1. University of Pennsylvania Vijay Kumar 1 Stability of Enveloping Grasps Vijay Kumar GRASP Laboratory University of Pennsylvania Philadelphia, PA 19104-6315 (Joint work with Hermann Bruyninckx and Stamps Howard)

2. University of Pennsylvania Vijay Kumar 2 Basic Definitions EQUILIBRIUM A grasped object is in equilibrium with an external wrench g iff (1) W c = g (2) FORCE CLOSURE A grasp is defined as force closed iff, for any arbitrary wrench there exists an intensity vector  satisfying (1), such that STABILITY A grasped object in equilibrium, in which all forces and moments can be derived from a potential function V(q), is defined to be stable if  V > 0 for every non zero virtual displacement,  q.

3. University of Pennsylvania Vijay Kumar 3 Key Questions Measure of stability What are suitable measures of stability? Does force closure imply stability? Do internal forces improve the stability of a grasp? Is there a trade off between Minimal actuator/contact forces Stability

4. University of Pennsylvania Vijay Kumar 4 Goal of this presentation Modeling, analysis, and stability of grasps (Howard, 1995) Enveloping grasps Compliance material properties control algorithms Measures of grasp stability (Bruyninckx, Demey and Kumar, 1997)

5. University of Pennsylvania Vijay Kumar 5 Modeling Geometry Curvature of contacting element (Cutkosky 85, Nguyen 88, Montana 91) Size (Cutkosky 85, Montana 91) Contact forces Compliance Contact stiffness Link flexibility Joint compliance Control algorithm

6. University of Pennsylvania Vijay Kumar 6 Assumptions Quasi-static framework Small displacements Method Model the effective stiffness of the grasp by a 6  6 Cartesian stiffness matrix  G.  F =  G  x Definition A grasp is stable if the Cartesian grasp stiffness matrix is positive definite. Stability all eigenvalues of the stiffness matrix are positive Approach

7. University of Pennsylvania Vijay Kumar 7 Key Questions Derivation of the grasp stiffness matrix? What is a suitable measure of stability? Does the stiffness matrix give us a measure of stability? < <

8. University of Pennsylvania Vijay Kumar 8 Enveloping Grasps Modeling Contact kinematics second order kinematics Arm/finger kinematics second order (derivative of the Jacobian matrix) Contact compliance continuum models Joint compliance actuator models control scheme External forces

9. University of Pennsylvania Vijay Kumar 9 Kinematics of Contact

10. University of Pennsylvania Vijay Kumar 10 Kinematics of Compliant Contact

11. University of Pennsylvania Vijay Kumar 11 Contact Model: Compliance

12. University of Pennsylvania Vijay Kumar 12 Planar Frictional Contacts

13. University of Pennsylvania Vijay Kumar 13 Contact Model (continued)

14. University of Pennsylvania Vijay Kumar 14  F c = -  c  X c L A curvature of the object M o torsional moment L B curvature of the finger k n normal stiffness F no normal force k t tangential stiffness F to tangential force k torsional stiffness Three Dimensional Frictional Contacts

15. University of Pennsylvania Vijay Kumar 15 Finger Compliance

16. University of Pennsylvania Vijay Kumar 16 Second Order Arm Kinematics

17. University of Pennsylvania Vijay Kumar 17  F cg = -  cg  X cg ORIGINAL POSITION DISPLACED POSITION External forces x y m g

18. University of Pennsylvania Vijay Kumar 18 1,1 O c 2,1 O c 1 O c 2,2 O c y x y y y x x x O cg x y O Coordinate Transformations The stiffness matrix can be transformed to any other frame, O, by:  F o = T T   c T  X o where

19. University of Pennsylvania Vijay Kumar 19 i C F The joint stiffness matrix i K struct Link stiffness matrix  c The contact stiffness matrix nf Number of fingers np Number of contacts with fixed surfaces FINGER STIFFNESS CONTACT STIFFNESS EXT. FORCES (HESSIAN) Grasp Stiffness Matrix

20. University of Pennsylvania Vijay Kumar 20 Example 1 A Whole Arm Grasp

21. University of Pennsylvania Vijay Kumar 21 Eigenvalues: 10094, 12636, 12620, 73.5, 43, 27

22. University of Pennsylvania Vijay Kumar 22 Eigenvalues: {6.4, 151, and 7876} Example 2 Not force closed by first or second order criteria Joint compliance makes the grasp stable

23. University of Pennsylvania Vijay Kumar 23  G = Eigenvalues: {-16.8, 4009, 190} Example 3 The grasp is “force closed” but unstable

24. University of Pennsylvania Vijay Kumar 24 Measures of Stability 1. Stability under a given disturbance Grasp 1 is more stable than Grasp 2 if the restoring wrench for a given disturbance twist is larger for Grasp 1 than for Grasp 2 if the resulting twist for a given disturbance wrench is smaller for Grasp 2 than for Grasp 1 2. Stability Grasp 1 is more stable than Grasp 2 if the minimum restoring wrench over all unit disturbance twists is larger for Grasp 1 than for Grasp 2 if the largest resulting twist over all unit disturbance wrenches is smaller for Grasp 2 than for Grasp 1 Difficulty need the definition of a norm on the space of all twists (wrenches)

25. University of Pennsylvania Vijay Kumar 25 Measures of Stability 3. Smallest eigenvalue of the stiffness matrix? Eigenvalues of Cartesian stiffness matrix make little sense K t = t (left hand side is a wrench, right hand side is a twist) Eigenvalues are not invariant with respect to rigid body transformations Congruence transformation does not preserve the eigenvalues Signs of eigenvalues are preserved - stability does not depend on the coordinate system

26. University of Pennsylvania Vijay Kumar 26 Measure of stability Basic idea Grasp 1 is more stable than Grasp 2 if the minimum restoring wrench over all unit disturbance twists is larger for Grasp 1 than for Grasp 2 Approach Define a metric on SE(3) Invent a suitable metric on se(3) Extend it to SE(3) by translation M - metric on the space of all twists M -1 - metric on the space of all wrenches Generalized eigenvalue problem SE(3) se(3)

27. University of Pennsylvania Vijay Kumar 27 Measure of stability se(3)se(3)* M, K M -1, C TWISTSWRENCHES Generalized eigenvalue problem for the stiffness matrix on se(3) K t =  M t Dual problem with the compliance matrix on se(3)* C w =  M -1 w Eigenvalues  is the eigenvalue of M -1 K  is the eigenvalue of M C

28. University of Pennsylvania Vijay Kumar 28 Metrics Problem No natural metric on SE(3) Choose metric Physical considerations Invariance Two types of metrics Left invariant metric (metric that is independent of inertial frame) that transforms as a tensor Energy metric Bi-invariant metric INERTIAL FRAME BODY-FIXED FRAME

29. University of Pennsylvania Vijay Kumar 29 Properties For any metric, M The eigenvalues of M -1 K are invariant with respect to changes of reference frames For any symmetric, non degenerate M All eigenvalues are real The signature of M -1 K is equal to the signature of M t T M t and have the same sign Eigenvectors corresponding to different eigenvalues are M-orthogonal M -1 K is positive definite if and only if M and K are positive definite

30. University of Pennsylvania Vijay Kumar 30 Bi-invariant, non degenerate metric M-orthogonality is the property of reciprocity Eigenvalues are dimensionless Signature (+, +, +, -, -, -) Disadvantage M is only a “pseudo-metric” Benefits Define eigenscrews for the three positive eigenvalues: Eigentwists and eigenwrenches (Patterson and Lipkin, 1993) Can define stability measures for a subset of disturbance twists or disturbance wrenches

31. University of Pennsylvania Vijay Kumar 31 Example 1 revisited Homogeneous cylinder ={709, 23, 254, 1425, 339, 161} Homogeneous cylinder ={2427, 15, 701, 8323, 250, 150} y z

32. University of Pennsylvania Vijay Kumar 32 Concluding Remarks The stability analysis of enveloping grasps A complete model of the grasp stiffness Stability analysis Force closure does not imply stability Internal forces can make a stable grasp unstable Measure of grasp stability based on a metric on SE(3) Energy metric Bi-invariant metric Kinematic metric A scale-dependent left invariant metric