A model of Caterpillar Locomotion Based on Assur Tensegrity Structures

1. 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.

2. The main idea. • Tensegrity. • Assur Graph (Group). • Singularity+ Assur Graph+tensegrity= Assur Tensegrity • Impedance control • Assur tensegrity+ Impedance control • Further applications. The outline of the talk:

3. Tensegrity= Tension + Integrity Tensegrity structures are usually statically indeterminate structures

4. Tensegrity AssurGraph Animal/Caterpillar- Soft and rigid robot The Main Idea Singularity

5. 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.

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

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

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

9. The Map of all Assur Graphs in 2D

10. Singularity and Mobility Theorem in Assur Graphs

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

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

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

14. DOF = 0 + 2 = 2 Example with two self-stresses (SS) The joint can move infinitesimal motion. Where is the other mobility?

15. All the three joints move together. Extended Grubler = 2 = 0 + 2 The Other Motion

16. 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.

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

18. 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.

19. ONLY Assur Graphs have this property!!! B A A B NO SS in All links. Joint A is not mobile. A A B B 2 DOF (instead of 1) and 2 SS (instead of 1). SS in All the links, but Joint A is not mobile.

20. 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=AssurTensegrity

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

22. 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. From Soft to Rigid Structure

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

24. Shortening the length of one of the cables

25. Shortening the length of one of the cables

26. Almost Rigid Structure

27. Almost Rigid Structure

28. At the Singular Position

29. Singular point The structure is Rigid

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

31. Assur tensegrity + Impedance control • Stability The self-stress of the AssurTensegrity is always maintained, and the structure stays in a singular configuration. 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. • Controllable softness The structure is reactive to external forces. Moreover, the degree of “softness” can be determined by the stiffness coefficient

32. Caterpillar model The model consists of triads connected in series Cable Bar Leg Strut

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

34. Results

35. Conclusions • Assurtensegrity robots together with an impedance control are useful for building soft robots and provide controllable degree of softness. • Because Assurtensegrity 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.

36. All the details of this work will appear in October 2011 in Orki’s thesis (in English). Thank You!