INVERSION OF OPERATOR PENCILS ON HILBERT SPACE

We consider a linear operator pencil with complex parameter mapping one Hilbert space onto another. It is known that the resolvent is analytic in an open annular region of the complex plane centred at the origin if and only if the coefficients of the Laurent series satisfy a doubly-infinite set of left and right fundamental equations and are suitably bounded. If the resolvent has an isolated singularity at the origin we propose a recursive orthogonal decomposition of the domain and range spaces that enables us to construct the key nonorthogonal projections that separate the singular and regular components of the resolvent and subsequently allows us to find a formula for the basic solution to the fundamental equations. We show that each Laurent series coefficient in the singular part of the resolvent can be approximated by a weakly convergent sequence of finite-dimensional matrix operators and we show how our analysis can be extended to find a global expression for the resolvent of a linear pencil in the case where the resolvent has only a finite number of isolated singularities.

On Representations for Joint Moments Using a Joint Coordinate System

In studies of the biomechanics of joints, the representation of moments using the joint coordinate system has been discussed by several authors. The primary purpose of this technical brief is to emphasize that there are two distinct, albeit related, representations for moment vectors using the joint coordinate system. These distinct representations are illuminated by exploring connections between the Euler and dual Euler bases, the “nonorthogonal projections” presented in a recent paper by Desroches et al. (2010, “Expression of Joint Moment in the Joint Coordinate System,” ASME J. Biomech. Eng., 132(11), p. 11450) and seminal works by Grood and Suntay (Grood and Suntay, 1983, “A Joint Coordinate System for the Clinical Description of Three-Dimensional Motions: Application to the Knee,” ASME J. Biomech. Eng., 105(2), pp. 136–144) and Fujie et al. (1996, “Forces and Moment in Six-DOF at the Human Knee Joint: Mathematical Description for Control,” Journal of Biomechanics, 29(12), pp. 1577–1585) on the knee joint. It is also shown how the representation using the dual Euler basis leads to straightforward definition of joint stiffnesses.

Expression of Joint Moment in the Joint Coordinate System

The question of using the nonorthogonal joint coordinate system (JCS) to report joint moments has risen in the literature. However, the expression of joint moments in a nonorthogonal system is still confusing. The purpose of this paper is to present a method to express any 3D vector in a nonorthogonal coordinate system. The interpretation of these expressions in the JCS is clarified and an example for the 3D joint moment vector at the shoulder and the knee is given. A nonorthogonal projection method is proposed based on the mixed product. These nonorthogonal projections represent, for a 3D joint moment vector, the net mechanical action on the JCS axes. Considering the net mechanical action on each axis seems important in order to assess joint resistance in the JCS. The orthogonal projections of the same 3D joint moment vector on the JCS axes can be characterized as “motor torque.” However, this interpretation is dependent on the chosen kinematic model. The nonorthogonal and orthogonal projections of shoulder joint moment during wheelchair propulsion and knee joint moment during walking were compared using root mean squares (rmss). rmss showed differences ranging from 6 N m to 22.3 N m between both projections at the shoulder, while differences ranged from 0.8 N m to 3.0 N m at the knee. Generally, orthogonal projections were of lower amplitudes than nonorthogonal projections at both joints. The orthogonal projection on the proximal or distal coordinates systems represents the net mechanical actions on each axis, which is not the case for the orthogonal projection (i.e., motor torque) on JCS axes. In order to represent the net action at the joint in a JCS, the nonorthogonal projection should be used.

PROJECTIVE NOISE CLEANING WITH DYNAMIC NEIGHBORHOOD SELECTION

In recent years, several methods of noise cleaning have been devised, of which projective methods have been particularly effective. In our paper, we explain in detail why orthogonal projections are nonoptimal and how the nonorthogonal projections suggested by Grassberger et al., naturally emerge from the SVD method. We show that this approach when combined with a dynamic neighborhood selection yields optimal results of noise cleaning.