Lagrange $(40$ pts$)-$ Using Lagrangian techniques, find the necessary expressions of torque for

each joint of the Robot Arm shown in figures $1$a$,$ and $1b.$

A$.(5$ pts$)$ Using the $3$ joint planar arm in figures $1$a and $1b,$ draw and label the arm with the

joints indicating the axis components for each joint. Assume that the center of mass is located

at the midpoint of the link. Let Link L$0$ has mass M$0,$ Link L$1$ has mass M$1,$ and Link L$2$ has mass

M$2.$

B$.(5$ pts$)$ Find the velocity square terms for the arm.

C$.(9$ pts$)$ Find the expression for the kinetic energy $K$ of the arm where $K=K0+K2+K3.$

D$.(6$ pts$)$ Find the potential energy $P$ of the arm where $P=P0+P1+P2.$

E$.(5$ pts$)$ Write the Lagrange term $L$ in terms of $K$ and $P.$

F$.(10$ pts$)$ Find the Torques necessary for each joint.

Note: For this problem, you may assume the following joint position configuration for simplicity

of drawing and labeling.