A physical interpretation of the rigidity matrix
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1 A physical interpretation of the rigidity matrix Hyo-Sung Ahn 1 (Collaborations with Minh Hoang Trinh, Zhiyong Sun, Brian D. O. Anderson, and Viet Hoang Pham) 1 Distributed Control & Autonomous Systems Laboratory (DCASL) School of Mechanical Eng., Gwangju Institute of Science and Technology (GIST), South Korea 2017 IFAC WC Workshop Rigidity Theory for Multi-agent Systems Meets Parallel Robots Towards the Discovery of Common Models and Methods 2017
2 Credits (a) Trinh Minh Hoang (b) Zhiyong Sun (d) Viet Hoang Pham (c) Prof. Brian D.O. Anderson
3 contents 1 Background 2 Rigidity Matrix 3 The symmetric rigidity matrix The symmetric rigidity matrix Physical meaning of the eigenvectors Further properties of the eigenvalues 4 Simulation 5 The new rigidity indices Motivation and definition Properties of the new indices 6 Conclusions
4 Multi-agent systems & Distributed formation control Agents and multi-agent systems: An agent is understood as a dynamical system. A multi-agent system is a collection, a group, or a team of dynamical systems. Distributed formation control: No centralized controller for a given multi-agent system. Each agent has its own controller based on interaction with its neighboring agents. Only the distances among agents are controlled by relative interactions; but a formation defined w.r.t a global coordinate frame is achieved, upto translations and rotations. Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Background Zhiyong Sun, Brian D. O. Anderson, and Viet 2017Hoang1 Pham) / 39
5 Formation with Distance Constraints Only distances (edges) are constrained Formation is fixed (rigid) or not-fixed (flex)? Rigid Graphs (Graph Rigidity) Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Background Zhiyong Sun, Brian D. O. Anderson, and Viet 2017Hoang2 Pham) / 39
6 Formation with Distance Constraints Only distances are constrained Formation is fixed (rigid) or not-fixed (flex)? Rigid Graphs (Graph Rigidity) Flex graphs Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Background Zhiyong Sun, Brian D. O. Anderson, and Viet 2017Hoang3 Pham) / 39
7 Formation with Distance Constraints Only distances are constrained Formation is fixed (rigid) or not-fixed (flex)? Rigid Graphs (Graph Rigidity) Flex graphs Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Background Zhiyong Sun, Brian D. O. Anderson, and Viet 2017Hoang4 Pham) / 39
8 Formation with Distance Constraints Only distances are constrained Formation is fixed (rigid) or not-fixed (flex)? Rigid Graphs (Graph Rigidity) Rigid graphs (Locally rigid) Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Background Zhiyong Sun, Brian D. O. Anderson, and Viet 2017Hoang5 Pham) / 39
9 Formation with Distance Constraints Only distances are constrained Formation is fixed (rigid) or not-fixed (flex)? Rigid Graphs (Graph Rigidity) Rigid graphs (Locally rigid) Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Background Zhiyong Sun, Brian D. O. Anderson, and Viet 2017Hoang6 Pham) / 39
10 Formation with Distance Constraints Only distances are constrained Formation is fixed (rigid) or not-fixed (flex)? Rigid Graphs (Graph Rigidity) Rigid graphs (Locally rigid) Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Background Zhiyong Sun, Brian D. O. Anderson, and Viet 2017Hoang7 Pham) / 39
11 Formation with Distance Constraints Only distances are constrained Formation is fixed (rigid) or not-fixed (flex)? Rigid Graphs (Graph Rigidity) Rigid graphs (Globally rigid) Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Background Zhiyong Sun, Brian D. O. Anderson, and Viet 2017Hoang8 Pham) / 39
12 Formation with Distance Constraints Only distances are constrained Formation is fixed (rigid) or not-fixed (flex)? Rigid Graphs (Graph Rigidity) Rigid graphs (Globally rigid) Unique!!! Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Background Zhiyong Sun, Brian D. O. Anderson, and Viet 2017Hoang9 Pham) / 39
13 Formation with Distance Constraints Only distances are constrained Formation is fixed (rigid) or not-fixed (flex)? Rigid Graphs (Graph Rigidity) Rigid graphs? Unique; but any point of epsilon neighborhood, the configuration is not unique! (infinitesimally rigid) Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Background Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 10 Pham) / 39
14 Consistency between the overall and the local tasks If all the agents complete their local task, then the overall task is achieved? What condition is required for G = (V, E) in order to satisfy (i, j) E, p i p j = pi pj i, j V, p i p j = pi pj? }{{}}{{} Equivalence (the local tasks) Congruence (the overall task) Rigidity or the persistence of G specifying the minimum number and the distribution pattern of edges. (e) Not rigid. (f) Rigid. (g) Not persistent. (h) Persistent. Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Background Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 11 Pham) / 39
15 Use of Graph rigidity for Consistency in Task Given an undirected graph G = (V, E), where V = {1,..., N}, let us assign p i R n to each vertex i for all i V. Realization: p = (p T 1,..., pt N )T R nn, Framework: (G, p) Equivalence: Two frameworks (G, p) and (G, q) are equivalent if (i, j) E, p i p j = q i q j. Congruence: Two frameworks (G, p) and (G, q) are congruent if i, j V, p i p j = q i q j. Definition (Rigidity) A framework (G, p) is rigid if there exists a neighborhood U p of p such that all frameworks equivalent to (G, p) are congruent in U p. If (G, p) is rigid, then the overall task and the local tasks is consistent. Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Background Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 12 Pham) / 39
16 contents 1 Background 2 Rigidity Matrix 3 The symmetric rigidity matrix The symmetric rigidity matrix Physical meaning of the eigenvectors Further properties of the eigenvalues 4 Simulation 5 The new rigidity indices Motivation and definition Properties of the new indices 6 Conclusions
17 Incidence rigidity & Similarity F = (G, p): a framework in R 2 G = (V, E), V = {1,..., n}, E V V, V = n, E = m p i = [x i, y i ] T, p = [p T 1,..., pt n ] T : a realization in R 2 H R m n : the incidence matrix Figure: A framework with four vertices and five edges. Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Rigidity Zhiyong Matrix Sun, Brian D. O. Anderson, and Viet 2017 Hoang 13 Pham) / 39
18 Incidence rigidity & Similarity F = (G, p): a framework in R 2 G = (V, E), V = {1,..., n}, E V V, V = n, E = m p i = [x i, y i ] T, p = [p T 1,..., pt n ] T : a realization in R 2 F = (G, p) and F = (G, p ) are similar iff. ζ > 0 such that ζ: the scale factor p i p j = ζ p i p j, i, j V (1) Figure: Similar frameworks: F 1, F 2, F 3. Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Rigidity Zhiyong Matrix Sun, Brian D. O. Anderson, and Viet 2017 Hoang 13 Pham) / 39
19 rigidity matrix Denote z ij = p j p i, (i, j) E: displacement vector Labeling m edges, we have z = [z T 1,..., zt m ] T = (H I 2 )p R 2m yo-sung Ahn (Collaborations with Minh Hoang Trinh, Rigidity Zhiyong Matrix Sun, Brian D. O. Anderson, and Viet 2017 Hoang 14 Pham) / 39
20 rigidity matrix Denote z ij = p j p i, (i, j) E: displacement vector Labeling m edges, we have z = [z1 T,..., zt m ] T = (H I 2 )p R 2m The distance function f G : R 2n R m, f G : p [ z 1 2,..., z m 2] T. The rigidity matrix R := 1 2 f G(p)/ p R m 2n yo-sung Ahn (Collaborations with Minh Hoang Trinh, Rigidity Zhiyong Matrix Sun, Brian D. O. Anderson, and Viet 2017 Hoang 14 Pham) / 39
21 rigidity matrix Denote z ij = p j p i, (i, j) E: displacement vector Labeling m edges, we have z = [z1 T,..., zt m ] T = (H I 2 )p R 2m The distance function f G : R 2n R m, f G : p [ z 1 2,..., z m 2] T. The rigidity matrix R := 1 2 f G(p)/ p R m 2n Figure: The rigidity matrix R. yo-sung Ahn (Collaborations with Minh Hoang Trinh, Rigidity Zhiyong Matrix Sun, Brian D. O. Anderson, and Viet 2017 Hoang 14 Pham) / 39
22 rigidity matrix Denote z ij = p j p i, (i, j) E: displacement vector Labeling m edges, we have z = [z T 1,..., zt m ] T = (H I 2 )p R 2m The distance function f G : R 2n R m, f G : p [ z 1 2,..., z m 2] T. The rigidity matrix R := 1 2 f G(p)/ p R m 2n A framework is infinitesimally rigid iff. rank(r) = 2n 3 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Rigidity Zhiyong Matrix Sun, Brian D. O. Anderson, and Viet 2017 Hoang 14 Pham) / 39
23 contents 1 Background 2 Rigidity Matrix 3 The symmetric rigidity matrix The symmetric rigidity matrix Physical meaning of the eigenvectors Further properties of the eigenvalues 4 Simulation 5 The new rigidity indices Motivation and definition Properties of the new indices 6 Conclusions
24 the symmetric rigidity matrix The symmetric rigidity matrix 1 where R is the rigidity matrix. M := R T R. (2) Figure: The symmetric rigidity matrix M resembles the graph Laplacian matrix. 1 Zelazo2012. Hyo-Sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 15 Pham) / 39
25 the symmetric rigidity matrix The symmetric rigidity matrix 1 M := R T R. (2) where R is the rigidity matrix. Some properties of M: M R 2n 2n is symmetric, positive semidefinite. N (M) = N (R), rank(m) 2n 3. M has at least three zero eigenvalues. F is infinitesimally rigid iff. rank(m) = 2n 3. 1 Zelazo2012. Hyo-Sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 15 Pham) / 39
26 eigenvalues and eigenvectors of M The symmetric rigidity matrix M has Eigenvalues: 0 λ 1 λ 2... λ 2n, (λ 1 = λ 2 = λ 3 = 0) Eigenvectors: v 1, v 2,..., v 2n, where v k = [(v k 1 )T,..., (v k n ) T ] T R 2n yo-sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 16 Pham) / 39
27 eigenvalues and eigenvectors of M The symmetric rigidity matrix M has Eigenvalues: 0 λ 1 λ 2... λ 2n, (λ 1 = λ 2 = λ 3 = 0) Eigenvectors: v 1, v 2,..., v 2n, where v k = [(v k 1 )T,..., (v k n ) T ] T R 2n v 1, v 2, v 3 associate with three zero eigenvalues 2 v 1 = [ ] T, v 2 = [ ] T, v 3 = [ y 1 x 1 y 2 x 2... y n x n ] T. 2 Sun2015a. yo-sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 16 Pham) / 39
28 eigenvalues and eigenvectors of M The symmetric rigidity matrix M has Eigenvalues: 0 λ 1 λ 2... λ 2n, (λ 1 = λ 2 = λ 3 = 0) Eigenvectors: v 1, v 2,..., v 2n, where v k = [(v k 1 )T,..., (v k n ) T ] T R 2n v 1, v 2, v 3 associate with three zero eigenvalues v 1 = [ ] T, v 2 = [ ] T, v 3 = v 3 + ȳv 1 xv 2, where x = 1 n n i=1 x i and ȳ = 1 n n i=1 y i. yo-sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 16 Pham) / 39
29 eigenvalues and eigenvectors of M The symmetric rigidity matrix M has Eigenvalues: 0 λ 1 λ 2... λ 2n, (λ 1 = λ 2 = λ 3 = 0) Eigenvectors: v 1, v 2,..., v 2n, where v k = [(v k 1 )T,..., (v k n ) T ] T R 2n v 1, v 2, v 3 associate with three zero eigenvalues v 1 = [ ] T, v 2 = [ ] T, v 3 = v 3 + ȳv 1 xv 2, Figure: v 1, v 2, v 3 correspond to infinitesimally rigid motions. yo-sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 16 Pham) / 39
30 eigenvalues and eigenvectors of M The symmetric rigidity matrix M has Eigenvalues: 0 λ 1 λ 2... λ 2n, (λ 1 = λ 2 = λ 3 = 0) Eigenvectors: v 1, v 2,..., v 2n, where v k = [(v k 1 )T,..., (v k n ) T ] T R 2n v 1, v 2, v 3 associate with three zero eigenvalues v 1 = [ ] T, v 2 = [ ] T, v 3 = v 3 + ȳv 1 xv 2, For λ k 0 physical interpretation of v k? Hyo-Sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 16 Pham) / 39
31 statics of frameworks Framework: rods and joints model 2 2 Roth1981. Hyo-Sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 17 Pham) / 39
32 statics of frameworks Framework: rods and joints model 2 Stress: a set of scalars w = [w ij ] (i,j) E defined for each edge Equilibrium stress: j N i w ij (p i p j ) = 0, i = 1,..., n. (3) A stress is trivial when w ij = 0, (i, j) E. Stress free: Only the trivial makes the equilibrium stress be satisfied Rigid graphs: Stress free Minimally rigid graphs: Stress free & All edges linearly independent (in the sense of rigidity matrix) Flex graphs: Edges linearly dependent (i.e., non-trivial w ij makes the equilibrium stress be satisfied) 2 Roth1981. Hyo-Sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 17 Pham) / 39
33 statics of frameworks Framework: rods and joints model 3 3 Roth1981. Hyo-Sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 18 Pham) / 39
34 statics of frameworks Framework: rods and joints model 3 Stress: a set of scalars w = [w ij ] (i,j) E defined for each edge Equilibrium stress: j N i w ij (p i p j ) = 0, i = 1,..., n. (4) A stress is trivial when w ij = 0, (i, j) E. Stress free: Only the trivial makes the equilibrium stress be satisfied Rigid graphs: Stress free Minimally rigid graphs: Stress free & All edges linearly independent (in the sense of rigidity matrix) Flex graphs: Edges linearly dependent (i.e., non-trivial w ij makes the equilibrium stress be satisfied) 3 Roth1981. Hyo-Sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 18 Pham) / 39
35 statics of frameworks F = [F1 T,..., F n T ] T R 2n is an equilibrium force if n F i = 0, (5) i=1 n p i F i = 0, (6) i=1 where denotes the cross product. Hyo-Sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 19 Pham) / 39
36 statics of frameworks F = [F1 T,..., F n T ] T R 2n is an equilibrium force if n F i = 0, (5) i=1 n p i F i = 0, (6) i=1 where denotes the cross product. F is resolvable if scalars w ij s.t. F i + w ij (p i p j ) = 0, (7) j N i for all i = 1,..., n. Hyo-Sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 19 Pham) / 39
37 physical meaning of eigenvectors of M For each eigenvector v k = [v kt 1,..., v kt n ] T, Let w = [w ij ] (i,j) E = Rv k R m, or λ k v k i Mv k = R T Rv k = λ k v k Mv k = R T (Rv k ) = R T w = λ k v k, k = 1,..., 2n. (8) + j N i w ij (p i p j ) = 0, i = 1,..., n; k = 1,..., 2n. (9) Hyo-Sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 20 Pham) / 39
38 physical meaning of eigenvectors of M For each eigenvector v k = [v kt 1,..., v kt n ] T, Let w = [w ij ] (i,j) E = Rv k R m, Mv k = R T Rv k = λ k v k Mv k = R T (Rv k ) = R T w = λ k v k, k = 1,..., 2n. (8) or λ k vi k + w ij (p i p j ) = 0, i = 1,..., n; k = 1,..., 2n. (9) j N i Theorem Given an infinitesimally rigid framework F in a plane. Then each vector F = λ k v k (k = 4,..., 2n 3) is a resolvable force, where v k is the eigenvector corresponding to a nonzero eigenvalue of the symmetric rigidity matrix M of F. Hyo-Sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 20 Pham) / 39
39 physical meaning of eigenvectors of M Figure: The eigenvectors of an equilateral triangular frameworks Hyo-Sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 20 Pham) / 39
40 further properties of eigenvalues of matrix M Figure: F has matrix M with eigenvalues λ 1 λ 2... λ 2n, and F has matrix M with eigenvalues λ 1 λ 2... λ 2n. Hyo-Sung Ahn (Collaborations with Minh Hoang The symmetric Trinh, Zhiyong rigidity Sun, matrix Brian D. O. Anderson, and Viet 2017 Hoang 21 Pham) / 39
41 contents 1 Background 2 Rigidity Matrix 3 The symmetric rigidity matrix The symmetric rigidity matrix Physical meaning of the eigenvectors Further properties of the eigenvalues 4 Simulation 5 The new rigidity indices Motivation and definition Properties of the new indices 6 Conclusions
42 Simulation Consider the triangular framework depicted in the case (d) in page 13 (i.e., k = 4). Stress forces along the edges w ij (p i p j ) j N i Forces to each nodes λ 4 vi 4 Each node is governed by double integrator dynamics (i.e., p i = u i ). 1) u i = λ 4 vi 4 + w ij (p i p j ) j N i 2) u i = λ 4 vi 4(0) + w ij (p i p j ) j N i 3) u i = λ 4 normalized(vi 4(0) + 0.1q4 i (0)) + w ij (p i p j ), where j N i (vi 4)T qi 4 = 0 4) u i = λ 4 normalized(vi 4(0) + 0.5q4 i (0)) + w ij (p i p j ) j N i 5) u i = λ 4 normalized(vi 4(0) + 1.0q4 i (0)) + w ij (p i p j ) j N i Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 22 Pham) / 39
43 Case 1 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 23 Pham) / 39
44 Case 2 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 24 Pham) / 39
45 Case 3 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 25 Pham) / 39
46 Case 4 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 26 Pham) / 39
47 Case 5 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 27 Pham) / 39
48 Simulation (Cont.) Each node is governed by double integrator dynamics (i.e., p i = u i ). 6) u i = λ 4 (0)vi 4(0) + w ij (p i p j ) j N i 7) u i = λ 4 (0)vi 4(0) + w ij (0)(p i p j ) j N i 8) u i = λ 4 (0)normalized(vi 4(0) + 0.1q4 i (0)) + w ij (p i p j ), where j N i (v 4 i )T q 4 i = 0 9) u i = λ 4 (0)normalized(v 4 i (0) + 0.5q4 i (0)) + j N i w ij (p i p j ) 10) u i = λ 4 (0)normalized(v 4 i (0) + 1.0q4 i (0)) + j N i w ij (p i p j ) Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 28 Pham) / 39
49 Case 6 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 29 Pham) / 39
50 Case 7 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 30 Pham) / 39
51 Case 8 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 31 Pham) / 39
52 Case 9 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 32 Pham) / 39
53 Case 10 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 33 Pham) / 39
54 contents 1 Background 2 Rigidity Matrix 3 The symmetric rigidity matrix The symmetric rigidity matrix Physical meaning of the eigenvectors Further properties of the eigenvalues 4 Simulation 5 The new rigidity indices Motivation and definition Properties of the new indices 6 Conclusions
55 Motivation Figure: λ 4 depends on the size of F. Figure: Which framework is more rigid? λ 4 (M) is usually used as a rigidity index 4 λ 4 > 0 F is infinitesimally rigid λ 4 depends quadratically on the scale factor ζ. cannot compare rigidity between different frameworks. 4 Zelazo2012. Hyo-Sung Ahn (Collaborations with Minh Hoang The Trinh, new rigidity Zhiyongindices Sun, Brian D. O. Anderson, and Viet 2017 Hoang 34 Pham) / 39
56 The new rigidity indices Consider a framework F = (G, p) in the plane, Definition The worst-case rigidity index of F is defined as Definition χ = λ 4 2n i=1 λ i The imbalance index of the framework F is defined as = λ 4 tr(m). (8) ξ = λ 4 λ 2n. (9) Hyo-Sung Ahn (Collaborations with Minh Hoang The Trinh, new rigidity Zhiyongindices Sun, Brian D. O. Anderson, and Viet 2017 Hoang 35 Pham) / 39
57 properties of the new rigidity indices Proposition Assume F 1 = (G, p) and F 2 = (G, p ) are two similar frameworks with the worst-case rigidity indices χ 1, χ 2 and the imbalance indices ξ 1, ξ 2. Then χ 1 = χ 2 and ξ 1 = ξ 2. The new rigidity indices: χ > 0 and ξ > 0 F is infinitesimally rigid scale-free depend only on the framework s shape. Figure: F 1, F 2 and F 3 have the same worst-case rigidity index: χ 1 = χ 2 = χ 3. Hyo-Sung Ahn (Collaborations with Minh Hoang The Trinh, new rigidity Zhiyongindices Sun, Brian D. O. Anderson, and Viet 2017 Hoang 36 Pham) / 39
58 examples Example 1: Triangular frameworks Hyo-Sung Ahn (Collaborations with Minh Hoang The Trinh, new rigidity Zhiyongindices Sun, Brian D. O. Anderson, and Viet 2017 Hoang 37 Pham) / 39
59 examples Example 2: Square frameworks Hyo-Sung Ahn (Collaborations with Minh Hoang The Trinh, new rigidity Zhiyongindices Sun, Brian D. O. Anderson, and Viet 2017 Hoang 38 Pham) / 39
60 contents 1 Background 2 Rigidity Matrix 3 The symmetric rigidity matrix The symmetric rigidity matrix Physical meaning of the eigenvectors Further properties of the eigenvalues 4 Simulation 5 The new rigidity indices Motivation and definition Properties of the new indices 6 Conclusions
61 conclusions Main results: Further analysis on the symmetric rigidity matrix M: Physical interpretation of the eigenvectors Further properties of the eigenvalues Two scale-free rigidity indices: The worst-case rigidity index χ The imbalance index ξ. Further studies: Find more properties of the rigidity indices Relationship between the rigidity matrix and the stiffness matrix Extend the results to 3D frameworks. Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Conclusions Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 39 Pham) / 39
62 Q & A Thank you! hyosung@gist.ac.kr Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Conclusions Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 39 Pham) / 39
63 References I
arxiv: v1 [cs.sy] 6 Jun 2016
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