Page - 59 - in Integration of Advanced Driver Assistance Systems on Full-Vehicle Level - Parametrization of an Adaptive Cruise Control System Based on Test Drives

5.1. Path Prediction system, which has its origin at CG and its βx axis parallel to the speed vector vk in the CG. The transformation from the (βx,βy) to the vehicle coordinate system at time step k is done using [ vkxˆ vkyˆ ] = [ cosβ −sinβ sinβ cosβ ] ︸ ︷︷ ︸ Tvk,β [ βxˆ βyˆ ] . (5.12) The second option is to estimate the trajectory using a circle reading βxˆ 2 + ( βyˆ− 1 κ )2 = 1 κ2 . (5.13) The circle has its centre at βx= 0 and βy= 1 κ with the radiusR= 1 κ. Figure 5.2 illustrates the differences between eqs. (5.11) and (5.13). The big advantage of the parabola (black dashed) is that every βxˆ coordinate has its βyˆ coordinate. For the circle (grey), eq. (5.13) could only be solved for−1κ≤ βxˆ≤ 1κ. For the transformation fromthe (βx,βy) to the (vkx,vky)-coordinate system, eq. (5.12) is alsoused for the circle. 5.1.2. Path Prediction Using Linear Single-Track Model For the prediction of the trajectory, the linear STM described in eq. (B.8) is used. The prediction is done at time step tk, and the output is a number of i predicted state vectors xˆi|k= [ βˆi|k vωˆz,i|k ]T . The index kmeans that the prediction is done at time tk, and i indicates the i-th time step within the prediction. The steering angle δk is held constant for the whole simulation. The initial condition for the integration reads x0|k= [ βk vωz,k ]T , whereβk is the estimated side slip angle at time tk, as described in chapter 4.2, and vωz,k is the measured yaw rate. The lateral vehicle speed is predicted as vvˆy,i|k= vvx,ktan ( βˆi|k ) , (5.14) where vvx,k is the measured longitudinal vehicle speed at tk. The heading of the vehicle can be found using vkψˆi|k= tk+i∆t∫ tk vωˆz,i|kdt, (5.15) where the index vkmeans the vehicle coordinate system at time step k, and ∆t is the time step size for the prediction. The transformation of the other state variables is done using the expression [ vk ˙ˆxi|k vk ˙ˆyi|k ] =   cos(vkψˆi|k) −sin(vkψˆi|k) sin ( vkψˆi|k ) cos ( vkψˆi|k )   [vvx,k vvˆy,i|k ] . (5.16) 59