dubins car system ode45

Hello people, I need some help to understand physically and then in Matlab how this system works.

I want to use differential equations to see how a Dubins car will draw a circle (measure the time it takes) when it moves with a constant speed and a constant radius. Then I want to solve it with ode45.

Normally, in polar equations,

x=r*cos(theta)

y=r*sinθ(theta)

and the state variables are x1=theta, x2=x, x3=y

If I derivate I would get

x1dot=thetadot=v/r

x2dot=xdot=-rsin(theta)

x3dot=ydot=rcos(theta)

In the literature I found that

xdot= v*cos(theta)

ydot = v*sin(theta)

Can somebody exmplain why we get that ? I'm sorry it's so obvious but I am really blocked right now. Thank you in advance!

[tammineni ravi]

first you have written is r*sin(theta) is wrong. That should be v*sin(theta). where v is velocity vector of the dubins vehicle.with heading angle theta. xdot means velocity in x direction.and, similarly ydot is velocity in y direction. therefore directly xdot=v*cos(theta).