Dynamic Movement Primitives (DMP) are an established and efficient method for encoding robotic tasks that require adaptation based on reference motions. Typically, the nominal trajectory is obtained through Programming by Demonstration (PbD), where the robot learns a task via kinesthetic guidance and reproduces it in terms of both geometric path and timing law. Modifying the duration of the execution in standard DMPs is achieved by adjusting a time constant in the model. This paper introduces a novel approach to fully decouple the geometric information of a task from its temporal information using an algorithm called spatial sampling, which allows parameterizing the demonstrated curve by its arc-length. This leads to the definition of the Geometric DMP (GDMP). The proposed spatial sampling algorithm guarantees the regularity of the demonstrated curve and ensures a consistent projection of the human force throughout the task in a human-in-the-loop scenario. GDMP exhibits phase independence, as its phase variable is no longer constrained to the demonstration's timing law, enabling a wide range of applications, including phase optimization problems and human-in-the-loop applications. Firstly, a minimum task duration optimization problem subject to velocity and acceleration constraints is formulated. The decoupling of path and speed in GDMP allows to achieve optimal time duration without violating the constraints. Secondly, GDMP is validated in a human-in-the-loop application, providing a theoretical passivity analysis and an experimental stability evaluation in co-manipulation tasks. Finally, GDMP is compared with other DMP architectures available in the literature, both for the phase optimization problem and experimentally with reference to an insertion task, showcasing the enhanced performance of GDMP with respect to other solutions.

18 pages, 18 figures