Conservative Congruence Transformation for Joint and Cartesian Stiffness Matrices of Robotic Hands and Fingers

Abstract

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.

On Cartesian stiffness matrices in rigid body dynamics: an energetic perspective

“…With the help of a well-known identity 3 for the axial vector of a product of the form Ω j Ω k − Ω k Ω j , we conclude that…”

“…The vectorx can also be expressed in terms of the fixed basis {E 1 , E 2 , E 3 } and this leads to the representation U 2 . Related comments apply for the two potential functions U 3,4 .…”

“…Here, Δs is defined by (25) 3 . Paralleling the developments which lead to (28), we find the following representation for K O :…”

“…We can think of the Euler basis {g 1 , g 2 , g 3 } and the dual Euler basis {g 1 , g 2 , g 3 } as basis vectors in the tangent and cotangent spaces respectively of the manifold SO (3). The connection coefficients associated with the Euler angles are defined as…”

“…Howard et al [9] and Žefran and Kumar [21] used a Lie group approach, considered the more general case of a rigid body in a potential field, and established several representations for the Cartesian stiffness matrix. Several others researchers, such as [3,10,19], extended the formulation of the Cartesian stiffness matrices to a range of mechanical systems. Our interest in this matrix stems from its potential biomechanical applications ranging from quantifying the motion of a knee joint using a stiffness parameter [1], to modeling the intervertebral disc of the spine using a stiffness matrix [7,16,17].…”