bg minimal and redundant bearing rigidity 4932

Minimal and Redundant Bearing Rigidity Dr Minh Hoang Trinh (Trịnh Hoàng Minh) Lecturer Department of Automation Engineering School of Electrical and Electronic Engineering (SEEE) Hanoi University of Science and Technology (HUST) Vietnamese Control Systems and Robotics Workshop June 2022 Contents Introduction Bearing rigidity theory Minimal and redundant bearing rigidity Conclusions Introduction What is “rigidity”? • The ability to maintain some geometric properties of a figure • Example: distance rigidity of 2D bar-and-joint framework • Non-rigid (or flexible): one or several distances may change due to forces applied to joints • Rigid: all distances are maintained non-rigid distance rigid joint bar or linkage virtual bars change lengths⟹ non-rigid What is “rigidity”? • Example: bearing rigidity of 2D framework (prismatic joints) • Non-rigid (or flexible): several directional vectors between joints may change • Rigid: all direction vectors are maintained Non-rigid joint Bearing/direction vector bar or linkage Lengths are changeable virtual bars change directions⟹ non-rigid rigid Rigidity theory: a short history • Euler’s conjecture (1862): “A closed spatial figure allows no changes, as long as it is not ripped apart.” • Cauchy (1905): “If there is an isometry between the surfaces of two strictly convex polyhedral which is an isometry on each of the faces, then two polyhedra are congruent” • Stenitz & Rademacher, 1934; Strictly convex polyhedra • Alexandrov, 1950; • Gluck (1975): “Almost all simply closed surfaces are rigid” • R Connelly (1977) gives a counter example, settled the conjecture Convex polyhedra • • • L Euler (1862) Opera Postuna I, Petropoli, 494-496 A L Cauchy (1905) Sur les polygons et les polyèdres J E Graver, B Servatius, H Servatius, “Combinatorial rigidty”, Graduate Studies in Mathematics Amer Math Soc, 1993, vol Nonconvex polyhedra Rigidity theory: a short history • Applications: • Before 2000: mostly interested by mathematicians (matroid theory), mechanical engineering (statics, robotics), chemical engineering (structure of mocules), civil engineering (structural analysis), architects & artists Stewart platform Tensegrity structure (source: Youtube - Ohio University Robotics) (source: Youtube – Bear with the architect) Rigidity theory: a short history • Applications: • After 2000: control systems society (networked control systems) • Formation control [Olfati-Saber & Murray (2002), Eren et al (2002)] • Network localization [Eren et al (2004)] Drone show in Pyongchan Winter Olympic, S Korea 2018 Wireless sensor network for monitoring infastructure, Toshiba • Olfati-Saber & Murray (2002) “Distributed cooperative control of multiple vehicle formations using structural potential functions.” IFAC 2002, Barcelona, Spain, 495-500 • Eren, T., Belhumeur, P.N., Anderson, B.D.O., & Morse, A S (2002) A framework for maintaining formations based on rigidity IFAC Proceedings 35(1), 499-504 • Eren, T, Goldenberg, O.K., Whiteley, W., Yang, Y R., Morse, A.S., Anderson, B.D.O., & Belhumeur, P N (2004) “Rigidity, computation, and randomization in network localization” In IEEE INFOCOM 2004, Vol 4, 2673-2684 Bearing rigidity theory: literature review • Development of bearing rigidity theory: • Tay & Whiteley (1985): direction rigidity as a special case of matroid theory • T Eren et al (2003): direction rigidity for formation control/sensor network localization in 2D • Zhao & Zelazo: Bearing rigidity theory in 𝑑-D (Arxiv-2014), generic bearing rigidity, Laman graphs (2017) • Tron: Bearing rigidity - an edge-based approach (cycle basis) (2015) • Trinh et al (2019): directed bearing-based Henneberg construction; Trinh, Tran & Ahn (2020): minimal & redundant bearing rigidity, generalized Henneberg construction • Tang et al (2022): relaxed bearing rigidity = connectivity + P E Walter Whiteley Daniel Zelazo • • • • • • • • Tay & Whiteley (1985) Structural Topology, Vol 11, 21–69 Eren, Whiteley, Morse, Belhumeur, Anderson (2003), IEEE CDC, Hawaii, USA, 3064–3069 Zhao & Zelazo (2016) IEEE TAC, 61(5):1255–1268 Zhao & Zelazo (2016) Automatica, 69, 334–341 Tron et al (2015) ACC, Chicago, IL, USA, 3911–3918 Zhao et al (2017) IEEE CDC, Melbourne, AU, 2017, 3356–3361 Trinh et al (2019 IEEE TAC, 64(2):598 – 613 Trinh, Tran, & Ahn (2020) TAC, 65(10): 4186 – 4200 • Tang et al (2022) Automatica 141 (2022): 110289 Shiyu Zhao Tolga Eren Roberto Tron Bearing rigidity theory 10 18 Generic bearing rigidity • 𝐺 is generically bearing rigid (GBR) in ℝ𝑑 if there exists a configuration 𝒑 such that the framework (𝐺, 𝒑) is infinitesimally bearing rigid in ℝ𝑑 • If rank 𝑹𝑏 𝒑 = 𝑑𝑛 − 𝑑 − 1, then 𝒑 is a generic configuration • The set of generic configurations of a GBR graph 𝐺 is dense A non-generic configuration of 𝐶3 (3 points are collinear) A generic configuration of 𝐶3 obtained by perturbing a non-generic configuration ⟹ Bearing rigidity depends on the graph 𝐺 than on an actual configuration 𝒑 Design of IBR frameworks = Design of GBR graph 𝐺 + select a configuration 𝒑 • Zhao et al (2017), IEEE 56th CDC, 3356-3361 Minimal and redundant bearing rigidity 19 𝑑 Henneberg construction in ℝ 20 • Henneberg construction for generating GBR graphs in 𝑑-D: • Start with two vertices 𝑣1 , 𝑣2 and an edge 𝑣1 , 𝑣2 • For each step, choose from operators: • Vertex addition: add a new vertex 𝑣𝑘 and edges 𝑣𝑘 , 𝑣𝑘1 and 𝑣𝑘 , 𝑣𝑘2 • Edge splitting: remove an existing edge 𝑣𝑘1 , 𝑣𝑘2 and add a new vertex 𝑣𝑘 and edges 𝑣𝑘 , 𝑣𝑘1 , 𝑣𝑘 , 𝑣𝑘2 , 𝑣𝑘 , 𝑣𝑘3 Henneberg construction of a GBR graph and a corresponding IBR configuration • Trinh, Oh & Ahn (2014), ICMC, China, 2268 – 2271 • Zhao et al (2017), IEEE 56th CDC, 3356-3361 • Trinh et al (2019), IEEE TAC 64(2):598 - 613 ⟹ A Laman graph of 𝑛 vertices and 2𝑛 − edges Ear decomposition 21 • Ear decomposition of graph: 𝐺 = 𝒫1 ∪ 𝒫2 ∪ ⋯ ∪ 𝒫𝑘 , • 𝑃𝑖+1 is a path whose end vertices belong to 𝒫1 ∪ 𝒫2 ∪ ⋯ ∪ 𝒫𝑖 • 𝑃𝑖 : an open ear of length ≥ • A graph 𝐺 = (𝑉, 𝐸) with 𝐸 ≥ is 2-vertex-connected if and only if it has an ear-decomposition starting with a cycle Graph decomposition: 𝒫1 = 𝒞5 , 𝒫2 , 𝒫3 are of length 2, 𝒫4 , 𝒫5 are of length • Whitney (1932) Trans Amer Math Soc, 43(2):339-362 • Lovasz (1985) IEEE Symp Foundations of Comp Sci, 464 - 467 22 Bearing rigidity of cycle graphs • 𝐶𝑛 is GBR in ℝ𝑑 if ≤ 𝑛 ≤ 𝑑 + But 𝐶𝑛 is non-rigid in ℝ𝑑 if 𝑛 > 𝑑 + ⟹ If the cycle is too long, it will be non-rigid; ⟹ Short cycles act as a rigid base in any GBR graphs 𝑧 y 𝑥 x • • 𝐶3 is GBR in ℝ2 and ℝ3 𝐶4 is not GBR in ℝ2 but GBR in ℝ3 • Trinh, Tran & Ahn (2020), IEEE TAC 65(10): 4186 – 4200 • Ko, Trinh, & Ahn (2020), IJRNC 30(12): 4789-4804 𝑦 𝑑 Methods to generate GBR graphs in ℝ 23 • Method 1: Add more edges to an existing GBR graph • Method 2: Generalized Henneberg construction: From a GBR graph, at each step: • Add an open ear of length 𝑙 ≤ 𝑑 into existing vertices in a GBR graph, or • Split an edge by inserting a cycle of length 𝑙 ≤ 𝑑 + Method 1: Add more edges to a GBR graph • Method 3: Merge existing GBR graphs: • 2D: add a Z-link between graphs • 3D: add non-adjacent edges between graphs (a) (b) Method 3: (a) Merge GBR graphs in 2D (b) Merge GBR graphs in 𝑑-D 𝑑 ≥ (a) (b) Method 2: 𝐶4 is GBR in ℝ3 : (a) Add open ear (b) Split an edge 24 Minimal bearing rigidity • Graph 𝐺 = 𝑉, 𝐸 of 𝑉 = 𝑛 vertices and 𝐸 = 𝑚 edges • 𝐺 is minimally bearing rigid (MBR) in ℝ𝑑 if and only if there does not exist any graph 𝐻 of 𝑛 vertices so that 𝐻 is GBR in ℝ𝑑 and has less than 𝑚 edges • Each bearing vector specifies 𝑑 − constraints on framework⟹ a MBR graph has at least • In ℝ2 : MBR graphs ≡ Laman graphs constructed from a Henneberg construction • In ℝ𝑑 , 𝑑 ≥ 3: Laman graphs 𝑚 = 2𝑛 − are not MBR 𝑧 𝑦 𝑑𝑛−𝑑−1 𝑑−1 𝑥 𝑥 𝑦 𝒞3 is GBR in ℝ3 and has 𝑚 = = × − 𝒞4 is GBR in ℝ3 and has 𝑚 = < × − = 2𝑛 − edges 25 Minimal bearing rigidity • If 𝐺 is GBR in ℝ𝑑 , then 𝐺 has an open ear-decomposition starting from a cycle • For 𝑛 > 𝑑 + 1, 𝐺 is MBR in ℝ𝑑 if and only if it has 𝑓 𝑛, 𝑑 edges, where 𝑛, 𝑛−2 𝑓 𝑛, 𝑑 = ቐ 1+ × 𝑑 + mod 𝑛 − 2, 𝑑 − + sign mod 𝑛 − 2, 𝑑 − , 𝑑−1 • Example: • For 𝑑 = 2: 𝑓 𝑛, 𝑑 = + 𝑛 − × = 2𝑛 − (Laman graphs) • For d = 3, 𝑛 = then: mod − 2, − = mod 6, = 0, 8−2 𝑓 8, = + × = + × = 10 3−1 𝑛≤𝑑+1 𝑛>𝑑+1 Algorithm for generating MBR graphs Algorithm 1: • Given: 𝑛 ≥ and 𝑑 ≥ • If 𝑛 ≤ 𝑑 + 1, 𝐺 = 𝒞𝑛 else: • • • ⟹ There can be many MBR graphs of 𝑛 vertices in ℝ𝑑 depending on the selection of attaching vertices at each step (like isomerism in organic chemistry) Start from 𝒞𝑑+1 At each step: add an open ear of length 𝑑 until the remaining vertex 𝑞 less than 𝑑 − Add open ear of length 𝑞 + Step • Adding an open ear of 𝑙 − vertices and 𝑙 edges 26 Step Step Step Example: 𝑛 = 9, 𝑑 = mod − 2, − = mod 9, = 9−2 𝑓 9, = + × = + × + + = 12 3−1 Algorithm for generating MBR graphs 27 • Example: Generate GBR graphs using Henneberg construction and Algorithm for 𝑛 = 8, 𝑑 = Constructing a GBR graph by Henneberg construction 𝑚 = 2𝑛 − = 13 edges, steps Constructing a MBR graph by Algorithm 𝑚 = 𝑓 8, = 10 edges, steps Redundant bearing rigidity 28 • A graph 𝐺 is 1-redundantly bearing rigid (1-RBR) in ℝ𝑑 if and only if removing any edge in 𝐺 leaves a GBR graph 1-redundantly rigid in ℝ3 Not 1-redundantly rigid in ℝ3 29 Methods to generate 1-RBR graphs • Method 1: robust vertex addition • Given 𝑛, 𝑑 • Start with the complete graph 𝒦4 • At each step: add vertex and edges to the graph • Method 2: robust merging • Type 1: merging by a double Zlink between existing 1-RBR graphs • Type 2: merging by distinct edges between existing 1-RBR graphs Method 1: robust vertex addition (a) (b) Method 2: (a) Add a double Z-link (b) Add distinct edges Conclusions 30 Conclusions • Summary: • Bearing rigidity theory in ℝ𝑑 • Minimal and redundant bearing rigidity • Applications: • Design control/sensing topologies in bearing-based formation control/ network localization with minimal or with redundant constraints 31 References 32 Oh, Park, & Ahn (2015) “A survey of multi-agent formation control,” Automatica, 53, 424-440 W Whiteley (1996) “Some Matroids from Discrete Applied Geometry”, vol 137 Providence, RI, USA: AMS Loizou & Kumar (2007) “Biologically inspired bearing-only navigation and tracking,” 46th IEEE CDC, LA, USA, 1386–1391 T Eren (2012) “Formation shape control based on bearing rigidity,” IJC, 85:9, 1361–1379, 2012 Trinh, Oh, & Ahn (2014) “Angle-based control of directed acyclic formations with three-leaders,” IEEE ICMC, CN, 2268–2271 Tron et al (2015) “Rigid components identification and rigidity control in bearing-only localization using the graph cycle basis,” ACC, Chicago, IL, USA, 3911–3918 Zhao & Zelazo (2016) “Bearing rigidity and almost global bearing-only formation stabilization,” IEEE TAC, 61(5):1255–1268 Zhao & Zelazo (2016) “Localizability and distributed protocols for bearing-based network localization in arbitrary dimensions,” Automatica, 69, 334–341 Zhao et al (2017) “Laman graphs are generically bearing rigid in arbitrary dimensions,” 56 IEEE CDC, Melbourne, AU, 2017, 3356– 3361 10 Trinh et al (2019) “Bearing-based formation control of a group of agents with leader-follower structure,” TAC, 64(2):598 - 613 11 Trinh, Tran, & Ahn (2019) "A method to generate generically bearing rigid graphs", VCCA, Hanoi 12 Trinh, Zelazo, & Ahn (2020) “Pointing consensus and bearing-based solutions to the Fermat-Weber location problem,” TAC, 65(6):23392354 13 Trinh, Tran, & Ahn (2020) “Minimal and redundant bearing rigidity: Conditions and Applications,” TAC, 65(10): 4186 – 4200 14 Tang, Cunha, Hamel & Silvestre (2022) Relaxed bearing rigidity and bearing formation control under persistence of excitation Automatica, 141, 110289