A rigid body, three of whose points are constrained to move on the coordinate planes, has three degrees of freedom. Bottema and Roth [
] showed that there is a point whose trajectory is a solid tetrahedron, the vertices representing corank 3 singularities. A theorem of Gibson and Hobbs [
] implies that, for general 3-parameter motions, such singularities cannot occur generically. However motions subject to this kind of constraint arise as interesting examples of parallel motions in robotics and we show that, within this class, such singularities can occur stably.
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