IEEE Robotics & Automation Magazine - June 2012 - 31

Figure 18. The robot is grasping an object with its left hand.
The six-DoF TCP trajectory is realized while multiple objectives
are reached reactively and simultaneously.

the "Redundancy Resolution Concept" section is not used
here, but it will be integrated in a future work.
Although only two major priority levels exist, a more
detailed hierarchy is achieved by a proper choice and the
design of the subtasks within one level. Moreover, the fact
is exploited that several subtasks do not have any intersections. Examples are the decoupling of the upper-body
singularity avoidance and the dynamic singularity avoidance of the mobile base, or the decoupling of the upperbody mechanical end-stop potentials and the platform
impedance. Hence, undesired and undefined competitions
on one priority level are avoided. See [26] for a detailed
discussion on all combinations of the involved tasks.
Several experiments are conducted to demonstrate the
performance of the control structure. First, the step
response of the right TCP in the case of a forward motion
(Dx ¼ 0:2 m) is evaluated. All subtasks are *
activated and the transThe integration of
lational stiffness of the
Cartesian impedance is
unilateral, physical
set to Kt ¼ 500 N/m. As
it can be seen in Figconstraints into the higher
ure 14, the actual settling
time is less than 0:5 s. In
levels of a task hierarchy
addition, the overshooting is mentionable which
causes additional problems
can be traced back to the
delayed behavior of the
in terms of discontinuities
platform due to the
admittance coupling [see
in the control law.
Figure 14(b)] and a
damping ratio of f ¼ 0:7 *
in the impedance. As the
impedance is basically a proportional-derivative (PD) controller and does not possess an integrating component, a
steady-state error may remain which is observable in the
upper plots. The excitation in the x direction also affects
the other two translational directions marginally. The
steady-state errors can be reduced by using a higher translational stiffness.
The second experiment shows the robot behavior in
the case of a continuous trajectory [see Figure 15(a)]. The
initial configuration is depicted in Figure 16(a). The right
TCP frame is commanded to move forward 1 m and then
back to the initial frame [see Figure 16(b)]. Apparently,
the controller leads to a totally different configuration
when approaching the initial frame again. Figure 15(b)
depicts a quadratic norm of the selected null space subtask torques to allow direct comparison. Obviously,
returning does not lead to the same subtask participation.
For example, the upper-body singularity avoidance is
more crucial while moving forward to prevent outstretched arms than it is when moving backward. In contrast, the avoidance of mechanical end stops is only active
during the backward motion (after 11 s). This complies
JUNE 2012

IEEE ROBOTICS & AUTOMATION MAGAZINE

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - June 2012