**2. Materials and Methods**

#### *2.1. Actuator Modeling*

The schematics of EHB, EMB, and EWB brakes are shown in Figure 1a–c.

EHB model consists of a high-pressure source (master cylinder), hydraulic lines, build and dump valves, a brake cylinder chamber, and brake pads. The master cylinder provides pressure into the high-pressure line, controlled by the build and dump valves. Build and dump valves are considered to have varying states between their fully open and fully closed states. This is as opposed to the valves that can either be fully open or fully closed at a given time. In practice, these are solenoid valves that can be controlled with pulse width modulation. For the sake of initial comparison between these smart brake actuators, a vehicle model containing only one-wheel is utilized. When the pressure increases in the brake cylinder chamber, this pressure will move the brake pad forward. This forward movement of the braking pad stops the brake disk as a result. Upon stopping, the dump valve opens, decreases the pressure, and releases the brake pads, bringing them back to their original position.

The EMB comprises a small electric motor, planetary gear set, ball-screw mechanism, brake pad, and caliper. A planetary gear set and a ball screw mechanism move the brake pad when the motor rotates. This movement will result in a clamp force that is denoted by *Fcl* as illustrated in Figure 1b.

The EWB actuator converts the motor's rotation to a linear force on the wedge by using a planetary gear set (not depicted in the schematic) and a roller screw. The motor shaft's axial stiffness and resistance are also considered in modeling this actuator. *Kcal* represents the combined caliper stiffness and the stiffness between the wedge and the disk. This is similar to the EMB configuration, except that the caliper is shaped like a wedge, which, by inserting it inside the brake casing, creates a self-reinforcing mechanism.

Bond graph is a graphical modeling approach for dynamical systems based on the flow/exchange of power, and therefore, energy. Among the many benefits of bond graphs, they are suitable for the systems with multiple energy domains such as mechatronic systems that usually include various electronic, electrical, mechanical, and hydraulic components [32]. Bond graphs are multi-energy domain and open architecture, which means one can easily add and expand the models with minimum effort compared to other modeling techniques. Furthermore, the monitoring and processing power and energy consumption of various components and parts are conducted with ease when using bond graphs. Given the mentioned benefits of this modeling technique, this method is adopted here to study and model BBW systems.

**Figure 1.** Schematics of brake-by-wire actuators. (**a**) Electro-Hydraulic Brake [33]; (**b**) Electro-Mechanical Brake; (**c**) Electronic Wedge Brake [9].

Figure 2a–c show the bond graph of EHB, EMB, and EWB, respectively. A one-wheel vehicle model is included in all the actuator bond graph models. The wheel has rotational inertia and is connected to a point mass. For the preliminary studies of brake actuators and their algorithms (for example, the Anti-Lock Braking System, ABS, and Traction Control System, TCS), this simple one-wheel model can be used and is easy to implement later on

(**c**)

a hardware-in-the-loop test. Models such as this can be used for studying longitudinal dynamics in the vehicle. Since it focuses only on the longitudinal dynamics of the vehicle, it is perfectly suited for studying brake-by-wire actuators and ABS technologies [34].

**Figure 2.** Bond graphs of brake-by-wire actuators.

Based on the bond graphs in Figure 2a–c, the equations of motion for EHB, EMB, and EWB can be written. Equations (1)–(4) represent the equations of motion for the EHB. *qcyl*, *pp*, *xcal*, *Pin*, *ub*, and *ud* are the volumetric displacement of the cylinder fluid, momentum

of the caliper, caliper displacement, pressure of the master-cylinder (high pressure input), duty ratio of build valve, and duty ratio of dump valve (between 0 and 1), respectively.

*Cd*, *Sb*, *Sd*, *ρ*, *βh f* , *Vcyl*, *Sp*, *bp* ,*mp*, *x*0, and *kcal* are the maximum flow coefficient of the valve, cross-sectional area of the build valve when fully open, cross-sectional area of the dump valve when fully open, density of the brake fluid, bulk modulus of the brake fluid, cylinder's volume, cylinder's cross-section surface, damping coefficient, brake pad's mass, brake clearance, and caliper stiffness, respectively [33]. Since these equations are highly nonlinear because of the valves, a linearized version, for the purpose of control development, is given in Equations (5)–(7). In this linearization, it is assumed that *ud* = 1 − *ub*, and this means that when one valve is open, the other is closed.

EHB equations of motion are as follows:

$$\dot{q}\_{cyl} = \mathbb{C}\_d \mathbb{S}\_b \mu\_b \sqrt{\frac{2}{\rho} (P\_{\rm in} - \frac{\beta\_{hf}}{V\_{cyl}} q\_{cyl})} - \mathbb{C}\_d \mathbb{S}\_d u\_d \sqrt{\frac{2}{\rho} (\frac{\beta\_{hf}}{V\_{cyl}} q\_{cyl})} - \frac{\mathbb{S}\_p}{m\_p} p\_p \tag{1}$$

$$\dot{p}\_p = S\_p \frac{\beta\_{hf}}{V\_{cyl}} q\_{cyl} - b\_p \frac{p\_p}{m\_p} - k\_{\text{cal}} \max(\mathbf{x}\_{\text{cal}} - \mathbf{x}\_{0\prime}, 0) \tag{2}$$

$$
\dot{x}\_{cal} = \frac{1}{m\_P} p\_p \tag{3}
$$

$$P\_{cyl} = \frac{\beta\_{hf}}{V\_{cyl}} q\_{cyl} \tag{4}$$

Linearized EHB equations are as follows:

$$A = \begin{bmatrix} \mathcal{C}\_d S\_b \mu\_0 \sqrt{\frac{1}{2\rho}} \frac{-\frac{\beta\_{hf}}{\mathcal{V}\_{cg}}}{\sqrt{(\mathcal{P}\_{in} - \frac{\beta\_{hf}}{\mathcal{V}\_{cg}} q\_{c0})}} - \mathcal{C}\_d S\_d (1 - \mu 0) \sqrt{\frac{1}{2\rho} \frac{\beta\_{hf}}{\mathcal{V}\_{cg}} \frac{1}{\sqrt{q\_{c0}}}} & \frac{-S\_p}{m\_p} & 0\\ & S\_p \frac{\beta\_{hf}}{\mathcal{V}\_{cg}} & -\frac{b\_p}{m\_p} & -k\_{cal}\\ & 0 & \frac{1}{\Delta} & 0 \end{bmatrix} \tag{5}$$

$$\begin{bmatrix} \frac{\partial \mathbf{v}\_p}{\partial \mathbf{v}\_p} & & & & \frac{1}{m\_p} & & \cdots\\ 0 & & & & & \frac{1}{m\_p} & & \mathbf{0} \end{bmatrix}$$

$$B = \begin{bmatrix} \mathcal{C}\_d \mathcal{S}\_b \sqrt{\frac{2}{\rho} (P\_{\rm in} - \frac{\mathcal{C}\_{hf}}{V\_{cyl}} q\_{c0})} + \mathcal{C}\_d \mathcal{S}\_d \sqrt{\frac{2}{\rho} (\frac{\mathcal{C}\_{hf}}{V\_{cyl}} q\_{c0})} & 0 & 0 \end{bmatrix} \tag{6}$$

$$
\begin{bmatrix}
\dot{q}\_{cyl} \\
\dot{p}\_p \\
\dot{x}\_{cal}
\end{bmatrix} = A \begin{bmatrix}
q\_{cyl} \\
p\_p \\
x\_{cal}
\end{bmatrix} + B \,\mu\_b
\tag{7}
$$

Similarly, equations of motion for the EMB can be written using Equations (8a)–(8d). Note that the same nonlinear friction model has been used for the EMB and EWB models. *Im*, *Vin*, and *ω<sup>m</sup>* are current, voltage input, and angular velocity of the shaft, respectively. *Lm*, *Rm*, *Kt*, *Jm*, *Dm*, *Ns*, *Np*, and *Kcal* are the inductance of the electric motor, electrical resistance, electromotive force constant, total moment of inertia of the rotational parts (including the shaft and gears), axial viscous friction, planetary gear reduction ratio, ball-screw gear reduction ratio, and caliper stiffness, respectively.

EMB equations of motion are as follows:

$$I\_m = \frac{1}{L\_m} \times \left(V\_{\rm in} - R\_m \times I\_m - K\_t \times \omega\_m\right) \tag{8a}$$

$$\dot{\omega}\_m = \frac{1}{J\_m} \times \left(\mathbf{K}\_t \times I\_m - D\_m \times \omega\_m - \tau\_f - \mathbf{N}\_p \times \mathbf{N}\_s \times \mathbf{K}\_{cal} \times \max(\mathbf{X}\_{cal} - \mathbf{x}\_{0\prime}\mathbf{0})\right) \tag{8b}$$

$$
\dot{X}\_{\text{cal}} = \mathbf{N}\_{\text{s}} \times \mathbf{N}\_{p} \times \boldsymbol{\omega}\_{\text{m}} \tag{8c}
$$

$$F\_{\rm Cal} = \begin{cases} \begin{array}{l} \mathcal{K}\_{\rm cal}(X\_{\rm cal} - \mathfrak{x}\_{0}), & \text{if} \quad X\_{\rm cal} \ge \mathfrak{x}\_{0} \\ 0, & \text{otherwise} \end{array} \end{cases} \tag{8d}$$

Equations (9a)–(9h) show the equations of motion for the EWB, where *N*, *qax*, *Kax*, *Dax*, *Xw*, *Vw*, *Fm*, *α*, and *μcal* are combined gear reduction, shaft axial displacement, shaft axial stiffness, shaft axial viscous resistance, wedge displacement, wedge velocity, motor force exerted to the wedge, wedge angle, and friction coefficient between the pad and the wheel, respectively.

EWB equations of motion are as follows:

$$\dot{I}\_{\text{fl}} = \frac{1}{L\_{\text{m}}} \times \left(V\_{\text{in}} - i\_{\text{m}} \times R\_{\text{m}} - K\_{\text{m}} \times \omega\_{\text{m}}\right) \tag{9a}$$

$$
\dot{q}\_{\text{ax}} = LN\omega - \frac{V\_w}{\cos(\alpha)}\tag{9b}
$$

$$F\_{\rm m} = K\_{\rm ax} q\_{\rm ax} + D\_{\rm ax} \dot{q}\_{\rm ax} \tag{9c}$$

$$
\dot{\omega} = \frac{1}{f\_m} \{ K\_m I\_m - D\_m \times \omega\_- - \tau\_f - L \times N \times F\_m \} \tag{9d}
$$

$$\dot{V}\_w = \frac{1}{m\_w(1 + \tan^2(a))} \times \left\{ \frac{F\_m}{\cos(a)} + (K\_{\rm cal} \times X\_w \times \tan(a) \times (\mu\_{\rm cal} - \tan(a))) \right\} \tag{9e}$$

$$
\dot{X}\_w = V\_w \tag{9f}
$$

$$F\_B = \mu\_{cal} K\_{cal} X\_w \tan(\alpha) \tag{9g}$$

$$F\_{\rm Cal} = \begin{cases} \ K\_{\rm cal}(X\_{\rm cal} - x\_0), & \text{if} \quad X\_w \ge x\_0 \\ 0, & \text{otherwise} \end{cases} \tag{9h}$$

*τ<sup>f</sup>* is the lumped nonlinear frictions present in the shaft, planetary gears, and worm gear. The Lugre friction model has been used to model this nonlinear friction. The Lugre model is used for modeling the frictions in actuators since it offers a dynamical model which captures the dynamics very well while needing a lower number of parameters. Other types of friction models can be used as well to represent the frictions. Equation (10a) represents the Lugre friction model [35] where *σ*0, *σ*1, *σ*2, *ωs*, j, *τc*, and *τ<sup>s</sup>* are the contact (bristle) stiffness, damping coefficient of the bristle, viscous friction coefficient, Stribeck velocity, shape factor, Coulomb friction, and static friction, respectively. Equation (10d) shows that there is a linear relationship between the Coulomb friction and the clamping force, which is usually derived through experiment. As clamping force increases, the normal forces inside the gears increase as well, which results in increasing the friction torque [6].

Lugre dynamic friction model for EMB and EWB is as follows:

$$g(v) = \tau\_c + (\tau\_s - \tau\_c) \times e^{-|\frac{\partial \omega}{\partial s}|^\dot{j}} \tag{10a}$$

$$\dot{z} = \omega - \sigma\_0 \times \omega \times \frac{Z}{g(v)}\tag{10b}$$

$$
\pi\_f = \sigma\_0 \times z + \sigma\_1 \times \dot{z} + \sigma\_2 \times \omega \tag{10c}
$$

$$
\pi\_{\mathbb{C}} = \mathbb{C} + \mathbb{G} \times F\_{\mathbb{C}al} \tag{10d}
$$
