The algorithms of mathematical programming in muscle recruitment and muscle wrapping problems

In spite of the enormous variety of life on Earth, muscle work in all animals is accomplished by the same physiological mechanism in which an electro-chemical process causes contraction of muscle tissue. Furthermore, muscle systems across species are equipped with a significant level of redundancy, in the sense that the number of muscles significantly exceeds the number of degrees of freedom of the body. Hence, infinitely many combinations of muscle force can balance the external loads, and selecting the best one is an optimization problem that the central nervous system solves instantly when a certain movement is called for. The mathematical modelling of these optimization processes is of vital importance to our basic understanding of human physiology and planning of surgical procedures, design of prostheses and implants, and planning of rehabilitation procedures. In our presentation, we shall consider in particular optimization problems which carry out muscle recruitment in the inverse dynamic simulations of body movements. Namely, we shall introduce those problems which could be formulated as linear or quadratic programming problems and which could be therefore solved very efficiently. The several different algorithms for the solution of such problems will be presented and results will be compared on model and realistic simulations. Another important part of the muscle recruitment problem is an identification of the path of a muscle which wraps over bones or other tissues. Therefore we will also present a geometric model of a muscle that wraps over a set of obstacles. The muscle is represented by a thin elastic string and the obstacles are represented by rigid surfaces such that the muscle wrapping problem is identified as a contact problem. The finite element (FE) method is used to discretize the string and the FE model is solved iteratively with the efficient scalable FETI based contact solver that was originally proposed for solution of the contact problems of elastic bodies as described in [1]. The several practical models built in AnyBody Modeling System will be presented.