Problem $2$:

Fig. $2$ shows a differential drive mobile robot having a pair of powered wheels and a frictionless caster. The radius of the wheels is $r=3cm,$ while the distance between the two wheels is $2d=20cm.$ The angular velocity of the right and left wheels are ${}_R$ and ${}_L$ respectively. Each powered wheel is equipped with a shaft encoder that measures the angular velocity of the wheel.

a$)$ Provide the Jacobian matrix $J$ that relates the wheels' angular velocities to the linear and angular velocities of the robot with respect to the inertial frame $\{s\}.$ I.e$.,$ provide $J$ in $p^{}=\left[\begin{array}{c}{x}^{}\\ y^{}\\ {}^{}\end{array}\right]=J\left[\begin{array}{c}{}_L\\ {}_R\end{array}\right]$

b$)$ A camera is positioned with respect to the inertial frame $\{s\}$ as shown in Fig. $3.$ Compute the Jacobian matrix $J_c$ that relates the wheels' angular velocities to the linear and angular velocities of the robot with respect to the camera frame $\{c\}.$ I.e$.,$ compute $J_c$ in $p_c^{}=\left[\begin{array}{c}{x}_c^{}\\ y_c^{}\\ {}_c^{}\end{array}\right]=J_c\left[\begin{array}{c}{}_L\\ {}_R\end{array}\right].$ A symbolic expression of the solution suffices.