IEEE Robotics & Automation Magazine - September 2011 - 99

nb
nc
X
1 X
U¼ k
(lcj À lcj0 )2 þ
mghj ,
2 j¼1
j¼1

where lc and lb are the lengths for cables and bars,
respectively; m is the mass of the bars, considered equal
for all of them; g is the gravitational constant; and
hj ¼ 12 lbj sinbj is the height of the bar's center of mass.
Note the dependency of the lc on the i, j nodes coordinates lc2 ¼ (pj À pi )T (pj À pi ).
At first glance, one may believe in getting proper equations by taking derivatives of the potential energy with
respect to the considered generalized coordinates (whether
the unitary vectors of the bars or the angles ai , bi ); however, this generally is not enough since we have to observe
some constraints to be at a tensegrity configuration. These
constraints are mainly given by maintaining the springs in tension and
complying (4).
Now, for the direct kinematic
problem, we provide u as well as the
external forces and solve an optimization problem. This way, if we discretize the actuator space, the workspace
of our robot can be computed. Figures 2 and 3 show the possible positions for the upper nodes as well
as the workspace for the proposed
robot considering the output as the
center of mass of the upper platform.
Bars were allowed to change length
between lb ¼ 67 cm and lb ¼ 92 cm
with 5 mm increments.
Implementation
The implemented robot is basically
composed of three actuated bars and
nine passive strings in the form of
springs. It was necessary to design an
electronic motion control unit for
the actuators as well as a force sensor
acquisition system. In this section,
we analyze three basic items of the
robot implementation: previous simulation, design of the mechanic parts,
and control hardware.
Simulation
As a previous step to the physical
implementation of the robot, it is necessary to simulate the full structure
with the aim of estimating its key
parameters: the length of the bars,
motor torque required to perform a
given bar elongation, and springs
stiffness constant and rest lengths.

A simulation starting point is a stable configuration for
the three-bar tensegrity structure. In this case, the six-node
coordinates were obtained by solving a form-finding
process [4] considering a bar length of 55 cm. Once the
coordinates of the nodes were obtained, a model was built
using Adams, a physical simulation program of mechanical assemblies. A screen shot of the complete robot is
shown in Figure 4.
The form-finding process did not take into account
the gravity field, so at the beginning, the structure is not
in a well-balanced configuration. The first performed
simulation was allowing the structure to freely oscillate
until it achieved the equilibrium state. This is shown in
Figure 5, where the force exerted on the three springs
entering a node during the transitory mode is also
shown. The final equilibrium state gives us enough

(a)

(b)

Figure 4. Preview of the prototype structure using a simulation software: (a) front view
and (b) side view.

(a)

(b)

Figure 5. (a) Transient simulation before arriving to equilibrium configuration and
(b) real force exerted on the three springs linking each node.

SEPTEMBER 2011

IEEE ROBOTICS & AUTOMATION MAGAZINE

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