Theory of Robotics and Mechatronics

Transcription

1 Theory of Robotics and Mechatronics Final Exam Question: Total Points: Score: Name: Legi-Nr: Department: Semester: Duration: 120 min 1 A4-sheet (double sided) of notes allowed No calculators allowed No dictionaries allowed Language: English Try to answer all questions within the spaces provided on the sheet. If you run out of space there, continue on the back of the page. More paper will be provided if necessary. Hand in all paper you wrote on, even scratch paper. Write your name on every sheet. Put your Legi in front of you. 1

2 Theory of Robotics and Mechatronics: Final Exam Page 2 of 5 Question 1.. (18 points) Points earned: (a) (4 points) Briefly explain the difference between the precision and the accuracy of a sensor. Would you prefer a sensor that is accurate or one that is precise? Why? (b) (6 points) Answer true or false in the following questions. If you think that the statement is false, give the correct one. 1. For a given dimension of the workspace, a kinematically redundant manipulator has more degrees of freedom than needed. 2. Revolute joints always change the orientation of the end-effector and never the position. 3. The joint space defines the region where the end-effector of the robot moves. 4. The matrix exponential of a skew-symmetric matrix is a rotation matrix. 5. The velocity relationships between the joint space and the task space are determined by the Jacobian of forward kinematics. 6. The major axis of manipulability ellipsoid shows the direction requires the most effort from the joint motors. (c) (8 points) In general, we would need 6 independent parameters to define the transformation between two neighboring coordinate frames. However, the Denavit- Hartenberg (D-H) convention reduces the problem to 4 parameters by introducing two rules for the choice of the origin and orientation for the coordinate frames. 1. Describe the two rules to choose the origin and orientation of the coordinate frames. 2. Consider the two-link manipulator which has joint 1 revolute and joint 2 prismatic. The coordinate frames are established following the two rules for D-H convention. Find the link parameters, link lengths, twists, offsets, and joint angles. x % x $ d % z % z$ θ $ a " link length a i twist α i offset d i Joint angle θ i x " 1 z " 2 Figure 1

3 Theory of Robotics and Mechatronics: Final Exam Page 3 of 5 Question 2.. (32 points) Points earned: Consider the following manipulator at its initial configurations: l 1 l 2 θ $ θ % θ " l 3 l 0 x z S θ # y y T z x Figure 2: Manipulator at its initial configuration (θ 1 = θ 2 = θ 3 = θ 4 = 0). Joint 1 rotates about the positive z-axis and joint 2 rotates about the positive x-axis. 1 and 2 have a length of l 0 and l 1. Joint 3 translates along the positive y-axis and joint 4 translates along the negative z-axis. 3 and 4 have a length of l 2 and l 3. (a) (1 point) How many degrees of freedom does the manipulator have? (b) (8 points) Write the abstract geometrical screw parameters, the twists, and the twist coordinates for each joint on the manipulator. (c) (11 points) Derive the forward kinematics of the tool frame with respect to the spatial frame (g st (θ)). (d) (8 points) Derive the spatial manipulator Jacobian and tease out the spatial velocity. (e) (4 points) The hybrid velocity has the same physical meaning as the body velocity, but expressed in the spatial frame, rather than in the body frame. This corresponds to our physical intuition of velocity better than the spatial velocity. If the tool frame is expressed as [ ] R(θ) p(θ) g st (θ) = 0 1 the hybrid velocity is simply computed as [ ] R(θ) 0 V h = V b. 0 R(θ) Compute the hybrid velocity, V h, when the manipulator is passing its initial configuration (i.e., θ 1 = θ 2 = θ 3 = θ 4 = 0).

4 Theory of Robotics and Mechatronics: Final Exam Page 4 of 5 Answer sheet for Question 2

5 Theory of Robotics and Mechatronics: Final Exam Page 5 of 5 Question 3.. (10 points) Points earned: (a) (6 points) Name, describe and compare three different image smoothing filters. (b) (4 points) Erosion is the mathematical morphology to gradually shrink the boundaries of a region, and the amount that they shrink depends upon the choice of the structuring element. What would be the output of an erosion using a 4-connected and a 8-connected structuring element? Draw the answer on the following image Original Image 4-connectivity 8-connectivity