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2017 | OriginalPaper | Chapter

4. Biomechanics of Human Iliopsoas and Functionally Related Muscles

Authors : Tijana Ivancevic, Leon Lukman, Zoran Gojkovic, Ronald Greenberg, Helen Greenberg, Bojan Jovanovic, Aleksandar Lukman

Published in: The Evolved Athlete: A Guide for Elite Sport Enhancement

Publisher: Springer International Publishing

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Abstract

One of the causes of pain in the lumbosacral spine is the overexertion of the following muscles: quadratus lumborum, rectus abdominis, erector spinae group, gluteus maximus et medius, and piriformis.

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previous chapter Multi-physical Structure of Human Fascia: Harmony of Biomechanics and Neurodynamics

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Lagrangian $L\left( q,\dot{q}\right) $ is called regular when the symmetric matrix $[\partial ^{2}L/\partial \dot{q}^{i}\partial \dot{q} ^{j}],~(i,j=1,\ldots ,n)$ is everywhere nonsingular.

Recall that Kalman filter was the algorithm that was steering the Apollo 11 spaceflight in July 1969, that took Neil Armstrong and Buzz Aldrin to the moon and brought them back home.

A product of two Gaussians is another Gaussian; a Fourier transform of a Gaussian is the same Gaussian.

This Gaussian restriction is a characteristic of Kalman filtering; giving up of this restriction leads to a family of particle filters, in which the result of measurements are represented as a weighted sum of Dirac-delta functions.

In general, someone’s belief in an event or statement A will depend on their body of knowledge K. Formally, this is a belief measure, or a degree-of-belief: p(A|K), where $(\cdot |\cdot )$ means ‘given’. When K remains constant, we simply write p(A) instead of p(A|K); however, any statement about p(A) is always conditioned on a context K.

The conditional probability p(A|B) of an event A given another event B is the probability that the event A will happen given that the event B has already occurred. Also, when we have a joint event A and B, we have a joint probability : p(A, B), where $(\cdot ,\cdot )$ means ‘and’, $\wedge $, or ‘joint’. Conditional probabilities can be defined via joint probabilities and vice versa: $ p(A|B)=p(A,B)/p(B) $, which implies: $p(A,B)=p(A|B)p(B)$.

The general Bayesian filter attempts to recover the information about the current system state $x_{t}$ (at time t) based on the available measurements $y_{1:t}$ (at previous times $1,\ldots ,t$). Formally, the Bayesian filter includes the dynamics-given time-update recursion for calculating the Prior PDF:

$$\begin{aligned} \overset{\mathrm {Prior}}{p(x_{t}|y_{1:t-1})}=\int _{\mathbb {R}^{n}}\overset{ \mathrm {Dynamics}}{p(x_{t}|x_{t-1})}\overset{\mathrm {Previous\,Prior}}{ \,p(x_{t-1}|y_{1:t-1})}dx_{t-1}\,, \end{aligned}$$

and the measurement-update recursion for calculating the Posterior (i.e., the filter PDF) using the Bayes rule:

$$\begin{aligned} \overset{\mathrm {Posterior}}{p(x_{t}|y_{1:t})}=\,\overset{\mathrm {Prior}}{ p(x_{t}|y_{1:t-1})\,\,}\overset{\mathrm {Likelihood}}{p(y_{t}|x_{t})} \,/\,Z_{t}, \end{aligned}$$

where $Z_{t}$ is the normalizing constant (i.e., the partition function) defined by:

$$\begin{aligned} Z_{t}=\int _{\mathbb {R}^{n}}p(x_{t}|y_{1:t-1})p(y_{t}|x_{t})dx_{t}\,. \end{aligned}$$

The exponential map is a map from the orthogonal rotational Lie group $ G=SO(3)$ to its Lie algebra $\mathfrak {g}=\mathfrak {so}(3)$, that is, $\exp :SO(3)\rightarrow \mathfrak {so}(3)$. It is sometimes called the Lie functor [40].

The logarithm map is a map from the Lie algebra $\mathfrak {g}=\mathfrak {so} (3)$ to its Lie group $G=SO(3)$, $\log =\exp ^{-1}:\mathfrak {so} (3)\rightarrow SO(3)$. It is sometimes called theinverse Lie functor [40].

A slightly modified form of the basic BGC-maneuver is also an essential part of thepole-vault maneuver.

The pole vault was introduced as a full medal Olympic event in 1896 Olympic Games.

We remark that there are no restrictions on the material the pole is made from (currently it is either carbon-fibre or fibreglass) or the length of the pole.

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Title: Biomechanics of Human Iliopsoas and Functionally Related Muscles
Authors: Tijana Ivancevic
Leon Lukman
Zoran Gojkovic
Ronald Greenberg
Helen Greenberg
Bojan Jovanovic
Aleksandar Lukman
Publisher: Springer International Publishing
Book: The Evolved Athlete: A Guide for Elite Sport Enhancement
Print ISBN: 978-3-319-57927-6

Electronic ISBN: 978-3-319-57928-3

Copyright Year: 2017
DOI: https://doi.org/10.1007/978-3-319-57928-3_4

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