9.1: Three-Wheel Cart System Modeling

This section introduces the modeling process of the target three-wheel cart shown in Figure 9.2. A high-quality rechargeable sealed lead-acid battery (12V 12Ah) is on the back of middle deck. The mechatronic system electronics (Arduino MKR 1000 Wi-Fi board, MKR motor carrier board, MPU 6050 three-axis acceleration and rate gyro sensor board, and the MSU interface board) are in the front of the middle deck. Two DC-motor wheel assemblies with associated hall-effect speed sensors are mounted on the bottom-front. Similar to the mini-Segway system, the mini-USB connector on the Wi-Fi board is extended to the top of the three-wheel cart for easy accessing and connection reliability. This connector can be easily ripped off the board. With the full-size USB connector on the top of the cart, the USB connector is secured, even with large pulling force applied to it. In addition, a power switch and a power supply fuse are added to the top-back side of the cart. On the back-bottom, a swivel caster wheel is installed with its direction to be automatically aligned with the cart’s driving direction. The swivel caster is passive without any actuator powering it.

The three-wheel cart in Figure 9.2 is a rigid-body two-actuator robot. It utilizes a low-cost microcontroller (Arduino MKR 1000 Wi-Fi), two motor-wheel sets, and a cart body. Figure 9.3 shows a free body diagram of the cart’s system, where the left and right motors provide the driving torques, \(τ_L\) and \(τ_R,\) respectively. In addition, \(x_{cart}\), \(y_{cart}\), and \(z_{cart}\) are the direction of the cart’s displacement and \(x_{abs}\), \(y_{abs}\), and \(y_{abs}\) are the direction of the cart’s absolute displacements. Also, \(θ_L\) (shown on Figure 9.3) and \(θ_R\) (not shown) are the left and right wheel rotational angles, with direction defined in Figure 9.3 for the left wheel and the right wheel in opposite direction, respectively; \(\phi\) is the cart body yaw angle, respectively. Finally, \(l\), \(l_{cfw}\), \(d_{cmg},\ \)and \(h\) are the longitudinal distance between the front and back wheel axes, the longitudinal distance between the front wheel axis and center of mass; the vertical distance from the center of mass to ground, and the yaw distance between the two front wheels, respectively.

With all parameters and variables defined in Figure 9.3, similar to the mini-Segway system in Chapter 8, we can conduct the following step-by-step modeling process.

We’ll start with the DC motor and wheel dynamics. The motor and wheel assembly is the same as the one shown in Figure 8.4 with a different wheel mass \(m_w\), axis moment of inertia \(I_w\), and radius \(r_w\). (See Table 9.1 for these numerical values.) Using the same approach as that in Section 8.1, we have the following, assuming that the DC-motor armature winding inductance \(L_a\approx 0\).

\[I_w\dfrac{dω_L}{dt}=I_w\dfrac{d{\dot{θ}}_L}{dt}=τ_{mL}-bω_L-τ_L \label{9.1}\]

\[\begin{array}{c} τ_{mL}=\dfrac{-K_τK_m}{R_a}ω_L+\dfrac{K_τ}{R_a}u_{aL}=-\dfrac{K_τK_m}{R_a}{\dot{θ}}_L+\dfrac{K_τ}{R_a}u_{aL} \end{array} \label{9.2}\]

where \(u_{aL}\) is the left DC-motor armature voltage; \(R_a\) is the DC-motor armature winding resistance; \(τ_{mL}\), \(τ_L\), and \(τ_{fL}=bω_L\) are the left DC-motor output torque, load torque applied to the left wheel due to vehicle motion, and the left motor axis and bearing friction torque; respectively; \(I_w\) is the total left combined moment of inertia of left wheel, wheel axis, and left motor rotor; and \(θ_L\) and \(ω_L={\dot{θ}}_L\) are the rotational angular displacement and speed of the left wheel, respectively. Finally, \(K_τ\) and \(K_m\) are the DC-motor torque and electromotive force (EMF) coefficients, respectively, where motor output torque \(τ_{mL}=K_τi_{aL}\) and electromagnetic field voltage \(e_{mL}=K_mω_L\). Table 9.1 lists numerical values of these parameters.

Consider the right wheel spinning in the opposite direction of the left one. Assuming that the right motor and wheel have the same parameters as the left one, we have the following equations for the right motor and wheel assembly.

\[I_w\dfrac{dω_R}{dt}=I_w\dfrac{d{\dot{θ}}_R}{dt}=τ_{mR}-bω_R-τ_R \label{9.3}\]

\[\begin{array}{c} τ_{mR}=\dfrac{-K_τK_m}{R_a}ω_R+\dfrac{K_τ}{R_a}u_{aR} \end{array} =-\dfrac{K_τK_m}{R_a}{\dot{θ}}_R+\dfrac{K_τ}{R_a}u_{aR} \label{9.4}\]

where the subscript \(R\) is used for the right motor and wheel model above. Now, consider the motion of wheel moving forward. In this case, motor torque \(τ_{mR}\) (\(τ_{mL}\)) works against motor friction torque \(τ_{fR}=b{\dot{θ}}_R=bω_R\) (or \(τ_{fL}=b{\dot{θ}}_L=bω_L\)) and drives wheel and three-wheel cart assembly by \(τ_R\) (\(τ_L\)). The friction torque is due to both bearing and gear friction. As a result, the following equation for the right wheel can be obtained.

\[\begin{array}{c} τ_R=τ_{mR}-τ_{fR}=τ_{mR}-b{\dot{θ}}_R=\dfrac{-K_τK_m}{R_a}{\dot{θ}}_R+\dfrac{K_τ}{R_a}u_{aR}-b{\dot{θ}}_R \end{array} \label{9.5}\]

For the left motor wheel assembly, we have the following. Assume motor and wheel parameters are the same.

\[\begin{array}{c} \begin{array}{c} τ_L=τ_{mL}-τ_{fL}=τ_{mL}-b{\dot{θ}}_L=\dfrac{-K_τK_m}{R_a}{\dot{θ}}_L+\dfrac{K_τ}{R_a}u_{aL}-b{\dot{θ}}_L \end{array} \end{array} \label{9.6}\]

During the preliminary driving tests of the three-wheel cart, we found that with synchronized left- and right-front wheel speeds, the three-wheel cart slides toward the low ground side when the driving surface is not leveled. This challenge is the main motivation for modeling the lateral motion of the three-wheel cart—due to the lateral gravity force applied to cartwheels when the ground is not leveled. Before diving into the model, however, let’s review some of the literature.

Vehicle dynamics and system modeling has been developed for decades, with applications to the intelligent transportation system (ITS) aspects [12], autonomous driving [13], and connected vehicles [14]. Tire dynamics, a core vehicle characteristic, has been extensively studied. Studies include the Fiala model in the 1950s [15], Brush model [16], Dugoff in the 1970s [17], Magic Formula [18], UniTire [19], LuGre [20], and FTire model in the 1990s [21]. In general, tire models can be divided into three groups: theoretical, semi-empirical, or pure empirical. A set of equations and lookup tables are used to map the tire operating parameters, e.g., slip-rate to these forces applied to the tire.

Almost all vehicle-controlled motions are through its tire forces, which should be accurately calculated to achieve the desired control performance [22], [23], [24]. Due to the high cost and challenge of direct measurement, tire lateral forces are normally estimated [25] using model-based or empirical approach. For example, we can estimate tire forces, vehicle lateral-slip angle, and wheel cornering stiffness by measurements such as yaw rate, longitudinal/lateral accelerations, steering angle, and angular wheel velocities. These are available from sensors installed on modern automotives [26]. They use estimation methods such as sliding mode observer, Kalman filter [27], extended Kalman filter [28], unscented Kalman filter [29], and empirical approach [30].

To model the tire lateral-slip of a three-wheel cart, we will use an empirical approach due to the lateral force caused by the lateral gravity component on unleveled ground. The lateral force is perpendicular to the direction the wheel is pointing (rotating) and happens in the center of the tire contact patch. See Figure 9.4, where \(τ_L\) is the left motor output torque; \(f_{axL}\) and \(f_{azL}\) are forces applied to the left wheel axes in \(x_{cart}\) and \(z_{cart}\) directions, respectively; and \(f_{xL}\) and \(f_{yL}\) are forces applied to the left wheel ground-contact point in \(x_{cart}\) and \(y_{cart}\) directions, respectively.

The left wheel torque balance around ground contact point (red dot in Figure 9.4) yields the following by applying Newton’s law.

\[τ_L-f_{axL}r_w=τ_{mL}-b{\dot{θ}}_L-f_{axL}r_w=(I_w+m_wr^2_w){\ddot{θ}}_L \label{9.7}\]

where \(r_w\) is the effective wheel-tire radius; and \(I_w\) and \(m_w\) are the wheel assembly (including gearbox and DC-motor rotor) moment of inertia around its axis and wheel assembly mass, respectively. Combining equations \ref{9.6} and \ref{9.7} leads to the following equation.

\[\left(I_ω+m_ωr^2_w\right){\ddot{θ}}_L=\dfrac{-K_τK_m}{R_a}{\dot{θ}}_L+\dfrac{K_τ}{R_a}u_{aL}-b{\dot{θ}}_L-f_{axL}r_w=τ_{mL}-b{\dot{θ}}_L-f_{axL}r_w \label{9.8}\]

Similarly, the following equation can be obtained using the same process for the right wheel, assuming all parameters are the same as the left one.

\[\left(I_ω+{m_ωr}^2_w\right){\ddot{θ}}_R=\dfrac{-K_τK_m}{R_a}{\dot{θ}}_R+\dfrac{K_τ}{R_a}u_{aR}-b{\dot{θ}}_R-f_{axR}r_w=τ_{mR}-b{\dot{θ}}_R-f_{axR}r_w \label{9.9}\]

Now, consider the force in the \(y_{cart}\) direction. Assume the three-wheel cart vertical axis \(z_{cart}\) is not in line with the reference axis \(z_{abs}\). Gravity force \(f_g\) (see the blue arrow in Figure 9.5), in the opposite direction of \(z_{abs}\), can be factorized into components in \(x_{cart}\), \(y_{cart}\), and \(z_{cart}\) directions, as shown in Figure 9.5.

The angle between \(z_{abs}\) and \(z_{cart}\) is defined as \(θ_z\). In addition, \(f_{gz}\) and \(f_{gxy}\) are the projected gravity forces of \(f_g\) in the three-wheel cart coordinate in \(z_{cart}\) direction and the plane formed by \(x_{cart}\) and \(y_{cart}\) axes, respectively. Next, let the angle between \(f_{gxy}\) and \(y_{cart}\) be defined as \(θ_y\). We can calculate the gravity force components \(f_{gx}\) and \(f_{gy}\) with known pair of \(θ_z\) and \(θ_y\) using the following equations and the relationships shown in Figure 9.5.

\[ \begin{aligned} f_{gx} &=f_g \sin θ_z \sin θ_y  \\[4pt] f_{gy} &=f_g \sin θ_z \cos θ_y \end{aligned} \label{9.10}\]

The relation between lateral force and slip angle is reciprocal. The lateral force can be considered as a result of the slip angle, and the slip angle as a result of lateral force. Figure 9.6 defines the slip angle, where \(v_L\) is the left wheel speed, \(v_{xL}\) is the projection of the left wheel speed to the \(x_L\) axis; \(v_{yL}\) is the projection of the left wheel speed to the \(y_L\) axis; and the slip-angle \(α\) is defined as the angle between the \(x_L\) axis and left wheel speed vector \(v_L\).

Slip-angle is proportional to the lateral force for a given tire property. That is, the larger the lateral force, the larger the slip-angle, where the lateral force could be caused by centrifugal force generated by the vehicle turning motion or by external force. Centrifugal force [32] increases by decreasing the turn radius or increasing the vehicle’s linear speed. The external force could be caused by unleveled road due to gravity or gust wind. As the slip angle \(α\) increases, some lateral sliding begins to occur in the tire tread. If the slip angle is further augmented, eventually the tire will break loose and slide over the contact surface, which could happen easily for a solid wheel. For the three-wheel cart used in this chapter, the wheel-tire surface is relatively hard without tread. As a result, the lateral-slip considered is mainly from the tire sliding over the contact surface.

In general, there are three ranges of tire behavior when slip-angle \(α\) is not zero. These are no sliding (deformation of tire), some sliding (transitional range), and total sliding (frictional range) [1]. Within the sliding ranges, the sliding between the tire and the ground is caused only by lateral force applied to the associated wheel. Figure 9.7 shows the relationship between slip-angle and lateral force for the three-wheel cart. In the transitional range, the lateral force increases as the slip angle goes up. In the frictional range, the lateral force peaks at its left boundary; instead, it goes down a bit as the slip-angle increases.

The lateral force caused by gravity due to unleveled ground is relatively small for the three-wheel cart. Thus, we assume in the modeling process that slip-angle \(α\) is a linear function of lateral force. Figure 9.7 indicates the relationship between slip-angle and lateral force is linear when the lateral force is small. Regarding slip angle \(α\), where \(f_{ygL}\) is defined in Figure 9.6, we can assume the following.

\[α=c_{sa}f_{ygL} \label{9.11}\]

In equation \ref{9.11}, \(c_{sa}\) is the slip-angle coefficient to be calibrated. The coefficient \(c_{sa}\) is also a function of the normal force applied to the wheel. In this study, because \(θ_z\) is small, we assume the normal force is a constant. Another advantage of modeling the lateral slip using the slip-angle is that the resulting lateral slip speed is proportional to the tire linear speed. This is natural because the higher the tire rotational speed, the lower the lateral friction coefficient. We assume that the rear wheel cannot generate any reaction force in \(y_{cart}\) direction because it is a swivel caster wheel; gravity force \(f_{yg}\) in \(y_{cart}\) direction is evenly distributed on both left and right wheels. Therefore, the lateral force \(f_{ygL}\) applied to the left wheel is \(f_{ygL}=0.5f_{yg}\), where \(f_{yg}\) is defined in equation \ref{9.10}. In turn, we can calculate the slip-angle below for one wheel. Parameter \(c_{sa}\) and gravity force \(f_g\) can be combined as a single calibration if \(f_g\) is fixed.

\[α=0.5c_{sa}f_g{\mathrm{sin} θ_z\ }{\mathrm{cos} θ_y\ } \label{9.12}\]

Because the tire lateral slip speed is modeled using the slip-angle, we will study forward and turning motion first. Based on the obtained linear wheel speed, we can calculate the cart lateral speed based upon the slip angle \(α\). In the following, we assume that there is no slip between wheel and ground in the tire directions \(x_L\) and \(x_R\). We have \(x_L=θ_Lr_w\) and \(x_R=θ_Rr_w\) that yield the following.

\[{\dot{x}}_L={\dot{θ}}_Lr_w,\ \ {\ddot{x}}_L={\ddot{θ}}_Lr_w,\ \ {\dot{x}}_R={\dot{θ}}_Rr_w,\ \ {\ddot{x}}_R={\ddot{θ}}_Rr_w \label{9.13}\]

Equations \ref{9.8} and \ref{9.9} can be rewritten as functions of left and right wheel axis linear displacements, \(x_L\) and \(x_R\), as below.

\[\begin{array}{c} f_{axL}=\dfrac{-K_τK_m}{R_ar^2_w}{\dot{x}}_L+\dfrac{K_τ}{R_ar_w}u_{aL}-\dfrac{b{\dot{x}}_L}{r^2_w}-\dfrac{I_ω{\ddot{x}}_L}{r^2_w}-m_ω{\ddot{x}}_L \end{array} \label{9.14}\]

\[\begin{array}{c} f_{axR}=\dfrac{-K_τK_m}{R_ar^2_w}{\dot{x}}_R+\dfrac{K_τ}{R_ar_w}u_{aR}-\dfrac{b{\dot{x}}_R}{r^2_w}-\dfrac{I_ω{\ddot{x}}_R}{r^2_w}-m_ω{\ddot{x}}_R \end{array} \label{9.15}\]

Axis forces, \(f_{axL}\) and \(f_{axR}\), in equations \ref{9.14} and \ref{9.15}, are applied to the three-wheel cart body as shown in Figure 9.8. Next, we find the equations of motion for the three-wheel cart with \(x_{cart}\) and \(\phi\) representing the displacement and turning angle at the three-wheel cart center of mass location, respectively.

Assume that the three-wheel cart mass is represented by \(m_{cart}\) (see Table 9.1). Wheel forces \(f_{axL}\) and \(f_{axR}\) in \(x_L\) and \(x_R\) directions, along with the gravity force \(f_{gx}\) in \(x_{cart}\) direction at three-wheel cart center of mass (dark-dot in Figure 9.8), can be used to develop the equation of motion in \(x_{cart}\) direction as below, utilizing Newton’s law.

\[m_{cart}{\ddot{x}}_{cart}=f_{axL}+f_{axR}+f_{gx} \label{9.16}\]

Utilizing the following relationships

\[ \begin{aligned} \dot{x}_{cart} &=0.5(\dot{x}_L + \dot{x}_R), \\[4pt] \ddot{x}_{cart} &=0.5(\ddot{x}_L+ \ddot{x}_R) \end{aligned} \label{9.17}\]

and substituting equations \ref{9.10}, \ref{9.14} and \ref{9.15} into equation \ref{9.16} yields the following equation.

\[m_{cart}{\ddot{x}}_{cart}= \begin{array}{c} \dfrac{-2K_τK_m}{R_ar^2_w}{\dot{x}}_{cart}+\dfrac{K_τ}{R_ar_w}(u_{aL}+u_{aR})-\dfrac{2b{\dot{x}}_{cart}}{r^2_w}-\dfrac{2I_ω{\ddot{x}}_{cart}}{r^2_w}-2m_ω{\ddot{x}}_{cart} \end{array} ~+f_{gx} \label{9.18}\]

where \(f_{gx}\) is the gravity force applied to three-wheel cart body, as defined in equation \ref{9.19} and also in equation \ref{9.10}.

\[f_{gx}=f_g \sin θ_z \sin θ_y \label{9.19}\]

Substituting equation \ref{9.19} into equation \ref{9.18}, reorganizing, and simplifying the resulting equation, yield the first equation of motion for the three-wheel cart as follows.

\[\left(m_{cart}+2m_ω+\dfrac{2I_ω}{r^2_w}\right){\ddot{x}}_{cart}+2\left(\dfrac{K_τK_m}{R_ar^2_w}+\dfrac{b}{r^2_w}\right){\dot{x}}_{cart}=\dfrac{K_τ}{R_ar_w}\left(u_{aL}+u_{aR}\right)+f_g{\mathrm{sin} θ_z\ }{\mathrm{sin} θ_y } \label{9.20}\]

Next, we consider the three-wheel cart turning motion. We start with turning motion \({\phi }_x\) due to left and right wheel speed difference in \(x_{cart}\) direction. Then we will consider the combined turning motion \(\phi\) of \({\phi }_x\) and lateral-slip turning \({\phi }_{ts}\) to provide the second equation of motion. Let the combined three-wheel cart moment of inertia \(I_{cart}\) in the \(z_{cart}\) direction be

\[\begin{array}{c} I_{cart}=2\left(l^2_{cfw}+r^2_v\right)m_w+\left(I_{zbat}+l^2_bm_{bat}\right)+I_{cz} \end{array} \label{9.21}\]

The first term in equation \ref{9.21} is the moment of inertia of two DC motor and wheel assemblies around the center of gravity. The second term is the moment of inertia of the battery around the center of gravity. The last term is the cart moment of inertia of the cart without the battery and two motor-wheel assemblies. Parameters \(m_w\), \(l_{cfw}\), and \(r_v\) are respectively the motor and wheel assembly mass, the distance from the center of the cart total mass to the front axis defined in Figure 9.3, and the half distance between the two front wheels defined in Figure 9.8. Parameters \(m_{bat}\), \(l_b\), and \(I_{zbat}\) are battery mass, distance between battery and cart centers of mass, and battery moment of inertia around its center of mass, respectively. Finally, parameter \(I_{cz}\) is the cart moment of inertia without battery and two motor-wheel assemblies. With the cart total moment of inertia \(I_{cart}\) defined, applying Newton’s law leads to the following equation of motion.

\[\begin{array}{c} I_{cart}{\ddot{\phi }}_x=f_{axL}r_v-f_{axR}r_v \end{array} \label{9.22}\]

Utilizing the following relationships

\[{\dot{\phi }}_x=\dfrac{({\dot{x}}_L-{\dot{x}}_R)}{r_v},\ \ \ \begin{array}{c} {\ddot{\phi }}_x=\dfrac{({\ddot{x}}_L-{\ddot{x}}_R)}{r_v} \end{array} ~~~ \label{9.23}\]

and substituting equations \ref{9.14} and \ref{9.15} into equation \ref{9.22} yields the following equation.

\[I_{cart}{\ddot{\phi }}_x=-\dfrac{r^2_v}{r^2_w}\left(b+\dfrac{K_τK_m}{R_a}\right){\dot{\phi }}_x-\dfrac{r^2_v}{r^2_w}\left(I_ω+m_ωr^2_w\right){\ddot{\phi }}_x+\dfrac{r_vK_τ}{r_wR_a}\left(u_{aL}-u_{aR}\right) \label{9.24}\]

The above equation is part of the second equation of motion for the three-wheel cart and can be simplified as the following.

\[\left(I_{cart}+\dfrac{r^2_v}{r^2_w}I_ω+m_ωr^2_v\right){\ddot{\phi }}_x+\dfrac{r^2_v}{r^2_w}\left(b+\dfrac{K_τK_m}{R_a}\right){\dot{\phi }}_x=\dfrac{r_vK_τ}{r_wR_a}\left(u_{aL}-u_{aR}\right)~ \label{9.25}\]

Next, we will investigate the cart rotation factor \({\dot{\phi }}_g\) due to tire lateral-slip based on slip-angle \(α\), and use it to find the lateral slip-speed. The three-wheel cart has two fixed wheels; considering individual wheel tire-slip is not feasible. Instead, we assume a virtual wheel in the middle of the left and right wheels, with the resulting wheel longitudinal speed of \({\dot{x}}_{cart}\). The rear swivel caster will not slide because its direction will be changed to follow the cart motion automatically. The following formula calculates the lateral gravity force applied to the virtual wheel, assuming that no lateral gravity force is applied to the rear caster wheel.

\[f_{gy}=f_g \sin θ_z \cos θ_y \label{9.26}\]

The resulting slip-angle can then be calculated as below.

\[α=c_{sa} f_{gy}=c_{sa} f_g \sin θ_z \cos θ_y \label{9.27}\]

The resulting lateral speed and associated angular speed can be calculated below based upon Figure 9.7.

\[{\dot{y}}_g= \dot{x}_{cart}{\tan α } = {\dot{x}}_{cart}{\tan \left(c_{sa}f_g{\sin θ_z\ } {\cos θ_y\ }\right)\ },\ \dot{\phi }_g=-\dfrac{\dot{y}_{ts}}{l_{cfw}}=-β_3{\dot{x}}_{cart} \label{9.28}\]

Let total angular speed \(\dot{\phi }\) be the summation of \({\dot{\phi }}_x\) and \({\dot{\phi }}_g\). That is, \(\dot{\phi }={\dot{\phi }}_x+{\dot{\phi }}_g\). Note that \(\ddot{\phi }={\ddot{\phi }}_x+{\ddot{\phi }}_g\). Assuming slow change of \(θ_z\) and \(θ_y\) (\({\dot{θ}}_z={\ddot{θ}}_z=0\) and \({\dot{θ}}_y={\ddot{θ}}_y=0\)) yields the following.

\[ \ddot{\phi}_g = -\dfrac{\ddot{y}_g}{l_{cfw}} = -\dfrac{\tan \left(c_{sa}f_g \sin θ_z \cos θ_y \right)}{l_{cfw}} \ddot{x}_{cart} = -β_3 \ddot{x}_{cart} \label{9.29}\]

Utilizing relations of \(\dot{\phi }= \dot{\phi}_x + \dot{\phi}_g\) and \(\ddot{\phi}= \ddot{\phi}_x + \ddot{\phi}_g\), equations \ref{9.27} and \ref{9.29} lead to the following equation (the second equation of motion).

\[β_1 \ddot{\phi} + β_2 \dot{\phi} + β_1β_3 \ddot{x}_{cart} + β_2β_3 \dot{x}_{cart} = \dfrac{r_vK_τ}{r_wR_a} \left(u_{aL} - u_{aR}\right) \label{9.30}\]

where

\[β_1=I_{cart}+\dfrac{r^2_v}{r^2_w}I_ω+m_ωr^2_v, \\[4pt] β_2=\dfrac{r^2_v}{r^2_w}\left(b+\dfrac{K_τK_m}{R_a}\right), \\ β_3=\dfrac

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Simplifying both equations of motion in \ref{9.30} and \ref{9.20} results in the following two equations of motion for the three-wheel cart considering unleveled road surface.

\[\begin{aligned} \ddot{x}_{cart} &=-\dfrac{2α_2}{α_1} \dot{x}_{cart} + \dfrac{K_τ}{α_1R_ar_w}{(u}_{aL} + u_{aR}) \end{array} + \dfrac{α_3}{α_1} \\[4pt] \ddot{\phi } &= -\dfrac{β_2}{β_1} \dot{\phi} - \left(\dfrac{β_2}{β_1} + \dfrac{2α_2}{α_1}\right) β_3 \dot{x}_{cart} + \dfrac{r_vK_τ}{β_1r_wR_a} \left(u_{aL} - u_{aR}\right) -\dfrac{β_3K_τ}{α_1R_ar_w}{(u}_{aL}+u_{aR})-\dfrac{α_3β_3}{α_1} \end{aligned} \label{9.32}\]

where \(α_1\), \(α_2\), and \(α_3\) are defined below.

\[ \begin{aligned} α_1 &= m_{cart} + 2m_ω + \dfrac{2I_ω}{r^2_w}, \\[4pt] α_2 &=\dfrac{K_τK_m}{R_ar^2_w} + \dfrac{b}{r^2_w}, \\[4pt] α_3 &=f_g \sin θ_z \sin θ_y \end{aligned}\label{9.33}\]

From equations \ref{9.31}, \ref{9.32}, and \ref{9.33}, we find that when \(θ_y=π/2\), then \(β_3=0\). That is, the lateral force is zero, leading to zero slip-angle. In this case, the turning motion equation in \ref{9.32} is shown in equation \ref{9.34}, indicating that the turning motion is caused only by the difference between the DC-motor armature voltages. When the difference is zero, the turning will be gradually stopped with nonzero initial turning speed or no turning motion with zero initial speed. That is, if \(θ_y=π/2\), the gravity force will not generate lateral motion due to zero tire slip in \(y_{cart}\) direction.

\[\begin{array}{c} \ddot{\phi }=-\dfrac{β_2}{β_1}\dot{\phi } + \dfrac{r_vK_τ}{β_1r_wR_a}\left(u_{aL}-u_{aR}\right) \end{array} \label{9.34}\]

When \(θ_y=π/2\), then \(θ_z\neq 0\), and \(α_3\neq 0\), resulting in additional force applied to the three-wheel cart to accelerate or decelerate the cart. If \(θ_y=0\), then \(α_3=0\). Thus, the cart speed is affected only by motor driving torques. As a result, if \(θ_y=0\), gravity force will not affect the movement in the \(x_{cart}\) direction.

Table 9.1 lists all model parameter definitions and associated values for the target three-wheel cart mechatronic system. It also lists key model parameters, including \(α_1\), \(α_2\), \(α_3\), \(β_1\), \(β_2\), and \(β_3\).

The model in equation \ref{9.32} is linear if \(θ_z=0\) or both \(θ_z\) and \(θ_y\) are constant. That is, the three-wheel cart system is linear if both \(θ_z\) and \(θ_y\) are constant, under the assumption there is no longitudinal tire slip for both driving wheels. When neither \(θ_z\) nor \(θ_y\) is constant, the three-wheel cart system is a linear-parameter-varying (LPV) system [33], [34]. LPV control techniques can be used for model-based control design. However, when the rates of variation for both \(θ_z\) and \(θ_y\) are slow compared with the system states, \(x_{cart}\), \({\dot{x}}_{cart}\), \(\phi\), and \(\dot{\phi }\), multiple model-based controllers can be designed for an LTI (linear time-invariant) system model with fixed \(θ_z\) and \(θ_y\), and control gain can be interpreted for a gain-scheduling control implementation [35]. For this application, LPV control is recommended; see Section 9.3.

Figure 9.9 shows a Simulink implementation of the mathematical three-wheel cart model described by equation \ref{9.32}. Two second-order differential equations are implemented with four integrators. Parameters \(α_1\), \(α_2\), \(α_3\), \(β_1\), \(β_2\), and \(β_3\) are calculated based upon equations \ref{9.31} and \ref{9.33}. The Simulink model in Figure 9.9 has four inputs. They are left and right motor armature voltage inputs (\(u_{aL}\) and \(u_{aR}\)) and the unleveled ground angles (\(θ_z\) and \(θ_y\)) with respect to the cart vertical \(z\)-axis and lateral \(y\)-axis (defined in Figure 9.7).

The Coulomb friction in the motor gear boxes is not modeled in the three-wheel cart model because there is no high-frequency forward and backward motions for the three-wheel cart. The Coulomb friction model in the mini-Segway model is used mainly to handle the forward and backward model for stabilizing the mini-Segway pendulum motion at high frequency. The simulation study of the Simulink model in Figure 9.9 will be conducted in Section 9.4 of this Chapter.