Grasping 3D Grasp Quality Computations

1. Grasping3D Grasp Quality Computations 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 c1 c1 c3 c3 c2

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-Line Point-Plane Point-Point

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 = fn+ ftwhere |ft| ≤ |μs*fn| 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 fy fx τz

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 Is this grasp stable? c1 d1 COM f1,1 f1,2 f2,1 f2,2 d2 c2

12. 2D Example • 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 c1 d1 COM f1,1 f1,2 f2,1 f2,2 d2 c2

13. 2D Example • Ignore fx for now c1 τout w1,1 w2,2 d1 Wrench hull w1,2 w2,1 COM f1,1 f1,2 -fy f2,1 f2,2 +fy d2 Does not contain origin, not stable c2 τin

14. 2D Example • What if there was a 3rd point? c1 τout w1,1 w2,2 d1 Wrench hull w1,2 w2,1 f1,1 f1,2 COM -fy f3,1 f3,2 f2,1 f2,2 +fy w3,2 d2 d3 Does contain origin, stable w3,1 c3 c2 τin

15. Grasp Quality • Both of these grasps are stable • But how stable are they? c1 c1 d1 d1 f1,1 f1,2 f1,1 f1,2 COM COM f3,1 f3,2 f3,1 f3,2 f2,1 f2,2 d2 d3 d3 c3 c3 c2

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 • 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 τout w1,1 w2,2 Hard to resist w1,2 w2,1 -fy +fy ε w3,2 w3,1 τin

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 • So are these equal? τout τout w1,1 w1,1 w2,2 w1,2 w1,2 w2,1 -fy +fy -fy +fy ε w3,2 ε w3,2 w3,1 w3,1 τin τin

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