Stability of Determinate Tensegrity Structures Daniel Cohen School of Mechanical Engineering Tel Aviv University Offer Shai School of Mechanical Engineering Tel Aviv University The outline of the talk Terminology determinate tensegrity truss, geometric/topological

self-stresses, Assur truss. Stability theorem based on minimum potential energy Stability theorem based on machine theory: checking the intersection between the curvature path and the reachable region ofthe tested element. Assur tensegrity truss. The special singularity of Assur tensegrity truss. Characterizing the singularity configuration. Practical applications. Terminology : grounded and floating determinate

tensegrity trusses Grounded determinate truss. Floating determinate truss. (also called - free standing structures). Strut elements (red) can carry only compression forces. Cable elements (blue) can carry only tension forces.

Self-stresses (prestress) in Tensegrity structures The stiffness of tensegrity systems is obtained due to the existence of self-stresses (inner forces), also called prestress. In indeterminate tensegrity structures the self-stress is due to the redundancy (topological self-stress). In determinate tensegrity structures the self-stress is due to the special geometry (geometric self-stress). Why is it stable?

Which one is stable? 1. 3. 2. 4. Structure is stable it

has the locally minimum potential energy In indeterminate tensegrity structures The self-stressed equilibrium (topological self-stress) is a necessary and sufficient condition for stability.

In determinate tensegrity structures the self-stressed equilibrium (geometric selfstress) is a necessary but insufficient condition for stability. Two known methods (equations) for checking prestress-stability (Connelly & Whiteley) - The velocity vector of the structure. The density force matrix.

(Zhang & Ohsaki) - The velocity vector of the infi mechanism of the structure. - The geometric stiffness matrix. Tensegrity stability through Machine theory Checking the stability of determinate tensegrity structures using methods from Machine theory: - Mechanism after removing the tested element. - Calculating the radius of curvature. - Calculating the normal acceleration.

The steps: Mechanisms + Tensegrity Systems 1. Remove a tested element, results in a mechanism. 2. Connecting joint the point where the tested element is connected to the mechanism. 3. The reachable region of the tested element (cable/strut inner/outer region of a circuit or sphere. 4. Calculate the normal acceleration. 5. Calculate the radius of curvature of the connecting joint.

The machine theory criteria for stability of Tensegrity structures Tensegrity structure is stable if and only if the intersection between the curvature path of the connecting joint and the reachable region of the tested element is a point.

Checking an unstable tensegrity dyad through machine theory A The cable is the tested element Reachable region of the tested element

A The mechanism A Curvature path The intersection is a segmentunstable

Checking the stability of a dyad tensegrity structure through machine theory A The strut is the tested element Reachable region A

A The mechanism Curvature path Intersection is a single point stable Stabilization of a determinate tensegrity structure 1. Choose a realization where there is an admissible selfstress (a singular configuration).

2. Choose a ground element to be the tested element. 3. Remove the tested element (mechanism) and calculate the normal acceleration of the connecting joint. 4. Calculate the radius of curvature of the connecting joint. 5. Locate the ground joint of the tested element on the ray whose initial point is the center of the curvature radius. If the tested element is a strut locate in the direction of the normal acceleration. If the test element is a cable locate in the opposite direction of the normal acceleration.

Locating the ground joint of the tested element The tested element is a cable. The ray is in the opposite direction of the normal acceleration. A

The tested element is a strut . The ray is in the direction of the normal acceleration. 2 = The velocity The acceleration

The normal acceleration The radius of curvature If the tested element is a strut THEN locate on the ray in the direction of the normal acceleration If the tested element is a cable THEN locate on the ray in the opposite direction of the normal acceleration Locating the ground joint of the tested element

A Tensegrity is stable iff the ground joint of the tested element (strut/cable) is located on the ray

whose initial point is the curvature radius and its direction is with/against the direction of the normal acceleration. The normal acceleration curvature radius and on which side of it. Stabilization of a Pentad tensegrity structure A The tested

element is a cable Stabilization of a Pentad tensegrity structure Curvature path The velocity Absolute acceleration. Normal acceleration indicates the location

of the curvature center A

The curvature radius. Reachable region Intersection is a single point stable Stabilization of a Pentad tensegrity structure Checking the stability by the second method

Normal acceleration indicates the location of the curvature center A The tested element is a cable

The curvature radius. The ground node of the tested element is in the correct side of the ray. Assur Truss (also called Assur group) Structure with zero DOF that does not contain any other substructure with zero DOF. Assur trusses are the building block of any mechanism and

determinate trusses in 2D and 3D. Special singularity of Assur truss: (Servatius et al., 2010) Structure is an Assur truss iff there exists a configuration in which there is a single self-stress in all the elements and all the nodes are mobile with one DOF. The types of tensegrity structures tensegrity structures Statically

determinate Redundant Assur Truss Mechanism Composition of Assur Truss

The tensegrity structures been taken from (Jing Yao Zhang Makoto Ohsaki 2015) Assur tensegrity trusses (Paul et al. 2005) - Triad (Sunny et al. 2013) - Tetrad (FRANKLIN. 2000) - Hexad (Correa. 2001) - Pentad

(Tibert and Pellegrino. 2003) N polygon The singular characterization of the Assur Trusses Each Assur Truss has its special singular characterization, both in 2D and 3D. There exists a combinatorial method that defines the characterization in an efficient way. In this talk we show the singular characterization of the Tetrad Assur truss.

The singular constraint rule: 2D - a line has to pass through three points. 3D a line is normal to three lines. The singular characterization using Dual Kennedy theorem 1. For each face (circuit without inner links) we associate a line dual to the absolute instant center of the corresponding link. 2. The line between two adjacent face is dual to the relative instant center of the corresponding two links. 3. In 2D, the sum of two lines is a line that passes through the

intersection point of the two line. 4. In 3D it can be a screw. 5. The singular constraint: a line that passes through three points. Characterizing the singularity of 2D Tetrad ( 0 , 2) (2 , 4) using Dual Kennedy theorem ( 4 , 0 ) = ( 0 , 3) (3 , 4 ) {

(0 , 1) (1 , 4) 4 1 2 3 0

= (2 , 4 ) } { ( 0 ,3 ) ( 3 , 4 ) } {( 0 ,1 ) ( 1 , 4 ) } { (0 , 2) Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem ( 0 , 2) (2 , 4) ( 4 , 0 ) = ( 0 , 3) (3 , 4 ) (0 , 1) (1 , 4) {

4 1 2 3 0 = (2

{ (0 , 2) (0 ,,2)4 ) } { ( 0 ,3 ) ( 3 , 4 ) } {( 0 ,1 ) ( 1 , 4 ) } Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem ( 0 , 2) (2 , 4) ( 4 , 0 ) = ( 0 , 3) (3 , 4 ) (0 , 1) (1 , 4) {

4 1 2 3 0 4 )}

= {(2 (0 , ,2)(2 , 4 ){}( 0 ,3 ) ( 3 , 4 ) } {( 0 ,1 ) ( 1 , 4 ) } { (0 , 2) Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem ( 0 , 2) (2 , 4) ( 4 , 0 ) = ( 0 , 3) (3 , 4 ) (0 , 1) (1 , 4)

{ 4 1 2 3 0

4 )} = {(2 (0 , ,2)(2 , 4 ){}( 0 ,3 ) ( 3 , 4 ) } {( 0 ,1 ) ( 1 , 4 ) } { (0 , 2) Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem ( 0 , 2) (2 , 4)

( 4 , 0 ) = ( 0 , 3) (3 , 4 ) (0 , 1) (1 , 4) { 4 1 2 3

0 ,3, 3) = (2 , 4 ) } { ( 0( 0 ) ( 3 , 4 ) } {( 0 ,1 ) ( 1 , 4 ) } { (0 , 2) Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem

( 0 , 2) (2 , 4) ( 4 , 0 ) = ( 0 , 3) (3 , 4 ) (0 , 1) (1 , 4) { 4 1 2

3 0 = (2 , 4 ) } { ( 0 ,3 ) ( 3( 3 , 4,)4 ) } {( 0 ,1 ) ( 1 , 4 ) } { (0 , 2) Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem

( 0 , 2) (2 , 4) ( 4 , 0 ) = ( 0 , 3) (3 , 4 ) (0 , 1) (1 , 4) { 4 1 2

3 0 ( 3, 4, )4 ,3)) (3 = (2 , 4 ) } { ( 0{ ( 0,3 { (0 , 2) } ) } {( 0 ,1 ) ( 1 , 4 ) }

Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem ( 0 , 2) (2 , 4) ( 4 , 0 ) = ( 0 , 3) (3 , 4 ) (0 , 1) (1 , 4) { 4

1 2 3 0 (0 , 2)(2 , 4 )} , 4 ){}( 0{ (,3

0 ,3) ) ((33, 4, )4} ) } {( 0 ,1 ) ( 1 , 4 ) } = {(2 { (0 , 2) Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem ( 0 , 2) (2 , 4) ( 4 , 0 ) = ( 0 , 3) (3 , 4 ) (0 , 1) (1 , 4)

{ 4 1 2 3 0

= (2 , 4 ) } { ( 0 ,3 ) ( 3 , 4 ){}( 0 , 1{) ( 0 ,1 ) ( 1 , 4 ) } { (0 , 2) Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem ( 0 , 2) (2 , 4) ( 4 , 0 ) = ( 0 , 3) (3 , 4 )

(0 , 1) (1 , 4) { 4 1 2 3

0 = (2 , 4 ) } { ( 0 ,3 ) ( 3 , 4 ) } {( 0 ( 1,1 ,4) (1 , 4 ) } { (0 , 2) Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem ( 0 , 2) (2 , 4)

( 4 , 0 ) = ( 0 , 3) (3 , 4 ) (0 , 1) (1 , 4) { 4 1 2 3

0 = (2 , 4 ) } { ( 0 ,3 ) ( 3 , 4 ) } {( 0 ,1 ) ( 1 , 4 ) } { (0 , 2) Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem ( 0 , 2) (2 , 4) ( 4 , 0 ) = ( 0 , 3) (3 , 4 )

(0 , 1) (1 , 4) { 4 1 2 3

0 = (2 , 4 ) } { ( 0 ,3 ) ( 3 , 4 ) } {( 0 ,1 ) ( 1 , 4 ) } { (0 , 2) Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem ( 0 , 2) (2 , 4) ( 4 , 0 ) = ( 0 , 3) (3 , 4 ) (0 , 1) (1 , 4)

{ 4 1 2 3 0

= (2 , 4 ) } { ( 0 ,3 ) ( 3 , 4 ) } {( 0 ,1 ) ( 1 , 4 ) } { (0 , 2) Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem ( 0 , 2) (2 , 4) ( 4 , 0 ) = ( 0 , 3) (3 , 4 ) (0 , 1) (1 , 4)

{ 4 1 2 3 0

= (2 , 4 ) } { ( 0 ,3 ) ( 3 , 4 ) } {( 0 ,1 ) ( 1 , 4 ) } { (0 , 2) Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem ( 0 , 2) (2 , 4) ( 4 , 0 ) = ( 0 , 3) (3 , 4 ) (0 , 1) (1 , 4) {

4 1 2 3 0 = (2 , 4 ) } { ( 0 ,3 ) ( 3 , 4 ) } {( 0 ,1 ) ( 1 , 4 ) }

{ (0 , 2) Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem ( 0 , 2) (2 , 4) ( 4 , 0 ) = ( 0 , 3) (3 , 4 ) (0 , 1) (1 , 4) {

4 1 2 3 0 = (2 , 4 ) } { ( 0 ,3 ) ( 3 , 4 ) } {( 0 ,1 ) ( 1 , 4 ) } { (0 , 2)

Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem ( 0 , 2) (2 , 4) ( 4 , 0 ) = ( 0 , 3) (3 , 4 ) (0 , 1) (1 , 4) { 4

1 2 3 0 = (2 , 4 ) } { ( 0 ,3 ) ( 3 , 4 ) } {( 0 ,1 ) ( 1 , 4 ) } { (0 , 2)

Characterizing the singularity of 2D Tetrad using Dual Kennedy theorem ( 0 , 2) (2 , 4) ( 4 , 0 ) = ( 0 , 3) (3 , 4 ) (0 , 1) (1 , 4) { 4

1 2 3 0 = (2 , 4 ) } { ( 0 ,3 ) ( 3 , 4 ) } {( 0 ,1 ) ( 1 , 4 ) } { (0 , 2) The proposed method is applied to 3D

determinate tensegrity structures (in the same way as in 2D) 1. Choose a realization according to the singular characterization. 2. Choose a ground element to be the tested element. 3. Remove the tested element (mechanism) and calculate the normal acceleration of the connecting joint. 4. Calculate the radius of curvature of the connecting joint.

5. Locate the ground joint of the tested element on the ray whose initial point is the center of the curvature radius. Checking the stability of 3D Triad tensegrity truss C A B c

a b C A Calculate the: curvature radius normal acceleration

B

c The tested element a b The ground node of the tested element is in the correct side of the ray.

The use of the rigidity matrix in the calculations B C B C E

E D A A F

D For each element (A,B) + =0

The derivation of the perpendicularity rule For each element AB. = Calculate of the radius of curvature 46

A practical application that our research group is working on: Tensegrity robots for tunnels and narrow places 48 49 50

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