Abstract-We present a task-parameterized probabilistic model encoding movements in the form of virtual springdamper systems acting in multiple frames of reference. Each candidate coordinate system observes a set of demonstrations from its own perspective, by extracting an attractor path whose variations depend on the relevance of the frame at each step of the task. This information is exploited to generate new attractor paths in new situations (new position and orientation of the frames), with the predicted covariances used to estimate the varying stiffness and damping of the spring-damper systems, resulting in a minimal intervention control strategy. The approach is tested with a 7-DOFs Barrett WAM manipulator whose movement and impedance behavior need to be modulated in regard to the position and orientation of two external objects varying during demonstration and reproduction.

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In the gauge natural bundle framework a new space is introduced and a first-order purely frame-formulation of General Relativity is obtained. In some of our recent works [1, 2, 3] a new geometrical framework for YangMills field theories and General Relativity in the tetrad-affine formulation has been developed.The construction of the new geometrical setting has been obtained quotienting the first-jet bundles of the configuration spaces of the above theories in a suitable way, resulting into the introduction of a new family of fiber bundles.In this letter we show that these new spaces allow a (covariant) first-order purely frame-formulation of General Relativity.The whole geometrical construction will be developed within the gauge natural bundle framework [4], which provides the suitable mathematical setting for globally describing gravity in the tetrad formalism.To start with, let M be a space-time manifold, allowing a metric tensor g with signature η = (1, 3): the manifold M will be called a η-manifold and the metric tensor canonical representation will be η µν := diag (−1, 1, 1, 1). Moreover, let L(M ) be the frame-bundle over M and P → M a principal fiber bundle over M with structural group SO (1, 3).The configuration space of the theory (the tetrad space) is a GL(4, ℜ) bundle π : E → M , associated to P × M L(M ) through the left-action λ : (SO(1, 3) × GL(4, ℜ)) × GL(4, ℜ) → GL(4, ℜ), λ(Λ, J; X) = Λ · X · J −1 (1) 1

A gauge-independent formulation of the theory developed in J. Phys. A: Math. Gen. 36 8341 ( 2003) is proposed.