2. Let us consider the RPR manipulator with 3 links represented in the schematic below. The schematic is drawn in the configuration 1 = 0,03 = 90 28 X2, X3 X4 Za (Out) Zo, Z dz TX, XI TTTT Luckily, you do not need to all of the forward kinematics, because the transformation matrices for the first 3 joints are given to you here (note that C13 = cos(01 +03)): ] 0 T = [-$1 001 c -810 lic _ si C 0 0 0 | sic T = Olisi 10 01 02 0 0 1 d2 10 001] 10 00 1] Oy : [cc C381 83 0 -83 81 -8183 - C30 0 0 lic 1181 de 1 (a) Calculate the position of the end effector (frame {4}) with respect to frame {0} (b) Compute the basic Jacobian of the end-effector (c) Find the linear velocity and angular velocity of the end-effector in frame {0} as a function of the joint variables. Additionally, compute the numerical values of the linear and angular velocities [y w of the end-effector when the joint angles and velocities are given by: q= [ -/6 rad] 0.5 m /3 rad = [ 0.4 rad/s] 0.1 m/s 1 -0.2 rad/s] Take l1 = 13 = 1 m. (d) Now, we attach a tool to the robot's hand. The tool starts at the origin of {4}, has a length of l = 20 cm, and is rotated about the 24 axis by a fixed angle 01 = -30 22 (320X2, X3 SX4 Zo, Z dz{ tool tip Ixo, Xi TITT Since we now care about the motion of the tool tip instead of the origin of {4}, compute the basic Jacobian of the tool tip State your answer using l1 = 13 = 1 m again, but leave all joint parameters as variables 2. Let us consider the RPR manipulator with 3 links represented in the schematic below. The schematic is drawn in the configuration 1 = 0,03 = 90 28 X2, X3 X4 Za (Out) Zo, Z dz TX, XI TTTT Luckily, you do not need to all of the forward kinematics, because the transformation matrices for the first 3 joints are given to you here (note that C13 = cos(01 +03)): ] 0 T = [-$1 001 c -810 lic _ si C 0 0 0 | sic T = Olisi 10 01 02 0 0 1 d2 10 001] 10 00 1] Oy : [cc C381 83 0 -83 81 -8183 - C30 0 0 lic 1181 de 1 (a) Calculate the position of the end effector (frame {4}) with respect to frame {0} (b) Compute the basic Jacobian of the end-effector (c) Find the linear velocity and angular velocity of the end-effector in frame {0} as a function of the joint variables. Additionally, compute the numerical values of the linear and angular velocities [y w of the end-effector when the joint angles and velocities are given by: q= [ -/6 rad] 0.5 m /3 rad = [ 0.4 rad/s] 0.1 m/s 1 -0.2 rad/s] Take l1 = 13 = 1 m. (d) Now, we attach a tool to the robot's hand. The tool starts at the origin of {4}, has a length of l = 20 cm, and is rotated about the 24 axis by a fixed angle 01 = -30 22 (320X2, X3 SX4 Zo, Z dz{ tool tip Ixo, Xi TITT Since we now care about the motion of the tool tip instead of the origin of {4}, compute the basic Jacobian of the tool tip State your answer using l1 = 13 = 1 m again, but leave all joint parameters as variables
