1. EMPLOYING ASSUR TENSEGRITY STRUCTURES METHODS FOR SIMULATING A CATERPILLAR LOCOMOTION

2. Biological Background The caterpillar is a soft-bodied animal and divided into three parts: head, thorax, and abdomen The thorax consists of three segments, each bearing a pair of true legs. The abdomen constitutes over a ¾ of the total caterpillar's length. It has 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 TS A7 A6 A5 A4 A3 A2 A1 Abdomen Thorax & Head Prolegs Anatomy

3. Biological Background The caterpillars' locomotion is primarily done by crawling. Caterpillars crawl via a wave of muscular contractions that start at the posterior and progress forward to the anterior. The two feet on both sides of each body segment move together. At least three segments are in varying states of contraction at the same time. Once a particular proleg pair has moved and has been "planted", there is no further movement by that proleg or body segment, until the next cycle. Lifted Touching the ground Left A6 Right A6 Left A5 Right A5 Left A4 Right A4 Left A3 Right A3 1 2 3 4 5 6 Time unit Locomotion

4. Biological Background The musculature is complex - about 70 muscles per segment. Most muscles are contained entirely within the body segment. The major abdomen muscles are the ventral longitudinal muscle (VL1) and the dorsal longitudinal muscle (DL1). Muscles VL1 DL1

5. Assur Tensegrity Structures A unique geometrical property of Assur trusses is that they have a configuration, in which there exists a self-stress in all the elements, although the structure is statically determinate. An Assur truss in this configuration is said to be in a singular configuration In a 2D triad the singularity is characterized by the intersections of the continuations of the three rods: (O 1 C), (O 2 A) and (O 3 B) at the same point. B C A O1O1 O2O2 O3O3 B C A O2O2 O3O3 O1O1 Singular configuration Not a singular configuration

6. Assur truss in a singular configuration can turn into tensegrity structure : Elements under Tension are replaced with Cables. Element under Compression replaced with Struts. Assur Tensegrity Structures Tensegrity structures are well known in the literature and were first patented by R. Buckminster Fuller in 1962. These structures are composed of cables and struts.

7. The Shape Change Algorithm Force exists in all elements The triad is in self-stress The triad is in a singular configuration Force exists in the force element Its length is changing In order to keep an Assur truss in a singular configuration, it is sufficient to maintain the internal Force of only one element. The control algorithm is based on the algorithm that is described and mathematically proven by our group in 2009 : B C A O1O1 O2O2 O3O3 Measuring the tension Length control

8. The Caterpillar Model In this model, the segment of the biological caterpillar is represented as a planar tensegrity triad. The triad is a modification of the standard triad. It consists of two cables and a linear actuator connected between two bars. The cables are connected one at each side of the bars, and the linear actuator is connected in between. The triads are connected in cascade to consist the caterpillar model. Four segments caterpillar

9. 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

10. The basic control algorithm of one triad consists of : Two elements under position control (cable + strut). One cable under force control. The Inverse solution depends on the external forces acting on the triad Without external forces The continuations of the cables and the linear actuator intersect at the same point With an external force acting on the C.M. 1.The action line of the external force and the cable force is found. 2.This line and the continuations of the other two elements intersects The Caterpillar Model However, the triad can be subjected to other external forces: Ground contact and the forces applied at each triad by its neighboring triads. The calculation of the exact inverse kinematics in all of these cases is impractical.

11. Increasing magnitude of control force The Caterpillar Model This seemingly disadvantage makes this model suitable for simulating the caterpillar’s soft body. The triad responses to external forces and demonstrates a behavior that is referred to as a structural softness. The degree of "softness" can be by altering the tension in the force controlled cable.

12. 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.

13. Impedance Control Problem … When a triad touches the ground, it becomes rigid and loses its softness. Force control element Almost any movement of the position controlled elements will be impossible and will cause a simulation error. The control algorithm has to be very exact. It has to take into account all the external forces: gravity, ground contact, forces applied at each triad by its neighboring triads – virtually impractical.

14. Impedance Control The current control algorithm being used was evolved from the former principle and based on the concept of impedance control. General control low of 1 DOF impedance control : The control low being used in the model: The main difference between the two control lows. Its role is to maintain the triad in self-stress. Negative for the cables (tension) and positive for the actuator (compression). This control low is applied to all elements – cables & linear actuator The stiffness coefficient. Low K means softer triad and high K means a stiffer triad. Damping coefficient

15. Impedance Control Cables Struts Is negative for the cables (tension forces) Low K means soft triad. High K means a stiffer triad. Is positive for the cables (compression forces). Its magnitude is constant. equals zero Is the same for cables and struts. Its value does not change during simulation The wanted position

16. Adding Legs Problem The cables and the linear actuator exert forces that are primarily horizontal. While lifting the segment – a vertical movement – large forces have to be exerted. Solution Like in the biological caterpillar, the solution is to add legs that can be lifted vertically. In this case the segment itself doesn’t have to be lifted.

17. When legs are at the end of the segments : When legs are at the middle of the segments : Adding Legs

18. Caterpillar Area While simulating the caterpillar the following collapse happened : What can be learned from the biological caterpillar to prevent this collapse ?

19. Caterpillar Area Caterpillars have a constant volume. Keeping the segment area by adding the following moment to the joints : Segment area :

20. Motor Activity Normalized abdominal motor activity DN L - dorsal motor activity DN P - ventral motor activity VN L - proleg motor activity In each abdominal segment dorsal motor activity was recruited before ventral motor activity, which was followed by proleg motor activity. Johnston, R. M. and Levine, R. B. (1996). Crawling motor patterns induced by pilocarpine in isolated larval nerve cords of Manduca sexta. J. Neurophysiol. 76, 3178-3195.

21. Lin, H. T. and Trimmer B. A. (2010). The substrate as a skeleton: ground reaction forces from a soft-bodied legged animal. The Journal of Experimental Biology. 213, 1133-1142 Six phases were identified in a crawl cycle. In the first stage there was a progressive loss of contact points as the prolegs were lifted and the posterior part of the body shortened. In the subsequent stage the abdomen was stretched out by anchoring the anterior part of the body. Caterpillar Locomotion

22. Conclusions  The main property of the caterpillar, from our point of view, is its capability of being soft.  The model achieves this property by using Assur tensegrity structure and the new shape change algorithm.  The softness degree can be controlled during simulation.  One outcome of this model is the terrain-compatibility behavior.