## 1 Introduction  
Active safety and intelligent control technologies in vehicles have been widely developed and applied, and most active safety systems will often benefit from a reliable and robust estimation of brake force. The brake force information is essential for a better tracking precision of potential trajectories, resulting in better vehicle control management. However, it is usually difficult to install sensors at contact points of the tires to collect the information directly due to the limited physical accessibility. Accordingly, it is more advantageous and beneficial for practical applications if we can develop methods to indirectly estimate or observe brake force based on the pre-existing sensor information. In line with this sense, previous studies of indirect tire force estimation are actively conducted and can be classified into two major approaches. As the first stream, the tire model-based methods used analytical models such as Burckhardt static tire/road friction model [1], Duggoff's model [2], and the most widely used, Pacejka model, also called \"Magic formula\" [3]. However, as all other static models, these models contain parameters that need to be estimated and disadvantages related to the inability to describe low slip effects. Meanwhile, as the second one, the sensor-based methods utilized EKF (Extended Kalmanfiltering) as a complex non-linear estimation technique to derive the tire force in [4]. Ref. [5] applied an EKBF (extended Kalman-Bucy filtering) and Bayesian hypothesis selection to estimate tire forces of vehicles on asphalt surfaces. An adaptive observer was constructed by combining a vehicle model and a tire force model by [6]. These approaches show accuracy and good robustness properties. Nonetheless, Refs. [4; 5; 6] generate inaccurate estimation results in some extreme road conditions and require a high computational complexity. Therefore, they are practically limited and often denied to be mounted in the commercial ECU (electronic control unit). Fundamentally, according to Equation (1), to achieve a better result of tire force \(F_{B}\) during brake, we should identify the precise static maximum road friction coefficient \(\mu_{max}\) and vertical tire forces \(F_{z}\). \[F_{B}=\mu_{max}F_{z} \tag{1}\]  
Consequently, many various studies of tire-road friction modeling and estimation have been investigated. [PERSON] [7] proposed a new predictive methodology for the estimation of \(\mu\) by using a camera and a microphone. [PERSON] [8] and [PERSON] [9] measured the wetness of the road by detecting reflected light using optical sensors. The disadvantages of these vision-based methods depend on the intensity and direction of light. Many commercial vehicles use a thermometer to warn drivers of icy roads. These cause-based methods detect the factors that change road friction. However, this method cannot reflect the vehicle driving conditions, such as variations in tire pressure, wear, or wheel dynamics. Moreover, the longitudinal dynamic based approaches are in general adopted for friction estimation. Many studies proposed friction coefficient identification strategies based on the concept \"slip-slope\" [10; 11; 12; 13]. In all the approaches discussed above, the research results only focused on the estimation of the average tire-road friction coefficient for the entire vehicle. On the other hand, the second element for brake force calculation is the vertical force at each tire as seen in Equation (1). These forces are determined by vehicle mass. [PERSON] et al. [14; 15] proposed an RLS (recursive least-square) mass and road grade estimation strategy that utilizes multiple forgetting factors to reflect a constant mass and time-varying road grade. RLS was also used in [16] and Ref. [17] combined with EKF to simultaneously estimate the electric bus mass and road grade. Meanwhile, Ref. [18] proposed a formulation to calculate the vehicle's mass essentially based on the knowledge of the vehicle's velocity and the engine torque, and Ref.