Robot modelling $1.$ Identify the configuration of the robot given in Fig$\mathrm{.}1$ and solve for inverse kinematics using the DH convention. $(3$ points$)$ Fig. $12.$ Identify the configuration of the robot given in Fig$\mathrm{.}2$ and solve for inverse kinematics using the DH convention. $(3$ points$)$ Fig$\mathrm{.}23.$ For the Cylindrical robot with spherical wrist given in Fig$\mathrm{.}3,$ find the transformation matrix using the DH convention. $(4$ points$)$Q$1)(30pts)$ The mobile robot given in Fig. $1$ reached its current location from the origin of world frame $(o_{11})$

by translating $5m$ along the $x-$axis of world frame, then rotating about $z-$axis of world frame by $90,$ then

translating $2m$ along the $x-$axis of mobile robot $(x_m),$ and finally rotating about z$-$axis of mobile robot framo

$(z_m)$ by $-90.$ The mobile robot has a camera and laser scanners to explore the environment. For this robot,

a$)(15$ pts$)$ Find homogeneous transformation between world frame and mobile robot's frame.

b$)(15$ pts$)$ The origin of camera frame $(o_c)$ is located at $(x,y,z)=(0\mathrm{.}2,0,0\mathrm{.}5)$ with respect to mobile robot's

frame and rotational relationship between the camera frame and mobile robot's frame is shown in Fig. $1.$

On the other hand, the origin of front laser scanner $(o_s)$ is located at $(x,y,z)=(0\mathrm{.}5,-0\mathrm{.}3,0)$ with respect

to mobile robot's frame. There is $$ rad difference between $z-$axes of mobile robot and laser scanner frames.

The laser scanner has detected an object $10m$ away from it$.$ The position of the object with respect to laser

scanner's frame $(p_{lt})$ is given in Fig. $1.$

i$.$ Find $"$a$"$ value.

ii$.$ Determine the position of the object with respect to world frame.

iil. Determine the position of the object with respect to camera frame.

Figure I