1. A model of Caterpillar Locomotion Based on Assur Tensegrity Structures
Orki Omer
In the supervision of:
Prof. Shai Offer
School of Mechanical Engineering, Faculty of engineering, Tel-Aviv University.
Dr. Ben-Hanan Uri
Department of Mechanical Engineering, Ort Braude College, Karmiel.
With collabaration of:
Prof. Ayali Amir
Department of Zoology, Faculty of Life Sciences, Tel Aviv University.

2. Overview Biological background Previous work
Assur Tensegrity structure
The caterpillar model
Results and discussion

3. Biological background
Caterpillars do not have rigid segments. Instead they have soft body.
The internal pressure of the hemolymph within the body provides a hydrostatic skeleton.
Abdomen
Prolegs
Thorax
True legs
Head
Abdomen consists of ¾ of the body size. 5 pairs of prolegs.

4. Biological background
The biological caterpillar has a complex musculature.
Each abdominal body segment includes around 70 discrete muscles !!
The major abdominal muscles in each segment are:
The dorsal longitudinal muscle - DL1
The ventral longitudinal muscle - VL1
DL1
VL1
Antecostae
Eaton, J. L., Lepidopteran Anatomy. 1st edition, John Wiley, New York.

5. Biological background
Caterpillars have a relatively simple nervous system.
Yet, caterpillars are still able to perform a variety of complex movements.
Mechanical properties of the muscles are also responsible for some of the control tasks. (Woods et al., 2008)
Brain
Ganglions
Woods, W.A., Fusillo, S.J., and Trimmer, B.A., "Dynamic properties of a locomotory muscle of the tobacco hornworm Manduca sexta during strain cycling and simulated natural crawling". Journal of Experimental Biology, 211(6), March, pp

6. Biological background
The primary mode of caterpillar locomotion is crawling.
Crawling is based on a wave of muscular contractions that starts at the posterior end and progresses forward to the anterior.
During motion, at least three segments are in varying states of contraction.

7. Previous work
A caterpillar robot that is assembled using two types of modules:
Joint actuation modules
Adhesion modules.
Wang et al. (2008)
Rigid bodies
No softness
A computer simulation of a multi-body, caterpillar like, robot.
The robot was assembled using a series of actuated Stewart-platform.
Stulce (2002)
Wang, W., Wang, Y., Wang K., Zhang, H., and Zhang, J., “Analysis of the Kinematics of Module Climbing Caterpillar Robots”. Proceeding of 2008 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, pp
Stulce J. R., “Conceptual Design and Simulation of a Multibody Passive-Legged Crawling Vehicle”. PhD Thesis, Virginia Polytechnic Institute and State University, Department of Mechanical Engineering

8. Previous work Trimmer et al. (2006)
A caterpillar model using soft and deformable materials (silicone) and actuated using shape memory alloys wires (SMA).
The authors did not report results.
Conventional control Methods are ineffective
Near-infinite degrees of freedom
Trimmer B., Rogers C., Hake D., and Rogers D., “Caterpillar Locomotion: A New Model for Soft-Bodied Climbing and Burrowing Robots,” 7th International Symposium on Technology and the Mine Problem.

9. Tensegrity Tension + Integrity Definition
“ Islands of compression inside an ocean of tension ” (Fuller, 1975).
Air – Compressed element
Envelope – Tensioned element
Fuller, R. B., Synergetics—Explorations in the Geometry of Thinking. Macmillan Publishing Co.

10. Tensegrity Tension + Integrity Definition
“ A tensegrity system is a system in a stable self-equilibrated state comprising a discontinuous set of compressed components inside a continuum of tensioned components ” (Motro, 2003).
Struts – Compressed elements
Cables – Tensioned elements
Motro, R., Tensegrity: Structural Systems for the Future. Kogan Page Science.

11. Tensegrity structures are usually statically indeterminate structures
In Nature
(a) Human spine (b) Cytoskeleton
Tensegrity structures are usually statically indeterminate structures
Ingber, D.E., "The Architecture of life", Scientific American, 278(1), January ,pp

12. Previous work Rieffel et al. (2010)
A 15-strut, highly indeterminate tensegrity model, inspired by the caterpillar structure.
Using of Artificial Neurons Networks (ANNs) for control.
Locomotion does not resemble caterpillar crawling.
Conventional control methods are ineffective
Indeterminate structure
Rieffel J.A., Valero-Cuevas, F. J., and Lipson, H., "Morphological communication: exploiting coupled dynamics in a complex mechanical structure to achieve locomotion". Journal of the Royal Society interface, 7(45), April, pp

13. Motion is divided into many steps
Shape Change
Self-stress forces must be maintained during motion
Sultan and skeleton (2003)
The method is based on the identification of an equilibrium manifold.
Motion is divided into many steps
Van de Wijdeven and de Jager (2005)
The nodal positions of the tensegrity structure are found at every sub-shape by solving a constrained optimization problem.
Sultan C. and Skelton R. E., 2003 "Deployment of tensegrity structures“, International Journal of Solids and Structures, 40(18), September, pp. 4637–4657.
van de Wijdeven J. and de Jager B., 2005 "Shape change of tensegrity structures: design and control“, in American Control Conference.

14. √ X Assur Trusses Definition
An Assur truss is a determinate truss, in which applying an external force at any joint, results in forces in all the rods of the truss. √ X
Triad
Assur truss
Not an Assur truss

15. Assur Trusses In general - determinate trusses cannot have self-stress
Assur trusses have a configuration in which there is:
An infinitesimal mechanism
A single self-stress in all elements.
This configuration is termed singular configuration (Servatius et al., 2010) B C A O1 O2 O3 B C A O2 O3 O1
Singular configuration
Generic configuration
Servatius, B., Shai, O., and Whiteley, W.,2010. “Geometric properties of Assur graphs”. European Journal of Combinatorics, 31(4), May, pp

16. Assur Tensegrity structure is a statically determinate structure
Assur truss in a singular configuration can turn into tensegrity structure.
Tensioned elements
Compressed elements
Cables
Struts
Assur Tensegrity structure is a statically determinate structure

17. Shape change of Assur Tensegrity
it is possible to make any Assur truss configuration into a singular one simply by changing the length of any one of its ground elements
Shai O., "Topological Synthesis of All 2D Mechanisms through Assur Graphs" in Proceedings of the ASME Design Engineering Technical Conferences.

18. Shape change of Assur Tensegrity
The algorithm:
(Bronfeld, 2010)
Activate the device controllers.
One ground element is defined a the force-controlled element.
All other elements are position-controlled elements.
For the force-controlled element select a desired force in.
For the position controlled elements generate a desired trajectory.
Bronfeld A., 2010 "Shape change algorithm for a tensegrity device," M.S. thesis, Tel-Aviv University, Tel-Aviv, Israel.

19. Caterpillar model
Each caterpillar segment is represented by a 2D tensegrity triad consisting of two bars connected by two cables and a strut.
Cables
Strut
Bars
Only simulation, actuated elements

20. Caterpillar model
The complete model consists of eight segments connected in series.
Leg
Parameters of each segment:
Mass : (g)
Height : 5 (mm)
Length : 4.55 (mm) (in rest)

21. Biological caterpillar
Caterpillar model
Caterpillar Model
Biological caterpillar
Upper cable
DL1
Strut
Hydrostatic skeleton
Lower cable
VL1
Bar
Antecostae
VL1
DL1

22. Area conservation
The internal air cavity that can be emptied constitutes 3-10% of body volume (Lin et al., 2011) g
𝜑 𝐵
𝜑 𝑆
Bending
Shearing
Lin H. T., Slate D. J., Paetsch C. R., Dorfmann A. L. and Trimmer B. A., 2011 "Scaling of caterpillar body properties and its biomechanical implications for the use of hydrostatic skeleton“, The Journal of Experimental Biology, 214(7), April , pp

23. Area conservation
The internal air cavity that can be emptied constitutes 3-10% of body volume (Lin et al., 2011)
Internal toque:
𝑀 𝑖 =−𝑐∙ 𝜑 𝑆
The internal torque enables a small (but not negligible) shear angle
Lin H. T., Slate D. J., Paetsch C. R., Dorfmann A. L. and Trimmer B. A., 2011 "Scaling of caterpillar body properties and its biomechanical implications for the use of hydrostatic skeleton“, The Journal of Experimental Biology, 214(7), April , pp

24. Control Scheme Level 1 Central Control Brain Ground contact sensor
localized control
Ganglia
High level control
Low level control
Leg Controllers
Cable Controllers
Strut controllers
Leg behavior
Muscle behavior
Hydrostatic pressure

25. Low Level Control 𝑇 𝑐𝑎𝑏𝑙𝑒 =− 𝐹 0 2 +𝑘 𝑙− 𝑙 0 −𝑏𝑣
Cables Controller Muscles behavior
The biological caterpillar muscles have large, nonlinear, deformation range and display viscoelastic behavior (Woods et al., 2008)
𝑇 𝑐𝑎𝑏𝑙𝑒 =− 𝐹 𝑘 𝑙− 𝑙 0 −𝑏𝑣
Initial tension
Static term
Damping term
Output tension
Impedance Control:
𝐹 0 , 𝑘 𝑙 0 , 𝑏
𝑙 , 𝑣
𝑇 𝑐𝑎𝑏𝑙𝑒
Woods, W.A., Fusillo, S.J., and Trimmer, B.A., "Dynamic properties of a locomotory muscle of the tobacco hornworm Manduca sexta during strain cycling and simulated natural crawling". Journal of Experimental Biology, 211(6), March, pp

26. Low Level Control Cables Controller Muscles behavior
Nerve Stimulation
High Level Command
Stiff and shrunken cable
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Cable controller
𝑙 , 𝑣
𝑇 cable
High Level Command (0 - 1)
Soft and relaxed cable

27. Causes slower cable reaction
Low Level Control
Cables Controller Muscles behavior
The caterpillar muscles develop force slowly (Woods et al., 2008)
High Level Command (0 - 1)
Low pass filter
Causes slower cable reaction
Emphasize that the muscles of the adult fly are 4 times faster.
Cable controller
𝑙 , 𝑣
𝑇 cable
Woods, W.A., Fusillo, S.J., and Trimmer, B.A., "Dynamic properties of a locomotory muscle of the tobacco hornworm Manduca sexta during strain cycling and simulated natural crawling". Journal of Experimental Biology, 211(6), March, pp

28. Low Level Control 𝑇 𝑐𝑎𝑏𝑙𝑒 = 𝐹 0 2 +𝑘 𝑙− 𝑙 0 −𝑏𝑣 𝐹 𝑠𝑡𝑟𝑢𝑡 = 𝐹 0 − 𝑏𝑣
Strut Controller Internal pressure
The internal pressure in caterpillars is not isobarometric and the fluid pressure changes do not correlate well with movement (Lin et al., 2011)
𝑇 𝑐𝑎𝑏𝑙𝑒 = 𝐹 𝑘 𝑙− 𝑙 0 −𝑏𝑣
𝑲=𝟎 → 𝑭 𝒔𝒕𝒓𝒖𝒕 ≈𝒄𝒐𝒏𝒔𝒕.
𝐹 𝑠𝑡𝑟𝑢𝑡 = 𝐹 0 − 𝑏𝑣
Initial force
Damping term
Output force
Lin H.T., Slate D. J., Paetsch C. R., Dorfmann A. L. and Trimmer B. A., “Scaling of caterpillar body properties and its biomechanical implications for the use of hydrostatic skeleton”. The Journal of Experimental Biology, 214(7), April, pp

29. Shape change Without external forces
the cables assume exactly the virtual lengths :
𝑙 1 = 𝑙 0,1 , 𝑙 2 = 𝑙 0,2
describing the shape of the triad by 𝜑 𝑏 and 𝑙 𝑠 :
𝜑 𝑏 = sin −1 𝑙 0,1 − 𝑙 0,2 2ℎ , 𝑙 𝑠 = 𝑙 0,1 + 𝑙 0,2 2

30. Shape change Softness Axial force Bending torque Bending force 𝐹 𝑒 𝑙 𝑠
𝜑 𝐵
𝑀 𝑒
As long as the external load is within its limits, the initial force within the segment doesn’t influence segment stiffness
𝑙 𝑠 = 𝑙 0 − 𝐹 𝑒 2𝑘
𝑙 𝑠 = 𝑙 0
𝑙 𝑠 ≈ 𝑙 0
𝜑 𝐵 =0
𝜑 𝐵 ≈ 𝑀 𝑒 𝑘 ℎ 2
𝜑 𝐵 ≈ 𝑙 0 𝐹 𝑒 2𝑘 ℎ 2
𝐹 𝑒 < 𝐹 0
𝑀 𝑒 < 𝐹 0 ℎ 2
𝐹 𝑒 < 𝐹 0 ℎ 𝑙 0

31. Trigger when touching the ground
Low Level Control
Leg Controller Leg behavior
Caterpillar legs are used as support rather than levers.
When a leg touches the ground it cannot be lifted until it is actively unhooked and retracted.
(Lin and Trimmer, 2010)
Leg controller
Leg position
High Level Command (0 , 1)
Trigger when touching the ground
Ground contact sensor
Leg locking
Lin H. T. and Trimmer B. A., "The substrate as a skeleton: ground reaction forces from a soft-bodied legged animal“, The Journal of Experimental Biology, 213(7), April, pp

32. High Level Control Level 1 control Brain (CPG) coordinates motion
activate relevant segments in each step
Step 1
Step 2
Crawling direction
Step 3
Step 4
Anterior side
H H H H H H H H1
Posterior side
Step 5
Step 6
Step 7
Swing phase
Stance phase
Step 8
Step 9

33. High Level Control Level 1 control Brain (CPG)
A new stride starts before the previous stride ends.
The transition from step to step is triggered by the contact of the legs with the ground.
In each step, several segments are in various stages of contraction
Each stride issues the same set of step commands.
Anterior side
H H H H H H H H1
Posterior side
L L L L L L L L L0

34. High Level Control Level 2 control Ganglia
responsible for fitting motion to the terrain shape:
Mode I: Adjusting a segment in stance phase to the terrain shape
Mode II: Adjusting a segment in swing phase to the terrain shape
Mode I:

35. High Level Control Level 2 control Ganglia
responsible for fitting motion to the terrain shape:
Mode I: Adjusting a segment in stance phase to the terrain shape
Mode II: Adjusting a segment in swing phase to the terrain shape
Mode II:
0.7
0.4 + =
0.9
0.5
Level 1 commands
Stance commands
Output commands

36. Results
Crawling stages
Stage 1.
Stage 2.
Stage 3. 𝑳𝟎 𝑯𝟏 𝑯𝟒 𝑳𝟖

37. Results Segment length Time (s) Segment length (mm) Segment
Stance Length (mm)
Min length (mm)
Length change (%)
Stance time (s)
Swing time (s)
Step time (s)
Duty factor (%) H1
4.55
3.07 32
1.40
0.76
2.16 35 H2
4.64
3.08 34
1.26
0.90 41 H3
4.38
3.06 30
1.17
0.99 46 H4
4.42 31
1.15
1.01 47 H5
4.65
3.05
1.20
0.96 44 H6
4.24 28
1.10
1.06 49 H7
4.36
1.14
1.02 H8
4.35 29
0.97
1.19 55
Average
4.45

38. Biological caterpillar
Results
Crawling parameters
Stride 𝑖
Stride 𝑖+1
Crawling direction
Stride Length (mm)
Duration of one crawl (s)
Velocity
(mm/s)
Biological caterpillar
8.52
2.78
3.03
Model caterpillar
4.64
2.71
1.93

39. Biological caterpillar
Results
Dynamics
 Caterpillar Model
Biological caterpillar
Activation of a model cable
Tetanic stimulus of a caterpillar muscle
0.27 s
50% of peak force
0.26 s
0.41 s
80% of peak force
0.56 s
Woods, W.A., Fusillo, S.J., and Trimmer, B.A., "Dynamic properties of a locomotory muscle of the tobacco hornworm Manduca sexta during strain cycling and simulated natural crawling". Journal of Experimental Biology, 211(6), March, pp

40. The change of cable forces in H3 while crawling
Results
Dynamics
The change of cable forces in H3 while crawling
The maximum change in cable forces is only 13.8% relative to resting force

41. Results Area conservation
g is only 6.14%
Minimum:
157.3𝑚 𝑚 2
Maximum:
167.6𝑚 𝑚 2
Time
Caterpillar area (mm2)
The change in caterpillar area while crawling 6.14%
Internal pressure
The model was tested with various levels of internal pressure.
As long as 𝐹 0 is above a certain threshold, crawling is independent of the magnitude of internal pressure

42. Results
Different terrains

43. Discussion Stability Softness Simplicity
Assur tensegrity + Impedance control
Stability
self-stress of the tensegrity structure is always maintained.
Softness
Tensegrity structures have natural high compliance (softness).
Using impedance control, this degree of “softness” can be changed and controlled.
Simplicity
Statically determinate structure & Independent controllers creates
simple and intuitive shape change.

44. Discussion Assur tensegrity + Impedance control Soft robot Rigid robot

45. Discussion Internal Pressure Crawl Stages Simple nerve system
The model exhibits several characteristics which are analogous to those of the biological caterpillar:
Internal Pressure
During growth, body mass is increased 10,000-fold while internal pressure remains constant. In the same way, our model is able to maintain near constant internal forces regardless of size.
Crawl Stages
The model has demonstrated that effective crawling requires three different stages. Trimmer et al. found kinematic differences between three anatomic parts of the caterpillar.
Simple nerve system
Mechanical properties of the muscles are also responsible for some of the control tasks. Our model shows that using impedance control for each cable also simplifies the high level control.

46. Discussion Slow Muscle reaction Stride Duration Stride Length
The model exhibits several characteristics which are analogous to those of the biological caterpillar:
Slow Muscle reaction
The caterpillar muscle develop force slowly. Our model show that adding the LP filter, which makes the cable to react slower, ease the high level control and makes the motion smoother
Stride Duration
The duration of one stride are comparable in both the model and the biological caterpillar.
Stride Length
There is a discrepancy between the stride length of the model and that of the biological caterpillar.

47. Discussion Locomotion energy Ground reactions
The model suggests some characteristics of the biological caterpillar:
Locomotion energy
The model shows that cables’ force doesn’t significantly change while the caterpillar is in motion. It suggests that Caterpillars don’t invest considerably more energy while crawling than while resting.
Ground reactions
The model schedule the motion using signals from the legs when they touch the ground. It suggests that the biological caterpillar also uses ground reaction to coordinate its movements.

48. Future Research Future research can be made in three directions:
Improving the existing model
Expanding the model to a three dimensional model.
Building a mechanical model.

49. Thank You!

50. Choosing triad configuration
Δ𝑈=− 𝑙 0 𝑙 𝑓 𝐹 𝑙 𝑑𝑙
potential energy of the system:
If the system is in stable equilibrium, the potential energy function is at its minimum point.
Therefore,
to get a stable system, any shift from equilibrium must result in
cable lengthening and\or strut shortening.
Two options:

51. Choosing triad configuration
4 bar mechanism
(In rigid lines) 𝑪 𝑩 𝑨
Tested element
(In dashed line) 𝜔 𝑶𝟏 𝑶𝟑 𝑶𝟐

52. Choosing triad configuration𝑪 𝑩 𝑨 𝜔 𝑶𝟏 𝑶𝟑 𝑶𝟐