# Tube-Based Robust Model Predictive Control for Autonomous Vehicle with Complex Road Scenarios
###### Abstract
In addition to these efforts, recent research has extended MPC applications to novel traffic paradigms, such as lane-free environments [15; 16] proposed an optimal path planning framework for connected and automated vehicles (CAVs) operating in lane-free traffic conditions, demonstrating the versatility of MPC in unstructured scenarios where traditional lane-based assumptions are not applicable. However, despite its demonstrated advantages in sensitivity control, the algorithm remains limited in its ability to handle uncertainties in model parameters. Data-driven methodologies have emerged as a powerful and efficient paradigm for modeling complex and partially unknown dynamic systems [17]. Among these, Koopman operator theory [18; 19; 20] has become a pivotal framework, enabling the representation of nonlinear system dynamics through linear evolution within an augmented observable space. In the domain of automated driving, this approach has been applied by leveraging deep Koopman-based modeling of vehicle dynamics, often integrated with MPC strategies for trajectory-tracking tasks [21]. Despite its advantages in facilitating global linear representations, the integration of Koopman operator theory with MPC introduces substantial computational complexity and demands extensive, high-precision data for effective implementation. This reliance on large-scale data can lead to potential safety risks and unintended behaviors in automated driving scenarios [18]. To enhance robustness and reduce reliance on precise, high-volume data, robust model predictive control (RMPC) frameworks [19; 20] such as min-max MPC [21] and Tube-based MPC [22; 23] have garnered significant attention. These approaches confine parameter variations within a predefined region, ensuring improved robustness and computational efficiency. Min-max MPC [24]--a widely studied RMPC approach--explicitly incorporates system uncertainties by formulating constraints on state variables and control inputs. However, its worst-case optimization strategy results in a computational complexity that scales with the input dimensionality and prediction horizon, posing challenges in multi-input multi-output systems. As a result, its real-time feasibility remains a key limitation in practical implementations. Considering the integration of predictive information and constraints, RMPC methods have been extensively studied and applied in automated vehicle (AV) systems. As previously discussed, to address uncertainties such as model inaccuracies under extreme conditions, two primary approaches have emerged: stochastic MPC, which employs probabilistic descriptions, and robust MPC (RMPC), which relies on bounded-set representations [25]. Due to the inherent challenges in accurately describing model uncertainties probabilistically, RMPC is generally regarded as a more promising solution for motion control. A more computationally efficient alternative is Tube-based RMPC (TRMPC) [26], which confines system uncertainties to a robust tube, reducing the online optimization problem to a nominal MPC problem with tightened constraints. TRMPC has found extensive applications in AV control, including lateral control [27], trajectory tracking in autonomous racing [28], fault-tolerant control [29], and vehicle platooning [30]. Despite its effectiveness in handling structured model uncertainties, TRMPC primarily focuses on system disturbances rather than external environmental variations. Consequently, the impact of road-tire adhesion variations on control performance remains insufficiently explored, posing challenges in adapting to diverse road conditions, such as wet, snowy, or dry asphalt surfaces. Ensuring robust control across varying road surface conditions remains a critical challenge that warrants further investigation. To address the limitations identified in existing methods, this study proposes a Tube-RMPC-based vehicle control approach to explicitly address model parameter uncertainties and variations in road-tire adhesion coefficients.