Robust Optimization-Based Calculation of Invariant Trajectory Representations for Point and Rigid-body Motion

Invariant representations of demonstrated motion trajectories provide context-independent motion models that can be used in motion recognition and generalization applications such as robot programming by demonstration. In practice, the use of invariant representations is still limited because their numerical calculation from a demonstrated trajectory is complicated by sensitivity to measurement noise and singularities, yielding inaccurate invariant functions that do not correspond well with the original trajectory. This paper improves the calculation of invariant representations for point and rigid-body motions by reformulating their calculation as an optimization problem that minimizes the error between the trajectory reconstructed from the invariant representation and the measured trajectory. Robustness against noise and singularities is ensured through the addition of regularization terms on the invariants. Simulations and real motion experiments show that the accuracy of the calculated invariant representations greatly improves with respect to standard smoothing methods. These results encourage future developments of motion recognition and generalization applications based on invariant trajectory representations.