Jane Li. Assistant Professor Mechanical Engineering Department, Robotic Engineering Program Worcester Polytechnic Institute

Transcription

1 Jane Li Assistant Professor Mechanical Engineering Department, Robotic Engineering Program Worcester Polytechnic Institute

2 We know how to describe the transformation of a single rigid object w.r.t. a single frame If we have many rigid object in serial connection, how to express and derive their spatial relations? RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/2018 2

3 Forward Kinematics Planar Robotic Systems, Representation of Serial Robots, Open Polygon Model, Denavit-Hartenberg Representation, Singularities Inverse Kinematics Kinematic Decoupling, Inverse Position: Geometric Approach, Inverse Orientation Kinematics in a Nut Shell RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/2018 3

4 For industrial robots, the main concern is the position and orientation of the end-effector or the attached tool Robot Joint Angles Inverse Kinematics Forward Kinematics End Effector Pose RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/2018 4

5 Tool Center Position (TCP) of a Planar Robotic Manipulator X d a cos A b cos B c cosc Y e asin A bsin B csin C RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/2018 5

6 Homogeneous transformations can be applied to all joints to get the end effector / tool position. However The transformation matrix depends on how the coordinate systems are set up and how the structural parameters are defined. Hence, how to make sure two people can develop same transformation matrices for the same robot? A standard method of Kinematics derivation RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/2018 6

7 Every coordinate frame is established following three rules: The z i 1 axis lies along the axis of motion of the ith joint RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/2018 7

8 Every coordinate frame is established following three rules: The x i axis is normal to the z i 1 axis, and points away from it to the z i axis RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/2018 8

9 Every coordinate frame is established following three rules: The x i axis forms the common perpendicular between the z i-1 and z i axis RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/2018 9

10 Base Frame You can choose any location for the coordinate frame 0 in the the robot base as long as the z 0 axis is aligned with the first joint End-effector Frame The last coordinate frame (nth frame) can be placed anywhere in the tool or end effector, as long as the x n axis is normal to z n 1 axis. RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

11 Relative pose between rigid bodies Position + Orientation How many parameters do you need to fully specify their relative pose? RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

12 θ i is the joint angle from the x i 1 to the x i axis about the z i 1 axis using the right hand rule Angle between two X-axes RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

13 d i is the offset distance from the origin of the (i 1)th coordinate frame to the intersection of the z i 1 axis with the x i axis along the z i 1 axis. Distance between two X-axes RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

14 a i is the distance from the intersection of the z i 1 axis with the x i axis to the origin of the ith frame along the x i axis (the shortest distance between the z i 1 and z i axes). Distance between two Z-axes RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

15 α i is the twisted angle from the z i 1 axis to the z i axis about the x i axis (using the right-hand rule). Angle between two Z-axes RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

16 RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

17 i A 1 Rot z, Trans 0,0, d Trans a,0,0 Rot x, i i i i i c s i i ai s c c s 0 i i i i d i s c 0 i i c s c s s a i i i i i ic i s c i c i c i s i a i is i 0 s c d i i i T Complete transformation from base frame to tool frame: A A A A A A RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

18 Recall: H nx sx ax dx ny sy ay d y n s a d nz sz az d z n = normal direction s = sliding direction a = approach direction RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

19 RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

20 # θ d a α 1 θ 1 0 a θ 2 0 a 2 0 RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

21 D-H table:, 0,0,,0,0, T A Rot z Trans d Trans a Rot x c s c s s a c c s 0 a c s c 1 c 1 c s a s s c 0 a s 1 0 s c d # θ d a α 1 θ 1 0 a θ 2 0 a 2 0 RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

22 # θ d a α 1 θ 1 0 a θ 2 0 a 2 0, 0,0,,0,0, T A Rot z Trans d Trans a Rot x c s c s s a c c s 0 a c s c 2 c 2 c s a s s c 0 a s 2 0 s c d RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

23 T T A c c s s c s s c 0 a c a c c s s s c c s c c s s 0 a s a s c c s A A A A c s 1 0 a 1 1c 1 s c 0 1 a 1 1s c s 0 a 2 2 2c 2 s c 0 a 2 2 2s RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

24 RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

25 # θ d a α 1 θ θ 2 0 a θ 3 0 a θ 4 0 a θ θ RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

26 T A A A A A A c c c c s s c c c c s c c c s c c a c a c a s1s5c 6 s1s5 s6 s1c 5 s c c c s s s c c c s c s c s + c1s 5s6 c1s5 s6 c1c 5 s234c5c6 c234s6 s234c5c6 c234c6 s234s5 s234a4 s23a3 s2a s1 c234a4 c23a3 c2a2 RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

27 There are 3 common singularities with serial robotics systems Wrist alignment joint 4 and 6 collinear axis Elbow singularity - Out-of-reach Alignment singularity wrist is as close to joint 1 as it can get RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

28 Degeneracy = Robot looses 1 DOF Physical Limits 2 similar joints become collinear Determinant of position matrix = zero Reduced dexterity Impossible to orient end effecter at a desired orientation, at the limits of robots workspace. RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

29 A spherical wrist A singular configuration when the vectors z3 and z5 are linearly dependent. The axes z3 and z5 are collinear, which happens when θ5 = 0 or π. RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

30 Unavoidable singularity for sphere wrist, unless The wrist is designed in such a way as to not permit this alignment. Not limited to a spherical wrist If any two revolute joint axes become collinear a singularity results. RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

31 RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

32 RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

33 The robot arm has two joints The joint space has 2 dimensions Theoretically, any position within the robot workspace is reachable by the end effector. However A singularity reduces the mobility of the robot. This will occur in two configurations what are they? RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

34 RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

35 RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

36 RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

37 Singularity due to aligned links Like the 2D case we just saw, there are two singularities due to the parallel Z1 and Z2 axes. RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

38 Singularity due to aligned rotational axes If the wrist center intersects with the axis of the base rotation, z 0, then there are an infinite number of solutions to the inverse kinematic equations. In other words, any value of θ1 will produce the same wrist position. We have again lost a degree of freedom RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

39 Singularity due to reach limit There are workspace volumes (shown in purple) where the end of arm tooling cannot reach. RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

40 Singularity due to self collision There are also configurations where the arm will collide with itself (another form of singularity). RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

41 RBE/ME 4815 Industrial Robotics Instructor: Jane Li, Mechanical Engineering Department & Robotic Engineering Program - WPI 3/22/

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