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

where V = vehicle’s lateral velocity, r = vehicle’s yaw velocity, ψ = vehicle’s yaw position, and Y_g = the y-axis coordinate of the vehicle’s center of gravity. K is a parameter to be varied depending upon trajectory changes. In a specific vehicle traveling at a speed of U = -10m/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 (Ünyeliogl̆ u, 1997).Assuming the vehicle will be controlled in closed loop:

a. Find the system’s characteristic equation as a function of K.

b. Find the system’s 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.

Transcribed Image Text:

bịK a12 -bK d a11 V bzK an -b,K d a21 1 U