Robotics Chapter 5 – Path and Trajectory Planning

1. RoboticsChapter 5 – Path and Trajectory Planning Dr. Amit Goradia

2. Topics • Introduction – 2 hrs • Coordinate transformations – 6 hrs • Forward Kinematics - 6 hrs • Inverse Kinematics - 6 hrs • Velocity Kinematics - 2 hrs • Trajectory Planning - 6 hrs • Robot Dynamics (Introduction) - 2 hrs • Force Control (Introduction) - 1 hrs • Task Planning - 6 hrs

3. Robot Motion Planning • Path planning • Geometric path • Issues: obstacle avoidance, shortest path • Trajectory planning, • “interpolate” or “approximate” the desired path by a class of polynomial functions • Generate a sequence of time-based “control set points” for the control of manipulator from the initial configuration to its destination.

4. The World is Comprised of… • Obstacles • Already occupied spaces of the world • In other words, robots can’t go there • Free Space • Unoccupied space within the world • Robots “might” be able to go here • To determine where a robot can go, we need to discuss what a Configuration Space is

5. Configuration Space • Notation: • A: single rigid object –(the robot) • W: Euclidean space where A moves; • B1,…Bm: fixed rigid obstacles distributed in W • FW – world frame (fixed frame) • FA – robot frame (moving frame rigidly associated with the robot) Configuration q of A is a specification of the physical state(position and orientation) of A w.r.t. a fixed environmental frame FW. Configuration Space is the space of all possible robot configurations.

6. Definitions • Configuration: Specification of all the variables that define the system completely • Example: Configuration of a dof robot is • Configuration space (C-space): Set of all configurations • Free configuration: A configuration that does not collide with obstacles • Free space ( F ) : Set of all free configurations • It is a subset of C

7. C Cfree qslug Cobs qrobot Configuration Space of a 2D Planer Robot Configuration Space of A is the space (C )of all possible configurations of A. Point robot (free-flying, no constraints) For a point robot moving in 2-D plane, C-space is

8. C y Cfree qgoal Z Cobs qstart x Configuration Space of a Robot Moving in 3D For a point robot moving in 3-D, the C-spaceis What is the difference between Euclidean space and C-space?

9. Configuration Space of a 2R Articulated Robot b b a a 2R manipulator Configuration space topology

10. Configuration Space 360 qrobot 270 b 180 b 90 a qgoal a 0 45 135 90 180 Two points in the robot’s workspace Torus (wraps horizontally and vertically)

11. Configuration Space If the robot configuration is within the blue area, it will hit the obstacle 360 qrobot 270 b 180 b 90 a qslug a 0 45 135 90 180 An obstacle in the robot’s workspace a “C-space” representation What is dimension of the C-space of puma robot (6R)? Visualization of high dimension C-space is difficult

12. Motion Planning Find a collision free path from an initial configuration to goal configuration while taking into account the constrains (geometric, physical, temporal) C-space concept provide a generalized framework to study the motion planning problem A separate problem for each robot?

13. Robot as a Point Object Expand obstacle(s) Reduce robot not quite right ...

14. Growing Obstacles C-obstacle Point robot

15. Minkowski Sums This expansion of one planar shape by another is called the Minkowski sum  Rectangular robot which can translate only P  R R P P  R = { p + r | p  P and r  R } (Dilation operation) Used in robotics to ensure that there are free paths available.

16. Additional Dimensions What would the C-obstacle be if the rectangular robot (red) can translateandrotate in the plane. (The blue rectangle is an obstacle.) y Rectangular robot which can translate and rotate x

17. C-obstacle in 3-D What would the configuration space of a 3DOF rectangular robot (red) in this world look like? (The obstacle is blue.) 3-D 180º y 0º x can we stay in 2d ?

18. One Slice ofC-obstacle Taking one slice of the C-obstacle in which the robot is rotated 45 degrees... P  R R y 45 degrees P x

19. 2-D Projection y x why not keep it this simple?

20. Projection problems qinit qgoal too conservative!

21. Motion Planning Methods The motion planning problem consists of the following: Input Output • geometric descriptions of a robot and its environment (obstacles) • initial and goal configurations • a path from start to finish (or the recognition that none exists) qrobot qgoal Problem Statement Compute a collision-free path for a rigid or articulated moving object among static obstacles What to do?

22. Motion Planning Methods (1) Roadmap approaches (2) Cell decomposition (3) Potential Fields (4) Bug algorithms Goal reduce the N-dimensional configuration space to a set of one-D paths to search. Goal account for all of the free space Goal Create local control strategies that will be more flexible than those above Limited knowledge path planning

23. Dijkstra’s algorithm Order(N^2) N = the number of vertices in C-space Roadmap: Visibility Graphs Visibility graphs: In a polygonal (or polyhedral) configuration space, construct all of the line segments that connect vertices to one another (and that do not intersect the obstacles themselves). • Formed by connecting all “visible” vertices, the start point and the end point, to each other. • For two points to be “visible” no obstacle can exist between them • Paths exist on the perimeter of obstacles From Cfree, a graph is defined Converts the problem into graph search.

24. Visibility Graph in Action • First, draw lines of sight from the start and goal to all “visible” vertices and corners of the world. goal start

25. Visibility Graph in Action • Second, draw lines of sight from every vertex of every obstacle like before. Remember lines along edges are also lines of sight. goal start

26. goal start The Visibility Graph • Repeat until you’re done. Since the map was in C-space, each line potentially represents part of a path from the start to the goal.

27. Visibility Graph Drawbacks Visibility graphs do not preserve their optimality in higher dimensions: shortest path shortest path within the visibility graph In addition, the paths they find are “semi-free,” i.e. in contact with obstacles. No clearance

28. Roadmap: Voronoi diagrams “official” Voronoi diagram (line segments make up the Voronoi diagram isolates a set of points) Generalized Voronoi Graph (GVG): locus of points equidistant from the closest two or more obstacle boundaries, including the workspace boundary. Property: maximizing the clearance between the points and obstacles.

29. Roadmap: Voronoi diagrams • GVG is formed by paths equidistant from the two closest objects • maximizing the clearance between the obstacles. • This generates a very safe roadmap which avoids obstacles as much as possible

30. Voronoi Diagram: Metrics • Many ways to measure distance; two are: • L1 metric • (x,y) : |x| + |y| = const • L2 metric • (x,y) : x2 +y2 = const

31. Voronoi Diagram (L1) Note the lack of curved edges

32. Voronoi Diagram (L2) Note the curved edges

33. Motion Planning Methods • Roadmap approaches • Visibility Graph • Voronoi Diagram • Cell decomposition • Exact Cell Decomposition (Trapezoidal) • Approximate Cell Decomposition (Quadtree) • Potential Fields • Hybrid local/global

34. Exact Cell Decomposition Trapezoidal Decomposition: Decomposition of the free space into trapezoidal & triangular cells Connectivity graph representing the adjacency relation between the cells (Sweepline algorithm)

35. Exact Cell Decomposition Trapezoidal Decomposition: Search the graph for a path (sequence of consecutive cells)

36. Exact Cell Decomposition Trapezoidal Decomposition: Transform the sequence of cells into a free path (e.g., connecting the mid-points of the intersection of two consecutive cells)

37. Optimality Trapezoidal Decomposition: 15 cells 9 cells Obtaining the minimum number of convex cells is NP-complete. Trapezoidal decomposition is exact and complete, but not optimal there may be more details in the world than the task needs to worry about...

38. Approximate Cell Decomposition Quadtree Decomposition: recursively subdivides each mixed obstacle/free (sub)region into four quarters... Quadtree:

39. Further Decomposition Quadtree Decomposition: recursively subdivides each mixed obstacle/free (sub)region into four quarters... Quadtree:

40. Even Further Decomposition Quadtree Decomposition: • The rectangle cell is recursively decomposed into smaller rectangles • At a certain level of resolution, only the cells whose interiors lie entirely in the free space are used • A search in this graph yields a collision free path Again, use a graph-search algorithm to find a path from the start to goal is this a complete path-planning algorithm? i.e., does it find a path when one exists ? Quadtree

41. Probablistic Roadmap Methods • What is a PRM (Probablistic Roadmap Method) • A probabilistic road map is a discrete representation of a continuous configuration space generated by randomly sampling the free configurations of the C-space and connecting those points into a graph • Complete path planning in high dimensional C- spaces is very complex • PRM methods boost performance by trading completeness for probabilistic completeness • Two phase approach: Learning phase, Query phase

42. s ~ s ~ g g Probabilistic Roadmap Methods • Probabilistic techniques to incrementally build a roadmap in free space of robot • Efficiency-driven • Robots with many dofs (high-dim C-spaces) • Static environments

43. Learning Phase • Learning phase: • Construction: • randomly sample free space and create a list of nodes in free space. • Connect all the nearest neighbors using a fast local planner. • Store the graph whose nodes are configurations and edges are paths computed by local planner • Expansion step: Find “Difficult” nodes and expand the graph around them using random walk techniques

44. Query Phase • Find a path from the start and goal positions to two nodes of the roadmap • Search the graph to find a sequence of edges connecting those nodes in the roadmap • Concatenating successive segments gives a feasible path for robot.

45. Motion Planning Methods • Roadmap approaches • Cell decomposition • Exact Cell Decomposition (Trapezoidal) • Approximate Cell Decomposition (Quadtree) • Potential Fields • Hybrid local/global

46. Potential Field Method Potential Field (Working Principle) – The goal location generates an attractive potential – pulling the robot towards the goal – The obstacles generate a repulsive potential – pushing the robot far away from the obstacles – The negative gradient of the total potential is treated as an artificial force applied to the robot -- Let the sum of the forces control the robot C-obstacles

47. Potential Field Method • Compute an attractive force toward the goal C-obstacles Attractive potential

48. Potential Field Method • Compute a repulsive force away from obstacles Repulsive Potential Create a potential barrier around the C-obstacle region that cannot be traversed by the robot’s configuration It is usually desirable that the repulsive potential does not affect the motion of the robot when it is sufficiently far away from C-obstacles

49. Potential Field Method • Compute a repulsive force away from obstacles • Repulsive Potential

50. Potential Field Method • Sum of Potential Attractive potential Repulsive potential C-obstacle Sum of potentials