Abstract
The distance between sets is a long-standing computational geometry problem. In robotics, the distance between convex sets with Minkowski sum structure plays a fundamental role in collision detection. However, it is typically nontrivial to be computed, even if the projection onto each component set admits explicit formula. In this paper, we explore the problem of calculating the distance between convex sets arising from robotics. Upon the recent progress in convex optimization community, the proposed model can be efficiently solved by the recent hot-investigated first-order methods, e.g., alternating direction method of multipliers or primal-dual hybrid gradient method. Preliminary numerical results demonstrate that those first-order methods are fairly efficient in solving distance problems in robotics.
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The Moore–Penrose pseudo-inverse exists and is unique for any \(A\in \mathbb {C}^{m\times n}\). Let \(\hbox {rank}(A)=r\) and \(A=U\Sigma V^*\) be the singular value decomposition with \(U\in \mathbb {C}^{m\times r}\), \(\Sigma \in {{\mathbb {R}}}^{r\times r}\) and \(V\in \mathbb {C}^{m\times r}\). Then \(A^\dagger =V\Sigma ^{-1}U^*\). Particularly, \(A^\dagger =(A^*A)^{-1}A^*\) when A is of full column rank, and \(A^\dagger =A^*(AA^*)^{-1}\) when A is of full row rank.
Let \(\mathrm {bd}({\mathcal {C}})\) and \(N_{\mathcal {C}}(x)\) denote the topological boundary and normal cone (see also Example 2.8 for definition) of a set \({\mathcal {C}}\subset {{\mathbb {R}}}^n\), respectively. The \({\mathcal {C}}\) is called normally smooth if, for any \(x\in \mathrm{bd}({\mathcal {C}})\), there exists an \(a_x\in {{\mathbb {R}}}^n\) such that \(N_{\mathcal {C}}(x)=\mathrm {cone}\{a_x\}\). The \({\mathcal {C}}\) is said to be round if \(N_{\mathcal {C}}(x)\ne N_{\mathcal {C}}(y)\) for any x, \(y\in \mathrm{bd}({\mathcal {C}})\) and \(x\ne y\).
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Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments, which helped improving the paper substantially. X. Wang was supported by NSFC 11871279 and STCSM 19ZR1414200. W. Zhang was supported by NSFC 11971003.
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Wang, X., Zhang, J. & Zhang, W. The distance between convex sets with Minkowski sum structure: application to collision detection. Comput Optim Appl 77, 465–490 (2020). https://doi.org/10.1007/s10589-020-00211-0
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DOI: https://doi.org/10.1007/s10589-020-00211-0