2.165 RoboticsSpring 2006Problem Set #1Issued : Tue 02/14/2006Due : Thu 02/23/2006Problem 1:Consider a two degree-freedom planar manipulator with two rotational joints with linklengths l1= 5 a nd l2= 3. The endpoint velocity is denoted by V = [vx, vy]T.(a) Given a desired endpoint velocity, find joint velocities that produce the desiredendpoint velocity.(b) Find singular configurations, and determine in which direction the endpoint can’tmove f or each singular configuration.Figure 1: Trajectory for Problem 1. OA=6, OB=4, OC=4, OD = 6. These dimensions havebeen chosen such that the trajectory lies within the workspace of the manipulator.2.165 Problem Set #1 Page 2(c) Plot profiles of joint velocities when the endpoint is required to track a specifiedtrajectory (shown in Figure 1) at a constant tangential speed.Problem 2:Consider a two degree-freedom planar manipulator with link lengths l1= l2= 2m.Measuring the ratio of joint torque to joint displacement, we identify the stiffness o feach joint:k1= 3 × 105Nm/rad k2= 2 × 105Nm/rad(a) Compute the endpoint compliance matrix for the configuration of θ1= 450andθ2= 600.(b) F ind the directions of maximum and minimum compliance at this configuration.(c) Plot the maximum and minimum stiffness values as t he function of θ1and θ2.Problem 3:For the same manipulator as above, consider the problem of inverse kinematics usingsliding variables as we have discussed in class. Explain why defining qe(t) by theequation:˙qr= ˙qe− λ(q − qe)leads to an explicit inverse kinematics solution for qe. Plot your result of qein simula-tion for xdbeing a circle with radius 1 and constant tangential