# Autonomous Controller-Aware Scheduling of Intra-Platoon V2V Communications
## 4 Problem Formulation  
Contrary to other works, rather than optimizing the controller while knowing the communications performance, we aim to minimize the amount of resources used for intra-platoon communications. In particular, we focus on the minimization of the total number of transmissions while maintaining the safe operation of the platoon using a known controller, understood as keeping the time of the emergency braking activation below a predefined level \(\epsilon\). More specifically, we assume that emergency braking is activated when the distance between two subsequent vehicles is below a specified threshold distance \(d_{Th}\). Thus, the optimization problem in terms of the information update period (\(\Delta t_{Tx}\)) and the initial latency (\(\tau\)) can be described as:  
\[\min_{\tau,\Delta t_{Tx}}\sum_{i=0}^{N-1}\sum_{k}\mathbf{q_{s}}^{(i)}(k), \tag{7}\]  
subject to:  
\[\forall_{i=0,\dots,N-1}\quad\frac{1}{T}\int_{0}^{T}I[e_{i}(t)>d_{d}-d_{Th}]dt<\epsilon, \tag{8}\]  
where \(I()\) is the indicator function, defined as:  
\[I[e_{i}(t)>d_{d}-d_{Th}]= \begin{cases}1,&\text{if $e_{i}(t)>d_{d}-d_{Th}$}\\ 0,&\text{otherwise}\end{cases} , \tag{9}\]  
Here, \(T\) is the observation interval (so with the discrete model, we sum over \(k=\frac{T}{\Delta t}\) samples), \(\epsilon\) is the acceptable fraction of emergency braking events, and \(e_{i}(t)\) is the distance error between two subsequent platoon vehicles at time \(t\), defined as \(e_{i}(t)=d_{d}-(x_{i-1}(t)-x_{i}(t))\). Figure 2: A single platoon vehicle’s transmission pattern, with the initial latency \(\tau\) of the first message and the update period \(\Delta t_{Tx}\) between the transmission of subsequent messages. The key element impacting upon the selection of resources is then the safety constraint defined in (8). However, the exact solution to (8) is nontrivial, due to the mutual dependencies between the accelerations of the consecutive platoon vehicles introduced by the controller, and the possibility of different latency introduced by the individual platoon vehicles. Such a dependency model between the acceleration disturbance \(z_{3}(k)\) and the distance error for the first pair of vehicles in the platoon \(e_{1}(k)\) has been found, assuming an information update at every time step \(\Delta t\), following the rationale presented in Appendix A, along with the formulas for the velocity difference in the first pair \(\Delta v_{0,1}(k)\) and the acceleration of the first follower \(a_{1}(k)\). Such relations as functions of the information update period (\(\Delta t\)) can be expressed as:  
\[ \begin{split}& e_{1}(k)=-\frac{\Delta t^{2}}{2}\sum_{j=0}^{k-1}(z_{3}(k- 1-j)\cdot\sum_{m=0}^{\lfloor\frac{k-1-j}{2}\rfloor\lfloor\frac{k-1-j}{2} \rfloor-m}\theta_{m,n}^{(e)}(k-j)\cdot((\alpha_{2}+\alpha_{3})\Delta t)^{m} \cdot(\alpha_{1}\frac{\Delta t^{2}}{2})^{n}),\\ &\Delta v_{0,1}(k)=\Delta t\sum_{j=0}^{k-1}(z_{3}(k-1-j)\cdot \sum_{m=0}^{\lfloor\frac{k-1-j}{2}\rfloor\lfloor\frac{k-1-j}{2}\rfloor-m} \theta_{m,n}^{(v)}(k-j)\cdot((\alpha_{2}+\alpha_{3})\Delta t)^{m}\cdot(\alpha _{1}\frac{\Delta t^{2}}{2})^{n}),\\ & a_{1}(k)=\sum_{j=0}^{k-1}(z_{3}(k-1-j)\cdot\sum_{m=0}^{\lfloor \frac{k-1}{2}\rfloor\lfloor\frac{k-1}{2}\rfloor-m}\theta_{m,n}^{(a)}(k-j) \cdot((\alpha_{2}+\alpha_{3})\Delta t)^{m}\cdot(\alpha_{1}\frac{\Delta t^{2}} {2})^{n}),\\ \end{split}  \tag{10}\]  
where the corresponding \(\theta_{m,n}^{(. )}(k)\) coefficients are calculated recursively as follows. \[ \begin{split}\theta_{m,n}^{(e)}(k)&=\theta_{m,n}^{( e)}(k)+2\sum_{i=1}^{k-1}\theta_{m,n}^{(v)}(i),\\ \theta_{m,n}^{(v)}(k)&=-\sum_{i=1}^{k-1}\theta_{m, n}^{(a)}(i),\\ \theta_{m,n}^{(a)}(k)&=-\theta_{m-1,n}^{(v)}(k)- \theta_{m,n-1}^{(e)}(k),\end{split} \]  
with \(\theta_{0,0}^{(e)}(k)=2k-1\), \(\theta_{0,0}^{(v)}(k)=1\), \(\theta_{0,0}^{(a)}(k)=1\), \(\theta_{m,0}^{(a)}(k)=-\theta_{m-1,0}^{(v)}(k-1)\), \(\theta_{0,n}^{(a)}(k)=-\theta_{0,n-1}^{(e)}(k-1)\).