# Truck Platoon Control Considering Heterogeneous Vehicles
## 2 Definitions and Assumptions  
Suppose that there are \(N\) heterogeneous trucks. \(p^{i}\) denotes the \(i\)-th truck (\(i\in\{1,2,\ldots,N\}\)). The predecessor of \(p^{i+1}\) is \(p^{i}\). \(p^{1}\) is the leader truck driven by a trained truck driver. One considers discrete-time systems. Let \(T\) denote the sampling interval. The subscript \(k\) denotes the time step \(k\). \(\mathbf{P}_{k}^{i}\) denotes the position of the \(i\)-th truck at time step \(k\). The global coordinate of a truck \(p^{i}\) is given by \(\mathbf{P}_{k}^{i}=[x_{k}^{i},y_{k}^{i}]^{T}\), while its orientation in the global frame is represented by \(\psi_{k}^{i}\). In the body frame of \(p^{i}\), \(\delta_{k}^{i}\) is the steering angle of the truck at time step \(k\). The truck's speed at time step \(k\) is \(s_{k}^{i}\) in the truck's x-direction (body frame), and zero in the y-direction (body frame), since this paper assumes that the wheels cannot slip sideways [28]. If the front wheel is located at distance \(L^{i}\) from the rear wheel along the orientation of the truck \(p^{i}\), then the Ackermann Steering model [29] is  
\[ \begin{array}{l}x_{k+1}^{i}=x_{k}^{i}+T*s_{k}^{i}*\sin(\psi_{k}^{i}),\\ y_{k+1}^{i}=y_{k}^{i}+T*s_{k}^{i}*\cos(\psi_{k}^{i}),\\ \psi_{k+1}^{i}=\psi_{k}^{i}+T*\frac{s_{k}^{i}}{L^{i}}\tan(\delta_{k}^{i}).\end{array}  \tag{1}\]  
In Ackermann steering model, each wheel has its own pivot point and the system is constrained in such a way that all wheels of the car drive on circles with a common center point, avoiding skid [29]. It is acknowledged that in the case where slip sideways are allowed, more complicated truck models in [30; 31; 32] are required. However, as we consider slip sideways, we need to set various parameters, such as the truck mass or inertia. Considering heterogeneous trucks, each truck has a distinct motion model from other trucks. Setting complicated models for each individual truck is time-consuming, and a truck's model may not represent the true dynamics of the truck as the truck runs a longer distance. For instance, the functionality of a truck's hardware (such as engine, accelerator, or braking system) gets worse as time goes on. In addition, various road (or tire) situations may make effects on slip sideways. Accurate motion modeling in various environments may be impossible in practice. Equation (1) can handle heterogeneous trucks by allowing \(L^{i}\
eq L^{i}\) for \(j\
eq i\). \(L^{i}\
eq L^{j}\) implies that the length of the \(i\)-th truck is distinct from that of the \(j\)-th truck. Thus, this paper uses (1) as the motion model of each heterogeneous truck. A car cannot arbitrarily change its speed within one time step. Thus, let \(a^{i}_{A}\) denote the maximum acceleration of a truck \(p^{i}\). Additionally, let \(a^{i}_{D}\) denote the maximum deceleration of \(p^{i}\). Here, \(a^{i}_{A}\) and \(a^{i}_{D}\) are determined by hardware information of \(p^{i}\), such as engine, accelerator, or braking system of \(p^{i}\). This implies that  
\[s^{i}_{k-1}-a^{i}_{D}*T\leq s^{i}_{k}\leq s^{i}_{k-1}+a^{i}_{A}*T. \tag{2}\]  
Equation (2) presents the feasible range of \(s^{i}_{k}\) considering the maximum acceleration and deceleration of \(p^{i}\). The maximum speed of \(p^{i}\) is \(s^{i}_{M}\), i.e., \(s^{i}_{k}\leq s^{i}_{M}\) for all \(k\). Considering heterogeneous trucks, \(a^{i}_{A}\
eq a^{i}_{A}\) (\(\forall j\
eq i\)) is feasible. In addition, \(a^{i}_{D}\
eq a^{i}_{D}\) (\(\forall j\
eq i\)) is feasible. This is due to the fact that the hardware information (such as engine or braking system) of \(p^{i}\) may be distinct from that of \(p^{i}\). A car cannot change its heading abruptly between adjacent time steps, due to the car's geometry such as the maximum steering angle of the car. Let \(\dot{\psi}^{i}_{M}\) present the maximum yaw rate of \(p^{i}\). One has  
\[\|\psi^{i}_{k+1}-\psi^{i}_{k}\|\leq\dot{\psi}^{i}_{M}T. \tag{3}\]  
Let \(\delta^{i}_{M}\) denote the maximum steering angle of a truck \(p^{i}\).