1. Analysis of Multifingered Hands Kerr and Roth 1986

2. Outline  1. contribution of this paper  1.1 Selection of Internal Grasp Forces  1.2 Manipulating Objects within the Hand  1.3 Hand Workspaces  2. limitation of this paper  3. future direction of this paper

3. 1.1 Selection of Internal Grasp Forces  Determining how hard to squeeze an object in order to ensure a secure grasp  -f=WC （ 1 ）  C=C p +C h （ 2 ）  （ 3 ）  C h ＝ N （ 4 ）

4. 1.1 Selection of Internal Grasp Forces  Determining how hard to squeeze an object in order to ensure a secure grasp  Equation 1 shows that all forces must be balanced  Then we consider friction at the contact points  the friction cannot exceed the maximum available force due to friction, determined by the coefficient of static friction at each point  and the tangential components of the friction must balance out the contact forces so that the object does not move  Then we set up 2 nd ~8 th equations

5. 1.1 Selection of Internal Grasp Forces  Friction constraints  For each contact,we have the constraint (5)  We must also ensure that the tangential components of the contact forces do not exceed the magnitude of the equilibrating friction force. Using a Coulomb friction model,this constraint is written as (6)

6. 1.1 Selection of Internal Grasp Forces  If we choose to use four planes, the linear approximation to equation 6 is written as the set of equations  similarly for torsional friction...  （ 7 ） (8)

7. 1.1 Selection of Internal Grasp Forces  joint torque limits  we also know the max torque that each finger can exert… so the other constraints are that this max torque cannot be exceeded and min torque is required.  (9)

8. 1.1 Selection of Internal Grasp Forces  Through a series of computation and substituting  (10)  (11)

9. 1.1 Selection of Internal Grasp Forces  Optimal selection of internal grasp force  An optimal selection of is one for which we are furthest from violating any one of the constraints.the distance of some point from the plane formed by the ith constraint of equation is given by (12)  reformulating the formula to pose this problem as a standard linear optimization problem,we get (13)

10. 1.1 Selection of Internal Grasp Forces  These inequalities are written in more conventional form if we create a vector which is the vector augmented with the distance variable, d. Equation 13 then becomes  (14)  the function we wish to maximize (15)  Equations 14 and 15 form a linear programming problem in its standard form, which is readily solved by the Simplex method.

11. 1.2. Manipulating Objects within the Hand  determining the motion of each finger joint required to achieve a desired motion of the object.  The motion of the contact point on each fingertip is identical to the motion of the corresponding contact position on the object.If the inverse kinematic solution for each finger is known, the required finger-joint motions are easily calculated.

12. 1.2. Manipulating Objects within the Hand  Here,we will consider only the case of pure rolling between the object and the fingertip.  In the meantime We will consider the minimum case of fingers with three joints.

13. Consider a simple case that a finger in contact with an object

14. 1.2. Manipulating Objects within the Hand Going through the computation of math (16)-(23) for first row,(24)-(29) for The second row

15. 1.2. Manipulating Objects within the Hand  Finally we got

16. 1.3. Hand Workspaces  4 assumptions:  The type of contact present between the fingertips and the object is considered to be point contact with friction,and the contact points do not change during the motion.  The hand workspace will be defined for a particular configuration of contact points on the object and for particular locations of the contact points on the fingertips.

17. 1.3. Hand Workspaces  The workspace for each fingertip5 is known and is considered fixed relative to the palm of the hand.  Collisions or intersections between the object and the finger links or between the finger links them- selves will be ignored.  with these four assumptions, determining the workspace for a hand can be reduced to the problem illustrated in figure 6 for a three-fingered hand.

18. 1.3. Hand Workspaces

19.  Rather than deal with the complete six- dimensional configuration space of the object, we will define the total workspace of the hand as the volume swept out by one particular point q fixed in the object as the object is moved through all possible configurations. For our purposes, we will choose q to be the spatial center of the contact points on the object.

20. 2. Limitations of this paper  Assumptions  1. for selection of internal grasp force, the paper assume there are only friction constraint and joint torque constraint.  2. for manipulating objects within the hand, the paper,Here,we will consider only the case of pure rolling between the object and the fingertip.  3. for hand workspaces, all the four assumptions are too restrictive for practical purposes.

21. 3. Future Direction of this paper  1.Adding other types of constraints,to make a real case  2. for hand workspaces,the treatment of this problem here has not ad- dressed many of the important issues relating to the workspace. The first of these is the range of orientations possible at each point in the total workspace.  3. the analysis here assumes that the contact points are fixed on both the fingertips and the object.  4. The last important point not addressed above is that of intersection between the object and the fingers and between the fingers themselves.

22. 3. Future Direction of this paper  There is no discussion about the influence of the size and shape on manipulation. And it’s important.  there is no discussion about how to decide a particular configuration and orientation of the end-effector to pick an object from many different possible configurations and orientations  In the meantime,The issue of singular configurations had not been covered.

23. Thank YOU!