IEEE Robotics & Automation Magazine - December 2015 - 45

State Feedback Damping Control
Many applications, such as manipulation, require an adjustable
damping behavior, e.g., for a fast positioning accuracy. Integrating an additional variable damping element in each joint would
further increase the mechanism complexity and size. For that
reason, we inject damping via control. Furthermore, the use of
an active control system is beneficial as the powerful actuators

Motor Position (°)

Continuous Task Hierarchies
via Null Space Projections
We have introduced a new method to impose dynamic task
hierarchies for torque-controlled robots by applying modified null-space projection techniques in [28]. The reason
behind the adaptation of classical null space methods is to
avoid discontinuities in the control law while (de)activating
tasks within the hierarchy, which could result in instability.
The proposed approach uses an intuitive, geometric representation of the null space to smoothly scale in/out the
respective control action from the lower levels in the critical
directions while keeping the remaining directions unaffected. Exemplarily, the method has been implemented on
the elbow joint of the DLR HASy, applying a hierarchy with
two levels.
● High Priority: A virtual, repulsive wall with a stiffness of
500 Nm/rad is designed, which starts at 42° of the elbow
joint. The wall is assigned to the motor position, a similar
link-side behavior can easily be realized by applying it to
the static link side equivalent qr (i) .
● Low Priority: A passive PD controller with a stiffness of
8,000 Nm/rad is supposed to lead the motor position from
0° to 80° on a smooth trajectory.
In Figure 9, four different cases are drawn. First, the passive PD controller (blue/dashed line) realizes the low priority
task in an unimpeded manner, i.e., with no virtual wall. Second, the two task torques are superposed so that wall repulsion and PD controller compete (green/dashed-dotted line).
This leads to an equilibrium, which is primarily determined
by the stiffnesses of the tasks. Up to this point, no welldefined hierarchy has been realized. The third plot (red/
dashed-dotted line) implements a two-level hierarchy by
applying modified null-space projections [28]. Although the
virtual wall has a significantly smaller stiffness than the PD
controller, the strict constraint is fulfilled. A smooth transition between 42 and 45 inserts the high-priority task in the
dynamic hierarchy. The notch in the respective torque signal
[Figure 9(b)] is due to the deceleration of the link side. Even
in the critical phase at about 1.4 s, the motor position
remains below 45°. The applied continuous null-space projection is not computational resource intensive, as it only
contains multiplications and no expensive (pseudo)inversions or decompositions have to be performed. More complex high-priority tasks can be realized analogously, e.g., a
motor position-based singularity avoidance to keep the
manipulability measure on a high level. The last plot (black/
solid) shows the behavior when using classical null-space
projections that do not properly deal with the discontinuities.
Chattering occurs, which can be observed in the zoomed
area. Moreover, the 45° limit is crossed with a frequency of
about 31 Hz. The corresponding torque signal is omitted in

the lower diagram, since from 1.4 s on, it only switches
between minimum and maximum feasible torque with 31
Hz and would conceal the
other curves.
It is possible to apply
In general, the goal is
null-space projections in
variable stiffness robots.
to design the dynamic
However, in general, this
is limited to task represenbehavior of the robot.
tations in collocated variables (see the "Basic
Control: Energy Shaping Techniques" section) as done in the
previous experiment. Due to the problems stated in the "State
Feedback Damping Control" section, a noncollocated feedback of the link position, which additionally passes through
one or more null-space projectors, tends to deteriorate the robustness. This can happen even if a well-parameterized state
feedback damping controller is applied as proposed in the
"State Feedback Damping Control" section. The null-space
projector may arbitrarily cut out essential contributions of the
damping torques.

Torque (Nm)

(879 N/m). Thus, deflections must be observable on the
motor side due to the active displacement of the motor that
concurs with the synchronous deflection of x i and x q , as
shown in Figure 7.

90
80 Repulsion from Virtual Wall
70 Started at 42°, Fully
0.5°°
60 Active at 45°
Frequency: ~31 Hz
50
40
PD, No Wall
30
PD Versus Wall
20
Continuous Hierarchy
10
Classical
Hierarchy
0
-10
0
0.5
1.0
1.5
2.0 2.5 3.0 3.5 4.0
Time (s)
(a)
20
18
16
14
12
10
8
6
4

PD, No Wall
PD Versus Wall
Continuous Hierarchy
0

0.5

1.0

1.5

2.0 2.5
Time (s)
(b)

3.0

3.5

4.0

Figure 9. A repulsive potential to avoid particular regions of the elbow
joint is used as a high-priority task in a dynamic hierarchy. The torques
are filtered for the plot due to sensor noise. (a) Motor positions.
(b) Torques.

DECEMBER 2015

IEEE ROBOTICS & AUTOMATION MAGAZINE

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