We propose a learning framework, named Multi-Coordinate Cost Balancing (MCCB), to address the problem of acquiring point-to-point movement skills from demonstrations. MCCB encodes demonstrations simultaneously in multiple differential coordinates that specify local geometric properties. MCCB generates reproductions by solving a convex optimization problem with a multi-coordinate cost function and linear constraints on the reproductions, such as initial, target, and via points. Further, since the relative importance of each coordinate system in the cost function might be unknown for a given skill, MCCB learns optimal weighting factors that balance the cost function. We demonstrate the effectiveness of MCCB via detailed experiments conducted on one handwriting dataset and three complex skill datasets.
Differential coordinates have been extensively used in the computer graphics community @cite_6 @cite_11 . Prior work in trajectory learning that incorporates differential coordinates includes the Laplacian trajectory editing (LTE) algorithm @cite_9 . Using Laplacian coordinates, the LTE algorithm adapts a single demonstration to new initial, target, and via points while preserving the shape. However, the LTE algorithm does not reason about the relative importances of multiple coordinates.
@cite_9 @cite_6 @cite_11
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Assuming that a robot trajectory is given from a high-level planning or learning mechanism, it needs to be adapted to react to dynamic environment changes. In this article we propose a novel approach to deform trajectories while keeping their local shape similar, which is based on the discrete Laplace---Beltrami operator. The approach can be readily extended and covers multiple deformation techniques including fixed waypoints that must be passed, positional constraints for collision avoidance or a cooperative manipulation scheme for the coordination of multiple robots. Due to its low computational complexity it allows for real-time trajectory deformation both on local and global scale and online adaptation to changed environmental constraints. Simulations illustrate the straightforward combination of the proposed approach with other established trajectory-related methods like artificial potential fields or prioritized inverse kinematics. Experiments with the HRP-4 humanoid successfully demonstrate the applicability in complex daily-life tasks. One of the main challenges in editing a mesh is to retain the visual appearance of the surface after applying various modifications. In this paper we advocate the use of linear differential coordinates as means to preserve the high-frequency detail of the surface. The differential coordinates represent the details and are defined by a linear transformation of the mesh vertices. This allows the reconstruction of the edited surface by solving a linear system that satisfies the reconstruction of the local details in least squares sense. Since the differential coordinates are defined in a global coordinate system they are not rotation-invariant. To compensate for that, we rotate them to agree with the rotation of an approximated local frame. We show that the linear least squares system can be solved fast enough to guarantee interactive response time thanks to a precomputed factorization of the coefficient matrix. We demonstrate that our approach enables to edit complex detailed meshes while keeping the shape of the details in their natural orientation. One of the challenges in geometry processing is to automatically reconstruct a higher-level representation from raw geometric data. For instance, computing a parameterization of an object helps attaching information to it and converting between various representations. More generally, this family of problems may be thought of in terms of constructing structured function bases attached to surfaces. In this paper, we study a specific type of hierarchical function bases, defined by the eigenfunctions of the Laplace-Beltrami operator. When applied to a sphere, this function basis corresponds to the classical spherical harmonics. On more general objects, this defines a function basis well adapted to the geometry and the topology of the object. Based on physical analogies (vibration modes), we first give an intuitive view before explaining the underlying theory. We then explain in practice how to compute an approximation of the eigenfunctions of a differential operator, and show possible applications in geometry processing.