In this paper, we develop the theoretical work on the properties and mapping of stiffness matrices between joint and Cartesian spaces of robotic hands and fingers, and propose the conservative congruence transformation (CCT). In this paper, we show that the conventional formulation between the joint and Cartesian spaces, K θ = J T θ K p J θ , first derived by Salisbury in 1980, is only valid at the unloaded equilibrium configuration. Once the grasping configuration is deviated from its unloaded configuration (for example, by the application of an external force), the conservative congruence transformation should be used. Theoretical development and numerical simulation are presented. The conservative congruence transformation accounts for the change in geometry via the differential Jacobian (Hessian matrix) of the robot manipulators when an external force is applied. The effect is captured in an effective stiffness matrix, K g , of the conservative congruence transformation. The results of this paper also indicate that the omission of the changes in Jacobian in the presence of external force would result in discrepancy of the work and lead to contradiction to the fundamental conservative properties of stiffness matrices. Through conservative congruence transformation, conservative and consistent physical properties of stiffness matrices can be preserved during mapping regardless of the usage of coordinate frames and the existence of external force.
A new theory in contact mechanics for modeling of soft fingers is proposed to define the relationship between the normal force and the radius of contact for soft fingers by considering general soft-finger materials, including linearly and nonlinearly elastic materials. The results show that the radius of contact is proportional to the normal force raised to the power of γ , which ranges from 0 to 1/3. This new theory subsumes the Hertzian contact model for linear elastic materials, where γ = 1/3. Experiments are conducted to validate the theory using artificial soft fingers made of various materials such as rubber and silicone. Results for human fingers are also compared. This theory provides a basis for numerically constructing friction limit surfaces. The numerical friction limit surface can be approximated by an ellipse, with the major and minor axes as the maximum friction force and the maximum moment with respect to the normal axis of contact, respectively. Combining the results of the contactmechanics model with the contact-pressure distribution, the normalized friction limit surface can be derived for anthropomorphic soft fingers. The results of the contact-mechanics model and the pressure distribution for soft fingers facilitate the construction of numerical friction limit surfaces, and will enable us to analyze and simulate contact behaviors of grasping and manipulation in robotics.
We propose a method for modeling dextrous manipulation with sliding fingers. The approach combines compliance and friction limit surfaces. The method is useful for describing how a grasp will behave in the presence of external forces (e.g., when and how the fingertips will slide) and for planning how to control the fingers so that the grasped object will follow a desired trajectory. The sliding trajectories are characterized by a transient and steady-state solution. The underlying theory is first dis cussed and illustrated with several single-finger examples. Experimental results are also presented. The analysis is then extended to grasps with multiple sliding and nonslid ing fingers. The multifinger analysis is illustrated with an example of manipulating a card with two soft-contact fingers.
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