The reliability of collision prediction for host vehicle-based collision warning systems relies heavily on the accuracy of estimated future paths of target vehicle kinematics. This paper presents a constrained filtering scheme that incorporates roadway information into a radar-based vehicle tracking process. Our proposed method makes use of the geometric structure of the road (the clothoid model) to form constraints on target vehicle kinematical variables. These constraints are used to generate an independent estimation of a state variable subset, and the inclusion of the constraints into a Kalman filter is materialized by a fusion process, which yields current state estimates. With the assistance of unconstrained filtering, a hypothesis testing scheme is performed every sampling instant to remove constraints when the target vehicle is not following the road or is changing lanes. A multiple constraint scheme is used to determine the current lane of the target vehicle, which enables a lane adaptive hypothesis testing procedure. Simulation shows that the proposed method can effectively reduce estimation errors and improve the prediction of in-lane collisions without degrading the prediction of cross-lane collisions.