1. BINARY MORPHOLOGY and APPLICATIONS IN ROBOTICS

2. Applications of Minkowski Sum 1.Minkowski addition plays a central role in mathematical morphology 2.It arises in the brush-and-stroke paradigm of 2D computer graphics (with various uses, notably by Donald E. Knuth in Metafont ), and as the solid sweep operation of 3D computer graphics 3.Motion planning 1.Minkowski sums are used in motion planning of an object among obstacles. 2.They are used for the computation of the configuration space which is the set of all admissible positions of the object. 3.In the simple model of translational motion of an object in the plane, where the position of an object may be uniquely specified by the position of a fixed point of this object, the configuration space are the Minkowski sum of the set of obstacles and the movable object placed at the origin and rotated 180 degrees. 4.NC machining In NC machining the programming of the NC tool exploits the fact that the Minkowski sum of the cutting piece with its trajectory gives the shape of the cut in the material.

3. Application of Morphological Method to Robot Motion Planning

4. Mathematics of Motion Planning

5. The Environment of Motion Planning and Obstacle Avoidance

6. The Concept of CONFIGURATION SPACE Notation:

7. Configuration space of object A 1.Object is a point 2.Space is 2- Dimensional

8. Configuration space of object A 1.Object is a point 3- 2.Space is 3- Dimensional

9. Possible Motions of the object in the Configuration space 1.Object is NOT a point 2.Object has shape

16. Some Other Examples of C-Space A rotating bar fixed at a point – what is its C-space? – what is its workspace A rotating bar that translates along the rotation axis – what is its C-space? – what is its workspace A two-link manipulator – what is its C-space? – what is its workspace? – Suppose there are joint limits, does this change the C-space? – The workspace?

17. Configuration Space for a simple robot arm

20. Disk (mobile robot with rounded base) in a Configuration Space

21. Example of a robot world with obstacles and a robot

22. Configuration space for the previous slide

24. Motion Planning Revisited

27. Configuration Space A key concept for motion planning is a configuration: – a complete specification of the position of every point in the system A simple example: a robot that translates but does not rotate in the plane: –– what is a sufficient representation of its configuration? The space of all configurations is the configuration space or Cspace. C-space formalism: Lozano-Perez ‘79

28. Formalization of Obstacles in Configuration Space (C-Space)

32. Obstacles Configuration Space example 1.The robot A (a triangle) can translate freely in the plane at fixed orientation. 2.Its configuration is represented as q=(x,y), the coordinates in F W of the vertex of A marked as a small circle (the origin of F A ). 3.Hence, A’s configuration space is C = R 2. 4.The C-obstacle Cbi (shown dark) is obtained by “growing” the corresponding workspace obstacle Bi (a rectangle) by the shape of A. marked vertex of A 5.Planning a motion of A relative to Bi is equivalent to planning a motion of the marked vertex of A relative to CBi

33. Notation for Free Space in Motion Planning This is expansion of one planar shape by another planar shape

34. Minkowski Sums in Robot Motion Planning

36. C-obstacles in 3D Space take into account possibility of rotations of the object (robot) This is twisted

37. C-obstacle in 3D Can we stay in 2D?

38. Assume now that the object can rotate while it moves Any reference point configuration Taking the cross section of configuration space in which the robot is rotated 45 degrees... x y 45 degrees How many sides does P ⊕ R have?

39. Taking only one slice of the C- obstacle This is only slice for 45 degrees

62. Why study the Topology –Extend results from one space to another: spheres to stars –Impact the representation –Know where you are –Others?

63. The Topology of Configuration Space – Topology is the “intrinsic character” of a space – Two space have a different topology if cutting and pasting is required to make them the same (e.g. a sheet of paper vs. a mobius strip) –– think of rubber figures --- if we can stretch and reshape “continuously” without tearing, one into the other, they have the same topology A basic mathematical mechanism for talking about topology is the homeomorphism.

73. More on dimension

79. What is the derivative of a rotation matrix? –– A tricky question --- what is the topology of that space

84. A Useful Observation The Jacobian maps configuration velocities to workspace velocities Suppose we wish to move from a point A to a point B in the workspace along a path p(t) (a mapping from some time index to a location in the workspace) – dp/dt gives us a velocity profile --- how do we get the configuration profile? – Are the paths the same if choose the shortest paths in workspace and configuration space?

85. Summary Configuration spaces, workspaces, and some basic ideas about topology Types of robots: holonomic/nonholonomic, serial, parallel Kinematics and inverse kinematics Coordinate frames and coordinate transformations Jacobians and velocity relationships T. Lozano-Pérez. Spatial planning: A configuration space approach. IEEE Transactions on Computing, C-32(2):108-120, 1983.

86. A Few Final Definitions A manifold is path-connected if there is a path between any two points. A space is compact if it is closed and bounded – configuration space might be either depending on how we model things – compact and non-compact spaces cannot be diffeomorphic! With this, we see that for manifolds, we can – live with “global” parameterizations that introduce odd singularities (e.g. angle/elevation on a sphere) – use atlases – embed in a higher-dimensional space using constraints Some prefer the later as it often avoids the complexities associated with singularities and/or multiple overlapping maps

87. Use in path planning with large object in 2D Projection

88. Problems with this approach to Projection The approach for the object shape from previous slide is too conservative

89. Howie Choset G.D. Hager, Z. Dodds, Dinesh MochaSources