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Re: Errors in BSP/error propagation

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  • Re: Errors in BSP/error propagation

    Dear Biomch-L readers:

    I enjoyed reading the debate concerning 'errors in the body segmental
    parameters (BSP)' last week. Especially, Brian Davis' last posting was
    instructional for me in that the discussion was quantitative (practical) as
    well as analytic. I think it was a good demonstration showing how to deal with
    'propagation of error' in calculation of a quantity (e.g., moment, M), which
    is a function of more than one variable. Whenever such a function involves the
    same variable more than once, some of the errors or uncertainties may cancel
    themselves. This effect sometimes is called 'compensating errors'.
    Neglecting compensating errors may result in overestimation of the final
    uncertainty. At this point, those who are familiar with this matter may
    want to skip the part (1) to the part (2) of this posting.

    (1) Such overestimation can be avoided using the concept of 'partial
    derivatives' in computing uncertainty in a functional quantity. Lets
    suppose that x, y, ..., z are measured with uncertainties dx, dy, ..., dz
    and the measured values are used to compute the function, q(x,y,...,z). Then

    dq = |@q/@x|dx + |@q/@y|dy + ...... + |@q/@z|dz, [1]
    where d and @ are symbols for uncertainty and partial derivative,
    respectively. Note the symbol of 'approximation' instead of 'equality' in
    [1]. Disregarding _independence_ and _randomness_ of uncertainties in the
    variables of q, the uncertainty in q is never larger than the ordinary sum of
    each term in [1], i.e.,

    dq < |@q/@x|dx + |@q/@y|dy + ...... + |@q/@z|dz. [2]

    However, if the uncertainties in the variables of q are _independent_ and
    _random_, then the uncertainty in q can be given:

    dq = sqrt of { [(@q/@x)dx]**2 + ...... + [(@q/@z)dz]**2 }. [3]

    This is the equation that you saw in Brian Davis' posting last week.

    (2) I would like to somewhat extend Brian Davis' argument over the
    effects of uncertainties in BSP and location of joint to calculation of
    moment, M. However, I'd like to simplify the inverse dynamics problem
    itself as much as he did without hurting the error issue. Only the difference
    I make here is replacing a GRF with a distal joint force, D. This allows
    reaction forces on the segment (near or at two joints) to be comparable. I
    think, however, it is legitimate to keep a net moment at the distal joint zero
    in this problem. Then an appropriate diagram of the problem should look
    something like below:

    ma /|\ P /
    | /|\ /
    | alfa(@) | /
    //|\ | CM M (@:ccw or positive
    / | | direction)
    / | \|/ mg
    / D

    Now the net torque about the estimated center of mass (CM) of the segment of
    interest is given:

    I*alfa = -D*cL + P*(1-c)L + M, [4]

    where c is the ratio of the distance between the distal joint and the
    segmental CM to the segmental length, L. The c is one of BSP's, which appear
    in [4], besides moment of inertia about the segmental CM, I. + The I (=mr**2)
    does not depend explicitly on the c. The r is the radius of gyration of the
    segment. Furthermore the c does not affect angular acceleration, alfa, as long
    as the CM is along on the line of the segment of interest. Thus the term,
    I*alfa, is regarded as constant with respect to c. Then changes in moment
    estimation to small changes in L and c are given:

    @M/@L = Dc - P(1-c) and

    @M/@c = (D+P)L,

    respectively. Let e(L) and e(c) be errors in estimation of moment due to
    uncertainties in L and c, respectively. ++ Let k be a 'given' fractional
    uncertainty both in L and c. Then

    e(L) = kLcD - KL(1-c)P and

    e(c) = kcL(D+P).

    Taking the ratio of e(L) to e(c),

    e(L)/e(c) = [ D - {(1-c)/c}P ] / (D+P).

    +++ Therefore,

    |e(L)| < |e(c)| if sign(D) = sign(P),

    |e(L)| > |e(c)| if sign(D) = -sign(P), and

    |e(L)| = |e(c)| if either |D| > |P|,
    where .4 < c < .6 for major segments of human body.

    (3) + Exclusion of the term, I*alfa, in computing e(L) and e(c) depends
    on how the r is obtained. If the estimation of the r is based on
    measuremnt, then the I*alfa can be treated independently of c. However, if
    the estimation of the r is based on an analytical relationship with the c,
    then partial derivative of the I*alfa should be taken with respect to the c.

    ++ I used the same k for fractional uncertainties in L and c with a clear
    reason. The point of the late debate was the _sensitivities_ in estimation
    of moment, M, to variation of L and c, respectively. Variations of L and c
    are uncertainties, i.e., dL and dc, around their best estimated or measured
    values. In practice, however, dc/c doesn't have to be the same as dL/L.
    Therefore, the corresponding uncertainty in estimating or measuring each
    variable (e.g., dc) should be used in calculating the net contribution of
    each variable (e.g., |@M/@c|dc) to the final error, dM. Which one is larger
    between dL and dc or how large they are is another matter to be asked in
    addition to the sensitivity of dM (e.g., |@M/@c|) to uncertainty in each

    +++ What I showed above is simply listing all possible cases under different
    conditions. I don't have enough variety of data to take examples for each
    condintion. Brian Davis suggested that the 2nd condition is where movemnts
    would normally occur. I wonder which of those conditions is pertinent as
    near the toe-off, especially, in kicking, and near the touch-down. I
    believe there are many who are familiar with such data among the readers.
    Resposes to this would be appreciated.

    Thanks for reading along this far.


    Seogyong Lee
    Department of Biomechanics
    University of Maryland
    College Park, MD 20742

    phone: (301) 405-2572