This paper presents a novel truss-shaped variable-stiffness deployable robotic grasper to grasp large unknown objects, the grasper comprises a series of basic modules and has the advantages of a large workspace, adjustable stiffness, and a high deploy/fold ratio. First, detailed mechanism designs of the grasper and variable-stiffness joint are introduced, and a mobility analysis and variable stiffness analysis are conducted. Second, the structural analysis of the basic module is carried out, by which several major indices, including deploy/fold ratio, grasping angle, deployment angle, grasping torque, and deployment torque, can be calculated. Third, kinematic analysis is presented to provide the workspace and kinematic simulation, and then the joint trajectory planning based on fifth-order polynomial is also conducted. Fourth, the condition of stable grasp is analyzed and a mathematical model of grasping motion is established. Adam optimization algorithm is then applied to optimization of the mathematical model and a grasping simulation is performed with objects of various sizes and for various working states of the variable-stiffness joint. Finally, a simple physical prototype is fabricated, and variable-stiffness experiments and grasping experiments confirm that the proposed grasper shows excellent grasping performance.