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Stability of Enveloping Grasps

Stability of Enveloping Grasps. Vijay Kumar GRASP Laboratory University of Pennsylvania Philadelphia, PA 19104-6315 (Joint work with Hermann Bruyninckx and Stamps Howard). Basic Definitions. EQUILIBRIUM A grasped object is in equilibrium with an external wrench g iff (1) W c = g

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Stability of Enveloping Grasps

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  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. Basic Definitions • EQUILIBRIUM A grasped object is in equilibrium with an external wrench giff (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. 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. 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. 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. Approach • 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

  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. 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. Kinematics of Contact

  10. Kinematics of Compliant Contact

  11. Contact Model: Compliance

  12. Planar Frictional Contacts

  13. Contact Model (continued)

  14. Three Dimensional Frictional Contacts Fc = -cXc • LA curvature of the object Motorsional moment • LB curvature of the finger knnormal stiffness • Fnonormal force kttangential stiffness • Ftotangential force ktorsional stiffness

  15. Finger Compliance

  16. Second Order Arm Kinematics

  17. External forces Fcg = -cgXcg ORIGINAL POSITION x DISPLACED POSITION y g m

  18. Coordinate Transformations • The stiffness matrix can be transformed • to any other frame, O, by: Fo = TTcTXo • where x 1,1 O c 2,2 O c y y x O cg x y y 2,1 O c x y x 1 O c O 18

  19. Grasp Stiffness Matrix CONTACT STIFFNESS EXT. FORCES (HESSIAN) FINGER STIFFNESS • iCF The joint stiffness matrix • iKstruct Link stiffness matrix • c The contact stiffness matrix • nf Number of fingers • np Number of contacts with fixed surfaces

  20. Example 1 • A Whole Arm Grasp

  21. Eigenvalues: • 10094, 12636, 12620, • 73.5, 43, 27

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

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

  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. Measures of Stability • 3. Smallest eigenvalue of the stiffness matrix? • Eigenvalues of Cartesian stiffness matrix make little sense • K t = l 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. se(3) SE(3) 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

  27. Generalized eigenvalue problem for the stiffness matrix on se(3) K t = l M t Dual problem with the compliance matrix on se(3)* C w = s M-1 w Eigenvalues l is the eigenvalue of M-1 K s is the eigenvalue of M C M, K se(3) se(3)* M-1, C TWISTS WRENCHES Measure of stability

  28. BODY-FIXED FRAME INERTIAL FRAME 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

  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 TM t and l 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. 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. Homogeneous cylinder l={709, 23, 254, 1425, 339, 161} Homogeneous cylinder l={2427, 15, 701, 8323, 250, 150} Example 1 revisited z y

  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

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