Classical grasp quality evaluation: New algorithms and theory

This paper investigates theoretical properties of a well-known L 1 grasp quality measure Q whose approximation Q − <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</inf> is commonly used for the evaluation of grasps and where the precision of Q − <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</inf> depends on an approximation of a cone by a convex polyhedral cone with l edges. We prove the Lipschitz continuity of Q and provide an explicit Lipschitz bound that can be used to infer the stability of grasps lying in a neighbourhood of a known grasp. We think of Q − <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</inf> as a lower bound estimate to Q and describe an algorithm for computing an upper bound Q + . We provide worst-case error bounds relating Q and Q − <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</inf> . Furthermore, we develop a novel grasp hypothesis rejection algorithm which can exclude unstable grasps much faster than current implementations. Our algorithm is based on a formulation of the grasp quality evaluation problem as an optimization problem, and we show how our algorithm can be used to improve the efficiency of sampling based grasp hypotheses generation methods.