<p>Planar linkages are a rich area of study motivated by practical applications in engineering mechanisms. A central result is Kempe’s Universality Theorem, which states that semi-algebraic sets can be realized by planar linkages. Polyhedral linkages are generalizations of planar linkages to higher dimensions, where the faces are required to be rigid. In this paper, we generalize Kempe’s Universality Theorem to polyhedral linkages with an embedded construction in dimension three and above.</p>