08/06/04

2 Dimensions

Consider a hand-held object grasped by a prismatic precision grip in which the tips of the fingers and the thumb oppose each other in static planar situation as shown in Figure 1.  We limit the consideration to planar static tasks. We assume that friction at the digit-object interface is sufficiently large to prevent the object from slipping at any exerted digit forces.  

Figure 1. Planar static prehension

For the system to be at rest, the sum of all forces and moments acting on the handle should be equal to zero. Hence, the following three requirements should be satisfied:

(1) The sum of the normal forces of the three fingers equals the normal force of the thumb

                       (1)

(2) The sum of the digit tangential forces equals the weight of the hand-held object

                       (2)

(3) The total moment produced by the digit forces M is equal and opposite to the external torque exerted on the objects. It equals

 (3)

where the subscripts th, i, m, and  r refer to the thumb, index, middle and ring finger, respectively; the superscripts n and t stand for the normal and tangential force components, respectively; L is load (weight of the object), and coefficients d and r stand for the moment arms of the normal and tangential force with respect to a pre-selected center, respectively. 

The moment of the normal finger forces Mⁿ about the point of force application of the thumb (selected as a pivot) is

               (4)

where the subscript f designates individual fingers (the index, middle or ring), is the normal virtual force (the resultant normal force exerted by the three fingers) and D is the moment arm of the resultant force with respect to the pivot point. D represents the location of the resultant of the three normal finger forces.  The location of D may vary due to changes in the sharing percentage of  among the fingers and/or due to the displacement of the points of finger force application with respect to the sensor centers. Note that the index and ring fingers work as a positive and negative moment producers, respectively, while the direction of the moment by the middle finger depends on its relative position with respect to the thumb.

The moment of the tangential forces M^t is proportional to the difference between the total tangential force of the three fingers combined (the tangential virtual force) and the tangential force of the thumb. Hence, the following equation is also valid

                  (5)

The equations (1)-(3) impose three constraints on the 12 variables (normal and tangential finger force components and the coordinates of the points of force application in the vertical direction). Therefore, the system has nine degrees of freedom (DoF) that can be manipulated by the performer in different ways. (The system has also two inequality constraints: (1) the fingers can only push but not pull on the sensors and (2) the sum of the normal finger forces should be sufficiently large to prevent slipping the handle from the hand. These constraints, however, do not change the number of DoF.)

Virtual forces and moments. 

The virtual finger (VF) is an abstract representation of all three fingers together acting as a functional unit to produce a force and a moment with respect to the thumb.  The VF tangential force and VF normal force were computed as the sums of the tangential and normal forces of the three fingers, respectively.  The moment of the tangential forces was computed from equation 5. The moment of the normal forces was computed with respect to the point of application of the thumb force, see equation 4. The moment arm of the normal VF was computed from the Varignon theorem

                                       (6)

where is the normal force of finger f (f = 1, 2, 3, 4) and  is the moment arm of the finger force with respect to the point of application of the thumb force (a projected distance from the point of application of a finger force to the point of application of the thumb force).

References

Jae K. Shim, Mark L. Latash, Vladimir M. Zatsiorsky (2003). Prehension synergies: Trial-to-trial variability and hierarchical organization of stable performance. Experimental Brain Research 152(2) pp.173-184.

Vladimir M. Zatsiorsky, Mark L. Latash, Fan Gao, Jae Kun. Shim (2004). The principle of superposition in human prehension, Robotica. [In press]

3 Dimensions

1. System of prehensile object in 3D

2. Variables

1)      Digit force: , j = {i, m, r, l, th}

2)      Virtual finger force: , j = {i, m, r, l}

3)      Digit finger moment arm: , where j = {i, m, r, l, th} and  is the displacement from the origin to the center of the transducer and  is the displacement from the center to the digit tip contact point

4)      , where j = {i, m, r, l, th } and  is the free moment on digit contacts at x-axis

a.      , where  and  are the moments of  and , respectively, at X-axis.

b.