Performing real-time receding horizon motion planning for autonomous vehicles while providing safety guarantees remains difficult. This is because existing methods to accurately predict ego vehicle behavior under a chosen controller use online numerical integration that requires a fine time discretization and thereby adversely affects real-time performance. To address this limitation, several recent papers have proposed to apply offline reachability analysis to conservatively predict the behavior of the ego vehicle. Reachable sets can be constructed by utilizing a simplified model whose behavior is assumeda priorito conservatively bound the dynamics of a full-order model. However, it can be challenging to meticulously construct this conservative bound. This paper proposes a framework named REFINE to overcome the limitations of these existing approaches. REFINE utilizes a parameterized robust controller that partially linearizes the vehicle dynamics even in the presence of modeling error. Zonotope-based reachability analysis is then performed on the closed-loop, full-order vehicle dynamics to offline compute the corresponding control-parameterized, over-approximate Forward Reachable Sets (FRS). Because reachability analysis is applied to the full-order model, the potential conservativeness introduced by using a simplified model is avoided. The pre-computed, control-parameterized FRS is then used online in an optimization framework to ensure safety. The proposed method is compared to several state-of-the-art methods during a simulation-based evaluation on a full-size vehicle model and is demonstrated on a$\frac{1}{10}$th race car robot in real hardware testing. In contrast to existing methods, REFINE is shown to enable the vehicle to safely navigate itself through complex environments.