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We present an immersed boundary projection method formulated in a body-fixed frame of reference for flow-structure interaction (FSI) problems involving rigid bodies with complex geometries. The body-fixed formulation is aimed at maximizing the accuracy of surface stresses on the FSI body (the target) on during spatial and temporal discretization. The incompressible vorticity equations and Newton's equations of motion are coupled implicitly so that the method remains stable for low solid-to-fluid mass ratios. The influence of fictitious fluid inside the rigid bodies is considered and the spurious oscillations in surface stresses are filtered to impose physically correct rigid body dynamics. Similar to many predecessors of the immersed boundary projection method, the resulting discrete system is solved efficiently using a block-LU decomposition. We then validate the method with two-dimensional test problems of a neutrally buoyant cylinder migrating in a planar Couette flow and a freely falling or rising cylindrical rigid body.