## 3 Literature Classification and Analysis
### Mathematical Models  
The mathematical model is created using the formalism of robotics based on the geometric description of the modified Denavit-Hartenberg method. The method considers that the autonomous vehicle is a multi-articulated system consisting of \(n\) bodies wherein the chassis is the movable base and the wheels are the terminals. Each body is connected to its antecedent by a joint, which represents a translational or a rotational degree of freedom. A body can be virtual or real; the virtual bodies are introduced to describe joints with multiple degrees of freedom or to introduce intermediate fixed frames. The mixed Euler-Lagrange model is obtained from two recurrences of the algorithm of Newton-Euler in the following way. The forward recursive equations (from the mobile base to the effectors) compute the total forces \({}^{j}F_{j}\) and moments \({}^{j}Mo_{j}\) on each link \(j\) by calculating the angular and the linear speeds and accelerations of each body. The backward recurrence (from the effectors to the mobile base) computes the forces \({}^{j}f_{j}\) and the moments \({}^{j}mo_{j}\) exerted on each body by its antecedent, considering the external forces applied to the robot. The torque \(\tau_{j}\) applied on the body \(C_{j}\) is calculated by projecting, according to the type of the joint \(j\), the vector of force \({}^{j}f_{j}\) or moment \({}^{j}mo_{j}\) on the axle of movement [8].  
\[\tau_{j}=\left(\sigma_{j}^{j}f_{j}+\overline{\sigma_{j}}^{j}mo_{j}\right)^{tj} a_{j}, \tag{31}\]  
where \({}^{j}a_{j}=\left[0\ 0\ 1\right]^{t}\), \(\sigma_{j}=1\) if the joint \(j\) is translational and \(\sigma_{j}=0\) if the joint \(j\) is rotational. If there is no degree of freedom between two frames that are fixed with respect to each other, \(\sigma_{j}=2\).  
The inverse dynamic model of a robot with a mobile base can be written as [8]  
\[\tau=f\big{(}q,\dot{q},\ddot{q},f_{e}\big{)}=A(q)\ddot{q}+H\big{(}q,\dot{q} \big{)}+J(q)f_{e} \tag{32}\]  
where \(\tau\) is the vector of the actuators torques or forces, q, \(\dot{q}\) and \(\ddot{q}\) are the vectors of positions, velocities, and accelerations of all the joints, including the variables of the chassis, and \(f_{e}\) is the vector of external forces. \(H\) is the vector of centrifugal, Coriolis, and gravity terms; \(J\) is the jacobien matrix; \(Jf_{e}\) is the vector of generalized efforts representing the projection of external forces and torques on the joint axis; and \(A\) is the inertial matrix of the system.  
The direct dynamic model is then given by [8]:  
\[\ddot{q}=\big{[}A(q)\big{]}^{-1}\big{(}\tau-H\big{(}q,\dot{q}\big{)}-J(q)f_{e }\big{)}. \tag{33}\]  
Once the expression of \(\tau\) is determined by the Newton-Euler algorithm, the direct dynamic model for calculating the matrices \(A\), \(H\), and \(J\) is as follows [8]. The column \(c_{a}\) of the matrix \(A\) is computed by  
\[A(;,c_{a})=\frac{\partial\tau}{\partial\ddot{q}(c_{a})},\ c_{a}\in\left[1,\ l\right] \tag{34}\]  
where \(l\) represents the number of degrees of freedom in the system, which is the dimension of the vector \(q\). The matrix \(J\) is computed similarly by  
\[J\big{(};,c_{j}\big{)}=\frac{\partial\tau}{\partial f_{e}(c_{j})},\ c_{j}\in \left[1,\ r_{f}\right], \tag{35}\]where \(r_{f}\) represents the dimension of the vector \(f_{e}\). The matrix \(H\) is obtained using \(H\big{(}q,\dot{q}\big{)}=\tau\) when \(\ddot{q}=f_{e}=0\). Then  
\[H=f\big{(}q,\dot{q},0,0\big{)} \tag{36}\]  
The adopted mixed Euler-Lagrange formalism allowed for directly elaborating the dynamic model of the autonomous mobile robot with minimum computation time. This formalism was used to develop a four-wheeled vehicle model. The model was validated using the Scanner-Studio simulator, and the results showed its validity for a large range of driving conditions [8].