We consider the path planning problem for a 2-link robot amidst polygonal obstacles. Our robot is parametrizable by the lengths of its two links, the thickness of the links, and an angle that constrains the angle between the 2 links to be strictly greater than. The case and corresponds to “thick non-crossing” robots. This results in a novel 4DOF configuration space where is the torus and the diagonal band of width. We design a resolution-exact planner for this robot using the framework of Soft Subdivision Search (SSS). First, we provide an analysis of the space of forbidden angles, leading to a s5oft predicate for classifying configuration boxes. We further exploit the T/R splitting technique which was previously introduced for self-crossing thin 2-link robots. Our open-source implementation in Core Library achieves real-time performance for a suite of combinatorially non-trivial obstacle sets. Experimentally, our algorithm is significantly better than any of the state-of-art sampling algorithms we looked at, in timing and in success rate.