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.
-
The main idea.
Tensegrity.
Assur Graph (Group).
Singularity+ Assur Graph+tensegrity= Assur
Tensegrity
- Impedance control
- Assur tensegrity+ Impedance control
- Further applications.
Tensegrity= Tension + Integrity
Tensegrity structures are usually statically
indeterminate structures
Tensegrity
Assur
Graph
Animal/CaterpillarSoft and rigid robot
Singularity
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.
Example of a determinate truss that is
NOT an Assur Group.
Removing this joint results in
Determinate truss with the same mobility
Example of a determinate truss that is an Assur Group – Triad.
TRIAD
We remove this joint
And it becomes a mechanism
The MAP of all Assur Graphs in 2d
is complete and sound.
Singularity and Mobility Theorem
in
Assur Graphs
First, let us define:
1. Self-stress.
2. Extended Grubler’s equation.
1
5
1
P
6
6
4
2
2
4
P
3
(a)
3
(b)
Self Stress – A set of forces in the links (internal
forces) that satisfy the equilibrium of forces around
each joint.
Extended Grubler’s equation =
Grubler’s equation + No. self-stresses
1
A
inf
DOF = 0
DOF = 0 + 1 = 1
2
DOF = 0 + 2 = 2
The joint can move infinitesimal motion. Where is the
other mobility?
All the three joints move together.
Extended Grubler = 2 = 0 + 2
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.
ASSUR GRAPHS IN SINGULAR POSITIONS
3
1
5
6
2
4
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.
1
A
inf
2
ONLY Assur Graphs have this property!!!
B
A
A
B
NO SS in All links.
Joint A is not mobile.
A
B
A
B
2 DOF (instead of 1) and
SS in All the links, but
Joint A is not mobile.
2 SS (instead of 1).
Assur Graph at the singular position 
There is a unique self-stress in all the links 
Check the possibility:
tension  cables.
compression  struts.
Combining the Assur triad with a tensegrity structure
4
A
1
5
6
4
A
C
3
1
4
C
5
A
1
6
B
B
2
2
C
3
5
3
6
B
2
(a)
(b)
(c)
A
A
Changing the singular
point in the triad
4
C
4
C
1
5
1
5
6
3
B
B
2
2
(a)
(b)
6
3
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.
Transforming a soft (loose) structure into Rigid Structure
Shortening the length of one of the cables
Shortening the length of one of the cables
Almost Rigid Structure
Almost Rigid Structure
At the Singular Position
The structure is Rigid
Singular point
Impedance control
The general control low
Output force
𝐹 = 𝐹0 + 𝑘 𝑙 − 𝑙0 − 𝑏𝑣
Damping term
Virtual force. Maintain the triad in selfstatic stress.
relation between output force and input
displacement
Assur tensegrity + Impedance control
Advantages :
» Stability
The self-stress of the Assur Tensegrity is always maintained, and the
structure stays in a singular configuration.
» 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 − 𝑘
Caterpillar model
The model consists of triads connected in series
Cable
Strut
Bar
Leg
Ground contact sensor
Control Scheme
Level 1
Central Control
Level 2
localized control
Leg
Controllers
Cable
Controllers
Muscle
behavior
CPG - Central
Pattern Generator
Ganglions
Strut
controllers
Hydrostatic
pressure
High level
control
Low level
control
Results
Conclusions
» 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.
All the details of this work will appear in October 2011 in Orki’s thesis
(in English).