Kinematics of a grasping articulated mechanical hand

The grasping of an object by a robotic hand is essentially a contact problem that may be represented by several closed loops, the object constituting the passive chain and the fingers the active one. Great developments of articulated mechanical hands have been achieved. The Stanford-JPL hand is a mechanism constituted of three modular fingers that has not been designed for hominoid ends, but rather and mainly for handling purposes. Salisbury [1] reported later that to be more effective, the hand needs a fourth finger. The development of a four-finger hand has been carried out by the Massachusetts Institute of Technology [2] and the University of Utah [3]. It possesses a thumb opposing three long fingers, and has been the subject of interest in the field of teleoperation [4]. The present investigation focuses on both geometric models and kinematics of a four fingered robotized hand. Closed loop constraints are integrated whenever there is an object to be grasped. The contact points are determined using the inverse geometric model of the active loop. Singularities as well as any closed loop kinematics movement are analyzed by considering two fingers in any permutation. The achieved simulation results are presented using object models build up by means of the superquadric approach developed by El-Khoury [5].


Introduction
The grasping of an object by a robotic hand is essentially a contact problem that may be represented by several closed loops, the object constituting the passive chain and the fingers the active one.
Great developments of articulated mechanical hands have been achieved.The Stanford-JPL hand is a mechanism constituted of three modular fingers that has not been designed for hominoid ends, but rather and mainly for handling purposes.Salisbury [1] reported later that to be more effective, the hand needs a fourth finger.The development of a four-finger hand has been carried out by the Massachusetts Institute of Technology [2] and the University of Utah [3].It possesses a thumb opposing three long fingers, and has been the subject of interest in the field of teleoperation [4].
The present investigation focuses on both geometric models and kinematics of a four fingered robotized hand.Closed loop constraints are integrated whenever there is an object to be grasped.
The contact points are determined using the inverse geometric model of the active loop.Singularities as well as any closed loop kinematics movement are analyzed by considering two fingers in any permutation.
The achieved simulation results are presented using object models build up by means of the superquadric approach developed by El-Khoury [5].

The articulated hand
The robotized hand is an articulated mechanical system constituted of a complex kinematics chains.The considered structure consists of four fingers (three long fingers opposing a thumb) having each four degrees of freedom (Fig. 1).

The direct geometrical model
To represent the geometric model of the fingers represented in Fig. 1, the parameters developed by Khalil and Kleinfinger [6], namely j , j , j d , j , j r allowing a uniform description and presented in Table 1 and Table 2 are adopted.
Fig. 1 Four fingered mechanical hand   The developed geometrical model in this paper is essentially based on an analysis performed earlier by Bouachari and Barkat [7].Its application needs to start determining the antecedent of j namely i a j before computing the transformation matrices i j T followed by their product 0 j T (i = 0 representing the center of the palm) to finally computing the position vectors corresponding to each of transformation matrix.
Fig. 3 shows three simulated configurations of our hand.They are represented by pinch grasp, normal grasp, and a grasp using the four fingertips.They have been achieved through the application of the direct geometrical model.The kinematical model of a finger is represented by the fingertip velocity as a function of its joint velocities jk q jk q .The basic Jacobian matrix is used to express the kin- ematics torsor n in the coordinate system 0 R : (1) The corresponding Jacobian matrix is noted 0 n J and its column k is expressed as: where the elements of each column are obtained when calculating the direct geometrical model of the matrices , , , J J J J ) is therefore derived.For the first fin- ger, it is expressed as: (3) The joint variable for each fingertip being equal to zero, the above expressed Jacobian matrices become J .This leads to expressing the basic Jacobian matrix (1) for the total number of fingers.
By expressing the Jacobian matrix of Eq. ( 4) as the product of two sub-matrices 0 0 4 4 JT JR , one can com- pute the fingertip translational velocity vector 0 5 V in the coordinate system 0 R linked to the hand palm through the sub-matrix 0 4 JT of dimensions (3×4): The sub-matrix 0 4 JR leads to the computation of the fingertip rotational velocity vector 0 5 in the coordinate system 0 R linked to the hand palm and is also of di- mensions (3×4): ,z q q JR .q q 1 q 1 q 1 q 2 q 2 q .

Singular configurations specification
To (4 3), it may be expressed as the product of three orthogonal matrices U, V and S j of dimension (4 4), (3 3) and (4x3) respectively.It comes: The computation of the rotation matrix for the first finger under Maple gives (where S and C stand for sine and cosine respectively): The singular positions of the first finger correspond to the nullification of the determinant of the matrix j h leading to 3 0 and therefore 2 0 cos q . The same procedure is applied to all fingers.
For the long fingers, this means: 7 0 cos q and 11 0.

cos q
For the thumb, the corresponding matrix can be shown to be: The computation of the corresponding singular positions for the thumb requires 16 0.

cos q
In this case, the thumb does not present any singularities since these are usually independent of the first joint.It is noticed that a unique singular configuration shows up for each finger, and leads to a singular layout for the whole hand.Four different simulations have been carried out and are presented in Fig. 4: , , 2 q q q : The long fingers are folded up over the palm in both directions (Fig. 4, a); 2 2 q : The first finger folded up over the palm (Fig. 4, b); 7 2 q : The second finger folded up over the palm in both directions (Fig. 4, c); 12 q : The third finger folded up over the palm in both directions (Fig. 4, d).

Hand-object kinematical movement processing
The processing of the system hand-object involves necessarily loops.When a number of N objects are grasped using n fingers for example, a number of (n-1) kinematics loops are involved as stated by El-Khoury [5] and Rosales et al [9].The equation needed for the closure of the system and incorporating loops taken in pairs in any permutations, leads to a vanishing torsor.This relationship is obtained through expressing the velocities relation at a reduction station.For a vanishing torsor, this condition can be expressed independently of any projection coordinate system as shown by El-Khoury [5]: A linear relationship links the kinematics torsor and the derivative of the configuration vector noted i, j Q i, j Q i .
Taking into account the straight path of the loop (i towards j, i and j indicating the fingers), this relationship may be obtained through expressing the kinematics torsor at station i.This leads to: where I 3 and J ij refer to the identity matrix and the (3×n) Jacobian sub-matrix used to compute the velocity of rotation respectively.The combination of Eqs. ( 14) and ( 15) requires the computation of the velocity for each contact point of fingers i and j.If it is traversed in the forward direction, the corresponding relative velocity should be computed at i through the application of the point switching classical relationship at this particular point.Eqs. ( 16) and ( 17) are obtained for the cases of the direct and indirect paths respectively: where x O x O x represents the vector pre-product matrix.
The first first-order approximation of the derivative of the configuration vector may be expressed as: , 1 , 1 Introducing a Boolean variable that equals unity or zero if it is traversed in the forward or reverse directions respectively, and combining the above Eqs.( 15)-( 18) leads to the exact variational formulation of the strain relationship which is linear with respect to the increment Q (Eq.19).
In addition to the Jacobian rotation sub-matrix J ij , the use of the sub-matrix of translation J vij allows for the reformulation of the strain relationship Eq. ( 19): The factorization of the Jacobian is justified in the literature by diverse contact models leading to the grasping of objects [10].Three basic contact types may be defined.Both the first and second types use the Jacobian rotation sub-matrix to formulate the kinematics torsor.However, while the first case does not allow for any fingertips slipping or rolling (Eq.20), the second one allows solely for fingertips rolling (Eq.21): The third and last contact type uses the Jacobian translation sub-matrix, and allows for fingertips slipping only.The kinematics torsor is expressed as:

Applications to object grasping
The closure constraint expressed by Eq. ( 20) may be applied to specific types of grips [11].In this case, the objects are held between the fingertips while the thumb is placed in opposition.
It has to be acknowledged that most studies involving the grasping of an object imply a simplified representation of it based on an extension of the superquadratics i.e. the superellipsoïdes which possess the following implicit form: where a 1 , a 2 and a 3 define the scale factor on the coordinates axes x, y and z respectively; ε 1 and ε 2 define the curvature in the (x, y) and (x, z) directions; f(x, y, z) = 1 if the point (x, y, z) is situated on the body surface; , , 1 f x y z 1 if the point , , x y z is situated inside the body; , , 1 f x y z 1 if the point , , x y z is situated out- side the body.
The above listed conditions enable the description of a wide spectrum of shapes ranging from simple ellipsoids ( 1 2 1 ), parallelepipeds ( 1 0 and 2 0 ) and cylinders ( 1 1 and 2 0 ).They have been utili- Table 3 Diverse modeled objects to be grasped Superellipsoïdes  3.Moreover, when the last condition , , 1 f x y z 1 is corroborated, the specific requirements to the first joint of each finger used in the inverse geometric model have to be integrated.
Three different grasping types have been investigated.They are represented in Figs.5-9.

Conclusions
The present paper investigates the theoretical approaches concerning the grasping of diverse objects using a robotized hand consisting of four fingers.The study of grasping leads us to the modeling of articulated systems working in closed loop.
The inverse geometrical model is applied to determining the contact points of the fingers with the object under consideration, while the kinematics model is used to highlighting the singular configurations of the fingers.The problem associated to singularities is treated first for the case of free motion.
The hand Jacobian matrix singular value decomposition approach is carried out to determining the singularities occurring close to the contacts considered.
Finally, the approach developed is applied to simulating the grasping of diverse objects using the robotized hand and the results obtained presented.
of the thumb need the addition of two new parameters to its description, namely j and j b .They are represented in Fig.2.

Fig. 3
Fig. 3 Simulated configurations that are specific to a hominoid hand: a -pinch grasp; b -normal grasp; c -four finger tips grasp 2.2.The direct kinematical model define the singular configurations of the hand, the theory of Singular Value Decomposition developed by Brown and Vardy [8] is applied.It implies the decomposition of the Jacobian matrix into two sub-matrices of dimensions (4×3) for each finger, one for the translation movement matrix can be decomposed fur- ther into a product of three matrices involving the singular values.The rotation matrix 0 diagonal matrix constituted by the non-zero singular values i of the Jacobian matrix of rotation arranged in decreasing order.The singular

Fig. 4
Fig. 4 Singular configurations: a -full hand; b -first finger; c -second finger; d -third finger the objects represented in Table

Fig. 7
Fig. 7 Graphical representation of a cylinder grasping

Table 1
Geometrical parameters of the prime long finger

Table 4
Inverse Geometrical Model (MGI) data for fingertips grasping of a sphere