Various reference frames can be employed for the modeling and analysis of the motion of vehicle, but in reality, three reference frames are used to derive the equations of motion for vehicle mechanics [8] . Now, the 3D motion of the thermal vehicle is then studied in three 3D reference frames, respectively called the mobile frame $R_G\left(G,X_G,Y_G,Z_G\right)$ associated with the vehicle’s center of gravity $G$; aerodynamic frame $R_a$ and the inertial frame fixed to the ground $R_O\left(O,X_O,Y_O,Z_O\right)$. The transition from the inertial frame $R_O$ to the moving frame $R_G$ is carried by a matrix $R$ known as the rotation operator [5] [6] .

The equations describing the dynamics of the thermal vehicle system are summarized in Isaac Newton’s two principles, to which we add the hypothesis of variation in the mass of the moving vehicle [1] [9] :

${\displaystyle \sum F_{ext}}-\frac{\text{d}m}{\text{d}t}{v}_a=m{\stackrel{\dot{}}{v}}_a,{\displaystyle \sum M_{F_{ext}}}=J\left(G,V_{eh}\right)\frac{\text{d}\Omega }{\text{d}t}$(1)

In the above system, $F_{ext}$ are the external forces acting on the moving vehicle, $m\left(t\right)$ is the mass of the vehicle, $v_a$ is the aerodynamic speed of the vehicle, $J\left(G,V_{eh}\right)$ is the inertial matrix of the vehicle and Ω is the rotation speed vector of the vehicle in three directions, as shown in Figure 1 .

The relations between the derivatives in the $R_G$ frame and $R_O$ are given by the following fundamental relation of dynamics or Poisson’s formula [9] [10] :

${\frac{\text{d}r}{\text{d}t}|}_{R_O}={\frac{\text{d}r}{\text{d}t}|}_{R_G}+{\Omega }_{R_G/R_O}\wedge r$(2)

where ${\frac{\text{d}r}{\text{d}t}|}_{R_O}$ is the derivative with respect to time of the vector $r$ in the inertial frame $R_O$, ${\frac{\text{d}r}{\text{d}t}|}_{R_G}$ is the derivative with respect to time of the vector $r$ in the vehicle frame $R_G$, $\Omega$ is the angular velocity of the vehicle and ${\Omega }_{R_G/R_O}$ is the angular velocity $R_G$ with respect to $R_O$ [9] .

The relation (2) means that the derivation is made considering an observatory attached to the $R_O$ frame but the equations are written in the $R_G$ frame. A second-time derivation and then multiplying the two members of the fundamental Equation (2) by the mass $m\left(t\right)$ of the vehicle, and under the additional assumption that the mass of the vehicle varies only in the $R_G$ frame, we obtain the following dynamic system:

$\{\begin{array}{l}\stackrel{\dot{}}{m}=-{\left(C_{cse}\times \tau \times \left[\frac{P_i{v}_e}{LH{V}_{fuel}\cdot {\eta }_m\cdot {\eta }_c}+\left(p_0-p_e\right)\cdot A_e\right]\right)}^{\frac{1}{2}}\hfill \\ {\stackrel{\dot{}}{v}}_a=\frac{1}{2m}\rho A_f{C}_a{\left(v_a+v_{wind}\right)}^2+C_{r}g\cos \theta \hfill \\ {\stackrel{\dot{}}{\gamma }}_a=-\frac{\stackrel{\dot{}}{m}}{2m^2}\rho A_f{C}_a{\left(v_a+v_{wind}\right)}^2+\frac{1}{m}\rho A_f{C}_a{\stackrel{\dot{}}{v}}_a\left(v_a+v_{wind}\right)-C_{r}g\stackrel{\dot{}}{\theta }\sin \theta \hfill \\ \stackrel{\dot{}}{u}=\frac{1}{m}\left[-mg\sin \theta -\frac{1}{2}\rho A_f{\left(u+u_w\right)}^2{C}_x-C_{r}mg\cos \theta +F_x-\stackrel{\dot{}}{m}u-m\left(qw-rv\right)\right]\hfill \\ \stackrel{\dot{}}{v}=\frac{1}{m}\left[mg\cos \theta \sin \varphi +\frac{1}{2}\rho A_f{\left(v+v_w\right)}^2{C}_y+F_y-\stackrel{\dot{}}{m}v-m\left(ru-pw\right)\right]\hfill \\ \stackrel{\dot{}}{w}=\frac{1}{m}\left[mg\cos \theta \cos \varphi +\frac{1}{2}\rho A_f{\left(w+w_w\right)}^2{C}_z+F_z-mg\cos \theta -\stackrel{\dot{}}{m}w-m\left(pv-qu\right)\right]\hfill \\ \stackrel{\dot{}}{p}=\frac{C}{AC-D^2}\left[\frac{1}{2}\rho A_{f}t{\left(u+u_w\right)}^2{C}_l+\left(B-C\right)qr+Dpq\right]\hfill \\ +\frac{D}{AC-D^2}\left[\frac{1}{2}\rho A_{f}L{\left(w+w_w\right)}^2{C}_n+\left(A-B\right)pq-Dqr\right]\hfill \\ \stackrel{\dot{}}{q}=\frac{1}{B}\left[\frac{1}{2}\rho A_{f}L{\left(v+v_w\right)}^2{C}_m+\left(C-A\right)pr-D\left(p^2-r^2\right)\right]\hfill \\ \stackrel{\dot{}}{r}=\frac{A}{AC-D^2}\left[\frac{1}{2}\rho A_{f}L{\left(w+w_w\right)}^2{C}_n+\left(A-B\right)pq-Dqr\right]\hfill \\ +\frac{D}{AC-D^2}\left[\frac{1}{2}\rho A_{f}t{\left(u+u_w\right)}^2{C}_l+\left(B-C\right)qr+Dpq\right]\hfill \\ \stackrel{\dot{}}{\varphi }=p+q\tan \theta \sin \varphi +r\tan \theta \cos \varphi ,\mathrm{<mi>\; </mi>}\stackrel{\dot{}}{\theta }=q\cos \varphi -r\sin \varphi ,\hfill \\ \stackrel{\dot{}}{\psi }=q\frac{\sin \varphi }{\cos \theta }+r\frac{\cos \varphi }{\cos \theta }\hfill \\ {\stackrel{\dot{}}{X}}_G^o=u\cos \psi -v\sin \psi ,\mathrm{<mi>\; </mi>}{\stackrel{\dot{}}{Y}}_G^o=u\cos \psi +v\sin \psi ,\mathrm{<mi>\; </mi>}{\stackrel{\dot{}}{Z}}_G^o=w\hfill \end{array}$ (3)

where $\psi$, $\theta$ and $\varphi$ are yaw, roll and pitch angles of the vehicle. $A=I_{xx}$, $B=I_{yy}$, $C=I_{zz}$ and $D=I_{xz}$ are the moments of inertia of the vehicle. $C_{cse}$ is the specific consumption coefficient of the vehicle. $\tau$ and $v_e$ are respectively the coefficient of performance and the gas ejection velocity relative to the engine. $P_i$ is the power injected into the engine, $LH{V}_{fuel}$ is the calorific power of the fuel. ${\eta }_m$ and ${\eta }_c$ are respectively the engine’s mechanical efficiency and the fuel’s combustion efficiency, $p_e$ and $p_0$ are respectively the exhaust gas pressure at the engine outlet and the ambient atmospheric pressure. $A_e$ is the exhaust gas surface area, $\rho$ is the air density, $A_f$ is the frontal area of the vehicle and $t$ is the vehicle width [11] [12] . $C_r$ is the rolling resistance coefficient, $C_a=C_{a,0}+k\cdot \left(A_f/A_{ref}\right)$ is the coefficient of aerodynamic force, $k$ is a correction factor depending on specific vehicle characteristics, $A_{ref}$ is a standard reference surface used to normalize $A_f$ [13] - [15] . $C_x=C_{x,0}+k\left(L/H\right)$, $C_y=C_{y,0}\cdot \left(t/L\right)\cdot \sin \psi$, and $C_z=C_{z,0}+k_1\cdot \left(A_{roof}/A_f\right)+k_2\cdot \left(A_{spoiler}/A_f\right)$ are longitudinal, lateral and vertical force coefficients. $k_1$ and $k_2$ are correction factors for roof shape and aerodynamic devices, $A_{roof}$ is the roof area and $A_{spoiler}$ is the spoiler area [1] [5] [14] [15] . $C_l=C_{l,0}+k_l\cdot \left(h/t\right)$, $C_m=C_{m,0}+k_m\cdot \left(d/L\right)$ and $C_n=C_{n,0}+k_n\cdot \left(L/t\right)$ are rolling, pitching and yawing moment coefficients, $h$ is the height of the vehicle’s center of gravity, $d$ is the distance between the center of pressure and the center of gravity, $k_m$ and $k_n$ are the pitching and yawing moment correction factors [5] [11] [15] .

The constraints arising from the modeling of our problem concern constraints on the safety of the vehicle, driver and passengers on board, technological constraints, operational constraints, speed, roll angle, yaw angle and control position. These constraints are expressed in terms of the following boundary conditions [9] :

1) Longitudinal vehicle position is bounded by $X_0^o\le X_G^o\le X_f^o$ where $X_0^o$, $X_f^o$ are longitudinal position limits. The lateral vehicle position is bounded by $Y_0^o\le Y_G^o\le Y_f^o$ where $Y_0^o$, $Y_f^o$ are lateral positions limits and the vertical vehicle position is bounded by $Z_0^o\le Z_G^o\le Z_f^o$ where $Z_0^o$, $Z_f^o$ are the vertical positions limits.

2) The aerodynamic velocity of the vehicle $v_a$ must be bounded by $v_{\text{min}}\le v_a\le v_{\text{max}}$ where $v_{\text{min}}$ and $v_{\text{max}}$ are limits of the vehicle aerodynamic velocity. Vehicle acceleration ${\gamma }_a$ is given by $a_{\text{min}}\le {\gamma }_a\le a_{\text{max}}$ where $a_{\text{min}}$ and $a_{\text{max}}$ are limits of the vehicle acceleration.

3) Constraints on longitudinal velocity of the vehicle $u_{\text{min}}\le u\le u_{\text{max}}$, lateral velocity of the vehicle $v_{\text{min}}\le v\le v_{\text{max}}$ and vertical velocity of the vehicle $w_{\text{min}}\le w\le w_{\text{max}}$.

4) Constraints on vehicle roll speed $p_0\le p\le p_f$, vehicle pitching speed $q_0\le q\le q_f$ and the vehicle yaw rate $r_0\le r\le r_f$ should also be taken into account. Nautical angles also vary: roll angle $\theta \in \left[{\theta }_0,{\theta }_f\right]$, pitch angle $\varphi \in \left[{\varphi }_0,{\varphi }_f\right]$ and yaw angle $\psi \in \left[{\psi }_0,{\psi }_f\right]$.

5) Command $\delta \left(t\right)$ of the vehicle is bounded between the limits ${\delta }_{x_0}$ and ${\delta }_{x_f}$ for thrust control, ${\delta }_{l_0}$ and ${\delta }_{l_f}$ for roll control, ${\delta }_{m_0}$ and ${\delta }_{m_f}$ for pitch control and ${\delta }_{n_0}$ and ${\delta }_{n_f}$ for yaw control.

6) The vehicle mass varies in the mobile frame: $m_f\le m\le m_0$. This constraint results in fuel consumption, the thrust of the vehicle and the addition of passengers or goods [9] .

On the whole, the constraints come together in the following connection:

$l_1\left(y,\delta \right)\le 0,l_2\left(y,\delta \right)\ge 0$(4)

where $l\left(t\right):{ℝ}^{15}\times {ℝ}^4\to {ℝ}^{15}\times \mathcal{U},\left(y,\delta \right)\mapsto l\left(y,\delta \right)$

$\begin{array}{c}{l}_1\left(y,\delta \right)=(m\left(t\right)-m_f,v_a\left(t\right)-v_{af},{\gamma }_a\left(t\right)-{\gamma }_{af},u\left(t\right)-u_f,v\left(t\right)-v_f,\\ w\left(t\right)-w_f,p\left(t\right)-p_f,q\left(t\right)-q_f,r\left(t\right)-r_f,\theta \left(t\right)-{\theta }_f,\varphi \left(t\right)-{\varphi }_f,\\ \psi \left(t\right)-{\psi }_f,X_G^o\left(t\right)-X_f^o,Y_G^o\left(t\right)-Y_f^o,Z_G^o\left(t\right)-Z_f^o\\ {\delta }_x\left(t\right)-{\delta }_x{}_f,{\delta }_l\left(t\right)-{\delta }_l{}_f,{\delta }_m\left(t\right)-{\delta }_m{}_f,{\delta }_n\left(t\right)-{\delta }_n{}_f)\end{array}$

$\begin{array}{l}{l}_2\left(y,\delta \right)=(m\left(t\right)-m_0,v_a\left(t\right)-v_{a0},{\gamma }_a\left(t\right)-{\gamma }_{a0},u\left(t\right)-u_0,v\left(t\right)-v_0,\\ w\left(t\right)-w_0,p\left(t\right)-p_0,q\left(t\right)-q_0,r\left(t\right)-r_0,\theta \left(t\right)-{\theta }_0,\varphi \left(t\right)-{\varphi }_0,\\ \psi \left(t\right)-{\psi }_0,X_G^o\left(t\right)-X_0^o,Y_G^o\left(t\right)-Y_0^o,Z_G^o\left(t\right)-Z_0^o\\ {\delta }_x\left(t\right)-{\delta }_x{}_0,{\delta }_l\left(t\right)-{\delta }_l{}_0,{\delta }_m\left(t\right)-{\delta }_m{}_0,{\delta }_n\left(t\right)-{\delta }_n{}_0)\end{array}$

The main goal of this work is to optimize the fluidity of road traffic. This flow traffic function is defined as the measure of a road network’s ability to allow for the efficient and uninterrupted movement of vehicles. This function promotes smoother traffic flow. High fluidity implies rapid and unimpeded movement, while low fluidity indicates congestion, frequent stops, and interruptions in the flow of traffic.

To model the traffic fluidity function, we introduce $V\left(y,t\right)$ as the total circulation speed of the flow of vehicles, which takes into account the contributions of the components of the aerodynamic speed transformed by the aerodynamic angles together with the effects of rotation [16] - [18] . The traffic fluidity function is then modeled as follows:

$J\left(y,\delta ,t\right)=\frac{1}{v_{\max }^2}{\Vert R\left(\varphi ,\theta ,\psi \right)v_a+\Omega \times r_G^o\Vert }^2+g\left(y,\delta \right),$(5)

where $R\left(\varphi ,\theta ,\psi \right)$ is the rotation matrix, $v_a={\left(\begin{array}{l}u,v,w\hfill \end{array}\right)}^{\text{T}}$ is the vehicle aerodynamic speed, $\Omega ={\left(\begin{array}{l}p,q,r\hfill \end{array}\right)}^{\text{T}}$ is the vehicle rotation speed vector and $r_G^o={\left(\begin{array}{l}{X}_G^o,Y_G^o,Z_G^o\hfill \end{array}\right)}^{\text{T}}$ is the vehicle position in the mobile frame $R_G$ relative to the observator frame $R_O$, $g\left(y,\delta \right)=\frac{{\Vert \delta \Vert }^2}{{\delta }_{\text{max}}^2}$ is the control price function [17] [19] , and $y=\left(m,v_a,{\gamma }_a,u,v,w,p,q,r,\theta ,\varphi ,\psi ,X_G^o,Y_G^o,Z_G^o\right)$ is the state vector.

The form (5) above is chosen as the most suitable in the concept of traffic optimization because it takes into account aerodynamics, which influences the dynamic performance of the vehicle in traffic; the vehicle orientation matrix, which allows for changes in direction and inclination, essential for smooth movement in varied conditions; and the costs associated with control decisions.

The conceptual definition of traffic fluidity and the mathematical form (5) allow for effective modeling of traffic dynamics while incorporating practical and theoretical considerations, making it a valuable tool for traffic optimization. To support this approach, we can refer to pioneering works that laid the foundations for traffic flow by linking speed to traffic density and mathematical models that link speed and density, reinforcing the idea that traffic flow is a function of these parameters [20] [21] .

By combining dynamic modelling (3), constraints formulation in section 1 and the objective function (5), the nonlinear mathematical traffic optimization model is given as follows:

$\underset{\left(y,\delta \right)\in {ℝ}_{ad}^{15}\times {\mathcal{U}}_{ad}}{\max }J\left(y,\delta ,t\right)=\frac{1}{v_{\max }^2}{\Vert R\left(\varphi ,\theta ,\psi \right)v_a+\Omega \times r_G^o\Vert }^2+\frac{{\Vert \delta \Vert }^2}{{\delta }_{\max }^2}$

$s.c:\{\begin{array}{l}\stackrel{\dot{}}{m}=-{\left(C_{cse}\times \tau \times \left[\frac{P_i\left(v_e+{\alpha }_1{\delta }_x\right)}{LH{V}_{fuel}\cdot {\eta }_m\cdot {\eta }_c}+\left(p_0-p_e\right)\cdot A_e\right]\right)}^{\frac{1}{2}}\hfill \\ {\stackrel{\dot{}}{v}}_a=\frac{1}{2m}\rho A_f{C}_a{\left(v_a+v_{wind}\right)}^2+C_{r}g\cos \theta \hfill \\ {\stackrel{\dot{}}{\gamma }}_a=-\frac{\stackrel{\dot{}}{m}}{2m^2}\rho A_f{C}_a{\left(v_a+v_{wind}\right)}^2+\frac{1}{m}\rho A_f{C}_a{\stackrel{\dot{}}{v}}_a\left(v_a+v_{wind}\right)-C_{r}g\stackrel{\dot{}}{\theta }\sin \theta \hfill \\ \stackrel{\dot{}}{u}=\frac{1}{m}\left[-mg\sin \theta -\frac{1}{2}\rho A_f{\left(u+u_w\right)}^2{C}_x-C_{r}mg\cos \theta +F_x-\stackrel{\dot{}}{m}u-m\left(qw-rv\right)\right]\hfill \\ \stackrel{\dot{}}{v}=\frac{1}{m}\left[mg\cos \theta \sin \varphi +\frac{1}{2}\rho A_f{\left(v+v_w\right)}^2{C}_y+F_y-\stackrel{\dot{}}{m}v-m\left(ru-pw\right)\right]\hfill \\ \stackrel{\dot{}}{w}=\frac{1}{m}\left[mg\cos \theta \cos \varphi +\frac{1}{2}\rho A_f{\left(w+w_w\right)}^2{C}_z+F_z-mg\cos \theta -\stackrel{\dot{}}{m}w-m\left(pv-qu\right)\right]\hfill \\ \stackrel{\dot{}}{p}=\frac{C}{AC-D^2}\left[\frac{1}{2}\rho A_{f}t{\left(u+u_w\right)}^2{C}_l+\left(B-C\right)qr+Dpq\right]\hfill \\ +\frac{D}{AC-D^2}\left[\frac{1}{2}\rho A_{f}L{\left(w+w_w\right)}^2{C}_n+\left(A-B\right)pq-Dqr\right]+{\alpha }_2{\delta }_l\hfill \\ \stackrel{\dot{}}{q}=\frac{1}{B}\left[\frac{1}{2}\rho A_{f}L{\left(v+v_w\right)}^2{C}_m+\left(C-A\right)pr-D\left(p^2-r^2\right)\right]+{\alpha }_3{\delta }_m\hfill \\ \stackrel{\dot{}}{r}=\frac{A}{AC-D^2}\left[\frac{1}{2}\rho A_{f}L{\left(w+w_w\right)}^2{C}_n+\left(A-B\right)pq-Dqr\right]\hfill \\ +\frac{D}{AC-D^2}\left[\frac{1}{2}\rho A_{f}t{\left(u+u_w\right)}^2{C}_l+\left(B-C\right)qr+Dpq\right]+{\alpha }_4{\delta }_n\hfill \\ \stackrel{\dot{}}{\varphi }=p+q\tan \theta \sin \varphi +r\tan \theta \cos \varphi ,\mathrm{<mi>\; </mi>}\stackrel{\dot{}}{\theta }=q\cos \varphi -r\sin \varphi ,\hfill \\ \stackrel{\dot{}}{\psi }=q\frac{\sin \varphi }{\cos \theta }+r\frac{\cos \varphi }{\cos \theta }\hfill \\ {\stackrel{\dot{}}{X}}_G^o=u\cos \psi -v\sin \psi ,{\stackrel{\dot{}}{Y}}_G^o=u\cos \psi +v\sin \psi ,{\stackrel{\dot{}}{Z}}_G^o=w\hfill \end{array}$ (6)

The simplified notation of NLP (6) above is expressed as follows:

$\underset{\left(y,\delta \right)\in {ℝ}_{ad}^{15}\times {\mathcal{U}}_{ad}}{\text{max}}J\left(y,\delta \right)$

$s.c:\{\begin{array}{l}\stackrel{\dot{}}{y}=f\left(t,y,\delta \right)\hfill \\ t_0\le t\le t_f\hfill \\ t_0=0,y\left(0\right)=y_0,\delta \left(0\right)={\delta }_0\hfill \\ l_1\left(y,\delta \right)\le 0,l_2\left(y,\delta \right)\ge 0\hfill \end{array}$(7)

where $y:\left[t_0,t_f\right]\to {ℝ}^{15}$ is the state vector, and $\delta :\left[t_0,t_f\right]\to \mathcal{U}\subset {ℝ}^4$, $f:\left[t_0,t_f\right]\to {ℝ}^{15}\times \mathcal{U}$.

Although this paper discusses a model that optimizes the trajectory of a single vehicle and focuses on individual parameters, its effects are felt throughout the overall traffic system. By improving the behavior of one vehicle, we contribute not only to its own efficiency but also to the traffic flow of other vehicles, creating a more fluid and organized traffic environment. This holistic approach is essential for developing sustainable and effective urban traffic management solutions.