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A model of Caterpillar Locomotion Based on Assur Tensegrity StructuresShai OfferSchool of Mechanical Engineering, Faculty of Engineering, Tel-Aviv University, Tel-Aviv, Israel.Orki OmerSchool of Mechanical Engineering, Faculty of Engineering, Tel-Aviv University, Tel-Aviv, Israel.Ben-Hanan UriDepartment of Mechanical Engineering, Ort Braude College, Karmiel, Israel.Ayali AmirDepartment of Zoology, Faculty of Life Sciences, Tel Aviv University, Tel Aviv, Israel.The outline of the talk:

The main idea.Tensegrity.Assur Graph (Group).Singularity+ Assur Graph+tensegrity= Assur TensegrityImpedance controlAssur tensegrity+ Impedance controlFurther applications.

Tensegrity= Tension + Integrity

Tensegrity structures are usually statically indeterminate structuresThe Main IdeaAnimal/Caterpillar-Soft and rigid robotAssur GraphTensegrity SingularityThe 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.TRIADWe remove this jointExample of a determinate truss that is an Assur Group Triad.And it becomes a mechanismThe MAP of all Assur Graphs in 2d is complete and sound.

The Map of all Assur Graphs in 2D Singularity and Mobility Theorem in Assur Graphs

First, let us define:

1. Self-stress.

2. Extended Grublers equation.Self stress

Self Stress A set of forces in the links (internal forces) that satisfy the equilibrium of forces around each joint. Extended Grublers equationExtended Grublers equation = Grublers equation + No. self-stresses

DOF = 0DOF = 0 + 1 = 1Example with two self-stresses (SS)DOF = 0 + 2 = 2The joint can move infinitesimal motion. Where is the other mobility? The Other MotionAll the three joints move together.Extended Grubler = 2 = 0 + 2Special 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 POSITIONS126543

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

ONLY Assur Graphs have this property!!!SS in All the links, butJoint A is not mobile. NO SS in All links.Joint A is not mobile. AAAABBBB2 DOF (instead of 1) and2 SS (instead of 1).Assur Graph at the singular position

There is a unique self-stress in all the links

Check the possibility: tension cables. compression struts.Assur Graph + Singularity + Tensegrity=Assur Tensegrity

Combining the Assur triad with a tensegrity structure

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 pointThe structure is RigidImpedance controlThe general control lowVirtual force. Maintain the triad in self-stress. static relation between output force and input displacementDamping termOutput forceAdvantages :Assur tensegrity + Impedance controlSimple shape changeSince the structure is statically determinate, any change in length of one element results in shape change. This in contrast to statically indeterminate structures.StabilityThe self-stress of the Assur Tensegrity is always maintained, and the structure stays in a singular configuration.Caterpillar modelThe model consists of triads connected in series

CableBarLegStrutControl SchemeLevel 1Central ControlLevel 2 localized controlLeg ControllersCable ControllersStrut controllersGround contact sensorHigh level controlLow level controlCPG - Central Pattern GeneratorHydrostatic pressureGanglionsMuscle behavior Results

ConclusionsAssur tensegrity robots together with an impedance control are useful for building soft robots and provide controllable degree of softness.Because Assur tensegrity is astatically determinate truss, shape change is very simple.The Control Scheme is relatively simple and inspired by the biological caterpillar anatomy and physiology.The caterpillar model can adjust itself to the terrain with only one type of external sensors ground contact sensors.The caterpillar can crossed curved terrains and can crawl in any direction.Thank You!All the details of this work will appear in October 2011 in Orkis thesis (in English).1

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