Robustness, compactness, and portability of tensegrity robots make them suitable candidates for locomotion on unknown terrains. Despite these advantages, challenges remain relating to ease of fabrication, shape morphing (packing-unpacking), and locomotion capabilities. The paper introduces a design methodology for fabricating tensegrity robots of varying morphologies with modular components. The design methodology utilizes perforated links, coplanar (2D) alignment of components and individual cable tensioning to achieve a 3D tensegrity structure. These techniques are utilized to fabricate prism (three-link) tensegrity structures, followed by tensegrity robots in icosahedron (six-link), and shpericon (curved two-link) formation. The methodology is used to explore different robot morphologies that attempt to minimize structural complexity (number of elements) while facilitating smooth locomotion (impact between robot and surface). Locomotion strategies for such robots involve altering the position of center-of-mass (referred to as internal mass shifting) to induce “tip-over.” As an example, a sphericon formation comprising of two orthogonally placed circular arcs with conincident center illustrates smooth locomotion along a line (one degree of freedom). The design of curved links of tensegrity mechanisms facilitates continuous change of the point of contact (along the curve) that results from the tip-over. This contrasts to the sudden and piece-wise continuous change for the case of robots with traditional straight links which generate impulse reaction forces during locomotion. The two resulting robots—the Icosahedron and the Sphericon Tensegrity Robots—display shape morphing (packing-unpacking) capabilities and achieve locomotion through internal mass-shifting. The presented static equilibrium analysis of sphericon with mass is the first step in the direction of dynamic locomotion control of these curved link robots.