Look-ahead information can be used to automatically steer

Chapter 6, Problem 62

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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

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

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 (Ö̈zgüner, 1995). Use the Routh-Hurwitz criterion to find the range of K for which the system is closed-loop stable.