Finding All Valid Hand Configurations for a Given Precision Grasp - PowerPoint PPT Presentation

Introduction Formulation Numerical solution Tests Conclusions Finding All Valid Hand Configurations for a Given Precision Grasp Carlos Rosales 1 , 2 , Josep M. Porta 2 , ul Suarez 1 and Llu´ ıs Ros 2 Ra´ 1 Institut d’Organitzaci´ o i Control de Sistemes Industrials (UPC) 2 Institut de Rob` otica i Inform` atica Industrial (CSIC-UPC) C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions Problem statement C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions Problem statement C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions Problem statement C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions Problem statement C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions Grasping and manipulation tasks Usually tackled in two steps: 1 Find the grasping points: Largely solved, e.g. force/form closure, etc. 2 Solving inverse kinematics: Previous work [Borst et al. , 2002] Unconstrained optimization, penalty terms [Gorce et al. , 2005] Neural networks, reinforcement learning [Rosell et al. , 2005] Fingertip-contact distance minimization C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions Grasping and manipulation tasks Usually tackled in two steps: 1 Find the grasping points: Largely solved, e.g. force/form closure, etc. 2 Solving inverse kinematics: Previous work [Borst et al. , 2002] Unconstrained optimization, penalty terms [Gorce et al. , 2005] Neural networks, reinforcement learning [Rosell et al. , 2005] Fingertip-contact distance minimization C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions Shortcomings of previous works Need an initial estimation May diverge Converge to only one solution Incomplete C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions Contribution over previous works The proposed approach is an inverse kinematic technique that: Does not require an initial estimation Is complete (converges to all solutions) Is applicable to other hand structures C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions Approach Formulation : formulate kinematic loop closure constraints algebraically Numerical solution : solve the resulting equations via a branch-and-prune technique based on linear relaxations C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions Formulation The formulation is tailored to the numerical solution adopted: Algebraic equations directly Involving monomials of linear, bilinear and quadratic type C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions System of equations 4 4 X X x j − q j,i = x k − q k,i (1) i =1 i =1 � o 1 � = 1 , � o 2 � = 1 and o 1 · o 2 = 0 (2) � r j,i � = 1 , � p j,i � = 1 and r j,i · p j,i = 0 (3) r j, 2 = r j, 3 = r j, 4 (4) r j, 1 · r j, 2 = 0 (5) x j = ( o 1 , o 2 , o 3 ) · ˆ (6) x j q j, 4 = ( r j, 4 , p j, 4 , t j, 4 ) · ˆ (7) q j, 4 ( r j, 4 , p j, 4 , t j, 4 ) ˆ m j = ( o 1 , o 2 , o 3 ) ˆ (8) n j C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions Loop closure constraints 4 4 X X o 1 x j − q j,i = x k − q k,i (1) object o 2 i =1 i =1 x j � o 1 � = 1 , � o 2 � = 1 and o 1 · o 2 = 0 (2) x k � r j,i � = 1 , � p j,i � = 1 and r j,i · p j,i = 0 (3) finger j r j, 2 = r j, 3 = r j, 4 (4) r j, 1 · r j, 2 = 0 (5) finger k x j = ( o 1 , o 2 , o 3 ) · ˆ (6) x j q j, 4 = ( r j, 4 , p j, 4 , t j, 4 ) · ˆ (7) q j, 4 ( r j, 4 , p j, 4 , t j, 4 ) ˆ m j = ( o 1 , o 2 , o 3 ) ˆ (8) n j h 1 h 2 palm C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions 4 4 X X x j − q j,i = x k − q k,i (1) i =1 i =1 � o 1 � = 1 , � o 2 � = 1 and o 1 · o 2 = 0 (2) � r j,i � = 1 , � p j,i � = 1 and r j,i · p j,i = 0 (3) r j, 2 = r j, 3 = r j, 4 (4) r j, 1 · r j, 2 = 0 (5) x j = ( o 1 , o 2 , o 3 ) · ˆ (6) x j q j, 4 = ( r j, 4 , p j, 4 , t j, 4 ) · ˆ (7) q j, 4 ( r j, 4 , p j, 4 , t j, 4 ) ˆ m j = ( o 1 , o 2 , o 3 ) ˆ (8) n j C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions Reference frame constraints o 1 o 2 object 4 4 X X x j − q j,i = x k − q k,i (1) x j i =1 i =1 � o 1 � = 1 , � o 2 � = 1 and o 1 · o 2 = 0 (2) fingertip � r j,i � = 1 , � p j,i � = 1 and r j,i · p j,i = 0 (3) q j, 4 r j, 2 = r j, 3 = r j, 4 (4) r j, 4 p j, 4 r j, 1 · r j, 2 = 0 (5) x j = ( o 1 , o 2 , o 3 ) · ˆ (6) x j r j, 3 r j, 1 q j, 4 = ( r j, 4 , p j, 4 , t j, 4 ) · ˆ (7) q j, 4 p j, 3 r j, 2 ( r j, 4 , p j, 4 , t j, 4 ) ˆ m j = ( o 1 , o 2 , o 3 ) ˆ (8) n j p j, 2 h 1 q j, 1 palm h 2 C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions Joint position constraints o 1 o 2 object 4 4 X X x j − q j,i = x k − q k,i (1) x j i =1 i =1 � o 1 � = 1 , � o 2 � = 1 and o 1 · o 2 = 0 (2) fingertip � r j,i � = 1 , � p j,i � = 1 and r j,i · p j,i = 0 (3) q j, 4 r j, 2 = r j, 3 = r j, 4 (4) r j, 4 p j, 4 r j, 1 · r j, 2 = 0 (5) x j = ( o 1 , o 2 , o 3 ) · ˆ (6) x j r j, 3 r j, 1 q j, 4 = ( r j, 4 , p j, 4 , t j, 4 ) · ˆ (7) q j, 4 p j, 3 r j, 2 ( r j, 4 , p j, 4 , t j, 4 ) ˆ m j = ( o 1 , o 2 , o 3 ) ˆ (8) n j p j, 2 h 1 q j, 1 palm h 2 C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions 4 4 X X x j − q j,i = x k − q k,i (1) i =1 i =1 � o 1 � = 1 , � o 2 � = 1 and o 1 · o 2 = 0 (2) � r j,i � = 1 , � p j,i � = 1 and r j,i · p j,i = 0 (3) r j, 2 = r j, 3 = r j, 4 (4) r j, 1 · r j, 2 = 0 (5) x j = ( o 1 , o 2 , o 3 ) · ˆ (6) x j q j, 4 = ( r j, 4 , p j, 4 , t j, 4 ) · ˆ (7) q j, 4 ( r j, 4 , p j, 4 , t j, 4 ) ˆ m j = ( o 1 , o 2 , o 3 ) ˆ (8) n j C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions Contact constraints 4 4 X X x j − q j,i = x k − q k,i (1) i =1 i =1 � o 1 � = 1 , � o 2 � = 1 and o 1 · o 2 = 0 (2) � r j,i � = 1 , � p j,i � = 1 and r j,i · p j,i = 0 (3) object o 1 o 2 r j, 2 = r j, 3 = r j, 4 (4) ˆ n j r j, 1 · r j, 2 = 0 (5) x j q j, 4 x j = ( o 1 , o 2 , o 3 ) · ˆ (6) x j m j ˆ q j, 4 = ( r j, 4 , p j, 4 , t j, 4 ) · ˆ (7) q j, 4 r j, 4 p j, 4 ( r j, 4 , p j, 4 , t j, 4 ) ˆ m j = ( o 1 , o 2 , o 3 ) ˆ (8) n j fingertip C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions Introducing joint limits constraints Joint angles are constrained by limiting their sine and cosine To limit φ to [ − α, α ] we define c = cos( φ ) , s = sin( φ ) , then, introduce two new constraints c = u · v , s · w = u × v , with u , v , w appropriate finger vectors, and finally set c ∈ [ c min , c max ] , s ∈ [ s min , s max ] . C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp

Introduction Formulation Numerical solution Tests Conclusions Numerical solution 1 System of polynomials to be solved 2 Note all monomials are of the form x i , x 2 i or x i x j 3 Change of variables q i = x 2 i and b k = x i x j 4 New system: L ( x ) = 0 (9) Q ( x ) = 0 (10) B ( x ) = 0 (11) 5 Search space: Rectangular box defined by the ranges of the variables C. Rosales, J. M. Porta, R. Su´ arez and L. Ros Finding All Valid Hand Configurations for a Given Precision Grasp