Uncertainty-propagated Cartesian coordinated human–robot collaboration on Riemannian manifold with hidden state-space model

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This paper proposes a novel Cartesian coordinated human–robot collaboration framework derived from the hidden state-space models, which are established on the behaviour cloning of both the human and robot in a nonparametric form by using dual quaternion on Riemannian manifold. Based on the Taylor linearisation, this framework could provide an analytical approximation of the posterior distribution and hence infer six DoF Cartesian hidden state variables of the collaborative manipulator given human observation and its uncertainties. As the full Cartesian pose (special Euclidean group, SE(3)) includes translation and rotation (special orthogonal group, SO(3)), which is not Euclidean, directly encoding the Cartesian motions with nonparametric regression in Euclidean space could cause significant errors. This paper addresses this issue by defining both human and robot behaviours using modified quaternion and cloning them in the tangent space of the Riemannian manifold. Not akin to other coordinated human–robot collaboration methods, the collaboration framework not only preserves the adaptation functionalities but also propagates full Cartesian state variables and their uncertainties during real-time coordinated collaboration implementation. Leveraging on the Gaussian mixture model on the Riemannian manifold, the multiple task recognition is further addressed to enhance the generalisation capability of the framework presented. The application feasibility is demonstrated in both theoretical comparison simulation and experiments.