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I am trying to derive a state space equation for human movement (i.e. walking, running, standing still) in some normal distribution sense. The following equations should not be so hard to follow:

If the state $\bar{x} = (x, \dot{x})$

The state equation will be $\bar{x}(t) = A\bar{x}(t-1)$

With the covariance matrix $C(t) = AC(t-1)A^{-1} + Q$

Where $A = \begin{bmatrix} 1 & T \\ 0 & 1 \end{bmatrix}$

where $T$ is the time period,

and where $Q$ a "process noise" covariance matrix.

$Q = \begin{pmatrix} \sigma_x^2 & \sigma_x \sigma_{\dot{x}} \\ \sigma_x \sigma_{\dot{x}} & \sigma_{\dot{x}}^2 \end{pmatrix}$

In my view, the key of a correct model lies in getting correct values of $\sigma_x$ and $\sigma_{\dot{x}}$.

According to wiki, the fastest human speed recorded is 44.72 km/h.

What do you think?

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What is the use of covariance matrix? It seems that it has no effect on the state. Also, if you are considering a periodic motion, then there is no need for the velocity in the state. –  hwlau Jan 13 '11 at 11:11
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I am not sure I follow. What is the $x$ variable supposed to represent? Distribution of traveled distance? Also, do I understand it correctly that $t$ denotes number of steps so this is just a Markov chain? Why are you working in a discrete setting? Also, where do $C$ and $Q$ matrices come from? –  Marek Jan 13 '11 at 11:18
    
And human is not a rigid body in context of motion analysis. –  mbq Jan 13 '11 at 14:26
    
This is a very important topic especially in the context of the rapidly developing field of robotics. –  user346 Jan 13 '11 at 18:21
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1 Answer

When I look at the approach, my first inclination is to think you are stating that each possible configuration of position of human arms, legs, etc combined with the vector velocity of those components, represent a state of the human body. In some sense then there should be some smooth mapping of one state to the next. It seems that you are saying that you should be able to predict within some level of certainty what the next state should be based on the last state.

The problem with the approach would be that you are suggesting that human motion is somehow not guided by a controller, so while there are errors introduced in the system that the controller (brain) can't compensate for, in general, the overall process is not probabilistic.

I think the question you are asking may be more closely related to questions in psychometrics than system dynamics. Where Bessel was the father of "personal equations" to correct for human induced error in astronomy.

http://books.google.com/books?id=4YtB63kdK0AC&lpg=PA4&ots=1HGQEnpw2E&dq=bessel%20astronomy%20human%20error&pg=PA4#v=onepage&q&f=false

If you can clarify the question it might help.

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