To overcome the non-convexity and non-differentiability of multiple eigenvalues in traditional truss topology optimization models, a unified semidefinite programming (SDP) model was established for truss topology optimization problems with various constraints. Equivalent semidefinite forms were first provided for volume, compliance, fundamental frequency, and global stability constraints. Since the stiffness matrix and the mass matrix of truss are both linear with respect to design variables, problems considering volume, compliance, and fundamental frequency constraints were reformulated as standard dual forms linear SDP. Demonstrated by the global stability constraint and the stress constraint, nonlinear semidefinite constraints and nonlinear scalar constraints were separately approximated by simpler SDP forms at the current design point, which converts the nonlinear model to a solvable approximate SDP model. An algorithm for sequentially solving the approximate problem was then introduced to deal with the nonlinear problem. When the model contains only linear semidefinite constraints, the resultant linear SDP is convex and numerically favorable. When it concerns nonlinear constraints, the sequential approximate scheme enjoys the numerical advantage of linear semidefinite forms and also maintains the ability to handle ordinary nonlinear constraints, which may contribute to a more practical design. Examples show that the proposed SDP model and algorithm could deal with various constraints in truss topology optimization, especially fundamental frequency constraints and global stability constraints with multiple eigenvalues, which verified the effectiveness of the model and the algorithm.