1. B659: Principles of Intelligent Robot Motion Spring 2013 David Tidd

2. Grasp Quality Given two different grasps, how can they be compared? – Are they stable? -> Force closure – How stable are they? -> Grasp quality metrics c1c1 c2c2 c3c3 c1c1 c3c3

3. Agenda Point force generalization Wrench space Grasp quality metrics Simulation method

4. Contact Types Type of contact determined by colliding geometries – Point: point on plane (stable), point on point or line (unstable) – Line: line on plane or nonparallel line (stable), line on parallel line (unstable) – Plane: plane on plane Unstable contacts ignored in analysis Point-Plane Point-Point Point-Line

5. Everything as a Point Contact Line contact -> 2 points Plane contact -> convex hull of points Any distribution of normal forces across a region can be represented as a weighted sum of point forces along that region’s convex hull

6. Point Contacts with Coulomb Friction A point contact with friction is able to apply more than just a normal force “Friction cone” is the vector space of all possible forces a point can apply due to friction f = f n + f t where |f t | ≤ |μ s *f n | n n

7. Approximating Friction Cones Pyramidal approximation converts vector space to finite set of vectors – 8-sided approximation used in simulation

8. Wrenches Each point force also applies torque – τ = d x f Wrench is a force-torque pair – The i-th point contact has m wrenches, one for each force in the pyramidal approximation – d is the vector from the point contact to the torque origin – λ is a constant relating force to torque for analysis λ = 1/r was chosen to make torque size invariant

9. Wrench Space For 3D objects, wrench space is 6D – 3D for force, 3D for torque – For 2D objects, it’s 3D fyfy fxfx τzτz

10. Wrench Hulls Set of wrenches from ONE point contact = boundary of what wrenches can be applied from that one point Set of wrenches from ALL point contacts = convex hull in wrench space, total possible range of wrenches that can be applied

11. 2D Example c1c1 c2c2 f 1,1 f 1,2 f 2,1 f 2,2 d1d1 d2d2 COM Is this grasp stable?

12. 2D Example c1c1 c2c2 f 1,1 f 1,2 f 2,1 f 2,2 d1d1 d2d2 COM 2 point contacts 4 wrenches Force closure? – Yes What about torque? Direction of d x f – All torque is in same direction, out of page

13. 2D Example c1c1 c2c2 f 1,1 f 1,2 f 2,1 f 2,2 d1d1 d2d2 COM Ignore f x for now -f y +f y τ out τ in w 1,1 w 1,2 w 2,2 w 2,1 Wrench hull Does not contain origin, not stable

14. 2D Example c1c1 c2c2 f 1,1 f 1,2 f 2,1 f 2,2 d1d1 d2d2 COM What if there was a 3 rd point? -f y +f y τ out τ in w 1,1 w 1,2 w 2,2 w 2,1 Wrench hull Does contain origin, stable c3c3 f 3,1 f 3,2 d3d3 w 3,2 w 3,1

15. Grasp Quality Both of these grasps are stable – But how stable are they? c1c1 f 1,1 f 1,2 d1d1 COM c3c3 f 3,1 f 3,2 d3d3 c1c1 c2c2 f 1,1 f 1,2 f 2,1 f 2,2 d1d1 d2d2 COM c3c3 f 3,1 f 3,2 d3d3

16. Grasp Quality Metrics Quality is how well a grip can resist disturbances Worst case scenario – How efficiently can a grip resist disturbance wrenches at its weakest point? Weakest means the direction (in wrench space) at which the sum normal force is converted to the desired wrench least efficiently – Grip a pencil at the end and try to resist torque – Now try it while gripping the center – The center requires much more normal force to get the same wrench

17. Worst Case Scenario -f y τ out τ in w 1,1 w 1,2 w 2,2 w 2,1 Hard to resist w 3,2 w 3,1 The point on the wrench hull that is closest to the origin is the weakest point Disturbances in the opposite direction are hardest to resist Metric ε = The radius of the largest ball that can be enclosed in the wrench hull – Varies from 0 to 1 due to normalization of wrenches +f y ε

18. Physical Meaning of ε In the worst case, the sum magnitude of the contact wrenches would need to be 1/ε times the disturbance wrench

19. Grasp Quality Metrics -f y τ out τ in w 1,1 w 1,2 w 3,2 w 3,1 So are these equal? +f y ε -f y τ out τ in w 1,1 w 1,2 w 2,2 w 2,1 w 3,2 w 3,1 +f y ε

20. Average Case Scenario How efficiently can a grip resist a disturbance wrench on average? Metric ν = Volume of the convex hull in wrench space The three point contact has more volume, so it is more stable on average

21. Grasp Simulation Method Set hand configuration except for distal links Iterate configuration of distal links and check for collisions with object Continue until all links have collided Only one solution found. There could be better solutions. How to determine initial configuration?

22. Grasp Analysis Method Decompose the collisions into point contacts Covert point contacts into sets of wrenches Construct wrench hull Compute quality metrics

23. Grasp Search Each hand configuration maps to one grasp via simulation The total possible grasp space is equivalent to the initial configuration space of the hand Explore a subset of C-space using finite steps Other methods?

24. Discussion