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Errors in BSP data

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  • Errors in BSP data

    Dear Biomechers

    I enjoyed reading Zvi Ladin's posting on the network, since I think this
    makes for an interesting discussion. He raised some points that I would
    like to address below.

    One such issue concerns the statement he made in his last paragraph
    "any errors in the location of the COM would have just as grave
    consequences as errors in the locations of the joint centers". At first
    glance this statement appears to be correct, since the equations
    of equilibrium do indeed contain terms that are the radii vectors from the
    segmental COM to the joint centers. However, after spending the morning
    thinking about it, I am still of the opinion that errors in the locations
    of joint axes are MUCH MORE damaging than errors in the location of the
    segment's COM. I will give my reasons in two ways; (i) a "logical"
    approach, and (ii) an "analytical" approach.

    (i). If one regards the foot, and for the moment assumes the GRF acts at
    the 2nd metatarsal head (MTH), and the COM is midway between the 2nd MTH
    and the ankle joint, then one can take moments about the COM and solve
    for the (unknown) joint moment. This equation will involve the distance
    between the 2nd MTH and COM (call this distance "A") as well as the
    distance between the COM and the ankle (call this distance "B"). If one
    makes a mistake in locating the COM, then either (a) A will be bigger and B
    smaller, or (b) A will be smaller and B will be bigger. The point is that
    in the equation, these errors will to some extent CANCEL each other.
    However, if there is an error in the location of the GRF (say A is bigger),
    then there is nothing in the equation that will cancel this. This is my
    first argument to support my opinion that errors in either joint axis
    location or location of GRF are more detrimental than errors in BSP
    parameters. (The second argument is much longer!)

    (ii) Analytical approach. Here I am going to try an do a complete error
    analysis of a somewhat simplified case--a foot in contact with the ground.
    (The foot does have an angular acceleration (alpha) and a vertical
    acceleration (Vacc).) F is the magnitude of the GRF, H is the reaction
    force at the ankle, mg is the weight of the foot (mass, m = 1.16kg,
    g = 9.81m/s/s) and the moment at the ankle = M. "O" represents the COM and
    "A" represnts the ankle joint. Fx, Ox and Ax are the x-coordinates of F, O
    and A respectively. "I" is the moment of inertia of the foot (about the
    COM). Some of these quantities are indicated in the sketch below.

    | |
    | |
    | |
    J \
    _______ / A @ ) @ represents a clockwise moment
    ----^ o /|\ / of magnitude M at the ankle joint
    (____T_______|____|___/
    /|\ \|/ |H
    |F mg
    |

    One can sum the vertical forces and take moments at O to obtain the
    following equation;

    M = -F*(Ox - Fx) + (m*Vacc + m*g - F)*(Ax - Ox) + I*alpha

    In performing an uncertainty analysis, one needs to know the partial
    derivatives of M with repect do each variable in the above equation:

    dM/dF = -(Ox - Fx)
    dM/dOx = -m*(Vacc + g)
    dM/dFx = F
    dM/dm = (Vacc + g)*(Ax - Ox)
    dM/dVacc = m*(Ax - Ox)
    dM/dg = m*(Ax - Ox)
    dM/dAx = m*(Vacc + g) - F
    dM/dI = alpha
    dM/dalpha = I

    One then needs to substitute actual values into the above equations and
    multiply by the errors associated with each variable. Then square the
    results, add up and then take the square root to find the overall
    uncertainty. i.e., overall uncertainty equals;
    square root of {([dM/dF]*error in F)^2 + ([dM/dOx]*error in Ox)^2 +.....}

    I did this for typical data used in gait analysis (using SI units);
    F = 830 N, Ox = 0.58 m, Fx = 0.5 m, m = 1.16 kg, Vacc = 5.54 m/s/s,
    g = 9.81 m/s/s, Ax = 0.66 m, I = 0.0099 kg/m2, alpha = 36 r/s/s.

    By substituting these values into the partial derivatives above, one can
    get the following values: (next to each is an assumed error for each
    variable)
    error
    dM/dF = -0.08 8.3 N (i.e., 1% of F)
    dM/dOx = -17.81 0.01 m (i.e., 1 cm)
    dM/dFx = 830 0.01 m "
    dM/dm = 1.228 0.116 kg (i.e., 10% of mass)
    dM/dVacc = 0.093 0.554 m/s/s (i.e., 10% of Vacc)
    dM/dg = 0.093 0 m/s/s
    dM/dAx = -812.2 0.01 m (i.e., 1 cm)
    dM/dI = 37 0.00099 kg/m/m (i.e., 10% of I)
    dM/dalpha = 0.0099 3.7 m/s/s (i.e., 10% of alpha)

    The calculated moment at the ankle is 131 Nm with an overall uncertainty
    of 11.634 Nm. Now for the "bottom line". If the error associated with Fx
    is reduced to zero, then the overall uncertainty becomes 8.15 Nm. If the
    error associated with Ax is reduced to zero, the overall uncertainty
    becomes 8.33Nm. (Both 8.15 and 8.33 are better than 11.6.) However, if
    the error associated with Ox is reduced to zero (perfect BSP data), the
    overall uncertainty is 11.632 Nm. (Hardly different to 11.634Nm) So, this
    tedious exercise has suggested that, for the data given above, COM location
    is not nearly as important as the location of the external force or the
    ankle joint center.

    Another issue that I would like to mention concerns the inclusion or
    exclusion of inertial components in the Inverse Dynamics Approach. I do
    not want to give readers the impression that I think inertial components
    should be excluded. Although I believe errors in either acceleration data
    or BSP parameters are not that serious (compared to errors in location of
    GRF and/or joint axes), I do not recommend that the dynamic terms
    should be neglected. That would be like saying "I am going to CONSISTENTLY
    overestimate (or underestimate) BSP and acceleration terms by 100%". This
    scenario is different to the situation where one might have large
    uncertainties in different variables--which results in some terms being
    overestimated, and some underestimated. Thus, errors in joint moments will
    only be linearly affected by errors in BSP parameters (e.g. limb masses) if
    the estimates are consistently too large or consistently too small.

    I hope this has added to what I consider an interesting question.

    Regards,
    Brian L. Davis, Ph.D.
    Dept. Biomedical Engineering (Wb3)
    Cleveland Clinic Foundation
    9500 Euclid Avenue
    Cleveland, Ohio 44195, U.S.A

    E-Mail: davis@bme.ri.ccf.org

    Ph: (216) 444-1055 (Work)
    Fax216) 444-9198 (Work)
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