The paper deals with the equilibrium configurations and their stability study of a new type of compliant tensegrity mechanism based on dualtriangle structures, which actuated by adjusting elastic connections between the triangle edges. For a single segment of such mechanism, the torque-deflection relation was obtained as a function of control inputs and geometric parameters. It was proved that a single section of the mechanism can has either a single or three equilibrium configurations that can be both stable and unstable. The derived conditions of stability were used to choose control inputs ensuring the mechanism controllability, which was illustrated efficient to give the corresponding control algorithm. The structure composed of two segments in serial was also analyzed and an equivalent serial structure with non-linear virtual springs in the joints was proposed. It was proved that the stiffness of such structure decreases while the external loading increases, which may lead to the buckling phenomenon. Then the control algorithm based on the above principle was carried on, and the obtained results are confirmed by the simulation examples.