Consider an RPR manipulator in its zero-configuration, as shown above on the left, with L1=3 and L2=2. This robot has a marker attached at the end-effector, which can draw on the board defined by the xy plane. You are tasked to control the robot to draw a straight vertical line from (2,3) to (2,0) within 3 seconds. Assume that the end effector of the robot is initially located at (2, 3) and moves downward till (2,0) to draw a straight line.

1.1. Generate a smooth trajectory of the end-effector position in the Cartesian coordinate system, considering the following: - The desired trajectory of the end-effector in the Cartesian coordinate is given by xd(t)=[xd(t);yd(t)]. - The initial and final velocities of the end-effector are x(0)=[0,0]T and x(3)=[0,0]T. answer: xd(t)= [ 2 ; 3-t ]

1.2. Using the end-effector trajectory obtained from Problem 1, generate the desired joint trajectories (d) numerically using the Newton-Raphson method and MATLAB. - Plot the joint trajectories in a single graph, with the time t on the x-axis and 1(t)(RED LINE) and 2(t)(BLUE LINE) on the y-axis. Save this as Figure 1. - Use different colors or line types for individual variables and provide a legend. Use h= 0.1[sec] for the time step size and the desired error, ed=0.001. - For the Newton-Raphson method, use the initial guess (0)0=[/6,/6]T; and for all following data point, use the previously calculated values as the initial guess for the next approximation, such that (ti)0=(ti1)n, where (ti1)n is the final approximation at the previous time step. MATLAB PLZ!!!

1.3 Velocity control MAKE PI controller. 2 joints use = 10, = 5 both. joint velocity (dth) Express it as a formula.

1.4 When t=0, the point value is 0 = [0.3948; 1.5708]. Using MATLAB Draw a graph of 1() and 2() by numerically integrating the velocity (dth) obtained from the controller designed in (1.3). - Draw h=0.1 at time intervals from t=0 to t=3. -1 is a solid red line, 1 is a red dotted line, 2 is a solid blue line, and 2 is a blue dot Add lines and appropriate legends. ( = [ 1; 2] obtained from (1.2))

1.5 Use Forward Kinematics and MATLAB to draw an end-effector trajectory graph according to 1() and 2() in (1.4). - Draw a trajectory by displaying the end-effector position on a 2D plane with a time interval h=0.1 from t=0 to t=3. - Add red solid lines and appropriate legend