# Truck Platoon Control Considering Heterogeneous Vehicles
## 2 Definitions and Assumptions
Then, \(\dot{\psi}^{i}_{M}\) is derived as  
\[\dot{\psi}^{i}_{M}=\frac{s^{i}_{k}}{L^{i}}\tan(\delta^{i}_{M}). \tag{4}\]  
This implies that as the length of a truck \(p^{i}\) increases, the maximum yaw rate of \(p^{i}\) decreases. In addition, as the maximum steering angle of \(p^{i}\) increases, the maximum yaw rate of \(p^{i}\) increases. Since we consider heterogeneous trucks, \(L^{i}\
eq L^{i}\) (\(\forall j\
eq i\)) is feasible. Thus, \(\dot{\psi}^{i}_{M}\
eq\dot{\psi}^{j}_{M}\) (\(\forall j\
eq i\)) is also feasible. We further present important notations in this paper. Whenever the leader travels \(Th\) meters, the leader's position is stored as a waypoint for the following trucks. Let \(W_{m}\) denote the \(m\)-th waypoint. This implies that waypoints are generated as \(W_{1},W_{2},\dots\), in this order. The distance between \(W_{m}\) and \(W_{m-1}\) is \(Th\) by its definition. The distance between \(p^{i}\) and \(p^{i-1}\) at time step \(k\) is \(D^{i-1,i}_{k}=\|\mathbf{P}^{i}_{k}-\mathbf{P}^{i-1}_{k}\|\). The reference platooning distance at time step \(k\) for a truck \(p^{i}\) is denoted as \(d^{i}_{k}\). This implies that the desired distance between \(p^{i}\) and \(p^{i-1}\) at time step \(k\) is \(d^{i}_{k}\). The main source of time delay between trucks is from the parasitic time delay induced by the mechanical systems and the communication delay generated by the communication and computing devices. It is assumed that the time delay in communication between the leader and a truck \(p^{i}\) is bounded by a certain constant, say \(t_{d}\). The travel distance of \(p^{i}\) within \(t_{d}\) seconds can be estimated as \(t_{d}*s^{i}_{k}\). In order to avoid collision due to delayed communication, the reference distance \(d^{i}_{k}\) is set as a bigger value than \(t_{d}*s^{i}_{k}\). Additionally, we set the lower bound for \(d^{i}_{k}\) for the safety of trucks. Let \(\mathit{mint}_{D}>0\) present the lower bound for \(d^{i}_{k}\). This paper uses  
\[d^{i}_{k}=t_{d}*s^{i}_{k}+\mathit{min}_{D}. \tag{5}\]  
Here, \(\mathit{min}_{D}\) is a positive constant. In simulations, this paper uses \(\mathit{min}_{D}=1\) in meters. As the speed of a truck converges to \(s^{1}\), \(d^{i}_{k}\) converges to \(t_{d}*s^{1}+\mathit{min}_{D}\), which is a positive constant. This separation change approach in (5) was inspired by the constant time headway spacing policy, which was also used in [16]. It is assumed that each truck can estimate the curvature of the lane using vision sensors [33]. A truck \(p^{i}\) moves along an arc lane with radius  
\[R(p^{i})=\tfrac{1}{C(p^{i})}, \tag{6}\]  
as the truck estimates the lane curvature as \(C(p^{i})\). As \(C(p^{i})\) goes to zero, \(R(p^{i})\) goes to \(\infty\) using (6). It is assumed that a truck \(p^{i}\) accesses both \(D_{k}^{i-1,i}\) and \(s_{k}^{i-1}\). To enable this, each truck needs to access the position and speed of its predecessor truck. These data are accessible using local sensors, such as radar, lidar, or local communication with its predecessor truck. Moreover, in practice, sensor noise exists in the measurements of \(D_{k}^{i-1,i}\) and \(s_{k}^{i-1}\). In simulations (Section 4), one added independent and identically distributed (i.i.d.) Gaussian noise in the measurements of \(D_{k}^{i-1,i}\) and \(s_{k}^{i-1}\).