# Non-Compliant Behaviour of Automated Vehicles in a Mixed Traffic Environment
## 3 Implementation of Automated Driving Functions in Microscopic Simulation
### Trajectory Planning Algorithm  
In recent years, numerous approaches have been proposed for the trajectory planning task in automated driving functions ([35; 36; 37; 38]). The method chosen in this paper is based on the work of [39; 40], with specific adaptations to support trajectory planning within microscopic traffic simulations. A key reason for selecting a polynomial-based approach lies in its low computational demands: the polynomial coefficients are directly derived from the start and end conditions of the vehicle trajectory, eliminating the need for iterative optimization. This efficiency is particularly important in microscopic simulations, where trajectories must be computed for several hundred vehicles at each time step. Moreover, the polynomial approach facilitates the generation of smooth, continuous trajectories in a computationally efficient manner. To reduce computational complexity, while maintaining real-time performance, vehicles in the simulation were modeled as point-mass systems, thus simplifying the representation of vehicle dynamics. While this paper focuses on the implementation of a non-compliant behavior module, the core components of the AV lane-change model were developed in a previous study [41]. For completeness and better comprehensibility, we present both the formulas of the base model from [41] and the newly developed non-compliant module. The lateral movement \(\mathrm{I}(t)\) of the vehicle is given by a fifth-degree polynomial. Lateral velocity, lateral acceleration, and jerk are obtained by continuous derivation of \(\mathrm{I}(t)\), which yields the equation system (1):  
\[ \begin{split}\mathrm{I}(t)&=-\frac{1}{120}\mathrm{ C}_{\mathrm{I}}\mathrm{t}^{5}+\frac{1}{24}\mathrm{C}_{2}\mathrm{t}^{4}-\frac{1}{6} \mathrm{C}_{3}\mathrm{t}^{3}+\frac{1}{2}\mathrm{C}_{4}\mathrm{t}^{2}+\mathrm{ C}_{5}\mathrm{t}+\mathrm{C}_{6}\\ \mathrm{v}_{\mathrm{lat}}(t)&=-\frac{1}{24}\mathrm{ C}_{1}\mathrm{t}^{4}+\frac{1}{6}\mathrm{C}_{2}\mathrm{t}^{3}-\frac{1}{2} \mathrm{C}_{3}\mathrm{t}^{2}+\mathrm{C}_{4}\mathrm{t}+\mathrm{C}_{5}\\ \mathrm{a}_{\mathrm{lat}}(t)&=-\frac{1}{6}\mathrm{ C}_{1}\mathrm{t}^{3}+\frac{1}{2}\mathrm{C}_{2}\mathrm{t}^{2}-\mathrm{C}_{3} \mathrm{t}+\mathrm{C}_{4}\\ \mathrm{j}_{\mathrm{lat}}(t)&=-\frac{1}{2}\mathrm{ C}_{1}\mathrm{t}^{2}+\mathrm{C}_{2}\mathrm{t}-\mathrm{C}_{3}\end{split}  \tag{1}\]  
The parameters C1-C6 can be determined together with the start (\(\mathrm{t}=\mathrm{t}_{0}\)) and end condition (\(\mathrm{t}=\mathrm{t}_{f}\)) of the trajectory. The start condition is given by the current lateral vehicle state and the end condition is given by the desired state at the end of the trajectory: \begin{table}
\begin{tabular}{l l} \hline \hline \multicolumn{3}{c}{**Car-Following Parameters**} \\ \hline \(\tau_{\mathrm{min}}\) & ISO 15622 [34]: minimum selectable time gap [s] \\ \hline \multirow{2}{*}{\(\mathrm{c}_{\mathrm{min}}\)} & ISO 15622: minimum clearance under steady state conditions \\  & for all speeds [m] \\ \hline \(\mathrm{d}_{\mathrm{safe}}\) & safe distance to lead vehicles [m] \\ \hline \(\mathrm{v}_{\mathrm{ego}}\) & ego vehicle velocity [m/s] \\ \hline \(\mathrm{v}_{\mathrm{lag}}\) & lag vehicle velocity [m/s] \\ \hline \multicolumn{3}{c}{Lane-change parameters} \\ \hline \(\mathrm{a}_{\mathrm{maxLag}}\,^{*}\) & accepted deceleration of the lag vehicle [m/s\({}^{2}\)] \\ \hline \(\mathrm{a}_{\mathrm{maxEgo}}\,^{*}\) & accepted deceleration of the ego vehicle [m/s\({}^{2}\)] \\ \hline \multicolumn{3}{c}{Non-compliant trajectories} \\ \hline \(\tau_{\mathrm{min,nc}}\) & minimum selectable time gap during non-compliant mode [s] \\ \hline \(\mathrm{T}_{\mathrm{max,nc}}\) & maximum allowed time for non-compliant mode [s] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters for automated driving functions.