 This paper deals with the following separability problem in 3D space: Given a rigid polyhedron P with n vertices , does a semi-rigid polyhedron G exist, such that both polyhedra can be transformed into an inseparable assembled state, where the fixture snaps on to P, by applying a linear force and exploiting the mild flexibility of G? If such a flexible snapping polyhedron exists, devise an efficient and robust algorithm that constructs it. In simple words, we are looking for s semi-rigid polyhedron G, such that when P and G are separate , we can push G towards P , slightly bending  G on the way  , and obtain a configuration, where G is back in its original shape , and both P and G are inseparable as rigid bodies. We define certain properties such a pair of polyhedron and its snapping fixture may possess, and prove two theorems related to the pair. We introduce an algorithm that produces a snapping fixture with such properties in O(n ^5) time , if a snapping fixture exists , and an efficient and robust implementation of this algorithm .
Fixtures for constraining the movement of parts have been extensively investigated in robotics, since they are essential for using robots in automated manufacturing. This paper deals with the design and optimized synthesis of a special type of fixtures, which we callsnapping fixtures. Given a polyhedral workpiece P with n vertices and of constant genus, which we need to hold, a snapping fixture is a semi-rigid polyhedron G , made of a palm and several fingers, such that when P and G are well separated , we can push P toward G , slightly bending the fingers of G on the way (exploiting its mild flexibility) , and obtain a configuration, where G is back in its original shape and P and G are inseparable as rigid bodies. We prove the minimal closure conditions under which such fixtures can hold parts, using Helly's theorem. We then introduce an algorithm running in O(n ^3) time that produces a snapping fixture, minimizing the number of fingers and optimizing additional objectives, if a snapping fixture exists . We also provide an efficient and robust implementation of a simpler version of the algorithm, which produces the fixture model to be 3D printed and runs in O(n^4) time. We describe two applications with different optimization criteria: Fixtures to hold add-ons for drones, where we aim to make the fixture as lightweight as possible, and small-scale fixtures to hold precious stones in jewelry, where we aim to maximize the exposure of the stones, namely minimize the obscuring of the workpiece by the fixture .
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Fixtures for constraining the movement of parts have been extensively investigated in robotics, since they are essential for using robots in automated manufacturing.
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following separability problem in 3D space: Given a rigid polyhedron
design and optimized synthesis of a special type of fixtures, which we call
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snapping fixtures
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. Given a polyhedral workpiece
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, does a
and of constant genus, which we need to hold, a snapping fixture is a
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exist, such that both polyhedra can be transformed into an inseparable assembled state, where the fixture snaps on to P, by applying a linear force and exploiting the mild flexibility of G? If such a flexible snapping polyhedron exists, devise an efficient and robust algorithm that constructs it. In simple words, we are looking for s semi-rigid polyhedron G,
, made of a palm and several fingers,
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separate
well separated
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G towards P
P toward G
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the fingers of
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(exploiting its mild flexibility)
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, and both
and
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define certain properties such a pair of polyhedron and its snapping fixture may possess, and prove two theorems related to the pair. We
prove the minimal closure conditions under which such fixtures can hold parts, using Helly's theorem. We then
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that produces a snapping fixture with such properties
running
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^5) time ,
^3) time that produces a snapping fixture, minimizing the number of fingers and optimizing additional objectives,
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, and
. We also provide
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this algorithm
a simpler version of the algorithm, which produces the fixture model to be 3D printed and runs in O(n^4) time. We describe two applications with different optimization criteria: Fixtures to hold add-ons for drones, where we aim to make the fixture as lightweight as possible, and small-scale fixtures to hold precious stones in jewelry, where we aim to maximize the exposure of the stones, namely minimize the obscuring of the workpiece by the fixture