A modelling approach to the dynamics of gait initiation

Gait initiation is an integral and complex part of human locomotion. In this paper, we present a novel compliant-leg model-based approach to understanding the key phases of initiation, the nature of the effective forces involved in initiation, and the importance of the anticipatory postural adjustments (APAs). The results demonstrate that in the presence of APAs, we observe a change in the characteristic of forcing required for initiation, and the energetic cost of gait initiation is also reduced by approximately 58%. APAs also result in biologically relevant leg landing angles and trajectories of motion. Furthermore, we find that a sublinear functional relationship with the velocity error from steady state predicts the required force, consistent with an open loop control law basis for gait initiation.

Regularisations of convex functions and slicewise suprema

For a number of years, there has been interest in the regularisation of a given proper convex lower semicontinuous function on a Banach space, defined to be the episum (=inf-convolution) of the function with a scalar multiple of the norm. There is an obvious geometric way of characterising this regularisation as the lower envelope of cones lying above the graph of the original function. In this paper, we consider the more interesting problem of characterising the regularisation in terms of approximations from below, expressing the regularisation as the upper envelope of certain subtangents to the graph of the original function. We shall show that such an approximation is sometimes (but not always) valid. Further, we shall give an extension of the whole procedure in which the scalar multiple of the norm is replaced by a more general sublinear functional. As a by-product of our analysis, we are led to the consideration of two senses stronger than the pointwise sense in which a function on a Banach space can be expressed as the upper envelope of a family of functions. These new senses of suprema lead to some questions in Banach space theorey.

Sublinear functionals ergodicity and finite invariant measures

By introducing a sublinear functional involving infinite matrices, we establish its connection with ergodicity and measure preserving transformation. Further, we characterize the existence of a finite invariant measure by means of a condition involving the above sublinear functional.