CS 4649/7649 Robot Intelligence: Planning

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1 CS 4649/7649 Robot Intelligence: Planning Differential Kinematics, Probabilistic Roadmaps Sungmoon Joo School of Interactive Computing College of Computing Georgia Institute of Technology S. Joo 1 *Slides based in part on Dr. Mike Stilman s lecture slides

2 Potential Fields S. Joo 2

3 Potential Fields *Simulation by Leng-Feng Univ. S. Joo 3

4 Summary: Navigation Algorithms S. Joo 4

5 Coordinates S. Joo 5

6 Homogeneous Transform 3D (1) Change the representation of a vector in frame B to frame A (2) Change a vector in a frame to another vector (rotated) in the (3) Unit vectors of frame B represented(expressed) in frame A S. Joo (sungmoon.joo@cc.gatech.edu) 6

7 Forward(Direct) Kinematics {0} y x S. Joo (sungmoon.joo@cc.gatech.edu) 7

8 Forward Kinematics S. Joo 8

9 Forward Kinematics Representation of frame2 in frame0 Representation of frame3 in frame0 S. Joo 9

10 Denavit-Hartenberg (DH) Parameters S. Joo 10

11 DH Parameters For a revolute joint link parameters joint variable *For a prismatic joint: d i is a joint variable S. Joo (sungmoon.joo@cc.gatech.edu) 11

12 DH Parameters S. Joo 12

13 DH Parameters *special links first, last *special cases eg. parallel links S. Joo 13

14 Kinematics S. Joo 14

15 Inverse Kinematics S. Joo 15

16 Configurations Space Work Space Configuration Space S. Joo 16

17 Configuration Space: IK Solution S. Joo 17

18 Configuration Space: IK Solution S. Joo 18

19 Configuration Space: IK Solution S. Joo 19

20 Analytical Methods S. Joo 20

21 Analytical Methods S. Joo 21

22 Analytical IK S. Joo 22

23 Infinitesimal Changes Forward Kinematics Differential (Forward) Kinematics S. Joo 23

24 Differential Kinematics S. Joo 24

25 Differential Kinematics S. Joo 25

26 How do we compute J? Method1 S. Joo 26

27 How do we compute J? Method1 S. Joo 27

28 How do we compute J? Method1 S. Joo 28

29 How do we compute J? Method2 Rigid body angular motion Rigid body linear + angular motion {B} {A} S. Joo (sungmoon.joo@cc.gatech.edu) 29

30 How do we compute J? Method2 Consider the propagation of velocities: linear velocity & angular velocity {i} {i-1} Linear velocity propagation Angular velocity propagation S. Joo 30

31 How do we compute J? Method2 Linear velocity propagation Prismatic Joint Angular velocity propagation Revolute Joint S. Joo 31

32 How do we compute J? Method2 S. Joo 32

33 How do we compute J? Method2 S. Joo 33

34 How do we compute J? Method2 S. Joo 34

35 How do we compute J? Method3 1. Simulate a small displacement Δθ i for each joint 2. Observe Δx and numerically compute J Less accurate, more time consuming Sometimes the only option S. Joo (sungmoon.joo@cc.gatech.edu) 35

36 Gradient IK - How do we use J? Workspace goal x 1x2 How do we get a joint space goal? q 1q2 x n q n Assuming 6 D.O.F and J is full rank: Δq = J 1 Δx Iterate until convergence Otherwise Still Possible (Pseudo-Inverse & Variants): J + = JT (JJ T ) 1 S. Joo (sungmoon.joo@cc.gatech.edu) 36

37 Gradient IK Generally applicable to n-dof kinematics Many existing implementations Slower and unstable around singularities S. Joo 37

38 Planning Goals for manipulators are really that complicated! Add what it takes to represent obstacles: But, does this remind you of anything? Δq = J 1 Δx S. Joo (sungmoon.joo@cc.gatech.edu) 38

39 Planning Manipulator + Potential Field S. Joo (sungmoon.joo@cc.gatech.edu) 39

40 Planning Manipulator + Potential Field First effective planning method for high-dof robot arms Very fast (potential fields) However: Local Minima Complex obstacle representation For Soundness must define obstacles with respect to each link S. Joo (sungmoon.joo@cc.gatech.edu) 40

41 Planning Manipulator + Potential Field Roadmap Methods: Combine Kinematics & Geometric Planning Visibility Graphs No polygons in C-space. How would you get polygons? Voronoi Diagrams Canny s Thesis: NP-hard Algorithms for Motion Planning Cell Decomposition: Exact Same troubles as Visibility Graphs. Approximate Ok for 2-3 dimensions Good luck in 4+ dimensions S. Joo (sungmoon.joo@cc.gatech.edu) 41

42 Kinematic Planning Algorithms Pre-1980 Mainly 2D, 3D planners In case of kinematics: Ignore robot geometry Robot is an ethereal entity that obeys kinematics 1980s Potential Fields! First feasible SOUND solution Locally optimal control strategy Not globally complete or optimal S. Joo 42

43 Can we solve these planning problems? Planning Algorithms, S. Lavalle S. Joo 43

44 Key Idea What did Visibility, Voronoi, Cells, Fields have in common? - Some form of explicit environment representation - Attempt at some form of optimality New concepts from 1990s: - Forget optimality altogether - Focus on Completeness - Think about Free Space S. Joo (sungmoon.joo@cc.gatech.edu) 44

points - Edges: Verified connections Probabilistic roadmaps for path planning in high-dimensional configuration

45 A New Kind of Roadmap Previous roadmaps used features related to actual obstacle features. Lydia Kavraki 94, 96 Present Mark Overmars 92, 96 - Present Probabilistic Roadmaps (PRM) - Features: Sampled free points - Edges: Verified connections Probabilistic roadmaps for path planning in high-dimensional configuration spaces By Kavraki, Svestka, Latombe, and Overmars, 1996, IEEE Transactions on Robotics and Automation S. Joo (sungmoon.joo@cc.gatech.edu) 45

46 Probabilistic Roadmap: Step 1 Randomly sample a configuration: P Keep only if P is in Free Space S. Joo (sungmoon.joo@cc.gatech.edu) 46

47 Probabilistic Roadmap: Step 1 Randomly sample a configuration: P Keep only if P is in Free Space S. Joo (sungmoon.joo@cc.gatech.edu) 47

48 Probabilistic Roadmap: Step 2 For each node P find k nearest neighbors: Q 1 Q k S. Joo (sungmoon.joo@cc.gatech.edu) 48

49 Probabilistic Roadmap: Step 3 For each node P find k nearest neighbors: Q 1 Q k Use a local planner to test connectivity between P and Qi S. Joo (sungmoon.joo@cc.gatech.edu) 49

50 Probabilistic Roadmap: Step 3 For each node P find k nearest neighbors: Q 1 Q k Use a local planner to test connectivity between P and Qi What could be a local planner? What happens next? S. Joo (sungmoon.joo@cc.gatech.edu) 50

51 Probabilistic Roadmap: Step 4 For each node P find k nearest neighbors: Q 1 Q k Use a local planner to test connectivity between P and Qi Find a path: Uniform Cost, A*, S. Joo (sungmoon.joo@cc.gatech.edu) 51

52 We can solve these planning problem S. Joo 52

CS 4649/7649 Robot Intelligence: Planning

CS 4649/7649 Robot Intelligence: Planning Roadmap Approaches Sungmoon Joo School of Interactive Computing College of Computing Georgia Institute of Technology S. Joo (sungmoon.joo@cc.gatech.edu) 1 *Slides