1. Configuration Space

2. Recap Represent environments as graphs –Paths are connected vertices –Make assumption that robot is a point Need to be able to use realistic representations of robot geometry and find safe paths

3. Rough idea Convert rigid robots, articulated robots, etc. into points –Expand obstacles to compensate Can now create graphs in this new space –If no collision along graph, no collision in real world

4. Configurations The configuration of a moving object is a unique specification of the position of every point on the object. –Usually a configuration is expressed as a vector of position & orientation parameters: q = (q 1, q 2,…,q n ). q=(q 1, q 2,…,q n ) q1q1q1q1 q2q2q2q2 q3q3q3q3 qnqnqnqn

5. Configurations The set of points defining the robot is then some function of q, such as R(q). q=(q 1, q 2,…,q n ) q1q1q1q1 q2q2q2q2 q3q3q3q3 qnqnqnqn

6. Configuration Example 2D translating circular robot Shape can be defined as q = (x,y) R(x,y) = {(x’,y’) | (x-x’) 2 + (y-y’) 2 ≤ r 2 } In this case, can think of q as a translation applied to the original robot

7. Another Example Two joint planar arm Does position of the end effector describe the configuration? (x,y)

8. Another Example Two joint planar arm Does position of the end effector describe the configuration? No! It does not uniquely specify all of the arm. –But joint angle does q = ( ,  )

9. Dimension of q Same as the degrees-of-freedom (DOFs) of the robot Can find DOFs by adding up free variables and subtracting constraints

10. DOF Example Planar translating, rotating triangle robot Triangle = {p1,p2,p3} = {(x1,y1,z1), (x2,y2,z2), (x3,y3,z3) } 9 free variables p1 p2 p3

11. DOF Example Planar: z1 = z2 = z3 = 0 9 – 3 = 6 free variables p1 p2 p3

12. DOF Example Rigid body constraints Fix p2 distance: –D(p1,p2) = r12 Fix p3 distance: –D(p1,p3) = r13 –D(p2,p3) = r23 6 – 3 = 3 free variables p1 p2 p3

13. DOF Example What’s left? Fix one point p1 = (x,y) Define rotation  of triangle relative to axis The three things we can choose were –(x,y,  ) Makes a 3 DOF robot p1 p2 p3 

14. Configuration space The configuration space C is the set of all possible configurations. –A configuration is a point in C. –Similar to a State space Parameter space The workspace is all points reachable by the robot (or sometimes just the end effector) C can be very high dimensional while the workspace is just 2D or 3D q=(q 1, q 2,…,q n ) q1q1q1q1 q2q2q2q2 q3q3q3q3 qnqnqnqn

15. C = S 1 x S 1 Topology of the configuration Space The topology of C is usually not that of a Cartesian space R n. Space is S 1 x S 1 = T 2 0 22 22      

16. Obstacles in the c-space A configuration q is collision-free, or free, if a moving object placed at q does not intersect any obstacles in the workspace. The free space F is the set of free configurations (also C free ). A configuration space obstacle (C- obstacle) is the set of configurations where the moving object collides with workspace obstacles.

17. C-Obstacles 2D translating robot Workspace robot and obstacle C-space robot and obstacle

18. Polygonal robot translating & rotating in 2-D workspace workspace configuration space

19. Polygonal robot translating & rotating in 2-D workspace x

20. Articulated robot in 2-D workspace workspace configuration space

21. Paths in the configuration space A path in C is a continuous curve connecting two configurations q and q ’ : such that  q and  q’. workspace configuration space

22. Free Paths in the configuration space A free path in C is a continuous curve connecting two configurations q and q ’ : such that  q and  q’. A semi-free path allows the robot and obstacles to contact (but not interpenetrate).

23. Two paths  and  ’  with the same endpoints are homotopic if one can be continuously deformed into the other: A homotopic class of paths contains all paths that are homotopic to one another. Homotopic paths

24. Connectedness of C-Space C is connected if every two free configurations can be connected by a path. Otherwise C is multiply-connected. Same as terminology for graphs

25. More C-space Obstacles Exercise