A technique to control the steering of a vehicle that follows a line located in the middle of a lane is to define a look-ahead point and measure vehicle deviations with respect to the point. A linearized model for such a vehicle is V_ r_ _ Y_ g 2 6 6 6 6 6 4 3 7 7 7 7 7 5 a11 a12 b1K b1K d a21 a22 b2K b2K d 01 0 0 1 0 U 0 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 V r Yg 2 6 6 6 6 4 3 7 7 7 7 5 where V vehicles lateral velocity,r vehicles yaw velocity, vehicles yaw position, and Yg the y-axis coordinate of the vehicles center of gravity. K is a parameter to be varied depending upon trajectory changes. In a specific vehicle traveling at a speed of U 10 m/sec, the parameters are a11 11:6842; a12 6:7632; b1 61:5789; a21 3:5143; a22 24:0257, and b2 66:8571: d 5 m is the look-ahead distance (nyelioglu , 1997). Assuming the vehicle will be controlled in closed loop: a. Find the systems characteristic equation as a function of K. b. Find the systems root locus as K is varied. c. Using the root locus found in Part b, show that the system will be unstable for all values K
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