Consider the SCARA manipulator shown in the figure. Xe Ye Ze Pe joint angles 0, i=1,2,4...

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Consider the SCARA manipulator shown in the figure. Xe Ye Ze Pe joint angles 0₁, i=1,2,4 and d3. The variable de is measured in reference to the fixed X axis. If oe = 0, then the gripper is aligned with the fixed X axis. Use Homogenous Transformation Matrices. (a) Perform the forward displacement analysis for the manipulator relating the the twist T Xe Ye (b) Find the Jacobian J of the manipulator relating the twist T = Oi, i=1,2,4 and d3. 2 ZA Y X Figure 1: SCARA Manipulator. to q = dz 84 0₁ 02 d3 04 in terms of the in terms of the joint angles (c) Suppose that the robot joint angle vector is q=[0 0 1 0]. i. Explain why this is a singular configuration. Find all the joint velocities that produce a zero twist (this set of joint velocities is know as the null space of J). ii. Find the end effector twist if q = [1 -1 10]. Consider the SCARA manipulator shown in the figure. Xe Ye Ze Pe joint angles 0₁, i=1,2,4 and d3. The variable de is measured in reference to the fixed X axis. If oe = 0, then the gripper is aligned with the fixed X axis. Use Homogenous Transformation Matrices. (a) Perform the forward displacement analysis for the manipulator relating the the twist T Xe Ye (b) Find the Jacobian J of the manipulator relating the twist T = Oi, i=1,2,4 and d3. 2 ZA Y X Figure 1: SCARA Manipulator. to q = dz 84 0₁ 02 d3 04 in terms of the in terms of the joint angles (c) Suppose that the robot joint angle vector is q=[0 0 1 0]. i. Explain why this is a singular configuration. Find all the joint velocities that produce a zero twist (this set of joint velocities is know as the null space of J). ii. Find the end effector twist if q = [1 -1 10].