Prof. Mark Yim University of Pennsylvania
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1 Robotics: Fundamentals Prof. Mark Yim University of Pennsylvania Week 5: Degrees of Freedom Robo1x-1.5 1
2 The Goal Understanding the position and orientation of robot links. Computing end-effector positions from joint angles. Computing joint angles from end effector positions. Robo1x-1.5 2
3 Robotics: Fundamentals Video 5.4 Mark Yim Robo1x-1.5 3
4 Universal Joint Example Robo1x-1.5 4
5 Degrees of Freedom Robo1x-1.5 5
6 Degrees of Freedom Robo1x-1.5 6
7 Robo1x-1.5 7
8 7 links: Base, spine, shoulder, shoulder twist, forearm, gripper 6 joints between the 7 links, all revolute 6(7-1) 6(5) = 6 degrees of freedom Robo1x-1.5 8
9 Robo1x-1.5 9
10 Robo1x
11 4-bar linkage Robo1x
12 Four-bar linkage Robo1x
13 Parallel Four-bar Robo1x
14 Robo1x
15 More complex linkage Robo1x
16 Pathological Exceptions By Van helsing - self-made largely based on an image at pergatory.mit.edu, CC BY 2.5, Robo1x
17 Robotics: Fundamentals Video 5.5 Mark Yim Robo1x
18 Forward Kinematics Robo1x
19 Labeling Conventions Robo1x
20 Labeling Conventions Joint 1 x 0 Joint 2 Link 1 x 1 θ 2 θ 1 y1 Link 2 x 2 y 0 y 2 Base (Link 0) Robo1x
21 Planar Forward Kinematics A " = R %" d %" 0 1 A ) = R ") d ") 0 1 Joint 1 x 0 Joint 2 Link 1 x 1 θ 2 θ y 1 1 Link 2 x 2 y 0 y 2 Base (Link 0) Robo1x
22 Planar Forward Kinematics A " = R %" d %" 0 1 A ) = R ") d ") 0 1 Joint 1 x 0 Joint 2 Link 1 x 1 θ 2 θ y 1 1 Link 2 x 2 y 0 y 2 Base (Link 0) Robo1x
23 Planar Forward Kinematics Joint 1 x 0 y 0 Joint 2 Link 1 x 1 θ 2 a 1 θ y 1 1 a 2 Link 2 y 2 x 2 Base (Link 0) Robo1x
24 Planar Forward Kinematics c " s " 0 a " c " s A " = " c " 0 a " s " A ) = c 1 = cos(θ 1 ) Joint 2 x 1 s 1 = sin(θ 1 ) Joint 1 x 0 θ 2 a 1 θ y 1 1 Link 1 a 2 c ) s ) 0 a ) c ) s ) c ) 0 a ) s ) Link 2 x 2 y 0 y 2 Base (Link 0) Robo1x
25 Planar Forward Kinematics T %) = A " A ) = R %) d %) 0 1 Joint 1 x 0 Joint 2 Link 1 x 1 θ 2 θ y 1 1 Link 2 x 2 y 0 y 2 Base (Link 0) Robo1x
26 3D 6DOF Links/Joints Robo1x
27 3D 6DOF Transformation T %0 = A " A ) A 1 A 2 A 3 A 0 = R %0 d %0 0 1 Robo1x
28 Robotics: Fundamentals Video 5.6a Mark Yim Robo1x
29 Denavit-Hartenberg Convention 4 parameters for each link/joint i a i is link length of link i α i is link twist of link i d i is the link/joint offset of link/joint i θ i is the joint angle of joint i Robo1x
30 Denavit-Hartenberg Convention 4 parameters for each link/joint i a i is link length of link i α i is link twist of link i d i is the link/joint offset of link/joint i θ i is the joint angle of joint i Robo1x
31 DH Example By Ollydbg (I generate this by Blender and Inkscape) CC BY-SA 3.0 Robo1x
32 DH Link Transformation Robo1x
33 DH Frame Placement Rules [DH1] The axis Z i-1 is the joint axis for joint i Axis of revolution for revolute joint Axis of translation for prismatic [DH2] The axis X i is perpendicular to the axis Z i-1 [DH3] The axis X i intersects the axis Z i-1 Robo1x
34 DH Parameters a i is distance between Z i and Z i-1 along X i α i is the angle between Z i and Z i-1 about X i d i is distance between X i and X i-1 along Z i-1 θ i is the angle between X i and X i-1 about Z i-1 Robo1x
35 DH Process 1. Label z i axes 2. Set base frame and end effector frame x 0 and y 0 as arbitrary 3. For i=1, n-1, A. Find common normal between z i and z i-1 (z i and z i-1 parallel is a special case) B. Establish x i on this normal C. Establish y i perpendicular to x i and z i to form a right handed coordinate frame 4. Create a table of all link parameters a i d i α i θ i 5. Form homogeneous transformation A i for each link 6. Form T %4 = A " A 4 Robo1x
36 Special Case If z i and z i-1 are parallel: Choose any d. Other parameters are the same as before Robo1x
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