2D vehicle is moving at a speed of u such that its velocity vector makes and ang

Question

2D vehicle is moving at a speed of u such that its velocity vector makes and angle with x-axis (called heading). The coordinates of its center are, x and y, and the steering angle is . The wheelbase L can be assumed to be 1m. Then kinematic equations of the vehicle can be written as: (x, y) v sin 7 Let's say you want to control the vehicle to move on a straight line (x axis) while it moves at a constant speed of v -1m/s. So, you want the y coordinate to be close to zero. The vehicle is controlled by prescribing its steering angle . we shall assume that the angle and steering angle are small enough (within 0.3 rad) so that sin and The angle of the vehicle 0 and its x and y coordinates are all available for measurement. Objective is to design a controller that uses any of the measurements and provides the correct steering angle to make the vehicle go in a straight line; assuming the vehicle started on that line with a wrong initial angle. What variable will be the input to the plant u? What variable will be the output of the plant y? What will be the transfer function of the plant from input to the output G(s)? (You'll have to linearize the differential equation around the relevant operating angle ). a.

Explanation / Answer

given 2D vehicle

velocity vector v, makes angel theta with x axis ( heading)

coordinates of center = (x,y)

steering angle phi

L = 1 m

x' = vcos(theta)

y' = vsin(theta)

phi' = (v*tan(phi))/L

assuming small angles

x' = v

y' = v*theta

phi' = v*phi/L

theta, x, y are measurable

a. the object needs to go ina astriaght line

input variable will be the values of x, y and theta

becauase this will let the bot know about ots v

output variable will be phi

now

phi' = v*phi/L

v = sqrt(x'^2 + y'^2)

hence

phi' = sqrt(x'^2 + y'^2)*phi/L = v*phi/L

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