Robotics: Fundamentals Prof. Mark Yim University of Pennsylvania Week 5: Degrees of Freedom 1
The Goal Understanding the position and orientation of robot links. Computing end-effector positions from joint angles. Computing joint angles from end effector positions. 2
Universal Joint Example 3
Degrees of Freedom 4
Degrees of Freedom 5
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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 7
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4-bar linkage 1
Four-bar linkage 11
Parallel Four-bar 12
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More complex linkage 14
Pathological Exceptions By Van helsing - self-made largely based on an image at pergatory.mit.edu, CC BY 2.5, https://commons.wikimedia.org/w/index.php?curid=2533845 15
Robotics: Fundamentals Prof. Mark Yim University of Pennsylvania Week 5: Forward Kinematics and DH Parameters 16
Forward Kinematics 17
Labeling Conventions 18
Labeling Conventions Joint 2 x1 θ2 Link 2 Joint 1 x θ1 y1 x2 Link 1 y y2 Base (Link ) 19
Planar Forward Kinematics Joint 2 x1 θ2 Link 2 Joint 1 y1 θ x 1 x2 Link 1 y y2 Base (Link ) 2
Planar Forward Kinematics Joint 2 c1 = cos(θ1) s1 = sin(θ1) Joint 1 θ x 1 θ2 Link 2 a1 y1 Link 1 y x1 a2 x2 y2 Base (Link ) 21
Planar Forward Kinematics Joint 2 x1 θ2 Link 2 Joint 1 y1 θ x 1 x2 Link 1 y y2 Base (Link ) 22
3D 6DOF Links/Joints 23
3D 6DOF Links/Joints Joint 3 k n i L 3 k n Li 2 Joint 2 Lin k1 Joint 4 Joint 1 Link 5 Joint 5 Lin k4 Joint 6 Link 6 Base (Link ) RRPRRR 24
3D 6DOF Transformation 25
Denavit-Hartenberg Convention 4 parameters for each link/joint i a is link length of link i i α is link twist of link i i d is the link/joint offset of link/joint i i θ is the joint angle of joint i i 26
DH Example By Ollydbg (I generate this by Blender and Inkscape) CC BY-SA 3. 27
DH Link Transformation 28
DH Frame Placement Rules [DH1] The axis Zi-1 is the joint axis for joint i Axis of revolution for revolute joint Axis of translation for prismatic [DH2] The axis Xi is perpendicular to the axis Zi-1 [DH3] The axis Xi intersects the axis Zi-1 29
DH Parameters a is distance between Z and Z i i i-1 along Xi α is the angle between Z and Z about Xi d is distance between X and X along Zi-1 i i i i i-1 i-1 θ is the angle between X and X i i i-1 about Zi-1 3
DH Example By Ollydbg (I generate this by Blender and Inkscape) CC BY-SA 3. 31
DH Process 32
Special Case If z and z i i-1 are parallel: Choose any d. Other parameters are the same as before 33
Robotics: Fundamentals Prof. Mark Yim University of Pennsylvania Week 5: Examples of Forward Kinematics 34
3link Cylindrical Robot DH Parameters Link ai αi di θi 1 d1 θ1 2-9 d2 3 d3 Bolded are joint variables 35
Link 1: revolute joint Link ai αi di θi 1 d1 θ1 2-9 d2 3 d3 36
Link 2: prismatic joint Link ai αi di θi 1 d1 θ1 2-9 d2 3 d3 37
Link 3: prismatic joint Link ai αi di θi 1 d1 θ1 2-9 d2 3 d3 38
End-effector Transform 39
Spherical Wrist DH Parameters Link ai αi di θi 4-9 θ4 5 9 θ5 6 d6 θ6 Bolded are joint variables 4
Link 4: revolute joint Link ai αi di θi 4-9 θ4 5 9 θ5 6 d6 θ6 41
Link 5: revolute joint Link ai αi di θi 4-9 θ4 5 9 θ5 6 d6 θ6 42
Link 6: revolute joint Link ai αi di θi 4-9 θ4 5 9 θ5 6 d6 θ6 43
End-effector Transform 44
End-effector Transform 45
Final Transform 46
Stanford Arm (RRP) DH Parameters Link ai αi di θi 1-9 θ1 2 9 d2 θ2 3 d3 4-9 θ4 5 9 θ5 6 d6 θ6 Bolded are joint variables 47
Link 1: revolute joint Link ai αi di θi 1-9 θ1 2 9 d2 θ2 3 d3 4-9 θ4 5 9 θ5 6 d6 θ6 48
Link 2: revolute joint Link ai αi di θi 1-9 θ1 2 9 d2 θ2 3 d3 4-9 θ4 5 9 θ5 6 d6 θ6 49
Link 3: prismatic joint Link ai αi di θi 1-9 θ1 2 9 d2 θ2 3 d3 4-9 θ4 5 9 θ5 6 d6 θ6 5
Link 1-3: prismatic joint Link ai αi di θi 1-9 θ1 2 9 d2 θ2 3 d3 4-9 θ4 5 9 θ5 6 d6 θ6 51
Link 1-3: prismatic joint Link ai αi di θi 1-9 θ1 2 9 d2 θ2 3 d3 4-9 θ4 5 9 θ5 6 d6 θ6 52
Links 4-6: 3-axis Spherical Joint Link ai αi di θi 1-9 θ1 2 9 d2 θ2 3 d3 4-9 θ4 5 9 θ5 6 d6 θ6 53
Stanford Arm Transform 54
Standard configurations 55
Standard configurations 56
SCARA robot arm Selective Compliance Articulated Robot Arm. By Nikola Smolenski - CC BY-SA 3. 57
SCARA Arm DH Parameters Link 1 2 3 4 ai a1 a2 αi 18 di d3 d4 θi θ1 θ2 θ4 Bolded are joint variables 58
Link 1: z-axis revolute joint Link ai αi di θi 1 a1 θ1 2 a2 18 θ2 3 d3 4 d4 θ4 59
Link 2: z-axis revolute joint Link ai αi di θi 1 a1 θ1 2 a2 18 θ2 3 d3 4 d4 θ4 6
Link 3: prismatic joint Link ai αi di θi 1 a1 θ1 2 a2 18 θ2 3 d3 4 d4 θ4 61
Link 4: z-axis revolute joint Link ai αi di θi 1 a1 θ1 2 a2 18 θ2 3 d3 4 d4 θ4 62
End-effector Transform 63