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.
Overview
» Biological background
» Previous work
» Assur Tensegrity structure
» The caterpillar model
» Results and discussion
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.
Head
Thorax
True legs
Abdomen
Prolegs
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:
1. The dorsal longitudinal muscle - DL1
2. The ventral longitudinal muscle - VL1
DL1
VL1
Antecostae
Eaton, J. L., 1988. Lepidopteran Anatomy. 1st edition, John Wiley, New York.
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., 2008. "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. 873-82.
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.
Previous work
Wang et al. (2008)
A caterpillar robot that is assembled using
two types of modules:
1. Joint actuation modules
2. Adhesion modules.
Rigid bodies
No softness
Stulce (2002)
A computer simulation of a multi-body, caterpillar like, robot.
The robot was assembled using a series of actuated Stewart-platform.
Wang, W., Wang, Y., Wang K., Zhang, H., and Zhang, J., 2008. “Analysis of the Kinematics of Module Climbing
Caterpillar Robots”. Proceeding of 2008 IEEE/ASME International Conference on Advanced Intelligent Mechatronics,
pp. 84-89.
Stulce J. R., 2002. “Conceptual Design and Simulation of a Multibody Passive-Legged Crawling Vehicle”. PhD Thesis,
Virginia Polytechnic Institute and State University, Department of Mechanical Engineering
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.
Near-infinite
degrees of freedom
Conventional control
Methods are ineffective
Trimmer B., Rogers C., Hake D., and Rogers D., 2006. “Caterpillar Locomotion: A New Model for Soft-Bodied
Climbing and Burrowing Robots,” 7th International Symposium on Technology and the Mine Problem.
Tensegrity
Tension + Integrity
Definition
“ Islands of compression inside an ocean of tension ” (Fuller, 1975).
Air – Compressed element
Envelope – Tensioned element
Fuller, R. B., 1975. Synergetics—Explorations in the Geometry of Thinking. Macmillan Publishing Co.
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., 2003. Tensegrity: Structural Systems for the Future. Kogan Page Science.
Tensegrity
In Nature
(a) Human spine
(b) Cytoskeleton
Tensegrity structures are usually statically
indeterminate structures
Ingber, D.E., 1998. "The Architecture of life", Scientific American, 278(1), January ,pp. 48-57.
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.
Indeterminate
structure
Conventional control
methods are ineffective
Rieffel J.A., Valero-Cuevas, F. J., and Lipson, H., 2010. "Morphological communication: exploiting coupled dynamics in
a complex mechanical structure to achieve locomotion". Journal of the Royal Society interface, 7(45), April, pp.613-621.
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.
Equilibrium manifold
Final configuration
Equilibrium path
Initial configuration
(a)
(b)
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.
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
Assur truss
Not an Assur truss
Assur Trusses
 In general - determinate trusses cannot have self-stress
Assur trusses have a configuration in which there is:
1. An infinitesimal mechanism
2. A single self-stress in all elements.
This configuration is termed singular configuration (Servatius et al., 2010)
C
B
C
A
O1
B
A
O2
O3
Singular configuration
O1
O2
O3
Generic configuration
Servatius, B., Shai, O., and Whiteley, W.,2010. “Geometric properties of Assur graphs”. European Journal of
Combinatorics, 31(4), May, pp. 1105-1120.
Assur Tensegrity
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
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., 2010. "Topological Synthesis of All 2D Mechanisms through Assur Graphs" in Proceedings of
the ASME Design Engineering Technical Conferences.
Shape change of Assur Tensegrity
The algorithm:
(Bronfeld, 2010)
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.
Activate the device controllers.
Bronfeld A., 2010 "Shape change algorithm for a tensegrity device," M.S. thesis, Tel-Aviv University, TelAviv, Israel.
Caterpillar model
Each caterpillar segment is represented by a 2D tensegrity triad
consisting of two bars connected by two cables and a strut.
Bars
Strut
Cables
Caterpillar model
The complete model consists of eight segments connected in series.
Leg
Parameters of each segment:
Mass : 0.182 (g)
Height : 5 (mm)
Length : 4.55 (mm) (in rest)
Caterpillar model
Caterpillar Model
Upper cable
Strut
Lower cable
Bar
Biological caterpillar
DL1
Hydrostatic skeleton
VL1
Antecostae
DL1
VL1
Area conservation
The internal air cavity that can be emptied constitutes 3-10% of body
volume (Lin et al., 2011)
g
g
𝜑𝐵
Bending
𝜑𝑆
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. 1194-1204.
Area conservation
The internal air cavity that can be emptied constitutes 3-10% of body
volume (Lin et al., 2011)
Internal toque:
𝑀𝑖 = −𝑐 ∙ 𝜑𝑆
Ф1
Ф2
M
M
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. 1194-1204.
Ground contact sensor
Control Scheme
Level 1
Central Control
Level 2
localized control
Brain
Ganglia
Leg
Controllers
Cable
Controllers
Strut
controllers
Leg
behavior
Muscle
behavior
Hydrostatic
pressure
High level
control
Low level
control
Low Level Control
Cables Controller
Muscles behavior
The biological caterpillar muscles have large, nonlinear, deformation
range and display viscoelastic behavior (Woods et al., 2008)
Impedance Control:
𝑇𝑐𝑎𝑏𝑙𝑒
𝐹0
= − + 𝑘 𝑙 − 𝑙0 − 𝑏𝑣
2
Output tension
Static term
Initial tension
Damping term
𝑙 ,𝑣
𝐹0 , 𝑘
𝑙0 , 𝑏
𝑇𝑐𝑎𝑏𝑙𝑒
Woods, W.A., Fusillo, S.J., and Trimmer, B.A., 2008. "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. 873-82.
Low Level Control
High Level
Command
Muscles behavior
Nerve
Stimulation
𝑙 ,𝑣
Cable
controller
𝑇cable
Stiff and shrunken cable
High Level Command (0 - 1)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Soft and relaxed cable
Cables Controller
Low Level Control
Cables Controller
Muscles behavior
The caterpillar muscles develop force slowly (Woods et al., 2008)
High Level Command (0 - 1)
Causes slower
cable reaction
Low pass
filter
𝑙 ,𝑣
Cable
controller
𝑇cable
Woods, W.A., Fusillo, S.J., and Trimmer, B.A., 2008. "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. 873-82.
Low Level Control
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)
𝐹0
𝑲 = 𝟎= → + 𝑭𝑘𝒔𝒕𝒓𝒖𝒕
𝑇𝑐𝑎𝑏𝑙𝑒
𝑙 −≈𝑙0𝒄𝒐𝒏𝒔𝒕.
− 𝑏𝑣
2
𝐹𝑠𝑡𝑟𝑢𝑡 = 𝐹0 − 𝑏𝑣
Output force
Damping term
Initial force
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. 1194-1204.
Shape change
 Without external forces
the cables assume exactly the virtual lengths :
𝑙1 = 𝑙0,1 ,
𝑙2 = 𝑙0,2
describing the shape of the triad by 𝜑𝑏 and 𝑙𝑠 :
𝜑𝑏 = sin−1
𝑙0,1 − 𝑙0,2
2ℎ
, 𝑙𝑠 =
𝑙0,1 + 𝑙0,2
2
Shape change
 Softness
Axial force
𝐹𝑒
Bending torque
𝜑𝐵
𝑙𝑠
Bending force
𝜑𝐵
𝑙𝑠
𝑙𝑠
𝐹𝑒 force within the
As long as the external load𝑀
is𝑒within its limits, the initial
segment doesn’t influence segment stiffness
𝐹𝑒
𝑙𝑠 = 𝑙0 −
2𝑘
𝑙𝑠 = 𝑙0
𝑙𝑠 ≈ 𝑙0
𝜑𝐵 = 0
𝑀𝑒
𝜑𝐵 ≈ 2
𝑘ℎ
𝑙0 𝐹𝑒
𝜑𝐵 ≈
2𝑘ℎ2
𝐹𝑒 < 𝐹0
𝐹0 ℎ
𝑀𝑒 <
2
𝐹0 ℎ
𝐹𝑒 <
𝑙0
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)
Trigger when touching the ground
High Level Command (0 , 1)
Ground contact sensor
Leg
controller
Leg position
Leg locking
Lin H. T. and Trimmer B. A., 2010. "The substrate as a skeleton: ground reaction forces from a softbodied legged animal“, The Journal of Experimental Biology, 213(7), April, pp. 1133-1142.
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
H8
H7
H6
H5
H4
H3
H2
H1
Posterior
side
Step 5
Step 6
Step 7
Swing phase
Stance phase
Step 8
Step 9
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.
H8
Anterior
side
L8
H7
L7
H6
L6
H5
L5
H4
L4
H3
L3
H2
L2
H1
L1
Posterior
side
L0
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:
+
+
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:
Level 1
commands
0.7
Stance
commands
0
+
0.7
Output
commands
0.5
=
0.4
0.9
Results
 Crawling stages
𝑯𝟒
𝑯𝟏
𝑳𝟖
𝑳𝟎
Stage 1.
Stage 3.
Stage 2.
Results
 Segment length
H1
H2
H1
5
H5
4
3
4
3
5
6
7
8
9
10
11
12
13
14
15
5
6
7
8
9
H3
H2
H3
11
12
13
14
15
11
12
13
14
15
11
12
13
14
15
11
12
13
14
15
5
H6
4
3
4
3
5
6
7
8
9
10
11
12
13
14
15
5
6
7
8
9
H5
10
H6
5
5
H7
4
3
4
3
5
6
7
8
9
10
11
12
13
14
15
5
6
7
8
9
H7
10
H8
5
H4
10
H4
5
Segment length (mm)
Segment length (mm)
5
5
H8
4
3
4
3
5
6
7
8
9
10
11
12
13
14
15
5
6
7
8
9
10
Time (s)
Time (s)
Segment
H1
H2
H3
H4
H5
H6
H7
H8
Average
Stance
Length
(mm)
4.55
4.64
4.38
4.42
4.65
4.24
4.36
4.35
4.45
Min length Length
(mm)
change (%)
3.07
3.08
3.06
3.06
3.05
3.05
3.06
3.07
3.06
32
34
30
31
34
28
30
29
31
Stance
time (s)
1.40
1.26
1.17
1.15
1.20
1.10
1.14
0.97
1.17
Swing time Step time
(s)
(s)
0.76
0.90
0.99
1.01
0.96
1.06
1.02
1.19
0.99
2.16
2.16
2.16
2.16
2.16
2.16
2.16
2.16
2.16
Duty
factor (%)
35
41
46
47
44
49
47
55
46
Results
 Crawling parameters
Stride
End of
stride 𝑖i +
(In red square)
C.M. position - Y axis (cm)
6
1
Start of phase 3
of the i th stride
Stride 𝑖
5
4
3
C.M position
Start of phase 1
Start of phase 2
Start of phase 3
End of stride
2
1
0
-50
-48
Start of phase
2
Crawling
of the i+1 th stride
(In blue circle)
-46
Start of phase 1
direction
of the i+1 th stride
-44
-42
-40
C.M. positoin - X axis (cm)
Biological
caterpillar
Model
caterpillar
Stride Length
(mm)
Duration of one
crawl (s)
Velocity
(mm/s)
8.52
2.78
3.03
4.64
2.71
1.93
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., 2008. "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. 873-82.
Results
 Dynamics
The change of cable forces in H3 while crawling
Cables' Force (mN)
70
60
50
40
30
Cable 1
Cable 2
Resting Force
Min/Max Forces
20
10
0
5
6
7
8
Maximum change of cable
forces relative to the resting
force is 13.8%
9
10
11
12
13
14
15
Time (s)
The maximum change in cable forces is only 13.8% relative to
resting force
Results
 Area conservation
2)
2)
Caterpillar area
area (mm
(mm
Caterpillar
200
150
MaxMaximum:
Area = 167.6 mm 2
Min area
= 157.3 mm2
Minimum:
The change in caterpillar
area while crawling 6.14%
The change in caterpillar area is 6.14%
50
0
167.6𝑚𝑚2
157.3𝑚𝑚2
100
g is only 6.14%
5
10
Time
Time
(s)
15
20
 Internal pressure
The model was tested with various levels of internal pressure.
As long as 𝐹0 is above a certain threshold, crawling is independent of
the magnitude of internal pressure
Results
 Different terrains
Discussion
 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.
Discussion
 Assur tensegrity + Impedance control
Soft robot
Rigid robot
Assur tensegrity
Discussion
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.
Discussion
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.
Discussion
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.
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.
Choosing triad configuration
potential energy of the system:
Δ𝑈 = −
𝑙𝑓
𝐹 𝑙 𝑑𝑙
𝑙0
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:
(a)
(b)
Choosing triad configuration
4 bar mechanism
(In rigid lines)
Tested element
(In dashed line)
𝟏
real
𝑐
1, 2,
𝑐
𝑣
critical
𝑣
𝑐
𝑣𝐵 =
2
𝐵
𝐵
𝑣
𝐵
𝐵
𝒄
𝒄𝒏
Choosing triad configuration
1, 2,
critical
real
𝐵
𝐵
𝟏
lengthen
𝐵
shorten