  Whenever a robotic task needs to be defined and adapted based on a reference
motion, Dynamic Movement Primitives (DMP) represent a standard and efficient
method for encoding it. The nominal trajectory is typically obtained through a
Programming by Demonstration (PbD) approach, where the robot is taught a
specific task through kinesthetic guidance. Subsequently, the motion is
reproduced by the manipulator in terms of both geometric path and timing law.
The basic approach for modifying the duration of the execution involves
adjusting a time constant characterizing the model. On the contrary, the goal
of this paper is to achieve a complete decoupling between the geometric
information of the task and the timing law governing the execution, thanks to a
new spatial sampling algorithm. This leads to a new DMP concept called
Geometric DMP (GDMP), which exhibits the property of being phase-free since the
phase variable is no longer constrained to the demonstration timing law. GDMP
open up to a variety of applications, including task duration optimization
subject to velocity and acceleration constraints and human-in-the-loop
applications in co-manipulation tasks. With reference to the latter
application, a co-manipulation activity where the robot assists the humans in
reproducing simple rehabilitation tasks is considered in this paper as a case
study. A custom phase law is designed and the system passivity and stability
analyses are carried out. The conclusions drawn through the system stability
analysis are validated by the proposed experimental results.
