# Simultaneous Trajectory and Speed Planning for Autonomous Vehicles Considering Maneuver Variants
## 5 Simulation
The speed limits under the sideslip condition are shifted to the maneuver's beginning movement to Lane 1 and are shifted to the final phase for passing to Lane 3. The restrictions for following Lane 2 are significantly higher than the speed limit \(V_{\zeta max}\) and are not reflected in the figure's boundaries. However, the speed in Lane 2 does not increase as the acceleration reaches the limit of the powertrain's potential, and the final state of AV requires zero longitudinal acceleration. Note that the curvatures in the maneuvers' final phases are approximately the same, which is reflected in the final values of lateral accelerations that are close in modules (Figure 5c). As seen in Figure 5d, not a single total acceleration curve exceeds the limit \(a_{max}\) by the adhesion properties. Figure 6 presents the kinematic and geometric parameters used to impose restrictions. As seen in Figure 6a, all the longitudinal jerk curves are well below the upper limit and do not exceed the lower limit. An important indicator is the smoothness and unambiguity of the yaw rate and acceleration curves (Figure 6b,c). This characterizes the strict stability of the yaw angle and the absence of uncontrolled dynamic phenomena. Figure 6d shows that AV increases the distance from vehicles located behind and reduces gaps with the vehicles ahead while keeping safe spaces. Figure 7 reflects the geometric parameters characterizing the maneuvers' trajectories. As seen in Figure 7a, all curvature curves are extremely smooth and concordant with the polynomial used for the trajectory model. The steering angle (Figure 7d) scales the curvature due to the ideal kinematic model. The curvature derivatives (Figure 7b,c) are also  
Figure 5: Results of speed and dynamic indicators of forecasts for the lane maneuvers at \(\varphi_{max}=0.85\): (**a**) longitudinal speeds and critical restrictions on adhesion conditions, (**b**) longitudinal accelerations, (**c**) lateral accelerations, (**d**) total accelerations. smooth without intense oscillations and sharp spikes, characterizing a consistent speed distribution along the trajectory's curvature. **Experiment 2**. Now, we consider the motion-planning variants (Figure 8) under the same conditions as for Experiment 1 but with worse adhesion \(\varphi_{max}=0.5\). In this case, more time and distance are needed to complete the maneuver since the traction potential is limited by an external factor. Therefore, the length of the required path \(s\) is closer to the upper limit of the reference \(s_{r}\) length. Figure 6: Results of forecasting kinematic parameters of maneuvers at \(\varphi_{max}=0.85\). (**a**) Longitudinal jerk, (**b**) yaw rate, (**c**) angular acceleration, (**d**) distance to obstacles. Figure 7: Results of geometric indicators for forecasting lane maneuvers at \(\varphi_{max}=0.85\): (**a**) curvature, (**b**) curvature’s first derivative, (**c**) curvature’s second derivative, (**d**) steering angle. As seen, the nature of the maneuvers differs in phase. Thus, on Lane 2, there is no significant change in curvature, and AV may be quickly accelerated. For the maneuver to Lane 1, the intensity is shifted to the initial phase, and for the maneuver to Lane 2, it is shifted to the final phase. Figure 9 depicts the results of forecasting speeds and accelerations under the condition of reduced tire-road adhesion. Accordingly, the critical speed values, caused by the locally increased curvature, also decrease and limit the increase in AV speed for maneuvers to Lanes 1 and 3 (Figure 9a). At the same time, the curvature of the trajectory for driving on Lane 2 changes little, which allows the speed to rise since the critical one, in this case, is greater than the preset maximum. The same is noted for longitudinal accelerations with a difference that the vehicle, naturally, cannot realize its full traction potential. As seen in Figure 9d, the maximum acceleration of the maneuver to Lane 1 is formed at the beginning of the lane change, and for Maneuver 3 at the end.