Chapter 6, Problem 62P

Look-ahead information can be used to automatically steer a bicycle in a closed-loop configuration. A line is drawn in the middle of the lane to be followed, and an arbitrary point is chosen in the vehicle’s longitudinal axis. A look-ahead offset is calculated by measuring the distance between the look-ahead point and the reference line and is used by the system to correct the vehicle’s trajectory. A linearized model of a particular bicycle traveling on a straight-line path at a fixed longitudinal speed is

$\left[\begin{array}{c}\stackrel{\dot{}}{V}\\ \stackrel{\dot{}}{r}\\ \stackrel{\dot{}}{\psi }\\ {\stackrel{\dot{}}{Y}}_g\end{array}\right]=\left[\begin{array}{cccc}-11.7& 6.8& 61.6K& 7.7K\\ -3.5& -24& -66.9K& 8.4K\\ 0& 1& 0& 0\\ 1& 0& -10& 0\end{array}\right]\left[\begin{array}{c}V\\ r\\ \psi \\ Y_g\end{array}\right]$

In this model, V = bicycle’s lateral velocity, r = bicycle’s yaw velocity, $\psi$ = bicycle’s yaw acceleration, and Y_g= bicycle’s center of gravity coordinate on the y -axis. K is a controller parameter to be chosen by the designer $\left(Özg\ddot{u}ner,\text{ 1995}\right)$ . Use the Routh-Hurwitz citerion to find the range of K for which the system is closed-loop stable.