Cartesian stiffness for wrist joints: analysis on the Lie group of 3D rotations and geometric approximation for experimental evaluation

Abstract

This paper is concerned with the analysis and the numerical evaluation from experimental measurements of the static, Cartesian stiffness of wrist joints, in particular the human wrist. The primary aim is to extend from Euclidean spaces to so(3), the group of rigid body rotations, previous methods for assessing the end-point stiffness of the human arm, typically performed via a robotic manipulandum. As a first step, the geometric definition of Cartesian stiffness from current literature is specialised to the group so(3). Emphasis is placed on the choice of the unique, natural, affine connection on so(3) which guarantees symmetry of the stiffness matrix in presence of conservative fields for any configuration, also out of equilibrium. As the main contribution of this study, a coordinate-independent approximation based on the geometric notion of geodesics is proposed which provides a working equation for evaluating stiffness directly from experimental measurements. Finally, a graphical representation of the stiffness is discussed which extends the ellipse method often used for end-point stiffness visualisation and which is suitable to compare stiffness matrices evaluated at different configurations.

Keywords::

• static Cartesian stiffness
• Lie group so(3)
• bi-invariant affine connection
• wrist robot

Acknowledgements

The author thanks Dr Domenico Formica for useful discussion and, in particular, for introducing his experimental study which inspired this theoretical development. This study is supported by the Academic Research Fund (AcRF) Tier1 (RG 40/09), Ministry of Education, Singapore.

Notes

^1. A reflex arc is heteronymous when one muscle is activated in response to the stretching of another muscle. A reciprocal reflex arc would connect the same two muscles, but activating the latter in response to the stretching of the former. Unequal gains for a reflex arc and its reciprocal might cause a path-dependent mechanical work, leading to a non-zero curl elastic field.

^2. In fact, is not a differential operator, but just a bilinear function equivalent to evaluated at the identity (R = I).

^3. Although dealing with torques, in this paper we shall use ‘wrench’ for elements of and ‘force’ for generic elements of .

^4. As shown in Zefran and Kumar (2002), are in fact the components of a (0, 2) tensor .

^5. For the specific example in Figure 1, any rotation about each would lead to one unstable equilibrium configuration.

^6. The pull-back mapping is only possible on covariant tensors (Boothby 1986), e.g. one-forms and the stiffness tensor, respectively, a (0,1) and a (0,2) tensor.