In this chapter we present a new formulation that exploits the principle of Support Vector Machine (SVM). This formulation - \emph{Augmented-SVM} (A-SVM) - aims at combining gradient observations with the standard observations of function values (integer labels in classification problems and real values in regression) within a single SVM-like optimization framework. The presented formulation adds onto the existing SVM by enforcing constraints on the gradient of the classifier/regression function. The new constraints modify the original SVM dual, whose optimal solution then results in a new class of support vectors (SV). We present our approach in the light of a particular application in robotics, namely, learning a non-linear dynamical system (DS) with multiple attractors. Non-linear DS have been used extensively for encoding robot motions with a single attractor placed at a predefined target where the motion is required to terminate. In this chapter, instead of insisting on a single attractor, we focus on combining several such DS with distinct attractors, resulting in a multi-stable DS. While exploiting multiple attractors provides more flexibility in recovering from unseen perturbations, it also increases the complexity of the underlying learning problem. We address this problem by augmenting the standard SVM formulation with gradient-based constraints derived from the individual DS. The new SV corresponding to the gradient constraints ensure that the resulting multi-stable DS incurs minimum deviation from the original dynamics and is stable at each of the attractors within a finite region of attraction. We show, via implementations on a simulated 10 degrees of freedom mobile robotic platform, that the model is capable of real-time motion generation and is able to adapt on-the-fly to perturbations.

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