Kinematic Car Model

The attached image depicts the kinematic car in the horizontal plane. Let us suppose that the Ackermann steering assumptions hold true, hence all wheels turn around the same point (denoted by P) which lies on the line of the rear axle. It follows that the kinematics of the car can be fully described by the kinematics of a bicycle fitted in the middle of the car (see attached image. The coordinates of the rear axle midpoint are given by $x$ and $y$. The orientation of the car with respect to the axis of $x$ is denoted by 9. The angle of the front wheel of the bicycle with respect to the longitudinal symmetry axis of the car is denoted by $φ$ . One may consider $φ$ or its time derivative $u_2=\dot{φ}$ as input. The longitudinal velocity of the rear axle midpoint is denoted by $u_1$ if it is a control input (two input case) and by $v_{car}$ if not (one input case). All lengths involved in the kinematic calculations, and in particular $l$, equal to one.

$$\begin{align*} \dot{x} &= u_1 \cos{\theta},\\ \dot{y} &= u_1 \sin{\theta},\\ \dot{\theta} &= u_1 \tan{\varphi},\\ \dot{\varphi} &= u_2. \end{align*}$$