IEEE Robotics & Automation Magazine - December 2015 - 46

allow for the fast adjustment of the damping behavior ranging
from low damping, as provided by the pure mechanical friction
in the joints, up to the practically highest damping, i.e., (over)
critical damping in the
case of emergency stops.
A vibration damping
The use of an active control
controller has to meet
several requirements. The
system is beneficial as the
dynamic VSA robot
model is highly nonlinear,
powerful actuators allow
conditioned by the inertial properties and the VS
for the fast adjustment of
elastic elements, resulting
in large parameter variathe damping behavior.
tions (such as in the
eigenfrequencies). Furthermore, there are several practical restrictions in controller
realization due to the technical implementation. Dominating
factors include actuator saturation, transmission delays, measurement noise, and real-time computation constraints. Main
consequences are practical upper and lower limits on controller gains, which are very challenging from the controller
design perspective.
Nonlinear controllers, which can cope with the VSA plant
in a mathematically correct manner, include feedback linearization and backstepping. Both approaches rely heavily on the
robot model, where small deviations may lead to large control
errors. Furthermore, the control action is computed by shaping or decoupling the differential equations of the model
through the torque inputs. Therefore, it is required to derive
sensor signals multiple times, which is practically difficult due
to sensor quantization and noise. A further possibility is linear
parameter varying controllers, which are, however, limited to
small parameter variations [43].
Feedback linearization uses a local diffeomorphism to
transform the nonlinear system into a linear dynamics [44].
The control of the linear dynamics is a standard issue in control theory. However, it is well known that the feedback linearization approach is very sensitive regarding modeling
errors. Furthermore, the resulting total feedback gains vary
significantly, and the given gain limitations of the system are
hard to meet. Backstepping controllers can be better

B
Ki

K
q

Figure 10. The sketch of the simplified model of one joint. Controlled,
active elements are depicted in dashed green line and mechanical
elements are depicted in solid black line. The controller acts to adjust the
joint position via K i and add the desired damping properties D i and D q .

IEEE ROBOTICS & AUTOMATION MAGAZINE

DECEMBER 2015

constrained even for higher plant parameter variations, but
the cascaded structure restricts to use link-side coordinates
on the highest control level. The elastic elements in VSAs,
nevertheless, also depend on the motor coordinates (1).
Another choice is gain scheduling controllers, which use a
linearization of the nonlinear dynamics obtained at a high
rate (typically above 1 kHz) [45], [46]. The benefit is that the
controller gains can be adjusted to the rapidly changing plant
parameters. Furthermore, the use of high state derivatives
can be avoided using a state feedback structure. Although
widely applied in research and industrial applications, the
stability of these controllers can only be guaranteed for limited parameter rates. Both the types of controllers have been
tested on the DLR HASy. For a comparison, please refer to
[47] and [48]. Independent of the exact controller structure,
it has been found that the choice of controller gains to
remain in a limited range is crucial and very challenging for
stability and practicability. In this section, we present our
approach to physically motivated gain design. Our approach
relies on designing the gains based on the physical robot
model, and, therefore, the smoothness of the gains can be
ensured, parameter variations can be accounted for, and
gains in a limited range result, simultaneously. Finally, the
application in a state feedback control structure is presented
and experiments are shown.
The gain design algorithm consists of the following steps.
1) Gain design model: The model used for gain design is
structurally similar to (2) with Coriolis/centrifugal and
gravity terms omitted. Gravity compensation is added in a
subsequent step. Furthermore, the spring stiffness is
assumed to be constant at each controller design time
instant. The design model is
c

B 0 ip
K -K i
xm
m e o +c
m c m =c
m.
-K K
0 M qp
q
x ext

K > 0 is a diagonal stiffness matrix. In addition, compare
the solid part in Figure 10 for one DOF. The external
torques x ext are considered to be the short-term disturbances, which are corrected by the controller.
2) Decoupling: The design system is in the multiple-input,
multiple-output (MIMO) format, where the coupling
originates from the inertia matrix M. The problem of
designing controller gains for the MIMO system is tackled
by transforming the n-dimensional system into a space
where the matrices become diagonal using the transformation matrix Q.
This is obtained from a generalized eigenvalue decomposition and it simultaneously diagonalizes the inertia
matrix and the stiffness matrix M = Q -T M Q Q -1 and
K = Q -T Q -1, where M Q is a diagonal matrix in the new
coordinates. The decoupled dynamic system becomes
c

Q
Q
I -I i Q
Q T BQ 0
xm
ip
me Q o = e Q o
Qm e Q o +c
-I I q
0
M
x ext
qp

(9)

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - December 2015