IEEE Robotics & Automation Magazine - September 2011 - 97

incorporate tensional elements in their muscleskeleton system such that they maintain the structure
integrity while acting it, storing, and distributing energy [9].
The purpose of this article is to demonstrate the feasibility of constructing robots using tensegrity structures.
Mobile robotics is our long-term objective, but first, to
understand how these structures can be actuated and
sensed, we face the problem of manipulators, i.e., with a
fixed base. Thus, we present the implementation, including design, and construction details of a robot based
on a three-bar prismlike tensegrity in a three-dimensional (3-D) minimal configuration. The designed mechanism is capable of following any desired trajectory inside
its workspace.
Robot Description
A schematic of the designed robot is given in Figure 1,
representing a minimal tensegrity configuration in a stable
equilibrium. In this context, minimal means that the number of cables used to link the rigid elements is the minimum required to give stability (in terms of rigidity) to the
structure in the 3-D space.
A fixed reference frame is considered to be located at
node n1 being the y axis oriented toward node n2 , as
depicted in the figure. The three lower nodes labeled as
n1 , n2 , n3 have been fixed to the ground, thus eliminating
possible rigid displacements of the whole structure in the
space. Therefore, being a the side of the lower base triangle,
the vectors of the position for these nodes with respect to
the chosen reference frame can be stated as
p1 ¼ (0, 0, 0)T
p2 ¼ (0, a, 0)T
p3 ¼ (Àsin(60 Ã pi=180) Ã a, cos(60 Ã pi=180) Ã a, 0)T :

(1)

the angle of the bar projection onto the xy plane with
respect to the x axis, ai .
0

1
cosai cosbi
bi ¼ @ sinai cosbi A:
sinbi

We have chosen to use springs instead of cables so as to
ensure the tensile elements of the structure passively adapt to
the required length when changing the bar lengths. In addition, we ensure that the
structure always stays in a *
minimum energy configThe implemented robot is
uration once the actuators
are locked. Hence, six
basically composed of three
springs are needed c4 -c9
that, respectively, link
actuated bars and nine
nodes n1 to n4 , n2 to n5 ,
and n3 to n6 for the vertipassive strings in the form
cal springs and n4 to n5 ,
n5 to n6 , and n6 to n4 for
of springs.
the horizontal springs
on the upper triangular *
platform. These springs
are considered to be massless and linear elastic with the
stiffness constant k equal for all of them and rest length lc0 .
The stiffness of the bars is assumed to be infinite with
respect to that of the springs.
The reference configuration for our robot corresponds
to, using a ¼ 57 cm and lc0 ¼ 38 cm, an elongation of
the bars of 67 cm. However, control inputs can be generated from lb 2 ½55, 92 cm. Note that for the range
lb 2 ½55, 67) cm, the robot is not a valid tensegrity since
the springs are not in tension. This range of control inputs
should not be considered here. Nevertheless, it is worth

Edges between the three lower nodes have been eliminated since they have no function at all. Upper nodes are
linked to the lower ones by three actuated bars that may
vary their lengths; in fact, this can be mathematically seen
as a strut with an upper elongation limit. Bar b1 links
nodes n2 and n4 , bar b2 nodes n3 and n5 , and bar b3 links
node n1 to n6 . Note that the ordering of bars and nodes
may be arbitrarily chosen provided there are correct connections between them. The position of the upper nodes
can, hence, be expressed as

n4
c7

n1
y

where lbi denotes the current longitude of bar bi and bi a
unit vector on the direction of the ith bar. This could
alternatively be expressed using two rotation angles for
each bar, pitch and yaw, which, respectively, represent
the angle of the bar with respect to the xy plane, bi , and

p5 ¼ p3 þ lb2 b2
(2)

p4 ¼ p2 þ lb1 b1
p6 ¼ p1 þ lb3 b3 ,

(3)

x
Figure 1. Minimal tensegrity in a stable configuration,
representing the designed robot. Cables are represented by thin
red lines, while bars are represented by the black lines. The
green lines signify the axes.

SEPTEMBER 2011

IEEE ROBOTICS & AUTOMATION MAGAZINE

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - September 2011