Keywords: bar-joint framework, cable braced structure, infinitesimal rigidity, configuration space, tensegrities, optimization, computation complexity.

Bar and joint frameworks have served as models of the engineering structures. The rigidity of bar-joint structures has received much attention in geometry, mathematics, engineering, and material science. The primary purpose is to find an efficient algorithm for deciding infinitesimal rigidity in cable braced three-dimensional Skeletal framework of Cubic Structure. Using the bar-joint structure's symmetry to determine the rigidity is a problem of long-standing interest in kinematics, statics, and optimization. The algorithm has applications in robotics as an actuator-controlled mechanism and in material science as meta-materials and reconfigurable materials. The rigidity and mobility of bar and joint framework have served as valuable models of the structure of metals, crystal states of matter, and biological systems. The simple preliminary analysis gives input to a more complicated consideration of a threedimensional frame structure than the mentioned disciplines in building science. Lattices, as repetitive objects, are helpful as preliminary structures of design. We considered the Cubic tiling as a bar-joint framework (the edges correspond to bars, the vertices correspond to ball joints). They are mechanisms. Applying some further bracing elements such as Cable, Strut, or Rod (CSR) along the diagonals of the faces of the Twist-free Cubic Tiling Framework (TCTF), the framework could move (disregarding the rigid-body like motions) or will be rigid. We give a characterization of the rigidity of TCTF using CSR bracing elements. We present a description of the system's motions if we insert CSR bracing elements. The given model describes the rigidity of the repetitive frameworks that produce smaller dimensional tensegrity structures as a configuration space that are solvable efficiently as the original structure.

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