IEEE Robotics & Automation Magazine - September 2016 - 79

system. Deviations from this approximate model will degrade
the overall system performance, but we currently treat them
as disturbances in the system. Although approximate, the
results of MPC for the models presented here show that this
rigid-body, decoupled model is sufficient for basic control of
the system.

τ 1(p 1,q )

Link Dynamics
The differential equation that we use to describe the motion
of a single link is that of an inverted pendulum:
(1)

where q, qo , and qp are the joint angle, velocity, and
acceleration, respectively; I is the moment of inertia of the
link about its joint; Kd is a damping coefficient; m is the
mass of the link; g is the gravity constant; L is the length of
the link; and τa is the actuation torque applied. Initial
system identification for the grub found that gravity
effects were three orders of magnitude less than pressure
effects, which include damping and stiffness. This led to
making the assumption that gravity terms are negligible
due to the low mass of the linkages, which reduced (1) to
Iqp + K d qo = x a .

(a)
120
100
80
Force (N)

Iqp + K d qo + mg L sin (q) = x a ,
2

τ 0(p 0,q )

60
-60°
-45°
-30°
-15°
0°
30°
60°

40
20

(2)
0

Adding a more significant link-side load to a single joint or
controlling a multijoint robot more accurately will require
reexamining this approximation. An important improvement
that we present in this article over our past work is a model
relating the actuation chamber pressure and the resultant
torque applied on the link.
The Link Torque Model
Our new torque model is developed with the idea that each
bladder produces an independent torque on a joint that is a
function of the pressure in the bladder and the current
angle of the arm [Figure 5(a)]. The difference of these
torques produces the total torque τa that is seen by the
links. To test this model and find the form of the torque
functions, we first measured the torque produced by a single bladder. For the test, we kept the base link of the grub
fixed while we filled a given bladder to different pressures
and measured the resultant force. Figure 5(a) shows a picture of the grub in the test rig and the force sensor, a setup
where x 1 was being characterized.
After taking the initial data, we realized that there was
another torque term that we had not taken into consideration.
This was the torque due to the stiffness of the fabric joint and
internal body bladder. We measured this stiffness using the
same setup as in Figure 5(a), but we left the actuation bladders empty and measured the passive torque output of the
link at different angles. The results suggested that the torque
from the joint stiffness (τstiffness) can be approximated as a
linear function of the joint angle, with its x and y intercept at
the origin.

0

20

40
60
80
Pressure (kPa-Gauge)
(b)

100

Figure 5. The torque model verification testing: (a) The testing
setup with a force sensor mounted at 0°. (b) The actuator force
at different pressures and angles.

We then reevaluated the data for the actuator while subtracting the force due to the joint stiffness and produced the
results shown in Figure 5(b). Once again, the results suggested a linear relationship, but this time between force and pressure while being essentially independent of angle. It is
important to note that while force was the value we measured,
the relationship between the force output measured and the
actual torque output of the system is just a scalar multiplier,
which is the moment arm. The final description for the new
torque models is shown in the following equations, where (3)
is the torque due to the actuators, (4) is the torque due to the
joint stiffness, and (5) is the full torque seen by the links.
= c 0 P0 - c 1 P1,
= K s q,
x a = x 0 - x 1 + x stiffness .

x0 - x1
x stiffness

(3)
(4)
(5)

We initially found values for γi in (3) by using the data in
Figure 5(b). In this case, γi was the slope of the line in this figure multiplied by a scalar. The scalar was the moment arm
between the joint and the force sensor shown in Figure 5(a),
and it converts the units on the slope from force (newtons) to
torque (newton meters). Ks in (4) was found a similar way
September 2016

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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