Formulate the mobile robot trajectory following problem as a control problem. A mobile robot

position at time $t$ is denoted as $(x(t),y(t))$ and its orientation at time $t$ is denoted as $(t),$ the

velocity and turn rate are denoted as $v(t)$ and $(t)$ respectively. The nonlinear motion model can

be simplified as

$x^{}=$vcos$,y^{}=$vsin$,{}^{}=$

Suppose the desired trajectory is

tilde$(x)(t)=5t,$tilde$(y)(t)=5\sqrt[\phantom{2}]{3}t,$tilde$()(t)=\frac{}{3}$

Assuming $v(t)$ is close to $10\frac{m}{s},(t)$ is close to $0$ and $(t)$ is close to $\frac{}{3},$ define the difference

between the actual trajectory and the desired trajectory as the state, $v(t)-10$ and $(t)$ as control

inputs, assume the output is the same as the state, derive the state$-$space model of the linearized

control system.