A physical interpretation of the rigidity matrix

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

Credits (a) Trinh Minh Hoang (b) Zhiyong Sun (d) Viet Hoang Pham (c) Prof. Brian D.O. Anderson

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 = [1 0 1 0... 1 0] T, v 2 = [0 1 0 1... 0 1] 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

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 = [1 0 1 0... 1 0] T, v 2 = [0 1 0 1... 0 1] 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

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 = [1 0 1 0... 1 0] T, v 2 = [0 1 0 1... 0 1] 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

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 = [1 0 1 0... 1 0] T, v 2 = [0 1 0 1... 0 1] 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

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

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

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

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

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

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

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

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

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

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

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

Case 1 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 23 Pham) / 39

Case 2 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 24 Pham) / 39

Case 3 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 25 Pham) / 39

Case 4 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 26 Pham) / 39

Case 5 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 27 Pham) / 39

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

Case 6 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 29 Pham) / 39

Case 7 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 30 Pham) / 39

Case 8 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 31 Pham) / 39

Case 9 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 32 Pham) / 39

Case 10 Hyo-Sung Ahn (Collaborations with Minh Hoang Trinh, Simulation Zhiyong Sun, Brian D. O. Anderson, and Viet 2017 Hoang 33 Pham) / 39

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

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

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

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

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

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

Q & A Thank you! email: 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

References I