Proceedings of the ASME 2010 International Design Engineering Technical Conferences &Computers and Information in Engineering ConferenceIDETC/CIE 2010August 15-18, 2010, Montreal, Quebec, Canada, DETC2010-28687

EMPLOYING ASSUR TENSEGRITY STRUCTURES FOR SIMULATING A

CATERPILLAR LOCOMOTION

Omer Orki, Offer Shai, Itay Tehori, Michael Slavutin, Uri Ben-HananSchool of Mechanical Engineering

Israel

The outline of the talk:- The main idea.- Tensegrity.- Assur Graph (Group).- Singularity in Assur Graph (main theorem).- Previous application: Adjustable Deployable Tensegrity Structures.- Caterpillar (various types of animals) robot.- Further applications.

The Main Idea

Animal/Caterpillar-

Soft and rigid robot

Assur Graph

Tensegrity

Singularity

Tensegrity = tension + integrity

Consist of:Cables – sustain only tension.Struts - sustain only compression

The equilibrium between the two types of forces yields static stability (structural integrity) of the system.

The definition of Assur Graph (Group) :

Special minimal structures (determinate trusses) with zero mobility from which it is not possible to obtain a simpler

substructure of the same mobility .

Another definition: Removing any set of joints results in a

mobile system.

Removing this joint results in

Determinate truss with the same mobility

Example of a determinate truss that is NOT an Assur Group.

TRIAD We remove this joint

Example of a determinate truss that is an Assur Group – Triad.

And it becomes a mechanism

The MAP of all Assur Graphs in 2d

is complete and sound.

The Map of all Assur Graphs in 2DThe Map of all Assur Graphs in 2D

Singularity and Mobility Theorem in

Assur Graphs

First, let us define:

1. Self-stress.

2. Extended Grubler’s equation.

Self stress1

4

3

2

65

(a)

1

3

26

(b)

4

P

P

Self Stress – A set of forces in the links (internal forces) that satisfy the equilibrium of forces around each joint.

Extended Grubler’s equation

Extended Grubler’s equation = Grubler’s equation + No. self-stresses

1 2A

inf

DOF = 0 DOF = 0 + 1 = 1

Example with two self-stresses (SS)

DOF = 0 + 2 = 2

The joint can move infinitesimal motion. Where is the other mobility?

The Other Motion

All the three joints move together.

Extended Grubler = 2 = 0 + 2

Special Singularity and Mobility properties of Assur Graphs:

G is an Assur Graph IFF there exists a configuration in which there is a unique self-stress in all the links and all the joints have an

infinitesimal motion with 1 DOF .

Servatius B., Shai O. and Whiteley W., "Combinatorial Characterization of the Assur Graphs from Engineering", European Journal of Combinatorics, Vol. 31, No. 4, May, pp. 1091-1104, 2010.

Servatius B., Shai O. and Whiteley W., "Geometric Properties of Assur Graphs", European Journal of Combinatorics, Vol. 31, No. 4, May, pp. 1105-1120, 2010.

ASSUR GRAPHS IN SINGULAR POSITIONS

1

2

6

5

4

3

Singularity in Assur Graph – A state where there is:1. A unique Self Stress in all the links. 2. All the joints have infinitesimal motion 1DOF.

1 2A

inf

ONLY Assur Graphs have this property!!!ONLY Assur Graphs have this property!!!

SS in All the links, but

Joint A is not mobile.

NO SS in All links.

Joint A is not mobile.

A

AA

AB

B

B

B

2 DOF (instead of 1) and

2 SS (instead of 1).

Assur Graph + Singularity + Tensegrity

Assur Graph at the singular position

There is a unique self-stress in all the links

Check the possibility: tension cables. compression struts.

1

2

3

4

5 6

A C

B

1

2

3

4

5 6

A C

B

(a) (b)

A C

B

(c)

3

2

1

4

65

Combining the Assur triad with a tensegrity structure

A C

B

(a)

B

(b)

1

2

3

4

5 6

1

3

2

65

4A

C

Changing the singular point in the triad

From Soft to Rigid Structure

Theorem: it is enough to change the location of only one element so that the Assur Truss is at the singular position.

In case the structure is loose (soft) it is enough to shorten the length of only one cable so that the Assur Truss is being at the singular position.

Transforming a soft (loose) structure into Rigid Structure

Shortening the length of one of the cables

Shortening the length of one of the cables

Almost Rigid Structure

Almost Rigid Structure

At the Singular Position

Singular pointSingular point

The structure is Rigid

The First type of robot that employs the three properties:

1. Assur Graph.2. Tensegrity.3. Singularity.

Adjustable Deployable Tensegrity Structure –

A structure that can deploy and fold but all the time is rigid, i.e., can sustain external forces.

This property is obtained by constantly maintaining the structure at the singular position!

Folded systemDeployed system

The Second type of robot that relies on these three properties:

1. Assur Graph.2. Tensegrity.3. Singularity.

Animal (caterpillar) robot.

Caterpillar robot based on Assur Tensegrity structure

Rigid – at the singular position.

Soft – not at the singular position.

Biological Background

The caterpillar is a soft-bodied animal.Divided into three parts: head, thorax, and abdomen

The thorax: three segments, each bearing a pair of true legs.The abdomen: eight segments- Segments A1-A7 and the Terminal Segment (TS). Segments A3 to A6 and TS have a pair of fleshy protuberances called prolegs.

Anterior side

Posterior side

Dorsal surface

Ventral surface TSA7A6A5

A4

A3

A2

A1 Abdomen

Thorax & Head

Prolegs

The cables can be thought of as representing the major longitudinal muscles of the caterpillar segments:

The upper cable represents the ventral longitudinal muscle (VL1) and the lower cable represents the dorsal longitudinal muscle (DL1).

The linear actuator, which is always subjected to compression forces, represents the hydrostatic skeleton.

Four segments caterpillar

The Caterpillar Model

The Caterpillar Model

In this simulation both cables in each triad were independently force controlled. The force in each cable was controlled with spring-like properties. When the cable stretches and becomes longer, the tension force increases and vice versa.

The fault tolerance of the robot – the robot has the ability to adjust itself to the terrain without any high level control.

New Possible Application

Crawling in Tunnels.

Caterpillar robot based on Assur Tensegrity structure

Rigid – at the singular configuration.

Soft – not at the singular position.