Hybrid Probabilistic Trajectory Optimization Using Null-Space Exploration

In the context of learning from demonstration, human examples are usually imitated in either Cartesian or joint space. However, this treatment might result in undesired movement trajectories in either space. This is particularly important for motion skills such as striking, which typically imposes motion constraints in both spaces. In order to address this issue, we consider a probabilistic formulation of dynamic movement primitives, and apply it to adapt trajectories in Cartesian and joint spaces simultaneously. The probabilistic treatment allows the robot to capture the variability of multiple demonstrations and facilitates the mixture of trajectory constraints from both spaces. In addition to this proposed hybrid space learning, the robot often needs to consider additional constraints such as motion smoothness and joint limits. On the basis of Jacobian-based inverse kinematics, we propose to exploit robot null-space so as to unify trajectory constraints from Cartesian and joint spaces while satisfying additional constraints. Evaluations of hand-shaking and striking tasks carried out with a humanoid robot demonstrate the applicability of our approach.