This paper is devoted to the study of the constrained consensus problem in second-order multi-agent systems, where the agents’ positions are subject to convex constraints, and their velocities are subject to nonconvex constraints. The nonlinearity caused by the coupling between the nonconvex velocity constraint sets, which include the origin, and the convex position constraint sets, which may not include the origin, presents a severe challenge. To overcome this challenge, we propose a novel constrained consensus algorithm that guarantees agents’ positions and velocities stay within their respective constraints. By applying multiple model transformations and leveraging the system’s convexity, it is proved that the constrained consensus problem with different state constraints can be solved. Finally, numerical examples validate the conclusions.