The path (its x- and y-positions) is time indepedent, but a path tracking control and its feed forward are time depedent.
A path can be described parametric from the path length . But the time deviation of the path length
is the driving velocity of the vehicle:
.
Here, flatness-based feedforward controls are derived, because for nonlinear systems it is more accurate.
For a pure kinematic model
.
The velocity can be formulated from the flat output and its derivation:
.
The orientation (yaw angle) can be formulated from the flat output and its derivation:
.
The yaw rate (deviation of the orientation) can also be formulated from the flat output and its derivation:
The steering angle can be calculated easly:
.
But transform into the time, the coherence
[1] B. Müller, J. Deutscher and S. Grodde, "Continuous Curvature Trajectory Design and Feedforward Control for Parking a Car," in IEEE Transactions on Control Systems Technology, vol. 15, no. 3, pp. 541-553, May 2007, doi: 10.1109/TCST.2006.890289.
[2] B. Müller, J. Deutscher and S. Grodde, "Trajectory generation and feedforward control for parking a car," 2006 IEEE Conference on Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control, 2006, pp. 163-168, doi: 10.1109/CACSD-CCA-ISIC.2006.4776641.
[3] B. Müller and J. Deutscher, "Orbital tracking control for car parking via control of the clock using a nonlinear reduced order steering-angle observer," 2007 European Control Conference (ECC), 2007, pp. 1917-1924, doi: 10.23919/ECC.2007.7068555.