How to avoid collisions in 3D-realizations for moving graphs

If we parameterize the positions of all vertices a given graph in plane such that distances between adjacent are fixed, obtain moving graph. An L-linkage is realization 3D-space, by representing edges using horizontal bars and vertical sticks. Vertical sticks parallel revolute joints, while links connecting them. We give sufficient condition for to have collision-free L-linkage. Furthermore, provide an algorithm guiding construction linkage when fulfills condition, via computing height function (horizontal bars). In particular, prove any Dixon-1 has no Dixon-2 graphs L-linkages, where two classic families graphs.