Projected gradient descent denotes a class of iterative methods for solving optimization programs. In convex optimization, its computational complexity is relatively low whenever the projection onto the feasible set is relatively easy to compute. On the other hand, when the problem is nonconvex, e.g., because of nonlinear equality constraints, the projection becomes hard and thus impractical. In this paper, we propose a projected gradient method for nonlinear programs that only requires projections onto the linearization of the nonlinear constraints around the current iterate, similar to sequential quadratic programming (SQP). The proposed method falls neither into the class of projected gradient descent approaches, because the projection is not performed onto the original nonlinear manifold, nor into that of SQP, since second-order information is not used. For nonlinear smooth optimization problems, we assess local and global convergence to a Karush Kuhn Tucker point of the original problem. Further, we show that nonlinear model predictive control is a promising application of the proposed method, due to the sparsity of the resulting optimization problem.