An Object-Based Approach to Map Human Hand Synergies onto Robotic Hands with Dissimilar Kinematics

This chapter introduces the main equations necessary to study hands controlled by synergies. Most of the results presented here are related to grasp and manipulation analysis of underactuated structures that present compliance at joint and contact level. The introduction of the synergies concept in robotics can be seen, in fact, as a possible reduction of the hand DoF space. Compliance is needed to solve static indeterminacy. 2.1 Quasi-static manipulation model Consider a generic hand as a collection of arbitrary numbers of robot “fingers” (i.e., simple chains of links connected through revolute or prismatic joints) attached to a common base “palm”, and an object, which is in contact with all or some of the links as sketched in Fig. 2.1. The hand and the object have nc contact points. The position of the contact point i in {N} is defined by the vector pi ∈R3. At contact point i, we define a frame {C}i, with axes {n̂i, t̂i, ôi}. The unit vector n̂i contains pi is normal to the contact tangent plane and is directed toward the object. The other two unit vectors are orthogonal and lie in the tangent plane of the contact. 12 2. Mathematics of hand synergies Table 2.1: Notation for Grasp Analysis Notation Definition u ∈R6 position and orientation of the object w ∈R6 external wrench applied to the grasped object nd system dimension nc number of contact points Co i reference system at the i-th contact point on the object ci ∈R3 position of the contact point i ĉi ∈Rnd position and orientation of reference frame Co i Ch i reference system at the i-th contact point on the hand ĉi ∈Rnd position and orientation of reference frame Ch i λi vector of forces (and moments) at the contact i λ ∈Rnl vector of contact forces (and moments) nl dimension of the contact force vector nq number of joints q ∈Rnq actual joint variables qre f ∈Rnq reference joint variables τ vector of joint forces and torques nz number of postural synergies z ∈Rnz synergy variables σ ∈Rnz generalized forces along synergies G ∈Rnd×nl grasp matrix J ∈Rnl×nq hand jacobian matrix S ∈Rnq×nz synergy matrix Let the joints be numbered from 1 to nq. Denote by q = [q1...qnq ] T ∈ Rnq the vector of joint displacement. Also, let τ = [τ1...τnq ] T ∈ Rnq represent joint loads (forces in prismatic joints and torques in revolute joints). Let u ∈ Rnu denote the vector describing the position and orientation of {B} relative to {N}. For planar systems nu = 3. For spatial systems, nu is three plus the number of parameter used to represent orientation, typically three for Euler angle and four for quaternion. Denote by ν = [vT ωT ] ∈ Rnν the twist of object described in {N}. It is composed of the translational velocity ν ∈ R3 of the point o and the angular velocity ω ∈R3 of the object, both expressed in {N}. A twist of a rigid body can be referred to any convenient frame fixed to the body. The components of the referred twist represent the velocity of the origin of the new frame and the angular velocity of the body, both expressed in the new frame. 2.1. Quasi-static manipulation model 13