The study on the human grip has inspired to the robotics over the past decades, which has resulted in performance improvements of robotic hands. However, current robotic hands do not have the enough dexterity to execute complex tasks. Recognizing this fact, the soft fingertips with hemispherical shape and deformation models have renewed attention of roboticists. A highfriction contact to prevent slipping and the rolling contribution between the object and fingers are some characteristics of the soft fingertips which are useful to improve the grasping stability. In this paper, the parallel distributed deformation model is used to present the dynamical model of the soft tip fingers with
From physiological point of view, the human hands are considered as a powerful tool whereby the human brain interacts with the world, that is, how it perceives and acts with the environment [
In general, the dexterous manipulation in robotics, to emulate pinching motions, have been formulated in terms of the object, that is, the forces/torques exerted on it to produce the desired movements and how they behave [
On the basis that human hands have fingers with soft tips, in this paper the grasping and the object manipulation is presented, using a pair of robotic fingers with
This paper is organized as follows. Section
Consider a pair of soft tip fingers, with three degree of freedom each one, grasping an object in the horizontal plane, as shown in Figure
Rigid object grasped by a pair soft tip fingers.
Deformation Model proposed by [
On the other hand, taking into account the curvature effects of the fingertip
Thus, the equations of motion for the fingers are given as
Summing the products between
As first step before to execute manipulation tasks, a stable grasping must be established. To grasp stably an object, the forceclosure is used to guarantee that the object should be held securely by the fingers. This mean that maintaining the contact between the fingers and the object, that is,
Inspired that humans can execute some manipulation tasks without any object information, in this paper a stable grasping and the object manipulation based only on the center position of the soft tip finger
To start the analysis and to ensure a stable grasping, it is considered that the joint torque is defined as
Expressing the closedloop system equations (
Taking the inner product between
Although
In order to define a constrained manifold where the trajectories of the system guarantees a stable grasping; a first step is to show the boundedness of solutions in the closed loop. Considering that
Now, if we multiply (
Now, we are in conditions to define the following.
Let the neighborhood
Let the neighborhood
Definition
Finally, we have the following result.
Considering that the desired reference state
Once the forces applied to the object have been compensated to hold it stably, the object can be rotated to the desired angle using the superposition principle, that is, the control law now is defined as
Now, taking the inner product between
Now, when the stable grasping and object orientation tasks has been established, the objective will be to move the object to desired coordinates
Now, taking the inner product between
In order to demonstrate usefulness of our scheme for stable grasping and object orientation, numerical simulations were carried out on a pair of deformable fingertips in the horizontal plane, see Figure
The physical parameters of the fingers and object are shown in Table
Physical parameters.
Parameter  Value 



















Parameters of the object.
Parameter  Value 







The simulation study is divided in three steps. As first step before any manipulation task is necessary to guarantee that the stable grasp is achieved through the control law defined as
In Figure
Convergence of
Convergence of
Convergence of
Convergence of
Finally, the Figures
Convergence of
Convergence of
Convergence of
Notice that the convergence to zero of
Once the all forces applied to the object have been compensated, it is possible to execute any manipulation task. In this case the rotation of the object, to a desired angle
Convergence of
Convergence of
As the final step of this study, a stable grasping and object manipulation which include the orientation and translation of the object to a desired reference is presented. Using a superposition principle, the control law for this step is defined as
The Figures
Convergence of
Convergence of
Convergence of
A scheme to grasp and manipulate an object using soft tip fingers is presented. In order to include more characteristics of the human fingertips a parallel deformation model is used. The control law proposed ensures stability on a constrained manifold. Furthermore, the control law avoids information of the radius of the tips and the convergence to the desired force value is guaranteed.
Numerical simulations, on a pair of deformable fingertips in horizontal plane, allow us to visualize the convergence of the closedloop trajectories to the desired point. In addition, the special role of
This work was partially supported by CONICYT, Departamento de Relaciones Internacionales, Programa de Cooperación Científica Internacional, CONICYT/CONACYT 2011380, and the Universidad de los Andes, Chile, FAI project ICI00211.