The paper presents an ongoing project aiming to build a robot, composed of Assur tensegrity structures, that mimics the caterpillar locomotion. Caterpillars are soft bodied animals capable of making complex movements with an astonishing fault-tolerance. In this model, a caterpillar segment is represented as a 2D tensegrity triad, consists of two cables and a linear actuator which are connected between two bars. The unique engineering properties of Assur tensegrity structures which were mathematically proved only this year, together with the suggested control algorithm share several analogies with the biological caterpillar. It provides each triad with an adjustable structural softness. Therefore, the proposed robot has a fault-tolerance and can adjust itself to the terrain roughness. This algorithm also reduces the control demands of the non-linear model of the triad by enabling simple motion control for the linear actuator and one of the cables, while the other cable is force controlled.