A mathematical introduction to robotics manipulation

ly, the derivation of the grasp kinematics is identical to thefixed contact case except that the grasp map and hand Jacobian dependon the instantaneous contact location, which in turn depends on ηi, thecontact coordinates for each finger. Thus, the contact kinematics havethe formJh(θ, xo, η)θ̇ = GT (η)V bpoη̇i = Ai(ηi)V bofi ,(5.40) where η = (η1, . . . , ηk) ∈ R is the vector of contact coordinates andAi(ηi) ∈ R5×6 encodes the contact kinematics for each finger, as givenin equation (5.28). One additional fact is needed to extend the previousanalysis to the moving contact case: BciV boci = 0. This condition is astatement that the contact point is not allowed to move in any of theconstrained directions. A complete derivation is left as an exercise.Equation (5.40) describes the grasp kinematics in terms of a set ofordinary differential equations. To analyze the properties of the grasp,we make use of the following result. Let W ⊂ SE(3) be the set of allconfigurations gof for which the two objects are in contact.Proposition 5.12. Smooth dependence of η on gofThere is a smooth local bijection between η ∈ R and gof ∈ W ⊂ SE(3)if the matrixKf +RψKoR Tψ