Sensorimotor Coordination in a Humanoid Robot: Building Intelligence on the iCub - Bioinspired Approaches for Human-Centric Technologies - page 178

5. Description of complex constraints. It is easy to add new constraints as linear
and/or nonlinear inequalities either in task or joint space. In the case of the iCub,
for instance, we added a set of constraints that avoid reaching the limits of the
tendons that actuate the three joints of the shoulder.
Once q d is determined as described above, there is still the problem of generating
a trajectory from the current robot configuration q to q d . Simultaneously, we would
like to impose suitable smoothness constraints to the trajectory. This has been
obtained by using the Multi-Referential Dynamical Systems approach (Hersch
and Billard 2008 ), whereby two dynamical controllers, one in joint space and
another in task space, evolve concurrently (Fig. 6.10 ). The coherence constraint,
that is, q , with J the Jacobian of the kinematics map, guarantees that at each

instant of time, the trajectory is meaningful. This is enforced by using the Lagrang-
ian multipliers method and can be tuned to modulate the relative influence of each
controller (i.e., to avoid joint angle limits). The advantage of such a redundant
representation includes the management of the singularities while maintaining a
quasi-straight trajectory profile of the end-effector in the task space—reproducing a
humanlike behavior (Abend et al. 1982 ).
Differently from the work of Hersch and Billard, we designed a feedback
trajectory generator instead of the VITE (Vector-Integration-To-Endpoint) method
used in open loop. A complete discussion of the rationale of the modifications to the
trajectory generation is outside the scope of this paper; the interested reader is
referred to Pattacini et al. ( 2010 ). Reasons to prefer a feedback formulation include
the possibility of smoothly connecting multiple pieces of trajectories and correcting
on-line for accumulation of errors due to the enforcement of the constraints of the
multi-referential method.

x _ ﾼ J _

6.5.1 Validation and Further Improvements

As earlier for the dynamics, we compared our method with other methods from the
literature. The comparison with the method of Hersch and Billard ( 2008 ) was
almost immediate since the work was developed on the iCub. This provides the
multi-referential approach together with the VITE trajectory generation at no cost.
Additionally, we included in the assessment another controller representing a more
conventional strategy that uses the damped least-squares (DLS) rule (Deo and
Walker 1992 ) coupled with a secondary task that comprises the joint angle limits
by means of the gradient projection method (Lee et al. 2007 ). This solution employs
the third-party package Orocos ( http://www.orocos.org/kdl ), a tool for robot control
that implements the DLS approach and whose public availability and compliance
with real-time constraints justified its adoption as one of the reference controllers.
In the first experiment, we put to test the three selected schemes in a point-to-
point motion task wherein the iCub arm was actuated in the “7-DoF mode” and
where the end-effector was controlled both in position and orientation. Results