Figure 1 shows a hand physiology. It can perform grasping, holding and pinching operations while manipulating objects of various sizes, weights and shapes. To mechanically simulate these functions, planar mechanisms with one DOF are generally used in mechanical hands^{ [14]}. Over the past several years trends in prosthetic hand research have dictated a move away from grippers having only two rigid fingers and no phalanges, focusing more on hands with at least three to five functional fingers, each with two to three phalanges^{[5]}. Several types of electric powered hand prosthesis with four functional fingers and a thumb have been created in an attempt to increase user acceptance and satisfaction. The idea to approach the spatial complement of the shape of an object to ensure a distributed grasp is rather common in biologicallyinspired robotics: E.g., snake robots or elephant trunks. They belong to what has been defined as the FrenetSerret manipulators^{[6]} intended for wholearm manipulation^{[7]}. General grasping processes have also been discussed in^{[8]}.
 Fig. 1: 
Physiology of a human hand 
In fact, for the users good hand prosthesis should be cosmetically attractive, comfortable enough to wear it all day long and be sufficiently controllable to execute easily with it daily task^{[9,10]}. The technology and expertise has crossed over into and benefited the area of prosthetic hand design^{[11]}, hands are available for industrial and nonindustrial applications. In order for the previously described parameters to be met optimum sizing of finger driving mechanisms by using fundamental characteristics regarding with the humanlike behavior, grasp efficiency and force transmission, identification solution based on the Particle Swarm Optimization (PSO) was proposed in this study. However, significant efforts have been made to find designs that are simple enough to be easily built and controlled in order to obtain practical systems, particularly in human prosthetics. MATERIALS AND METHODS
Force properties of underactuated fingers: Underactuation in robotic fingers
is different from the concept of underactuation usually presented in robotic
systems and both notions should not be confused. An underactuated robot is generally
defined as a manipulator with one or more unactuated joints. On the other hand,
underactuated fingers generally use elastic elements in their ‘‘unactuated’’
joints. Thus, one should rather think of these joints as uncontrollable or passively
driven instead of unactuated. In an underactuated finger, the actuation wrench
ta is applied to the input of the finger and is transmitted to the phalanges
through suitable mechanical elements, e.g., fourbar linkages. Since underactuated
fingers have many degrees of freedom and fewer actuators, passive elements are
used to kinematically constrain the finger and ensure the shapeadaptation of
the finger to the object grasped. To this end, springs and mechanical limits
are often used. An example of underactuated twophalanx finger using linkages
and its closing sequence are shown in Fig. 2. The actuation
torque ta is applied to the first link which transmits the effort to all phalanges.
Notice the mechanical limit that allows a preloading of the spring to prevent
any undesirable motion of the second phalanx and also to prevent hyperextension
of the finger. Springs are useful for keeping the finger from incoherent motion,
but when the grasp sequence is complete, they still oppose the actuation. Thus,
springs shall be designed with the smallest stiffness possible, however sufficient
to keep the finger from collapsing. With practical prototypes, one has to ensure
that grasps are stable in the sense that ejection is prevented. Indeed, an ideal
grasping sequence as shown in Fig. 2 does not always occur.

Fig. 2: 
Ideal grasping sequence of a threephalanx finger with linkage
transmission 

Fig. 3: 
Example of an ejection sequence for a threephalanx finger
with linkage transmission 
For in the final configuration some phalanx forces may be negative. If onephalanx force is negative the corresponding phalanx will loose contact with the object. Then, another step in the grasping process will take place: the remaining phalanges corresponding to positive forces will slide on the object surface. This sliding process will continue until either a stable configuration is achieved, or the last phalanx will curl away and loose contact with the object (Fig. 3). Static equilibrium: A particular design of underactuated finger will be simplified version of the finger that was used in the Mars and Sarah M1 prototypes^{[12]}. Figure 4 shows the tow models. The actuation torque ta is applied to the link a1 (or pulley r1) which transmits the effort to the phalanges. A rotational springs t_{2}, t_{3} in O_{2}, O_{3} are used to keep the finger from incoherent motions. In order to determine the configurations where the finger can apply forces to the object grasped, we shall proceed with a quasistatic modeling of the finger. The latter will provide us with the relationship between the input actuator torque and the forces exerted on the object.
 Fig. 4: 
Model of underactuated threephalanx finger using (a): Linkages
(b): Tendons 
Equating the input and the output virtual powers, one obtains: where, f = [f_{1}, f_{2}, f_{3}]^{T} the vector of normal contact forces and t is the input torque vector exerted by the actuator and the springs, i.e., t = [T_{1}, T_{2}, T_{3}]^{T}. Matrix J is a lower triangular matrix characteristic of the contact locations and friction, if modeled. That can be expressed analytically. Neglecting friction, one has: where, C_{θ} = Cos θ symbols are indicated in Fig. 5. It is observed that this matrix can also used with fullyactuated fingers. Matrix T is characteristic of underactuation. It becomes the identity matrix for fullyactuated fingers) and, more precisely, of the transmission mechanism used. For a finger using linkages as shown in Fig. 4 or have:
is the signed distance between point O_{i} and the geometric intersection of lines (O_{i1}O_{i}) and (P_{2i2}P_{2i3}). This value can he negative if the intersection point is on the same side as O_{i1} with respect to O_{i} .Angle Ψ_{i} is the angle between O_{i}P_{2i2} and O_{i+1}O_{i} for i> 1, i.e.: Hence, For a linkagedriven finger, the expressions of the contact forces are: Where: and for tendondriven fingers, the expressions are simpler, i.e., one has: where, r2i1 and r2i for i > 0 are respectively the radius of the pulley located at the base and at the end of the ith phalanx (cf. Fig. 4b). Hence, for tendondriven fingers, the expressions of the contact forces are:
Optimization of the design: Because of the complexity of the system,
it is very difficult, in the static model (Eq. 110),
to isolate each parameter. To solve the problem, a Particle Swarm Optimization
(PSO) algorithm was used. The PSO algorithm used was developed by Source Code
Library for the software Matlab.
PSO algorithm is similar to that of the evolutionary computation techniques
in which a population of potential solutions to the optimal problem under consideration
is used to probe the search space. Each potential solution is also assigned
a randomized velocity and the potential solutions, called particles, correspond
to individuals. Each particle in PSO flies in the Ddimensional problem space
with a velocity dynamically adjusted according to the flying experiences of
its individuals and their colleagues. The location of the ith particle is represented
as D where
u are the lower and upper bounds for the dth dimension, respectively. The best
previous position (which gives the best fitness value) of the ith particle is
recorded and represented as ,
which is also called P_{best} . The index of the best particle among
all the particles in the population is represented by the symbol g . The location
P_{g} is also denoted by g_{ best}. The velocity of the ith
particle is represented by and
is clamped to a maximum velocity ,
which is specified by the user. The particle swarm optimization concept consists
of, at each time step, regulating the velocity and location of each particle
toward its P_{best} and g_{best} locations according to the
Eq. 23, respectively:
where, w is the inertia weigh; c_{1}, c_{2} are two positive
constants, called cognitive and social parameter respectively; i
= 1, 3, …, m and m is the size of the swarm; r_{1}, r_{2} are
two random numbers, uniformly distributed in [0, 1]; and n = 1, 3, …, N denotes
the iteration number, N is the maximum allowable iteration number.
Criteria of optimization: Because the main task of this finger is to grasp
objects (so to apply forces to them), it’s normal to do the optimization in
function of forces criteria for the static model, presented previously. Those
criteria were defined to found the parameters and then those criteria are derived
from the static model (Eq. 110).
A power grasp uses the both phalanxes in comparison with a tip grasp which uses only the distal phalanx. One would like that a power grasp could be possible for all positionorientation of the finger. Mathematically, this means: Because of the contact forces, if those forces are negative, the associated phalanx will move in clockwise directions which get away from the object. The pinching force is the sum of f_{1}, f_{2} and f_{3} that will be applied on an object. Because this pinching force is generated by the user, the force should be preferably constant, no matter of the positionorientation of the finger. Then to be assuring the stability of the grasp, one needs a certain pinching force. This force should be as high as possible, so this criteria could be mathematically represent as:
With
Parameters to optimize: The fingers of the hand prosthesis optimization are a function of the size of the hand. Here, a glove^{[12]} of the company Otto Buck was used to define the parameters boundaries. RESULTS The parameters that had been defined by the optimization are shown in the Fig. 4 and the Table 1 shows the numerical values of the boundaries used for those parameters. (Both have the exact same condition for positiveness! a tendonactuated finger with pulley radii equivalent to link lengths, i.e., r_{2i1} = a_{i} and r_{2i} = c_{i}). DISCUESSION
As said before, the goal of this optimization is to find a good solution. Although
the PSO method seems to be sensitive to the tuning of some weights or parameters,
according to the experiences of many experiments, the following PSO and GA parameters
can be used.
PSO method:
Population size 
= 
100 
Generations 
= 
40 inertia weight factor w 
where, w_{max} 
= 
0.7 
w_{1} 
= 
0.4 
The limit of change in velocity of each member in an individual was:
Acceleration constant c_{1} = 2 and c_{2} = 2.
Table 1: 
Boundaries of parameters to optimized. 

GA method:
Population size 
= 
100 
Generations 
= 
40 
Crossover rate P_{C} 
= 
0.6 
Mute rate P_{m} 
= 
0.05 
Crossover parameter a 
= 
0.5 
The optimizations used a variation of θ_{1} and θ_{2} from 200° to 200°. This seems to be a reasonable workspace for this application of the finger. Table 2, the parameters found by three optimizations method and the absolute error, Mean Absolute Error (MAE) and Maximum error (Max) is also compared for Largescale Unconstrained Nonlinear (indicated as LSUN) and Genetic Algorithm (indicated as GA) and Particle Swarm Optimization (indicated as PSO) estimator in Table 1 and 2. From the analysis of the results in Table 2, it is observed that the accuracy of the (PSO) algorithm is slightly superior when compared with the (GA) algorithm on account of Mean Average Error (MAE) this comparison is 2.75<2.79 for parameter's. The computational time is the least, for the (PSO), the GA computational time is less as compared with the SLUN method as indicated in Table 2.
The constant pinching force was evaluated using the standard deviation (sd).
Figure 57 shows the statistical data of
the forces. One can see that solution by PSO has the smallest s.d. More over
this solution has interesting parameters.
Finally, for all those reasons, PSO solution was preferred and declared "the optimal solution.
Table 2: 
Optimal parameters of by using three methods (for a linkagedriven
finger) 

 Fig. 5: 
The force distribution (SLUN optimization method 
 Fig. 6: 
The force distribution (GA optimization method) 

Fig. 7: 
The force distribution (PSO optimization method) 
CONCLUSION The new underactuated finger seems to be very interesting for a hand prosthesis use. The simplicity of the design and its self adaptation to different shapes of objects are some qualities that give it a good chance to be successful in prosthetics. This study has presented and analyzed the force capabilities of underactuated fingers of a threephalanx finger considering geometry of the contact and optimal phalanx force distribution, two different methods, a genetic algorithm and a Particle swarm optimization method. An optimization was done to find a good configuration of the parameters of the finger. The design problem has been formulated as a multiobjective optimization problem. The numerical procedure is characterized by fairly simple formulations for the optimality criteria and no great computational efforts in order to achieve practical optimal design solutions. To ensure a stable grasp, ejection must be prevented. The future work is to study the controllability of an underactuated hand based on these results. ACKNOWLEDGEMENT The researchers thank the department of Machines and Equipments Engineering, of WHO that funded the project with resources received for research from university of Technology. " target="_blank">View Fulltext
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