance of any unbiased location estimator [73]. The estimator can use received signal strength, time of arrival, or angle of arrival. Moreover, CRB has been shown to accurately predict performance [93]. To calculate the CRB in two dimensions, we first calculate the Fisher Information Matrix F r , a2× 2 matrix whose elements are given by F r [i,j]= 1 (cσ T ) 2 X k∈Cr∪D (x ir − x ik )(x jr − x jk ) d k 2 (3.7) where d k = p (x 1r − x 1k ) 2 +(x 2r − x 2k ) 2 and we assume time of arrival measurements. In the above equation,c is the speed of light,σ T is the variance of the time delay error, and (x 1k ,x 2k ) are the coordinates of robotk.F r and the symmetric rigidity matrix are closely related [88]. The CRB matrix is the inverse of F r and gives a lower bound on the covariance (uncertainty) of any unbiased estimator. We want this bound to be small, and so following [88] we constrain the trace of this matrix, using notation CRB r =tr(F − 1 r ). 31 We define a virtual queue Z r (t) with the following dynamics: Z r (t+1)=max[Z r (t)+(CRB r (α r (t))− θ CRB ),0] (3.8) where CRB r (α r (t)) denotes the CRB for robotr reflecting location decision α r (t), andθ CRB is a desired threshold on this bound. Stabilizing this virtual queue will result in a desirable time-average expectation. Formally, CRB r = lim t→∞ 1 t P t− 1 τ =0 E[CRB r (τ )]≤ θ CRB . Note that for certain configurations the CRB is unbounded [92]. To fit our optimization framework, in implementation we enforce CRB r =min(CRB r , 1 ϵ ). We can choose arbitrarily smallϵ , where the fraction of time the system spends in undesirable configura- tions is bounded byϵθ CRB . During this time, state estimation algorithms which perform filtering can rely on odometry and features in the environment until a desirable configuration is recovered. 3.3.5 LyapunovOptimization We can now succinctly formulate an optimization problem. Minimize: Y (3.9) Subject to: λ 2 ≥ θ λ 2 CRB≤ θ CRB D(t) is mean rate stable α (t)∈A(t) ∀t∈{0,1,...} whereY =lim t→∞ 1 t P t− 1 τ =0 E[Y(τ )],Y(t) is the distance to new information (Eq. (3.2)), and the expecta- tion is overS(t) andα (t). The goal is to choose an action in the discrete, finite action space to minimize the distance to new information subject to constraints on connectivity and localizability, all while keeping 32 the delay queue stable. We assume throughout that limits are well-defined, and have dropped the subscript r for readability throughout this and the following subsections. Let Θ( t) = [D(t),Q(t),Z(t)] be a concatenated vector of the queues defined in Eq. (3.5), (3.6), and (3.8). We assume throughout that the initial queue values are 0. We define the Lyapunov function, L(Θ( t))= 1 2 k q D(t) 2 + 1 2 k Q Q(t) 2 + 1 2 k Z Z(t) 2 (3.10) as a scalar indicator of the size of this vector, where k q , k Q , and k Z are weights on the priority of each objective. The one-step conditional Lyapunov drift is given by∆(Θ( t))=E[L(Θ( t+1))− L(Θ( t))|Θ( t)]. Greedily minimizing∆(Θ( t))+k Y E[Y(t)|Θ( t)], the drift-plus-penalty expression, will lead to mean rate stability for all queues [10]. Here, k Y ≥ 0 is a weight on the priority of exploration. Note that k Y = 0 leads to the max-weight algorithm of Tassiulas-Ephremides [127]. Substituting our queue dynamics gives ∆(Θ( t))+k Y E[Y(t)|Θ( t)]≤ B+k Y E[Y(t)|Θ( t)]+E[k q D(t)(q(t)− θ d b1(t)) +k Q Q(t)(θ λ 2 − λ 2 (t))+k Z Z(t)(CRB(t)− θ CRB )|Θ( t)] (3.11) where1(t) indicates a communication path exists for this robot at timet, and a constantB upper bounds the expectation of (θ λ 2 − λ 2 (t)) 2 2 + (CRB(t)− θ CRB ) 2 2 + q 2 max 2 + (θ d b) 2 2 (3.12) givenΘ( t) and holds for allt since we have enforced a bound on CRB and the Fiedler value of a graph is bounded. We opportunistically minimize the expectations on the right-hand side of Eq. (3.11) which allows our algorithm to be online. Each robot r assumes the probabilities of decisions α i̸=r (t) are uniformly 33 distributed, thereforeE[x i (t+1)|x i (t)] = x i (t). At timet, our distributed controller observesS(t),q(t), D(t),Q(t), andZ(t) and choosesα (t)∈A(t) to minimize k Y Y(α (t),S(t))+k q D(t)(q(t)− θ d bp rD (α (t),S(t)) +k Q Q(t)(θ λ 2 − λ 2 (α (t),S(t)))+k Z Z(t)(CRB(α (t),S(t))− θ CRB ) (3.13) where the expectation that a path exists fromr toD isp rD and we have reintroduced the inputsα (t),S(t) to emphasize which variables are functions of the decision and state. We have derived a distributed controller which captures each of the four objectives enumerated at the beginning of this section, and can flexibly focus on one or more of these goals at a given time depending on which term dominates Eq. (3.13). 3.3.6 Analysis The action space at timet depends on previous decisions, and we cannot expect that greedy decisions will lead to an optimal solution to the problem defined in Eq. (3.9). In particular, to claim that the controller presented in Eq. (3.13) can achieve performance arbitrarily close to optimal based on its derivation using the drift plus penalty method, we would first need to show that for all time t there exists a decision that causes all queues to decrease, or remain constant, in expectation [10, 121]. To illustrate that this assumption does not hold, consider a stylized version of the problem. A single robot explores a one-dimensional discretized world made up of Ω cells such thatx(t) ∈ {0,1,2,...,Ω − 1}∀t. The data sink is located in the left-most cell, D = 0. The robot has two objectives: explore and stabilize the data queue. Consider a simplified, linear connectivity model where the probability of successful communication is given by p=(1− x(t) Ω ). Our controller now choosesα (t)∈{x(t)− 1,x(t),x(t)+1} to minimize: k Y (|x f (t)− α (t)|)+k q D(t)(q(t)− θ d b(1− α (t) Ω )) (3.14) 34 where x f (t) is the location of the frontier at time t. Note that if x(t) = 0, the action space is limited to {x(t),x(t)+1} and ifx(t)=Ω − 1, the action space is limited to{x(t)− 1,x(t)}. It is possible that choosing to prioritize connectivity (i.e., move closer to the baseα (t)=x(t)− 1) still results in an increase in the delay queue in expectation; this occurs when the data queue is sufficiently large or the robot is sufficiently far from the data sink: q(t)>θ d b(1− x(t)− 1 Ω ). This stylized setting allows several useful observations. There is an upper bound on size of the queue given the size of the world, q(t) < Ω . If there exists a finite time limit T max , the maximum size of the delay queue is upper bounded by Ω T max . This is because the delay queue can grow by at most q(t) in each step. The rate at whichq(t) increases is upper bounded by 1, because there is at most one new cell discovered in one-dimensional exploration. The expected rate at which q(t) decreases is lower bounded b Ω ; the probability of data transfer is(1− x(t) Ω ) so the expectation of capacity isb(1− x(t) Ω ) > b Ω . Let us now consider a deterministic system where the transfer rate is exactly b Ω . The maximum size of the delay queue is( 1 2b Ω 3 + 1 2 Ω 2 ). This maximum occurs when the robot solely prioritizes exploration, fully exploring the environment before transferring data. To see this, consider a dummy controller which greedilytriestomaximizethedelayqueue by selectingα (t)=x(t)+1 whenever possible. We can visualize the size of the queue as linear growth until reachingΩ and subsequently linear decline with rate b Ω , as shown in Fig. 3.1 (top). Given that the delay queue increases at each time step by q(t), the size of the delay queue is upper bounded by the area under this curve. Note that size of the queue can only remain constant atq =0, which will not change the size of the delay queue and can be ignored. We can prove that this greedy controller achieves the largest possible delay queue by considering that an optimal controller can be described by the sequence of points{t +1 = 0,t − 1 ,t +2 ,t − 2 ,t +3 ,t − 3 ,...} wheret +i indicates the start time of an interval where the queue increases andt − i indicates the start time of an interval where the queue decreases. Note that initially the queue and delay queue are both empty 35 0 2 4 6 8 10 q(t) 0 20 40 60 80 100 Time step 0 2 4 6 8 10 q(t) Figure 3.1: Given an upper bound on q(t) and minimum rate of data transfer, the maximum size of the delay queue is achieved by the controller which results in the largest area under theq(t) curve (top). and the robot will always explore. There is a constraint on feasible sequences such that P i t − i − t +i <Ω . LetU (t +1 =0,t − 1 ,t +2 ,t − 2 ,t +3 ,t − 3 ,...) indicate the area under the curve. Now consider an alternate controller given by{t +1 = 0,t ′ − 1 ,t +3 ,t − 3 ,...}, where the first decrease is pushed later (see Fig. 3.1). Lett ′ − 1 =t − 1 +(t − 2 − t +2 ) so the constraint is still satisfied. The area under the new controller is given by U (t +1 ,t ′ − 1 ,t +3 ,t − 3 ,...) =U (t +1 ,t − 1 ,t +2 ,t − 2 ,t +3 ,t − 3 ,...) +U ′ (3.15) where U ′ ≥ 0 is the area of the shaded region, given by the magnitude of the cross product < (t ′ − 1 − t − 1 ),(t ′ − 1 − t − 1 )>× <(t +2 − t − 1 ),− b Ω (t +2 − t − 1 )>. Given that U (t +1 ,t ′ − 1 ,t +3 ,t − 3 ,...) ≥ U (t +1 ,t − 1 ,t +2 ,t − 2 ,t +3 ,t − 3 ,...) , it is clear that pushing the first decrease later will never decrease the area under the curve. Doing so until we have pushed the first decrease as late as possible, which describes our greedy controller, will satisfy the constraint and maximize the delay queue. We have given a conservative upper bound on the maximum size of delay queue, as in reality the transfer rate will increase as the robot returns to the data sink. 36 While our queue-stabilizing controller does not guarantee optimal performance or bounds on limiting behavior ast→∞,Ω →∞, in practice exploration is limited by the size of the region or lifetime of the robot and these observations may be useful for practitioners. For the remainder of this chapter we return to the generalized queue-stabilizing controller presented in Eq. (3.13). 3.3.7 LocalOptimaandLimitCycles If α r (t) = x r (t) strictly minimizes Eq. (3.13), the robot can get stuck. When this occurs we implement the following recovery control strategy: WhileE[f c r,D ] ≤ θ c , instead choose α r (t) ∈ A r (t) to minimize ∥α r (t)− D∥ mr(t) . This moves the robot along the shortest path towards the data sink until it is close enough for direct communication, at which point the robot switches back to the nominal controller. We also switch to this recovery controller whenq r (t) = q max , so the robot necessarily recovers connectivity when the data queue is full. Whenk Q > 0 ork Z > 0, the robot may sacrifice exploration even when the data queue is empty in order to increase connectivity or localizability. In fact, as we increase the gainsk Q andk Z , we expect the robots to converge to or oscillate within configurations which meet the constraints, much like a strictly constrained exploration strategy would [2]. These stable limit cycles can be avoided by loosening the constraints or decreasing k Q ,k Z at the cost of reduced connectivity and localizability. In the following section, we observe this in simulation and discuss the performance of our queue-stabilizing controller. 3.4 SimulationResults Fig. 3.2 shows two simulation environments with obstacles which prevent communication and ranging. We assume robots start in the vicinity of the data sink and can move into adjacent free cells. Robots have a finite sensing radius of 1.5 with perfect sensing ( Z i = X i ) and an expected transmit radius d θ c = 4 37 Figure 3.2: Simulated environments with obstacles (purple), robots (yellow), and data sink (red). Left: 4 robots in 30x30 space with 20 obstacles. Right: With 75 obstacles. Additional experiments considered 8 robots and 50x50 space, and results are summarized in Table 3.1. (all units in robot widths). Accurate localization is key to exploration and in the absence of GPS, bea- cons, or easily detectable landmarks in the environment, relative localization performance is important. We use the Cramer-Rao bound to evaluate uncertainty of relative localization, but for simplicity assume perfect/deterministic movement in simulation. We assumePr(X i =0)=Pr(X i =1), so the number of cells visited is the reduction in map entropy. We useγ =100,θ d =10,θ CRB =200, andθ λ 2 =0.1. Source code, other parameter values, and accompanying videos for all experiments are available on GitHub [128]. We introduce two simple approaches and two competitive approaches and compare these to our queue- stabilizing (QS) controller. • Unconstrained/information-theoretic(UN): Robots chooseα r (t)∈A r (t) in order to minimize Y(α r (t)). This is a purely information-theoretic controller. • StrictlyConstrained(SC): Robots limit their action space toA r (t)={α r (t): CRB r (α r (t))≥ θ CRB andλ 2r (α r (t))≥ θ λ 2 }. Then robots chooseα r (t)∈A r (t) to minimizeY(α r (t)), and do not move ifA r (t) is empty. 38 Figure 3.3: Map cells known at data sink over time, averaged over six independent trials, with the following parameters: QS (our novel approach): Solid - k q = 0.005,k Q = 0,k Z = 0,k Y = 100, Dashed - k q = 1000,k Q = 0,k Z = 0,k Y = 100, TP: Solid - ρ = 0.35, Dashed - ρ = 0.99, MO: w 1 = 10,w 2 = 0.1,w 3 =0.1,w 4 =100. • Time Preference (TP): This controller presented by Spirin et al. uses a target ratio ρ comparing the queue to the map size [107]. If 1− q r (t)/|m r (t)| ≥ ρ , where|m r (t)| is the number of visited cells, robots chooseα r (t)∈A r (t) to minimizeY(α r (t)). Otherwise robots chooseα r (t)∈A r (t) to minimize∥α r (t)− D∥ mr(t) . In the absence of a direct communication path to the data sink, robots transfer the contents of q r (t) to a neighboring robot i if f c r,i > θ c and i is closer to D than r. To provide a fair comparison, we implement this relaying for q r (t) and D r (t) in our QS approach as well. • Multi-objective (MO): We modify the approach of Benavides et al. [108], which weighs connec- tivity against exploration, to consider our objectives and be suited to local decisions. Robots choose α r (t) ∈ A r (t) to minimize w 1 Y(α r (t))− w 2 p rD (α (t))− w 3 λ 2 (α (t)) + w 4 CRB r (α (t)), where w 1 ,w 2 ,w 3 ,w 4 >0 are priority weights. 39 Figure 3.4: Average percent of time spent with CRB below θ CRB per robot against number of map cells known at data sink for 4 robots in 30x30 space with 20 obstacles. Each point is the average performance of 6 trials, and ideal is at the top right. We show more points for the more parameterized approaches (MO and our novel QS), while SC and UN are captured by a single point because they do not depend on parameters. 3.4.1 Performance: ExplorationCoverage Fig. 3.3 shows the average map size at the data sink over time for the set of parameters which maximizes final coverage (solid lines) for each approach in the world shown in Fig. 3.2 (left). QS outperforms UN, SC, and MO and slightly outperforms TP with respect to map coverage. We depict QS here withk Q =k Z =0, i.e. only the delay queue is stabilized. This causes QS and TP to have similar emergent behavior [107], and allows us to closely compare our contribution to the state of the art. One subtle difference between the two is that QS maximizesp rD , explicitly considering the location of our neighbors and taking multi- hop communication into account, whereas TP simply minimizes∥α r (t)− D∥ mr(t) . This is likely why QS performs better. Note that as we increaseρ,k q (Fig. 3.3, dashed lines), we see smoother curves caused by incremental data transfer. In cluttered and larger environments where communication is severely limited, increasing connectivity to allow incremental data transfer significantly improves performance relative to UN (by up to 99% and 88% respectively). This finding emphasizes that the choice of k q ,k Q ,k Z ,k Y which maximizes coverage depends on characteristics of the environment (see Table 3.1). 40 Figure 3.5: QS connectivity and localizability performance with and without stabilizing the relevant virtual queues. Left: Averageλ 2D against coverage. Right: Average percent of time spent with CRB belowθ CRB per robot against coverage. Notation¬k Q indicatesk Q =0, and in both plots ideal is at the top right. 3.4.2 Performance: Localizability In Fig. 3.4, we plot localizability against coverage. We use the time spent below the uncertainty constraint as a localizability metric. Since we have bounded the CRB, this is a more meaningful metric than average CRB but follows the same trend. We see that high localizability can hurt coverage, as discussed in Sec. 3.3.7. QS can improve localizability over TP and UN without sacrificing coverage, and improves coverage over SC without sacrificing localizability. MO can achieve the highest localizability (when minimizing CRB is the sole or primary objective), but suffers with respect to coverage from not adapting to the amount of untransferred data. 3.4.3 RoleofVirtualQueues Actions which stabilize D r (t) and Q r (t) both result in increased connectivity. In Fig. 3.5 we plot aver- age λ 2D against map coverage, noting which of these virtual queues was stabilized. Interestingly, here connectivity appears more correlated withk q (associated with the delay queue) thank Q (associated with 41 Figure 3.6: Average percent of time spent with CRB below θ CRB per robot against number of map cells known at data sink for 8 robots in 30x30 space with 20 obstacles. Each point is the average performance of 6 trials using 8-hop-connectivity, and ideal is at the top right. connectivity). This indicates that stabilizing the delay queue, which explicitly consider multi-hop commu- nication, does encourage overall network connectivity. We also plot localizability against coverage, noting whetherZ r (t) is stabilized. As expected,k Z >0 is highly correlated with localizability performance. 3.4.4 AdditionalExperiments In order to scale to larger networks, since exact calculation of p rD grows intractable as the size of the network increases, we suggest a closely related metric. The k-hop-connectivity of vertices i and j in a graph is the weighted number of paths i → j of k hops or fewer, where weights indicate probability of success along the path [97]. In Fig. 3.6 we demonstrate results for eight robots using8-hop-connectivity as a heuristic, and QS again slightly outperforms TP with respect to coverage and localizability. We have also tested our queue-stabilizing controller in a cluttered environment, and an environment with much larger area. For the latter, we loosened the delay bound toθ D = 50. Our findings comparing the best coverage achieved by QS relative to TP, the leading competitive approach, are briefly summarized in Table 3.1. 42 Environment TP: Max Cov. QS: Max Cov. size # rob. # obs. ρ cov. k q k Q k Z k Y cov. 30x30 4 20 0.35 608 0.005 0 0 100 685 30x30 8 20 0.5 786 0.005 250 0 100 813 30x30 4 75 0.7 602 0.01 5 0 100 635 50x50 4 50 0.85 689 0.005 125 0 100 691 Table 3.1: Summary of results: maximum coverage Figure 3.7: Team of mobile robots during field-testing for the DARPA Subterranean Challenge [8]. 3.4.5 Implementation Thus far, we have considered simulated robots with abstract queues. When implementing queue-aware autonomy on a real-world multi-robot system, the primary challenges include determining which data to store in a queue for reliable transfer and how to implement queue-size monitoring within a publish- subscribe paradigm (which is common for robotic systems) in a manner that exposes this transport-layer metric to the decision-making application layer. In [129], we have presented a multi-layer networking so- lution which addresses these challenges. We demonstrate this solution on a network of 3 quadruped robots and 3 wheeled robots (see Fig. 3.7) in the context of underground exploration in the DARPA Subterranean Challenge. 43 3.5 Summary In this chapter, we have presented a novel distributed controller for multi-robot exploration which uses ideas from queue-based stochastic network optimization to autonomously decide at each time step, based on the current state of the system, how to weigh network reliability, connectivity, localizability, and infor- mation gain. We have demonstrated that this controller can achieve better coverage than the state of the art, reduce localization uncertainty, and result in desirable emergent behavior. Thus far, we have assumed a stochastic connectivity model which depends only on distance and com- plete connectivity outage when an obstacle intersects the line-of-sight path. In a real-world setting, the geometry and materials of the environment play a significant role in signal attenuation. In Chapter 4, we present a more realistic approach to modeling connectivity which first learns certain characteristics of the propagation environment. This enables accurate signal strength prediction, which is key to the success of our queue-stabilizing controller. The controller presented in this chapter is derived from a time-average constraint on the Cramer-Rao Bound, and indicator of the performance of an unbiased location estimator. Thus, it is agnostic to the type of localization used in practice. In Chapters 5 and 6 we present two approaches to achieve accurate, robust, and collaborative localization for a robotic network. 44 Chapter4 AccurateSignalStrengthPrediction Accurate radio signal strength prediction enables communication-aware exploration. Models which ig- nore the effect of the environment on signal propagation or rely on a priori maps suffer in unknown, communication-restricted environments. In this chapter, we present Propagation Environment Modeling and Learning (PropEM-L), a framework which leverages real-time sensor-derived 3D geometric repre- sentations of an environment to extract information about line of sight between radios and attenuating walls/obstacles in order to accurately predict received signal strength (RSS). Our data-driven approach combines the strengths of well-known models of signal propagation phenomena (e.g. shadowing, reflec- tion, diffraction) and machine learning, and can adapt online to new environments. ∗ 4.1 ProblemFormulation Our goal is to accurately predict the strength of a signal transmitted at a known location and received at an arbitrary location, which has many applications in multi-robot exploration. Predicting whether robot- to-base communication is available (through one or more hops) affects decisions made by the robot, for example when and where to move to transfer data. Additionally, understanding where connectivity is available improves situational awareness for the remote human supervisor and can assist in centralized ∗ This chapter is adapted from Clark et al. [130]. 45 Figure 4.1: Left: Network of exploring robots, stationary relay radios, and human supervisor base station. The network provides communication during exploration, disaster mitigation, or search and rescue and is visualized over a ground truth map of the 2021 DARPA Subterranean Challenge course. Right: 3D map of connectivity predicted by PropEM-L, learned from sparse point cloud data, which can be used to enable communication-aware exploration. planning and task allocation [9, 131]. Because the signal strength depends on distance and the environ- ment, accurate prediction can also aid in radio signal source-seeking [132], multi-robot localization [122], and distributed task-planning [133]. However, the mobility of the robots and lack of a priori maps present new challenges for accurate RSS prediction. Let PL dB = P tx,dB − P rx,dB be the ratio of transmit power to receive power (in dB this ratio is a difference). Our objective is to estimate d PL dB to minimize the error: | d PL dB − (P tx,dB − P rx,dB )|. (4.1) 4.2 RelatedWork Existing work simplifies the inter-robot communication model to a deterministic communication radius [134], or predicts a probability of connectivity based solely on distance [97]. These simple models break 46 down if, for instance, there is a metal wall within the communication radius which blocks the signal. For this reason, Miyagusuku et al. seek to capture the role of the environment through a learned model of signal attenuation [135, 136]. Their data-driven approach is well-suited for repeated operation in the same environment, but for exploration of an unknown and dynamic environment an online method is advantageous. The authors of [132] demonstrate an online method for modeling RSS during single-robot exploration. While they present a model which captures the size and location of obstacles (for example, a 1m thick wall 4m from the transmitter), they defer the discussion of estimating these parameters. Quattrini Li et al. propose a Gaussian Process (GP)-based method to build a communication map from measurements taken by multiple robots [137]. These works also focus on 2D operations, and give little attention to the challenge of modeling the propagation environment given partial maps in the form of 3D point clouds. Previous work in the ecology community has considered extracting relevant information from 3D point cloud data. The authors of [138] and [139] use point clouds to determine the location and thickness of trees in a dense forest, which could theoretically feed into the model presented in [132]. Recently, the authors of [140] extend this idea to use learning-based methods on 3D point cloud data to predict the affect of tree canopies on signals propagating between static wireless communication towers. Precise digital terrain models have also been used to inform signal propagation models. [112] extends a network simulator with awareness of the topography, while [141] validates terrain-aware signal strength models with experimental data. Recently, the authors of [142] demonstrate the capability of neural networks to learn from known digital terrain models to improve cellular network design. Our work extends this method to be suitable for a dynamic network of mobile, exploring robots. 47 4.3 Approach To enable communication-aware multi-robot exploration in unknown environments, we propose Propaga- tion Environment Modeling and Learning (PropEM-L), a framework for signal strength prediction which learns the effect of the environment on attenuation. In Sec. 4.3.1, we propose PropEM which leverages sparse 3D geometric representations of the environment (e.g. LiDAR point clouds) to extract features of the physical space which affect signal propagation, including line-of-sight visibility, shadowing due to obstacles, reflection, and diffraction. In Sec. 4.3.2, we validate PropEM in conjunction with conven- tional data-driven approaches to RSS prediction which rely on linear regression. We then propose a neural network-based approach, PropEM-L, which significantly improves prediction accuracy relative to a log- distance path loss model and can estimate RSS within a few decibels. 4.3.1 PropEM:PropagationEnvironmentModeling PropEM, depicted in Fig. 4.2, serves as an encoding layer from raw point cloud data to physically-relevant context of the propagation environment between two radios. In spacious environments with multiple robots equipped with 3D LiDAR taking frequent measurements (e.g. 10Hz), the aggregated point cloud capturing the physical space around them is constantly changing in size and can quickly exceed millions of points. Learning on the raw data would be challenging; the reduction in feature space offered by PropEM is advantageous. PropEM creates a discretized representation of the world via OpenVDB, an open source library for the efficient storage and manipulation of sparse volumetric data in three-dimensional grids [143]. Each voxel in this grid is associated with the probability that the space it represents is occupied. We store probabilities as log-likelihoods so that performing Bayesian updates on a voxel amounts to addition and subtraction. At each timet that a robot takes a LiDAR scan, we associate the robot’s estimated pose,x r (t), and the scan, s r (t). Localization is performed via a pose-graph SLAM algorithm called LAMP, designed for the 48 Propagation Environment Modeling sparse point clouds transmitter position receiver position log-distance LOS visibility occupied space (obstacles) ground reflection loss diffraction loss Learning RSS prediction transmit power radio frequency Figure 4.2: PropEM-L is a framework for predicting the signal strength between two radios, given their positions, which learns from the geometry of their environment. PropEM acts as an encoding layer from raw sensor data to specific features which are relevant to propagation modeling. Figure 4.3: 3D occupancy grid constructed for predicting line of sight. Figure 4.4: Left: Occupied (red), maybe occupied (yellow) and free (green) voxels intersected during ray tracing. Right: Voxels which comprise the first Fresnel zone between two radios (black), where colors help illustrate perspective. 49 large-scale exploration of perceptually-degraded environments [144, 145]. PropEM performs ray tracing from the robot’s estimated position to each point in the point cloud representation of the scan. Voxels along these rays have an increased likelihood of being unoccupied (as the light ray did not intersect an obstacle) while voxels at the points have an increased likelihood of being occupied. Following the approach presented in [146], PropEM performs this Bayesian update for all robots and all scans in a stationary, shared frame. The result is a 3D occupancy grid as shown in Fig. 4.3. Likelihoods continue to update as robots re-visit an area, such that we can account for dynamic obstacles. Given an arbitrary pair of positions representing a transmitter and receiver, we can use this occupancy grid to determine the propagation environment which would affect the signal passing between them. To trace the LOS ray, PropEM uses digital differential analyzers (DDAs), a technique from computer graphics used to interpolate between two points. We traverse the voxels between the transmitter and receiver, and classify them as free, occupied, maybe occupied, or unknown. This classification is done by thresholding the log-likelihoods associated with each voxel (see Fig. 4.4). The number of total and occupied voxels informs our calculation of free space path loss and shadowing. To consider third-order effects like reflection and diffraction, we follow the approach presented in [112] to consider a model of the terrain (in our case the occupancy grid representing the partially known environment) as it causes reflections from the ground and as it intersects with the first Fresnel zone. We calculate the reflection loss RL given by Eq. 2.5 as a function of the transmitter and receiver locations. Then, incrementally along the line-of-sight path, we traverse voxels horizontally and vertically in the robot’s frame of reference to check the first Fresnel zone clearance (see Fig. 4.4). Along each direction we calculate the Fresnel-Kirchoff diffraction parameter v determined from the occupancy grid, and use the worst casev to approximate the diffraction loss DL given by Eq. 2.7. By maintaining and updating the occupancy grid representation internally, PropEM can determine for an arbitrary pair of radio positions the distance, visibility, and obstacles between them as well as the 50 ground reflection loss and diffraction loss from the cave/tunnel walls. In the next section, we discuss how these features are used in RSS prediction. 4.3.2 PropEM-L:Learning We performed extensive field testing in an underground limestone mine in Kentucky and captured about 80 minutes of autonomous exploration with 1 base station radio, 6 ground robots (3 legged Boston Dynamics Spot robots and 3 wheeled Clearpath Robotics Husky robots), and 13 communication relay radios which were autonomously deployed by the ground robots when signal-to-noise ratio (SNR) fell below a desired threshold. All radios were MIMO StreamCaster 4000-series radios from Silvus Technologies [147, 129]. Exploration spanned about 300 meters in each of thex andy directions, and we collected 1.3 million data points capturing transmitter position, receiver position, frequency, noise, transmitter power, RSS, SNR, loss rate, etc [8]. Although we can only collect signal strength data between pairs of radios which have connectivity, we have location estimate data for all radios. For pairs of radios which have certainly lost connectivity, meaning neither of the robots report RSS measurements at a given location for a two-minute period, we augment the dataset following the approach presented in [135], where the received signal strength is inferred to be the noise floor of the radios (here -94dB). To ensure synchronization between the location estimate and the signal strength measurement, we remove any data for which the two were observed more than one second apart. Additionally, we observe a limited number of outliers caused by localization failure which are easily removed for static radios whose location estimates should never vary more than a few meters. As a preliminary step in validating PropEM, we note the correlation between the features PropEM outputs and the RSS measurements. Table 4.1 captures the Pearson correlation coefficient of data collected during field testing, which illustrates that signal attenuation is correlated with distance and the number of 51 Table 4.1: Correlation between PropEM output features and RSS. Feature Correlation with RSS (dB) Log distance (m) -0.84 Line of sight 0.39 Not-free voxels (#) -0.78 Reflection loss (dB) 0.58 Diffraction loss (dB) 0.34 � �� ��� ��� ��� ��� ��� ������������ ��� ��� �� �� ������������������������ ��� ����N Figure 4.5: Signal attenuation plotted against distance. Measurements at each distance are averaged. Line- of-sight (LOS) links see less degradation than non-line-of-sight (NLOS) links. not-free voxels (occupied, maybe occupied, or unknown), followed by the additional loss predicted by the two-ray ground reflection model. To demonstrate PropEM as part of an RSS prediction framework, and as baselines for comparison, we implement the standard log-distance path loss prediction model and several extensions of this model. Table 4.2 presents a summary of the accuracy of each model, and Fig. 4.6 offers a snapshot of their performance. • Simple Path Loss: Using linear regression, we fit the parameters of the simple path loss model given by Eq. 2.1. We abbreviatePL(d 0 ) dB atd 0 = 1m toPL d0 in this and the following sections. This regression givesPL d0 =14.84 andη =4.73. • Visibility: Motivated by the distinctly different curves observed in the data, as demonstrated in Fig. 4.5, we implement a model which incorporates visibility. We use linear regression to fit two distinct 52 lines to Eq. 2.1 depending on whether there is line of sight, which givesPL LOS d0 =36.5,η LOS =2.75, PL NLOS d0 =13.72 andη NLOS =4.81. • Shadowing Heuristic: Considering not just visibility but the amount of occupied space, we use linear regression to approximate the dB of attenuation per meter of not-free voxels. Similar to the approach presented in [148], this model adds an additional attenuation of 0.16 dB/m to the output of the path loss model with LOS parameters. • Two-Ray: This ground reflection model subtracts RL dB , given by Eq. 2.5, from the simple path loss model using LOS parameters. • Knife-Edge: This diffraction model subtracts DL dB , given by Eq. 2.7, from the simple path loss model using LOS parameters. • Reflection-Diffraction: As presented in [112], this model combines the effects of two-ray ground reflection and knife-edge diffraction. For a maximum v≤− 0.8, the Fresnel zone is considered clear and this model reduces to the two-ray model. Otherwise, the diffraction loss is subtracted from the two-ray model with the modification that any constructive reflective interference is ignored. As implemented, the conventional models make a few simplifying assumptions: the simple path loss model assumes no obstacles, the visibility and shadowing heuristic models assume all obstacles have the same material properties, the two-ray model assumes a flat reflective terrain, and the knife-edge model assumes the worst-case from obstructions. While these models have the advantage of being interpretable and grounded in the physics of how radio signals propagate, non-linear combinations of these features, learned via stochastic gradient descent, can capture subtleties in the environmental dependence which are inadequately captured in the simpler models. We introduce five fully-connected neural networks (NN), each with the same simple structure: an input layer, a sixteen-unit hidden layer with restricted linear activation function, and an output layer which 53 predicts attenuationPL dB . Each NN learns to minimize the L2 loss (to reduce large errors) between the predicted attenuation and the measured attenuation, using the Adam optimizer to train for 100 epochs or until convergence. The field test data from the limestone mine is normalized such that all features have zero-mean and unit standard deviation. 30% of the data is withheld from training, and the remainder is divided randomly into batches of size 2048. Each NN takes a subset of the features which PropEM outputs, as described below: • NN-vis uses logarithmic distance and LOS, which is broken into two boolean input features: strictly visible indicates all the voxels between the transmitter and receiver are free; strictly not visible indi- cates there is at least one occupied voxel. • NN-vox uses logarithmic distance and the number of occupied, maybe occupied, free, and unknown voxels. • NN-ref uses logarithmic distance and reflection loss. • NN-diff uses logarithmic distance and diffraction loss. • NN-all combines the features of NN-vox, -ref, and -diff. 4.4 ExperimentalResults We experimentally evaluate the performance of PropEM-L on a dynamic network of 13 autonomously- deployed static radios, one base station, and six robots exploring an underground environment. Via online learning, we demonstrate that PropEM-L can adapt to challenging new environments and construct signal strength maps. 54 5 10 15 20 25 30 120 100 80 60 40 Rx power - Tx power (dB) PL (true) Pathloss Visibility Shadow Two-Ray Knife-Edge Refl-Diff 5 10 15 20 25 30 Time (mins) 120 100 80 60 40 Rx power - Tx power (dB) PL (true) NN-vis NN-vox NN-ref NN-diff NN-all Figure 4.6: Example performance of the learning models visualized against ground-truth attenuation (PL). Top: conventional data-driven propagation models. Bottom: neural-network based prediction. These plots depict the link between a single exploring robot and the base station. 55 Table 4.2: Mean absolute error (dB) for each model (offline) Learning Model Field Test Day 1 Day 2 Day 3 Simple path loss 6.73 15.87 15.37 13.0 Visibility 6.5 13.93 14.14 12.57 Shadowing Heuristic [148] 6.61 14.12 14.55 12.88 Two-Ray 12.07 17.59 17.86 16.57 Knife-edge 10.18 13.54 14.69 13.36 Reflection-Diffraction [112] 10.15 13.93 15.45 14.11 NN-vis 6.11 12.63 13.42 12.03 NN-vox 3.87 11.87 12.70 11.21 NN-ref 6.14 13.00 13.56 12.58 NN-diff 6.36 12.64 14.99 12.43 NN-all 3.73 11.42 12.95 11.70 30 20 10 0 10 20 30 Error (dB) 0.00 0.02 0.04 0.06 0.08 Probability NN-vis NN-vox NN-ref NN-diff NN-all Figure 4.7: Error probability density functions for each NN model. 4.4.1 Performance: Accuracy Comparing the conventional signal propagation models, the visibility model achieves the most accurate prediction, as shown in Fig. 4.6 (top). The two-ray, knife-edge, and reflection-diffraction models are very sensitive to small differences in antenna height, distance between transmitter and receiver, and precise location of neighboring walls. Because PropEM relies on sparse point clouds, uncertain location measure- ments, and discretized space (voxels are up to 1.5m wide), we see rapid changes in the predicted diffraction loss as new voxels are explored or the robot’s position changes even slightly. This sensitivity is likely the reason the more parsimonious models actually perform better. 56 Table 4.2 compares the performance of each NN model, and Fig. 4.7 depicts the probability density function of errors observed using each model. NN-all saw the best performance of any method with an average error of only 3.72 dB, and is less likely to produce errors above 10dB. To contextualize this performance, the standard deviation ofPL dB for a pair of static radios (at a constant distance) observed in the limestone mine was up to 3.43 dB, indicating that errors in this range may be attributed to inevitable channel noise. 4.4.2 Performance: Generalizability One of the most important aspects of a prediction framework is the ability to generalize and transfer across different, complex environments. We collected data from two 30-minute (days 1 and 2) and one 60-minute (day 3) analog exploration missions in the course constructed for the DARPA Subterranean Challenge [8]. This environment, shown in Fig. 4.1 was characterized by narrow, winding passageways and had three distinct subsections: an urban environment similar to a subway station, a mine-like environment, and a subterranean cave-like environment. We had no a priori knowledge of the course or the materials from which it was constructed. Table 4.2 summarizes the results of each model for each day of the competition. The average error for each model was more than 10dB, which could be due to a number of things. The scale of this environment was drastically different from that of the field test, with tunnels of 2m width rather than 20m width. Additionally, the environment was constructed of different materials (e.g. thin metal panel dividers in place of thick limestone columns). These materials attenuated signal more significantly than expected, as illustrated in Fig. 4.8. As a result, a simple path loss model was unable to accurately predict signal strength, with more than 15dB of error on average during each 30-minute experiment. To adapt to drastically different environments, PropEM-L can perform online learning to leverage the partial 3D geometric representations available during exploration. Note that for simplicity we assume in 57 0 20 40 60 x (m) 0 10 20 30 40 50 60 y (m) branch 1 branch 2 branch 3 Simple prediction 4m 8m 16m 120 110 100 90 80 70 60 50 Rx power - Tx power (dB) Figure 4.8: Spatial plot of signal strength from a static transmitter (red) in the environment depicted in Fig. 4.1. A simple path loss model predicts the signal will attenuate with distance (circles around transmitter). However, branch 2 sees more degradation than predicted due to barriers between branches. this section RSS measurements from all radios are available at the base station immediately, ignoring any networking latencies. Using the visibility model and re-fitting the parameters η LOS ,PL LOS d0 ,η NLOS ,PL NLOS d0 every k minutes, we can achieve a mean absolute error of 10.99dB (21% improvement over the offline- learned model) on day 1, 12.47dB (12% improvement) on day 2, and 11.89dB (5% improvement) on day 3 for k = 1. These improvements are due to the fact that the model updates the attenuation it associates with a given number of occupied voxels between a transmitter and receiver. This reflects a change in the model’s belief about the RF properties under identical geometries. Perhaps more notably, the final path loss parameters learned via linear regression can shed insight on the environment itself. The learnedη LOS in the new environment was between 1.59-1.94, which indi- cates indoor areas and is significantly lower than the initial estimate of 2.75 which is typical for outdoor urban environments. Without any other information about the map, we can conclude the competition environment was more confined. Similarly, re-fitting the heuristic model increased our approximation of 58 Table 4.3: Mean absolute error (dB) for each NN model trained online. Model Day 1 Day 2 Day 3 NN-vis 11.20 11.77 10.48 NN-vox 8.65 9.60 7.28 NN-ref 10.55 11.81 10.70 NN-diff 9.57 11.69 9.51 NN-all 5 8.76 9.70 7.61 0 10 20 30 40 50 60 70 Time (mins) 4 6 8 10 12 14 Mean Absolute Error (dB) Figure 4.9: Performance of NN-vox trained online in the competition environment on day 3. By the final minute of exploration, mean absolute error is 7.54 dB. attenuation over not-free space from 0.16 to 0.43-0.66 dB/m. This lets us infer that the competition en- vironment had material which blocked radio waves on average up to 4 times more than expected. This interpretability is an advantage of the conventional linear regression models over the NN models. To train the neural network component of PropEM-L online during exploration, every 60 seconds we train for 10 epochs (selected empirically) on data collected in the last minute. The number of samples used to retrain is on average 2644 (with a minimum of 498 and a maximum of 5414). Starting with the model trained offline prevents overfitting to initial measurements, while over time the new environment is learned. Fig. 4.9 shows how the mean absolute error decreases and converges over time. Over the entire duration of the competition run (day 3), NN-vox trained online can achieve a mean absolute error of 7.28 dB, improving performance over the simple path loss model trained offline almost two-fold. 59 4.5 Summary In this chapter, we presented Propagation Environment Modeling and Learning (PropEM-L), a framework for signal strength prediction which learns the effect of complex, communication-restricted environments on attenuation. Our approach leverages sparse sensor-derived 3D geometric representations to model features of the propagation environment between a transmitter and receiver, including line-of-sight visi- bility, shadowing due to obstacles, reflection, and diffraction. We compare our neural network-based on- line learning with conventional approaches to RSS prediction, and demonstrate PropEM-L on a dynamic network of exploring robots and stationary radios in multiple communication-restricted, underground en- vironments. Our findings indicate that PropEM-L can significantly improve RSS prediction and adapt to new environments, which will be an important aspect of transferring communication-aware exploration strategies from analog missions to real exploration missions. Given the location of static radios and their transmit powers, we can use our framework to construct a signal strength map, as shown in Fig. 4.1. PropEM-L can update this map as areas are explored or the net- work topology changes. Connectivity maps have a number of uses in communication-aware exploration, for example robots can autonomously identify areas to deploy additional relay radios [129]. In the previous chapter, we presented an approach to exploration which relies on connectivity model- ing to predict the probability of successful data transfer. The accurate signal strength prediction presented in this chapter enables our queue-stabilizing framework to make communication-aware decisions even in unknown environments with non-line-of-sight links. In this chapter, we leverage LiDAR sensors onboard the robots to inform our model of the environment and predict whether a link is NLOS. In the next section, we will see that it is possible to infer whether a link is NLOS even without LiDAR, given a large network with rich data on pairwise signal strength measurements. 60 Chapter5 RobustNetworkLocalization Thus far, we have focused on connectivity as it is one key component of remote exploration: new data needs to be communicated. For the remainder of this dissertation we will turn to localization, another key component of remote exploration; new data needs to be associated with its location to build accurate and useful maps. In this chapter, we examine the network localization problem in the presence of noisy signals where pairwise measurements may be NLOS or missing illustrated in Fig. 2.2. We make two key observations: (1) the Euclidean distance matrix of the true node positions (ind dimensions) will have rankd+2, which we show in Sec. 5.1, and (2) the matrix capturing the additional NLOS attenuation is commonly sparse and non-negative, i.e. there are a limited number of walls which can only degrade signal strength. Given these observations, our approach draws on recent advances in sparse plus low-rank decomposition to first extract the positively biased NLOS attenuation, which we refer to as sparse matrix inference. After extracting the NLOS attenuation, we leverage a popular method for dimensionality reduction, locally linear embedding (LLE), to recover a set of weights which allow linear reconstruction to determine the exact positions of all nodes given the anchors’ positions. 61 Specifically, we extend the discrete optimization approach for sparse matrix recovery which is pre- sented in [12] to (1) handle missing pairwise measurements by imputation and (2) approximate the un- known sparsity structure of the sparse component of the matrix. Additionally, we provide details for solving for node positions using the LLE weight matrix directly (using least squares method) without the need for additional eigen-decomposition and coordinate frame alignment used conventionally. We demonstrate that SMILE significantly improves localization accuracy over baseline methods like classical multidimensional scaling (MDS) which ignore the affect of NLOS attenuation. Further, we draw parallels between this approach and another method for graph signal processing which has recently been demonstrated as promising, namely Graph Convolutional Networks (GCNs) [13]. We compare the per- formance of SMILE and a state-of-the-art GCN implementation on large-scale simulated networks and demonstrate an improvement in localization accuracy and reduced computation time for network of more than 1000 nodes. Finally, we evaluate performance on two real-world networks: an outdoor wireless sen- sor network with 11 nodes, and 5 mobile robots and 13 static radios in a GPS-denied limestone mine with significant NLOS attenuation. ∗ 5.1 ProblemFormulation LetA be the set of anchors whose positions are known, where|A|=n A . LetB be the set of agents whose positions are unknown, where|B|=n B . LetC =A∪B be the set of nodes which includes both anchors and agents, withn A +n B =n. We define the (n× n) true distance matrixD as D[i,j]=||p i − p j || (5.1) ∗ This chapter is adapted from Clark et al. [149]. 62 wherep i = [x i ,y i ] T is the position of node i. We establish the convention that the first n A rows and n A columns pertain to the anchors. The Eucliden Distance Matrix (EDM) is given by D ◦ 2 , where this notation indicatesD is squared entry-wise. This matrix is zero along the diagonal and symmetric [150]. LetP∈R n× 2 withP[i,:]=p T i be the position matrix. Given the discussion of EDM rank in Chapter 2.4, we know thatD ◦ 2 has rank4. Let O be the matrix that contains the distances that we observe, where O[i,j] is a function of the strength of radio signal transmitted by nodei and received by nodej. In general, our observation can be captured as the Hadamard product O=Ω ◦ [D+N+S] (5.2) whereΩ is the observation mask and takes on values of 1 when a measurement is available and 0 otherwise (for instance, when the transmitter and receiver are out of range). N is an asymmetric matrix capturing noise in the observations, which we assume is Gaussian and zero-mean. S captures the additional NLOS attenuation, and as in [87] we assumeS is non-negative and sparse. We also assumeS is symmetric which relies on assumptions that an attenuating obstacle will affect a radio link in both directions equally. We make the realistic assumption that nodes are in general position, meaning that in 2D they do not lie on a straight line. We also assume that an upper bound on the distance between any two nodes, d max , is known or can be approximated. Our objective is finding an estimate for the locations of all nodes ˆ P=[ˆ p 1 ,...,ˆ p n ] T which is consistent with the prior information - (1) the observations inO as well as (2) the known anchor positionsP A =[p 1 ,...,p n A ] T . 5.2 RelatedWork Recent works in network localization extend semi-definite programming (SDP) methods to consider non- Gaussian noise [84] and, more specifically, NLOS noise [85, 86, 87]. Jin et al. demonstrate that when 63 the NLOS noise has a certain structure, namely non-negative and sparse, a sparsity-promoting term in the objective function can improve the performance of SDP approaches [87]. However, SDP methods suffer with respect to complexity and are intractable for very large networks [13]. Yanetal. recently demonstrated promising results in the application of Graph Convolutional Networks (GCNs) to the network localization problem. Their approach maintains accurate localization despite NLOS attenuation, and is scalable to large-scale networks at an affordable computation cost. However, we will see that the learned model is unable to exactly recover positions for a completely observed distance matrix in the absence of noise. In this work we propose a novel network localization algorithm which builds on the principles of multidimensional scaling for exact recovery in the absence of noise, exploits the sparsity of NLOS attenuation for improved localization accuracy, and scales to very large networks at an affordable computation cost. 5.3 Approach In this section we propose Sparse Matrix Inference and Linear Embedding, our novel large-scale network localization algorithm, which is illustrated in Fig. 5.1. Details of the algorithm are presented in Algorithm 1. In subsection 5.3.1, we discuss our method of extracting NLOS attenuation via sparse matrix inference, which results in a low-rank approximation of the Euclidean distance matrix. Then in subsection 5.3.2, we discuss our method of transforming this matrix into an estimate of the locations of all nodes. 64 Anchor positions low-rank X sparse NLOS Y Observations Complete the Euclidean distance matrix Sparse matrix inference Construct locally linear weight matrix Solve sparse sub-problem Solve low-rank sub-problem E Predicted locations W Figure 5.1: An overview of Sparse Matrix Inference and Linear Embedding (SMILE). 5.3.1 SparseMatrixInference For a given input matrixE ∈ R n× n , for whichE = X+Y, sparse matrix inference seeks to find the low-rank componentX∈R n× n and sparse componentY∈R n× n which solves: min X,Y f(X,Y)=||E− X− Y|| 2 F +λ ||X|| 2 F +µ ||Y|| 2 F s.t. rank(X)≤ α, ||Y|| 0 ≤ β (5.3) where||X|| F denotes the Frobenius norm. To apply this to the problem defined in 5.1, we first square the observed matrix (line 2 in Algorithm 1). Temporarily assumingN=0 and no missing observations, i.eΩ =11 T , E=O ◦ 2 =(D+S) ◦ 2 = D ◦ 2 |{z} low-rank +S ◦ 2 +2D◦ S | {z } sparse (5.4) where the expression in parentheses is non-negative and sparse (becauseS is non-negative and sparse), andD ◦ 2 is low-rank. Thus the problem is amenable to the sparse matrix inference framework. 65 Let us consider the case with noiseN ̸=0 and no missing observations E=O ◦ 2 =( ˜ D z }| { D+N+S) ◦ 2 =( ˜ D+S) ◦ 2 = ˜ D ◦ 2 +S ◦ 2 + ˜ D◦ S | {z } sparse (5.5) The last expression is exactly the same as the formulation in 5.4 except that ˜ D ◦ 2 =D ◦ 2 +N ◦ 2 +2N◦ D may no longer be low rank due to the addition ofN ◦ 2 +2N◦ D where entries of the first term are now i.i.d from a Chi-Squared DistributionX 2 1 . However, in our ablation study, we empirically verify that as long as the standard deviation of the normal noise in entries ofN is not too large, using the estimated low- rank matrix from the sparse matrix plus low-rank inference, SMILE is able to recoverP fairly faithfully under RMSE loss. The approach presented in [12] assumes a complete input matrix. While more complex approaches to matrix completion exist, for example by finding the sum total length of the shortest path between two nodes [151, 152], we take a simpler approach. WhenΩ ̸= 11 T , we complete the observation matrix by filling in missing values with d max (line 1). Even in the face of missing observations, we show that the problem remains amenable to the sparse matrix inference framework as long as the missing observations are not too large of a fraction of the total number of observations inO. 66 E= ˜ O ◦ 2 =[Ω ◦ O] ◦ 2 =[Ω ◦ (D+N+S)+(1− Ω )d max ] ◦ 2 =[Ω ◦ D+Ω ◦ (N+S)+(1− Ω )d max ] ◦ 2 =[D− (1− Ω )◦ D+Ω ◦ (N+S)+(1− Ω )d max ] ◦ 2 =[D+Ω ◦ N+Ω ◦ S+(1− Ω )◦ (d max I− D) | {z } ˜ S (sparse) ] ◦ 2 =[D+Ω ◦ N | {z } ˜ D + ˜ S] ◦ 2 =[ ˜ D+ ˜ S] ◦ 2 = ˜ D ◦ 2 + ˜ S ◦ 2 +2 ˜ D◦ ˜ S | {z } sparse (5.6) Thus, ˜ O decomposes similarly toO as long as(1− Ω )+S is still sparse, i.e. the total number of NLOS and missing measurements is low. ˜ D ◦ 2 is still amenable to SMILE like in the previous case. As presented in [12], we alternate between solving two sub-problems. For a givenX, we estimateY by composing a sparse matrix with non-zero entries at the largest indices of(E− X). This is described further in Algorithm 2. Then for a givenY, we estimateX by reducing the rank of(E− Y). This repeats until the value of the objective function in Eq. 5.3 has converged, corresponding to the inner loop (lines 11-18). We initializeX with a random, low-rank matrix and repeat this process forT > 0 random initialization, ultimately selecting the decomposition which minimizes the objective function (lines 5-19). The approach presented in [12] assumes the sparsity of Y is known, however in realistic settings this may not be true. We observed empirically that for a matrix with known components, solving the sparse matrix inference problem for increasing estimate ˆ β causes the objective function to decrease, and around the true value of β it converges (Fig. 5.2). This motivates the outer loop (lines 3-21). To reduce 67 Algorithm1: SMILE Input: O (observation matrix),P A (anchor positions),α (desired rank) Output: ˆ P B (agent location estimates) 1: ˜ O=O+(1+Ω )d max 2: E= ˜ O ◦ 2 3: ˆ β ← ˆ β i 4: while∆ f/f >ϵ β do 5: fort in{1,...,T}do 6: X ′ ∈R n× n ← random 7: U,Σ ,V T = PSVD(X ′ ,α ) 8: X=UΣV T 9: Y∈R n× n ← 0 10: f =||E− X− Y|| 2 F +λ ||X|| 2 F +µ ||Y|| 2 F 11: while∆ f/f >ϵ do 12: Y ′ = compose_sparse(E− X, ˆ β ) 13: Y = 1 1+µ Y ′ ◦ (E− X) 14: X ′ = 1 1+λ (E− Y) 15: U,Σ ,V T = PSVD(X ′ ,α ) 16: X=UΣV T 17: f =||E− X− Y|| 2 F +λ ||X|| 2 F +µ ||Y|| 2 F 18: endwhile 19: endfor 20: ˆ β ← ˆ β +η 21: endwhile 22: W∈R n× n ← 0 23: for each nodeido 24: find k nearest neighborsNN(i) 25: for each pair of neighbors (j,l)do 26: H∈R k× k whereH[j,l]= 1 2 (X[i,l]+X[j,i]− X[j,l]) 27: solveHw =1 forw 28: endfor 29: W i ← w/ P w at indices of neighbors 30: endfor 31: m=(I− W) A P A 32: solve(W− I) B ˆ P B =m for ˆ P B 33: return ˆ P B Algorithm2: compose_sparse Input: M∈R n× n (matrix),β (sparsity) Output: B (binary matrix) 1: sorted indices = argsort(M) 2: one indices = sorted indices[-β :] 3: B∈R n× n ← 0 4: B[one indices] = 1 5: return B 68 Table 5.1: SMILE parameters Parameter Symbol k Number of neighbors ˆ β i Initial sparsity estimate η Step size T Number of random initializations λ Low-rank matrix regularizer µ Sparse matrix regularlizer ϵ Inner loop tolerance ϵ β Outer loop tolerance 0 20000 40000 60000 80000 100000 120000 140000 0 1 2 3 4 5 Objective function f 1e8 Figure 5.2: The final value of the sparse matrix inference objective function f (Eq. 5.3) when we assume different levels of sparsity ˆ β , plotted alongside the true sparsityβ (dotted line) for various simulated ma- tricesE∈R 500× 500 . 69 computation time, we can initialize ˆ β with the approximate sparsity, increase the search step size η , or increase the converge toleranceϵ β . However this may come at the cost of localization accuracy. 5.3.2 LinearEmbedding Next we find the set of k nearest neighbors NN(i) for each nodei using the estimated EDM gotten from sparse matrix inference. For each node, we compute ak× k matrixH which captures the pairwise similarity between the nodes inNN(i)∪{i} (line 25) using the corresponding sub-matrix in the estimated EDM.H is calculated using the same approach used for classical MDS, but for local neighborhoods [68]. FromH we can recover the desired weight matrixW (lines 26-27), first by solving for w and then inserting it at the row position corresponding to the node index ofi with the entries inw aligned with the column belonging to the nearest neighbours and zeros for non-neighbours. At this point, each row ofW corresponds to a node and captures how the node’s position can be expressed as a linear combination of its neighbors. This is a meaningful representation of the positions of all nodes, but is not yet a node embedding. Typically, LLE then computes a sparse matrix(I− W) T (I− W) whose eigenvectors corresponding to the two smallest non-zero eigenvalues result in a solution up to rotation and translation. In this setting, we can compute a solution using W and the anchor location P A directly. This is possible because, by construction,WP = P. This is essentially a system of n equations with n B < n unknowns. TreatingW− I ∈ R n× n as a block matrix, let (I− W) A = (I− W)[:,:n A ] ∈ R n× n A be the sub-matrix corresponding to the anchors and(W− I) B = (I− W)[:,− n B :]∈R n× n B be the sub- matrix corresponding to the agents, whereI∈R n× n is the identity matrix. Similarly treatingP ∈R n× 2 70 as a block matrix, letP A = P[: n A ,:]∈ R n A × 2 andP B = P[− n B ,:]∈ R n B × 2 correspond to the block matrices for positions of anchors and agents respectively. With some manipulation we get, WP=P =⇒ T z }| { (W− I)P=0 =⇒ TP=0 =⇒ T A | T B P A − P B =0 =⇒ T A P A +T B P B =0 =⇒ T A P A =− T B P B =⇒ (I− W) A P A | {z } m, known =(W− I) B | {z } known P B |{z} unknown (5.7) (lines 29-30) and from the last expression we can estimate a solution ˆ P B for the unknown agent positions using the least squares minimization. 5.3.3 ComparisonwithGraphConvolutionalNetworks We highlight several interesting parallels between SMILE and GCN. Briefly, the approach presented by Yan et al. computes the position estimates ˆ P=A ′ ϕ (A ′ (A◦ O)Z (1) )Z (2) (5.8) whereA is the thresholded adjacency matrix for a given thresholdθ GCN ,A ′ is the row-normalized aug- mented thresholded adjacency matrix, ϕ (·) is a nonlinear activation function, andZ (1) ,Z (2) are learned weights. Note that the graph signal is the observed matrix. 1. θ GCN andk: The GCN-based approach introduces a threshold such that edges between nodes in the graph are present only if the observed distance is less than the threshold. Decreasing this threshold is 71 similar to decreasing the number of nearest neighbors, and both have the benefit of noise truncation. However, as the threshold is increased, GCN experiences over-smoothing due to the aggregation step. Intuitively, if every node is connected to every other node, the aggregation step causes all node embedding to collapse at a single point. This means that more information, i.e. additional pairwise measurements, actually hurts the algorithm. SMILE does not experience this issue, and we will see in Fig. 5.5 that increasingk improves performance at the cost of increased runtime. 2. Low-passfilteringandPSVD: Repeated multiplication by the normalized adjacency matrix acts as a low pass filter [118]. This reduces the rank of the graph signal, similar to the process of extracting the low-rank component of the noisy euclidean distance matrix. However, repeated multiplication by the normalized adjacency matrix reduces the rank of the observed matrix by some amount, which corresponds to distances (rather than squared distances). Our approach applies rank reduction to exactly reduce the Euclidean distance matrix to rankd+2, informed by the principles of the problem formulation. 3. Convolutionandlinearembedding: Each convolution layer multiplies the adjacency matrix and input matrix by a set of weights. Intuitively, this makes the node features at the next layer a weighted sum of the neighbor features. The learned GCN weights are analogous to the set of weightsW in SMILE which allow linear reconstruction of agents from the anchors positions. However, the re- peated convolutions and nonlinear activation prevent a straightforward analysis. The interpretabil- ity ofW is an advantage of our approach. Qualitatively, we expect SMILE to outperform GCN because (1) it is not subject to oversmoothing and makes careful and productive use of additional pairwise measurements, (2) it finds a Euclidean distance matrix with the exact expected rank, and (3) it relies on principled methods to determine a weight matrix 72 relating nodes to their neighbors, which is thus interpretable. In the next section, we compare these approaches quantitatively. 5.4 SimulationResults In the following two sections we evaluate the performance of SMILE with respect to localization accuracy and computation time. We also consider performance under different noise settings, as an ideal method is both robust to high-levels noise and able to exactly recover positions when possible. Localization accu- racy is measured by the root-mean-squared-error (RMSE) given by||P B − ˆ P B || F . Robustness considers the affect of Guassian noise, NLOS attenuation, and missing pairwise measurements. The experiments consider networks in 2D, thus the desired rank of EDM is α = 4. The SMILE parameters are set to k = 50, ˆ β i = 5 n 2 100 ,η = n 2 100 ,T = 1,λ = 0.01,µ = 0.1,ϵ = 0.001, andϵ β = 0.01. For comparison, we train a two-layer GCN according to [13] with distance threshold 1.2. More details of our implementation are available online † . Our simulated scenarios consider n nodes randomly placed over a 5m× 5m square area, with the first n A nodes considered anchors. NoiseN is drawn from a zero-mean Gaussian, N[i,j] ∼ N (0,σ 2 ). NLOS noise is drawn from a uniform distribution,S[i,j] =S[j,i]∼U [0,10] with probabilityp NLOS and 0 otherwise. Both matrices are zero along the diagonal,N[i,i]=S[i,i]=0. The observation mask limits our observed distance measurements by a thresholdθ such thatΩ [i,j]=1 ifO[i,j]≤ θ and 0 otherwise. 5.4.1 Performance First, we consider the setting where n = 500,n A = 50,σ = 0.1,p NLOS = 1 10 , and θ = d max , for which an example dataset is available ‡ . Fig. 5.3 illustrates the performance of SMILE, which achieves an RMSE of 0.06 in 5.22 seconds, and GCN which achieves an RMSE of 0.11 in 5.73 seconds. GCN’s performance † https://github.com/ANRGUSC/smile-network-localization ‡ https://github.com/Yanzongzi/GNN-For-localization 73 0 1 2 3 4 5 0 1 2 3 4 5 GCN (0.11) true anchor pred anchor true node pred node 0 1 2 3 4 5 SMILE (0.06) true anchor pred anchor true node pred node Figure 5.3: Localization with GCN (Yan et al. [13]) and novel SMILE for a 500 node network with 50 anchors. GCN achieves RMSE 0.11 and SMILE achieves RMSE 0.06. is consistent with that reported in [13], which reports RMSE for various other methods, including SDP with sparsity promoting regularization [87] which achieves an RMSE of 0.26. SMILE achieves the highest reported localization accuracy, and Fig. 5.4 illustrates that not only is the error low on average but the error density has a smaller tail. In Fig. 5.5, we investigate the localization accuracy and computation time as we varyk, the number of neighbors used for linear embedding. We observe that RMSE remains consistent forn A ≥ 50. While the minimum of 0.048 is reached atk = 130, this comes at the expense of increased computation time. Note that selecting k is analogous to setting the GCN threshold, however we do not observe the degradation in performance that GCN is prone to when the threshold is too high. This comes from the aggregation component of convolution, which causes over-smoothing if a node has too many neighbors and results in the position estimates collapsing to a point. Fig. 5.6 demonstrates the performance of SMILE and GCN asp NLOS varies from0 to 1 2 . From this, we observe that SMILE outperforms GCN for up to 30% NLOS links. Beyond this, the assumption thatS is sparse is no longer true, and performance suffers as expected. The same is true when θ falls below 3.5, and(1− Ω ) is no longer sparse, i.e not enough entries in the observations matrix ˜ O. Notably, we do not 74 0.0 0.1 0.2 0.3 0.4 0.5 Error 0 20 40 60 80 100 120 Frequency GCN SMILE Figure 5.4: Error density for GCN and SMILE on the data from Fig. 5.3, where SMILE results in a more desirable distribution. 0 20 40 60 80 100 120 140 Number of neighbors 0.1 0.2 0.3 0.4 0.5 RMSE 8 9 10 11 12 Runtime (sec) Figure 5.5: RMSE (solid line) and runtime (dashed line) trade-off as we vary the number of neighbors k. Each point is the average of 10 trials. 75 0 10 20 30 40 50 p NLOS 0.1 0.2 RMSE SMILE GCN 0.0 0.1 0.2 0.3 0.4 0.5 0.10 0.15 RMSE SMILE GCN 3 4 5 6 7 8 0.0 0.5 1.0 RMSE SMILE GCN Figure 5.6: Performance of SMILE and GCN as the probability of NLOS links, standard deviation of Gaus- sian noise, and threshold for distances which can be measured are varied (T =2). observe a clear trend in the performance of GCN asσ increases. We posit that this is because the learned approach does not rely on exact decomposition, even in the absence of noise. To test this, we compare the performance of GCN and SMILE on an ideal dataset and consider the results in Fig. 5.7. While the performance of both methods is better in this ideal setting, we observe that GCN cannot exactly recover positions. While being robust to Gaussian noise and sparse non-Gaussian noise, SMILE is also accurate in ideal scenarios. As we increase the number of nodes, accuracy of both methods increases (accuracy similarly increases with the percentage of anchors). However, complexity increases with the size of the network. Because our 76 0 1 2 3 4 5 0 1 2 3 4 5 GCN ideal (0.12) 0 1 2 3 4 5 0 1 2 3 4 5 SMILE ideal (0.0) 0 1 2 3 4 5 0 1 2 3 4 5 GCN noisy (0.19) 0 2 4 0 1 2 3 4 5 SMILE noisy (0.1) actual a predicted a actual predicted Figure 5.7: Performance of SMILE and GCN in an ideal setting (p NLOS = 0,σ = 0) and a noisy setting (p NLOS = 1 10 ,σ =0.5). SMILE is robust to noise without sacrifice performance in ideal settings. approach is iterative, we measure complexity numerically (computation time) in lieu of analytically as it is difficult to predict when the sparse matrix inference will converge. Fig. 5.8 shows that the computation time of both SMILE and GCN remain reasonable asn increases. For least squares optimization and SDP, we have seen that the compute time for very large networks becomes intractable [13]. For the 1000 node network, the time to predict ˆ P using SMILE is 29.84 seconds while the time to train and predict ˆ P using GCN is 46.90 seconds. Note that this is a fair comparison because the learned approach requires training a specific model for each network; the same learned weights are not applicable to a new network, and neither is the SMILE weight matrixH. The low compute times for SMILE are likely because the linear embedding component depends on the number of neighbors (rather than the number of nodes) and the sparse matrix inference component adapts the step size to the number of nodes. We highlight that scalability is not at the cost of localization accuracy. Forn=1500, SMILE achieves RMSE of 0.05 while GCN achieves RMSE of 0.08. 77 0 200 400 600 800 1000 1200 1400 Number of Nodes 0.2 0.4 0.6 0.8 RMSE SMILE GCN 0 200 400 600 800 1000 1200 1400 Number of Nodes 0 10 20 30 40 Runtime (sec) SMILE GCN Figure 5.8: Compute time for different sizes of networks for GCN and SMILE. Previous work has shown that learned approaches (GCN, multi-layer perception, neural tangent kernel) scale better than optimization approaches (LS, expectation conditional maximization, SDP) [13]. SMILE outperforms GCN for very large networks. 78 0 2 4 6 6 4 2 0 2 4 6 MDS (1.49) 0 2 4 0 1 2 3 4 5 PSVD + MDS (0.25) 0 2 4 0 1 2 3 4 5 SMI + MDS (0.19) 2 0 2 4 6 2 0 2 4 6 8 LLE (1.16) 0 2 4 0 1 2 3 4 5 PSVD + LLE (0.19) 0 2 4 0 1 2 3 4 5 SMILE (0.12) actual a predicted a actual predicted Figure 5.9: Ablation study with σ = 0.3,θ = 5. Top row, left to right: classical MDS achieves RMSE 1.49, de-noising via PSVD improves RMSE to 0.25, and sparse matrix inference further improves RMSE to 0.19. Bottom row, left to right: LLE with direct position recovery achieves RMSE 1.16, de-noising via PSVD improves RMSE to 0.19, and sparse matrix inference further improves RMSE to 0.12. 79 Thus far we have compared our approach to an existing solution, but it is also interesting to consider the role of each component of SMILE and, in particular, whether simpler existing approaches from the literature are sufficient. Fig. 5.9 considers a dataset with σ = 0.3,p NLOS = 1 10 , and θ = 5. We compare the performance of (A) no rank reduction, (B) rank reduction via PSVD, and (C) sparse and low-rank matrix recovery. In conjunction, we consider (1) classical MDS with the Kabsch algorithm for coordinate system registration [68] and (2) LLE-based embedding using anchor positions. Specifically, the top left plot represents a naive baseline: multidimensional scaling assuming Gaussian zero-mean noise. The bottom right plot represents SMILE. We observe several key takeaways from this figure. 1. De-noising: Moving from left to right, each column of this plot contains increasing more sophisti- cated noise reduction and decreases the final RMSE. In particular, sparse matrix inference increases localization accuracy by almost an order of magnitude. 2. Nearest neighbors: Methods in the top row use all available links in eigen-decomposition, while methods in the bottom row use only the nearest neighbors to determine weights for linear recon- struction. This local-neighborhood approach consistently improves localization accuracy. Note that if θ ≥ d max , i.e. no measurements are missing, the performance of (1) and (2) are similar. This is likely because LLE’s strength comes from relying more on nearby measurements, and our ap- proach to matrix completion puts missing measurements at effectively long distances. In setting where long-distance measurements are unreliable, which is commonly the case in realistic wireless communication [110], using nearest neighbors is advantageous. 3. SMILE: The combination of both sparse matrix inference and linear embedding is more robust to NLOS attenuation and missing measurements than either component in isolation. 80 5.5 ExperimentalResults In this section we apply SMILE to two real-world small scale datasets, and discuss its performance, our findings, and directions for future work. For the second dataset, we use PropEM-L (proposed in Chapter 4) to offer "ground truth" estimates of whether links are truly line-of-sight. First we consider a network of 11 Mica2 motes placed randomly in a parking lot [153] § . The maximum distance between any two nodes is 10.57m. We consider received signal strength (RSS) measurements between pairs (RSS is averaged over 20 packets). To estimate distance, we use the following formulation of the log-distance path loss model introduced in Eq. (2.1): RSS(d)=P tx − PL(d 0 )− 10γ log 10 ( d d 0 )+N(0,σ 2 ) (5.9) where P tx is the transmit power, PL(d 0 ) is the path loss at a reference distance, and γ is the path loss exponent [110]. For this data, use the modelγ = 2.9 andP tx − PL(d 0 ) =− 49.12dB ford 0 = 1. Given the ground truth location estimates, Fig. 5.10 shows that the noise in this dataset has mean 0.35m with standard deviation 2.94. Note that even though we believe this data to be in open space (all LOS), seven links see a measurement error of greater than 5m. Figure 5.11 shows the performance of GCN and SMILE on this network, where we update the SMILE parameters toλ =0.05,µ =0.05, andT =10 because the network is smaller. If we assume four of these nodes are anchors, GCN achieves RMSE of 1.67 with threshold 5.2m, and SMILE achieves a comparable RMSE of 1.63 withk =3, guessingβ =11. These results are the average performance of 10 trials. Secondly, we consider measurements from a robotic network with 18 nodes, 13 of which are stationary beacons at known locations, and 5 of which are mobile (legged and wheeled) robots ¶ . Each robot carries a Streamcaster 4200 from Silvus Technologies, which are also used as beacons. We consider a set of pairwise § http://anrg.usc.edu/www/download_files/RSSLocalizationDataSet_11nodes.txt ¶ https://github.com/NeBula-Autonomy 81 5 0 5 10 15 Measurement error (m) 0 2 4 6 8 10 Frequency Figure 5.10: Measurement error for 11 node wireless sensor network outdoors, comparing distance from RSS (Eq. 5.9) with ground truth. 0 2 4 6 8 0 2 4 6 8 10 12 14 GCN (1.6) true anchor pred anchor true node pred node 0 2 4 6 8 SMILE (1.63) true anchor pred anchor true node pred node Figure 5.11: Localization accuracy of GCN (RMSE of 1.6) and SMILE (RMSE of 1.63) on the sensor network. Edges in the left graph represent observed distances less thanθ GCN . Edges in the right graph connect each nodei to its nearest neighborsNN(i). 82 100 50 0 50 100 150 200 250 Measurement error (m) 0 2 4 6 8 10 12 14 Frequency LOS NLOS Missing Figure 5.12: Measurement error for 18 node robotic network in GPS-denied environment. RSS measurements from a single timestamp of robotic exploration. They nodes are spread over a large area in an underground limestone mine with distances between nodes of up to 341 meters. Ground truth positions are available via a simultaneous localization and mapping algorithm, and we assume these are accurate [44]. Estimates of whether links are LOS are available via our predictive model presented in Chapter 4. Fig. 5.12 plots the error density for LOS, NLOS, and missing links from this data, where we note that unfortunately only 14% of measurements are available and LOS. We augment this real dataset with simulated data such that missing measurements are sampled from a zero-mean Gaussian with standard deviation equivalent to the observed LOS noise (σ =36.13). Fig. 5.13 show the results of our approach on this realistic dataset, where edges indicate error in distance. GCN achieves RMSE of 45.8 on this simulated data with threshold 110, while SMILE achieves an improved RMSE of 34.22 fork = 17. These results are the average performance of 10 trials. We observe that the average localization error for unknown nodes is roughly the standard deviation of the noise on LOS links, while the standard deviation of NLOS error is 59.60 (mean 30.14m). This indicates that SMILE is able to achieve localization accuracy comparable to a distance-based model on a single LOS link for all nodes, even those which have many NLOS neighbors, and shows promising performance. 83 100 50 0 50 100 150 200 200 150 100 50 0 50 100 GCN (45.8) 100 50 0 50 100 150 200 SMILE (34.22) Figure 5.13: Localization accuracy of GCN (RMSE 45.8) and SMILE (RMSE 34.22) on the robotic network. Edges indicate error in distance. 5.6 Discussion One advantage of SMILE is that we computeY which approximates the sparsity structure ofS. Therefore, we can use this method to estimate which links are NLOS and where walls, obstacles, or other potentially adversarial sources of attenuation may be present. This means SMILE could be useful as a complementary modality for simultaneous localisation and mapping, as it provides useful approximations of the geometries of the environment. While it is impossible to unique localize a network in the absence of anchors (given the possibility of translations and rotations), anchor-less localization problems have significant overlap with other useful signal processing problems. In face, SMILE could be directly applied to anchor-less localization problems by replacing lines 29-30 in the algorithm with the embedding step of conventional LLE, as LLE typically does not require anchors [11]. Thus, our approach may be applicable to problems outside the sensor networks domain. For instance, consider a matrix of user data, where each row corresponds to a specific user and a small number of users obfuscate their data with the addition of non-negative noise. SMILE could be applied to the problem of determining how similar users are, by extracting the sparse obfuscation matrix and finding an embedding of users in low-dimensional space. 84 We considered a modification to LLE in which we replace the k nearest neighbors with then A anchors as reference points. We expected this would minimize the risk of errors propagating, as each of the an- chors is at a known location. Interestingly, we did not see an improvement in performance. Similarly, we considered a modification to the sparse matrix inference approach in which we project X andY on to the space of matrices which are non-negative and symmetric. We expected this would improve performance asD andS are non-negative and symmetric. Preliminary experiments did not demonstrate a significant improvement in localization accuracy, and in some cases accuracy suffered. Following [13], we considered the probability of each link being NLOS as independent. However, in realistic settings there will be some correlation based on the structure of the environment itself. For example, in Fig. 2.2 we illustrate 20 nodes in an indoor setting with three rooms and one hallway. Nodes in the smaller rooms are more subject to NLOS attenuation, given their proximity to two walls. Currently, we estimate that links with the weakest RSS are most likely to be NLOS. However it may instead be the case that certain nodes are more likely to have NLOS links, is in Fig. 2.2. Exploiting patterns like this in the structure ofS is an interesting direction for future work. 5.7 Summary In this chapter we presented Sparse Matrix Inference and Linear embedding, a novel network localization algorithm which is robust to noisy, NLOS, and missing measurements. Our approach outperforms the state of the art on large, simulated networks, achieving high localization accuracy with low computation times. During the experiments presented in this chapter, anchors were deployed autonomously [129, 154]. This means anchors are only located in places the robots have already visited, while the exploration objec- tive encourages the robots to move away from these anchors. In fact, the exploring robots then tend to be outside the convex hull defined by the anchors. This configuration appears to stress network localization, especially for smaller networks with significant noise. In Chapter 3, we proposed a controller which seeks 85 to constrain relative localization uncertainty. We expect our queue-stabilizing controller would encourage the robots to return to the area inside the convex hull periodically, improving performance further. SMILE is a centralized approach which leverages the redundancy in information that comes with es- timatingn node positions given up ton(n− 1) pairwise measurements. Implementing SMILE in practice requires data collection at the base station and careful synchronization. To use the position estimates in decision-making, they also need to be shared between the base and the team of robots. While this has ben- efits with respect to localization accuracy, and is of clear interest to the sensor network community, we are also interested in approaches which do not require centralization. In the next chapter, we will propose a distributed approach which relies on trilateration instead of matrix manipulation. 86 Chapter6 CollaborativeLocalizationandMapping In this chapter, we present Trilateration for Exploration and Mapping (TEAM), a novel algorithm for low- complexity localization and mapping with robotic networks. TEAM is designed to leverage the capability of commercially-available ultra-wideband (UWB) radios on board the robots to provide range estimates with centimeter accuracy and perform anchorless localization in a shared, stationary frame. It is well- suited for feature-deprived environments, where feature-based localization approaches suffer. We pro- vide experimental results in varied Gazebo simulation environments as well as on a testbed of Turtlebot3 Burgers with Pozyx UWB radios. We compare TEAM to the popular Rao-Blackwellized Particle Filter for Simultaneous Localization and Mapping (SLAM). We demonstrate that TEAM requires an order of mag- nitude less computational complexity and reduces the necessary sample rate of LiDAR measurements by an order of magnitude. These advantages do not require sacrificing performance, as TEAM reduces the maximum localization error by 50% and achieves up to a 28% increase in map accuracy in feature-deprived environments and comparable map accuracy in other settings. ∗ ∗ This chapter is adapted from Clark et al. [155]. 87 6.1 ProblemFormulation Our objective is to demonstrate the feasibility of low-complexity localization and mapping with robotic networks using ultra-wideband radios, explicit communication, and coordinated mobility. We consider a distributed approach which enables each robot to accurately estimate its own location in a shared, global frame. UWB radios configured to perform two-way ranging (TWR), i.e. exchanging packets and measuring round-trip time to estimate distance, have become a popular solution for localization in GPS-denied envi- ronments (e.g. indoors) [156, 157, 158]. UWB positioning allows a robot to determine its position relative to the known positions of three other UWB radios, called anchors or beacons [157, 158, 53]. In the absence of existing UWB anchors or other positioning references (e.g. GPS), we seek to demonstrate that mobile robots can act as an instantaneous anchor for cooperative positioning. 6.2 RelatedWork A popular and well-known implementation of SLAM is the improved Rao-Blackwellized Particle Filter (referred to in this chapter as simply SLAM), a Monte Carlo localization algorithm which is available via the GMapping library [15, 159]. In brief, this approach uses a particle filter in which each particle represents a hypothesis of where the robot is and carries an individual map of the environment. Each particle is updated based on odometry when the robot moves, then the particles are resampled based on how well measured LiDAR data matches the predicted state. Eventually, the particles shift towards the true position of the robot. Designed for single robot localization, this approach does not rely on inter-robot ranging or communication. In a closely related work to ours, Guleretal. experimentally evaluate the use of three UWB transceivers on a single robot and one UWB transceiver on a second robot to perform accurate relative localization 88 Figure 6.1: TEAM leverages collaborative localization and coordinated mobility. Top left: robots initiate a shared coordinate frame. Top right: the first robot moves and maps while using its neighbors to trilaterate. Bottom left and right: subsequent robots take turns moving and mapping. without explicit inter-robot communication [160]. In another closely related work, Kurazume and Hirose propose an approach to cooperative positioning in which a group of robots is divided into two groups who alternate acting as reference landmarks, which they demonstrate on three robots using a laser rangefinder for inter-robot ranging [50]. Inter-robot detection in this manner introduces the correspondence problem of matching the detected robot with its identifier, which is difficult to address for large swarms. Using UWB with unique device addresses solves this issue. Our work combines the accurate relative localization of UWB demonstrated by Guler et al. [160] and coordinated mobility for cooperative positioning demon- strated by Kurazume and Hirose [50]. Furthermore, we introduce explicit inter-robot communication and consider the application of mapping unknown environments. 6.3 Approach A high-level overview of Trilateration for Exploration and Mapping is shown in Fig. 6.1. TEAM is described in Algorithm 3. After initializing position estimates, each robot iterates through lines 2-12 wherein they map their environment and exchange their coordinates and maps. During a robot’s 89 TDMA window (see Fig. 6.3), the robot drives autonomously, initiates two-way ranging, and trilaterates. The following paragraphs present each component of the algorithm. Algorithm3: TEAM running on roboti coords, neighborCoords = Initialization() while Truedo neighborCoords = RadioReceive() if currentTime is in TDMAwindow i then odom = DriveAutonomously() neighborDistances = UWBRange() coords = TrilateratePosition(odom, neighborCoords, neighborDistances) endif scan = LiDARScan() map = UpdateMap(coords, scan, map) RadioTransmit(coords, map) endwhile Initialization: Initialization occurs in line 1. If initial positions are unknown, TEAM can determine initial positions given that each robot has a known and unique identifier. Robot 0’s position defines the map origin and the forward direction. Robot 1’s position defines the y axis and robot 2’s position defines the positive direction of the x axis. Robot 3 and any additional robots can then trilaterate as discussed below. If initial orientations are unknown, each robot can move a known distance in its local forward direction and then trilaterate to determine orientation. If initial positions are known, TEAM leverages this data to perform sensor auto-calibration, estimating the sensor offset from the truth and using that bias to correct future measurements. Trilateration: Robots receive position estimates from their neighbors via RadioReceive in line 3, and collect range measurements via UWBRange in line 6. Given the positions and distances from the three closest instantaneous anchors, TrilateratePosition (line 7) determines a position estimate as shown in Fig- ure 6.2. The grey rings represent uncertainty in the two-way ranging estimate, and the estimated position is the centroid of the curved triangle formed by their intersection points. If fewer than three anchors are within the communication range, TrilateratePosition relies on the robot’s previous position estimate and 90 Figure 6.2: Robot 3 trilateration, using the received position estimates of robots 0, 1, and 2 as anchors. Figure 6.3: Time-division multiple access scheme for 4 robots performing TEAM. Each robot initializes two-way ranging (TWR) during its TDMA window, and this cycle repeats. odometry to determine an updated position estimate. For a robotic network of at least five robots, TEAM can trilaterate in three dimensions. Time division multiple access: To prevent UWB signal interference, TEAM uses a time-division multiple access (TDMA) protocol such that lines 5-7 are only executed during a specified window as illus- trated in Fig. 6.3. We selected TDMA because it offers high channel utility and has extensions suitable for large teams of cooperating robots [53, 161]. Mobility: TEAM is suitable for both autonomous exploration and teleoperation. In this work we are focused on the resource-constrained setting and elect to use simple drive control (line 5) in which robots are given a target direction and perform autonomous collision avoidance based on LiDAR data. To improve positioning accuracy, robots are stationary outside of their TDMA window. Note that with four robots, this scheme takes four times as long to cover a sum total distance when compared to all robots driving simultaneously. This highlights an important system design trade-off between prioritizing quick 91 or accurate map coverage. For larger networks, the three robots required to act as instantaneous anchors are a smaller fraction of the network and the reduction in sum total distance covered is less significant. Mapping: Each time the LiDAR takes a measurement (line 9), this data is associated with the cur- rent position estimate. The occupancy grid map representation is then updated (line 10) with the LiDAR data. The resulting map is shared with the data sink via RadioTransmit in line 11. To reduce the burden on the resource-constrained robots, the data sink is responsible for merging the maps received from all sources. Using the map merging algorithm from [162], the data sink is able to perform feature mapping and calculate the appropriate frame transforms to merge all received maps. This process does not require syn- chronization between robots, and the data sink can receive a map as long as any multi-hop communication path is available, as discussed below. Communication: Under the assumption that all robots are within communication range, the pub- lish/subscribe paradigm provided by the Robot Operating System (ROS) is sufficient for sharing position- ing and map data (lines 3 and 11). However, lava tubes are characterized by branches which challenge connectivity [163]. Previous work exploring tunnels has considered a variety of approaches including data tethers and droppable network nodes [164]. This work builds on the approach of [165] with mobile network nodes which can forward or re-route data. The robots and data sink form an ad hoc mesh net- work, in which robots can act as relays to the data sink when needed, using Optimized Link State Routing (OLSR) to determine network neighbors and appropriate routes [166]. 92 6.4 SimulationResults We use the Gazebo robotics simulator to conduct experiments in various environments. We implemented UWB ranging in simulation according to the following model: d pozyx = d true +N(µ =0,σ =10cm), if1 LOS None, otherwise (6.1) whered true is the true distance between two robots and1 LOS is an indicator function which evaluates to true if and only if line of sight is available. Table 6.1 presents relevant parameters used in the simulation experiments. We assume UWB mea- surements have zero mean and a standard deviation of 10cm [52]. Each call to UWBRange returns the average of 10 independent measurements. To ensure synchronization between the LiDAR scans and UWB ranging measurements, a delay of more than 100 ms between the two sensors prevents TEAM from up- dating the map with this scan data. This timeout was empirically selected. Note that with time-division mobility in the drive module, our approach generates maps at a slower pace. In simulation, we choose to not model the UWB interference constraint and allow the robots to range simultaneously and therefore drive simultaneously. This allows us to compare SLAM and TEAM across the same time scale. 6.4.1 QualitativePerformance The first environment we consider challenges SLAM but is well-suited for TEAM: a long, featureless, obstacle-free corridor. This environment is difficult for a particle filter because the particle distribution becomes spread as the robots navigate along the hallway which lacks discernible features for localization [15]. The results of this experiment are shown in Figure 6.4; they indicate that TEAM can significantly improve map accuracy in featureless environments. 93 Parameter Value Number of robots 4 Number of UWB measurements averaged 10 Std dev of UWB measeurements 10 cm [52] LiDAR/Trilateration synchronization timeout 100 ms Map publish rate 1 Hz Position estimate publish rate 10 Hz Max drive speed 0.22 m/s 360 ◦ LiDAR sample rate 5 Hz 360 ◦ LiDAR resolution 1 ◦ 360 ◦ LiDAR standard deviation 0.01 RBPF number of particles 50 Odometry variance 0.01 mm Table 6.1: Relevant parameters for simulation experiments Figure 6.4: The top image resulted from SLAM with 50 particles, the bottom image resulted from TEAM, each after 25 simulated minutes. The true environment dimensions are overlaid in red. Figure 6.5: The left image resulted from SLAM with 50 particles, the right image resulted from TEAM. The true environment dimensions are overlaid in red. The second environment we consider challenges TEAM but is well-suited for SLAM: an environment with lava tube-like characteristics, including branches that prevent line of sight. The Gazebo model for this environment comes from [14]. TEAM relies heavily on odometry data in this environment, and the results of this experiment are shown in Fig. 6.5 and strengthen our claim that TEAM results in accurate maps. We demonstrate the performance in the presence of noisy odometry data in Sec. 6.5.1. One significant advantage of TEAM is that it decouples localization from LiDAR sensor measurements. This allows us to reduce the frequency of LiDAR scanning in order to save power; this is useful for robots that have severe energy constraints or memory limitations. Fig. 6.6 shows the result of an experiment in 94 Figure 6.6: The left image resulted from SLAM with 50 particles and scans throttled to the low frequency of 0.1Hz. The right image resulted from TEAM with the same low LiDAR sample rate. Figure 6.7: The left image resulted from SLAM with 1 particle. The right image resulted from TEAM, which is computationally comparable. which scans are throttled to 0.1Hz from 5Hz used in the previous experiments. This shows that reduced Li- DAR frequency significantly deteriorates the performance of SLAM, while leading to topologically correct albeit sparse maps with TEAM. As discussed in Sec. 6.5.2, the algorithmic complexity of SLAM is a function of the number of particles used to capture the belief distribution of the location estimate. For TEAM, the algorithmic complexity is comparable to SLAM with a single particle. In Fig. 6.7 we illustrate the performance of SLAM with 1 particle and TEAM side by side, to highlight the difference in map accuracy despite similarly reduced computation time. 6.4.2 QuantitativePerformance While the merged map presents a clear picture of the capabilities of TEAM, it is also useful to consider the localization error of the robots over time. For SLAM, the localization uncertainty varies as the belief distribution changes over time. In feature-deprived environments like the one shown in Fig. 6.4, this error can grow up to 25m. Fig. 6.8 compares the localization error as a function of time during exploration of 95 Figure 6.8: These graphs show the localization error over time for SLAM and TEAM for the environment depicted in Fig. 6.5. the lava tube-like environment. We observe that errors in TEAM are less than 3m and these errors are infrequent, however errors in SLAM can grow up to 6m and are higher on average. It is similarly useful to quantify the accuracy of the merged map created by each algorithm over time. We measure the absolute pixel error of map image files which were manually aligned with and compared to the image file of the true environment. We plot the map accuracy over time and show that while SLAM and TEAM both result in decreasing map error, TEAM achieves a lower final absolute pixel error (Fig. 6.9). We observe that TEAM achieves a 28% decrease in error relative to SLAM for the environment depicted in Fig. 6.4, and a 27% decrease in error for the environment depicted in Fig. 6.5. 96 Figure 6.9: This graph shows the map error as a function of time for the environment depicted in Fig. 6.5. 6.5 ExperimentalResults Our testbed is comprised of four Turtlebot3 Burgers, each equipped with a RaspberryPi 3B+ running Rasp- bian, an OpenCR1.0 control board, a 360 Laser Distance Sensor, and differential drive. We have extended each platform with a Pozyx UWB Creator series Anchor, and a USB Wireless Adapter Mideatek RT5370N with 2dBi antenna. The wireless adaptor allows each robot to designate one wireless interface for internet connectivity, and one wireless interface for joining our ad hoc mesh network, TurtleNet. The robots and data sink, a PC running Ubuntu 16.04, all implement Optimized Link State Routing (OLSR), a proactive routing protocol for mobile ad hoc mesh networks [166, 167]. We provide a tutorial for mesh networking Turtlebot3 Burgers [168]. Table 6.2 presents parameters used in the testbed experiments in addition to those previously listed in Table 6.1. The size of the TDMA window should be larger than the time to collect 10 ranging measurements (0.5 sec) and smaller than the time it would take a robot to drive beyond the maximum communication range (90 sec). Delays between the robots and the data sink do not affect the quality of the resulting merged map, as it is processed asynchronously; however, delays or dropped packets between robots affect the location estimate accuracy. We introduce a 0.3 second buffer period at the end of each TDMA window 97 Figure 6.10: Turtlebot3 Burger with Pozyx UWB anchor. Parameter Value TDMA window size 5 s UWB ranging rate 20 Hz Worst case expected communication delay 400 ns Table 6.2: Additional relevant parameters for testbed experiments during which the robot stops driving but continues to perform trilateration. Note that for consistency, our time-division approach to driving autonomously was used across all experiments. 6.5.1 Performance: ResultingMaps Fig. 6.11 shows the merged map created by our testbed in a typical hallway environment using SLAM. Fig. 6.12 shows the map created by a single robot, illustrating the effect of inaccuracy in the odometry readings. We noticed that the drift in odometry was not consistent across the four robots, but the map merging algorithm mitigated this. For better location estimate accuracy, the robots should localize within the merged map rather than their individual maps. Fig. 6.13 shows the merged map created using TEAM. We observe that the wall edges are slightly less well-defined. This is due to noise in the UWB ranging measurements causing small jumps in the position estimate. This noise is primarily attributed to multipath effects, and further characterizing the UWB per- formance will help improve map accuracy [51]. Fig. 6.14 shows the individual map of a single robot and 98 Figure 6.11: SLAM in the hallway: merged map Figure 6.12: SLAM in the hallway: map created by a single robot Figure 6.13: TEAM in the hallway: merged map Figure 6.14: TEAM in the hallway: map created by a single robot highlights the effect of loss of line of sight. This causes the robot to rely on its odometry until UWB signals are available again, at which point its location estimate may jump, leading to discontinuities in the map. Distributed strategies to maintain connectivity as presented in Sec. 3 can prevent these discontinuities. 6.5.2 ComplexityAnalysis In TEAM, selecting the three nearest anchors from the anchors within range can be done inO(NlogN) where N is the size of the network. Calculating the location estimate is then a constant number of op- erations and several closed-form and approximate algorithms exist [169]. Associating scan data with a location estimate and updating the map can also be done in constant time, as new scans are integrated independent of the size of the existing map. Following the complexity analysis in [15], SLAM introduces complexityO(P) each time the location estimate is updated, whereP is the number of particles. This computation is associated with computing the proposal distribution, computing the particle weights, and testing if a resample is required. SLAM also introduces complexityO(P) for each map update, and complexityO(PM) each time a resample occurs, 99 Computation SLAM, P=60 SLAM, P=1 TEAM Trilaterate - - 0.001 s Update map 0.40 s 0.1 s 0.1 s Process scan 2.25 s 0.05 s < 0.001 s Table 6.3: Runtime comparison where M is the size of the map. For an optimized system, the number of particles required is typically between 8 and 60 [15], depending on the size and features of the environment. Thus, TEAM can provide up to a 60x reduction in computational complexity. Table 6.3 contains the average runtimes for relevant computations on our hardware. 6.6 Summary In this chapter we have presented the design, implementation, and evaluation of TEAM, a novel algorithm for localization and mapping in unknown environments with a robotic network. We demonstrate the ability of TEAM to leverage ultra-wideband positioning to generate maps of various environments with high accuracy. Our algorithm significantly reduces the computational complexity and the required rate of LiDAR samples, making it suitable for resource-constrained multi-robot systems, and performs well in feature-deprived environments where SLAM struggles. The feasibility of low-complexity collaborative location, in conjunction with an exploration method which improves connectivity and localizability (the primary concerns observed in this chapter and ad- dressed in Chapter 3), allow us draw promising conclusions about the role of robotic networks in advanc- ing the state of the art in harsh environment exploration. These conclusions are discussed in the next and final chapter. 100 Chapter7 Conclusion It is an exciting time for networked, robotic autonomous exploration. Advancements in mobility plat- forms, sensing capabilities, and the miniaturization of computing devices has made the transition from expensive monolithic robots to distributed robot systems possible. These advancements allow us to use multi-robot teams in applications where sending humans is not safe or practical, and this is exciting for the future of space and planetary exploration and impactful for the future of robotic first response and disaster mitigation. The robotic networks community is still faced with many challenges when it comes to making harsh environment exploration a reality, however several of these challenges are addressed in this disser- tation. Specifically, we focused on the need for timely data transfer despite the lack of communication infrastructure, and accurate relative and global positioning despite the lack of localization infrastructure. We considered the competing objectives of exploration (which necessitates robots moving far away from a stationary base station) and communication (which is inherently limited by distance). To address this, we presented a queue-stabilizing framework for exploration which was derived from time-average constraints. The key innovation here was that working with time averages gives robots the flexibility to sacrifice connectivity for exploration for a period and then sacrifice exploration to meet the needs of timely data transfer. Similarly, we showed it is possible to prioritize other objectives with the same framework, including relative localization uncertainty. In Chapter 3, we demonstrated improved exploration coverage 101 and promising emergent behavior from this approach in simulation, and discussed implementation details of queue-monitoring for a real-world robotic network. A key insight from Chapter 3 was that connectivity prediction is an integral component of queue- aware autonomy. In particular, the probability that a movement will allow data transfer depends on the quality of the signal at the final location. We addressed the challenge of accurate signal strength prediction in unknown environments with line-of-sight obstructions through the design of Propagation Environment Modeling and Learning. The key innovation here was that leveraging real-time data about the geometry of the environment significantly improved the accuracy of connectivity models, especially when com- bined with the computational strengths of machine learning. In Chapter 4, we illustrated the advantage of data-driven prediction, and discussed practical ways of extending our approach to completely new envi- ronments. Given that we can accurately model the relationship between distance and signal strength, we consid- ered using radio measurements for network localization. We proposed Sparse Matrix Inference and Linear Embedding, an approach to network localization which is robust to the severe attenuation from obsta- cles which we observed in Chapter 4. The key insight was that a matrix of pairwise distance estimates is separable into sparse NLOS attenuation and low-rank true Euclidean distances, via recent advances in ma- trix decomposition. In Chapter 5, we demonstrated improved performance over state-of-the-art network localization algorithms without sacrificing computational efficiency. We observed in Chapter 5 that centralized network localization can introduce implementation chal- lenges with respect to data transfer and synchronization. We proposed Trilateration for Exploration and Mapping which provided an alternative approach to address these challenges via UWB two-way ranging and a distributed localization approach. The key innovation here was that controlled mobility of the sys- tem allows the robots to take turns acting as reference points for trilateration. We demonstrated TEAM 102 on a small network of mobile robots in Chapter 6 and analyzed the exploration performance of the system through the resulting maps. Together, a team of networked mobile robots with inter-robot communication and ranging can use TEAM to localize in a shared, stationary frame with low-complexity; rely on centralized, highly accu- rate location estimates when they are available via SMILE; use PropEM-L to predict whether an arbitrary location will provide connectivity; and use our queue-stabilizing framework to make autonomous deci- sions which reduce localization uncertainty and enable timely data transfer, thereby improving exploration coverage at the stationary data sink. Thus, this dissertation provides a thorough treatment of remote ex- ploration with robotic sensor networks. 7.1 FutureDirections In general, we have focused on widening the horizon of algorithms and frameworks available for resource- constrained robots, with an emphasis on low complexity. Because of this, we have primarily considered myopic decision-making and localization at a single moment in time. As the field advances, it may be advantageous to consider a treatment of these problems which focuses on longer time scales. For exam- ple, we could consider the problem of selecting optimal trajectories for exploration which both maximize information gain and minimize queue sizes over a finite time horizon. It would be interesting to compare this extension of our queue-stabilizing framework with existing work which guarantees connectivity peri- odically [170]. Similarly, both SMILE and TEAM could benefit from leveraging time-series data, at the cost of additional complexity. In particular, fusing odometry data with the inter-robot ranging measurements and considering Bayesian updating of the position estimates may lead to solutions which converge with high accuracy [171, 172]. This trade between accuracy and complexity is worthy of further study. Aside from queue-aware autonomy and collaborative localization, remote exploration with robotic networks could benefit from increased emphasis on predictable, explainable, and assuredly safe autonomy. 103 An improved understanding of the range of possible behavioral outcomes for a team of mobile robots, despite limited understanding of the operational conditions, will be key to ensuring trust and reducing risk. Trustworthy and safe teams of mobile robots will be key to the future of remote exploration. 7.2 FinalRemarks This dissertation made two primary recommendations: the use of queue size as a metric for decision- making, and the use of inter-robot ranging as a method of localization. As such, we demonstrated the advantages of queue-aware autonomy and collaborative localization, and addressed the practical concerns in queue-size monitoring, connectivity modeling, and recovering node positions from pairwise measure- ments. We note that while the solutions proposed in this work are intended for robotic networks, some of our methods and findings have applications for a single robot exploring a remote environment as well. Transferring data from the robot to the base is still a key concern in any environment where Wi-Fi is not guaranteed. Meeting communication requirements, especially reliable data transfer, is a challenge for multi-robot systems and single robot systems alike and both can benefit from queue-aware autonomy. Ad- ditionally, some of our methods and findings have applications outside of exploring remote environments. 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Clark, Lillian Mary
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Remote exploration with robotic networks: queue-aware autonomy and collaborative localization
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Electrical Engineering
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2023-05
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multi-robot systems
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