# Autonomous Controller-Aware Scheduling of Intra-Platoon V2V Communications
## Appendix A Derivation of the Distance Error for the First Platoon Pair
Thus, assuming that \(z_{3}(0)\
eq 0\) and \(z_{3}(k>0)=0\), the information update interval is \(\Delta t_{Tx}=\Delta t\), and that there are no acceleration constraints (\(a_{max}=\infty\) and \(a_{min}=-\infty\)), we can derive the distance error values for the first platoon pair (between the leader and the first following vehicle) as follows:  
\[\text{For }k=0:\] \[e_{1}(0)=0,\Delta v_{0,1}(0)=0,a_{0}(0)=z_{3}(0),a_{1}(0)=0;\] \[\text{For }k=1:\] \[e_{1}(1)=-\frac{\Delta t^{2}}{2}z_{3}(0),\Delta v_{0,1}(1)= \Delta tz_{3}(0),a_{0}(1)=z_{3}(0),a_{1}(1)=z_{3}(0);\] \[=d_{d}-(x_{i-1}(k-1)+\Delta tv_{i-1}(k-1)+\frac{\Delta t^{2}}{2} a_{i-1}(k-1)-x_{i}(k-1)-\Delta tv_{i}(k-1)\]0 \[=d_{d}-(x_{i-1}(k-1)+\Delta tv_{i-1}(k-1)+\frac{\Delta t^{2}}{2} a_{i-1}(k-1)-x_{i}(k-1)-\Delta tv_{i}(k-1)\]1 \[=d_{d}-(x_{i-1}(k-1)+\Delta tv_{i-1}(k-1)+\frac{\Delta t^{2}}{2} a_{i-1}(k-1)-x_{i}(k-1)-\Delta tv_{i}(k-1)\]2 \[=d_{d}-(x_{i-1}(k-1)+\Delta tv_{i-1}(k-1)+\frac{\Delta t^{2}}{2} a_{i-1}(k-1)-x_{i}(k-1)-\Delta tv_{i}(k-1)\]3 \[=d_{d}-(x_{i-1}(k-1)+\Delta tv_{i-1}(k-1)+\frac{\Delta t^{2}}{2} a_{i-1}(k-1)-x_{i}(k-1)-\Delta tv_{i}(k-1)\]4 \[=d_{d}-(x_{i-1}(k-1)+\Delta tv_{i-1}(k-1)+\frac{\Delta t^{2}}{2} a_{i-1}(k-1)-x_{i}(k-1)-\Delta tv_{i}(k-1)\]5 \[=d_{d}-(x_{i-1}(k-1)+\Delta tv_{i-1}(k-1)+\frac{\Delta t^{2}}{2} a_{i-1}(k-1)-x_{i}(k-1)-\Delta tv_{i}(k-1)\]6 \[=d_{d}-(x_{i-1}(k-1)+\Delta tv_{i-1}(k-1)+\frac{\Delta t^{2}}{2} a_{i-1}(k-1)-x_{i}(k-1)-\Delta tv_{i}(k-1)\]7 \[=d_{d}-(x_{i-1}(k-1)+\Delta tv_{i-1}(k-1)+\frac{\Delta t^{2}}{2} a_{i-1}(k-1)-x_{i}(k-1)-\Delta tv_{i}(k-1)\]8  
Assuming that \((\alpha_{4}+\alpha_{5})=1\) (following the controller constraint given in [6]) and substituting: \(\phi=\alpha_{1}\frac{\Delta t^{2}}{2}\) and \(\xi=(\alpha_{2}+\alpha_{3})\Delta t\)), the equations can be rewritten as:  
\[=d_{d}-(x_{i-1}(k-1)+\Delta tv_{i-1}(k-1)+\frac{\Delta t^{2}}{2} a_{i-1}(k-1)-x_{i}(k-1)-\Delta tv_{i}(k-1)\]9 \[\qquad-\frac{\Delta t^{2}}{2}a_{i}(k-1))=\]0 \[\qquad-\frac{\Delta t^{2}}{2}a_{i}(k-1))=\]1 \[\qquad-\frac{\Delta t^{2}}{2}a_{i}(k-1))=\]2 \[\qquad-\frac{\Delta t^{2}}{2}a_{i}(k-1))=\]3 \[\qquad-\frac{\Delta t^{2}}{2}a_{i}(k-1))=\]4 \[\qquad-\frac{\Delta t^{2}}{2}a_{i}(k-1))=\]5 \[\qquad-\frac{\Delta t^{2}}{2}a_{i}(k-1))=\]6 \[\qquad-\frac{\Delta t^{2}}{2}a_{i}(k-1))=\]7 \[\qquad-\frac{\Delta t^{2}}{2}a_{i}(k-1))=\]8 \[\qquad-\frac{\Delta t^{2}}{2}a_{i}(k-1))=\]9 \[=d_{d}-(x_{i-1}(k-1)-x_{i}(k-1))-\Delta t(v_{i-1}(k-1)-v_{i}(k-1))\]0 \[=d_{d}-(x_{i-1}(k-1)-x_{i}(k-1))-\Delta t(v_{i-1}(k-1)-v_{i}(k-1))\]1Denoting \(\zeta_{0}^{(e)}(k)=2k-1\), \(\zeta_{\xi}^{(e)}(k)=(k-2)^{2}\), \(\zeta_{\phi}^{(e)}(k)=(k-2)^{2}+2\sum_{l=1}^{k-3}l^{2}\), \(\zeta_{\xi}^{(v)}(k)=k-2\), we can rewrite the equations as follows:  
\[=d_{d}-(x_{i-1}(k-1)-x_{i}(k-1))-\Delta t(v_{i-1}(k-1)-v_{i}(k-1))\]2 \[=d_{d}-(x_{i-1}(k-1)-x_{i}(k-1))-\Delta t(v_{i-1}(k-1)-v_{i}(k-1))\]3 \[=d_{d}-(x_{i-1}(k-1)-x_{i}(k-1))-\Delta t(v_{i-1}(k-1)-v_{i}(k-1))\]4 \[=d_{d}-(x_{i-1}(k-1)-x_{i}(k-1))-\Delta t(v_{i-1}(k-1)-v_{i}(k-1))\]5  
Following the approach presented above and conducting further calculations, we can obtain the general formulas for \(e_{1}(k)\), \(\Delta v_{0,1}(k)\), and \(a_{1}(k)\):  
\[=d_{d}-(x_{i-1}(k-1)-x_{i}(k-1))-\Delta t(v_{i-1}(k-1)-v_{i}(k-1))\]6 \[=d_{d}-(x_{i-1}(k-1)-x_{i}(k-1))-\Delta t(v_{i-1}(k-1)-v_{i}(k-1))\]7 \[=d_{d}-(x_{i-1}(k-1)-x_{i}(k-1))-\Delta t(v_{i-1}(k-1)-v_{i}(k-1))\]8  
with the corresponding \(\theta_{m,n}^{(\cdot)}(k)\) coefficients calculated recursively, as follows:  
\[=d_{d}-(x_{i-1}(k-1)-x_{i}(k-1))-\Delta t(v_{i-1}(k-1)-v_{i}(k-1))\]9 \[\qquad-\frac{\Delta t^{2}}{2}(a_{i-1}(k-1)-a_{i}(k-1))=\]0 \[\qquad-\frac{\Delta t^{2}}{2}(a_{i-1}(k-1)-a_{i}(k-1))=\]1  
where \(\theta_{0,0}^{(e)}(k)=2k-1\), \(\theta_{0,0}^{(v)}(k)=1\), \(\theta_{0,0}^{(a)}(k)=1\), \(\theta_{m,0}^{(a)}(k)=-\theta_{m-1,0}^{(v)}(k-1)\), \(\theta_{0,n}^{(a)}(k)=-\theta_{0,n-1}^{(e)}(k-1)\).