# A Combined Dynamic-Kinematic Extended Kalman Filter for Estimating Vehicle Sideslip Angle
## 1 Introduction  
In modern automotive engineering, the quest for enhanced vehicle safety, performance, and driver comfort is constantly evolving. Among the critical parameters influencing vehicle dynamics, sideslip angle stands out as a key factor, essential for the development and implementation of advanced driver-assistance systems (ADAS), electronic stability control (ESC) systems, and autonomous driving technologies. Direct measurement of sideslip angle, which involves sophisticated and expensive instrumentation such as optical or GPS-based systems, is not always feasible for production vehicles due to cost, installation complexity, and environmental limitations. Consequently, estimation techniques represent a valuable alternative that can provide reliable sideslip angle estimation using standard onboard sensors. Data-driven estimators, or black-box methods, such as those using neural networks, fit experimental data without requiring a vehicle model. These methods can generate very accurate estimations. Their main drawbacks are that they depend heavily on the comprehensiveness and accuracy of the training dataset and need retraining if the vehicle or driving conditions change [1]. The use of an artificial neural network (NN) for estimating sideslip angle represents one of the most popular trends among data-driven methods. In general, the inputs to NNs are commonly measured signals, such that lateral and longitudinal acceleration, yaw rate and wheel speeds, while sideslip angle is the output of the network [2, 3]. A thorough mathematical description of NNs is reported in [4]. On the other hand, model-based estimation techniques such as the Kalman filter (KF) have emerged as a powerful tool in this context. The KF ability to fuse commonly recordeddata from multiple sensors, account for dynamic system behaviors, and provide real-time estimates makes it an ideal candidate for sideslip angle estimation. The Kalman filter can integrate measurements from accelerometers, gyroscopes, wheel speed sensors, and steering angle sensors to estimate the sideslip angle accurately. Two main Kalman filter approaches can be distinguished in the literature, i.e., kinematic and dynamic model-based KF. Kinematic KFs are based on the integration of vehicle longitudinal and lateral acceleration and the kinematic relationships among them, yaw rate and vehicle velocity [5; 6]. The implementation is not demanding, in general, and they perform well in transient conditions. Moreover, kinematic equations do not involve any vehicle or tyre-related parameter. However, they have observability issues when the yaw rate approaches zero and the estimates may tend to drift [7]. Dynamic KFs use a dynamic model of the vehicle, combined with onboard sensor measurements. This method is less sensitive to sensor inaccuracies and is able to compensate for them [8]. Its main drawback is that it requires a thorough characterisation of both the vehicle and the tyres, otherwise significant estimation biases can occur [9]. To address this issue, parameters, such as tyre cornering stiffness, can be added in the state vector to obtain an estimation of them too, but this increases the problem complexity, and tuning covariance matrices may prove to be difficult [10]. A third possible strategy is represented by the combination of the two aforementioned strategies. Many researches have tried to exploit the benefits of both kinematic and dynamic Kalman filters for estimating vehicle sideslip angle, by combining them in several different ways. [PERSON] et al. [11] propose a combined kinematic-dynamic estimator by augmenting a standard kinematic model with a feedback part. Longitudinal and lateral velocities are the two state variables, while the feedback state observer incorporates a dynamic model. [PERSON] et al. [12] combine dynamic and kinematic approach, by feeding the kinematic model with part of the output of the dynamic model, and vice versa. The resulting sideslip angle is given by weighting kinematic and dynamic estimations.