On smooth relaxations of obstacle sets
We consider the problem of avoiding obstacle sets described by finitely many smooth convex inequality constraints, as it frequently occurs in, for example, trajectory or location planning. We present and discuss a general method to relax such sets by the upper level set of a single smooth convex function, covering different smoothing approaches like hyperbolic and entropic smoothing.
Based on error bounds and Lipschitz continuity, special attention is paid to the computations of the maximal geometric approximation error and of a guaranteed safety margin. Our results thus allow to safely avoid the obstacle by obeying a single nonconvex smooth constraint. Numerical results indicate that our technique gives rise to smoothing methods which perform well even for smoothing parameters very close to zero.
KeywordsRelaxation Error bound Lipschitz continuity Hyperbolic smoothing Entropic smoothing Obstacle problem
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