2. A model of Caterpillar Locomotion Based on Assur Tensegrity Structures Shai Offer School of Mechanical Engineering, Faculty of Engineering, Tel-Aviv University, Tel-Aviv, Israel. Orki Omer School of Mechanical Engineering, Faculty of Engineering, Tel-Aviv University, Tel-Aviv, Israel. Ben-Hanan Uri Department of Mechanical Engineering, Ort Braude College, Karmiel, Israel. Ayali Amir Department of Zoology, Faculty of Life Sciences, Tel Aviv University, Tel Aviv, Israel.

3. -The main idea. -Tensegrity. -Assur Graph (Group). -Singularity+ Assur Graph+tensegrity= Assur Tensegrity -Impedance control -Assur tensegrity+ Impedance control -Further applications.

4. Tensegrity structures are usually statically indeterminate structures

5. Animal/Caterpillar- Soft and rigid robot Assur Graph Tensegrity Singularity

6. 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.

7. Removing this joint results in Determinate truss with the same mobility Example of a determinate truss that is NOT an Assur Group.

8. TRIAD We remove this joint Example of a determinate truss that is an Assur Group – Triad. And it becomes a mechanism

9. The MAP of all Assur Graphs in 2d is complete and sound.

11. Singularity and Mobility Theorem in Assur Graphs

12. First, let us define: 1. Self-stress. 2. Extended Grubler’s equation.

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

14. Extended Grubler’s equation = Grubler’s equation + No. self-stresses DOF = 0 DOF = 0 + 1 = 1

15. DOF = 0 + 2 = 2 The joint can move infinitesimal motion. Where is the other mobility?

16. All the three joints move together. Extended Grubler = 2 = 0 + 2

17. 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.

18. ASSUR GRAPHS IN SINGULAR POSITIONS 1 2 6 5 4 3

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

20. 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 A A A B B B B 2 DOF (instead of 1) and 2 SS (instead of 1).

21. Assur Graph at the singular position  There is a unique self-stress in all the links  Check the possibility: tension  cables. compression  struts.

22. Combining the Assur triad with a tensegrity structure Changing the singular point in the triad

23. 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.

24. Transforming a soft (loose) structure into Rigid Structure

25. Shortening the length of one of the cables

27. Almost Rigid Structure

29. At the Singular Position

30. Singular point The structure is Rigid

31. The general control low Virtual force. Maintain the triad in self- stress. static relation between output force and input displacement Damping term Output force

32. Advantages : » Simple shape change Since the structure is statically determinate, any change in length of one element results in shape change. This in contrast to statically indeterminate structures. » Stability The self-stress of the Assur Tensegrity is always maintained, and the structure stays in a singular configuration.

33. The model consists of triads connected in series CableBar Leg Strut

34. Level 1 Central Control Level 2 localized control Leg Controllers Cable Controllers Strut controllers Ground contact sensor High level control Low level control CPG - Central Pattern Generator Hydrostatic pressure Ganglions Muscle behavior

36. » Assur 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.

37. All the details of this work will appear in October 2011 in Orki’s thesis (in English).