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# Grasp and Manipulation of Objects with a Multi-Fingered Hand ...

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CHAPTER 2. State of the art
grasp geometry, coupling between joints or fingers, different contact types, changes
of contact locations and so on.
The work in [24] is aimed to control the gains of actuators in order to have
the desired compliance or the desired stiffness matrix. Authors noticed that the
stiffness matrix is decoupled into two terms: in the first one it is pointed out the
dependence of the matrix with respect to the changes in the grasp's configuration,
while the second term expresses the restoring forces at contact points caused by
the movements of the object. A method to calculate these sub-matrices is shown:
hence, the actual stiffness matrix is the sum of these two components. It is worth
noticing that if the eigenvalues of the stiffness matrix are all positive, then the grasp
is stable. After calculating the actual stiffness matrix, a sort of error between the
actual and the desired matrix is computed and the servo gains are tuned respecting
contacts and joints constraints.
A decentralized stiffness control scheme is presented in [16], where the stiffness
matrix is evaluated as a sum of two terms like before, and where it is pointed
out the dependence also with respect to the changes in the configuration of joints.
Rolling and sliding contacts are not included in the formulation, as well as fingers
and transmissions couplings. The decentralized object's stiffness control allows
the system to reach the desired stiffness in two steps: in the first step the stiffness
matrix at fingers level and its reflection at the joint level are calculated with a
least square algorithm; in the second step the component in the null space of the
previous computed matrix is summed to the previous contribution, and then the
gravity forces are compensated.
The stiffness control has obviously some limitations which are pointed out
in [91]. Many of these limitations in performance are due to mechanical proprieties
of the robotic hands caused by backlash, difficulty in coordination of so many
degrees of freedom, tactile and force sensors inadequacy. Other limitations are
underlined studying the case of changing the center of compliance of a grasped
object. Using the Cartesian stiffness control method, the controller measures the
difference between the actual and the desired position of the center of compliance.
A restored force is then calculated to carry back the center of compliance in the
right position. Through the use of the grasp matrix, this force applied to the
object is mapped on a finger force and then in a joint force, trough the finger
Jacobian. In this simple case, errors can be present in the computation of the
kinematics of the hand and of the grasp matrix, due to the presence of rolling
and sliding contacts. Moreover, friction constraints put a lower limitation in the
feasible stiffness, while the grasp geometry and fingers compliance put an upper
limitation.
To overcome some problems pointed out in the previous mentioned work, the
same authors proposed in [91] a method in which tactile sensors can be helpful
to know the initial pose of the object in order to track the fingers in rolling and
sliding movements. The control law is the same proposed in [91], but now it is
18

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Summary :

## Authors noticed that the stiffness matrix is decoupled into two terms: in the first one it is pointed out the dependence of the matrix with respect to the changes in the grasp's configuration, while the second term expresses the restoring forces at contact points caused by the movements of the object. The decentralized object's stiffness control allows the system to reach the desired stiffness in two steps: in the first step the stiffness matrix at fingers level and its reflection at the joint level are calculated with a least square algorithm;

Tags : control,fingers,grasp,force,desired,compliance,two,actual,out,pointed,then,object,changes
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