IEEE Robotics & Automation Magazine - September 2016 - 81

presented in [22], although most similar to ours, is still a
very different type of actuator and does not provide
dynamic performance for joint position tracking but
instead uses a learning algorithm to attain desired positions in a plane. Therefore, we compare our recent
advances against our own past work described in [10]. To
differentiate, we will refer to the older controller as a twostate MPC controller and to the new controller presented in
this article as a four-state torque-based MPC controller.
By including pressures in the actuation chambers as
states, we expected the performance of our soft robot
control to improve when compared with our past work.
We now describe at a high level the formulation of the
new model predictive controller and then present the
results of this comparison.
The Controller Formulation
The model predictive controller solves an optimization by
predicting states over a time horizon that is T steps long while
varying the pressure inputs to produce a trajectory resulting
in the smallest cost subject to constraints. The discretized
matrices Ad and Bd; the current states qo [k], q [k], P0 [k], and
P1 [k]; the previous inputs P0,d [k - 1] and P1,d [k - 1]; the
final goal joint angle qgoal; and the model constraints and
weights are fed into an MPC solver at every time step. The
cost function minimized across the horizon T is

k =0

+ P0 [k] - PT

2
Q

+ qo [k]
2
S

2
R

+ P1 [k] - PT 2S ,

P 0,d , P 1,d

(8)

subject to the system model (7) as a constraint, as well as the
following additional constraints:
q $ q min ,
q # q max ,
Pmin # Pd # Pmax ,
DPd # DPmax ,

The Results for a Single DoF
In our preliminary modeling and control design, we focused
on results for the single-DoF platform or grub. To compare
performance, we commanded a series of 30°
step-angle commands The soft robot platform
ranging from −60° to
6 0 ° , c h a n g i n g t h e is capable of repeatedly
command at time increments of 10 s. The resul- picking up an unknown
tant q of both controllers
and commanded qgoal object from the same
are plotted over time in
Figure 8.
location and placing it
Compared to the
previous two-state con- at a desired location.
troller, utilizing press u re s t ate s i n t h e
four-state torque-based MPC controller significantly
improved overall performance. As described in Table 1,

(9)
(10)
(11)
(12)

where Q, R, and S are manually tuned scalar weights, PT is a
target pressure, qmin and qmax are joint limits, Pmin and Pmax are
bladder pressure limits, DPd is the change in desired pressure
from the previous time step, and DPmax is the maximum change
allowed in the desired pressure per time step. While simplified
pressure dynamics are included in our model, the slew rate
constraint on commanded pressure in (12) serves to prevent
valve chatter.
We generated a solver for the MPC problem using CVXGEN [12], a web-based tool for developing convex optimization solvers. The optimization solver written in C and
subsequent Python code that calls the solver can be run at up
to 300 Hz, predicting a trajectory horizon of T = 20 time steps
(or 0.067 s) into the future. Once solved, the first time step
from the optimized trajectory of desired pressures is made
available over ROS. These desired pressures are received by
the underlying pressure PID controller, and valve position

q,q

CMD0, CMD1

Pressure
Controller

MPC
Controller

q goal

Plant

P 0, P 1

q,q

Joint-Angle
Estimation

Figure 7. An MPC control diagram for joint control.

Step-Angle Tracking

80
60
40
Angle (°)

minimize / q goal - q [k]

commands are then sent to the individual valves. As shown in
Figure 7, the current pressure states are measured and fed
back into the pressure controller, while both the current estimated joint angle states and pressure states are fed into the
MPC controller.

20
0
-20
θ goal
θ Two-State MPC
θ Four-State MPC

-40
-60
-80

40 50 60
Time (s)

Figure 8. The controller comparison results for a single joint.

September 2016

IEEE ROBOTICS & AUTOMATION MAGAZINE

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - September 2016