The inverse kinematic problem at the position level for a three$-$link robot is to find the joint angles ${}_1,{}_2$ and ${}_3$ that allow the end$-$effector $(P)$ at the third link to be at a specified point characterized by the coordinates $(x,y,),$ where $(x,y)$ is the Cartesian position and $$ the angle of the end$-$effector axis with respect to the $x-$axis. In this example, we assume that the link lengths are $a_1=6,a_2=5$ and $a_3=4(cm).$ The first link makes an angle ${}_1$ with the horizontal axis, the second link makes an angle ${}_2$ with the direction defined by the first link, and the third link makes an angle ${}_3$ with the direction defined by the second link as depicted in Fig. $1.$

$(1)-(20$ pts$)$ By following the schematic representation illustrated in Fig. I, give the expression of the coordinates $(x,y,)$ as a function of the joint angles ${}_1,{}_2$ and ${}_3.$

$(2)-(20$ pts$)$ Determine the Jacobian of the system of three equations found in $(1).$

$(3)-(20pts)$ Give the conditions under which the Jacobian of the robot is singular. Give a physical interpretation of the singularity conditions of the robot.

$(4)-(40$ pts$)$ We wish to find the angles so that the arm will move to the position $(x,y)=(10,4)(cm)$ with an orientation $=0\mathrm{.}4(rad),$ while starting with initial joint angles of ${}_1=0\mathrm{.}5,{}_2=0\mathrm{.}7$ and ${}_3=0\mathrm{.}6(rad).$ Use the Newton$-$Raphson formula to find the values of the requested joint angles ${}_1,{}_2$ and ${}_3.$ To proceed, write a sub$-$program implementing the Newton$-$Raphson algorithm. Perform up to six iterations or until obtaining an estimated error $||||=\sqrt[\phantom{2}]{|{}_1{|}^2+|{}_2{|}^2+|{}_3{|}^2}$ less than $0\mathrm{.}001$